V I' H E AND A V V L I R \) M AT II EMATIC S
A Scries of Monographs and Textbooks
INVERTI ILIT
'NDSINGUL RITY
F R UN ED
LINE'R E 'T RS
i
Robin Harte

Invertibility and
Singularity for
Bounded Linear Operators

PURE AND APPLIED MATHEMATICS
A Program of Monographs, Textbooks, and Lecture Notes
EXECUTIVE EDITORS
Earl J. Taft
Rutgers University
New Brunswick, New Jersey
Zuhair Nashed
University of Delaware
Newark, Delaware
CHAIRMEN OF THE EDITORIAL BOARD
S. Kobayashi
University of California, Berkeley
Berkeley, California
Edwin Hewitt
University of Washington
Seattle, Washington
EDITORIAL BOARD
M. S. Baouendi
Purdue University
Jack K. Hale
Brown University
Marvin Marcus
University of California, Santa Barbara
W. S. Massey
Yale University
Leopoldo Nachbin
Centro Brasileiro de Pesquisas Fisicas
and University of Rochester
Anil Nerode
Cornell University
Donald Passman
University of Wisconsin-Madison
Fred S. Roberts
Rutgers University
Gian-Carlo Rota
Massachusetts Institute of
Technology
David Russell
University of Wisconsin-Madison
Jane Cronin Scanlon
Rutgers University
Walter Schempp
Universitat Siegen
Mark Teply
University of Wisconsin-Milwaukee

MONOGRAPHS AND TEXTBOOKS IN
PURE AND APPLIED MATHEMATICS
1. K. Yano, Integral Formulas in Riemannian Geometry (1970) (out of print)
2. S. Kobayashi, Hyperbolic Manifolds and Holomorphic Mappings (1970)
(out of print)
3. V. S. Vladimirov, Equations of Mathematical Physics (A. Jeffrey, editor;
A. Littlewood, translator) (1970) (out of print)
4. B. N. Pshenichnyi, Necessary Conditions for an Extremum (L. Neustadt,
translation editor; K. Makowski, translator) (1971)
5. L. Narici, E. Beckenstein, and G. Bachman, Functional Analysis and
Valuation Theory (1971)
6. D. S. Passman, Infinite Group Rings (1971)
7. L. Dornhoff Group Representation Theory (in two parts). Part A:
Ordinary Representation Theory. Part B: Modular Representation Theory
(1971,1972)
8. W. Boothby and G. L. Weiss (edsj, Symmetric Spaces: Short Courses
Presented at Washington University (1972)
9. Y. Matsushima, Differentiable Manifolds (E. T. Kobayashi, translator)
(1972)
10. L. E. Ward, Jr., Topology: An Outline for a First Course (1972) (out of
prin t)
11. A. Babakhanian, Cohomological Methods in Group Theory (1972)
12. R. Gilmer, Multiplicative Ideal Theory (1972)
13. /. Yeh, Stochastic Processes and the Wiener Integral (1973) (out of print)
14. /. Barros-Neto, Introduction to the Theory of Distributions (1973)
(out of print)
15. R. Larsen, Functional Analysis: An Introduction (1973) (out of print)
16. K. Yano and S. Ishihara, Tangent and Cotangent Bundles: Differential
Geometry (1973) (out of print)
17. C. Procesi, Rings with Polynomial Identities (1973)
18. R. Hermann, Geometry, Physics, and Systems (1973)
19. N. R. Wallach, Harmonic Analysis on Homogeneous Spaces (1973)
(out of print)
20. /. Dieudonne, Introduction to the Theory of Formal Groups (1973)
21. /. Vaisman, Cohomology and Differential Forms (1973)
22. B. -Y. Chen, Geometry of Submanifolds (1973)
23. M. Marcus, Finite Dimensional Multilinear Algebra (in two parts) (1973,
1975)
24. R. Larsen, Banach Algebras: An Introduction (1973)
25. R. O. Kujala and A. L. Vitter (edsj, Value Distribution Theory: Part A;
Part B: Deficit and Bezout Estimates by Wilhelm Stoll (1973)
26. K. B. Stolarsky, Algebraic Numbers and Diophantine Approximation (1974)
27. A. R. Magid, The Separable Galois Theory of Commutative Rings (1974)
28. B. R. McDonald, Finite Rings with Identity (1974)
29. /. Satake, Linear Algebra (S. Koh, T. A. Akiba, and S. lhara, translators)
(1975)

30. /. S. Golan, Localization of Noncommutative Rings (1975)
31. G. Klambauer, Mathematical Analysis (1975)
32. M. K. Agoston, Algebraic Topology: A First Course (1976)
33. K. R. Goodearl, Ring Theory: Nonsingular Rings and Modules (1976)
34. L. E. Mansfield, Linear Algebra with Geometric Applications: Selected
Topics (1976)
35. N. J. Pullman, Matrix Theory and Its Applications (1976)
36. B. R. McDonald, Geometric Algebra Over Local Rings (1976)
37. C. W. Groetsch, Generalized Inverses of Linear Operators: Representation
and Approximation (1977)
38. /. E. KuczkowskiandJ. L. Gersting, Abstract Algebra: A First Look (1977)
39. C. O. Christenson and W. L. Voxman, Aspects of Topology (1977)
40. M. Nagata, Field Theory (1977)
41. R. L. Long, Algebraic Number Theory (1977)
42. W. F. Pfeffer, Integrals and Measures (1977)
43. R. L. Wheeden and A. Zygmund, Measure and Integral: An Introduction to
Real Analysis (1977)
44. /. /-/. Curtiss, Introduction to Functions of a Complex Variable (1978)
45. K. Hrbacek and T. Jech, Introduction to Set Theory (1978)
46. W. S. Massey, Homology and Cohomology Theory (1978)
47. M. Marcus, Introduction to Modern Algebra (1978)
48. E. C. Young, Vector and Tensor Analysis (1978)
49. S. B. Nadler, Jr., Hyperspaces of Sets (1978)
50. S. K. Segal, Topics in Group Rings (1 978)
51. A. C. M. van Rooij, Non-Archimedean Functional Analysis (1978)
54. L. Corwin and R. Szczarba, Calculus in Vector Spaces (1979)
53. C. Sadosky, Interpolation of Operators and Singular Integrals: An
Introduction to Harmonic Analysis (1979)
54. /. Cronin, Differential Equations: Introduction and Quantitative Theory
(1980)
55. C. W. Groetsch, Elements of Applicable Functional Analysis (1980)
56. /. Vaisman, Foundations of Three-Dimensional Euclidean Geometry (1980)
57. H. I. Freedman, Deterministic Mathematical Models in Population Ecology
(1980)
58. S. B. Chae, Lebesgue Integration (1980)
59. C. S. Rees, S. M. Shah, and C. V. Stanojevic, Theory and Applications of
Fourier Analysis (1981)
60. L. Nachbin, Introduction to Functional Analysis: Banach Spaces and
Differential Calculus (R. M. Aron, translator) (1981)
61. G. Orzech and M. Orzech, Plane Algebraic Curves: An Introduction
Via Valuations (1981)
62. R. Johnsonbaugh and W. E. Pfaffenberger, Foundations of Mathematical
Analysis (1981)
63. W. L. Voxman and R. H. Goetschel, Advanced Calculus: An Introduction
to Modern Analysis (1981)
64. L. J. Corwin and R. H. Szcarba, Multivariate Calculus (1982)
65. V. I. Istratescu, Introduction to Linear Operator Theory (1981)
66. R. D. Jarvinen, Finite and Infinite Dimensional Linear Spaces: A
Comparative Study in Algebraic and Analytic Settings (1981)

67. /. K. Beem and P. E. Ehrlich, Global Lorentzian Geometry (1981)
68. D. L. Armacost, The Structure of Locally Compact Abelian Groups (1981)
69. /. W. Brewer and M. K. Smith, eds., Emmy Noether: A Tribute to Her Life
and Work (1981)
70. K. H. Kim, Boolean Matrix Theory and Applications (1982)
71. T. W. Wieting, The Mathematical Theory of Chromatic Plane Ornaments
(1982)
72. D. B. Gauld, Differential Topology: An Introduction (1982)
73. R. L. Faber, Foundations of Euclidean and Non-Euclidean Geometry (1983)
74. M. Carmeli, Statistical Theory and Random Matrices (1983)
75. /. H. Carruth, J. A. Hildebrant, and R. J. Koch, The Theory of
Topological Semigroups (1983)
76. R. L. Faber, Differential Geometry and Relativity Theory: An
Introduction (1983)
77. S. Barnett, Polynomials and Linear Control Systems (1983)
78. G. Karpilovsky, Commutative Group Algebras (1983)
79. F. Van Oystaeyen and A. Verschoren, Relative Invariants of Rings: The
Commutative Theory (1983)
80. /. Vaisman, A First Course in Differential Geometry (1984)
81. G. W. Swan, Applications of Optimal Control Theory in Biomedicine (1984)
82. T. Petrie and J. D. Randall, Transformation Groups on Manifolds (1984)
83. K. Goebel and S. Reich, Uniform Convexity, Hyperbolic Geometry, and
Nonexpansive Mappings (1984)
84. T. Albu and C. Nastasescu, Relative Finiteness in Module Theory (1984)
85. K. Hrbacek and T. Jech, Introduction to Set Theory, Second Edition,
Revised and Expanded (1984)
86. F. Van Oystaeyen and A. Verschoren, Relative Invariants of Rings: The
Noncommutative Theory (1984)
87. B. R. McDonald, Linear Algebra Over Commutative Rings (1984)
88. M. Namba, Geometry of Projective Algebraic Curves (1984)
89. G. F. Webb, Theory of Nonlinear Age-Dependent Population
Dynamics (1985)
90. M. R. Bremner, R. V. Moody, and J. Patera, Tables of Dominant Weight
Multiplicities for Representations of Simple Lie Algebras (1985)
91. A. E. Fekete, Real Linear Algebra (1985)
92. S. B. Chae, Holomorphy and Calculus in Normed Spaces (1985)
93. A. J. Jerri, Introduction to Integral Equations with Applications (1985)
94. G. Karpilovsky, Projective Representations of Finite Groups (1985)
95. L. Narici and E. Beckenstein, Topological Vector Spaces (1985)
96. /. Weeks, The Shape of Space: How to Visualize Surfaces and Three-
Dimensional Manifolds (1985)
97. P. R. Gribik and K. O. Kortanek, Extremal Methods of Operations Research
(1985)
98. J.-A. Chao and W. A. Woyczynski, eds., Probability Theory and Harmonic
Analysis (1986)
99. G. D. Crown, M. H. Fenrick, and R. J. Valenza, Abstract Algebra (1986)
100. /. H. Carruth, J. A. Hildebrant, and R. J. Koch, The Theory of
Topological Semigroups, Volume 2 (1986)

101. R. S. Doran and V. A. Belfi, Characterizations of C*-Algebras: The
Gelfand-Naimark Theorems (1986)
102. M. W. Jeter, Mathematical Programming: An Introduction to
Optimization (1986)
103. M. Altman, A Unified Theory of Nonlinear Operator and Evolution
Equations with Applications: A New Approach to Nonlinear Partial Differential
Equations (1986)
104. A. Verschoren, Relative Invariants of Sheaves (1987)
105. R. A. UsmanU Applied Linear Algebra (1987)
106. P. BlassandJ. Lang, Zariski Surfaces and Differential Equations in
Characteristic p > 0 (1987)
107. /. A. Reneke, R. E. Fennell, and R. B. Minton. Structured Hereditary
Systems (1987)
108. H. Busemann and B. B. Phadke, Spaces with Distinguished Geodesies
(1987)
109. R. Harte, Invertibility and Singularity for Bounded Linear
Operators (1988).
110. G. S. Ladde, V. Lakshmikantham, and B. G. Zhang, Oscillation
Theory of Differential Equations with Deviating Arguments (1987)
111. L. Dudkin, I. Rabinovich, and I. Vakhutinsky, Iterative Aggregation
Theory: Mathematical Methods of Coordinating Detailed and Aggregate
Problems in Large Control Systems (1987)
112. T. Okubo, Differential Geometry (1987)
113. D. L. Stand and M. L. Stand, Real Analysis with Point-Set Topology
(1987)
114. T. C. Gard, Introduction to Stochastic Differential Equations (1988)
115. S. S. Abhyankar, Enumerative Combinatorics of Young Tableaux (1988)
116. H. Strade and R. Farnsteiner, Modular Lie Algebras and Their
Representations (1988)
Other Volumes in Preparation

Invertibility and
Singularity for
Bounded Linear Operators
ROBIN HARTE
University College
Cork, Ireland
MARCEL DEKKER, INC.
New York and Basel

Library of Congress Cataloging-in-Publication Data
Harte, Robin
Invertibility and singularity for bounded linear
operators.
(Monographs and textbooks in pure and applied
mathematics ; 109)
Bibliography: p.
Includes index.
1. Linear operators. 2. Singularities (Mathematics)
I. Title. II. Series: Monographs and textbooks in pure
and applied mathematics ; v. 109
QA329.2.H38 1987 515.7'246 87-9148
ISBN 0-8247-7754-9
COPYRIGHT © 1988 by MARCEL DEKKER, INC. ALL RIGHTS
RESERVED
Neither this book nor any part may be reproduced or transmitted in any
form or by any means, electronic or mechanical, including photocopying,
microfilming, and recording, or by any information storage and retrieval
system, without permission in writing from the publisher.
MARCEL DEKKER, INC.
270 Madison Avenue, New York, New York 10016
Current printing (last digit):
10 987654321
PRINTED IN THE UNITED STATES OF AMERICA

for EMMA
with love and squalor

Preface
Suppose T : X —> Y is a bounded linear operator between normed spaces:
then T is said to be invertible if there is another bounded linear operator
S : Y -► X for which
ST = I :X —► X and TS = I: Y —► Y (0.1)
To be invertible an operator must obviously be nonsingular in various ways:
for example, it must be one-one in the sense that
x, x' e X and Tx' = Tx => x' = x (0.2)
and it must be onto in the sense that
y e Y => y = Tx for some xeX . (0.3)
It will be familiar that the conditions (0.2) and (0.3) are together sufficient
for (0.1) to hold for some (unique) linear 5, which may or may not be
bounded: to ensure that S is also bounded we may expect to strengthen
one or other of the conditions (0.2) and (0.3). This book is about the
various kinds of nonsingularity that arise in this way, and their relationships
with one another and with invertibility, together with "relative" analogs
in which one operator is compared with another, "essential" analogs which
hold "to within finite dimensions," and "algebraic" analogs, aimed at
ensuring that for example an inverse S should belong to some specified subspace
of operators from Y to X. We have a story to tell which involves perhaps
three deep theorems and two major constructions: we attempt to bring
the reader in totally elementary fashion from the definition of a normed
space to the construction of the Taylor spectrum of a system of commuting
operators.
v

VI
Preface
We have tried to write an "introduction to operator theory,"
accessible to students meeting the definitions for the first time; in all honesty
the joke will probably be better appreciated by those who have already
read, or written, a more orthodox course of functional analysis. In a sense
we are perpetrating two confidence tricks in this account: for as long as
possible we are discussing Banach spaces without making the completeness
assumption, and spectral theory without defining the spectrum. To explain
what we mean, recall that if the normed spaces X and Y are complete then
the conditions (0.2) and (0.3) are together sufficient for the condition (0.1)
- but only because of the first of our deep theorems, the "open mapping
theorem." The proof of the open mapping theorem consists in showing, for
complete spaces, that an onto mapping is "open"; the only way of doing
this passes through the auxiliary concept of an "almost open" mapping.
We are thus in the presence of two variations of the condition (0.3), of
which we can be quite unaware on complete spaces: our experiment in this
book is to introduce them, and explore their properties, in an incomplete
environment. It turns out that we are able to describe most of the
consequences of the open mapping theorem before we assume completeness: of
course the description involves three different kinds of "ontoness" instead
of one. In a similar way the operator T : X —► Y can sometimes represent
either a system of operators on the space X or a system of operators on
the space Y, and provides us with a vehicle for discussing about half of
the "multiparameter spectral theory" of such a system. The other half of
course depends on the third of our deep theorems, Liouville's theorem from
complex analysis. The second of our deep theorems is the Hahn-Banach
theorem, which accompanies the important dual space construction, and
almost reduces normed spaces to coordinate geometry. Duality enables us
to interchange conditions of the form (0.2) and (0.3): our other major
construction is a process of "enlargement," which enables us to use for example
the condition (0.2) in its simplest form as a substitute for its more subtle
variants, and similarly with conditions of the form (0.3).
In detail, Chapter 1 and Chapter 2 introduce normed spaces and
bounded operators respectively, including the enlargement process; we note
in particular a group of three deceptively simple results in Section 1.5 which
we have collectively called "the Riesz lemma." Chapter 3 is perhaps the
core of the book: we introduce here two variants of condition (0.2), three
variants of condition (0.3), and several kinds of variant of condition (0.1).
Both the "regular" operators of Section 3.8 and the "essentially invertible"
operators of Section 3.9 offer premonitions of the Fredholm theory to
follow; the "algebraically invertible" operators of Section 3.10 are a preview
of Banach algebra theory. All this is developed without the help of
completeness, which comes with the open mapping theorem in Chapter 4, or of
duality, which comes with the Hahn-Banach theorem in Chapter 5.
Chapter 6 begins with what amounts to classical linear algebra, the theory of

Preface
vn
finite dimensional spaces and operators, together with their opposite, "Fred-
holm operators." We offer a very simple algebraic framework for Fredholm
theory, and also the "essential" analogs of the enlargement process: these
enable us to work with conditions (0.2) and (0.3) in place of their own
essential analogs. The normed algebras of Chapter 7 deserve a book of their
own: we meet them as the logical generalization of operators T : X —► Y
for which Y = X, and also of spaces of continuous functions; in a sense
spectral theory consists of the attempt to use continuous functions to
represent operators. The Hilbert spaces of Chapter 8 may well be the only
normed spaces that matter: for us their significance is the way in which
invertibility and nonsingularity coalesce for operators on Hilbert spaces. At
the same time a Hilbert space is "self-dual," giving operators the chance to
interact with their own adjoints; we see this most clearly in the
corresponding normed algebras which, in total defiance of standard terminology, we
have insisted on calling "Hilbert algebras." Spectral theory proper comes in
Chapter 9, with the extension to operators of Liouville's theorem from
complex analysis; in a sense our account has now converged to a more standard
treatment. In Chapter 10 however we turn to the "relative" nonsingular-
ities, and normed space versions of "exactness": this is in preparation for
the "multi-parameter" spectral theory of Chapter 11. There can be no
doubt that the Koszul complex construction of Section 11.9, discovered by
Joseph Taylor, is the right "spectrum" for a system of commuting
operators on a Banach space; we offer the conceptually simple "left" and "right"
spectrum of Section 11.1 as a sort of halfway stage, but find that we can do
enough with them to achieve some of the simpler applications of Taylor's
theory, although of course not a functional calculus. We end the book with
a pastiche of the functional calculus for commutative algebras, leaving the
reader on the brink of Taylor's construction.
We have found and corrected hundreds of silly mistakes in this writing:
we know that there remain as many uncorrected, and offer them as a series
of unspoken exercises for the reader. More substantially, we have certainly
not said the last word at several places in the book: for example we invite
the reader to see if he can formulate a working concept of "almost regular,"
fusing Definition 3.7.1 and Definition 3.8.1. It would have been nice to have
been able to do enough Fredholm theory in incomplete spaces using only
the essential enlargement functor P of Definition 6.7.4 to achieve the analog
of Theorem 6.11.3 for almost Fredholm operators, without having to use
the auxiliary Px. We would expect to use the argument of Theorem 10.3.9
to prove that the Euler number is a continuous function of a chain, at least
if it terminates with zeros; we also believe that Theorem 10.5.6 could be
improved to show that almost invertible chains between complete spaces
are invertible, and that the "(left,right) bounded below" chains of
Definition 10.7.1 form open sets. All this and more is offered to the reader to see
if he or she can succeed where we have failed. The reader will also probably

viii
Preface
find our obsession with incomplete spaces tedious: there would have been
a case for working entirely within complete spaces from Chapter 5 onward.
Thus as an exercise we invite the reader to see how much more neatly the
Fredholm theory of Chapter 6 can be written out if completeness is built
in from the start.
It is impossible adequately to acknowledge all the help and advice we
have had both during and before this writing: but Edward Bach of Trinity
College looked hard at an early version of the first half of the book, and led
us to a clearer picture of the relationship between enlargement and
completeness, while Raul Curto in Iowa prevented us from making many silly
mistakes toward the end. Others among those whose unpublished ideas
appear in these pages are Ernst Albrecht, Manuel Gonzalez, Graeme
Jameson, Martin Mathieu, Gerard Murphy, Miceal O'Searcoid, Roger Smyth,
Timothy Starr, Trevor West, Tony Wickstead, Tommy Wilansky, and Wie-
slaw Zelazko. I am more grateful than I can say for the bemused tolerance
extended to me over the years by the mathematical Faculty in Cork, for a
year's hospitality in Iowa, and for the extraordinary patience of the
editorial staff at Marcel Dekker, Inc. My wife and daughter, Chris and Emma,
have had to live through the destruction of three typewriters, at all hours
of the day and night: forgive me. The title of the book is a tribute to the
smooth operators of West Cork, who occupy that subtle territory between
the invertible and the singular.
Robin Harte

Contents
PREFACE
1. NORMED LINEAR SPACES
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
1.10
1.11
Norms or length functions
Metric and topology
Translation invariance
Subspaces and quotients
The Riesz lemmas
Cartesian products
Isometry and equivalence
Sequence and function spaces
Enlargements
Normed linear algebras
Partially ordered spaces
2. BOUNDED LINEAR OPERATORS
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
2.10
2.11
Continuity of linear operators
The normed space of bounded operators
Subspaces and quotients
Cartesian products
Projections
Sequence and function spaces
Enlargements
Shift operators
Composition operators
Normed linear algebras
Partially ordered spaces
1
3
4
5
7
10
12
14
17
19
21
25
25
26
28
31
33
39
41
43
46
49
51
ix

X
Contents
INVERTIBILITY AND SINGULARITY
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
3.10
3.11
3.12
Invertibility and isomorphism
Monomorphisms and epimorphisms
Boundedness below
Openness
Boundary mappings
Left and right invertibility
Almost invertible operators
Regular operators
Essential invertibility
Algebraic invertibility
Subspaces, quotients, and products
Sequence and function spaces
53
53
56
60
65
70
73
76
80
86
91
98
102
BANACH SPACES AND COMPLETENESS
4.1 Cauchy sequences
4.2 Completeness
4.3 Spaces of functions and operators
4.4 Extension by continuity
4.5 Completions
4.6 The open mapping theorem
4.7 Almost open and onto mappings
4.8 Complemented subspaces
4.9 Uniform boundedness
107
107
110
113
115
123
127
130
132
134
LINEAR FUNCTIONALS AND DUALITY
5.1
5.2
5.3
5.4
5.5
5.6
5.7
5.8
5.9
5.10
5.11
5.12
The dual space and the dual operator
Poles and polars
The Hahn-Banach theorem
Duality theory
The separation theorem
Composition operators
Enlargements
Sequence and function spaces
The second dual
An uncomplemented subspace
Extreme points
Differential calculus
137
137
139
141
143
145
150
154
156
160
163
165
166
6. FINITE DIMENSIONAL SPACES AND COMPACTNESS 173
6.1 Linear dependence and independence 173
6.2 Finite dimensional spaces 178

Contents
6.3 Operators of finite rank 183
6.4 Fredholm operators 187
6.5 Weyl operators and the index 192
6.6 Compactness and total boundedness 197
6.7 Essential enlargement 202
6.8 Compact operators 206
6.9 Semi-Fredholm operators 208
6.10 Almost Fredholm operators 214
6.11 Completeness 220
6.12 Duality theory 222
6.13 Composition operators 229
7. OPERATOR ALGEBRA AND COMMUTIVITY 237
7.1 Commutants and double commutants 237
7.2 Maximal ideals and the radical 240
7.3 Regularity 245
7.4 Quasinilpotent elements 251
7.5 Polar and quasipolar elements 256
7.6 Homomorphisms and Fredholm theory 261
7.7 Browder operators 265
7.8 Ascent and descent 271
7.9 Semi-Browder operators 276
7.10 Connectedness and homotopy 280
7.11 Generalized exponentials 290
7.12 Continuous functions 296
7.13 • Linear functional and states 304
8. INNER PRODUCTS AND ORTHOGONALITY 309
8.1 Inner products 309
8.2 Orthogonality 313
8.3 The nearest point theorem 317
8.4 Completeness 320
8.5 Duality 323
8.6 Positive operators 325
8.7 Regularity 329
8.8 Hilbert algebra 332
8.9 Enlargements 334
9. LIOUVILLE'S THEOREM AND SPECTRAL THEORY 337
9.1 Liouville's theorem 337
9.2 The spectrum 341
9.3 The spectral boundary 345

xii
Contents
9.4 Subalgebras and quotients 348
9.5 The spectral radius 352
9.6 Gelfand's theorem 356
9.7 The functional calculus 362
9.8 Essential spectra 369
9.9 Hilbert algebra 378
9.10 States and representations 383
10. COMPARISON OF OPERATORS AND EXACTNESS 391
10.1 Majorization and factorization 391
10.2 Mixed interpolation 394
10.3 Exactness 401
10.4 Composition operators and duality 414
10.5 Enlargement and completion 419
10.6 Essential exactness 425
10.7 Algebraic exactness 435
10.8 Hilbert spaces 441
10.9 Skew exactness 448
11. MULTIPARAMETER SPECTRAL THEORY 457
11.1 Left and right spectra 457
11.2 Polynomials 461
11.3 The spectral mapping theorem 466
11.4 Many variables 470
11.5 The Silov boundary 474
11.6 Composition operators 481
11.7 Tensor products 493
11.8 Quasicommuting systems 509
11.9 The Taylor spectrum 517
11.10 Algebraic and essential spectra 526
11.11 Functional calculus 532
NOTES, COMMENTS, AND EXERCISES 541
REFERENCES 559
NOTATION 581
INDEX 585

Normed Linear Spaces
A normed linear space is both a linear space and a metric space. Because
the distance function for the metric structure is introduced through the
medium of vector "length" there is a certain compatibility between the two
structures.
1.1 NORMS OR LENGTH FUNCTIONS
Let K be either the real field R or the complex field C—much of what we
have to say applies equally in both cases—and recall that a linear space or
vector space X over K is a system consisting of a set X together with a zero
element 0 G X and mappings of addition x, y —► x + y from X x X to X,
subtraction x —► — x from X to X and scalar multiplication t,x —► tx from
K x X to X, subject to the usual axioms: for each x,y,z G X and each
s,te K,
and
(x + y) +z = x+ (y + z) y + x = x + y
x + 0 = x x+(-x)=0
(st)x = s(tx) (s + t)x = (sx) + (tx)
s(x + y) = [sx) + (sy) lx = x
(1.1.0.1)
(1.1.0.2)
It is then familiar that if x + y = x we must have y = 0, and that if x + y = 0
we must have y = — x\ also,
*0 = 0 = Ox (-i)x = -(tx) = t(-x) (-t)(-x)=tx (1.1.0.3)
As a matter of notation we should make an early apology for certain per-
l

2
1. Normed Linear Spaces
sistent bad habits: for example, the same letter X is being used for the
set of vectors X as well as for the whole system consisting of the set and
its additional structure, while the same letter 0 is being used for the zero
elements of different vector spaces X and Y as well as for the number 0
in the field K. We will (without apology) write x — y = x + (—y), and
sx + ty = (sx) + (ty) for a linear combination of x and y. If H and K are
subsets of X and C/CKwe shall write
(1.1.0.4)
sH + tK = {sx + ty: xe H,y e K}
K + x = K + {x} = {y + x:y e K}
and
UK = {sx: seU,xeK} Ux = {sx: seU} (1.1.0.5)
The reader is invited to amuse himself by discovering how much of (1.1.0.1)
and (1.1.0.2) remain valid with subsets H and K in place of vectors x and
y; they should also distinguish clearly between
H - K = {x - y: x e H,y e K}
and (1.1.0.6)
H\K = {xeH:x£K}
A norm on a vector space is essentially a definition of length:
1.1.1 DEFINITION A norm on the linear space X is a mapping x —►
||x|| : X —► [0, oo] such that, for each x, y G X and each iGK,
||x||<oo (1.1.1.1)
||x|| =0=^x = 0 (1.1.1.2)
11**11 = 1*111*11 (1-1-1-3)
ll* + y||<IWI + l|y|| (1.1.1.4)
Here [0,oo] is obtained by adjoining an element oo to the positive
half-line [0, oo[ C R and making the obvious extensions of addition,
multiplication, and inequality: in particular we define Ooo = 0 = ooO. If in
Definition 1.1.1 we relax condition (1.1.1.2) we get what we call a semi-
norm; if instead we relax (1.1.1.1) we get a quasinorm; if we relax both
(1.1.1.1) and (1.1.1.2) we get what we shall call a quasi-seminorm. The
remaining conditions (1.1.1.3) and (1.1.1.4) are together equivalent to
ll« + *y||< WN + I*ll|y|| (1.1.1.5)
for each x, y and each s,t. The condition (1.1.1.4) is perhaps the most

1.2 Metric and Topology
3
important component of Definition 1.1.1: we shall sometimes refer to it as
the triangle inequality.
1.2 METRIC AND TOPOLOGY
If X is a normed linear space, in the sense of a linear space X with a norm
||-|| (we are falling into more bad habits of speech), then the distance
between two points is defined as the norm or length of their vector difference:
for each x, y G X,
dist(x,y) = ||y - x|| (1.2.0.1)
The usual metric space conditions are satisfied: if x, y, z G X then
0<dist(x,y) <oo (1.2.0.2)
dist(x, y) = 0 <=> y = x (1.2.0.3)
dist(y, x) = dist(x, y) (1.2.0.4)
dist(x, z) < dist(x, y) + dist(y, z) (1.2.0.5)
For example, we verify the "triangle inequality" (1.2.0.5) by observing
dist(x,*) = ||* - x|| = ||(y - x) + (z - y)\\ < \\y - x|| + ||* - y||
using (1.1.0.4).
For the record, the topology associated with the metric dist(-,-) is
described by declaring that a set U C X is a neighborhood of a point x G X,
written
U G Nbd(x) (1.2.0.6)
if and only if there is 6 > 0 for which
Disc(x ; 6) = {y G X: dist(x, y) < 6} C U (1.2.0.7)
The interior and the closure of a set K C X are the sets int(if) of points
x G X for which K G Nbd(x), and c\(K) of points x G X for which
U G Nbd(x) <= KC\U ^ 0; then a set K C X is called open iff K = int{K),
and is called closed iff if = cl(if). It is easily checked that the family of
open sets satisfies the usual conditions for a "topology": the empty set and
the whole space, and all finite intersections and all unions of open sets, are
open. The disc Disc(x ; S) of (1.2.0.7) is an example of a closed set, called
the closed disc of center x and radius 6] we can also check that
int(Disc(x ; 6)) = {y G X: dist(x, y) < 6} (1.2.0.8)
As a matter of choice, we shall work with closed discs rather than open
discs.

4
1. Normed Linear Spaces
As in any topological space, a sequence (xn) in X is said to converge
to an element y G X iff for every neighborhood U G Nbd(x) there is a
natural number N G N for which n > N =>• xn G U. In any metric space
the topology can be completely described by convergent sequences: for
example, if K C X then its closure c\(K) is the set of those points t/Gl
for which (xn) converges to y for some sequence (xn) in X with all its terms
xneK.
If x G X and if 0 ^ X C X we shall write
dist(x,if) = inf {||y - x\\:y G X} (1.2.0.9)
Here inf denotes the usual "greatest lower bound" of a set of real numbers.
From this, it is clear
c\{K) = {xeX: dist(x, K) = 0} (1.2.0.10)
If X is a normed linear space and if H is a topological space then a
mapping x : H —► X is said to be continuous at a point £ G H iff for
each neighborhood U G Nbdxx(£) there is a neighborhood V G Nbdn(£)
for which x(V) C £/. For example, the norm ||-|| is a continuous mapping
from the topological space X into the scalar field K: this follows from the
inequality
|||y||-||x|||<||y-x|| (1.2.0.11)
which is just the triangle inequality (1.1.1.4) turned inside out. The reader
is invited to verify that, more generally, the mappings x —► dist(x,iC) :
X —► K are continuous for each nonempty K C X.
1.3 TRANSLATION INVARIANCE
The most immediate impact of the compatibility between metric and linear
structure is "translation invariance": for each x,y,z G X,
dist(x + z, y + z) = dist(x, y) (1.3.0.1)
"Geometrically" this means that all the mappings x —► x + z move sets
of points around X in a rigid manner. The neighborhoods of each point
x G X can be described in terms of the neighborhoods of the origin:
Nbd(x) = x + Nbd(0) = {x + U: U G Nbd(0)} (1.3.0.2)
The process of taking either the interior or the closure is translation
invariant:
mt{K + x) = mt(K) + x and c\{K + x) = c\{K) + x (1.3.0.3)

1.4 Sub spaces and Quotients
5
It follows at once that all the translates of an open set are open, and of
a closed set are closed. Continuity and convergence are also translation
invariant: the sequence (xn) converges to the element y iff the sequence
(xn — y) converges to 0, while the mapping x : ft —► X is continuous at the
point t G ft iff the mapping y = x — x{i) : ft —► X is continuous there.
A more comprehensive summary of the compatibility between linear
and topological structures is that the defining maps for the linear structure
are continuous:
1.3.1 THEOREM If X is a normed space then the mappings x —> x + y
and x —> tx and t —► tx are continuous at each point of X or of K. Further,
the mappings x,y —► x + y and t,x —► tx are jointly continuous at each
point of X x X or K x X.
Proof: These are simple exercises using the properties of the norm. If
x G X and U G Nbd(x + y) then U = V + y for some V G Nbd(x): thus the
mapping x —► x + y is continuous at x G X. If instead Disc(£x ; e) C £/ G
Nbd(*x) we may take V = Disc(x;£) with \t\6 < e to find V G Nbd(x) with
tV C £/: thus the mapping x —► £x is continuous at x G X. Alternatively,
we may take V = {s G K: \s -1\ < 6} with ||x||£ < e to find V G Nbdx(*)
with Vx C U: thus the mapping t —► £x is continuous at £ G K. For the joint
continuity we are working with the cartesian product topology of Y x X,
with Y = X or Y = K: here W G Nbd(y,x) means that V x U C IV for
some V G Nbd(y) and U G Nbd(x). Thus if Disc(x+y;e:) C W G Nbd(x+y)
we may take C/ = Disc(x ; 6) and V = Disc(y ; 6) with 2£ < e to find
£/ G Nbd(x), V G Nbd(y), and U + V C W, which shows that addition is
continuous at (x,y) G XxX. If instead Disc(£x;e) C W G Nbd(£x), we may
take U = {s G K: \s - t\ < 6} and V + Disc(x ; 6) with (|*| + ||x|| + 6)6 < e
to get U G Nbd|<(0> V = Nbd(x), and UV C W, which shows that scalar
multiplication is continuous at (t, x) G K x X. Probably the simplest choice
of 6 in this last calculation is 6 = Min(l,e/(l + \t\ + ||x||)). ■
1.4 SUBSPACES AND QUOTIENTS
If Y is a linear subspace of a normed space X, in the sense that Y ^ 0 and
57 + f7 C 7 for each «,i G K, then it is not hard to use Theorem 1.3.1 to
show that the closure of Y is also a subspace. The interior of Y is almost
always empty:
1.4.1 THEOREM If Y C X is a substance of the normed space X then
cl(y) is also a subspace, and there is implication
Y ^ X =^ int(y) = 0
(1.4.1.1)

6
1. Normed Linear Spaces
Proof: For the first part we must show that if x,y G cl(Y) and s,t G K,
then for each e > 0 there is z €Y for which \\sx + ty — z\\ < e: and we can
see this by taking z = sx1 + ty1 with x',t/' G y satisfying ||x' — x|| < 6 and
\W -y\\<6 with <5 > 0 and (|s| + \t\)S < e. For (1.4.1.1) we show that
int(y) ^ 0 =^ o e int(y) (1.4.1.2)
and
o g int(y) =^ y = x (1.4.1.3)
Indeed if x G int(y) then also —x G int(y), which means that there is 6 > 0
for which
Disc(x ; S) C y and Disc(-x ;^)C7
Thus, if y G Disc(0 ; ^) is arbitrary then y = |(x + y) + |(—x + y) G
y + y C y, which means that 0 G int(y). Further, if this holds, then
0 ^ x G X implies x1 + (£/||x||)x G Y, since ||x'|| = 6, and hence also
x=(||x||/^'er. ■
Approximately two thirds of Theorem 1.4.1. extends to the absolutely
convex subsets of X, in the sense of those Y C X for which
\s\ + \t\ < i =^ sY + *y c y (1.4.1.4)
The argument above shows that if y is absolutely convex then so are cl(y)
and int(y). In particular (1.4.1.2) holds if Y C X is absolutely convex.
If y is a subspace of X, then the norm on X is still a norm when
restricted to the subset Y C X: thus Y becomes a normed space in its own
right. The quotient space
X/Y = {x + Y:xeX}
is another linear space, with linear combination defined by setting
s{x + y) + t{y + Y) = {sx + ty) + Y
for each x,y G X and 5,<6K. If we now define
||x + y|| =dist(x,y) = inf{||x-y||:t/Gy} (1.4.1.5)
then we obtain a well-defined mapping ||-|| : X/Y —► [0,oo]. We claim that
this is a seminorm:
1.4.2 THEOREM If Y is a subspace of the normed space X then the
mapping x + y —► dist(x,y) is a seminorm on X/Y, and is a norm if and
only if y = cl(y).

1.5 The Riesz Lemmas
7
Proof: The condition (1.1.1.1) is trivial, and so is (1.1.1.3) for t = 0. If* ^
0 then dist(*x,y) = infzey \\tx - z\\ = |*|infzey \\x- z/t\\ = |*| dist(x,y),
giving (1.1.1.3). If x,y G X then dist(x + y,Y) = infz€y ||x + y — z\\ =
infz/jZ„€y ||x—z'+y—z"\\ < infz/ey ||x—^/||+infz„€y \\y—z"\\ = dist(x,y) +
dist(t/,y), giving (1.1.1.4). Finally (1.2.0.9) says
||x + y || = o <=> x e ci(y) (1.4.2.1)
so that (1.1.1.2) holds if, and only if, Y = c\{Y). m
Evidently there is a strong case for replacing a subspace Y C X by its
closure cl(y) before forming the quotient X/Y. If ||«|| : X —► [0,oo] is a
quasi-seminorm on a linear space X then the subsets
y = {x e X: \\x\\ < 00} and Z = {x G X: x = 0}
are linear subspaces of X, and the mapping y+Z —* dist(y, Z) = mfzeZ \\y—
z\\ will be a norm on the quotient Y/Z in the full sense of Definition 1.1.1.
Conversely, if Z C Y C X are subspaces, then every norm on the quotient
Y/Z can be lifted to a seminorm on the subspace Y, and then extended
to a quasi-seminorm on X, taking ||x|| = ||x + Z\\ if x G Y and ||x|| = oo
if x G X \ Y. Later we shall see that many specific normed spaces can be
obtained by starting with a quasi-seminorm on some more primitive space.
1.5 THE RIESZ LEMMAS
It is time we did some work. We have grouped together three results which
superficially resemble one another: the second of these is usually called the
"Riesz lemma."
1.5.1 THEOREM If Y C X is a subspace of a normed space X and
x0 G X then there is a sequence (yn) in Y for which
\\yn — x0|| —► dist(x0,y) as n —► oo
and (1.5.1.1)
hnW < \\xo\\ + dist(x0,y) < 2||x0||
Proof: By definition of dist(x0,y) there is (yfn) in Y for which
||*o ~ y'n\\ ^ dist(x0,y) + J!^ for each neN (1.5.1.2)
n

8
1. Normed Linear Spaces
which implies
\\y'n\\ < ll*oll + Wv'n ~ xo\\ < U + M ll*oll + dist(x0,y) (1.5.1.3)
If we now take
yn = ——- y'n for each neN (1.5.1.4)
n + 1
then
Ill/nil < ll*oll + "XT dist(x0,y) < ||x0|| + dist(x0,y) (1.5.1.5)
and
Iko-ynll ^ H*0-2/nll + ll2/n-2/nll
1 ii / n (1.5.1.6)
-||x°"yJI + ^TT)l|yJI_"° asn-+°° ■
If the subspace Y is closed and if e > 0 is arbitrary, then for each
i0Glwe can find yeG7 for which
ll*0-y«ll<(i + e)dist(x0,y)
and (1.5.1.7)
||yc||<||x0|| + dist(x0,y)<2||x0||
Whether or not Y is closed, (1.5.1.7) holds with
Ve = *o (1.5.1.8)
if x0 G y, while if x0 £ cl(y) so that dist(x0,y) > 0, we can replace
(1.5.1.2) by
Iko-I/nll < dist(x0,y) + -dist(x0,y) (1.5.1.9)
n
and then put
3+ ||x0||/dist(x0,y)
ye=yN with " oll/ V 0? } <e (1.5.1.10)
where yN is derived from yfN by the formula (1.5.1.4).
1.5.2 THEOREM If Y C X is a subspace of a normed space X and if
x0 G X then for each real number t G ]0, l[ there is xt G X for which
cl(y) + Kx0 = cl(y) + Kxt and dist(xt,y) > t||xt|| (1.5.2.1)

1.5 The Riesz Lemmas
9
If x0 £ cl(y) we can also arrange ||xt|| = 1.
Proof: If x0 G cl(y) then (1.5.2.1) holds with xt = 0. By definition of
dist(-,y) there is yt G Y with
Now take
llxo-2/tll < -dist(x0,y)
xt = xo~ Vt
(1.5.2.2)
(1.5.2.3)
It is clear that the first part of (1.5.2.1) is satisfied, while for arbitrary
y G Y we have
llz*-y|| = llzo- {y + vM > <list(x0,y) >t\\x0-yt\\ = *||xt||
This gives (1.5.2.1), whether or not x0 G cl(y). If x0 £ cl(y) then certainly
x0 — yt ^ 0: thus if we want ||xt|| = 1 we may take instead
(*o ~ Vt)
\\xo-yt\\
(1.5.2.4)
1.5.3 THEOREM If Y C X is a subspace of a normed space X and if
x0 G X is not in cl(y) then there is M > 0 such that
||2/ + sx0\\ < \\y\\ + |«s|||x0|| < M\\y + sx0|
for each t/67 and each 6 G K.
Proof: If x0 G X is not in cl(y) write
(1.5.3.1)
and note that
||y + 6X0|| = \s\
giving
*(*o)
xo~y
dist(x0,y)
> |6|dist(x0,y) if 6 ^0
(1.5.3.2)
k|||x0|| < S(x0)\\y + sx0\\ for each y G Y,s G K (1.5.3.3)
But now also
||y|| < \\y + «x0ll + \\sxo\\ < (i + *(*o))lly + sxo\\ (1.5.3.4)
giving (1.5.3.1) with M = 1 + 26(x0). m

10
1. Normed Linear Spaces
Theorem 1.5.3 is rather a crude result: in (1.5.3.1) we must have M > 3
since S(x0) > 1. Ewe take advantage of Theorem 1.5.2 to replace x0 with an
"equivalent" xt then we will ensure that 6(x0) < l/t and hence Mt < 1+2/1.
We shall see later that there are circumstances in which (1.5.2.1) can be
obtained with t = 1: this happens if the subspace Y is finite dimensional, or,
alternatively, if the norm on X has a certain "quadratic" property. It seems
to be an interesting problem to try and obtain (1.5.1.1) with ||yn|| < ||x0||
instead of ||yn|| < 2||x0||. We do get this in one particular case:
x0 G cl(y) ^ ||yn - x0 II—+0
and
WVnW < \\xo\\ +dist(x0,y) = ||x0
In particular, if the closure of Y is X then also the closure of Discy(0 ; 1)
in X is Discx(0; 1).
1.6 CARTESIAN PRODUCTS
If Y and Z are normed linear spaces then the cartesian product Y x Z =
{{y>z): V € Y,z € Z} is a linear space, where linear combination is defined
by setting
*(</, z) + t{yf, z') = [sy + ty\ sz + tz')
for each y,yf G Y, z,zf G Z and s,t G K. There are several obvious ways
to define a norm on Y x Z: for example put, for each yG7 and z G Z,
ll(y.*)lli = ll»ll + H (1-6.0.1)
ll(y.*)IL=n««(ll»ll. 11*11) (1-6-0.2)
ll(y,z)ll2 = (lly|l2 + NI2)1/2 (1-6.0.3)
We claim that each of these is indeed a norm:
1.6.1 THEOREM If Y and Z are normed spaces then the mappings
||-|| are norms for each p = l,2,oo, and give the same cartesian product
topology for Y x Z.
Proof: For the first part there are twelve assertions to be checked, only
one of which is not immediately clear. This is the condition (1.1.1.4) for
(1.5.3.5)

1.6 Cartesian Products
11
the mapping ||«||2: we argue
lis/ + y'll2 + h + z'f < (||y|| + ||y'||)2 + (||«|| + ||*'||)2
= NI2 + l|y'll2 + W2 + lk'll2
+ 2(||y||||y'|| + H||*'||)
<l|y||2 + l|y'll2 + IN2 + l|z'll2
+ 2(||y||2 + INI2)1/2(||yT + ll*T)1/2
where the last inequality follows from the equality
(llyll2 + IN2)(lly'll2 + Ik'll2) - (llylllly'll + Nllk'll)2 (1611)
= (l|y||lk'll-WI|y'll)2>o
For the second part we note that the norm IHIoq certainly induces the
usual cartesian product topology, as described in the proof of Theorem
1.3.1: if 6 > 0 is arbitrary then
DisCoo((t/,2) ;«) = Disc(y ; «) x Disc(* ; 6) (1.6.1.2)
We claim that the norms ||-1| x and ||-||2 induce the same topology as the
norm H"!^: for if y G Y and z G Z are arbitrary then
||(y,*)lloo < ll(y.*)lla < ll(y.*)lli < 2||(y,*)IL (i-6.i.3)
and hence if 6 > 0 then
Discoo((0,0) ;S) C Disc^OjO) ; 2d) C Disc2((0,0) ; 26) C Discoo((0,0) ; 26)
(1.6.1.4)
This means that Nbdp(y,^) is the same for each p = l,2,oo, provided
(y,z) = (0,0), and hence by (1.3.0.2) for arbitrary {y,z). m
If Y and Z are normed spaces then, unless otherwise stated, we shall
assume that Y x Z is normed in such a way as to have the cartesian product
topology. Occasionally we shall be more specific: if p G {l,2,oo} we shall
write Y 0p Z to denote the space Y x Z with the norm ||-||p from the
appropriate (l.6.0.q).
More generally, if X is a normed linear space, a pair of supplemented
subspaces of X consists of closed subspaces Y C X and Z C X which satisfy
Y + Z = X and Y n Z = {0} (1.6.1.5)
For example if X = Xx x X2 is a cartesian product of normed spaces Xx and
X2 then the subspaces Y = Xx x {0} and Z = {0} x X2 satisfy (1.6.1.5).

12
1. Normed Linear Spaces
Conversely, if subspaces Y and Z satisfy (1.6.1.5) then there is a one-one
correspondence
j/ + ^M:7xZ^I (1.6.1.6)
in the sense of a mapping which is one-one and onto, and "linear" in the
sense that it respects linear combination on either side. Whether or not it
also respects topology turns out to be rather a delicate matter, to which
we shall return later: here we shall make a further definition, and say that
the subspaces Y C. X and Z C. X are complemented subspaces iff they are
supplemented in the sense of (1.6.1.5), while the topology of X is the same
as that induced on it by the mapping (1.6.1.6) and the cartesian topology
of the product Y x Z. We shall sometimes write
X=Y@Z (1.6.1.7)
to indicate that this is so.
If Y C X is a closed subspace then we shall say that it is supplemented
in X if there is a closed subspace Z C X for which (1.6.1.5) holds, and
that it is complemented in X if there is a closed subspace Z C X for
which (1.6.1.7) holds. Just occasionally we shall be prepared to speak of a
nonclosed subspace Y C X as supplemented if there is a closed subspace
Z C X satisfying (1.6.1.5).
1.7 ISOMETRY AND EQUIVALENCE
Normed spaces X and Y are called isomorphic, written
X~Y (1.7.0.1)
iff there is a one-one correspondence
x <—► y : X <—► Y (1.7.0.2)
between them which preserves linear algebra and topology:
x <—> y and xf <—> yf =>- sx + tyf <—> sy + ty
and (1.7.0.3)
Nbdx(x) <—>Nbdy(y)
X and Y will be called isometric, written
X^Y (1.7.0.4)
if in addition length is preserved:
' — y => \\x\\x = llvlly (1-7-0.5)

1.7 Isometry and Equivalence
13
Thus if, for example, Y and X are "supplemented" subspaces of X in
the sense (1.6.1.5) then X and Y x Z are isomorphic, or if Y and Z are
normed spaces then all three products Y ©p Z are isomorphic. If normed
spaces X and Y are isomorphic, or in particular isometric, then there is
a point of view that they are "the same space." We will be conscious
however of a certain schizophrenia: sometimes isomorphism and sometimes
isometry will be relevant. Indeed, certain key theorems in the sequel come in
two strengths—a basic "topological" version, and a more detailed "metric"
version. Actually the "topological" version will usually be the important
result, although sometimes the only way of reaching the topological version
will be through the "metric" version.
The difference between isomorphism and isometry appears in an acute
form when we compare the topologies of two different norms on the same
vector space, and in particular ask under what conditions they will be the
same. If ||-|| and ||-||; are norms on a vector space X we shall describe
||-||; as weaker, or coarser, than ||-||, if it generates a "weaker" or "coarser"
topology, in the sense that
Nbd'(x) C Nbd(x) for each x e X (1.7.0.6)
Synonymously we shall describe ||-|| as stronger, or finer, than ||-||.
Equivalent conditions are that all the ||-||'-open sets are ||-||-open, or that the
|| ^-closure of an arbitrary subset is contained in the ||«||'-closure. In terms
of sequences, (1.7.0.6) is equivalent to the implication, for arbitrary (xn) in
X and arbitrary t/Gl, that
\\xn ~ y\\ —* o asn —* °° => \\xn ~ y\\f —* o2LS n —* °° (1.7.0.7)
It turns out that there is a nice condition expressing (1.7.0.6) in terms
of the norms ||«|| and ||«||;:
1.7.1 THEOREM If ||-|| and ||-||' are the norms on a linear space X then
necessary and sufficient for ||«||; to be weaker than ||«|| is that there is k > 0
for which
||x||' < Jfc||x|| for each x G X (1.7.1.1)
Necessary and sufficient for ||«|| and ||«||; to generate the same topology on
X is that there are k > 0 and k' > 0 for which
fc'||x|| < ||x||' < Jfc||x|| for each xeX (1.7.1.2)
Proof: The condition (1.7.1.1.) is obviously sufficient for (1.7.0.6), since
if x e X and U e Nbd'(x) there is 6 > 0 for which Disc'(x ; 6) C U, from

14
1. Normed Linear Spaces
which (1.7.1.1) gives
Disc(x ; S/k) C Disc'(x ; S) C U (1.7.1.3)
putting U G Nbd(x). Conversely, if we assume (1.7.0.6) then
Disc'(0 ; 1) G Nbd(0) (1.7.1.4)
so that there is 6 > 0 for which Disc(0 ; 6) C Disc'(0 ; 1). Now if 0 ^ x G X
then £x/||x|| G Disc(0 ;£), so that ||(^x/||a:||)||' < 1, giving
W-H
£x
SH
Thus (1.7.1.1) holds with A; = l/S. Interchanging the roles of ||-|| and ||-||;
then gives (1.7.1.2). ■
The reader will recall that we used the easy half of this argument for
the second part of Theorem 1.6.1, to show that three norms on the product
Y x Z give the same topology. He will also see how Theorem 1.5.3 imposes
the condition (1.7.1.2) for two norms on a subspace Y + Kx0 C X provided
*0*ci(y).
1.8 SEQUENCE AND FUNCTION SPACES
The most trivial examples of a normed space are the scalar field K, normed
with the absolute value |-|, and the zero space 0 = {0} with only one
element. Each of the norms ||-||p of (1.6.0.1)-(1.6.0.3) makes K2 = K x K
into a normed space: for example the absolute value |«| on C is just the norm
||«||2 on R2. By induction we obtain norms ||«|| on Kn for each n G N.
More generally, suppose that H is a nonempty set and that X is a
normed linear space: then the set XQ = Map(H,X) of all mappings from
H to X is another linear space, where we define
(rx + sy)(t) = rx{i) + sy{i) for each t G H (1.8.0.1)
for each x,y G Xn and each r,6 G K. For each p = l,2,oo we define a
quasinorm ||«|| on Xn: if x G Xn we define
||x||00 = sup{||x(t)||:tGn} (1.8.0.2)
where this is the usual "least upper bound" of the set of real numbers
{||x(£)||:£ G H} provided that set is bounded, and is oo otherwise. Ep=l

1.8 Sequence and Function Spaces
15
or p = 2 put
Nip = £ ll*(')llp)1/p = sup{( £ ll^Wir) ": finite«' £ n}
ten ^ ^ ten' ' >
(1.8.0.3)
The conditions (l.l.l.2)-(l.1.1.4) are almost all easily verified: if p = 2 we
must follow the argument of Theorem 1.6.1 for the condition (1.1.1.4). If
we now put
/p(H, X) = {xe Xn: \\x\\p < oo} (1.8.0.4)
then we get a normed space. When X = K we shall write /p(H, K) = /p(H):
we might notice the inclusion
{<£©x:<t>e /p(n),x e x} c /p(n,x) (1.8.0.5)
where for each <j> G K and each iGlwe define 0 © x G Xn by setting
[<t> © x)(*) = <£(*)s for each t G H (1.8.0.6)
1.8.1 THEOREM If H is a nonempty set and X is a normed linear space
then
Nloo ^ Ilxll2 < llxlll for each ^ ^ X° (1.8.1.1)
and
/i(n,x) c /2(n,x) c un,*) (1.8.1.2)
If and only if H is finite there is equality throughout (1.8.1.2) (if X ^ 0).
Proof: The inequalities (1.8.1.1) were essentially observed in Theorem
1.6.1, and (1.8.1.2) follows at once. If H has cardinal number #H = m
then also
||z||i < m||z||oo for each x e XQ (1.8.1.3)
giving equality throughout (1.8.1.2). If H is infinite then it contains a copy
of N: thus it is enough to see that neither equality holds if fi = N and
X ^ 0. In fact if 0 ^ x G X then the constant function x = 1 © x is in
/^(NjX) and not in /2(N,X), while if y{n) = n~1x for each n G N then y
is in /2(N,X) and not in /1(N,X). ■
Theorem 1.8.1 extends to the situation in which {Xt)teQ is a family of
normed spaces associated with the points of H, and Xn is replaced by the
cartesian product Ilten ^t: we ^° n°t 6° m*° details. If H is not just a
set but a topological space then we shall write C(Cl,X) for the set of those

16
1. Normed Linear Spaces
mappings x G Xn which are continuous at each point iGf], and
c^x) = c/ootn.x) = c(n,x) n/oo(n,x) (i.s.i.4)
for the bounded continuous functions on H. It is rather easy to see that
C(H,X) and hence also C^Q, X) is a linear subspace of Xn:
1.8.2 THEOREM If H is a topological space and X is a normed space
then C^tfl, X) is a closed subspace of /^(f^X).
Proof: For each iGfi the set
{x G ^(HjX^x is continuous at t} (1.8.2.1)
is (by translation invariance as in Section 1.3) a linear subspace of/^(n, X):
we claim that it is also closed in the IHIoq topology. To see this suppose
y G /oo(n,X) and ||y — xn||00 ->0asn-^oo, with xn in the set (1.8.2.1)
for each n€N: then for arbitrary e > 0 there are N G N and U G Nbdn(£)
for which
n > N ==> IK - ylloo ^ Ie a114 5 e U => llXtf(5) - xtfWII < |*
It follows
«€P=>||yM-y(t)||
< IlyM -**MII + IKrM -**WII + ll**W -y(OII < *
The claim is substantiated, and now C^H, X) is the intersection of all the
closed subspaces (1.8.2.1). ■
Theorem 1.8.2 records a familiar fact: the uniform limit of continuous
functions is continuous. If H is again a nonempty set and X is a linear
space then we can look at the "finitely additive set functions" from H to
X, meaning the set M(H,X) of those mappings \i : P{ti) —► X defined on
the set P(H) of all subsets of H, for which
m(0) = o
and (1.8.2.2)
H{K UH) + n{K HH)= n{K) + n{H) for each K,HC{1

1.9 Enlargements
17
If X is a normed space then we obtain a quasinorm on M(H, X) by setting
II^Hj = / d\fi\ = sup< Y^ ||/x(iir)||:/C is a finite partition of H >
(1.8.2.3)
By a finite partition we mean a finite, pairwise disjoint, set of subsets whose
union is H. If we now put
Af^n.x) = {neM{n,X):\\n\\1 <oo} (1.8.2.4)
then we get another kind of normed space. A refinement of the idea is to
restrict \i to certain "measurable subsets" of H.
1.9 ENLARGEMENTS
If H is a nonempty set, then a bounded structure, or homology, on H will be
a set B of its subsets for which
{t} e BQ for each t G H (1.9.0.1)
and
K e B whenever K C K'U K" and K', K" e B (1.9.0.2)
We shall say that the homology B is nontrivial provided
H £ B (1.9.0.3)
For example, the set of all finite subsets of H is a homology. If Q has
a bounded structure and X is a normed linear space then the finite or
terminating functions from H to X form the subspace of l^Q^X),
c00(n,x) = {xe i^in^xy.it e n:*(t) ^ 0} e BQ} (1.9.0.4)
The null functions are given by
c0(H,X) = jxe/TO(n,X):i*en:||x(*)|| > - [ e Bn for each n G l\li
(1.9.0.5)
and the convergent functions are given by
c^n^X) ={xe l^n^Xy.x- y e c0(H,X) for some yeX} (1.9.0.6)

18 1. Normed Linear Spaces
1.9.1 THEOREM If H has a bounded structure and X is a normed space
then, in the topology of the norm H-H^,
c0(n,x) = cic00(n,x) (1.9.1.1)
For arbitrary x G l^Q^X),
dist(x,c0(H,X)) = inf sup||x(*)|| (1.9.1.2)
KeBci t£K
If BQ is nontrivial and x = 1 © y is constant then
dist(x,c0(fi,X)) = ||y|| (1.9.1.3)
Proof: To see that c0(H,X) is closed we follow the srgument of Theorem
1.8.2. Thus if y G l^iX) and \\y - xj^ -+ 0 with xn G c0(H,X) then
for each e > 0 there is N G N and if G Sq for which
n > N => \\y - xJn < §e and t£K^ \\xN{t)\\ < \e
(1.9.1.4)
so that \\y(t)\\ < e if t g K. To see that c00(n,X) is a dense subset of
c0(H,X) suppose that x G c0(H,X) is arbitrary and find K G Bn for which
H^MII <s'&t$.K\ now define xx : H —► X by setting for each t G H,
1°
Evidently
xK G c00(n,X) and
Since e is arbitrary, this proves (1.9.1.1). Toward (1.9.1.2) we have for each
inf sup ||x(t) || = inf ||x-xx|| > dist(x,c0(H,X)) (1.9.1.7)
K t(£K K
and if y G c0(H,X) is also arbitrary,
infsup||x(*)|| =infsup||s(*)-y(*)|| < ||x-y||oo (1.9.1.8)
K t(£K K t(£K
Taking the inf over y G c0(H,X) gives (1.9.1.2). Finally it is clear, toward
(1.9.1.3), that dist(x,c0(n,X)) < Hx^ = ||y|| if x = 1 © y is constantly
equal to y; conversely, provided Bn is nontrivial, if z G c00(n, X) is arbitrary
then there will always be t G H for which \\x(t) — z(i)\\ = \\y — 0|| = ||y||.
Since c00(n,X) is dense in c0(H,X) we have proved (1.9.1.3). ■
\iteK
tit&K
(1.9.1.5)
x — x
klloo
<e
(1.9.1.6)

1.10 Normed Linear Algebras
19
When H = N with the bounded structure of finite subsets, then the
expression in (1.9.1.2) is familiar: if x G l^X) = /^(NjX) then
dist(x,c0(X)) = limsup||xj| (1.9.1.9)
n—nxi
The space cx(X) is also recognizable: if as in (1.9.0.6) i-t/G co(^0 *nen
the vector y is unique:
y = lim xn (1.9.1.10)
n—►oo
The "enlargements" of a normed space X are built out of the quotients
of the spaces 1^(0, X) by the subspaces c0(H,X) for nontrivial H:
1.9.2 DEFINITION If H has a nontrivial bounded structure and X is a
normed space then
Qn(x)=ioo(n,x)/c0(n,x) (1.9.2.1)
If in particular H = N with the homology of finite subsets then
Q(x) = UX)Ao(x) (1-9-2-2)
We are entitled to call Qn(X) an "enlargement" because it contains
a copy of the space X, made up of the cosets x + c0(H,X) of constant
functions x = l©y. For each y G X this coset consists of exactly the
sequences x G c1(H,X) which "converge to y."
1.10 NORMED LINEAR ALGEBRAS
Many of the normed spaces of Section 1.8 have an additional element of
structure, in that the underlying vector space X = A is also a ring, and in
fact a linear algebra: there is a multiplication
a, b —> a • 6 = ab : Ax A —> A (1.10.0.1)
for which, if a, 6, c G A,
a • (b - c) = (a- b) • c
K J K J (1.10.0.2)
a • (6 + c) = (a • 6) + (a • c) (a + 6) • c = (a • c) + (6 • c)
and if also t G K then
(*a) • b = t(a • 6) = a • (*6) (1.10.0.3)

20
1. Normed Linear Spaces
We will always assume that a linear algebra has an identity, in the sense of
an element 1 G A for which
1. a = a = a • 1 for each a G A (1.10.0.4)
1.10.1 DEFINITION A normed linear algebra A is a normed linear space
and a linear algebra for which
lla* b\\ < Nlll&ll for each a, 6 € A (1.10.1.1)
and, unless 1=0,
||1|| = 1 (1.10.1.2)
We have not quite ruled out the possibility that 1 = 0: it occurs if
and only if A = 0. For example if H is a nonempty set then /^H, K) is a
normed linear algebra if we put, for each a,b,
(a • b)(t) = a(t)b{t) for each t G H (1.10.1.3)
More generally, if A is a normed linear algebra then so is /^(n, A). If
B C A is a linear subspace for which
leB and B-BCB (1.10.1.4)
then B is a subalgebra of A, and becomes a normed algebra in its own right.
If J C A is a linear subspace for which
A-JCJ and J-AC A (1.10.1.5)
then J is two-sided ideal of A, and the quotient A/ J is a linear algebra.
Here the quotient seminorm of (1.4.1.5) satisfies the compatibility condition
(l. 10.1.1): thus if also J is closed in the norm topology then A/J becomes
a normed algebra. For example if H is a topological space then C^Q, A)
is a subalgebra of /^(n, A); if n has a bounded structure then c0(fi,A) is
a two-sided ideal of /^(n, A).
If A is a linear algebra and a G A then we define an for each n G N by
induction, setting
a1 =a and an+1 =a-an for each nGN (1.10.1.6)
Then if k G N and c0, cl9 ..., ck G K and / = c0 + c^ H + c^* is a
polynomial we put
/(a) = c0l + cxa + • • • + ^a* (1.10.1.7)

1.11 Partially Ordered Spaces
21
In this way we can treat a polynomial as a mapping / : A —> A. We can
even interpret rational functions: if a G A we write a~l for the inverse of
a, in the sense that
a-a"1 = 1 = a"1-a and a"1 G A (1.10.1.8)
It is a familiar exercise that (1.10.18) determines a~l uniquely, if it exists
at all: thus we get a mapping z~l : A~l —> A defined on a certain subset
A'1 C A.
One final comment: if A is a normed linear algebra, with multiplication
a, 6 —► a • 6, then the mapping
a, 6 —>b-a = a-'b:AxA —► A (1.10.1.9)
will also satisfy the compatibility condition (1.10.1.1) as well as the
algebraic conditions (1.10.0.2) and (1.10.0.3). What we obtain in this way is
called the algebra formed by reversal of products: this process sometimes
enables us to prove two theorems at once.
1.11 PARTIALLY ORDERED SPACES
Many of the normed spaces of Section 1.8 have another element of structure:
the underlying set also carries a partial order. By a partial order on a
nonempty set H we shall understand a "relation" < —which the reader
may or may not care to identify with its "graph,"
graph(^) = {(x,t/) G n2:x < y} C H x H (1.11.0.1)
which satisfies three conditions: for each x,y,z,
x<x (1.11.0.2)
x<y <x => y = x (1.11.0.3)
x<y <z => x<z (1.11.0.4)
Variants are possible: both the "reflexivity" (1.11.0.2) and the
"symmetry" (1.11.0.3) are dispensable. The crucial condition is the "transitivity"
(1.11.0.4). If H is a partially ordered set then so is every subset of H, by
the same relation <. We call a partially ordered set totally ordered if there
is implication, for each x,y G H,
x<y or y < x (1.11.0.5)
A weaker requirement is that H be directed, in the sense that
if x,y £ H then there is z G H for which x < z and y < z (1.11.0.6)

22
1. Normed Linear Spaces
For example the "homologies" of Section 1.9 are directed if we interpret
UK < H" to mean UK C #."
If H is a partially ordered set and K C H and i6fi then we say that
x is an upper bound for K provided
y<x for each y G K (1.11.0.7)
Thus, in a directed set every finite subset has an upper bound.
1.11.1 DEFINITION The partially ordered set H satisfies Zorn's
condition if
every totally ordered subset of H has an upper bound in H (1.11.1.1)
If x E H is an upper bound for H then we call it a maximum element:
it is clear from (1.11.0.3) that a maxium element is uniquely determined.
A weaker condition is to describe x G H as maximal if there is implication
x^yGH=^y = x (1.11.1.2)
We are now in a position to state
1.11.2 ZORN'S LEMMA If the partially ordered set H satisfies Zorn's
condition then
H has maximal elements (1.11.2.1)
We offer no proof of Zorn's lemma: in the presence of "the usual axioms
of set theory" it is logically equivalent to "the axiom of choice." The reader
may care to use Zorn's lemma to prove the following sharpened version:
1.11.3 THEOREM If H satisfies Zorn's condition then
every totally ordered subset of H has a maximal upper bound.
(1.11.3.1)
Proof: If K C H is totally ordered, observe that
KA = {xetlix is an upper bound of K} (1.11.3.2)
also satisfies Zorn's condition. ■

1.11 Partially Ordered Spaces
23
The reader should note that the condition (1.11.3.1) implies Zorn's
condition. In applications we usually specialize (1.11.3.1) to single elements
K = {x}: thus if H satifies Zorn's condition then each element x G H
satisfies x <y for some maximal element y G H.
EJisa linear space then we are interested in partial orders which are
compatible with the linear structure: if x,y,z G X we ask that
x <y => x + z<y + z (1.11.3.3)
and
x<y=>tx<ty ifO<*GR (1.11.3.4)
If we now write
X+ = {xeX:0<x} (1.11.3.5)
then we find that, for each x,y G X,
x<y <=> y - x G X+ (1.11.3.6)
and also that X+ is a cone, in the sense that it has the following three
properties:
X+ + X+CX+ (1.11.3.7)
tX+ C X+ if 0 < t e R (1.11.3.8)
(X+) H (-X+) = {0} (1.11.3.9)
Conversely, if X+ is a cone in X then the relation < defined by (1.11.3.6)
satisfies the conditions (1.11.3.3) and (1.11.3.4), as well as (1.11.0.2)-
(1.11.0.4).
1.11.4 DEFINITION A partially ordered normed space X is a normed
space X with a cone X+ for which
X+ = cl(X+) (1.11.4.1)
If X is a partially ordered normed space we shall usually write
x < y ^=> y-xeX+ (1.11.4.2)
For example, if H is a nonempty set and X is one of the spaces /p(H, K) we
may take
X+ = {x e X: 0 < x(t) e R for each t G H} (1.11.4.3)

2
Bounded Linear Operators
The compatibility between the algebraic and the topological structures of a
normed space is reflected in the theory of linear operators between them: it
is particularly easy to decide whether or not a linear operator is continuous.
2.1 CONTINUITY OF LINEAR OPERATORS
A mapping T : X —► Y between linear spaces is called linear if it respects
linear combination: for each x,y G X and each 5,iGK
T(sx + ty) = sTx + tTy (2.1.0.1)
We recall from Section 1.2 that T is continuous at a point x G X iff for
each e > 0 there is 6 > 0 for which
yeX and \\x - y\\ < 6 ==> \\Ty - Tx\\ < e (2.1.0.2)
If this holds for each x in a set K C X then T is said to be continuous on
K: note that the number 6 > 0 satisfying (2.1.0.2) will in general depend
both on e > 0 and on x G K. If for each £ > 0 there exists 6 > 0 for which
(2.1.0.2) holds for each x G if, then T is said to be uniformly continuous
on K. The remarkable fact about a linear operator T : X —► Y between
normed spaces is that if T is continuous at any single point x G X, then it is
uniformly continuous on the whole of X. To see this we begin by observing,
with the aid of translation invariance, that if x G X and T : X —> Y is linear,
then
T continuous at x <=> T continuous at 0 (2.1.0.3)
To go further, we make a definition:
25

26
2. Bounded Linear Operators
2.1.1 DEFINITION A linear mapping T : X -> Y between normed
spaces is called bounded if there is k > 0 for which
||Tx|| < ifc||x|| for each xeX (2.1.1.1)
The greatest lower bound of the set of k > 0 for which (2.1.1.1) holds is
called the operator bound of T:
\\T\\ = inf {A; > 0: llr-H < Jk||-|| on X} (2.1.1.2)
The literal-minded reader should note that a bounded linear operator
T : X —► Y need not belong to the space /^(X, Y) of (1.8.0.4): indeed he
may like to check that the only linear mapping in ^{X, Y) is the constant
mapping 0. For linear operators, boundedness is necessary and sufficient
for continuity:
2.1.2 THEOREM If T : X -► Y is linear between normed spaces then
the following are equivalent:
T is continuous at 0 (2.1.2.1)
T is bounded (2.1.2.2)
T is uniformly continuous on X (2.1.2.3)
Proof: It is trivial that (2.1.2.3) implies (2.1.2.1) and rather easy to see
that (2.1.2.2) implies (2.1.2.3): for if k > 0 satisfies (2.1.1.1) and e > 0 then
(2.1.0.2) holds for all x G X with 6 = e/k. We claim that (2.1.2.1) implies
(2.1.2.2): for suppose that T is continuous at 0 and that 6 > 0 is such that
||jTx|| < e = 1 whenever ||x|| < 6; then if 0 ^ x G X the vector £x/||x|| has
norm 6 and hence
||rx|| = (||x||/*)||r(*x/||x||)||<||x||/« (2.1.2.4)
But this is (2.1.1.1) with k = 1/6. Since (2.1.1.1) always holds for x = 0,
the proof is complete. ■
The reader should remember this argument from Theorem 1.7.1.
2.2 THE NORMED SPACE OF BOUNDED OPERATORS
If X and Y are normed spaces over the field K we shall write
L(X,Y) = LK(X,Y) (2.2.0.1)
for the set of all linear mappings from X to Y. If S and T are in 1,{X,Y)
and 6, t G K, then the mapping sT+tS : x —> sTx + tSx from X to Y is also

2.2 The Normed Space of Bounded Operators
27
linear, and the process of forming the linear combination sT + tS satisfies
the rules (1.1.0.1) and (1.1.0.2): thus L{X, Y) is another vector space over
K. If, in particular, S and T are continuous then Theorem 1.3.1 shows
that all the linear combinations sT + tS are also continuous: thus the
continuous linear operators form a linear subspace of the space of linear
operators. We shall prefer to derive this "topological" statement from a
more detailed "metric" statement: we begin by finding several expressions
for the operator bound.
2.2.1 THEOREM If X ^ 0 and if T : X -> Y is bounded and linear
then
llTsll
||r|| =supI1n-nli = sup ||Tx|| = sup ||Tx|| (2.2.1.1)
x^O ||*|| ||x|| = l ||x||<l
Proof: If 0 ^ x G X then ||Tx|| < ifc||x|| if and only if ||rx||/||x|| < k,
which establishes the first equality, and the second follows if we divide x
by ||x||. The last expression is certainly no smaller than its predecessor.
Conversely, whenever 0 < ||x|| < 1 there is xf = x/||x|| for which ||x'|| = 1
and ||2V|| > \\Tx\\, giving the final equality. ■
In the degenerate case X = 0, two out of three expressions in (2.2.1.1)
are meaningless, bu the last is meaningful, and confirms ||T|| = 0. The
literal-minded reader will now see that a linear operator T G L{X, Y) is
"bounded" if and only if its restriction to the set H = Discx(0 ; l) is in the
space l^n^Y) of (1.8.0.4): indeed the operator bound ||-|| then coincides
with the norm H-^ of (1.8.0.2).
The operator bound is a quasinorm on the space L{X, Y):
2.2.2 THEOREM If X and Y are normed linear spaces and S,T G
L{X,Y) are bounded then so are sT + tS for each s,t G K, with
\\sT + tS\\ < \s\\\T\\ + |*|||S|| (2.2.2.1)
If Z is another normed space and U G L{Y, Z) is also bounded, then so is
UT, with
||C/r||<||C/||||r|| (2.2.2.2)
Proof: If x G X is arbitrary, then
||(*T + tS)x\\ < \s\\\Tx\\ + \t\\\Sx\\ < (H||m + |t|||5||)||x||
so that (2.1.1.1) holds for sT + tS with A; = |s|||r|| + |*|||S||, giving (2.2.2.1),

28
2. Bounded Linear Operators
and confirming that sT + tS is bounded. Also
||prx||<||P||||rx||<||P||||r||||x||
so that (2.1.1.1) holds for UT with k = ||l/||||r||, giving (2.2.2.2) and
confirming that UT is bounded. ■
The "topological" fact that the composition of two continuous
mappings is continuous tells us that if U and T are both bounded then so is UT:
the second part of Theorem 2.2.2 is just a more detailed metric version of
this. We shall write
||r|| = oo if T e L{X,Y) is not bounded (2.2.2.3)
and
BL(X,y) = BLK(X,y) = {Te L{X,Y): \\T\\ < oo} (2.2.2.4)
for the normed space of bounded operators from X to Y.
In the special case Y = X, Theorem 2.2.2 tells us that BL(X,Y) =
BL(X,X) is a normed linear algebra, in the sense of Definition 1.10.1.
In particular, if T G BL(X,X) then we define the powers Tn of T as in
(1.10.1.6), and more generally f(T) for a polynomial / = c0 + cxz + h
ckzk : K -> K, as in (1.10.1.7).
2.3 SUBSPACES AND QUOTIENTS
If X and Y are normed linear spaces then the mapping 0 : X —► Y defined
by setting
0(x) = 0 e Y for each xeY (2.3.0.1)
is always linear, and bounded with bound ||0|| =0. If Y = X then the
mapping / = Ix : X —► X defined by setting
I{x) =xeX for each xeX (2.3.0.2)
is also linear, and bounded with bound ||/|| < 1. Unless X = 0 in fact
||/|| = 1, as we see from the second expression in (2.2.1.1): indeed the "sup"
can here be replaced by "max". More generally, if X1 is the space obtained
from X by giving it a new norm ||-||;, then the mapping /' : X —► X1 defined
by (2.3.0.3) will be continuous if and only if ||-||; is weaker than ||-|| in the
sense of (1.7.0.6): thus Theorem 1.7.1 is actually a special case of Theorem
2.1.2.
If Y C X is a subspace, normed with the norm of X, then the mapping

2.3 Subspaces and Quotients
29
J = JY :Y —► X defined by setting
J{y) =yeX for each y eY (2.3.0.3)
is known as the natural injection of Y in X; evidently it is linear and
bounded, with \\J\\ < 1 and \\J\\ = 1 unless Y = {0}. When the subspace
Y is closed, so that the quotient X/Y has a norm, then the mapping K =
KY : X -+ X/Y defined by setting
K{x) = x + Y G X/Y for each xeX (2.3.0.4)
is again linear and bounded, with bound < 1: we shall call it the quotient
map induced by Y. If for example Y = X, so that X/Y = 0, then K = 0
has bound 0. If Y = c\(Y) ^ X then we claim that ||iir|| = 1, although
we must appeal to the Riesz lemma: in fact (1.5.2.1) tells us that ||jK"|| > t
whenever 0 < t < 1.
If subspaces and quotients give rise to bounded linear mappings, then
too, bounded linear mappings give rise to subspaces and quotients. If T G
BL(X,y), then
T~l(0) = {x G X: Tx = 0 G Y} (2.3.0.5)
is a linear subspace of X, called the null space of T. We claim that it is
closed: for if \\xn - x\\ -+ 0 as n -+ oo with xn G T_1(0) then Tx = 0 by
continuity. We shall sometimes refer to the natural injection of T_1(0) in
X as the kernel of T:
ker(T) = J : r_1(0) —> X (2.3.0.6)
The range of T is the linear subspace
T{X) = {Tx: x e X} C Y (2.3.0.7)
of the "codomain" Y of X: it will turn out to be quite difficult to know
whether or not T(X) is closed in Y. To be rather pompous, we will take
the quotient map induced by the closure of the range of T, and call it the
cokernel of T:
coker(T) = K :Y —► *7 cl{TX) (2.3.0.8)
The kernel and cokernel mappings can be used to decide whether or not
the product of two operators is zero:
2.3.1 THEOREM If X, Y and Z are normed spaces and T G BL(X, Y),
S G BL(y, Z), then the following are equivalent:
ST = 0eBL(X,Z)
(2.3.1.1)

30
2. Bounded Linear Operators
cl(rX) C 5_1(0) C Y (2.3.1.2)
T = (kerS)U for some U e BL(X,5_1(0)) (2.3.1.3)
S = y(cokerT) for some V G BL{Y/cl{TX),Z) (2.3.1.4)
Proof: The equivalence of (2.3.1.1) and (2.3.1.2) is clear, recalling that
5_1(0) is closed in Y, and it is also clear that if either (2.3.1.3) or (2.3.1.4)
holds then so does (2.3.1.1). If we assume (2.3.1.1) then we can obtain U :
X —*■ 5_1(0) to satisfy (2.3.1.3) by the simple expedient of writing Ux = Tx
for each x € X. There is also a well-defined mapping V : Y/ cl(TX) —> Z
defined by setting V{y + cl(TX)) = Sy for each y EY, which is evidently
linear, and satisfies VcokerT = 5. To see that V is bounded we go back
to the Riesz lemma again: (1.5.2.1) tells us that ||V|| < ||S*||/£ whenever
0 < t < 1, and hence that ||V|| < ||5||. ■
As special cases of Theorem 2.3.1 we can take either T = J : X —> Y
for a subspace XCYoiS = K:Y^Z = Y/W for a subspace W C Y:
we learn that X C 5"1 (0) iff S can be factored through V : Y/X -> Z, and
that TX C W iff T can be factored through U : X -> W. This suggests
certain isomorphisms between spaces of bounded operators:
2.3.2 THEOREM If Z C X and V7 C y are closed subspaces there is
isometry
BL{X,W) = {76 BL(X,y):TX C^}C BL(X,y) (2.3.2.1)
and
BL(X/Z,y) ^{TGBL^yjiZCr-^O)} CBL(X,y) (2.3.2.2)
Proof: The correspondences indicated by the notation (1.7.0.4) are those
derived from (2.3.2.3) and (2.3.1.4), and are certainly one-one and onto,
and linear; we must see that they preserve operator bounds. This is trivial
for (2.3.2.1), since it is rather hard to distinguish between an operator
T : X -> y for which TX C W and the induced mapping T : X -> W; for
(2.3.2.2) however we need the Riesz lemma, as in the proof of (2.3.1.4). ■
The reader may at this point be tempted to guess that the spaces
BL(X, Y/W) and BL(Z, Y) are isomorphic to certain quotients of the space
BL(X, y). He is invited to read on. For the moment, we will see how near,
and yet so far, the kernel and the cokernel can get to the operator T:
2.3.3 THEOREM ETG BL(X,Y) there is a bounded linear operator
core(T) : X/T'^O) —► cl{TX) (2.3.3.1)

2.4 Cartesian Products
31
for which
T = ker(coker(T)) o core(T) o coker(ker(T)) (2.3.3.2)
Proof: This can be built up from Theorem 2.3.1. Since Tker(T) = 0
there must be V : X/r_1(0) -► Y for which T = V o coker(ker(T)), and
then, since coker(r)T = 0 it follows that coker(r)V = 0, giving V =
ker(coker(T))C/ for some U : X/T'1^) -► cl(TX): thus take core(T) = U.
Alternatively, simply define
core(T)(x + T_1(0)) = Tx e cl(Tx) for each xeX (2.3.3.3)
and confirm that core(T) is well defined, linear, and bounded, and satisfies
(2.3.2.2). ■
Theorem 2.3.3 is sometimes referred to as the canonical factorization
of T:
Y/cl(TX)
coker(T)^^^
ker(coker (T)) ^V^ ^^coker(ker(T))
core(T)
cl(TX) *+ X/T"1(0)
2.4 CARTESIAN PRODUCTS
If X = Y x Z is a cartesian product of normed spaces, normed so as to
have the cartesian product topology, then the mappings
7rx : X —► Y tt2 : X —► Z ttJ : Y —► X <k'2 : Z —► X
defined by setting, for each y EY and z G Z,
^i(y>«) = y and 7r2(y^) =z (2.4.0.1)
and
*i(y) = M) and *£(*) = (M (2.4.0.2)
are linear and continuous. If, in particular, X = Y ©p Z with pG {l, 2,00}
as in (1.6.0.l)-(l.6.0.3), then each of these operators is of bound < 1, with
equality unless Y = 0 or Z = 0. If W is another normed space we shall
define
col{S,T){w) = {Sw,Tw) eY xZ for each w G W (2.4.0.3)

32
2. Bounded Linear Operators
if S G BL{W, Y) and T G BL(W, Z), and define
row(5, T) (y, z) =Sy + TweW for each {y,z) eY x Z (2.4.0.4)
if S G BL(y,V7) and T G BL(Z,V7): mentally we are viewing the product
Y x Z as a space of "column vectors."
2.4.1 THEOREM If X = Y xZ and W are normed spaces then there is
isomorphism
BL{W,X) = {col(5,r):5 G BL{W,Y),T G BL(W,Z)}
- BL(VP,y) x BL(VP, Z) (2.4.1.1)
and
BL(X,W) = {row(5,r):5GBL(y,V^),rGBL(Z,^)}
~ BL(Y,W) x BL{Z,W) (2.4.1.2)
Proof: If 5 G BL(VP,y) and T G BL(VP,Z), then certainly col(5,T) =
7r[ • S + 71*2 • T is linear and bounded; conversely, if U G BL(W, X), then
U = col(5,r) with S = 7rxU G BL(VP,y) and T = tt2U e BL{W,Z). If
instead 5 G BL(y,V7) and r G BL(Z,W), then row(5,T) = Sirx + Ttt2
is linear and bounded; conversely, if V G BL(X,W), then V = row(5,T)
with 5 = Vtt; G BL(y, W) and T = Vir'2 G BL(Z, W). This establishes the
equality in each of (2.4.1.1) and (2.4.1.2), and the linear isomorphisms at
the end. To see that these linear isomorphisms preserve topology we shall
be more specific, and show that there is isometry
BL{W, y ©^ Z) = BL{W, Y) ©^ BL{W, Z) (2.4.1.3)
and
BL(y ©! Z,W) s BL[Y,W) ©^ BL(Z,W) (2.4.1.4)
Indeed, if S G BL(W,Y) and T G BL{W,Z) are arbitrary, then it is
elementary that
sup max(||Sw||,||rw||) = max( sup \\Sw\\, sup ||Tti;||) (2.4.1.5)
IMI<i IMI<i|l IMI<i
which is (2.4.1.3), while if instead S G BL{Y,W) and T G BL(Z,V7) then
for each y €Y,z G Z
\\Sy + r*|| < ||S||||y|| + ||r||||s|| < max(||5||, ||r||)(||y|| + \\z\\) (2.4.1.6)
which says that || row(5, T)|| < max(||5||, ||T||). Conversely, by taking z = 0
and varying y G y it is clear that ||5|| < || row(5,T)||; similarly ||r|| <
||row(5,r)||. ■

2.5 Projections
33
The reader is invited to ponder which norms are induced on the
products BL{W, Y) x BL{W, Z) and BL(y, W) x BL(Z, W) by their isomorphism
with BL{W,Y ©! Z) and BL{Y ©^ Z,W), respectively.
If X = XY x X2 and Y = Yx xY2 are both cartesian products then
two applications of Theorem 2.4.1 show that each T G BL(X,Y) can be
represented as an "operator matrix":
(2.4.1.7)
where for each i and j we have Tty G BL(Xy,yt). In the special case in
which T12 = 0 and T21 = 0 we shall sometimes write
r„
rM
r»
^22.
xl
.X2.
T11x1 + .r12x2
^21Xl + ^22X2
r = r11er22 = diag(r11,r22)
(2.4.1.8)
If X = Y x Z is a cartesian product then Xj = 7x {0} and X2 =
{0} x Z are subspaces of X, both closed, hence giving rise to quotient spaces
X/X1 and X/X2:
2.4.2 THEOREM IfX = yxZisa product of normed spaces, then
X/{Y x {0}) c± Z and X/({0} x Z) ~ y (2.4.2.1)
Proof: We claim, more specifically, that
(ye»We.{0})S7 (2.4.2.3)
with correspondence (y, 2) + y x {0} <-► 2: in fact if y, 2 are arbitrary then
inf maxdly-y'lUI*- 0||) = ||'ll ■ (2.4.2.4)
y'EY
Theorem 2.4.2 shows that the "topological" part of Theorem 2.4.1 is
contained in Theorem 2.3.2.
2.5 PROJECTIONS
If y and Z are a pair of supplemented closed subspaces of a normed space
X, in the sense (1.6.1.5) that Y + Z = X and Y [\Z = {0}, then every
vector xGl has a unique representation in the form x = y + z with y EY
and z G Z. The mapping 7Ty : x —> y from X to X which implements this
is called the projection of X upon Y in the direction of Z: formally
if x G X then 7rf (x) G y and x - 7rf (x) G Z
(2.5.0.1)
The conditions (1.6.1.5) ensure that (2.5.0.1) indeed defines a mapping,

34
2. Bounded Linear Operators
which is evidently linear, and also idempotent in the sense that, if
E = tt§ (2.5.0.2)
then
E2 = E (2.5.0.3)
If (2.5.0.2) holds then the subspaces Y and Z each have two expressions in
terms of E:
Y = {E{x):x eX} = {xeX: E{x) = x} (2.5.0.4)
and
Z = {xeX: E{x) = 0} = {x - E{x):x e X} (2.5.0.5)
Conversely, if E : X —* X is linear and idempotent in the sense (2.5.0.3)
and the subspaces Y and Z are defined by (2.5.0.4) and (2.5.0.5), then Y
and Z satisfy the conditions (1.6.1.5), and (2.5.0.2) holds. Further, if we
only assume half each of the conditions (2.5.0.4) and (2.5.0.5), then the
other halves each follow. Also if (2.5.0.3) holds, then
[I-E)2 = I-E (2.5.0.6)
and if (2.5.0.2) holds then
J - E = tt| (2.5.0.7)
If linear subspaces satisfy the condition (1.6.1.5) and if the mapping
7Ty defined by (2.5.0.1) is continuous, then both Y and Z must be closed,
since by (2.5.0.4) and (2.5.0.5) we have Y ={I-E)~1(0) and Z = £_1(0),
where E = 7Ty. If, conversely, the subspaces Y and Z are both closed,
then it becomes a surprisingly delicate matter whether or not E = 7Ty is
bounded. We begin by observing that the boundedness of 7Ty is equivalent
to the condition that the subspaces Y and Z are complemented, in the
sense of (1.6.1.7):
2.5.1 THEOREM If closed subspaces Y and Z of the normed space X
are supplemented, in the sense that Y + Z = X and Y f]Z = {0}, then the
following are equivalent:
YxZ~X (2.5.1.1)
X/Z-Y and X/Y c* Z (2.5.1.2)
7Ty is bounded (2.5.1.3)
7r£ is bounded (2.5.1.4)

2.5 Projections
35
Proof: It is to be understood that in (2.5.1.1) and (2.5.1.2) the
isomorphisms are those induced by the natural mappings. If closed subspaces
Y C X and Z C X are supplemented in the sense of (1.6.1.5) then the
mappings
row(JY, Jz) : (y, 2) —► y + z from Y x Z to X (2.5.1.5)
and
co\(Kz, KY) : y + z —► (y + z + Z, y + z + Y)
(2.5.1.6)
= (y + Z, z + y) from X to (X/Z) x {X/Y)
are linear, of bound < 1, as well as being one-one and onto. Thus, the
topology of the space X is weaker than that induced by the product Y x Z,
and stronger than that induced by the product {X/Z) X (X/Y). Thus the
condition (2.5.1.2) implies the condition (2.5.1.1); conversely if (2.5.1.1)
holds, then Theorem 2.4.2 shows that (2.5.1.2) does. If (2.5.1.1) holds then
the mapping E = 7Ty is bounded, since it is the composition
JY o<KloT<m(JY,Jz)-1 :y + z—► [y,z) —>y—>y (2 5 17)
from X to X
where the mapping 7r1 is given by (2.4.0.1). Conversely, if E = 7Ty is
bounded then (2.5.1.2) holds, since for each y €Y and z G Z we have
||y|| < l|£||||y + *ll and ||*|| < ||J-£||||y + 2|| < (l + ||£||)||y + z\\
(2.5.1.8)
and hence
||y|| < \\E\\dist{y + z,Z)
and (2.5.1.9)
||*|| < ||/-E||dist(y + *,y)
We have proved (2.5.1.1) => (2.5.1.3) => (2.5.1.2); this with a similar
argument for (2.5.1.4) finishes the proof. ■
The reader should never forget that the projection E = 7Ty of (2.5.0.1)
depends on the subspace Z just as much as on the subspace Y. If E is
bounded and Y ^ {0} then
\\E\\ > 1 (2.5.1.10)
and it may be quite difficult to arrange Z in such a way that ||i5|| = 1.
If E = E2 and F = F2 are projections on the same space X then the
operator EF may or not be a projection:

36
2. Bounded Linear Operators
2.5.2 THEOREM If E = E2 and F = F2 are projections on X for which
EF = FE (2.5.2.1)
then EF and E + F — EF are projections, with
{EF){X) = (EX)n(FX)
and (2.5.2.2)
(EF)-10 = E~10 + F-10
and
[E + F- EF){X) = {EX) + {FX)
and (2.5.2.3)
{E + F - EF)-1!) = E^O + F~x0
Proof: If EF = FE then
{EF)2 = EFEF = E2F2 = EF (2.5.2.4)
so that EF is also a projection; also
I - {E + F - EF) = {I - E){I - F) = {I - F){I - E) (2.5.2.5)
is similarly a projection, and hence E-\-F—EF is. Towards (2.5.2.2) observe
EF{X) C E{X) and EF{X) = FE{X) C F{X) (2.5.2.6)
while, conversely, if x G EX f] FX then x = £"x and x = Fx and hence
x = F-Fx G EF{X). This gives the first equality: for the second we have
E-x0 C E"1^-^) = (FE)"^ = {EF)-1!)
and (2.5.2.7)
F_10 C F-^E^O) = {EF)-1!)
while conversely if x G {EF)-1!) then x = {x-Fx)-\-Fx with x-Fx G F_10
and Fx G £'~10. This gives the second equality in (2.5.2.2), and now both
parts of (2.5.2.3) follow by application of (2.5.2.2) to I - E and I- F. m
There is a connection between the "algebraic" and the "spatial"
relation of an operator to a projection:

2.5 Projections 37
2.5.3 THEOREM If T G BL[X,Y) and E = E2 e BL(X,X) and F =
F2 eBL{Y,Y), then
T{EX) CFY <=>TE = FTE (2.5.3.1)
and
T(E~l0) C i^O <=> FT = FTE (2.5.3.2)
Hence
[T{EX) C Fy and T{E~l0) C i^O) <=>TE = FT (2.5.3.3)
Proo/: For (2.5.3.1) argue
T(£X) Cfy = (J- F)"^ <=»(/- F)2\E = 0 (2.5.3.4)
and for (2.5.3.2)
F_10 2 r(E_10) = T(J - E)X <=> FT{I -E)=0 (2.5.3.5)
Forward implication in (2.5.3.3) now follows immediately: conversely,
observe
TE = FT=^TE = FTE = FT. m (2.5.3.6)
It is not necessary for projections E and F to commute for EF to be
a projection:
2.5.4 THEOREM If E = E2 and F = F2 are projections on X for which
{I - F){I - E) = 0 (2.5.4.1)
then EF = (EF)2 is a projection on X for which
E(EF) = (EF)E = EF (2.5.4.2)
and
F{X) =E-1{0) + {EF){X)
and (2.5.4.3)
E^^niEF^X) ={0}

38
2. Bounded Linear Operators
Proof: To see that EF is a projection note, using (2.5.4.1),
{EF)2 = E{FE)E = E{F + E- EF)F = EF2 + E2F - E2F2 = EF
(2.5.4.4)
For (2.5.4.2) note
E{EF) = E2F = EF
and (2.5.4.5)
{EF)E = E{F + E - I) = EF + E2 - E = EF
Towards (2.5.4.3) the second equality follows from (2.5.4.2), using Theorem
2.5.2; inclusion E'1^) C F(X) is just a restatement of (2.5.4.1), while
inclusion {EF)X C F{X) follows from
(J - F)EF = EF-{F + E-I)F = 0 (2.5.4.6)
Conversely, inclusion F(X) C £'~1(0) + (EF)(X) follows from the
implication
x e F{X) => x = Fx => x = (x - Ex) + Ex
= {I- E)x + [EF)x e E_1(0) + [EF)X
-i,^ . ,„^v _ (2-5-4-7)
The reader should note carefully that we do no£ claim
F=[I-E)+EF (2.5.4.8)
if (2.5.4.1) holds, then (2.5.4.8) is equivalent to (2.5.2.1). The reader should
also write out the three other theorems to be obtained by interchanging E
and I — E, or F and I — F, or both.
With no restriction at all on the projections E and F we have
VU = I - [E - F)2 = UV (2.5.4.9)
where
U = FE + [I-F){I-E)
and (2.5.4.10)
V = EF+{I-E){I-F)
Also
U{EX) C FX and U(E~l0) C F"1^) (2.5.4.11)
and
V{FX) C EX and ^(F"^) C E"1^) (2.5.4.12)

2.6 Sequence and Function Spaces
39
(2.5.4.9) does not quite tell us that U and V are mutually inverse. The
reader is invited to see what happens if, for example, E and F have the
same range, or instead the same null space.
2.6 SEQUENCE AND FUNCTION SPACES
Suppose that H is a nonempty set that T : X —> Y is a linear mapping:
then the mapping Tn : Xn -» Yn defined by setting
{Tnx)(t) = T[x[t)) for each t G H for each x e Xn (2.6.0.1)
is again linear. If S : X —> Y and U : Y —> Z are also linear then
{sT + tS)n = sTn + tSn for each s,teK (2.6.0.2)
and
{UT)Q = UQTn (2.6.0.3)
For bounded linear mappings between normed spaces it is relevant to
restrict Tn to the normed spaces /p(H,X) of (1.8.0.4).
2.6.1 THEOREM If T G BL(X,y) is a bounded linear mapping of
normed spaces and H is a nonempty set, then for each p G {l,2,oo} there
is inclusion
T%(n,x)cip(n,Y) (2.6.1.1)
and inequality
\\TnX\\p < \\T\\\\x\\p for each x G /p(H,X) (2.6.1.2)
If H is a topological space then also
rncoo(n,x)cc„(0!y) (2.6.1.3)
Proof: We begin by establishing (2.6.1.2) for arbitrary x G Xn, and the
quasinorms ||-||p (p = 1,2,00). For p = 1 and p = 2 let H' be an arbitrary
finite subset of H: then
Yl llr(*W)llP < ll^ir Yl llxWHP for each xeXn (2.6.1.4)
ten' ten'
Raising to the power l/p, and taking suprema over finite H' C H gives
the extended version of (2.6.1.2) for p = 1 and p = 2; taking p = 1 and
H' = {t} in (2.6.1.4) and then the supremum over all t G H, gives the
extended version of (2.6.1.2) for p = 00. Now both (2.6.1.1) and (2.6.1.2)

40
2. Bounded Linear Operators
follow. For (2.6.1.3) we combine (2.6.1.1) with the inclusion
rnc(n, x) c c(n, y) (2.6.1.5)
which holds because the composition of continuous functions is
continuous. ■
Strictly speaking, we should write something like / (H,T) for the
restriction of the mapping T^: we shall however continue to write Tn. We
can be quite explicit about its bound:
2.6.2 THEOREM If T G BL[X,Y) and H is a nonempty set, then for
each p G {1,2,00}
||/p(n,r)|| = ||rn||p = ||r|| (2.6.2.1)
If H is a topological space, then also
11^(0,^11 = 1^11 (2.6.2.2)
Proof: It is clear from (2.6.1.2) that the restriction /p(n,T) of Tn to
Zp(n,X) is bounded, with
||/p(n,r)||<||r|| (2.6.2.3)
KteH and iGl consider the mapping 6t 0 x : H —> X defined by setting
[ x if 6 = £
&©*)(«) = {n .r ^ (2.6.2.4)
for each s G H: 6t is the "Kronecker delta," and the notation is (1.8.0.6).
Evidently
||£ 0 x||p = ||x|| for each p G {1,2,00} (2.6.2.5)
and
Tn {St Gx) = 6teTxeYn (2.6.2.6)
We can how reverse the inequality (2.6.2.3): for a fixed iGfi and arbitrary
x G X,
||Tx|| = ||*t0rx||p = ||rn(*t©*)||p < ||rn||||*t©*||p = ||rn||||x||
This argument is liable to break down for (2.6.2.2), since it is most
unlikely that the mappings St © x : H —* X will be continuous. Instead,
however, we have the constant mappings 1 0 x : H —* X which send each

2.7 Enlargements
41
t G H into the same vector x: thus
||Tx|| = ||1 © Tx\\„ = \\T*(1 0 x)!^ < ||rn||||l © x\\„ = ||Tn||||x|| ■
If H is again a set and T G BL(X, Y), then there is also an operator Tn:
M(H, X) —> M(H, Y) between the spaces of "finitely additive set functions"
of (1.8.2.2), defined by setting
(r°M)(*) = W0) (2627)
for each K C H for each \i G M(fi, X)
The reader may like to verify that, in the notation of (1.8.2.3) and (1.8.2.4)
r^M^n,*) cM^n.y)
and (2.6.2.8)
l|rnM|li<||r|||Hli ifMGM^ax)
Also the bound of the operator Tn on M1(H,X) is equal to ||T||.
2.7 ENLARGEMENTS
If H has a bounded structure, or bornology, and T G BL(X, Y), then the
operator Tn respects the spaces c00(n,X), c0(H,X) and c1(H,X) of Section
1.9.
2.7.1 THEOREM If ft has a bounded structure and T G BL(X, y), then
rnc+(n,X) C c+(H,y) for each c, = Cqo,^^! (2.7.1.1)
There is equality
||c.(n,T)|| = ||r|| (2.7.1.2)
in each case.
Proof: If x G Xn and t G 0, then
x(i) = 0 ==> (Tnx)(*) = T{x{t)) = 0 (2.7.1.3)
it follows that if x G c00(n,X) then Tnx G c00(n,y). If fc > 0, then also
{t G H: ||x(t)|| < A:} C {t G H: ||r(x(t))|| < ||r||*} (2.7.1.4)
This implies that if x G c0(H,X) then Tx G c0(0,y). Finally, if x G
c^H,*), with z G X for which x- ze c0(n,X), then Tx - Tz = T{x -
z) G c0(n,y), so that Tx is in c^f^y). This completes the proof of

42
2. Bounded Linear Operators
(2.7.1.1). Inequality ||c+(n,r)|| < ||T|| follows from (2.6.1.2) with p = oo,
and is reversed by observing that the functions 6t © x of (2.6.2.4) are all in
c00(n,X). ■
Since the operator Tn : 1^(0,,X) —> l^Cl^Y) has the property that
Tnc0(fi,X) C c0(n,y), there is by Section 2.3 an operator induced from
/«„(*), X)/c0(n,X) to/^n^/co^y). We shall call it an "enlargement"
of the operator T:
2.7.2 DEFINITION If H has a nontrivial bornology and T G BL(X, y),
then the operator
Qn(r) : Qn(X) —> Qn(y) (2.7.2.1)
defined by setting
Qn(T)(x + c0(n,x)) = Tnx + c0(n,Y) (2722)
for each x G /^(njX)
is called an enlargement of T. When H = N we write
QN(T)=Q(T) (2.7.2.3)
Explicitly, if x = (x1,x2,x3)...) G ^(X)
Q(r)((x1,x2,x3,...) + c0(x)) = (rx1,rx2,rx3,...) + c0(y) (2.7.2.4)
It is clear that Qq(T) is bounded and linear, with bound < ||T||. If H
is nontrivial then there is equality:
2.7.3 THEOREM If H has a nontrivial bornology and T G BL(X,y),
then
||Qn(T)|| = ||r|| (2.7.3.1)
If also S G BL(X,y) and U G BL{Y,Z), then
Qn(sT + tS) = sQn(T) + *Qn (S) for each s, * G K (2.7.3.2)
and
Qn(^) = Qn(U)Qn(T) (2.7.3.3)

2.8 Shift Operators
43
Proof: We know already that ||Qn(T)|| < ||T||. Conversely, if x e X is
arbitrary write
qn(x) = {ze i^xy.z-1 o x e c0(n,x)} (2.7.3.4)
for the set of sequences z which "converge to x": then by (1.9.1.3)
||rx|| = ||qn(rx)|| = ||Qn(r)qn(x)|| < ||Qn(r)||||qn(*)ll = IIQn(r)IIMI
This proves (2.7.3.1), and the rest is clear. ■
One observation will be useful later: if Y = X and T = I : X —> X,
then Qn(T) is the identity on the space Qn(X):
QM = I (2.7.3.5)
2.8 SHIFT OPERATORS
Suppose, in contrast to Section 2.6, that X is a linear space and that
<f>: A —> H is a mapping of sets: then the mapping X^ : Xn —> XA defined
by setting
{X*x){t) = x{<t>{i)) for each * G A, for each x e Xn (2.8.0.1)
is well defined and linear. More generally, if <f> : A0 —> H is defined only on
a subset A0 C A, we shall define X^ : Xn —> XA by setting, for each tGA
and x e Xn,
. (xUii)) if * e A0
{X+x){t) = \ m" ° (2.8.0.2)
[ 0 if tg A0
If X is a normed space then we again restrict to the spaces lp(Q,X):
2.8.1 THEOREM If X is a normed space and <j> : A0 C A -> H is a
mapping of sets then there is inclusion
X^00(n,X)Ci00(A,X) (2.8.1.1)
and inequality
ll***lloo < Iklloo for each x e /oo(n.^0 (2.8.1.2)
If in particular <£ is one-one, then the analogous results hold for p = 1 and
p = 2. If <f> is an everywhere-defined continuous mapping of topological
spaces, then also
X+C^X) C CTO(A,X) (2.8.1.3)

44
2. Bounded Linear Operators
Proof: If x G Xn and t G A0 then
||(X**)(*)|| = ll*(*M)ll < sup ||x(*)|| = \\x\U (2.8.1.4)
sen
and then (2.8.1.2) follows if we take the supremum over t G A0, since the
t G A \ A0 are powerless to affect the result. We have in fact proved an
extended version of (2.8.1.2), from which (2.8.1.1) follows. If <f> is one-one
and p = 1 or p = 2, then for arbitrary finite A' C A0 and x G Xn,
£||(x**)(*)||p = £N^))||p
tEA' t€A'
(2.8.1.5)
= E ll*Mllp<£ll*MP
*e<t>(A') flGA
Raising to the power 1/p and taking the supremum over all finite A' C A0
gives the (extended) analogue of (2.8.1.2), and hence also the analogue of
(2.8.1.1). For (2.8.1.3) we observe
X*C(H, X) C C(A, X) (2.8.1.6)
by the composition of continuous mappings. ■
If A and H have bounded structures then we shall call a mapping
<f> : A —* H cofinal if
for each K G Bn there is H G SA for which <f>{A \ H) C H \ K (2.8.1.7)
If for example H = A = N the condition is that
<f>(n) —► oo as n —► oo (2.8.1.8)
2.8.2 THEOREM EXisa normed space and <j> : A -» H is cofinal, then
X+c+{n,X) C c,(A,X) for each c, = Coq,^,^ (2.8.2.1)
Proof: If x G c00(H, X), then {t G H: x(*) = 0} = H \ K is the complement
of a "bounded" subset if C H, and by (2.8.1.7) there is a "bounded" subset
iZ" C A for which
5 e A \ H => <£(s) G H \ X => x(<£(s)) = 0
so that
{5 G A: {X*x){s) ^ 0} C H is "bounded"

2.8 Shift Operators
45
This proves the first of the inclusions of (2.8.2.1), and the same
argument, with {t G H: \\x(t)\\ < 1/n} in place of {t G Vl:x(t) = 0}, gives the
second. For the third inclusion suppose x G c1(H, X), so that there is y G X
with x — 1 © y G c0(n,X), and
X+x.-X+{l © y) = X*{x -y)e c0(A,X)
by the previous result, giving X^x G c1(A,X). ■
The most celebrated example of (2.8.0.1) is the simple backward shift
(x1,x2,x3,...) —► (x2,x3,x4,...) (2.8.2.2)
on the space lp = Zp(N, K): the mapping <j> is given by <j>{n) = n + 1. For
the forward shift
(x1,x2,x3,...) — (0,x1,x2,...) (2.8.2.3)
on lp we need (2.8.0.2), defining <t>(n) = n - 1 if n G N + 1 = {2,3,4,....}
The superficially similar operators given by <f>{n) = n ± 1 for each n G Z =
{... — 1,0,1,2,...} turn out to behave quite differently.
If H = N, there is inclusion, for arbitrary normed spaces X,
c00(X) C l^X) C l2(X) C c0(X) C ^(X) (2.8.2.4)
and we might observe that, if ^(X), /2(X) and c0(X) are each given their
appropriate norms, then each of the natural injections
j : /i(X) —► c0(X) J : /2(X) —, c0(X) J : lx{X) — /2(X)
(2.8.2.5)
is bounded, with bound < 1; this is clear from the inequalities (1.8.1.1). In
each case we also have \\J\\ = 1, as is clear by looking at the vectors 6t © x
of (2.6.2.5). Finally, since in each case the closure of the subspace c00(X)
is the whole space, the same is true of the closure of the range of J. In
the reverse direction we have a linear mapping J^ : XN —> XN defined by
setting
(Ei*)n = Si + *2 + --- + Zn
(2.8.2.6)
for each n G N, for each x G X
The reader may like to verify the inclusion
Ei(<iP0)CU*) (2-8.2.7)
and also that the induced mapping J^ : /X(X) —> /^(X) is bounded, with
II Ex || = i.

46
2. Bounded Linear Operators
If H C R is an interval and a G H then there is a well-defined linear
mapping
/a:C(n,K)—>C(n,K) (2.8.2.8)
defined by setting
t
(Sax)(t)= J x(s)ds (2.8.2.9)
s=a
the reader may interpret this either as a limit of Riemann sums or as an
antiderivative. When the interval H is bounded (closed or not) then there
is also inclusion
/^(n^CC^K) (2.8.2.10)
and the resulting linear mapping is again bounded. In the sequel we will
see that such a mapping can be induced on the spaces C(H,X) for certain
kinds of normed spaces X (those which are "complete"). Perhaps the most
famous linear mapping of analysis is differentiation: it is disturbing to
report that it is not bounded. If, for example, Y = C[0, l] = C^fO, l] and
X is the subspace of Y consisting of those x G Y for which the derivative
x1 exists and belongs to Y, then the mapping D : x —+ xf is linear from X
to Y. We claim that, relative to the usual norm of Y, D is not bounded.
Indeed if zn G X is denned by setting zn(t) = tn for each t G [0, l] then
||;gn||00 = l and H-D^Hoo = ||nsn~1||00 = n for each n G N
(2.8.2.11)
Thus (2.1.1.1) cannot hold for any k > 0.
It is perhaps fortunate for the relevance of the theory of bounded linear
operators that much of the theory of the operator D can be carried out in
terms of the operators Ja, which are bounded.
2.9 COMPOSITION OPERATORS
If T G BL(X, Y) and if W is another normed space then we shall write
LT = BL{W,T):U —► TU from BL(W, X) to BL(W, Y) (2.9.0.1)
and
RT = BL(r,W) : V —► VT from BL{Y,W) to BL{X,W) (2.9.0.2)
for the left and right compositions, or multiplications, associated with T.
Evidently LT is obtained by further restricting the operator Tw : Xw -»
Yw of (2.6.0.1), while RT is a special case of the operator W* of (2.8.0.1).
As we might expect, LT and RT are also bounded linear operators:

2.9 Composition Operators
47
2.9.1 THEOREM If T G BL(X,y) and W is a normed space then LT
and RT are bounded and linear, with ||LT|| < ||r|| and \\RT\\ < \\T\\. If
also S G BL(X,y) and U G BL{Y,Z) then
■^aT+ts = sLj,-\-tL§ and RsT+ts = ^Rt^^^s ^or eacn s,£ G K
(2.9.1.1)
and
ut = ^u t and. Rut = Rj>Ru ^2.9.1.2]
Proo/: All this is clear from Theorem 2.2.2. ■
We shall see later—with some difficulty, in fact—that provided W ^ 0
there is equality
||LT|| = ||r|| = \\RT\\ for each T G BL(X,y) (2.9.1.3)
The reader may like to verify the first equality here if W = X and if
W = K, and to verify the second equality if W = Y. When W = K the
space BL(W, X) is just X:
2.9.2 THEOREM If X is a normed space then there is isometric
isomorphism
x <—► Lx : X ^ BL(K,X) (2.9.2.1)
where
Lx{t)=txeX foreachJGK (2.9.2.2)
Proof: This is clear. ■
If X and Y are normed spaces, and x G X, we shall also write
£x : r —► Tx from BL(X, y) to Y (2.9.2.3)
and call it an evaluation map.
When W = K the left multiplications LT of (2.9.0.1) are
indistinguishable from the operators T; by contrast the right multiplications RT =
BL(T, K) of (2.9.0.2) will play a crucial role in the theory to follow. When
y = K the elements of the space BL(X,Y) = BL(X, K) are known as
bounded linear functionals. Finally, if X and Y are normed spaces and if
y G y is a vector and / G BL(X, K) is a linear functional then the mapping
fQy.x —► f{x)y from X to Y (2.9.2.4)

48 2. Bounded Linear Operators
as in (1.8.0.5), is bounded and linear: the reader may like to verify that
||/0y|| = ||y||||/|| for each yG7and/G BL(X,K) (2.9.2.5)
For later reference, we shall make a distinction between this "rank one
operator" and a linear functional on a space of operators, writing
/ ® y : S —► /(St/), from BL(Y,X) to K (2.9.2.6)
If X, Y and Z are normed spaces then we can look at mappings from
X into the operators from Y to Z, and from Y to the operators from X
to Z.
2.9.3 THEOREM If X, Y and Z are normed spaces then there is
isometric isomorphism
$ <—► # : BL[X,BL{Y,Z)) = BL{Y,BL{X,Z)) (2.9.3.1)
given by the formula
$(x)(y) = #(y)(x) for each xe X and each yG7 (2.9.3.2)
Proof: In the notation (2.9.2.3) we have
¥(y) = £y o $ for each yeY (2.9.3.3)
so that the correspondence (2.9.3.2) gives a linear mapping $ —> \£ of bound
< 1. Also
$(x)=£zo# for each xeX (2.9.3.4)
giving the isometry. ■
In the sequel (Section 11.7) we shall see that we can also write
BL(X,BL(y,Z)) = BL{W,Z) (2.9.3.5)
with a normed space W = X ® Y depending only on X and Y.
2.9.4 THEOREM If H is a nonempty set and X and Y are normed spaces
then there is isometric isomorphism
Q^+TzBLiXJ^Y)) s/^n.BLpT.y)) (2.9.4.1)
given by the formula
$(*)(*) = Tt(x) for each x G X and each * G H (2.9.4.2)

2.10 Normed Linear Algebras
49
There are also bounded linear mappings
T —> T.{x.) : /^(n.BL^y)) — ^(11, y) for each x G l^X)
(2.9.4.3)
and
x —> r.(x.): /oo(n,x) —> /oo(n,y) for each r g /^(n.BLpr.y))
(2.9.4.4)
given by the formula
(T.[x.))(t) = Tt(xt) for each * G H (2.9.4.5)
Proof: The argument for the first part is the same as for Theorem 2.9.3,
and the second part can be left to the reader. ■
2.10 NORMED LINEAR ALGEBRAS
If A is a normed linear algebra in the sense of Definition 1.10.1, then each
element a G A gives rise to two multiplication operators on the normed
space A,
La:b —► a • b from A to A (2.10.0.1)
and
Ra:b —> 6 • a from A to A (2.10.0.2)
2.10.1 THEOREM If A is a normed linear algebra then for each a G A
we have
\\La\\ = \\a\\ = \\Ra\\ (2.10.1.1)
If also 6 G A then
Lsa+tb = sL + tLb and ^*a+t& = sRa + *^6 for each 5^K
(2.10.1.2)
and
La6 = LaLb and £a6 = £6£a (2.10.1.3)
Proof: We already know that
||La|| < ||a|| and ||#a|| < ||a|| (2.10.1.4)
This is simply a restatement of (1.10.1.1). To reverse the inequalities,
evaluate at the identity 1: provided 1^0
|M| = ||L0(1)|| < ||L0||||1|| = ||LJ| and ||a|| = ||J2a(l)|| < ||fi8||
(2.10.1.5)

50
2. Bounded Linear Operators
This argument does break down if 1 = 0, but if that happens, then A = 0
and ||a|| = 0. We have proved (2.10.1.1), and everything else is clear. ■
Theorem 2.10.1 says, in particular, that there is isometric isomorphism
a <—► La : A ^ {La: a e A} C BL(A, A) (2.10.1.6)
Here we intend the symbolism to record the preservation of multiplication,
as well as everything else. In a sense, therefore, we have no need of
Definition 1.10.1: we need only consider subalgebras of the algebras BL(X, X).
If A and B are normed linear algebras then it is relevant to look at
those operators T G BL(A, B) which are also homomorphisms, in the sense
that
T(l) = 1 and T{ab) = T{a)T{b) for each a, 6 G A
(2.10.1.7)
An example occurs in (2.10.1.6). More generally, T may satisfy
T(l) = 1 and T(ab) + T(ba) = T(a)T(b) + T(b)T(a)
(2.10.1.8)
for each a, b G A
We sometimes call this the Jordan property. The reader should be warned
that the product of two homomorphisms is a homomorphism, but not in
general the sum: the homomorphisms from one normed algebra to another
do not even form a linear space. We shall write
HBL(A,£) ={Te BL(A,B) : T is a homomorphism} (2.10.1.9)
If T is a homomorphism of normed algebras, then its range is a
subalgebra in the sense of (1.10.1.4), while its null space is an ideal in the sense
of (1.10.1.5):
2.10.2 THEOREM If A and B are normed algebras and T G HBL(A, B)
is a homomorphism, then T(A) is a subalgebra of B and T_1(0) is a closed
two-sided ideal. If A is a subalgebra of B and if J is a closed two-sided
ideal of A, then the injection J : A —+ B and the quotient K:A—+A/J
are homomorphisms.
Proof: This is left to the reader. ■
If a G A, then the mappings La and Ra are usually not
homomorphisms. Their ranges and null spaces are however "one-sided ideals" of
A: specifically L~ 1(0) and La(A) are right ideals of A, while R'1^) and

2.11 Partially Ordered Spaces
51
Ra{A) are left ideals. As usual both La1(0) and Ra 1(0) are closed, while
La(A) and Ra{A) need not be.
2.11 PARTIALLY ORDERED SPACES
If X and Y are partially ordered linear spaces, with cones X+ and Y+ as
in (1.11.3.4), then a mapping T : X —* Y will be called positive if it has the
property
r(x+) c y+ (2.H.0.1)
and will be called increasing, or monotonic, if for each x,y € X
y - x e X+ =^ Ty - Tx eY+ (2.11.0.2)
Evidently if T : X -> Y is linear then
T positive <=> T monotonic (2.11.0.3)
It is clear also that sums and products, and positive scalar multiples, of
positive linear mappings are positive. Thus, the positive linear mappings
nearly form a cone in the space of all linear mappings. The only condition
that is not clear is (1.11.3.8). We can say that if T : X —> Y is linear then
T positive and - T positive <=> T(X+) = {0} (2.11.0.4)
For this to ensure that T = 0 we have to know that the cone X+ is
generating, in the sense that the linear subspace of X generated by X+ is the
whole of X. This can be simplified: if X is a real vector space the condition
is
X+ - X+ = X (2.11.0.5)
while if X is a complex vector space the condition is
X+ - X+ + iX+ - iX+ = X (2.11.0.6)
2.11.1 THEOREM If X and Y are partially ordered normed spaces and
X has a generating cone X+, then BL(X, Y) is a partially ordered normed
space with cone
BL+(X,y) = {TeBL{X,Y):T{X+) C Y+} (2.11.1.1)
Proof: All that remains to be checked is that BL+(X, Y) is closed: we
leave this to the reader. ■

52
2. Bounded Linear Operators
E X and Y have generating cones, so that BL(X, Y) is a partially-
ordered normed space, it ip not clear that also BL(X, Y) should have a
generating cone.

3
Invertibility and Singularity
A bounded linear operator T : X -» Y between normed spaces is called
"invertible" if it has a bounded linear inverse, in the sense of S G BL(Y, X)
for which ST = I on X and TS = I on Y. If T is to be invertible then it
will have to be "nonsingular" in various ways; conversely, if T is nonsingular
in enough of these ways then it is invertible.
3.1 INVERTIBILITY AND ISOMORPHISM
If normed linear spaces X ~ Y are isomorphic in the sense (1.7.0.1), then
we can write the one-one correspondence X <-► Y in the form
x <—► Tx : X <—► Y (3.1.0.1)
thus defining a mapping T : X —* Y which will be one-one and onto, and
linear in the sense of (2.1.0.1). To preserve topology it must also satisfy
Nbdy {Tx) = T(Nbdx(x)) for each x e X (3.1.0.2)
In particular the mapping T will have to be continuous. By symmetry we
can equally well write the correspondence X <-► Y in the form
y <—> Sy : Y <—► X (3.1.0.3)
thus the mapping T : X —► Y must have an inverse S : Y —► X which is
both linear and continuous. Such mappings T are called invertible.
3.1.1 DEFINITION T G BL(X,Y) is said to be invertible if there is
SeBL{Y,X) for which
ST = I eBL{X,X) and TS = IeBL{Y,Y) (3.1.1.1)
53

54
3. Invertibility and Singularity
We shall write BL *(X, Y) for the set of invertible mappings in BL(X, Y),
and
aBL(X,y) = BL(X,y)\BL_1(X,y) (3.1.1.2)
We can summarize the connection between invertible mappings and
isomorphism:
X ~ Y <^> BL_1(X,y) ^ 0 (3.1.1.3)
The identity / : X —► X is always invertible, and provides its own
inverse. The zero 0 : X —► Y is almost never invertible, unless in fact
X = Y = 0, in which case 1 = 0; more generally if an idempotent E = E2 :
X —► X is invertible, then E = I. The inverse of an invertible mapping is
unique, and the product of invertible mappings is invertible:
3.1.2 THEOREM If T e BL(X,y) is invertible, then its inverse T'1 is
unique. If also S G BL(y, Z) is invertible then so is ST, with
(ST)'1 =T~1S-1 (3.1.2.1)
Proof: If U and V are inverses to T G BL(X, Y), then
U = UI = U(TV) = (UT)V =IV = V (3.1.2.2)
If S and T are invertible, with inverses 5_1 and T_1, then
[T^S'^iST) = r-1(5"15)T = T~lT = I
and (3.1.2.3)
{ST){T~1S-1)=SS-1 = 1 m
The formula (3.1.2.1) is the famous "reversal of product" for inverses.
In general the sum of two invertible operators need not be invertible:
3.1.3 THEOREM If T e BL(X,y) and S e BL(y,X), then
i-STeBL-1 (x,x)^^i-ts eBL-1 (y,y) (3.1.3.1)
If T e BL_1(X,y) is invertible and U e BL(X,y) then
J - T~lU e BL_1(X,X) =^ T - U e BL_1(X,y) (3.1.3.2)
and
/ - ut~1 e BL-1 (y, y)=^t-u e bl_1 (x, y ) (3.1.3.3)

3.1 Invertibility and Isomorphism
55
Proof: Suppose that J — ST is invertible with inverse U, so that
U{I - ST) = I = {I - ST)U (3.1.3.4)
Then we claim
(J + TUS){I-TS) = I={I- TS){I + TUS) (3.1.3.5)
For example
(J + TUS){I -TS) = I + T(U-I- UST)S = 1 + 0 (3.1.3.6)
This proves forward implication in (3.1.3.1), and the reverse is obtained by
interchanging S and T. For (3.1.3.2) use (3.1.2.1) with the observation that
T-U = T{I-T~1U)
The proof of (3.1.3.3) is similar; alternatively it follows from (3.1.3.2) and
(3.1.3.1). ■
The significance of the second part of Theorem 3.1.3 will be that if
we add a "sufficiently negligible" operator to an invertible operator we get
another.
In (3.1.1.2) we characterize topological isomorphism X ~Y: for isom-
etry X = Y as in (1.7.0.4) we require
T G BL_1(X,y) with ||Tx|| = ||x|| for each xeX (3.1.3.7)
Necessary and sufficient is
T G BL_1(X,y) with ||r|| < 1 and {{T^W < 1 (3.1.3.8)
We can extend the concept of isomorphism between spaces to one of
isomorphism between operators: this if T G L{X, Y) and Tf G L{Xf,Yf) are
linear we shall write
T ~T' (3.1.3.9)
to mean that there are invertible operators U G BL_1(X, X1) and V G
BL"1 {Y,Y') for which
VT = T'U (3.1.3.10)
and write
T = Tf (3.1.3.11)
if, in addition, the operators U and V satisfy (3.1.3.7). Evidently if

56
3. Invertibility and Singularity
(3.1.3.10) holds then
T bounded <^ T' bounded (3.1.3.12)
while if (3.1.3.11) holds then also
||r'|| = ||r|| (3.1.3.13)
We conclude with the remark that the formation of inverses is itself a
continuous process:
3.1.4 THEOREM If X and Y are normed spaces, then the mapping
T —-> T~l : BL_1(X,y) —► BL{Y,X) (3.1.4.1)
is continuous at each point S G BL_1(X, Y).
Proof: If S and T are both in BL_1(X,y), then
T~l -S~l =T-1{S-T)S~1
= (T-1 - 5_1)(5 - r)5_1 + S'^S - r)5_1 (3.1.4.2)
and hence
(i - \\s - rmi^-^Diir-1 - s_1|| < lls^Hlls - r||||-s,-1|| (3.1.4.3)
This forces T"1 -> S"1 as T -> 5: specifically, as soon as
lis-rims"1!! < | (3.1.4.4)
we have
||r-i_5-i|| <2||5-1||2||r-5||—>0 as ||r-5|| —>0 ■ (3.1.4.5)
3.2 MONOMORPHISMS AND EPIMORPHISMS
We recall the various subspaces associated with T G BL(X, Y) in Section
2.3:
3.2.1 DEFINITION T G BL(X, Y)
and is said to be onto if
T{X)
is said to be one-one if
= {0} (3.2.1.1)
= Y (3.2.1.2)

3.2 Monomorphisms and Epimorphisms
57
More generally T is said to be dense if
cl(TX) = Y (3.2.1.3)
We shall write
7rleftBL(X,y) = {Te BL(X,Y):T is not one-one} (3.2.1.4)
and
7rrightBL(X,y) = {Te BL(X,y):T is not dense} (3.2.1.5)
ETG BL(X, Y) is invertible in the sense of Definition 3.1.1 then it will
certainly be both one-one and onto; conversely, in the presence of certain
background conditions, if T is one-one and onto then it is also invertible.
To see this, however, requires a deep theorem: much of what follows will
be devoted to just that. We begin with a certain "monomorphic" property
of one-one operators, and find that the corresponding "epimorphisms" are
the dense operators rather than the onto operators. These notions use the
left and right multiplications LT and RT of Section 2.9:
3.2.2 THEOREM If T G BL(X,y), then
T one-one «<=> LT = BL(W, T) one-one for all normed spaces W
(3.2.2.1)
and
T dense <=> RT = BL(T,W) one-one for all normed spaces W (3.2.2.2)
Proof: If T G BL(X,y) is one-one, then for arbitrary W and U G
BL{W,X)
TV = 0 ==> TUw = 0 for each w e W ==> Uw = 0 for each w e W
(3.2.2.3)
which is that we mean by U = 0. Conversely, if T is not one-one then the
null space T_1(0) is not {0}, and hence the natural injection ker(T) = J :
T_1(0) —► X is not the zero operator. Thus
U = ker(T) =^TU = 0^U (3.2.2.4)
Towards (3.2.2.2), if T is onto and V : Y -► W is linear, then VT = 0 =>
V = 0. If we only assume that T is dense, but insist that V is bounded,
then
VT = 0 ==> VTx = 0 for each xeX
=^Vy= lim VTxn = 0 for each y eY (3*2-2*5)

58
3. Invertibility and Singularity
Conversely, if T is not dense then the quotient Y/ cl(TX) is not 0, and the
quotient mapping coker(T) = K : Y —► Y/ c\(TX) is not the zero operator.
Thus,
V = coker(T) ==> VT = 0 ^ V m (3.2.2.6)
There is an alternative derivation for the first part of Theorem 3.2.2:
we can take the space W to be the scalar field K. Indeed, if T is not
one-one there will be x G T_1(0) for which x ^ 0, and now we can take
U = Lx : t -► tx as in (2.9.2.2):
0 ^ x e T_1(0) =*► TLX =0^LX (3.2.2.7)
Very much later in Section 5.6 we shall see that we can also use W = K for
the second part of Theorem 3.2.2, but only with the help of another deep
theorem.
The reader is invited to speculate under what conditions on T the
operators LT will all be dense, or the operators RT. The conditions under
which a product ST is either one-one, or dense, are interesting:
3.2.3 THEOREM If T e BL(X,Y) and S e BL{Y,Z) then there is
implication
S one-one and T one-one => ST one-one => T one-one (3.2.3.1)
and
S dense and T dense =*► ST dense =*► S dense (3.2.3.2)
Proof: If S and T are both one-one and if STx = 0 then Tx = 0 and hence
x = 0, which is the first implication of (3.2.3.1). Also if ST is one-one and
Tx = 0, then also STx = 5(0) = 0, giving x = 0. The same argument,
using the operators RT and Rs, gives (3.2.3.2): alternatively the reader
may like to supply the direct argument. ■
We remark that also
S onto and T onto =*► ST onto =*► S onto (3.2.3.3)
ETG BL(X, Y) and if W is a closed subspace of X then whether or
not T is one-one on X can be tested by looking at the spaces W and X/W.
3.2.4 THEOREM If T e BL(X, Y) and if^CXandZCF are closed
subspaces then
T one-one <=> r_1(0) n V7 = {0} and T_1(0) C W (3.2.4.1)

3.2 Monomorphisms and Epimorphisras
59
and
T dense <=► c\{Z + cl(TX)) = Y and Z C cl{TX) (3.2.4.2)
Proo/: The right-hand side of (3.2.4.1) is equivalent to r_1(0) = {0}. If
T is dense then the conditions on the right-hand side of (3.2.4.2) certainly
hold. Conversely, if they are assumed, then for each y £ Y there are (xn)
in X and (zn) in Z for which ||y — Txn — zn\\ < l/n, and (xfn) in X for
which \\zn - Tx'J < l/n, giving \\y - T{xn + x(J|| < 2/n. ■
One way round, one-oneness and denseness can be tested by
enlargement.
3.2.5 THEOREM If T e BL{X,Y), then
Q(r) one-one =^ T one-one (3.2.5.1)
and
Q(r) dense =J> T dense (3.2.5.2)
Proof: If T is not one-one then there is x G X with i^0 = Tx, and hence
q(x)^0 = Q(r)q(x) (3.2.5.3)
where q(x) = qN{x) is given by (2.7.3.4). This proves (3.2.5.1). If Q(T) is
dense then for each y G Y and each e > 0 there is xe = (x*) in /00(X) for
which
limsup \\y - Txen\\ < e (3.2.5.4)
n—*oo
Thus, for each e > 0 there is ze = xeN for which ||y - Tze\\ < 2e. ■
The analogues of (3.1.3.1) hold:
3.2.6 THEOREM If T e BL{X,Y) and S e BL{Y,X), then
I- ST one-one <=> I-TS one-one (3.2.6.1)
and
J - ST dense <=> I-TS dense (3.2.6.2)
Proof: If J — ST is one-one and y G Y, then
(/ - TS)y = 0=^{I- ST)Sy = S{I - TS)y = 0
=j> Sy = 0 =}> y = T(5y) = 0

60
3. Invertibility and Singularity
This proves (3.2.6.1). Towards (3.2.6.2) suppose that I - ST is onto, so
that if x G X then x = (I — ST)x' for some x' G X: then it follows
yeY=^y = {I-TS){y + T{Sy)') where Sy = {I - ST){Sy)'
(3.2.6.3)
Thus we have shown
J - ST onto <=► I-TS onto (3.2.6.4)
To prove (3.2.6.2) suppose that ||x - {I - ST)x'€\\ < e and argue
\\y-{I-TS){y + T{Sy)'t\\
= \\T{Sy-{I-ST){Sy)'e)\\<\\T\\e ■
The gulf between invertibility, and being one-one and dense, can be
seen in the canonical factorization (2.3.3.2): evidently the mapping
core(T) : X/T~10 —► c\[TX) is always both one-one and dense, and, in
the sense of (3.1.3.10)
T one-one and dense <=>► T = core(T) (3.2.6.6)
Indeed we have
T one-one <*=> ker(T) invertible (3.2.6.7)
and
T dense <*=> coker(r) invertible (3.2.6.8)
In a sense, the kernel and the cokernel mappings perform a preliminary,
crude analysis of the invertibility or otherwise of T G BL(X, Y), and if we
clear them away, we are left with the mapping core(T), which in a sense
performs more delicate experiments:
3.2.7 DEFINITION T G BL(X, Y) will be called proper iff
core(T) is invertible (3.2.7.1)
Evidently
T invertible <$=> T one-one, dense and proper (3.2.7.2)
3.3 BOUNDEDNESS BELOW
In general, it turns out not to be sufficient for T G BL(X, Y) to be invertible,
that it be one-one and dense. To bridge the gap we can either strengthen
one-oneness, or strengthen denseness. Here we strengthen one-oneness:

3.3 Boundedness Below 61
3.3.1 DEFINITION T e BL(X,Y) is said to be bounded below if there
is k > 0 for which
||x|| < Jfc||rx|| for each xeX (3.3.1.1)
and will be called closed if
T is bounded below and TX = c\(TX) (3.3.1.2)
We shall write
rleft BL(X, Y) = {Te BL(X, Y): T is not bounded below} (3.3.1.3)
and
rleft BL(X,y) = {Te BL{X,Y):T is not closed} (3.3.1.4)
For example if ||Tx|| = ||x|| as in (3.1.3.6) then T is bounded below.
In general, boundedness below lies somewhere between invertibility and
one-oneness, and has the same kind of semigroup property as one-oneness.
3.3.2 THEOREM If T e BL(X,y) and S e BL{Y,Z) then there is
implication
5, T bounded below ==> ST bounded below ==> T bounded below
(3.3.2.1)
and
5, T closed =^ ST closed =^ T closed (3.3.2.2)
Also
T invertible =*► T closed =*► T bounded below =*► T one-one (3.3.2.3)
Proof: If ||x|| < Jfc||rx|| and ||y|| < k'\\Sy\\ for each x e X and each yeY,
then
||x|| < Jfc||rx|| < A;'A;||5rx|| for each x e X
giving the first part of (3.3.2.1). If instead ||x|| < A;"||5rx|| then
||x|| < A;"||5rx|| < A;"||5||||rx|| = Jfe||rx|| with k = k"\\S\\
giving the second part of (3.3.2.1). Toward (3.3.2.2) we claim that if T e
BL(X,y) is closed and K C X then
K = c\{K) =}> T{K) = c\{TK) (3.3.2.4)

62
3. Invertibility and Singularity
for if ||2/ — Txn\\ —► 0 with y £ Y and xn G K, then, in particular, y G
cl(rX) = TX, so that there is x G X for which y = Tx: now
||z - xj < k\\Tx - TxJ = k\\y - TxJ — 0
so that x e K and hence y = Tx e T(K). Applying (3.3.2.4) with S in
place of T and TX in place of K gives the first part of (3.3.2.2). If ST is
closed and \\y - Txn\\ -► 0 then \\Sy - STxn\\ -► 0, so that Sy = STx for
some iGl: now
\\y - Tx\\ < k'\\Sy - STx\\ = 0 (3.3.2.5)
For (3.3.2.3) the first part follows from the second part of (3.3.2.2) with
S = T_1, the second part is obvious, and the third part follows from the
argument of (3.3.2.5). ■
The set of bounded below operators has an interesting topological
property:
3.3.3 THEOREM If X and Y are normed spaces then
{J G BL(X,Y):T is bounded below} is an open set (3.3.3.1)
Proof: If T G BL(X,Y) is bounded below, with k > 0 satisfying (3.3.1.1),
and if T' e BL(X,F) satisfies A;||r'-r|| < 1, then (1.2.0.11) gives, for each
xeX,
\\T'x\\ > \\Tx\\ - \\{T' - T)x\\ > i(l - JfeHT' - r||)||x|| ■ (3.3.3.2)
Boundedness below can be tested by left composition operators:
3.3.4 THEOREM If T e BL{X,Y), then
T bounded below <$=>► LT bounded below for all normed spaces W
(3.3.4.1)
Proof: If T G BL(X,F) is bounded below, with k > 0 satisfying (3.3.1.1),
then for arbitrary W and U G BL(W,X) we have ||0ti;|| < Jfc||r0ti;|| for
each w €iW, hence
\\U\\= sup \\Uw\\ < sup ife||r^ti;||=ife||ri7||
IMI<i IIHI<i
Thus, each LT = BL(W,T) also satifies (3.3.1.1). Conversely, if LT =
BL(K,T) is bounded below, where W = K is the scalar field then there is

3.3 Boundedness Below
63
k > 0 for which, for each iGl,
||x|| = ||£J| < k\\LT(Lx)\\ = k\\LTx\\ = k\\Tx\\
Here we are using the same operators Lx of (2.9.2.2) as in (3.2.2.7). ■
Boundedness below can be tested by enlargement:
3.3.5 THEOREM If T G BL(X,y), then
T bounded below «$=>• Qq{T) one-one for all bornological spaces 0
(3.3.5.1)
In particular, taking 0 = N,
Q(r) one-one =>• T bounded below =>• Q(T) bounded below (3.3.5.2)
Proof: Suppose T is bounded below, with k > 0 satisfying (3.3.1.1), and
suppose that x G l^Q^X): we claim
Tnx e c0(n, y) =j> x e c0(n,x) (3.3.5.3)
This is because, for each e > 0,
{t e 0: ||x(t)|| > e}C{te n:ife||rx(t)|| > e} bounded in 0
Conversely, if T is not bounded below, then (3.3.1.1) must fail for each
k = n G N, so that there are vectors x'n G X for which HTxJJI < rc-1!!^!!'
and now if xn = x'n/\\xfn\\ then ||xn|| = 1 and ||T£n|| < l/n, giving
x = ixn) € /oo(X) with x £ c0{X) and TNx G c0{Y) (3.3.5.4)
This finishes the proof of (3.3.5.1), and gives the first part of (3.3.5.2). If
T is bounded below, with k > 0 satisfying (3.3.1.1), then for arbitrary
dist(z,c0(X)) = limsup||a;n|| < limsupfc||ra;n|| = fcdist(TNz,c0(F))
n—»-oo n—^oo
which is the second part of (3.2.5.2). ■
Boundedness below and closedness can be tested on a subspace and its
quotient:

64
3. Invertibility and Singularity
3.3.6 THEOREM If T e BL(X,Y) and W C X is a subspace then T is
bounded below iff there are k' > 0 and k" > 0 for which
HI < k'\\Tw\\ for each w e W
and
dist(x, W) < k"\\Tx\\ for each x e X
Also T is closed iff in addition
(3.3.6.1)
(3.3.6.2)
TW = c\{TW) and c\{TX) CTX + TW (3.3.6.3)
Proof: If T is bounded below, with k > 0 satisfying (3.3.1.1) then (3.3.6.1)
and (3.3.6.2) hold with k' = k" = k. Conversely, if x G X is arbitrary then
there is (wn) in W with ||x — tun|| —► dist(x, W), and now using (3.3.6.1)
lire < llx — tu_|| + l|wj|
" -" n" " n" (3.3.6.4)
< \\x-wn\\+k\\T{wn-x)\\+k'\\Tx\\ foreachnGN
Letting n —► oo and using (3.3.6.2) gives
||x|| < (1 + A;,||r||)dist(x,^) + A;,||rx|| < (A;'+A;"+A;V||r||)||rx|| (3.3.6.5)
If also (3.3.6.3) holds and ||y-rxn|| -»• 0 then y = Tx + Tw = T(x + w). ■
We conclude by noting that boundedness below, and closedness, do
their duty as strengthened versions of one-oneness:
3.3.7 THEOREM If T e BL(X, Y) then T is invertible iff
T is bounded below and onto (3.3.7.1)
and also iff
Also
T is closed and dense (3.3.7.2)
T closed <=> T one-one and proper (3.3.7.3)
Proof: If T is invertible then both (3.3.7.1) and (3.3.7.2) are satisfied, and
indeed are obviously equivalent to one another. Conversely, if (3.3.7.1)
holds, with k > 0 satisfying (3.3.1.1), then, in particular, T is one-one and
onto, hence has a linear inverse S : Y —► X. We claim that S is bounded:
for if y G Y is arbitrary then by (3.3.1.1)
\\Sy\\ < k\\TSy\\ = k\\y\\ (3.3.7.4)

3.4 Openness
65
Towards (3.3.7.3) we observe, using the canonical factorization (2.3.3.2),
T one-one =► T = J coie(T) with J closed (3.3.7.5)
also T and core(T) have the same range TX. Thus, if T is closed then it
is one-one, giving (3.3.7.5), which combined with (3.3.2.2) makes core(T)
closed, therefore by (3.3.7.2) invertible, which is what (Definition 3.2.5) we
mean by calling T proper. Conversely, if T is one-one and proper then again
(3.3.7.5) holds, and this time (3.3.7.2) combines with (3.3.7.5) to make T
closed. ■
3.4 OPENNESS
It would be satisfying to find that the strengthened version of "dense" was
"onto." At this stage the situation is much more complicated:
3.4.1 DEFINITION T e BL(X,Y) is said to be open if there is k > 0
for which
y e {Tx: \\x\\ < k\\y\\} for each y eY (3.4.1.1)
and is said to be almost open if there is k > 0 for which
y e cl {Tx: \\x\\ < k\\y\\} for each y eY (3.4.1.2)
We shall write
rrightBL(X,y) = {Te BL{X,Y):T is not open} (3.4.1.3)
and
rright BL(X, Y) = {T e BL(X, Y): T is not almost open} (3.4.1.4)
By translation invariance, T G BL(X, Y) is open if, and only if for each
KCX
K = int(ff) =}> T{K) = int(rfT). (3.4.1.5)
Openess lies somewhere between invertibility and ontoness, while almost
openess lies somewhere between openness and denseness:
3.4.2 THEOREM If T e BL(X,F) and S e BL{Y,Z) then there is
implication
5, T open => ST open =^ S open (3.4.2.1)
and
S,T almost open ==> ST almost open ==> S almost open (3.4.2.2)

66
3. Invertibility and Singularity
Also
T invertible =► T open =► T onto =► T dense (3.4.2.3)
and
T open =► T almost open =► T dense (3.4.2.4)
Proo/: If y G {Tx: ||x|| < k\\y\\} and 0 G {Sy: \\y\\ < k'\\z\\} for each y eY
and z €: Z then
0 G {STx: ||x|| < *||y|| < kk'\\z\\} for each zeZ
giving the first part if (3.4.2.1). If instead z G {STx: \\x\\ < k"\\z\\} then
« e {firiri Hirll = ||r*|| < lirilH^H < *##||r||||*||>
giving the second part of (3.4.2.1). The proof of each part of (3.4.2.2) is
almost the same, word for word, putting in the closure operator. The first
part of (3.4.2.3) follows from the second part of (3.4.2.1), with S = T"1,
and the rest is clear. ■
The almost open operators form an open subset of BL(X, Y):
3.4.3 THEOREM If X and Y are normed spaces, then
{T G BL(X,Y):T is almost open} is an open set (3.4.3.1)
Proof: If T G BL(X, Y) is almost open, with k > 0 satisfying (3.4.1.2), and
if &||T'—T|| < 1, we claim that T' is almost open, with k! = k/(l—S) in place
of A; whenever A;||r'-r|| < 6 < 1. Indeed if e > 0 withe+A;||r'-r|| = 8 < 1,
then for each y G Y there is xx G X for which ||y — Txx|| < e||y|| with
Ikill <*l|y||, so that also Wy-T'x^ < (e+*||r-r'||)||y|| = %||, and then
x2 G Xfor which \\y-T'x1-Tx2\\ < eHy-T'xJ with ||x2|| < JfcHy-T'xJ,
so also ||y — T'xx — Tfx2\\ < £2||y||- Inductively there is a sequence (xn) in
X for which
||y-r'(*1+*2+ • •• + *„)II <<nyII with||xn||<W',-1||y|| (3.4.3.2)
which gives
||*1+*2 + ... + *n||<(l + «5+.-. + 5n-1)<^M ■ (3.4.3.3)
One way around almost openness can be tested by composition
operators:

3.4 Openness
67
3.4.4 THEOREM If T G BL{X,Y), then
T almost open ==> RT bounded below for all normed spaces W (3.4.4.1)
Proof: If T G BL(X,Y) is almost open, with k > 0 satisfying (3.4.1.2),
then for arbitrary W and V G BL(Y,W)
||Vr||=sup{A;||Vry||:y€cl{rx:||x||<l}} = A; sup \\VTx\\ = k\\VT\\
IMI<i
which says that (3.3.1.1) holds for RT = BL(T, W). ■
Much later, in Section 5.6, using a deep theorem, we will find that if
RT = BL(T,W) is bounded below with W = K, then T has to be almost
open. We can also test for almost openness using enlargements.
3.4.5 THEOREM If T G BL{X,Y), then
T almost open ^=> Q^(^) almost open for all bornological spaces 0
(3.4.5.1)
In particular, taking 0 = N,
Q(r) almost open =^ T almost open => Q(T) open (3.4.5.2)
Proof: Suppose T G BL(X,Y) is almost open, with k > 0 satisfying
(3.4.1.2): then for arbitrary y G ^(OjY) and arbitrary <f> : 0 —► ]0, oo[
there is x G Xn for which
\\x[t)\\ < k\\y{t)\\ and \\y{t) - Tx{t)\\ < <f>{t) for each t G 0
(3.4.5.3)
In particular
IHIco < *ll»lloo < «> (3-4-5-4)
ensuring x G l^QjX). By taking <£(•) = e for arbitrary e > 0 we obtain
the forward implication in (3.4.5.1); by taking, if 0 = N with the finite
homology,
4>{n) = 1/n for each neN (3.4.5.5)
we also obtain the second implication of (3.4.5.2). For the first implication
of (3.4.5.2), which will also reverse the implication (3.4.5.1), take 0 = N
and suppose that Q(T) is almost open, with k > 0 satisfying the analogue
of (3.4.1.2): then for each y G Y and each e > 0 there is xe = (x£) in
l^iX) for which
limsup ||x* || < &||y|| and limsup ||y - Txen\\ < e. (3.4.5.6)
n—KDo n—*oo

68
3. Invertibility and Singularity
Thus if k < kf, and e>Owe can find ze = xeN for which
\\ze\\ < k'\\y\\ and \\y - Tze\\ < 2e ■ (3.4.5.7)
In section 5.7, using duality, we can improve (3.4.5.2) to the analogue
of (3.3.5.2).
Openness and almost openness can be tested on a subspace and its
quotient.
3.4.6 THEOREM If T G BL(X,Y) and Z C Y is a subspace, then T is
open iff there are k' > 0 and k" > 0 for which
y G {Tx: \\x\\ < k"dist{y,Z)} + Z for each y eY (3.4.6.1)
and
z G {Tx: \\x\\ < k'\\z\\} for each z G Z (3.4.6.2)
Also T is almost open iff there are k' > 0 and k" > 0 for which
y e cl {{Tx: \\x\\ < k" dist{Y, Z)} + Z) for each y eY (3.4.6.3)
and
z e cl {Tx: \\x\\ < k'\\z\\} for each z e Z (3.4.6.4)
Proof: If T is open, with k > 0 satisfying (3.4.1.1), then (3.4.6.1) and
(3.4.6.2) hold whenever k" > k! = k. Conversely, if y G Y is arbitrary,
then by (3.4.6.1) there are x G X and z G Z for which y = Tx + z with
||x|| < k"dist(y,Z) < k"\\y\\, and by (3.4.6.2) x' e X with Tx' = z and
||x'||<A;'||2||. Thus
y = T(x + *') with ||x + x'|| < ||x|| + k'\\y - Tx\\
<(l + *W|)*ly|| + *'||y||
This proves the first part. If instead T is almost open, with k > 0 satisfying
(3.4.1.2), then (3.4.6.3) and (3.4.6.4) hold with k" = k' = k. Conversely, if
y G Y is arbitrary, then by (3.4.6.3) there is (xn) in X for which
dist(y - Txn,Z) < 1/n and ||xn|| < k"dist{y,Z) < k"\\y\\
(3.4.6.6)
and by the Riesz lemma result (1.5.1.1) there is (zn) in Z for which
\\y-Txn-zn\\<dist(y-Txn,Z) + ± and \\zj < 2||y - TxJ
(3.4.6.7)

3.4 Openness
69
Finally by (3.4.6.4) there is (x'J in X for which
|K - 2VJ| < \ and ||x'J| < k'\\zn\\ (3.4.6.8)
n
Putting these together gives
||y-rxB-2VB||<3/n—►(>
and (3.4.6.9)
\\xn + x'n\\<{k" + 2k' + 2k'k"\\T\\)\\y\\. ■
We conclude by noting that openness does its duty as a strengthened
version of denseness:
3.4.7 THEOREM If T e BL(X, Y) then T is invertible iff
T is one-one and open (3.4.7.1)
Also
T open <$=>• T dense and proper (3.4.7.2)
Proof: If T is invertible then (3.4.7.1) holds. Conversely, if (3.4.7.1) holds,
with k > 0 satisfying (3.4.1.1), then in particular T is one-one and onto,
hence has a linear inverse S : Y —> X. We claim that S is bounded, for if
y G Y there is x G X with
T{Sy-x) =y-Tx = 0 and ||x|| < *||y|| (3.4.7.3)
Since T is one-one this gives
||5y|| = N|<*||y|| (3.4.7.4)
Towards (3.4.7.2) we claim
T dense =^ T = coie(T)K with K open (3.4.7.5)
This uses the canonical factorization (2.3.3.2), and the Riesz lemma
(1.5.2.1) which says that the quotient mapping K = coker(ker(T)) : X —►
X/r_10 satisfies (3.4.1.1) with k = l/t whenever 0 < t < 1. Now if T
is open, then it is also dense, giving (3.4.7.5), and (3.4.7.5) together with
(3.4.2.1) says that core(T) is open, therefore by (3.4.7.1) invertible. This
gives forward implication (3.4.7.2). Conversely, if T is dense and proper
then core(T) is open and now (3.4.7.5) together with (3.4.2.1) tells us that
T is open. ■

70
3. Invertibility and Singularity
The precise relationship between almost openness and invertibility
seems more elusive. As part of the argument for (3.4.7.2) we observed
that the quotient K : X —> X/Y associated with a closed subspace Y —► X
is always an open mapping. Another example is a nonzero linear functional
(bounded or not). Indeed, if 0 ^ / G L{X, K) then there be x1 G X for
which f{x1) = 1 (take xx = x0/f(x0) if f(x0) ^ 0): now
if t G K then t = /(toj and HtxJ < Wx^t] (3.4.7.6)
which is (3.4.1.1) with k = {{x^.
3.5 BOUNDARY MAPPINGS
Theorem 3.4.3 tells us that the almost open mappings form an open subset
of BL(X,y). Thus if T G BL(X,Y) is in the topological boundary of the
set of almost open mappings, then T is not itself almost open. It turns out
that T is not bounded below either. The auxiliary results needed for this
are of interest in their own right:
3.5.1 THEOREM If T e BL(X, Y) and Tn e BL(X, Y) for each neN,
then
T bounded below, Tn dense, and ||T - Tn|| —► 0 =^ T dense (3.5.1.1)
and
T bounded below and dense ==> T almost open (3.5.1.2)
Proof: Towards (3.5.1.1) suppose y G Y is arbitrary: then there is (xn) in
X for which
||y-TBa:J|<||r-rJ|—0 asn^oo (3.5.1.3)
which gives, using (3.3.1.1),
||y-rxn||<||y-rnxn|| + ||r-rn||||xn||<||y-rnxn|| + fc||r-rn||||rxj|
< \\y - Tnxn\\ + k\\T - Tj\\Txn - y\\ + k\\T - Tn\\\\y\\
so that
(i-*||r-rj|)||y-r*B|| < ||y-rB«j|+*||r-rj|||y|| < (i+fc||y||)||r-rj|
This forces ||y — Txn|| —► 0, and since y G Y is arbitrary, makes T dense.
Towards (3.5.1.2) we can apply the Riesz lemma result (1.5.1.1) to see
that
Y C cl(Z) ==> y G c\{z G Z: \\z\\ < \\y\\} for each y G Y (3.5.1.4)

3.5 Boundary Mappings
71
whenever Z is a linear subspace of Y. The natural injection J : Z —► Y
is almost open whenever it is dense. Alternatively we can prove (3.5.1.4)
directly by finding (zn) in Z for which ||y — zn\\ —► 0 and then taking
if|M<W . .
(llvll/ll*»ll)*» if|l*»ll>llvll
The reader should verify that ||y — 2JJI —► 0. Now we obtain (3.5.1.2) by
taking Z = TX in (3.5.1.4): if k > 0 satisfies (3.3.1.1) then
Y C cl(rX) =>ye c\{Tx: \\x\\ < k\\Tx\\ < k\\y\\} for each y G Y ■
(3.5.1.6)
The reader may, alternatively, like to verify that the sequence (xn)
of (3.5.1.3) satisfies \\xfn\\ < kf\\y\\ for some k' > 0. The boundary result
follows easily:
3.5.2 THEOREM If T e BL(X,Y) is in the boundary of the almost
open operators, then T is not bounded below.
Proof: Suppose that T G BL(X, Y) is in the closure of the almost open
operators, so that there is (Tn) in BL(X,Y) for which
Tn is almost open and ||T - Tn\\ —► 0 (3.5.2.1)
Then by (3.4.2.4) Tn is dense. By (3.5.1.1) T would be dense if it were
bounded below, and hence (3.5.1.2) almost open: but this would exclude T
from the boundary. ■
The product of two of these "boundary operators" is a boundary
operator:
3.5.3 THEOREM If T e BL(X,Y) and 5 G BL(Y,Z) are boundary
operators, then so is ST.
Proof: By (3.4.2.2) there is implication,
S,T G closure (almost open) ==> ST G closure (almost open)
=>• S G closure (almost open) (3.5.3.1)
and, hence, if S and T are both in the boundary of the almost open
operators then ST is in the closure of the almost open operators. If ST is
actually almost open, then by (3.4.2.2) S is almost open, hence by (3.4.3.1)
not in the boundary. ■

72
3. Invertibility and Singularity
We cannot expect that if ST is a boundary operator then so is 5, or
that if ST is a boundary operator then so is T: for if either S = 0 or T = 0
then ST = 0, and if Y = X ^ 0 then .0 G cl{e:J:e: >:0} is a boundary
operator. In the notation of (3.2.1.5), (3.3.1.5), and (3.4.1.4), Theorem
3.3.3 and Theorem 3.4.3 tell us that rleft BL(X, Y) and rright BL(X, F) are
closed subsets of BL(X, Y), and Theorem 3.5.1 gives inclusion
fright BL(X, Y) C fleft BL(X, Y) U int 7rright BL(X, Y)
C fleft BL(X, y) U int tright BL(X, F)
Then Theorem 3.5.2 says that
af right BL(X? y) g f left gj^ y}
and Theorem 3.5.3 says that
afright BL(X?y) . afright BL(X?y) g <9fright BL(X,y)
Theorem 3.5.1 has an application to enlargements.
3.5.4 THEOREM If T e BL{X,Y) then
Q(T) one-one and dense ==> T bounded below and dense
=^ Q(r) invertible (3.5.4.1)
Proof: If Q(T) is one-one and dense then by (3.3.5.2) and (3.4.5.8) T is
bounded below and dense, hence by (3.5.1.2) bounded below and almost
open, which by (3.3.5.2) and (3.4.5.2) makes Q(T) bounded below and
open, hence invertible. ■
Theorem 3.5.1 has an application to proper mappings. We begin with
a definition.
3.5.5 DEFINITION T e BL(X,Y) is called relatively open iff the
induced mapping
x —► Tx : X —> TX is open (3.5.5.1)
and is called relatively almost open iff the induced mapping
x —>TX:X —> TX is almost open (3.5.5.2)
This induced mapping is liable to be bypassed in the canonical factorization
(2.3.3.2). To catch it we should look at
core0(r) : x + T_1(0) —> Tx from X/r_1(0) to TX (3.5.5.3)
(3.5.3.2)
(3.5.3.3)
(3.5.3.4)

3.6 Left and Right Invertibility
73
This "precore" of T is always one-one and onto, and we have
core(r) = Jcore0(r) (3.5.5.4)
with
J : TX —► cl(rX) bounded below and dense (3.5.5.5)
Using the arguments of (3.3.7.3) and (3.4.7.2) it is clear that
T relatively open «*=>• core0(T) open ^=> core0(T) bounded below
(3.5.5.6)
and
T relatively almost open <*=>> core0(T) almost open (3.5.5.7)
3.5.6 THEOREM If T 6 BL(X,Y), then
T relatively open <$=>► core(T) bounded below (3.5.6.1)
T relatively almost open <=> core(T) almost open (3.5.6.2)
T proper <=► core(T) open <=> core(T) closed (3.5.6.3)
Proof: By (3.3.2.1) and the factorization (3.5.5.4), core(T) is bounded
below if and only if core0(T) is bounded below. This proves (3.5.6.1). By
Theorem 3.5.1 the mapping J of (3.5.5.5) is almost open: now by (3.4.2.2)
and the factorization (3.5.5.4), core(T) is almost open if and only if core0(T)
is almost open. The last part (3.5.6.3) is only (3.3.7.2) and (3.4.7.1) applied
tocore(r). ■
3.6 LEFT AND RIGHT INVERTIBILITY
A left or a right inverse satisfies only half the conditions for an inverse:
3.6.1 DEFINITION T 6 BL{X,Y) is said to be left invertible if there is
T' GBL(y,X) for which
r'r = jgbl(x,x) (3.6.1.1)
and is said to be right invertible if there is T" G BL(Y,X) for which
TT" = IGBL(Y,Y) (3.6.1.2)
We shall write
aleft BL(X, Y) = {Te BL(X, Y):T is not left invertible} (3.6.1.3)

74
3. Invertibility and Singularity
and
aright BL(X, Y) = {T e BL(X, Y): T is not right invertible} (3.6.1.4)
We emphasize that it is part of the definition of a left or a right inverse
that it is bounded, and everywhere defined on Y.
3.6.2 THEOREM If T e BL(X, Y) and S e BL(F, Z) then
5,T left invertible => ST left invertible =>• T left invertible (3.6.2.1)
and
S, T right invertible => ST right invertible => S right invertible
(3.6.2.2)
Also
T invertible => T left invertible => T closed (3.6.2.3)
and
T invertible =>• T right invertible => T open (3.6.2.4)
Proof: (3.6.2.1) and (3.6.2.2) are simple algebra, and the first parts of
(3.6.2.3) and (3.6.2.4) are trivial, while the second parts follow from
(3.3.2.2) and (3.4.2.1). ■
We might recall that if T is both left and right invertible then it is
actually invertible: we did this in (3.1.2.2). We can prove more:
3.6.3 THEOREM If T e BL{X,Y) then
T left invertible and dense =^ T invertible (3.6.3.1)
and
T right invertible and one-one ==> T invertible (3.6.3.2)
Proof: If T' G BL(F, X) satisfies T'T = I and if T is dense then (3.2.2.2)
gives
(/ - TT')T = 0=>I = TT' (3.6.3.3)
If instead T" e BL(Y,X) satisfies TT" = I and T is one-one then (3.2.2.1)
gives
T{I - T"T) =0=^1 = T"T ■ (3.6.3.4)
Of course, (3.4.7.1) is already an improvement on (3.6.3.2).

3.6 Left and Right Invertibility
75
3.6.4 THEOREM If T e BL(X, Y) and S e BL(F, X) then
J - ST left invertible <=> I-TS left invertible (3.6.4.1)
and
J - ST right invertible <=> I-TS right invertible (3.6.4.2)
Proof: As in Theorem 3.1.3, if U e BL(X,Y) is a left inverse for J - ST
then J + TUS is a left inverse for J — TS\ the rest of the argument is left
to the reader. ■
If T is either left or right invertible then so are all the left composition
operators LT = BL(W,T), and also the right composition operators RT,
but the other way round: indeed by (2.9.1.2)
T'T = I=> LT,LT = I: BL{W,X)
and (3.6.4.3)
RTRT, = I: BL{Y,W) —> BL{Y,W)
and
TT" = I=> LTLT„ = I: BL{W, Y) —^ BL(W, Y)
and (3.6.4.4)
RT,,RT = I: BL{X,W) —> BL{X,W)
If we choose the space W carefully then we get something much more
dramatic.
3.6.5 THEOREM If T e BL(X, Y) then, with LT = BL(F, T) and RT =
BL(r,X),
RT onto ==> T left invertible ==> RT open (3.6.5.1)
and
LT onto ==> T right invertible ==> LT open (3.6.5.2)
Proof: If RT = BL(T,X) is onto then there must be T' e BL(F,X) for
which
/ = RT{T') = T'T (3.6.5.3)
which is (3.6.1.1). By (3.6.4.3) RT is right invertible, hence by (3.6.2.4)
open. If instead LT = BL(Y,T) is onto then there must be T" G BL(F,X)

76
3. Invertibility and Singularity
for which
I = LT{T") = TT" (3.6.5.4)
which is (3.6.1.2). By (3.6.4.4) LT is right invertible, hence by (3.6.2.4)
open. ■
If T is either left or right invertible then so are all its enlargements:
for if VI is a bornological space then by (2.7.3.3)
rT = I=> Qn(r')Qn(r) = I : QnPO — QQ(X) (3.6.5.5)
and
TT" = I =► Qn(T)Qn(n = I : Qn(F) — Qn(F) (3.6.5.6)
3.7 ALMOST INVERTIBLE OPERATORS
The almost invertible operators bear the same relation to the invertible
operators as the almost open operators do to the open operators.
3.7.1 DEFINITION T e BL(X,Y) is said to be almost left invertible if
there is [T'n) in BL{Y,X) for which
\\I-T^T\\ —>0asn—> oo with supn||r^|| < oo (3.7.1.1)
and is said to be almost right invertible if there is (T^) in BL(Y,X) for
which
III-rZ^H —> 0 as n —» oo with supn||r^|| < oo (3.7.1.2)
If T is almost left and almost right invertible it is called almost invertible.
We shall write
<7left BL(X,y) = {Te BL(X,y):T is not almost left invertible}
(3.7.1.3)
and
aright BL(X,y) = {Te BL(X,Y):T is not almost right invertible}
(3.7.1.4)
It is not hard to see that almost left and right invertibility obey the
same product rules as left and right invertibility, and is transmitted to left
and right compositions, and to enlargements. We begin however with the
analogue of Theorem 3.6.5.

3.7 Almost Invertible Operators
77
3.7.2 THEOREM If T e BL(X, Y) then, with LT = BL(F, T) and RT =
BL(r,X),
RT dense ==> T almost left invertible ==> RT almost open (3.7.2.1)
and
LT dense => T almost right invertible ==> LT almost open (3.7.2.2)
Proof: If RT = BL(r,X) is dense then there must be T£ e BL(Y,X) for
which
||J-^r|| = ||J-i2r(^)||<l (3.7.2.3)
Now with
U = I-T'QT and T'n = (/ + U + • • • + Un)T'0 for each n e N
(3.7.2.4)
we have
\\I-T'nT\\ = HCr+Ml < \\U\\»+1 ^0 and ||l£|| < ^Jj^
(3.7.2.5)
giving (3.7.1.1). Further, if this holds, then for arbitrary V € BL(X, X) we
have, with k= ||^||/(1-||^||),
\\v-RT{vrn)\\ = \\v{i-rnT)\\ < \\u\\n+1\\v\\^o
and (3.7.2.6)
\\vrj < k\\v\\
which says that RT is almost open. The argument for (3.7.2.2) is the
same. ■
The reader will observe that the definition of the almost left invertible
operators can be simplified by dropping the requirement that supn \\T^\\ <
oo in (3.7.1.1), with a similar simplification of the definition of almost right
invertibility. In the sequel, however, he will see that our version is the one
to work with.
Using Theorem 3.7.2, we can systematically derive the properties of
the almost left and right invertible operators from those of the almost open
operators.

78
3. Invertibility and Singularity
3.7.3 THEOREM If T e BL(X, Y) and S e BL(F, Z) then
S, T almost left invertible ==> ST almost left invertible
(3.7.3.1)
T almost left invertible
and
5, T almost right invertible ==> ST almost right invertible
(3.7.3.2)
=>» T almost right invertible v * * * '
Also
T left invertible => T almost left invertible ==> T bounded below
(3.7.3.3)
and
T right invertible ==> T almost right invertible ==> T almost open
(3.7.3.4)
Proof: For (3.7.3.1) and (3.7.3.2) apply (3.4.2.2) for RT and for LT.
Alternatively, it is not hard to argue directly. The first part of each of (3.7.3.2)
and (3.7.3.3) is trivial. Towards the second part of (3.7.3.3), suppose that
||J-2^r||—0 withsupJftH*
Then if k < k' we may take U = T'N in such a way that
\\I-UT\\<6 with k = (1 - 6)k' (3.7.3.5)
Now for arbitrary x G X
\\x\\ < ||(/-C/r)x|| + ||C/rx|| < «||x|| + A;||rx|| =► ||x|| < k'\\Tx\\ (3.7.3.6)
Finally, for the second part of (3.7.3.4), suppose that
||/_rz*||_o with supn\\T':\\<k
Then for arbitrary y G Y we have, with xn = T„y,
l|y-r*j|<||i-rz3|y||->o
and (3.7.3.7)
w<raii»ii<%ii ■
The almost left and right invertible operators form open sets:

3.7 Almost Invertible Operators
79
3.7.4 THEOREM If X and Y are normed spaces then
{T e BL(X,Y)):T is almost left invertible} is an open set (3.7.4.1)
and
{T e BL(X,Y):T is almost right invertible} is an open set (3.7.4.2)
Proof: By Theorem 2.9.1 the mappings T —> RT and T —► LT are both
continuous, and by Theorem 3.4.3 the almost open operators between two
normed spaces form an open set. Thus the sets in (3.7.4.1) and (3.7.4.2)
are continuous counterimages of open sets. ■
Temporarily, we shall describe T G BL(X, Y) as a left topological zero
divisor iff
LT = BL(Y,T) : BL(Y,X) —► BL(Y, Y) is not bounded below (3.7.4.3)
and as a right topological zero divisor iff
RT = BL(r,X) : BL(Y,X) —► BL(X,X) is not bounded below (3.7.4.4)
Evidently both these form closed sets of T G BL(X, Y).
3.7.5 THEOREM If X and Y are normed spaces then
d{T e BL(X,Y):T almost left invertible}
C {J e BL(X,Y):T topological right zero divisor}
and
d{T e BL(X, Y): T almost right invertible}
C{T e BL(X,Y):T topological left zero divisor}
Proof: If T G BL(X, Y) is in the closure of the almost left invertible
operators then RT : BL(Y, X) —► BL(X, X) is in the closure of the almost
open operators, and if T is not almost left invertible then RT is not almost
open. It follows that if T is in the boundary of the almost left invertible
operators, then RT is in the boundary of the almost open operators. By
Theorem 3.5.2 it follows that RT is not bounded below, which is what we
mean (3.7.4.4) by calling T a topological right zero divisor. This proves
(3.7.5.1), and the argument for (3.7.5.2) is the same. ■

80
3. Invertibility and Singularity-
It is easy to see that if T G BL(X,Y) is almost left invertible then
so are all the left compositions LT = BL(W, T) and all the enlargements
Qn(T), while all the right compositions RT = BL(T,W) are almost right
invertible. The reader can write this out for himself, together with the
corresponding result for almost right invertible operators. It is clear also
that the almost invertible operators form an open set, the intersection of
the open sets of almost left and almost right invertible operators. We might
observe a certain "uniqueness of almost inverse" :
3.7.6 THEOREM If T e BL{X,Y) and if {T'n) and (T%) in BL{Y,X)
satisfy (3.7.1.1) and (3.7.1.2), then
11^-^11—^0 (3.7.6.1)
Proof: Observe
K - r^|| < K(/- m?)II + \\{T'nT - jjrj'H
<(supnK||)||/-rr:i| + Kr-/||suPnK|| —o .
3.8 REGULAR OPERATORS
For a common generalization of left and right invertibility, assume the
existence of a "generalized inverse":
3.8.1 DEFINITION T e BL(X,Y) is called regular or relatively Fred-
holm, if there is V G BL(F,X) for which
T = TT'T (3.8.1.1)
We shall write
rBT(X,y) = {Te BL{X,Y):Te T • BL(y,X) . T} (3.8.1.2)
If (3.8.1.1) holds we shall call T' a generalized inverse, or pseudo-
inverse, for T. For example if T' is either a left inverse or a right inverse
for T, then it is a generalized inverse. If Y = X and T = T2 is a projection,
as in (2.5.0.2), then (3.8.1.1) holds with T' = T, or alternatively with T = I.
In a sense, regular operators are the common generalization of invertibles
and idempotents.

3.8 Regular Operators
81
3.8.2 THEOREM If X and Y are normed spaces, then T e BL(X, Y) is
regular iff
T is proper and T_1(0) is complemented and cl(TX) is complemented
(3.8.2.1)
Proof: Suppose T is regular, with T = TT'T: then
T'T = (rT)2 = P and TT' = [TT')2 = Q (3.8.2.2)
are projections on X and Y, respectively, with
T-^O) = P_1(0) and Q{Y) = T{X) = c\{TX) (3.8.2.3)
Thus, T~l (0) and z\{TX) are both complemented, and also T(X) = c\(TX)
is closed. Further, the mapping
TA : P{X) —► Q{Y) (3.8.2.4)
induced by T is invertible, with inverse
(rA)_1 = T'A : Q{Y) —► P{X) (3.8.2.5)
induced by T', and finally there is isomorphism
TA - core(r) (3.8.2.6)
induced by the isomorphism (2.4.2.1)
X/T-^O) = X/P-^O) c* P{X) (3.8.2.7)
It follows that core(T) is invertible, so that T is proper.
Conversely, if T G BL(X,Y) satisfies (3.8.2.1), then we may choose
projections P = P2 on X and Q = Q2 on Y satisfying (3.8.2.3), and use
(3.8.2.6) to see that the mapping TA of (3.8.2.4) is invertible. Now, if we
define T'A : Q{Y) -► P{X) by (3.8.2.5), then (3.8.1.1) will be satisfied if
T' : Y —► X is defined by setting
T'{y) = T'AQ{y) e X for each y eY ■ (3.8.2.8)
If T is regular, and T" is a generalized inverse of T, then there is nothing
to tell us that T' is regular, or that if T' is regular then T is a generalized
inverse of T". However, this can be arranged: if (3.8.1.1) holds, then
T" = T'TT' => T = TT"T and T" = T"TT" (3.8.2.9)
If T G BL(X, Y) is regular then so are all the composition operators LT =
BL(W,T) and RT = BL(T,W), and all the enlargements Qn(T). Indeed,

82
3. Invertibility and Singularity
if T' satisfies (3.8.1.1) then, by (2.9.1.2),
Jbrp = L/fpL/fpiL/rp and Rrp = RrpRrpiRrp ^3.8.2.10]
while, by (2.7.3.3),
Qn(T) = Qn(T)Qn(T')Qn(T) (3.8.2.11)
In general, the product of regular operators need not be regular, and it
is not easy to know what one can add to a regular operator without losing
regularity:
3.8.3 THEOREM If T e BL(X, Y) and T' e BL(Y, X), then
T - TT'T regular => T regular (3.8.3.1)
and
J - T'T regular ^=> I - TT' regular (3.8.3.2)
If also S e BL{Y,Z) and S' e BL(Z,Y), and if V and S' are generalized
inverses for T and 5, respectively, then
ST regular <<=> S'STT' regular (3.8.3.3)
Proof: If U e BL(y, X) is a generalized inverse for T - TT'T then
T = T{T' + (/ - T'T)U{I - TT'))T (3.8.3.4)
giving (3.8.3.1). If instead U is a generalized inverse for J — T'T then
I-TT' = {I- TT') {I + TUT') {I - TT') (3.8.3.5)
giving forward implication in (3.8.3.2). The argument for backward
implication is similar. Finally, if ST is regular, with generalized inverse
UeBL{Z,X), then
S'STT' = S'S{TUS)TT' = {S'STT')TUS{S'STT') (3.8.3.6)
giving forward implication in (3.8.3.3). Conversely, if S'STT' is regular,
with generalized inverse V G BL(Y, Y), then
ST = S{S'STT')T = S{S'STT'VS'STT')T
= ST{T'VS')ST m (3.8.3.7)

3.8 Regular Operators
83
It is sufficient for S'STT' to be regular that
[S'S){TT') = {TT'){S'S) (3.8.3.8)
since by (2.5.2.2) S'STT' is then a projection. Sufficient for (3.8.3.8) is
that either S'S = I or TT' = I. Thus,
S left invertible, T regular => ST regular (3.8.3.9)
and
S regular, T right invertible => ST regular (3.8.3.10)
Conversely, looking at the canonical factorization (2.3.2.2), every regular
operator T can be written as the product of a left and a right invertible
operator:
T = UV with U left invertible and V right invertible (3.8.3.11)
We can also observe
T left invertible <=> T regular and one-one (3.8.3.12)
and
T right invertible <=> T regular and dense (3.8.3.13)
The first part of Theorem 3.8.3 begins to tell us how we can sometimes
add to a regular operator without disturbing regularity:
3.8.4 THEOREM If T e BL(X,Y) is regular, and T' e BL{Y,X) is a
generalized inverse for T, and if K G BL(X, Y) satisfies
I - T'K is invertible and (J - TT')K{I - T'K)-1 {I - T'T) is regular
(3.8.4.1)
then
T - K is regular (3.8.4.2)
Proof: Writing K' = {I - T'K)'1 we have
(/ - TT')KK'{I - T'T) = {{T{I - T'K) + [K - T))K'{I - T'T)
= T{I - T'T) + {K- T)K'{I - T'T)
= 0+(K-T)K'(I-T'K) (3.8.4.3)
+ {K-T){K'T'){K-T)
= K-T+{K- T)K'T'{K - T)

84
3. Invertibility and Singularity
It follows from (3.8.4.1) that the last expression is regular, and hence by
(3.8.3.1) that (3.8.4.2) holds. ■
It is clear that (3.8.4.2) will also hold if (3.8.4.1) is replaced by
J - KT' is invertible
and (3.8.4.4)
(J - TT'){I - KT')'1^ - T'T) is regular.
The details are left to the reader. Regular operators with invertible
generalized inverses are special:
3.8.5 DEFINITION T G BL(X,Y) is called decomposably regular, or
relatively Weyl, if there is T' G BL(Y,W) for which
T = TT'T and T' G BL"1 [Y, X) is invertible (3.8.5.1)
For example, idempotents are decomposably regular, if T = T2 G
BL(X,X) then (3.8.5.1) holds with T' = I. More generally,
{SP:Se BL"1 [X,Y),P = P2 e BL(X,X)}
(3.8.5.2)
= {QS-.se BL^p^y),*? = q2 e BL(y,y)}
is the set of decomposably regular operators from X to Y. In particular,
unless it is actually invertible, a decomposably regular operator must be a
"boundary operator" in the sense of Section 3.5: if T G BL(X,Y) then
T decomposably regular => T e cl BL"1 (X, Y) (3.8.5.3)
Indeed if S G BL(X, Y) is invertible and P G BL(X, X) is a projection then
sp-s(p + i(/-p)j
and (3.8.5.4)
s(p + -{i-pyj GBL-^y)
Decomposably regular operators can be characterized "spatially":
3.8.6 THEOREM T G BL(X,y) is decomposably regular if and only if
T is regular and T"1 (0) ~ Yj cl(TX) (3.8.6.1)

3.8 Regular Operators
85
Proof: If T is regular, and P = P2 e BL(X,X) and Q = Q2 e BL(Y,Y)
satisfy (3.8.2.4), then by (2.4.2.1) the second part of (3.8.6.1) is equivalent
to
^(O)^"1^)
(3.8.6.2)
If this holds, and SfV : Q x(0) —► P x(0) is invertible, then we can define
T' : Y -+ X by setting
T'{y) = T'AQ{y) + S'V{I - Q){y) e X for each y eY (3.8.6.3)
where T'A = (rA)_1 is given by (3.8.2.5): evidently T' satisfies (3.8.5.1).
Conversely, if T' satisfies (3.8.5.1), and P and Q are given by (3.8.2.2), then
T'{QY) C PX and T'{Q-l0) C P~l0 (3.8.6.4)
so that the restriction of T" to Q_10 supplies an isomorphism (3.8.6.2). ■
If T G BL(X, Y) is regular then we have a matrix representation, as in
(2.4.1.7),
r = diag(rA,o)
"rA o"
0 0
p(xy
.p~1q.
-^
Q(Y)'
(3.8.6.5)
in which T ^ core(T) is invertible. If, in particular, T is decomposably
regular then there is also an invertible operator
5 = diag(rA,5v)
Evidently
"rA o "
0 5V
:
p(xy
P-!Q
-^
Q(Y)'
T = SP = QS
(3.8.6.6)
(3.8.6.7)
as in (3.8.5.2).
The decomposably regular operators are in a sense "orthogonal" to the
left and right invertible operators in 'BL'(X, Y): the reader can verify that
T decomposably regular and left invertible ^=> T invertible (3.8.6.8)
and
T decomposably regular and right invertible ^=> T invertible (3.8.6.9)
We conclude with a sort of converse to (3.8.5.3):

86
3. Invertibility and Singularity
3.8.7 THEOREM If T G BL{X,Y) is regular, with T = TT'T, and if
there is U G BL(Y,X) for which
C/GBL_1(y,X) and /+ {U - T)T' G BL_1(X,X) (3.8.7.1)
then T is decomposably regular.
Proof: Since
(J + {U - T)T')T = {T- TT'T) + UT'T = UT'T
we have
T = SP (3.8.7.2)
with
P = T'T and S = {I + {U - r)r')_1C/ (3.8.7.3)
Evidently P is idempotent and S is invertible. ■
3.9 ESSENTIAL INVERTIBILITY
An operator T G BL(X,Y) which is not invertible may still be left and
right invertible "modulo" certain subspaces of operators.
3.9.1 DEFINITION If X and Y are normed spaces and A C BL(X,X)
and B C BL(Y, Y) are linear subspaces, then we shall say that T G
BL(X,y) is left invertible modulo A if there is T' G BL(F,X) for which
I-T'T e A (3.9.1.1)
and that T is right invertible modulo B if there is T" G BL(Y,X) for which
j _ TTn e B (3.9.1.2)
We shall write
alff BL(X,y) ={Te BL{X,Y):T is not left invertible mod A}
(3.9.1.3)
and
<rfj>ht BL(X,y) = {Te BL(X,Y):T is not right invertible mod B}
(3.9.1.4)
For example, if A = {0} then (3.9.1.1) reduces to left invertibility
(3.6.1.1), while if A = BL(X,X), then (3.9.1.1) holds for all T G BL(X,y):

3.9 Essential Invertibility
87
3.9.2 THEOREM If A C A! C BL(X,X) and B C B' C BL(y,y) and
rGBL(X,y), then
T left invertible ==> T left invertible mod A' ==> T left invertible mod A
(3.9.2.1)
and
T right invertible => right invertible mod Bf
(3.9.2.2)
=>► T right invertible mod B
Proof: If T e BL(y, X), then
I - T'T = 0 =► J - T'T e A =}> / - T'T e A' (3.9.2.3)
giving (3.9.2.1), and similarly (3.9.2.2). ■
If the product of two operators is essentially invertible then so is one
factor:
3.9.3 THEOREM If X, Y and Z are normed spaces and A C BL(X, X),
B C BL(y,y), D C BL(Z,Z) are linear subspaces, then, if T e BL(X,y)
andSGBL(y,Z):
ST left invertible mod A => T left invertible mod A (3.9.3.1)
and
ST right invertible mod D => S right invertible mod D (3.9.3.2)
If A, B and D satisfy
BL(y, X) • B • BL(X, y) C A and BL(y, Z) • B • BL(Z, Y) C D
(3.9.3.3)
then
5 left invertible mod B.T left invertible mod A
(3.9.3.4)
=^ ST left invertible mod A
and
S right invertible mod jD, T right invertible mod 5
(3.9.3.5)
==> ST right invertible mod D

88
3. Invertibility and Singularity
Proof: If I-UST e A, then (3.9.1.1) holds with T' = US, giving (3.9.3.1).
Conversely, if J - T'T e A and J - S'S G B, then
I-T'S'ST = I + T'{I-S'S)T+{I-T'T) e I + T'BT + A C A (3.9.3.6)
where the inclusion at the end uses (3.9.2.3). This proves (3.9.3.4), and
similarly (3.9.3.5). ■
If, for example, X = Y = Z and A = B = D is a two-sided ideal of
BL(X,X) then (3.9.3.3) holds. For the analogue of Theorem 3.6.5 we ask
that A and B are closed ideals, and write
RT/A : T' —► T'T + A e BL(X, X)/A for each T' e BL(F, X)
(3.9.3.7)
and
LT/B : T' —► TT' + B e BL(F, Y)/B for each T' e BL(F, X)
(3.9.3.8)
3.9.4 THEOREM If A is a closed left ideal of BL(X, X) and B is a closed
right ideal of BL{Y,Y), and if T e BL{X,Y), then
RT/A onto =^ T left invertible mod A => RT/A open (3.9.4.1)
and
LT/B onto => T right invertible mod B => LT/B open (3.9.4.2)
Proof: If RT/A is onto, then
/ G BL(X, X) = RT BL(y, X) + A (3.9.4.3)
giving T' G BL(y,X) to satisfy (3.9.1.1). Conversely, if (3.9.1.1) holds,
then provided A is a left ideal of BL(X,X),
U e BL(X, X) =}> U - UT'T eA=>U + A= {RT/A) [UT') (3.9.4.4)
so that RT/A is onto. Further, if U' e BL(X,X), then
U'-UeA=>U- U'T'T eA=>U + A = {RT/A) [U'T') (3.9.4.5)
and finally, if A is closed and 0 < t < 1 then by the Riesz lemma (1.5.2.1)
we can choose U' e U + A with \\U'\\ < (l/t) dist(Z7, A), giving
U + A= {RT/A){U'T') with WU'T'W < (1/011^1111^ +A|| (3.9.4.6)

3.9 Essential Invertibility 89
This means RT/A is open, and finishes the proof of (3.9.4.1). The argument
for (3.9.4.2) is the same. ■
If, instead of a left ideal, A is a right ideal of BL(X, X), then
T left invertible mod A => T not a left zero divisor mod A (3.9.4.7)
in the sense that if U G BL(X, X), then
TV = 0 =>» U e A (3.9.4.8)
We can prove a lot more:
3.9.5 THEOREM If A C BL(X,X) is a closed two-sided ideal, and if
T G BL(X,Y) is almost left invertible mod A, in the sense that there is
[T'n) inBL(y,X) for which
dist(/-r^r,A) —► 0 and supn||r£|| < oo (3.9.5.1)
then T is not a topological left zero divisor mod A in the sense that there
is k > 0 for which
dist(*7, A) < k\\TU\\ for each U e BL(X,X) (3.9.5.2)
Proof: If (3.9.5.1) holds, then there is (U^) in BL(X,X) and k' > 0 for
which
||/-r;r-ET;||—.0 and U'neA and supnK|| = *' < oo
(3.9.5.3)
Then, if U e BL(X,X) and U" e A are arbitrary, we have
U-U" = {I-T^T-U^iU-U^ + U^U-U^-T^TU^ + T^TU (3.9.5.4)
and hence
\\U - (U" + U'n(U - U") + rnTU")\\
< 11/-i^r-tr;i|||tr-tr"||+ 11^1111^11
Since A is both a left and a right ideal,
U" + U'n{U - U") + T'nTU" eA + A{U - U") + {T'nT)A C A (3.9.5.6)
so that
dist(CT,A) < \\I-T'nT-U%\\U-U'% + \\T'n\\\\TU\\<6\\U-U''\\ + k'\\TU\\
(3.9.5.7)

90
3. Invertibility and Singularity
for arbitrary S > 0, by choice of n: finally allowing \\U — U"\\ —► dist(C/, A)
gives (3.9.5.2) with k = k'/(l - S). m
The reader should state and prove the corresponding result for almost
right invertibility modulo B C BL(Y,Y). If A C A! C BL(X,X), then
combining Theorem 3.9.5 and Theorem 3.9.2 gives, using notation based
on (3.7.1.3) and (3.3.1.3),
TlftBL(X,Y)Cfl$BL(X,Y)
(3.9.5.8)
C a)f, BL(X, Y) C o~)f BL(X, y)
In particular, if it were possible to reverse the implication of Theorem 3.9.5
for a particular ideal A C BL(X, X), then this would also hold for all larger
ideals A'DA
If T £ BL(X,y) is regular in the sense of Definition 3.8.1, and the
ideal A C BL(X,X) satisfies a regularity condition, then we can actually
reverse the implication (3.9.4.8):
3.9.6 THEOREM If A C BL(X, X) is a right ideal and satisfies
BL(X, Y) • A C ^X, Y) (3.9.6.1)
then there is implication for T G BL(X, Y),
T left invertible mod A ^=> T regular with ^(O) C A (3.9.6.2)
If B C BL(Y, Y) is a left ideal and satisfies
B • BL(X, Y) C rBT(X, Y) (3.9.6.3)
then there is implication, for T £ BL(X, Y),
T right invertible mod B <*=> T regular with JKy^O) C B (3.9.6.4)
Proof: If T = rrT is regular, then
I-T'Te L?1 (0) and I-TT'G R^fi) (3.9.6.5)
giving backward implication in each of (3.9.6.2) and (3.9.6.4). Conversely,
if T is left invertible mod A then L? (°) £ A bv (3.9.4.8), and if T' £
BL(Y,X) satisfies (3.9.1.1), then
T - TT'T = T{I - T'T) £ BL(X, Y) • A (3.9.6.6)

3.10 Algebraic Invertibility
91
is regular by (3.9.6.1), so that (3.9.3.1) makes T regular. This finishes the
proof of (3.9.6.2), and the argument for (3.9.6.4) is the same. ■
If, in particular, A = {0}, then (3.9.6.2) reduces to (3.8.3.12), while if
B = {0}, then (3.9.6.4) reduces to (3.8.3.13). If N C BL{X,Y) is a linear
subspace for which
BL(y, Y)-N + N- BL(X, X) C N (3.9.6.7)
then, by an abuse of language, we shall sometimes call N a two-sided ideal
of BL(X, y); if in addition N satisfies a regularity condition,
iVCrBL1(X,y) (3.9.6.8)
then by the argument of (3.9.6.6) it follows
rBL1(X, y) + N C rBL1(X, Y) (3.9.6.9)
Indeed if T = TT'T e rBL1(X, Y) and K C N then
T+K- [T + K)T'{T + K) = K-KT'T-TT'K-KT'K e N C ^^y)
(3.9.6.10)
so that (3.8.3.1) makes T + K regular. We can also improve on Theorem
3.8.4: if T = TT'T e "W{X,Y) is regular and K e BL(X,y), then it
is sufficient, for T — K G 'BL'(X, Y) to be regular, that there is a two-
sided ideal A C BL(X, X), satisfying the regularity condition (3.9.6.1) such
that J — T'K is both left and right invertible mod A and has an "essential
inverse" K' e BL(X,X) for which
(/ - TT')KK'{I - T'T) e rBT(X, Y) (3.9.6.11)
Indeed, if these conditions are satisfied then the argument of Theorem 3.8.4
shows that there must be U G BL(Y,X) for which
T - K - [T - K)U{T - K) e BL(X, Y) • A C rBL1(X, Y) (3.9.6.12)
which makes T - K regular by (3.8.3.1).
3.10 ALGEBRAIC INVERTIBILITY
For the first time we meet a condition that is actually stronger than
invertibility:
3.10.1 DEFINITION If M e BL(y,X) is a subspace, then T e BL[X,Y)
is said to be left M-invertible if there is T' G BL(y,X) for which
T'T = I and T'e M (3.10.1.1)

92 3. Invertibility and Singularity
and is said to be right M-invertible if there is T" G BL(Y,X) for which
TT" = I and T" e M (3.10.1.2)
We shall write
a£ft BL(X, Y) = {T G BL(X, Y): T is not left M-invertible} (3.10.1.3)
and
a^ght BL(X,y) ={Te BL{X,Y):T is not right M-invertible}
(3.10.1.4)
For example, if M = BL(Y, X), then left M-invertibility reduces to left
invertibility (3.6.1.1):
3.10.2 THEOREM IfMCM'C BL(F,X) and T G BL(X,Y), then
T left M-invertible =>» T left M'-invertible => T left invertible
(3.10.2.1)
and
T right M-invertible ==> T right M'-invertible ==> T right invertible
(3.10.2.2)
Proof: If T' G BL(F, X) then
J' G M =}> T' G M' =>T' G BL(y, X) ■ (3.10.2.3)
3.10.3 THEOREM If M C BL(y,X), N C BL(Z,y) and MiV C L C
BL(Z,X), and T G BL(X,y),
5 left iV-invertible, T left M-invertible =► ST left L-invertible
(3.10.3.1)
and
5 right iV-invertible, T right M-invertible => ST right L-invertible
(3.10.3.2)
Proof: If 5;5 = I and T'T = I and T G M and 5' G iV then
T'S'ST = I and T'S'eMN<ZL (3.10.3.3)
If 55" = J and TT" = I and T" G M and 5" G JV then
STT"S" = I and T"S" eMN CL ■ (3.10.3.4)

3.10 Algebraic Invertibility
93
To argue that
ST left L-invertible =► T left M-invertible (3.10.3.5)
and that
ST right L-invertible =► S right iV-invertible (3.10.3.6)
we must know that
LS CM and TLQN (3.10.3.7)
Thus, for example, if X = Y = Z and M = N = L is a subalgebra
A C BL(X, X) and we restrict to T G A and S G A, then the conditions
for Theorem 3.10.3, as well as (3.10.3.7) will hold. In this situation the
restrictions of the composition operators
LT : A —► A and RT : A —► A (3.10.3.8)
coincide with the operators of (2.10.0.1) and (2.10.0.2).
If A is a normed linear algebra then the analogues of Theorem 3.6.5
and Theorem 3.7.2 hold:
3.10.4 THEOREM If A is a normed linear algebra and a £ A, then
Ra onto ==> a left invertible ==> Ra open (3.10.4.1)
and
La onto => a right invertible => La open (3.10.4.2)
Also
Ra dense ==> a almost left invertible ==> Ra almost open (3.10.4.3)
and
La dense ==> a almost right invertible ==> La almost open (3.10.4.4)
Proof: If Ra is onto, then 1 G A = Ra{A), so that there is af G A for
which
1 = a'a (3.10.4.5)
Conversely, if this holds then for arbitrary 6 G A
b = [ba')a with ||6a'|| < ||6||||a'|| (3.10.4.6)

94
3. Invertibility and Singularity
so that Ra is open. This proves (3.10.4.1), and the argument for (3.10.4.2)
is the same. For (3.10.4.3) we follow the argument for (3.7.2.1): if Ra is
dense then there is af0 G A for which ||l — a'Qa\\ < 1, and now with
u = 1 — af0a
and a!n = (1 + u + \- un)a! for each nGN
(3.10.4.7)
we have
||l-a'a|l = lkn+1||<||u|
n+l
and
lla'J|<
i-W
(3.10.4.8)
so that a is "almost left invertible." Further, if this is so, then for arbitrary
6 e A we have, with k = ||ao||/(l - ||u||),
in+li
0
with||6a'n||<A;||
\\b-Ra(ba'n)\\ = \\b(l-a'na)\\<\\u\
(3.10.4.9)
so that Ra is almost open. The argument for (3.10.4.4) is the same. ■
Theorem 3.10.4 can be extended to more general "algebraic
invertibility" provided there is some compatibility between T G BL(X, Y) and M C
BL(y,X). We need, in fact, T to belong to some subspace N C BL(X, Y)
for which there are subalgebras A C BL(X, X) and B C BL(Y, Y) for which
A M
N B
is a subalgebra of
BL(X,X) BL(y,X)
BL(x,y) BL(y,y)J
X
=-([?]•[?])
(3.10.4.10)
Then for the analogue of Theorem 3.10.4 we work with the composition
operators
[Rt)m : M —► A and (LT)
M
M
B
(3.10.4.11)
The condition for all this to be possible is simply stated, and reminiscent
of (3.8.1.1):
MTM CM
If A is a normed algebra and cGA then
a left invertible ==> a not a left zero divisor
in the sense that La : A —► A is one-one:
u £ A and au = 0 ==> u = 0
(3.10.4.12)
(3.10.4.13)
(3.10.4.14)

3.10 Algebraic Invertibility 95
We can prove more:
3.10.5 THEOREM If A is a normed linear algebra and cGA, then
a almost left invertible ==> a not a topological left zero divisor
(3.10.5.1)
and
a almost right invertible ==> a not a topological right zero divisor
(3.10.5.2)
Proof: We follow the argument for (3.7.3.3): if
||1 - a'na\\ —► 0 and supn||a'n|| = k' < 0 (3.10.5.3)
then for arbitrary u £ A
||u|| < (1 - a'na)u\\ + \\a'J\\au\\ < S\\u\\ + k'\\au\\ (3.10.5.4)
as soon as ||1 — a'na\\ < 6, giving if 0 < 6 < 1,
k'
\\u\\ < k\\au\\ with k = - (3.10.5.5)
l — o
This proves (3.10.3.1), and the argument for (3.10.5.2) is the same. ■
Of course we are calling a G A a topological left zero divisor if La is not
bounded below. We may now adapt all the notation of (3.2.1.4), (3.2.1.5),
... (3.7.1.4).
3.10.6 THEOREM If A is a normed linear algebra, then
aleft {A) aright {A) aleft {A) aright {A) (3.10.6.1)
are closed subsets of A, each of whose complements are closed under
multiplication, and there is inclusion
7rleft(A) C fleft(A) C <7left(A) C aleft(A) D rleit{A) D fleft(A) (3.10.6.2)
and
7rright(A) C fright(A) C aright(A) C aright(A) D rright(A) D fright(A)
(3.10.6.3)
Also
7rleft(A) Uaright(A) = 7rright(A) Ualeft(A)
= rleft(A) U aright(A) = A \ A"1 (3'10-6-4)

96
3. Invertibility and Singularity
Finally
daleft(A) C fright(A) and daright(A) C fleft(A) (3.10.6.5)
and, hence
d(aleft(A) U aright(A)) C fright(A) n fleft(A) (3.10.6.6)
Proof: Each of the sets in (3.10.6.1) is the counterimage, under one of the
continuous mappings a —► La or a —► i2a, of the complement of the set
of almost open, or bounded below, operators on A, each of which is open
(3.4.3.1), (3.3.3.1), and closed under multiplication (3.4.2.2), (3.3.2.1). In
each of (3.10.6.2) and (3.10.6.3) the first inclusion is (3.3.2.3), the second
is Theorem 3.10.5, the third is (3.4.2.4), the fourth is (3.6.2.3), and the
fifth is (3.3.2.3). If a is right invertible and not a left zero divisor then
1 = aa" and a(l — a"a) = 0, giving part of (3.10.6.4). If instead a is almost
right invertible and closed then La is dense and closed, hence invertible by
(3.3.7.2), which means that a is right invertible and not a left zero divisor.
For (3.10.6.5) apply Theorem 3.5.2 to Ra and to La. Finally, for (3.10.6.6)
argue
dK C H <md dH C K => d{K UH)C (dK) n {dH) ■ (3.10.6.7)
If A is a subalgebra of a larger normed algebra B and a e A then we
can compare A invertibility and B invertibility for La and Ra:
3.10.7 THEOREM If A is a subalgebra of the normed algebra B and
a G A then
u{B) nAC W(A) for each w G {aleft,aright,aleft,aright} (3.10.7.1)
and
u{B) n A D u{A) for each w G {7rleft, 7rright, fleft, fright} (3.10.7.2)
There is inclusion
d(aleft(A) U aright(A)) C (d(aleft(E)) U aright(5)) n A (3.10.7.3)
Proof: The first two inclusions of (3.10.7.1) are Theorem 3.10.2, and the
second two follow from the analogue of Theorem 3.10.2 for almost
invertibility. The first two inclusions of (3.10.7.2) are given by (3.2.4.1) and
the second two by Theorem 3.3.6. Combining (3.10.7.1), (3.10.7.2), and
(3.10.6.5) gives
d(aleft(A) U aright(A)) C rleft(E) n fright(5) n A (3.10.7.4)

3.10 Algebraic Invertibility
97
Now for (3.10.7.3) argue that if K and H are subsets of A then
dKCHQK=>dK CdH ■ (3.10.7.5)
Detailed proofs of (3.10.7.5) and (3.10.6.7) will be given in Theorem
7.10.3 below. In particular, from (3.10.7.1) and (3.10.7.2) we get implication
A C B and aleU{A) = fleft(A) =► aleft(E) n A = fleft(E) n A (3.10.7.6)
and
A C B and aright(A) = fright(A) =► aright(E) n A = fright(E) n A
(3.10.7.7)
As part of the argument for (3.10.6.4) we learn that a G A is invertible
if and only if La £ BL(A, A) is invertible.
a e A-1 <=> La e BL_1(A, A) <=> Ra e BL_1(A, A) (3.10.7.8)
Indeed, if af = a-1 is an inverse for a in A, then La, and Ra, are inverses
for La and Ra in BL(A, A). Conversely, if either La or i?a is invertible then
either a is right invertible and not a left zero divisor, or a is left invertible
and not a right zero divisor.
If A is a normed linear algebra and 0 is a bornological space then the
enlargement Qq{A) of (1.9.2.1) is a normed algebra, and the mapping qn
of (2.7.3.4) is a homomorphism in the sense of (2.10.17). For each a G A
there is equality
Qn(£.) = ^qn(a) and QQ(*B) = iJqo(a) (3.10.7.9)
When 0 = N then the topological zero divisors of A are represented
by zero divisors in Qn(A):
3.10.8 THEOREM If A is a normed linear algebra and a e A, then
q(o) e fleftQ(A) =^ae fleft(A) =► q(o) € ^leftQ(A) (3.10.8.1)
and
q(o) e frightQ(A) => a G fright(A) =► q(o) € 7rrightQ(A) (3.10.8.2)
Also
q(o) € aleftQ(A) ^aG aleft(A) =► q(o) € aleftQ(A) (3.10.8.3)

98
3. Invertibility and Singularity
and
q(o) e <7rightQ(A) ^cG <7right(A) =► q(o) € arightQ(A) (3.10.8.4)
Hence
q(a) almost invertible ==> a almost invertible =>• q(a) invertible
(3.10.8.5)
Proof: For (3.10.8.1) and (3.10.8.2) apply (3.3.5.2) with, respectively, T =
La and T = Ra\ for (3.10.8.3) and (3.10.8.4) apply (3.4.5.2) with,
respectively, T = Ra and T = La. The implication (3.10.8.5) follows at once from
(3.10.8.3) and (3.10.8.4). ■
3.11 SUBSPACES, QUOTIENTS, AND PRODUCTS
Suppose X and Y are normed spaces, with T G BL(X, Y), and suppose
that closed subspaces W C X and Z CY satisfy
T{W) C Z (3.11.0.1)
Then T induces operators
Tw,z :W —> z and T/w,z : xlw —> Ylz (3.11.0.2)
We shall not always be so careful about how we write them down. The
invertibility and singularity of T, Twz and Ttw^z are constrained by one
another:
3.11.1 THEOREM If T e BL(X, Y) and T{W) CZ CY, then
-^W z one~°ne smd T/yy % one-one
m m (3.11.1.1)
==> T one-one ==> Tw z one-one
Tw z bdd below and T/w zbdd below
' (3.11.1.2)
=► T bdd below =► Twz bdd below
Twz closed and T,w^z closed ==> T closed ==> Tw z closed (3.11.1.3)
T/w,z onto and Twz onto => T onto => T/WtZ onto (3.11.1.4)
T/w,z dense and Twz dense => T dense => T/WtZ dense (3.11.1.5)
T/WiZ open and Twz open ==> T open ==> T;^ open (3.11.1.6)
T/ix/ ^ almost open and TVy ^ almost open
», , ' m (3.11.1.7)
=*• T almost open =*• Tiw z almost open

3.11 Subspaces, Quotients, and Products
99
Proof: (3.11.1.1) is (3.2.4.1), (3.11.1.2) and (3.11.1.3) are Theorem 3.3.6,
(3.11.1.4), (3.11.1.6) and (3.11.1.7) are Theorem 3.4.6 and (3.11.1.5) is
(3.2.4.2). ■
We have also some "mixed" results:
3.11.2 THEOREM If T G BL(X, Y) and T(W) CZCY, then
Tw z onto and T one-one ==> T/W^z one-one (3.11.2.1)
T\v,z dense and T bounded below ==> T;^ bounded below (3.11.2.2)
Twz onto and T closed => T/W}Z closed (3.11.2.3)
T/w,z one-one and T onto ==> Twz onto (3.11.2.4)
T/w,z one-one and T open ==> Twz open (3.11.2.5)
TiWjz hounded below and T dense ==> Tw z dense (3.11.2.6)
T/w z bounded below and T almost open
' ' (3.11.2.7)
=> Tw z almost open
Proof: If Tw z is onto and T is one-one and if Tx G Z, then Tx = Tw
with w G W, giving x = w G W and making T/WfZ one-one. If Twz is
dense and T is bounded below, with k > 0 satisfying (3.3.1.1), then for
each x G X
inf llx -w\\<k inf \\T(x - w)\\ = k inf \\Tx - z\\ (3.11.2.8)
making T/W^z bounded below. If in particular T is also closed then
\\y-Txn-zj -^ o =► ||y-rxn-ru>n|| — o =* y = Tx 2 Q)
with \\T{x - xn - wn)\\ —» 0 =► ||z - xn - Wn\\ — 0
making Ttw^z closed. If T/W^z is one-one and T is onto then for each z G Z
there is x G X, for which Tx = z € Z, forcing x G W, making Tw z
onto. If T is open, with A; > 0 satisfying (3.4.1.1), then we can insist on
ll^ll < ^II2I|j making Twz open. If T is dense and TiWjZ is bounded below,
with dist(x,W) < kdist(Tx,Z), then for each z G Z there is (xn) in X for
which ||z — Txn\\ —► 0, giving
dist(xn,W) < fcdist(rxn,Z) < k\\Txn - z\\ —► 0 (3.11.2.10)
and hence there is also a sequence (wn) in W for which ||xn — tun|| <
dist(xn,W) + l/n-+0. It follows ||z-rti;n|| < ||*-rxn|| + ||r||||xn-ti;n|| -+

100
3. Invertibility and Singularity
0, giving Twz dense. If in particular T is almost open, so that we can
arrange ||xn|| < &'||z||, then we can use the Riesz lemma argument (1.5.1.1)
to ensure ||u;n|| < 2||xn|| < 2fc'||2||, making Twz almost open. ■
The invertibility of T is also constrained by that of Twz and T,w^z:
3.11.3 THEOREM If T e BL(X, Y) and T{W) QZ CY, then
Twz invertible and Ttwz invertible ==> T invertible (3.11.3.1)
T invertible and Twz invertible ==> T/^z invertible (3.11.3.2)
T invertible and T/WfZ invertible ==> Twz invertible (3.11.3.3)
Proof: We have proved this many times over: for example (3.11.3.1)
follows from (3.11.1.2) and (3.11.1.4), with (3.3.7.1), or alternatively from
(3.11.1.1) and (3.11.1.6), with (3.4.7.1). For (3.11.3.2) we can use (3.11.1.4)
and (3.11.2.2), with (3.3.7.1); for (3.11.3.3) we can use (3.11.1.1) and
(3.11.2.5), with (3.4.7.1). ■
If W = Wx and Z = Zx are complemented subspaces of X and Y,
then the left and right invertibility of T, Twz and T/WfZ are mutually
constrained. If W2 and Z2 are closed complements for W = W1 and Z = Z2,
respectively, then (2.4.2.1)
X/W1 ~ W2 and Y/Z1 ~ Z2 (3.11.3.4)
Recalling that T(W1) C Zu without the assumption that T(W2) C Z2,
gives an "upper triangular" representation for T : X —> Y:
o r99
3.11.4 THEOREM If T e BL{X,Y) satisfies T{W1 <ZZ1<ZY, for
complemented Wx C X and Zx CY, then in the notation (3.11.3.5)
^i 15 Too (almost) left invertible
11 22 V * (3.11.4.1)
==> T (almost) left invertible ==> Tn (almost) left inv.
^in^92 (almost) right inv.
11 22 V ' (3.11.4.2)
=>» T (almost) right inv. =>» T22 (almost) right inv.
T left invertible, Tn dense =>» T22 left invertible (3.11.4.3)
T right invertible, T22 one-one ==> Tn right invertible (3.11.4.4)
w1
Zi
(3.11.3.5)

3.11 Subspaces, Quotients, and Products
101
T almost left invertible, Tn almost open
T22 almost left invertible
T almost right invertible, T22 bounded below
==> Tn almost right invertible
Proof: If SnTn = In and S22r22 = J22 then
If
^11 ^11^12^22
0 S22
^ii ^ii
Tn
L12
^22
hi
'22
^21
^22
T„
0
r12"
^22.
"Ai
0
o "
A2.
(3.11.4.5)
(3.11.4.6)
(3.11.4.7)
(3.11.4.8)
then S^T^ = I11. This proves both implications of (3.11.4.1) for left
invertibility, and the corresponding implications for almost left invertibility
are obtained if we replace each 5X- • by a sequence (Sft). We leave the details
to the reader. The arguments for (3.11.4.2) are exactly similar, and can
also be left to the reader. Towards (3.11.4.3) suppose (3.11.4.8) holds and
note that
^21-^11 — 0 an(* ^21-^12 "+" ^22-^5
22-* 22
z22
(3.11.4.9)
If also Tn is dense, then (3.11.4.9) gives S21 = 0 and hence S22T22 = ^22?
which proves (3.11.4.3). If we assume that T is almost left invertible, then
11^
ul
and \\S^T12 + 52n2r22 - J22|| —> 0 (3.11.4.10)
If also Tn is almost open then (3.4.4.1) HSJJ -► 0, giving 11^22^22-^22II -+
0, which proves (3.11.4.5). The arguments for (3.11.4.4) and (3.11.4.6) can
now safely be left to the reader. ■
If the subspaces W = Wx and Z = Zl for which T(W1) C Zx have
complements W2 Q X and Z2 CY for which also T(W2) C Z2 then we can
say even more:
3.11.5 THEOREM If T e BL{X,Y) satisfies T{WX) C Z1 C Y and
T(W2) C Z2 C y, where Wx and W2 are a complemented pair in X and Zj
and Z2 are a complemented pair in Y, then, in the notation of (3.11.3.5),
T nonsingular <$=> Tn nonsingular and T22 nonsingular (3.11.5.1)

102
3. Invertibility and Singularity
for each "nonsingularity" among one-one-ness, boundedness below, closed-
ness, denseness, ontoness, almost openness, openness, left and right
invertibility, almost left and right invertibility.
Proof: Much of this can be obtained by applying Theorem 3.11.1
separately for WUZX and for W2,Z2; Theorem 3.11.2 and Theorem 3.11.4
between them account for anything left over. ■
3.12 SEQUENCE AND FUNCTION SPACES
We begin by seeing how much "nonsingularity" is inherited from T G
BL(X,y) by operators /p(0,r) : /p(0,X) -► /p(0,y):
3.12.1 THEOREM If 0 is a nonempty set and T G BL(X,Y) then for
each p G {1,2, oo} there is implication
T nonsingular <$=>► /p(0,T) nonsingular (3.12.1.1)
where "nonsingular" means: one-one; bounded below; closed; almost open;
open.
Proof: Begin with the case p = oo: if T is one-one and Tnx = 0 with
x G ^(OjX), then T(x(t)) = 0 for each t G 0, giving x(t) = 0 for each
t G 0, so that x = 0. Conversely, if ^(H,!1) is one-one and x = 1 0 x0 is
the constant function xQ then
Tx0 = 0 =>» Tn{l 0 x0) = 0 =>» 1 0 x0 = 0 =>» x0 = 0 (3.12.1.2)
If T G BL(X,y) is bounded below, with k > 0 satisfying (3.3.1.1), then if
xel^X),
IHL = 8uPt||x(t)|| < 8uPt*||r(x(0)|| = tnr^iL (3.12.1.3)
which makes ^(OjT) bounded below. The converse may be left to the
reader. If T is closed and ||Tnxn — t/H^ —► 0 then for each t G 0 we have
||r(a:„(*)) - w(*)|| < ||rQ*„ - »||«x, —► 0 asn^oo (3.12.1.4)
giving x^t) G X for each t satisfying y(t) = T(xOQ(t)): also
ll*oolloo < ll*oo - *tf lloo + Halloo < k\\V - T^XnWn + Hx^ll^ < OO
(3.12.1.5)
so that Xco G ^(OjX). This proves that /^(ftjT) is closed. The converse
needs a moment's thought: if x G XN and ||Txn - y|| -► 0, then \\Tn(l ©

3.12 Sequence and Function Spaces
103
xn) — 1 ® y||oo ~~* ° an(* nence there is z G 1^(0,,X) for which
l©y = Tn(z) (3.12.1.6)
If we choose t G 0 and 2^ = 2(£) G X it follows that y = T{zOQ).
If T G BL(X, Y) is open, with k > 0 satisfying (3.4.1.1), and if y G
/^(n, Y) is arbitrary, then there is x : 0 —► X for which
for each t G Q,T{x{t)) = y(t) and ||x(*)|| < *||y(0H (3.12.1.7)
Necessarily x G 1^(0,,X), since H^H^ < &||y||oo < °°» an(* indeed this is
the analogue for ^(OjT) of the condition (3.4.1.1). The converse is clear,
arguing as in (3.12.1.6). The reader can easily make the modifications
needed to do the same thing for almost openness.
The corresponding results for p = 1 and p = 2 can also be left to the
reader. We work with "delta functions" St © x instead of constants 1 © x
throughout. ■
The analogue of Theorem 3.12.1 is valid for the sequence spaces
associated with a bornological space 0:
3.12.2 THEOREM If 0 is a bornological space and T G BL(X, Y) and
if c+ G {c0o>co>ci}> then there is implication
T nonsingular <$=> c+(n,T) nonsingular (3.12.2.1)
where "nonsingular" means one-one, bounded below, closed, open, almost
open.
Proof: By (3.11.1.l)-(3.11.1.3) the one-one, bounded below, and closed
components of (3.12.2.1) extend from l^Cl^T) to each c+(0,r). If y G
c+(0,T) and T is open, with k > 0 satisfying (3.4.1.1), then there is x G
/oo(0,r)with
TnX = y and ||x(*)|| < &||y(*)|| for each t G 0 (3.12.2.2)
If y G c00(0, Y) then necessarily x G c00(Q,X), and if y G c0(0, Y), then
necessarily x G c0(0, X). If more generally y G cx (0, Y), with y — 1 © y^ G
c0(0, Y), then there are z G c0(0, Y) and x^ G X for which
7^ = 2/oo with HxJI < *||yj|
and (3.12.2.3)
J"* = y - 1 © Voo with 11*11^ < k\\y - 1 © yJI^

104
3. Invertibility and Singularity
It follows
Tn(z + 1 0 xj = y with ||* + 1 0 xJL < 3*11^11^ (3.12.2.4)
This gives (3.12.2.1) for openness, and the argument for almost openness
is the same. ■
If 0 is a topological space and T 6 BL(X, Y) then
T nonsingular <*=> £7^(0, T) nonsingular (3.12.2.5)
for each nonsingularity among one-one-ness, boundedness below, and
closedness. For openness and almost openness the argument of Theorem
3.12.2 only gives us the analogue for the larger subspaces of l^Q^X) and
/^(n, Y) consisting of the functions which are continuous at a particular
point t G 0.
For certain special topological spaces, the invertibility of a G A =
£7^(0, K) is interesting:
3.12.3 DEFINITION The topological space 0 is called normal if, for
arbitrary K0,K1 C 0,
c\(K0) C int(^) =► c\(K0) C int(if1/2) C c\(K1/2)
, x (3.12.3.1)
C mtyKi) for some K1/2 Q 0
For example, if 0 is a metric space then (3.12.3.1) is easily checked. If
cl(ff0) C in^ffj C 0, then repeated application of (3.12.3.2) throws up a
family {Kx)x G 0 of subsets of 0, indexed by a dense subset AC [0, l] C R,
for which
for each A,/xGA:A</x=^ c\(Kx) C int^) (3.12.3.2)
We shall call {Kx)XeA a relief map of 0. The Urysohn function of a relief
[Kx) is the mapping ifA : 0 —► [0, l] defined by setting
KA(t) = inf {A e A:t e Kx} for each t € (UA6AiTA) \ ((UA6AXA)
(3.12.3.3)
with
{0 if * € nA6AiCA
Urysohn's lemma says that the Urysohn function of a relief is
continuous:

3.12 Sequence and Function Spaces
105
3.12.4 THEOREM If 0 is a normal space and {Kx)XeA is a relief map
of 0, then the Urysohn function KA : 0 —> [0, l] is continuous.
Proof: For each A G A the set
{t G 0: KA{t) <X} = UM<AffM is open (3.12.4.1)
and
{t G 0: KA{t) >X} = nA<MffM is closed ■ (3.12.4.2)
If 0 is a normal topological space, and cl(ff0) C intf-K^) C 0, then
we can construct a relief (Kx) and then a Urysohn function for the relief
KA = u, giving
u G C(0, [0,1]) for which u(t) = 0 if t G K0 and u(t) = 1 if tg Kx
(3.12.4.3)
We shall call the topological space 0 separated if
{t} = cl {t} for each t G 0 (3.12.4.4)
3.12.5 THEOREM If 0 is a normal separated space and A = 0^(0, K)
then
A\A~X =rle{t{A) =fright(A) = (aG A: inf |a(t)| = o} (3.12.5.1)
and
7rleft(A) = 7rright(A) = {a G Aiinto^O £ 0} (3.12.5.2)
Proof: If infn |a(-)| > 0, then 6 = l/a : t —► l/a(t) is continuous on 0,
bounded with H&Hoo = l/infn |a(-)| < °°> an<^ since 6 • a = 1 = a • 6, it
is clear that a G A-1. Conversely, if infn |a(-)| = 0 then for each e > 0
there is t G 0 for which \a(t)\ < ^e, and then U G Nbd(i) for which
5 G £7 => |a(s)| < e. Now by (3.12.4.3) with K0 = {t} and KX = U there
is u G A for which Hu^ < 1; u(t) = 1; u(s) = 0 if s G 0 \ £/. Evidently
||u|| = 1 and \\a • u\\ = \\u • a\\ < e, so that a G rleft(A) and a G fright(A).
Towards (3.12.5.2), if the interior of the set a_10 is empty, then for arbitrary
u G A there is implication a - u = 0 => u(t) = 0 for each £ G 0, extending
by continuity from t G 0 \ a_10, excluding a from 7rleft(A) = 7rright(A).
Conversely, if t G inta_10, then by (3.12.4.3) there is u G A for which
u(i) = 1 and u{s) = 0 if 5 ^ a_10. Evidently a-u = u-a = Q^u. ■

4
Banach Spaces and Completeness
A normed space which as a metric space is complete is called a Banach
space. From our point of view the most dramatic effect of this is that
"open," "almost open," and "onto" mappings are the same thing.
4.1 CAUCHY SEQUENCES
Suppose that 0 is a bornological space, in the sense of (1.9.0.1) and (1.9.0.2),
and consider the product OxO. If if C 0 x 0 is declared to be "bounded"
whenever
H C K x K' with bounded K, K' C 0 (4.1.0.1)
then OxO becomes another "bornological space." For the present
discussion we need a different definition: we shall say that H C 0 x 0 is
"bounded" provided
H C (K x 0) U (0 x K') with bounded K,K' C 0 (4.1.0.2)
If also X is a linear space and x : 0 —► X is a mapping we shall define
xv : 0 x 0 —► X by setting
xw(s,t) = x(s) - x(t) for each s,t e 0 (4.1.0.3)
When X is a normed space and x G /^(0,-X") then xv G /^(O x 0,X) with
Halloo <2|Wlc„ (4.1.0.4)
and the mapping x —► xv is also linear. If 0 is a bornological space and
107

108
4. Banach Spaces and Completeness
0 x 0 is given the bounded structure (4.1.0.2) then
xe c+(0,X) =>• xv e c+(n x 0,X) for each c* e {coojCojCi} (4.1.0.5)
4.1.1 DEFINITION If X is a normed space and 0 is a bornological space
then x e /oo(n,X) is called Cauchy iff xw e /^(n x 0,X) is null:
c(n,x) = {xe zoo(n,x):xv e c0(n x n,x)} (4.1.1.1)
Necessary and sufficient is that there should be (Ke)e>0 in Bn for
which
if e > 0 and s,t e 0 \ if£ then ||x(s) - x(*)|| < e (4.1.1.2)
When 0 = N with the finite homology, the condition is that there should
be {Ne)e>0 in N for which
if e > 0 then m>n>Ne=3> \\xm - xn\\ < e (4.1.1.3)
4.1.2 THEOREM If 0 is a bornological space and X is a normed space
then
Cl(0,X) C c{n,X) = cl(c(0,X)) C l^i^X) (4.1.2.1)
If <t>: A —> 0 is cofinal then
X*c(0,X) Cc(A,I) (4.1.2.2)
Also if x G l^QyX) then, provided A is nontrivial,
xec(Q,X) andxo^GCifA,!) =J>xGc1(0,X) (4.1.2.3)
Proo/: By (4.1.0.4) and (1.9.1.1), c(0,X) is a continuous counterimage of
a closed subspace, therefore closed. If x G c1(0,X) then there is z G X and
(ife) in Bn for which
s,t e 0 \ Ke =>» ||x(s) - x(t)|| < ||z(s) - *|| + ||* - x(*)|| < 2e (4.1.2.4)
so that x e c(Q,X). The inclusion (4.1.2.2) follows from (2.8.2.1) with
c+ = c0, noting that the induced mapping <j> x <j> : A x A —>• 0 x 0 is also
cofinal. If, for (4.1.2.3), x is Cauchy and x o <j> is convergent then for each
e > 0 there are bounded Xe C H, Fe C A with x o <f>[A \ He) C Disc(*,e)
and t,t' e 0 \ Ke => ||x(*) - x(*')|| < £• With <£(A \ H'e) C 0 \ ff£ it follows
(A \ [He U H'e)) cn\Ke and x(<£(A \ {He U #£)) C Disc(*, e) and hence
iGO\iC£ =>» ||x(*)-*|| <e ■ (4.1.2.5)

4.1 Cauchy Sequences
109
Theorem 4.1.2 says that subsequences of Cauchy sequences are Cauchy,
and that if a Cauchy sequence has a convergent subsequence then it must
itself be Cauchy.
Bounded linear operators map Cauchy sequences into Cauchy
sequences:
4.1.3 THEOREM If T G BL(X,Y) is bounded and 0 is a bornological
space then
Tnc{n,X) Cc(0,y) (4.1.3.1)
If T is bounded below, then
iGUOJ) andrnxGc(0,y) =^xGc(0,X) (4.1.3.2)
Also
T bounded below <=> c(0,T) bounded below (4.1.3.3)
Proof: If x G ^(OjX) is arbitrary then
{Tnxy = {Tnxn)xw e /^(n x n,y) (4.1.3.4)
where xv is defined by (4.1.0.3), and now (4.1.3.1) follows from (2.7.1.1)
together with Definition 4.1.1. Alternatively it is rather easy to verify
(4.1.3.1) directly. Similarly (4.1.3.2) follows from (3.3.5.3). Forward
implication in (4.1.3.3) follows from (3.3.6.1) and (3.12.1.1). To reverse it just
note that all the constants are in c(Q,X). Alternatively, use (3.12.2.1) and
Definition 4.1.1. ■
It is clear that the analogue of (4.1.3.3) holds with either "one-one" or
"closed" in place of "bounded below." If T is either open, or almost open,
and y G c(0, Y), then Theorem 3.12.2 says that we can approximate to y
with Tnxnz where z G c0(O x 0,X). It is not clear that we can do it with
z = xw and x G c(0,X). In the special case O = N we do have something:
4.1.4 THEOREM If T G BL(X,F) then
T open => c(T) onto (4.1.4.1)
and
T almost open =^ c(T) dense (4.1.4.2)

110 4. Banach Spaces and Completeness
Proof: Suppose T G BL(X, Y) is open, with k > 0 satisfying (3.4.1.1), and
suppose (em) is a sequence of positive real numbers for which
oo
E £m < °° (4-1-4.3)
m=l
Then if y G c(0, Y) is arbitrary we may define <j> : N —► N in such a way
that, for each m,n,nf in N,
<j>(m) <n<n' =>• ||yn, - yn|| < em and <£(m) < <f>{m + l) and <f>{l) = 1
(4.1.4.4)
Now choose x = (xn) in X with Tx = Y in such a way that, for each m,n
inN
*M < n < <£(m + l) =► ||xn-x^(m)|| < *||yn-y^(m)|| and M < k\\yi\\
(4.1.4.5)
We claim
4>{m) <n< <t>{m + l),<t>{rri) < ri < <j>(m' + l),n < n'
=>» ||xn, - xn|| < 2fcem + em, —> 0 as m —► oo (4.1.4.6)
since
ll*n' - *nll < ll*n' - X4>{m')\\ + ll^(m') " X<f>{m)\\ + ll^(m) " *nll
< *l|yn' -^(m')ll +*H^(m') -^(m)ll +*l|y^(m) -Vnll
This proves (4.1.4.1), and the argument for (4.1.4.2) is similar. ■
4.2 COMPLETENESS
A normed space is said to be complete if all its Cauchy sequences converge:
4.2.1 DEFINITION The normed space X is said to be complete iff
c{X) =c1{X) (4.2.1.1)
More generally, a subset K C X will be called complete iff
{x e c{X):{xn:n e N} C K} C K + c0{X) (4.2.1.2)
The reader should check that, if K C X, then
if complete =► if closed (4.2.1.3)

4.2 Completeness
111
and
X complete, K closed ==> K complete (4.2.1.4)
If X is complete then (4.2.1.1) extends to arbitrary bornological 0:
4.2.2 THEOREM If X is a Banach space and 0 is bornological, then
c(n,X) = Cl(n,X) (4.2.2.1)
Proof: If 0 is itself a bounded set then (4.2.2.1) is trivial. If not then for
each x G c(0,X) there is [Kn)neU in Bn for which, for each n G N,
t,t'en\Kn=^ \\x{t') - x{t)\\ < 1/n and Kn C fTn+1 (4.2.2.2)
Now choose <£ : N —► 0 so that
for each neN, <j>{n) eVl\Kn (4.2.2.3)
The argument now resembles Theorem 4.1.2:
xo<j>ec{x) = c1{x) =^xec1{n,x) ■ (4.2.2.4)
Theorem 4.2.2 has some "practical" consequences. Many "limit"
constructions can only be justified by showing that some mapping is "Cauchy."
For example, if X is a Banach space and x : J —► X is continuous on an
interval JCR then
J3
f x{t) dt = lim{£(*y-*,-_!)*(*;) : * — [a,P}\ if [a,/?] C J (4.2.2.5)
can be interpreted by treating the set of "partitions" t = (t0>*i>* • • >*n)
of [a,^] as a bornological space 0, and using the uniform continuity of x
on [a,/3] to prove that the induced mapping from 0 to X is "Cauchy."
There are indeed two different ways of defining the "bounded subsets"
of 0: as the subsets of the complements of sets of the form {t G 0:
max(£ • — tj_i) < 8} for some S > 0, or the subsets of the complements of
sets of the form {t e 0: {*y} D E} for some finite E C [a,/?].
Completeness in normed spaces can be tested by means of series
convergence. We recall the mapping £i : XN -+ XN of (2.8.2.6):
4.2.3 THEOREM If X is a normed space, then
£*!(*) Cc(X) (4.2.3.1)

112
4. Banach Spaces and Completeness
Necessary and sufficient for X to be complete is that
Y/1h(X)Cc1(X) (4.2.3.2)
Proof: If xel^X), then
m < n ■
y=m+i
11, x3 - 11, WxjW—>0asm—¥<x> (4.2.3.3)
y=m+i
giving (4.2.3.1). The reader should check that if the partial sum of the
||xy|| does not converge to 0 then the series JZy llxyll cannot possibly be
finite. From (4.2.3.1) it is clear that the condition (4.2.3.2) is necessary for
completeness. Conversely, if X is not complete there will be x G c(X) \
Ci(X), and then {Nk)keti for which, for each k G N,
m > n > Nk =► \\xm - xn\\ < 2~k and Nk < Nk+1 (4.2.3.4)
Define </> : N -► N by setting </>(/:) = Nk for each A; G N: then by (4.1.2.2)
and (4.1.2.3)
x' = X<f> = (xHn)) e c(X) \ Cl(X) (4.2.3.5)
Finally define y G XN by setting
y1 = x\ and yn+1 = x'n+1 - x^ for each n eN (4.2.3.6)
Evidently
yel^X) and x' = ^y^cl{X) ■ (4.2.3.7)
If X is complete then the sum of an absolutely convergent series is
"rearrangement independent": if x G li{X), then
oo oo
y^ £^n) = ^inGl if </> : N —► N is one-one and onto (4.2.3.8)
n=l n=l
One way to see this is to observe that the associated mapping x : 0 —>• X
is Cauchy, therefore convergent, where 0 is the set of all finite subsets of N
in the natural boundedness, and x(K) = ^2neK xn for each finite K C N.
We shall sometimes encounter a generalization of the idea of a complete
subset of a normed space X. Recall that K C X is called convex if
0 < t < 1 =J> tK + (1 - *)ff C K (4.2.3.9)

4.3 Spaces of Functions and Operators
113
and write
A= j*G[0,l]N:fSy = l] (4.2.3.10)
L i=i }
Then we shall say that K C X is completely convex iff there is implication
oo
t e A and x e KN =► 3 ^ *yxy G if (4.2.3.11)
y=i
Evidently, if K C X,
if convex and complete =>» if completely convex =>» if convex
(4.2.3.12)
4.3 SPACES OF FUNCTIONS AND OPERATORS
Certain standard spaces are complete: for example the scalar field K in the
norm |-|, and more generally Kn for each nGN:
4.3.1 THEOREM If Y and Z are Banach spaces then so is their product
Y x Z.
Proof: We recall that Y x Z is supposed to be normed in such a way
as to have the usual cartesian product topology. If (xn) = [yn,zn) is
Cauchy in Y x Z, then [by (4.1.3.1)] (yn) is Cauchy in Y and (zn) is
Cauchy in Z. By assumption therefore there are y^ G Y and z^ G Z for
which ||yn — y^W —► 0 and \\zn — z^W —► 0, so that also (xn) converges to
(y«»*oo)- ■
Theorem 4.2.1, and induction, says that Kn is complete for each nGN,
provided we know that K is complete, and indeed C ~ R2 is complete
because R is. We may assume that the completeness of R is known: a proof
can however be extracted from Theorem 4.5.3 below.
Closed subspaces and quotients of complete spaces are complete:
4.3.2 THEOREM If Y C X is a closed subspace of a Banach space X,
then Y and X/Y are complete.
Proof: The completeness of Y is (4.2.1.2): if (yn) in Y is Cauchy then it
is Cauchy, therefore convergent, in X\ if (yn) in Y is convergent in X then
it must be convergent in Y. The completeness of X/Y is more substantial:
we use (4.2.3.2). If z e (X/y)N is in lx{X/Y) then by the Riesz lemma
(1.5.2.1) there is (xn) in X with
xnezn and ||xn||<||*n|| + 2-n for each n € N (4.3.2.1)

114
4. Banach Spaces and Completeness
By the completeness of X it follows
£x x G cx{X) =^Y,,ze ciWy) (4'3-2-2)
By (4.2.3.2) it follows that X/Y is complete. ■
If X is complete then so are the sequence spaces Z (0,X):
4.3.3 THEOREM If X is a Banach space and 0 is a nonempty set then
Zp(0,X) is complete for each p G {1,2, oo}. If 0 is a topological space then
(7^(0, X) is complete.
Proof: If (xn) is Cauchy in /^(fyX) then by (4.1.3.1) {xn{t)) is Cauchy
in X for each t G 0, hence converges to some element x^t) G X. We claim
that x^ : 0 —► X is bounded: for each £ G 0
11*^(011 =lim||*n(*)ll <sup||xn(0|| ^supllsJL (4.3.3.1)
n n n
Thus we have an element x^ G 1^(0,,X) for which xn(t) —► £^(2) as
n —>• oo for each £ G 0: we have to show that \\xn — x^W^ —► 0. To do this
we must go back to the Cauchy condition on (xn): there is {Ne)e>0 in N
for which m > n> N. => \\xm — x„ II „ < e, so that
m>n>Ne=3> \\xm{t) - xn{i)\\ < e for each t G 0 (4.3.3.2)
Allowing m —> oo in (4.3.3.2) says that ||xn(i) — rc00(^)|| < e if t G 0 and
n> Ne, hence
n>JV,=H|*B-*JL<e (4-3.3.3)
The argument for Zj (0, X) is very similar: if (xn) is Cauchy in Zx (0, X)
then again there is x^ : 0 —► X for which \\xn(t) — x^t)]] —► 0 for each
£ G 0. This time the Cauchy condition says that there is {Ne)e>0 for which
m>n> Ne => \\xm — xJli < e, so that
m>n>Ne=^^2 HxmM - xnWII < e for arbitrary finite J C N
(4.3.3.4)
and hence, allowing m —> oo,
» > We =► ||xn - zj^ = sup J£ ||x„(t) - Xoo{t)\\ finite J C N J < e
(4.3.3.5)

4.4 Extension by Continuity
115
In particular it now follows
ll'oolli < ll*oo - *«lli + ll*«lli < e+ Wxm\\i < <*> if m>Ne (4.3.3.6)
so that x^ G /1(0,X).
The argument for l2{ft,X) can at this stage be left to the reader. If 0
is a topological space then, since £7^(0, X) is a closed subspace of /^(n, X)
by Theorem 1.8.2, the completeness of £7^(0, X) is just (4.2.1.4) again. ■
By the same argument it is clear that, if 0 is a bornological space and
X is complete, then c0(O,X), c1(0,X) and c(0,X) are all complete, being
closed subspaces of l^ (0, X), and hence by Theorem 4.3.1 the enlargements
Qn(X) are also complete. We shall see below that Qn(X) is still complete
even when X is not. If 0 is a nonempty set and X is complete then the
reader may like to check that the measure space M1(0,X) of (1.8.2.4) is
also complete.
4.3.4 THEOREM If X and Y are normed spaces and Y is complete then
so is BL(X,y).
Proof: If (rn) is Cauchy in BL(X,Y) then by (4.1.3.1) (Tnx) is Cauchy
in Y for each iGl, therefore converges to some element T^x G Y. The
mapping x —► T^x is evidently linear, and we claim also bounded: for each
xeX
WT^xW = iim||rBx|| < suP||rnx|| < (sup||rn||)||z|| (4.3.4.1)
n n n
Thus we have an element T^ G BL(X,Y) for which Tnx -► T^x. To see
that \\Tn — ^ 11 -^Owe must go back to the Cauchy condition for (Tn): if
m > n > Ne => \\Tm - Tn\\ < e then
m>n>Ne=^ ||rmx-Tnx|| < e||x||,=> WT^x-T^W < e||x|| (4.3.4.2)
for each x G X, allowing m —► oo, giving ||Tn — T^H —► 0. ■
Alternatively we can regard BL(X, Y) as isometric to a closed subspace
of /^(^y), where 0 = Discx(0 ; l).
4.4 EXTENSION BY CONTINUITY
Bounded linear operators can be extended from dense subspaces to
complete spaces:

116
4. Banach Spaces and Completeness
4.4.1 THEOREM Suppose X and Y are normed spaces, and that Y is
complete: then each T0 G BL(X0,Y) for which cl(X0) = X has a unique
extension T G BL(X,Y):
BL(X0, Y) = BL(X, Y) (4.4.1.1)
Proof: Suppose x G X is arbitrary: then there is (xn) in X for which
||xn-x||—>0 and x G X0 (4.4.1.2)
By (4.1.2.1) the sequence (xn) is Cauchy, therefore by (4.1.3.1) the sequence
(T0xn) is Cauchy in Y, therefore by Definition 4.2.1 converges: write
Tx = limroxn if (4.4.1.2) holds (4.4.1.3)
n
The mapping x —> Tx is well denned in two senses: the limit exists by
what we have just said, and does not depend on the particular choice of
sequence (xn).
x = limxn = limx'n =* \\T0x'n - T0xJ — 0 (4.4.1.4)
n n
Now that it is well denned, it is easy to see that T is linear, and also
bounded, since
||T*|| = lim||rosn|| < ||T0||limK|| = ||ro||||x|| (4.4.1.5)
n n
Thus T0 G BL(X0,y) has at least one extension T G BL(X,Y). The
uniqueness follows at once from (3.2.2.2): any two continuous mappings
T : X —► Y and T' : X —► Y which agree on the dense subset X0 C X must
be equal, whether or not Y is complete. ■
We have written equality "=" in (4.4.1.1) to emphasize that the
induced isomorphism is derived from the restriction mapping: formally, if
J : X0 —► X is the natural injection, so that the composition operator
Rj : BL(X,y) —► BL(X0,y) is defined, we have proved that Rj is invert-
ible.
When a bounded linear operator has been "extended by continuity,"
then certain kinds of nonsingularity are transmitted:
4.4.2 THEOREM Suppose X and Y are normed spaces, and that T G
BL(X,y) satisfies T(XQ) C Y0, where X = c\{X0) and Y = cl(Y0): then
T0 bounded below <=> T bounded below (4.4.2.1)

4.4 Extension by Continuity
117
and
T0 almost open «*=>• T almost open (4.4.2.2)
Proof: If T0 is bounded below, with k > 0 satisfying (3.3.1.1), and if x G X
is arbitrary, then
x = limxn =► ||x|| = lim||xn|| < lim*||roxn|| = *||Tx|| (4.4.2.3)
n n n
This is implication one way in (4.4.2.1), and the reverse is (3.3.6.1). If T0 is
almost open, with k > 0 satisfying (3.4.1.2), then for arbitrary y G Y there
is by (3.5.1.4) (yn) in Y0 for which ||yj| < ||y|| and \\yn-y\\ -+ 0, and (xnm)
in X0 for each nGN such that ||xnm|| < fc||yn|| and ||T0xnm — yn\\ < 1/m.
It follows
||xnB|| < *||y|| and ||rxnn - y\\ < \\Txn - yn\\ + \\yn - y\\ —. 0
(4.4.2.4)
Conversely, if T is almost open and y67 then (3.5.1.4)
y € cl {Tx: ||x|| < *||y||} C cl {Tx0: ||x0|| < ||x|| < *||y||} ■ (4.4.2.5)
The analogue of (4.4.2.2) holds with "dense" in place of "almost open,"
but obviously not with "open" or "onto." It also is not clear that we can
replace "bounded below" by "one-one" in (4.4.2.1).
If X is complete and T G BL(X, Y) is either bounded below or open,
then the range T(X) is also complete:
4.4.3 THEOREM If T e BL{X,Y) and X is complete, then
T bounded below =^ T(X) complete (4.4.3.1)
and
T open =^Y = T(X) complete (4.4.3.2)
Proof: Suppose y = (yn) G c(Y) is Cauchy, with yn G T(X) for each n:
then there is (xn) in X for which yn = Txn, and necessarily (4.1.3.2) (xn)
is Cauchy:
\\xm ~ xn\\ < khm ~ Vn\\ —* ° as m,n —v oo (4.4.3.3)
Since X is complete it follows that (xn) is convergent, with limit x^ say:
now
ll»» - Tx^W < k\\Txn - TXoo\\ < \\T\\\\xn - xj\ — 0 (4.4.3.4)

118
4. Banach Spaces and Completeness
For (4.4.3.2) suppose y = (yn) G c(Y) is Cauchy: then by (4.1.4.2) there is
x = (xn) G c(X) for which TNx = y. Since X is complete it follows that
x G cx(X) is convergent, and hence (2.7.1.1) y G ^(Y). ■
Alternatively, for (4.4.3.2), we can use Theorem 4.2.3. When X is
complete then certain kinds of nonsingularity coalesce for T G BL(X, Y):
\AA THEOREM If T G BL(X, Y) and X is complete, then
T bounded below => T closed (4.4.4.1)
and
T almost open => T open (4.4.4.2)
Proof: The implication (4.4.4.1) is almost immediate, using (4.4.3.1) and
(4.2.1.3). For (4.4.4.2) we follow the argument for Theorem 3.4.3, taking
T' = T. If y G Y is arbitrary and if x = (xn) G XN is constructed as in
(3.4.3.2), taking T' = T and 6 = e, then by (3.4.3.3) the sequence x = (xn)
is in /i(X). From (3.4.3.2) again it follows
V = T"'£ix€T"c1(X)Cc1(Y) (4.4.4.3)
using also (4.2.3.2). ■
Notice that in Theorem 4.4.4 we do not assume that the space Y is
complete; if however T G BL(X,Y) is almost open, then by (4.4.4.2) and
(4.4.3.2) it follows that the space Y is complete.
4.4.5 THEOREM If T G BL(X, Y) and X is complete, then
T almost left invertible =^ T left invertible (4.4.5.1)
and
T almost right invertible ==> T right invertible (4.4.5.2)
Proof: If X is complete then (Theorem 4.3.4) so is BL(Y, X), and hence
(4.4.4.2)
RT almost open ==> RT open (4.4.5.3)
and
LT almost open => LT open (4.4.5.4)

4.4 Extension by Continuity
119
where RT = BL(T,X) and LT = BL(Y,T) as in Theorem 3.6.5 and
Theorem 3.7.2. But now (4.4.5.3), (3.7.2.1), and (3.6.5.1) give (4.4.5.1), while
(4.4.5.4), (3.7.2.2), and (3.6.5.2) give (4.4.5.2). ■
If X is complete then it is clear from Theorem 4.4.4, Theorem 3.3.3, and
Theorem 3.4.3 that the closed and the open operators form open subsets
of BL(X,y). In the notation (3.3.1.3), (3.3.1.4), (3.4.1.3), and (3.4.1.4),
rleft BL(X, Y) = rleft BL(X, Y) is closed (4.4.5.5)
and
rright BL(X, Y) = rright BL(X, Y) is closed (4.4.5.6)
Similarly, using Theorem 4.4.5 and Theorem 3.7.4, the left and right
invertible operators, and the invertible operators, form open sets:
aleft BL(X, Y) = aleft BL(X, Y) is closed (4.4.5.7)
aright BL(X, Y) = aright BL(X, Y) is closed (4.4.5.8)
and
BL_1(X,y) = BL(X,y) \ (aleftBL(X,y) UarightBL(X,y)) is open
(4.4.5.9)
If X is complete and A and B are closed ideals of BL(X, X) and
BL(y,y) and T e BL(X,y) then, applying (4.4.4.2) to RT/A and LT/B
from (3.9.3.7), (3.9.3.8) gives
T almost left invertible mod A => T left invertible mod A (4.4.5.10)
and
T almost right invertible mod B ==> T right invertible mod B (4.4.5.11)
If instead X = A is a normed algebra, and is complete, and a G A, then
applying (4.4.4.2) to Ra and La gives (Theorem 3.10.4)
a almost left invertible => a left invertible (4.4.5.12)
and
a almost right invertible => a right invertible (4.4.5.13)
Also
A-1 is open in A (4.4.5.14)

120
4. Banach Spaces and Completeness
In particular
oo
||a|| < 1 =► 3(1 - a)-1 =l + ^cnGA (4.4.5.15)
n=l
By Theorem 3.1.4 the mapping a —► a-1 : A-1 —► A is continuous. We
shall see later that in a sense it is also "differentiate." We conclude with
the observation that for certain incomplete normed algebras the invertible
elements A-1 can still form an open subset of A. An example would be the
algebra A = c00 of all terminating numerical sequences: more generally if
there is a sequence (An) of complete subalgebras of A for which
oo
A = |J An and 1 G An C An+1
n=1 (4.4.5.16)
and
{a~l:ae A~l n An} C An for each neN
then it will follow from (4.4.5.14) that A-1 is open in A. To make the
argument, begin with an observation about the range of an operator on an
incomplete normed space:
4.4.6 THEOREM If X is a normed space and T € BL(X, X), then
2/GX and (Y,Try) € cx(X) =► y € (J-T)(X) (4.4.6.1)
and the following are equivalent:
(Tnx) e c0(X) for each xeX (4.4.6.2)
(/-r)-1(o) = {o>
and
(I-T)(X) C^yeX: (l>r2/) €Cl(X)}
(4.4.6.3)
Proo/: If ?/ G X and
?/ + ^ Tr?/ —^xGX asn —>oo (4.4.6.4)
r=l

4.4 Extension by Continuity
121
then necessarily Tny —> 0 as n —► oo, giving
y = lim(y - rn+1</) = (J - T) lim(y + Ty + • • • + rn</)
(4.4.6.5)
= (/-r)(x)e(/-r)(x)
which proves (4.4.6.1). If (4.4.6.2) holds and x G X is arbitrary then
(J - T)(x) = 0 => x = Tx = T2x = • • • = Tnx —► 0 as n —► oo
(4.4.6.6)
so that (I - r)_1(0) = {0}, while for each neN
y={I- T){x) =>x = y + Tx
= y + T{y + Tx) = --- = {y + Ty + --- + Tny) + Tn+1x (4.4.6.7)
giving
y + Ty + • • • + Tny = x- rn+1x —► x as n —^ oo (4.4.6.8)
Thus, both parts of (4.4.6.3) hold. Conversely, if the second part of (4.4.6.3)
holds, then for arbitrary x £ X there is x' G X for which
(7 + r + --- + rn)(/-r)x—►*' asn—^ oo (4.4.6.9)
giving
Tn+1x —> x - x' e {I - T)-1 (0) asn-^oo
noting, for the inclusion at the end, that
(J - T)rn+1x —► 0 asn —► oo
If also the first part of the condition (4.4.6.3) holds then x
(4.4.6.10) gives (4.4.6.2). ■
If the condition (4.4.6.2) is strengthened to
||rn||—►() asn—>oo (4.4.6.12)
then the operator I — T is almost invertible in the sense of Definition 3.7.1,
and hence by (3.7.3.3) I— T is also bounded below. To find out when A-1
is open in A we apply Theorem 4.4.6 with T = Lx or T = Rx for elements
xe A:
(4.4.6.10)
(4.4.6.11)
= x1 and now

122
4. Banach Spaces and Completeness
4.4.7 THEOREM If A is a normed algebra then the following are
equivalent:
A-1 is open in A (4.4.7.1)
there is 8 G ]0, l] for which {1 - x: \\x\\ < 8} C A"1 (4.4.7.2)
there is 8 G ]0,l] for which ||x|| < 8 =>» (^zr ) € cx(A) (4.4.7.3)
v=o '
aleft {A) = aleft {A) and aright {A) = aright {A) (4.4.7.4)
Proof: If (4.4.7.1) holds then 1 G A-1 lies in the interior of A-1, giving
(4.4.7.2). Conversely, if (4.4.7.2) holds then
{a(l - x): \\x\\ <8}C A'1 for each a G A'1 (4.4.7.5)
giving (4.4.7.1). If (4.4.7.2) holds then also
||x|| < 8 =>» ||2£(1)|| —► 0 as n —> oo and 1 G (/ - LX)(A) (4.4.7.6)
So that also
||x|| < 8 => \\L"(a)\\ —> 0 as n —► oo for each aeA (4.4.7.7)
The condition (4.4.6.2) of Theorem 4.4.6 is therefore satisfied, and hence
also the condition (4.4.6.3). Since (4.4.7.6) 1 is in the range of I — Lx this
gives (4.4.7.3). Conversely, if (4.4.7.3) holds then (4.4.6.1) with y = 1 and
T = Lx gives
||x|| < 8 =► 1 G (1 - x)A =► 1 - x £ aleft(A) (4.4.7.8)
while (4.4.6.1) with y = 1 and T = Rx gives
||x|| < 8 =► 1 G A(l - x) =► 1 - x £ aright(A) (4.4.7.9)
Together (4.4.7.8) and (4.4.7.9) give (4.4.7.2). Finally if (4.4.7.2) holds
and if a G A is almost left invertible then there is (an) in A for which
|| 1 — ana\\ —► 0 as n —> oo, so that for sufficiently large nGN
ana = 1 - x with ||x|| < 8 =J> ana G A-1 =J> a £ aleft(A) (4.4.7.10)
which is the first part of (4.4.7.4). The argument for the second part of
(4.4.7.4) is the same; conversely, if both parts of (4.4.7.4) hold then
A-1 = A \ (aleft(A) U aright(A)) is open in A (4.4.7.11)
giving (4.4.7.1). ■

4.5 Completions
123
We might remark that by the same argument
aleft(A) closed in A <=> aleft(A) = <rleft(A) (4.4.7.12)
and
aright(A) closed in A <=> aright(A) = aright(A) (4.4.7.13)
Hence if A-1 is open then each of the sets of left invertible and of right
invertible elements are separately open in A.
4.5 COMPLETIONS
It is familiar that every metric space can be regarded as a dense subset
of some complete space. For a normed space X we can find this in the
enlargement Q(X). We begin by comparing Cauchy sequences in l^X)
with Cauchy sequences in Q{X):
4.5.1 THEOREM If X is a normed space and K : /^(X) -► Q{X) is the
quotient, then
ifNo(lM(I))Cc1(Q(I)) (4.5.1.1)
Proof: Suppose x G XN is arbitrary and write
(xA)n = {K"x)n = xn + c0{X) for each neM (4.5.1.2)
It is clear from (2.6.1.1), (2.7.1.1) and (4.1.3.1) that if x is bounded, or
convergent, or Cauchy, then so is xA. We claim that if x is Cauchy then xA
is convergent: if {Nm)meN is such that for each n,nf,m in N
n' > n > Nm => \\xn, - xj.,, < ^ and Nm < Nm+1 (4.5.1.3)
then define x^ G XN by setting
(*oo)m = (*Njm &>r each m € N (4.5.1.4)
Since
n > Nm =* ||(xjm - (x0O)m|| < ||xB -*WJI~ < ± (4.5.1.5)
m m
we have
limsup||(xn)m - (XoJ Jloo —► 0 as n —► oo (4.5.1.6)

124
4. Banach Spaces and Completeness
In particular
*oo € U*) (4-5.1.7)
and now (4.5.1.7) reads
IK-*IIq(x)—>0 where 2 = xoo + c0(X) ■ (4.5.1.8)
Theorem 4.5.1 and Theorem 4.1.4 tell us that enlargements are
complete.
4.5.2 THEOREM If X is a normed spaced then Q(X) is complete.
Proof: Suppose y G c(Q(X)) is a Cauchy sequence, and recall, as we
remarked following the proof of (3.4.7.2), that the natural quotient K :
/^(X) —y Q{X) is open: thus by (4.1.4.1) we have y = xA for some x G
c^X)), and hence by (4.5.1.1) y G c^QpT)). ■
If we are prepared now to regard a normed space X as a subset of its
enlargement Q(X), then the closure of X in Q(X) will be its "completion."
As is familiar, this closure will be precisely the quotient c(X)/c0(X) C
Q(X), while the dense subspace which we are identifying with X is really
the quotient c1(X)/c0(X):
4.5.3 THEOREM If X is a normed space, then
q(X) = c1{X)/c0{X) is dense in c{X)/c0{X) (4.5.3.1)
and
c(X)/c0(X) is complete (4.5.3.2)
If X- = c{X)/c0{X) then also
Q(X) ^ Q(X~) is complete (4.5.3.3)
Proof: Towards (4.5.3.1) suppose x G c(X) is arbitrary, so that there is
{Ne)e>0 in N for which
n' > n > Ne =^ \\xn, - xj < e (4.5.3.4)
and now define z G X by setting
z = xNe (4.5.3.5)
Evidently the coset q(z) G q(X) is within distance e of the coset x-\-c0(X) G
c(X)/c0(X).

4.5 Completions
125
Towards (4.5.3.2) suppose that x = (xn) is a sequence in c(X) whose
coset sequence xA = (xn + c0(X)) is Cauchy in c(X)/c0(X), so that there
is {Ne)e>0 in N for which
n' > n > Ne =>• limsup \\xn,m - xnm\\ < e (4.5.3.6)
m—*<x>
Then by (4.5.3.1) there is y = (yn) in X such that
limsup ||xnm — yn\\ < — for each n G N (4.5.3.7)
m—*oo 71
We claim now that
yec(X) and \\xn - y + c0(X)||Q(x) —► () as n—► oo
(4.5.3.8)
For if m G N is arbitrary then
llVn- " y»ll < l|y»- - *»'mll + K'm - *»mll + ll*»m " »»ll (4-5-3.9)
and hence
\\yn' ~ 2/nll < —, + limsup ||xn,m - xnm|| + - —> 0 as n' > n —► oo
ft m—oo »
(4.5.3.10)
This shows that the sequence y = (yn) is Cauchy, and then (4.5.3.7) shows
that the coset y + c0(X) is the limit of the Cauchy sequence x.
Finally, for (4.5.3.3), observe that the mapping q : X —> X~ is bounded
below and dense, so that by (3.5.4.1) Q(q) : Q(^0 —► Q(X~) is invertible;
also Theorem 4.3.3 and Theorem 4.3.2 tell us that Q(X~) is complete. ■
The space X~ = c(X)/c0(X) is the familiar way of constructing the
real field R from the rationals Q. We have given the proof of (4.5.3.2) for
sentimental reasons, since we already know the result from Theorem 4.5.2;
at the same time the reader may perhaps consider the deduction of (4.5.3.3)
from (4.5.3.2) to be easier than the proof of Theorem 4.5.2. As so often in
mathematics, it is if anything counterproductive to know exactly how to
construct the completion X~:
4.5.4 DEFINITION A completion of the normed space X is a complete
normed space X~ together with J G BL(X, X~) for which
J is isometric and dense (4.5.4.1)
A completion of the bounded linear operator T G BL(X, Y) is T~ G

126
4. Banach Spaces and Completeness
BL(X~,y~) for which
T~JX = JYT (4.5.4.2)
where X~ and y~ are completions of X and Y and Jx G BL(X, X~) and
JY G BL(y,y~) are isometric and dense.
Completions are essentially unique:
4.5.5 THEOREM If X~ and X~' are completions of X, then X~ ^ X~',
and if T~ and r~' are completions of T, then T~ S r~'.
Proof: This is Theorem 4.4.1: each of the mappings J : X —► X~ and
J' : X —► X~' has unique extensions J : X~' —► X~ and J' : X~ —► X~;,
whose products J J' : X~ —► X~ and J'J : X~' —>• X~; are extensions of
the identity I: X -+ X. Thus,
J J' = / : X~ —► X~ and J'J = I: X~' —> X~' (4.5.5.1)
This shows that X~ and X~' are isomorphic. For the second part observe
that the operator T~ of (4.5.4.2) is determined uniquely by X~ and Y~, or
at least by Jx and Jy. ■
We have of course given a "metric" definition of completion. More
generally for the "topological" definition, relax (4.5.4.1) to
J is bounded below and dense (4.5.5.2)
Some of the nonsingularity of T G BL(X, Y) can be expressed in terms
of its completion T~ G BL(X~,y~):
4.5.6 THEOREM If T~ G BL(X~,y~) is a completion of T G
BL(X,y),then
T bounded below <$=> T~ bounded below <$=> T~ closed (4.5.6.1)
and
T almost open <4=^ T~ almost open <4=^ T~ open (4.5.6.2)
Proof: For (4.5.6.1) the first equivalence is (4.4.2.1) and the second
(4.4.4.1), while for (4.5.6.2) the first equivalence is (4.4.2.2) and the second
(4.4.4.2). ■

4.6 The Open Mapping Theorem
127
We also have one way implications.
T almost left invertible =► T~ left invertible (4.5.6.3)
and
T almost right invertible ==> T~ right invertible (4.5.6.4)
The reader is invited to ponder what is preventing us from reversing them.
In spite of (4.5.6.3) and (4.5.6.4), almost invertibility in a normed
algebra is exactly the same as invertibility in the completion:
4.5.7 THEOREM If A is a normed algebra, then
aleft(A) =Analeft(A~) (4.5.7.1)
and
aright(A) = A n aright(A~) (4.5.7.2)
Proof: If a e A is arbitrary, apply (4.5.6.2) with T = Ra for (4.5.7.1) and
withT = La for (4.5.7.2). ■
4.6 THE OPEN MAPPING THEOREM
We have already, in Theorem 4.4.4, proved half of the "open mapping
theorem"; the other half rests on Bairt 's theorem:
4.6.1 THEOREM If X is a Banach space and if (Kn) in X satisfies
oo
|J int(cl(tfj) = 0 (4.6.1.1)
n=l
then
\jKn?X (4.6.1.2)
n=l
Proof: If (4.6.1.1) holds we shall construct a Cauchy sequence x = (xn) in
X which has no limits in \J<^>_1Kn, so that if X is complete then (4.6.1.2)
holds. In fact, if X is complete, then we claim that (4.6.1.1) implies
oo
int |J Kn = 0 (4.6.1.3)
n=l .
Indeed if (4.6.1.1) holds and (4.6.1.3) does not then there is xx G

128
4. Banach Spaces and Completeness
intU^Lji^ and £± for which
oo
Disc(x1 jeJC [j Kn\ cl^) and 0 < ex < \ (4.6.1.4)
n=l
For if not then we would have in^c^if x)) D int U^L1Kn ^ 0, contradicting
(4.6.1.1). Next choose x2 G X and e2 for which
Disc(x2 ; e2) — intDisc(x1 ; e^) \ cl{K2) and 0 < e2 < ^ (4.6.1.5)
which is possible because intDisc(x1 ; e^) g cl(if2)- Inductively there are
sequences (xn) in X and (en) > 0 for which, for each nGN
0 < en < 2~n and Disc(xn+1 ; en+1) C int Disc(xn ; sn) \ cl(tfn+1)
(4.6.1.6)
The sequence (xn) is Cauchy in X because
m>n=> \\xm -xj< ||xn+1 - xj| + • ■ ■ + ||*m - xm_1 \\
< 2_n_1 + • • • + 2_m < 2_n —► 0(n —► oo)
(4.6.1.7)
but if the sequence (xn) converges to x^ G X then
*oo € H Disc(*n : £n) C (int( Q ifB)) \ Q «•» ■ (4-6.1.8)
n=l ^ Si=l ' ' ^ n=l
Baire's theorem is true for any nonempty complete metric space, and
can be used in unlikely places: for example if X = C[0,l] then the
subset Y of those functions x G X for which there exists t G ]0, l[ for which
the derivative x'{t) exists can be put in the form \J™=1Kn where [Kn)
satisfies (4.6.1.1). Thus we deduce that there exist continuous functions
which are nowhere differentiable. When a topological space X has the
property expressed by Theorem 4.6.1 then subsets of the form \J™=1Kn
for which (4.6.1.1) can only make up very small subsets of X. In a
sense, therefore, a Banach space is "large," and also it is very much the
exception rather than the rule for a continuous function to be
differentiable.
We may now proceed with the other half of the "open mapping
theorem."
4.6.2 THEOREM If Y is complete and T G BL(X, Y), then
T onto =*► T almost open (4.6.2.1)

4.6 The Open Mapping Theorem
129
If X and Y are both complete, then
T onto =► T open (4.6.2.2)
Proof: If T is onto, so that TX = Y, then
oo
Y={J{Tx:\\x\\<n} (4.6.2.3)
n=l
so that if Y is complete then by (4.6.1.1) there exists n G N for which
int cl T Disc(0 ; 1) = - int cl {Tx : ||x|| < n} ^ 0 (4.6.2.4)
n
Since TDisc(0 ; 1) is absolutely convex it follows from (1.4.1.2) that
0 e int cl T Disc(0 ; 1) (4.6.2.5)
But this means that there is k > 0 for which (3.4.1.2) holds. This proves
(4.6.2.1), which together with (4.4.4.2) gives (4.6.2.2). ■
Theorem 4.6.2 gives an immediate characterization of invertibility:
4.6.3 THEOREM If X and Y are Banach spaces and T e BL(X, Y) then
T one-one and onto ==> T invertible (4.6.3.1)
Proof: This is (4.6.2.2) and (3.4.7.1). ■
Theorem 4.6.3 is satisfying in the sense that if T G BL(X, Y) is
algebraically capable of being inverted then its inverse is automatically
bounded. We might observe that (4.6.2.2) can be deduced from (4.6.3.1):
for if T is onto then
T = coTe(T)K with K = coker(ker(r)) open (4.6.3.2)
and if core(T) is invertible then T is open. Theorem 4.6.3 simplifies the
characterization of the "proper" mappings of Definition 3.2.7:
4.6.4 THEOREM If X and Y are Banach spaces and T e BL{X,Y),
then
T(X) closed =^ T proper (4.6.4.1)

130
4. Banach Spaces and Completeness
Proof: If X and Y are complete then so are X/r_1(0) and cl(rX), and
if TX is closed then core(T) is one-one and onto, therefore invertible by
(4.6.3.1). ■
For the record, we note that if X is complete and T G BL(X, Y) then
T bounded below and dense => T invertible (4.6.4.2)
This is just (4.4.4.1) and (3.3.7.2). Once again it now follows that Y must
also be complete.
4.7 ALMOST OPEN AND ONTO MAPPINGS
It has been conspicuous that, for general normed spaces X and Y and T G
BL(X, y), we have been unable to combine the two chains of implication
(3.4.2.3) and (3.4.2.4) in a single line. To see why we are unable to do so,
we make an auxiliary definition:
4.7.1 DEFINITION If X and Y are normed spaces then Y is strictly
weaker than X if there exists T G BL(X,y) for which
T is one-one and onto but not invertible (4.7.1.1)
If y is obtained from X by imposing a new norm ||-||' then the condition
is that (1.7.1.1) holds but not (1.7.1.2). In the same spirit, we shall say
that X is a dense proper subspace of Y if there is T G BL(X, Y) for which
T is bounded below and dense but not invertible (4.7.1.2)
4.7.2 THEOREM If X is a dense proper subspace of Y and T = J :
X —> Y is the natural injection then
T is almost open but not onto (4.7.2.1)
Proof: Since J is isometric it is bounded below, therefore almost open by
(3.5.1.2). ■
For example if X is any incomplete normed space we may take Y = X~
to be a completion of X. More specifically, take Y = C[0, l] with X QY the
subspace of continuously different iable functions, or of polynomials. The
reader might like to use Theorem 4.7.2 to see that the implication (4.4.4.2)
serves to characterize complete spaces X.

4.7 Almost Open and Onto Mappings
131
4.7.3 THEOREM If X is complete and Y is strictly weaker than X, and
T = I: X -► Y is the identity, then
T is onto but not almost open (4.7.3.1)
Proof: If J were almost open then it would be open by (4.4.4.2), and
therefore invertible by (3.4.7.1). ■
For example if X = C[0, l] in its usual norm H-H^ then we may obtain
Y by imposing the norm ||-1| j_ defined by setting
l
Nil = / lxMI * for each x € x (4.7.3.2)
It is easily verified that J : X —► Y is bounded, and also that Y is not
complete: alternatively if we define
zn(t) =tn{0<t<l) for each neN (4.7.3.3)
then
||*n||i = —^— and ||*n||oo = 1 for each nGN (4.7.3.4)
n + 1
which shows that J : X —► Y is not bounded below.
There is still a possibility to be considered; perhaps "almost open" and
"onto" together imply "open":
4.7.4 THEOREM If Y is complete and strictly weaker than X and T =
I-.X-+Y then
T is almost open and onto, but not open (4.7.4.1)
and
T is one-one and almost open, but not bounded below (4.7.4.2)
Proof: Trivially J is onto, and therefore almost open by (4.6.2.1). If it
were either bounded below or open then it would also be invertible, by
either (3.3.7.1) or (3.4.7.1). ■
Regrettably we are not, at this stage, able to follow up with a concrete
example; to prove that the situation of Theorem 4.7.4 can occur we must
wait until (5.5.6.2) and duality theory. It will then be clear (4.7.4.2) that
a most plausible "dual" to the implication (3.5.1.2) fails.

132
4. Banach Spaces and Completeness
4.8 COMPLEMENTED SUBSPACES
In a Banach space, supplemented subspaces are complemented:
4.8.1 THEOREM If X is a Banach space and if E = E2 e L(X, X) is a
linear idempotent for which
E{X) = c\{EX) and E'1^) = cl^"1^)) (4.8.1.1)
then
EeBL{X,X) (4.8.1.2)
Proof: Write Y = E(X) and Z = E_1(0): then the mapping
row( JY,JZ) :{y>z) —>y + zeX for each (y,z) e Y x Z (4.8.1.3)
is bounded and linear, one-one and onto. If X is complete and Y and Z
are closed then (Theorem 4.3.2) Y and Z are both complete, hence also
(Theorem 4.3.1) Y x Z: it therefore follows from (4.6.3.1) that
co\{E,I-E) =TOVt{JY,Jz)-1 :X—>Y x Z (4.8.1.4)
is bounded, and hence also
E = JY o^ocoliEJ-E) :X—>Y x Z—>Y —>X m (4.8.1.5)
A related result says that certain complemented subspaces must be
closed:
4.8.2 THEOREM If X and Y are complete and T e BL(X,Y), and if
there is a closed subspace Z C Y for which
T{X) + Z = Y and T{X) n Z = {0} (4.8.2.1)
then
T{X) = cl(rX) (4.8.2.2)
Proof: Define 5 : X/T^O x Z -+Yby setting
5(x + T~x0,z) =Tx + zeY for each x e X <ind z e Z (4.8.2.3)
Then S is bounded and linear, one-one and onto, between complete spaces,
therefore by (4.6.3.1) invertible. Now
E = TS'1 :Y —> Y (4.8.2.4)

4.8 Complemented Subspaces
133
is the projection of Y upon T(X) in the direction of Z, is bounded, and
has null space
T{X) =E~1{0) closed ■ (4.8.2.5)
Theorem 4.8.1 can be used to show that, if X is complete and / : X —►
K is linear and has closed null space /_1(0) then it must be continuous:
for if f{x0) = 1 and E = / © x0 then E = E2 : X —► X is linear with
E{X) = Kx0 and ^(O) = /^(O) (4.8.2.6)
which form a pair of supplemented closed subspaces. By Theorem 4.8.1
E and hence also / are bounded. In fact however the result survives for
incomplete X:
4.8.3 THEOREM If / G L(X, K) has closed null space f~l (0) then / G
BL(X,K).
Proof: We show that if / is not bounded then
clf-^O) =X (4.8.3.1)
Indeed, if x £ X is arbitrary and e > 0, then there is x' G X for which
||*'||<e and |/(*')|>|/(*)| (4.8.3.2)
Now
t = y^- =}> ||*x'|| < e and x - tx' € /_1(0) ■ (4.8.3.3)
We conclude with what is known as the closed graph theorem:
4.8.4 THEOREM If X and Y are complete and T e L(X,Y) is such
that
graph(r) = {(x,Tx):xe X} is closed inlx7 (4.8.4.1)
then T e BL(X,Y) is bounded.
Proof: Since graph(T) is a linear subspace of X x Y, which is complete
by Theorem 4.3.2, the condition (4.8.4.1) ensures by Theorem 4.3.2 that
graph(T) is a Banach space in its own right. Evidently the mapping
Sx : {x,Tx) —> xeX for each {x,Tx) e graph(T) (4.8.4.2)
obtained by restricting the projection 71^ : X x Y —► X is bounded and

134
4. Banach Spaces and Completeness
linear, one-one and onto, therefore invertible by Theorem 4.6.3: but now
T = tt2 • S'1 : X —> Y is bounded ■ (4.8.4.3)
The condition that graph(T) C X x Y is closed says that if (xn) G XN
then
\\*n ~ *JI — 0 ^d l|rxn - 2/ooH — 0 =► Voo = TXoo (4.8.4.4)
Thus, if T e BL(X,Y) then graph(T) is closed (whether or not X and Y
are complete). If T G BL(X,y) is one-one and onto then the linear inverse
T-1 : Y —> X always has closed graph. Thus we can derive the open
mapping theorem from the closed graph theorem. One more example: if
Y and Z are a pair of closed supplementary subspaces of a normed space
X then graph(E) is closed in X, where E is the projection of X on Y in
the direction of Z.
4.9 UNIFORM BOUNDEDNESS
If X and Y are Banach spaces, then the boundedness of a subset of
BL(X,Y) can be tested "pointwise":
4.9.1 THEOREM Suppose that X is a Banach space, that Y is a normed
space and that H is a nonempty set, then if (Tt)ten is a family in BL(X, Y)
for which
then also
sup ||rtx|| < oo for each x G X (4.9.1.1)
ten
sup||rt|| < oo (4.9.1.2)
ten
Proof: if (4.9.1.1) holds then
oo
X={jKn (4.9.1.3)
n=l
where, for each nGN,
Kn = I x e X:sup ||rtx|| <n\ (4.9.1.4)
I ten )
and hence if X is complete then by Baire's Theorom 4.6.1 there must be
n G N for which
0^int(tfn)=intcl(tfn) (4.9.1.5)

4.9 Uniform Boundedness
135
noting that each of the sets Kn is already closed. Since also each Kn is
absolutely convex, (1.4.1.2) again says
Oe'mtKn (4.9.1.6)
which means that there is S > 0 for which
||x|| < 6 =>» sup ||rtx|| < n (4.9.1.7)
ten
But this means
sup||rt||<5<oo ■ (4.9.1.8)
ten o
For an alternative derivation of Theorem 4.9.1, observe that as in
(2.9.4.3) the family {Tt)ten induces a mapping TA : x -+ T.(x) from X
to /00(n,y~), where Y~ is a completion of Y and the boundedness is given
by (4.9.1.1): now the reader should verify that
graph(rA) is closed in X x l^Q^Y^) (4.9.1.9)
One kind of application of the uniform boundedness principle is to
show that certain kinds of linear mapping, if they are defined at all, must
be bounded. For example if x G K and we write
Lx:y—►x-?/eKN for each y e KN (4.9.1.10)
where
(x • y)n = xnyn for each nGN (4.9.1.11)
then the reader should use Theorem 4.9.1 to show that
£x('i) Ql1=^xelOQ=^Lxe BLil^lJ (4.9.1.12)
Another application is known as the Banach-Steinhaus theorem:
4.9.2 THEOREM If X and Y are Banach spaces and H is a bornological
space, and if {Tt)ten in BL(X, Y) satisfies
T.x e c(H, Y) for each x e X (4.9.2.1)
then there is T^ e BL(X,Y) for which
T.x - T^x e c0(H, Y) for each xeX (4.9.2.2)

136
4. Banach Spaces and Completeness
Proof: We recall from (4.1.2.1) that
c^yjG/oo^y) (4.9.2.3)
so that by (4.9.2.1) the condition (4.9.1.1) is satisfied. Since Y is complete
we have by (4.2.2.1)
c(n,y) = c1{n,Y) (4.9.2.4)
Thus, for each x G X there is T^x G Y for which (4.9.2.2) holds. Evidently
the mapping T^ : x —> T^x is linear: what remains is to show that it is
bounded. If X is complete then by Theorem 4.9.1 we argue, for each x G X,
WT^xW = lim ||3>|| < sup ||3>|| < (sup ||Tt||) ||x|| (4.9.2.5)
* ten ^ten '
using the condition (4.9.1.2), so that
IIT^H <sup||Tt|| < oc ■ (4.9.2.6)
ten
The reader should note that we do not prove that
r.-r^ecofn.BLpr.y)) (4.9.2.7)

5
Linear Functionals and Duality
We have seen how the behavior of a linear operator is to some extent
determined by its interaction with other bounded operators through the medium
of all the composition operators LT = BL(W,T) and RT = BL(T,W)
associated with normed spaces W. We are ready to improve the precision with
which this can be done, and to do this using only the scalar field W = K.
We can do this only with the aid of a deep theorem, the Hahn-Banach
theorem.
5.1 THE DUAL SPACE AND THE DUAL OPERATOR
If X is a real or a complex normed space, with scalar field K = R or K = C,
then the dual space is the space
X1" = BL(X, K) = BLK(X, K) (5.1.0.1)
of bounded linear functionals on X. More traditional notation is to write
X* or X1. Like all bounded linear operators, the elements of the dual space
carry norms:
||/||= sup |/(x)| for each / G X1" (5.1.0.2)
ll*ll<i
If T G BL(X, Y) then for each g G Yt the product gT : X -> K is bounded
and linear, therefore belongs to the space X^: thus we have a well-defined
mapping T^ : Y^ —> X^ given by the formula
T*(g)=g-T for each g G Ff (5.1.0.3)
Of course this is RT = BL(T,W) with W = K. We shall refer to Tf as the
dual of T: it is more usually called the adjoint or conjugate, written T*
orT'.
137

138
5. Linear Punctionals and Duality
5.1.1 THEOREM If T G BL(X, Y), then T* G BL{Y^X^), with
||rt||<||r|| (5.1.1.1)
If also S G BL(X, Y), then for each s,t G K
{sT + *S)f = sT1" + tS* (5.1.1.2)
If also 17GBL(Y,Z), then
(l/r)1" = r^1" (5.1.1.3)
Proof: This is obtained by substituting W = K in Theorem 2.9.1. ■
Evidently the mapping T —> J1* is linear and bounded, with bound
< 1. We shall see that it is actually one-one, and isometric, though not
necessarily onto: this will use the Hahn-Banach theorem.
If K = C, so that X is a complex normed space, then the alert reader
is entitled to a certain anxiety: since every complex space is automatically
also a real space, there are two definitions of X*. Fortunately they are
isomorphic, and indeed isometrically isomorphic.
5.1.2 THEOREM If X is a complex normed space then there is isometric
isomorphism
* —> / : BLR(X, R) = BLC(X,C) (5.1.2.1)
given by the formulas
g(x) = Re f(x) for each x G X (5.1.2.2)
and
f(x) = g(x) - ig{ix) for each x G X (5.1.2.3)
Proof: If / : X —> C is complex linear, then the formula (5.1.2.2) defines
a mapping g : X —> R which is certainly real linear, and bounded if / is
bounded, with \\g\\ < \\f\\. We claim that if g is derived from / by means of
(5.1.2.2) then (5.1.2.3) also holds: for we certainly write f{x) = g(x) + ih(x)
for each x G X, and then observe that h : x —> Im/(x) is another well-
defined, real linear mapping from X to R; but now
g{ix) + ih{ix) = f{ix) = if{x) = ig{x) — h{x) for each x G X (5.1.2.4)
and by equating real and imaginary parts we find h(x) = — g(ix).
Conversely, suppose g : X —> R is real linear, and consider the mapping
/ : X —> C defined by (5.1.2.3). It is clear that / is real linear, and

5.2 Poles and Polars
139
evidently also f(ix) = i/(x), so that / is complex linear. Finally g also
satisfies (5.1.2.2). We have thus a linear isomorphism between LR(X, R) and
LC(X, C). We claim that it preserves boundedness, and indeed respects the
value of the norm. We have already seen that if / is bounded then so is g,
with ||<7|| < ||/||; conversely, if g is bounded then so is /, with ||/|| < 2\\g\\,
as the reader can check. To see more precisely that ||/|| < ||^|| write, for
each x G X,
/(*) = |/(x)|e" (5.1.2.5)
where iGR depends on x, without of course being uniquely determined:
we make a choice. Now
\f(x)\=e-itf(x) = f(e-itx) (5.1.2.6)
using complex linearity, and then—this is the subtle step—
f(e-itx)=g(e-itx) (5.1.2.7)
since by (5.1.2.6) it is already real. Now for arbitrary x G X we have, using
(1.1.1.3) for complex scalars,
|/(x)| = ff(e-"*x) = |*(e-«x)| < ||ff||||e-"*x|| = IWIINI ■
There is a genuinely larger space associated with a complex normed
space X, namely the space BLR(X, C) of real linear maps from X to C,
and then the dual has a complementary subspace BL£(X, C) in this space,
where we write
BLtpf,Y) = {T g BLR{X,Y):T(sx) = sTx
for each x G X and 5 G C}
in which 5 denotes the complex conjugate of a complex number s. The
mappings of (5.1.2.8) are called conjugate linear, or sesquilinear.
5.2 POLES AND POLARS
There is a correspondence between subspaces of a normed space and sub-
spaces of its dual. If Y is a subspace of X we shall write
Y° = {fe Xf: Y C f-1!)} (5.2.0.1)
and call it the annihilator of Y. If Z is a subspace of X^ we shall write
Z0 = f]{r10:f€Z} (5.2.0.2)

140
5. Linear Functionals and Duality
and call it the back annihilator of Z. Before recording some simple
properties of these constructions we find it convenient to extend to more general
subsets, in perhaps not the obvious way:
5.2.1 DEFINITION If K C X is nonempty then its polar is the subset
K° = \f e Xf: sup |/(x)| < 11 C X* (5.2.1.1)
I xGK )
If H C X^ is nonempty then its pole is the subset
H0 = \ x e X: sup |/(x)| <l|CX (5.2.1.2)
It is clear that the pole, or polar, of a linear subspace is just its back
annihilator, or annihilator. Thus, we are entitled to use the same notation.
The pole or polar of a nonempty set is always absolutely convex in the sense
(1.4.1.4), and further
K0 = (cyxk(K))° = (cl(cvxK(X)))° (5.2.1.3)
where
cvxK{K) = f){K':KC X'abscvx}
fv^ A, , 1 (5.2.1.4)
S=i y=i }
is the absolutely convex hull of the set K. Thus we are entitled to restrict
our discussion of poles and polars to absolutely convex sets, or to closed
absolutely convex sets. In particular the annihilator, or back annihilator,
of a subspace is always a closed subspace. We record the most elementary
facts about poles and polars:
5.2.2 THEOREM If K, K' are nonempty subsets of X and H, H' of Xf,
then
K C [K°)0 and H C (H0)° (5.2.2.1)
and
KCK'=^> (X')° £ K° and H C H' => (H')0 C H0 (5.2.2.2)
Hence also
((K°)0)° = K° and ((H0)\ = H0 (5.2.2.3)

5.3 The Hahn-Banach Theorem
141
Finally
{0}° = Xf, X° = {0} and {0}0 = X (5.2.2.4)
Proof: This is very straightforward, and left to the reader.
Theorem 5.2.2 remains valid if the dual space X* is replaced by the
space of operators BL(X,W) for a more general normed space W. The
reader is entitled to a feeling of relief at (5.2.2.3), and should contemplate
the deafening silence at the end of (5.2.2.4).
5.3 THE HAHN-BANACH THEOREM
Every normed space X has at least one bounded linear functional—the
zero mapping 0 : X —> K. As yet we have no evidence that there need
be any more. Such evidence, and much more, will be furnished by the
Hahn-Banach theorem: we begin with a preliminary version.
5.3.1 THEOREM If X is a real normed space and if gx : Xx —► R is
bounded and real linear on a real subspace Xx C X, then for each x2 C X
there is a bounded real linear extension g2 : X2 —> R of gx to the subspace
X2 = Xx + Rx2, for which \\g2\\ < \\9l\\.
Proof: If x2 G Xx (we have not excluded this possibility), then we can
(and must) take g2 = gx> If x2 £ Xx then Xx fl Rx2 = {0} and so each point
x G X2 can be written in the form x = y + sx2 with y G Xx and 5 G R. The
most general linear extension ht of gx to X2 can therefore be written as
ht(x) = gi(y) + ts if x = y + sx2 with y G Xx and 5 G R (5.3.1.1)
for some iG R: the problem is to show that there exists t G R for which
\\ht\\ < \\gi\\. It is evidently necessary that
-lltfilllly + *2ll <9i{y)+t< \\9i\\\\y + *2II for each y G Xx (5.3.1.2)
This is obtained from looking at (5.3.1.1) with 5 = 1. Conversely, if (5.3.1.2)
holds then \ht{y + sx2)\ < ||<7i||||2/ + <sx2|| for each 5 > 0, and for each 5 < 0:
thus (5.3.1.2) is necessary and sufficient, as is the condition
-II01IIII2/ + Z2II -9\{y) <* < Il0illl|y + *2ll+0i(y) for each t/GXx
(5.3.1.3)
Since the middle expression is independent of y G Xx this is the same as
-Ikilllly + Z2II - 0i(y) < * < ll^illll2 + «2II + 9i(z) for each y>z e xi
(5.3.1.4)

142
5. Linear Punctionals and Duality
On the face of it this is a step backwards, but actually we have moved
forward: necessary and sufficient that there should exist t G R for which
(5.3.1.3) holds is that
-II01IIII2/ + S2II ~9i{y) < l|0illl|2 + x2|| +0i(*) for each t/,2 G Xx
(5.3.1.5)
Obviously (5.3.1.5) is necessary for (5.3.1.4). Conversely, if (5.3.1.5) holds
then the left-hand side of (5.3.1.5) is bounded above by each representative
of the right-hand side, therefore has a least upper bound, which in turn is
a lower bound for the right-hand side. Thus the sup of the left-hand side
is less than or equal to the inf of the right-hand side of (5.3.1.5), leaving
room for t G R in the middle of (5.3.1.4). It remains to establish (5.3.1.4):
but this reduces to the defining properties of the norm and of the operator
bound; for each y,z G Xx
9iiz)-9i{y) =9i{*-y) < ll0illll*-y||
= Ik II IK* + *2) - iv + *2)ll (5.3.1.6)
<lkilllk + *2ll + lkilll|y + *2ll
This is just a rearrangement of (5.3.1.5). ■
Theorem 5.3.1 might be referred to as the "one-step real" Hahn-Banach
theorem: ordinary mathematical induction converts it to a "finite-step real"
Hahn-Banach theorem, which can then be made into a complex theorem
using Theorem 5.1.2. To get the general Hahn-Banach theorem from
Theorem 5.3.1 needs Zorn's lemma Theorem 1.11.3:
5.3.2 THEOREM If X is a real or complex normed space and f0 G [Xoy
is a bounded linear functional on a subspace X0 C X, then there is an
extension / G X* of f0 to X with ||/|| < ||/0||.
Proof: We consider the real and the complex cases separately. If X is a
real normed space and k > 0 consider the partially ordered set
Vk = {/o € (Xoy-.X0 C X and ||/0|| < k}
in which the partial order is extension of mappings:
fx ^ /2 ^^ xx C X2 and f2(x) = /x(x) for each x G Xx (5.3.2.2)
We must verify that £lk satisfies Zorn's condition (1.11.1.2): if {f\)\£\ is
a totally ordered family in flk then XA = UAGAXA is a linear subspace of
X-the reader should carefully check that it is closed under addition-and
we may define fA : XA —> R as the common extension of all the fx to XA.
(5.3.2.1)

5.4 Duality Theory
143
By (1.11.3.1) it now follows that each f0 in Qk has a maximal extension
f = fi in £lk: all that remains is to see that
if fx G nk is maximal then Xx = X (5.3.2.3)
But this is Theorem 5.3.1: if Xx ^ X then there is /2 G £lk for which
We have thus established the real case of Theorem 5.3.2: of course if
f0 G {X0)^ 1S prescribed we choose £lk with k = ||/0||. For the complex
case we combine the real case with Theorem 5.1.2: indeed if f0 G {Xoy
for a complex linear subspace X0 of a complex normed space X then we
define g0 : X —► R by means of (5.1.2.2), with f0 in place of /, use what
has just been proved to determine an extension g : X —► R of g0 with the
same norm as g, and finally define / by means of (5.1.2.3). ■
The reader is invited to derive a "one-step complex" Hahn-Banach
theorem from Theorem 5.3.1 and Theorem 5.1.2 without Zorn's lemma:
he is warned to "watch his steps." One other remark is that the entire
argument is valid for seminomas: we have at no stage used the condition
(1.1.1.2).
5.4 DUALITY THEORY
With the aid of Theorem 5.3.2, the dual space can really get to work on
a real or complex normed space. Essentially we are entitled to treat the
bounded linear functional on X as a "system of coordinates" for X: while
perhaps we do not have a "system," at least we have "enough." We have
for example "enough" bounded functionals to distinguish between elements
of the space, to calculate their norms, and to determine closed subspaces:
5.4.1 THEOREM The dual X* separates points of a normed space X,
in the sense that if y ^ x in X then there is / G X^ for which f(y) ^ /(x).
Further
||x|| = sup |/(x)| for each x G X (5.4.1.1)
ll/ll<i
and
\\T*\\ = \\T\\ for each T G BL{X,Y) (5.4.1.2)
Also
(Y°) = Y for each closed subspace Y C X (5.4.1.3)

144
5. Linear Functionals and Duality
Proof: The idea of the proof is to first find f0 on a suitable subspace X0
and then extend, using Theorem 5.3.2. Thus, if y ^ x take X0 = K(y — x)
and define f0 by setting fo{t(y — x)) =t for each iGK: evidently f0 is well
defined and linear, and bounded with ||/0|| = l/\\y — x\\. By Theorem 5.3.2
there is an extension / G X* of f0: evidently f(y — x) = f0(y — x) ^ 0,
so that f(y) ^ f{x). Towards (5.4.1.1) it is clear that the right-hand side
is always less than or equal to the left, and equality is trivial if x = 0. If
i^O define f0 : X0 —► K by taking X0 = Kx, and f0(tx) = t\\x\\ for each
t G K; then again f0 is well defined and linear, with this time ||/0|| = 1.
Theorem 5.3.2 gives an extension / G X^ with ||/|| = ||/0|| = 1, and of
course /(x) = f0(x) = ||x||. This gives equality in (5.4.1.1), and hence also
in (5.4.1.2): for if x G X then
||rx||= sup |<7(rx)|<||x|| sup ||^r|| = ||x||||rt||
IWI<i llffll<i
Finally, for (5.4.1.3), suppose that Y is a closed subspace of X and that
x0 G X \Y: since Y is closed this implies dist(x0, Y) > 0. If we define
X0 = Y+Kx0 then vectors in X0 can be written uniquely in the form y+sx0
with y G Y and 5 G K, giving a well-defined linear mapping f0 : y + sx0 —> 5
from X0 to K. We claim that /0, which is of course linear, is bounded: for
if 0 ^ s G K then
ll y ll
||t/ + 5x0|| = |5| - + x0 > |s|dist(x0,Y)
II 5 II
which makes ||/0|| < l/dist(x0, Y). Now by Theorem 5.3.2 again there is
/ G X^ extending f0: evidently
feY° and /(x0)^0 (5.4.1.4)
which proves that x0 is not in (Y°)0. ■
The first part of Theorem 5.4.1 breaks the "deafening silence" at the
end of (5.2.2.4): it is now clear that
(Xt)0 = {0} C X (5.4.1.5)
This is in a sense the "topological" content of the "metric" (5.4.1.1).
With the aid of the uniform boundedness principle, we can use the
dual space to decide whether or not a subset is bounded:
5.4.2 THEOREM A subset K C X is bounded if and only if f{K) is
bounded in K for each / G X^.

5.5 The Separation Theorem
145
Proof: If ||x|| < M for each iGif, then obviously |/(x)| < M||x|| for each
x G K: thus, if / G X^ then f{K) is bounded in K. Conversely, recall
(2.9.1.5) the operators
Rx : / — /(*) : *t _^ K (5.4.2.1)
If each set /(if) is bounded in K then {Rx)x^k 1S a family in BL(Xt,K)
satisfying the condition (4.9.1.1) of Theorem 4.9.1, while of course X is
complete by Theorem 4.2.4. Theorem 4.9.1 says that the condition (4.9.1.2)
is satisfied, which gives with (5.4.1.1) inequality
||x|| = HtfJI <M for each xeK ■ (5.4.2.2)
5.5 THE SEPARATION THEOREM
So far our "system of coordinates" X^ has succeeded in distinguishing
between points of X, evaluating their norms, as well as in delineating closed
subspaces and characterizing bounded subsets. We claim that X^ can also
delineate closed absolutely convex subsets, although we have to work a little
harder:
5.5.1 THEOREM If 0 ^ K C X is closed and absolutely convex then
(K°)o = K.
Proof: Of course K C (K°)0 by (5.2.2.1); if x0 G X \ K we must find
/Glf for which
|/(x0)|>sup|/| = sup|/(x)| (5.5.1.1)
K xGK
We begin by replacing the set K by a larger set: if 0 < 8 < dist(x0,if)
then
x0 £ K' = K + Disc(0 ;6) = {y + z:y€ K, \\z\\ < 6}
and then define a mapping q : X —> [0, oo] by setting
q{x) = inf {k > 0: x G kK'} for each x G X (5.5.1.2)
Evidently q satisfies the conditions (1.1.1.3) and (1.1.1.4) of Definition 1.1.1,
and is also finite since q(x) < ||x||/£ for each x G X. Thus q is a seminorm
on X, sometimes called the Minkowski functional of the absolutely convex
set K'. We might think of the set K' swelling up under scalar multiplication
until it swallows the vector x. For us the important property of q is the
implication, for each x G X,
q(x) < 1 =^ x G K' =J> q(x) < 1 (5.5.1.3)

146
5. Linear Punctionals and Duality
Now with X0 = Kx0 define f0 : X0 —► K by setting f0{tx0) = t for each
iGK. Evidently f0 is linear, and satisfies |/o(^o)l — tf(*xo) f°r eacn * £ K.
This holds for t = 1 by (5.5.1.3), and hence for all t G K. The Hahn-Banach
theorem 5.3.2 holds with the seminorm q instead of the norm ||-||, which
means that there is a linear extension / : X —► K of f0 which satisfies
|/(x)| < q{x) < ^ for each x G X
6
Thus / G X*. Noting that (3.4.7.6) a linear functional is always open, we
have finally
f{x0) = 1 > sup? > sup |/| = sup \f{y + z)\ ><5' + sup|/|
K' k' yGK,\\z\\<6 K
with 8' > 0, giving (5.5.1.1). ■
The reader might like to note a passing resemblance between the
Minkowski functional q and the "Urysohn function" KA of (3.12.3.3).
If T G BL(X, Y) then it is elementary that
r(X)° = (Tt)-1(0) (5.5.1.4)
The null space of the dual operator is the annihilator of the range. This
would be true with any normed space W in place of K. By the Hahn-
Banach theorem it follows (5.4.1.3) that the closure of the range is the back
annihilator of the null space of the dual:
clTX= (r^-1^ (5.5.1.5)
5.5.2 THEOREM If T G BL(X, Y) then there is implication
T dense <^ Tf one-one (5.5.2.1)
and
T almost open ^^ Tf bounded below (5.5.2.2)
Proof: The implication (5.5.2.1) is part of the equality (5.5.1.5). If T is
almost open then the dual J1*, like all the compositions RT = BL(T,W), is
bounded below, by Theorem 3.4.4: indeed if k > 0 satisfies (3.4.1.2) then
for each g G Y^
\\gT\\= sup |jrx| = Bup{|j(»)|:yecl{rx:||i||<l}}>||j||/*
11*1161

5.5 The Separation Theorem
147
Conversely, we need the separation Theorem 5.5.1: if \\g\\ < fc||<7T|| for each
geY* then
{Tx: \\x\\ < k}° C {y G Y: \\y\\ < l}°
and hence
Disc(0;l) C (Disc(0 ; l)°)0 C ((T(Disc(0 ; A;))°)0 = clTDisc(0 ; k)
using the separation theorem for the last equality. ■
In the notation of (3.2.1.4), (3.2.1.5), (3.3.1.3), and (3.4.1.4) we have
t e 7rrightBL(x,y) <^ r1" g 7rleftBL(yt,xt) (5.5.2.3)
and
T e pight BL(x,y) ^^ r1" g fleft BL(yt,xt) (5.5.2.4)
When the spaces X and Y are complete then we can also say that T^
is bounded below iff T is onto, or equivalently iff T is open. In a sense
Theorem 5.5.2 is useful, because it is easier to decide whether or not a
mapping is one-one than to decide whether or not a mapping is dense, or
onto. The dual result would appear to be not so useful: it is also not so
exact.
5.5.3 THEOREM If T G BL(X,y) then there is implication
T' dense => T one-one (5.5.3.1)
and
T* almost open => T bounded below => T^ open (5.5.3.2)
Proof: If rt(yt) is a dense subspace of X* and x G T_10 then for
arbitrary/ G X*
/(x)=lim<7nrx = 0
n
giving (5.4.1.4) x = 0, so that T is one-one: this proves (5.5.3.1). If T* is
almost open, with k > 0 satisfying the analogue of (3.4.1.2) then (5.4.1.1)
gives,
||x|| = sup \f{x)\<suv{\gTx\\\\g\\<k}<k\\Tx\\ for each x G X
ll/ll<i
which is (3.3.1.1): this is the first implication of (5.5.3.2). Conversely, if
t is bounded below, with k > 0 satisfying (3.3.1.1), then if / G X1" is

148
5. Linear Punctionals and Duality-
arbitrary there is a well-defined mapping g0 : TX —> K defined by setting
g0(Tx) = f{x) for each x G X
Evidently g0 is linear, and bounded with \\g0\\ < k\\f\\. By Theorem 5.3.2
there is an extension g G Y* of g0 with the same bound: thus
f = gT and ||$|| < *||/||
which means that T satisfies the condition (3.4.1.1). ■
Since X* and Y* are both complete (Theorem 4.3.4) we already know
by Theorem 4.4.3 the implication T^ almost open =>• open; it is
interesting that we get the same information independently. In the notation of
(3.2.1.4), (3.2.1.5), (3.3.1.3) and (3.4.1.4) we have shown that
T G 7rleftBL(X,Y) => Tf G 7rrightBL(Yt,Xt) (5.5.3.3)
and
T G rleftBL(X,Y) <^ T G rright BL(Yt,Xt) (5.5.3.4)
By the argument for the second implication of (5.5.3.2) we can identify the
range of J1*:
T*Y* = {fe X*:3k > 0 such |/(x)| < *||T(x)|| for each x e X}
(5.5.3.5)
We shall see below that the implication (5.5.3.3) cannot in general be
reversed.
5.5.4 THEOREM If X and Y are complete and T G BL(X,Y) then
there is implication
T invertible <=> Tf invertible (5.5.4.1)
Proof: If T is invertible then (5.1.1.3) shows that T* is invertible, with
(Tt)-i = (T-i)t (5.5.4.2)
Conversely, if T^ is invertible then it is bounded below and almost open,
so that T is almost open and bounded below, therefore open and bounded
below (using Theorem 4.4.3), therefore invertible. ■
If X and Y are complete then T G BL(X, Y) is proper if and only if
Tt is:

5.5 The Separation Theorem
149
5.5.5 THEOREM If T G BL(X, Y) then
core(rf) = (coreT)1" (5.5.5.1)
If X and Y are complete then
T proper <=> T'proper
Proof: If Z C X is a closed subspace then it is (2.3.2.2) that
[XjZ^ = ^°Clt
while by the Hahn-Banach Theorem 5.3.2 we have
Zf = X^/Z0 (5.5.5.4)
It follows that the canonical factorization (2.3.3.2) is obtained for the dual
J1* by taking duals in the canonical factorization of T; in particular (5.5.5.1)
holds, and then (5.5.4.1) gives (5.5.5.2). ■
The dual of Theorem 3.5.2 holds:
5.5.6 THEOREM If T G BL(X, Y) is in the topological boundary of the
bounded below operators then T is not almost open.
Proof: If T is in the boundary of the bounded below operators then there
is (TJ in BL(X, Y) for which \\T - Tn\\ -> 0 with each Tn bounded below,
while by Theorem 3.3.3 T is not itself bounded below. By (5.1.1.1) \\T* -
Tn\\ —► 0 while by (5.5.3.2) each T^ is almost open and T^ is not itself
almost open. Theorem 3.5.2 now says that T^ is not bounded below, which
by (the easy part of) (5.5.2.2) says that T is not almost open. ■
In the notation of (3.3.1.3) and (3.4.1.4)
dfleft BL(X, Y) C fright BL(X, Y) (5.5.6.1)
While the dual of Theorem 3.5.2 holds, the dual of Theorem 3.5.1
fails, as we noted in (4.7.4.2). To see that that situation actually can arise
suppose Z is a Banach space and that g : Z —> K is a discontinuous linear
functional. If z0 G Z satisfies g(z0) = 1 then by the Hahn-Banach Theorem
5.3.2 there is / G Z^ for which f(z0) = 1, and we now define a mapping
U : /_1(0) -> X by setting U(z) = z - g{z)z0 for each z G /_1(0). The
space Y of Theorem 4.7.4 is now obtained by renorming X as follows:
II^MIIy = \\z\\z for each z € /_1(°) (5.5.6.2)
(5.5.5.2)
(5.5.5.3)

150
5. Linear Punctionals and Duality
5.6 COMPOSITION OPERATORS
Theorem 5.5.2 and Theorem 5.5.3 show that the dual operator T* tells us
as much about an operator T as all the operators RT = BL(T,W) put
together. With the help of the Hahn-Banach theorem 5.3.2, and the "rank
one" operators of (2.9.2.4), it turns out that ifW^O then the composition
RT = BL(T,W) can do nearly as well. It is appropriate to review the
simplest properties of the rank one operators / © y G BL(X,Y) associated
with y G Y and / G X* in (2.9.2.4):
5.6.1 THEOREM If X and Y are normed spaces and y G Y and /elf
then
11/0 2/11 = 112/1111/11 (5.6.1.1)
In particular /©y = 0 iff either y = 0G7or/ = 0Glt. If U G BL(Y,Z)
and V GBL(Z,X) then
U{fOy) = f(D{Uy) and {f<Dy)V = [fV)®y = [V^f)ey (5.6.1.2)
Proof: If x G X is arbitrary then ||(/ © y){x)\\ = \\f{x)y\\ = \\y\\\f{x)\
giving (5.6.1.1). Also U(f © y)(x) = U{f{x)y) = f{x)Uy and (/ © y)Vz =
{fVz)y, giving (5.6.1.2). ■
Suppose now that W ^ 0 is a fixed normed space, and write
LT = BL {W, T) RT = BL (T, W)
Rx = BL(x,W) : BL(X,W) —► W (5.6.1.3)
for each T G BL(X,Y) and each x G X, as in (2.9.2.3): then
5.6.2 THEOREM If W ^ 0 and X is a normed space then BL{X,W)
separates points of X, with
||x|| = \\RX\\ for each x G X (5.6.2.1)
If TGBL(X,Y) then also
||r|| = ||Lr|| = ||iEr|| (5.6.2.2)
Proof: Towards (5.6.2.1) it is clear that Rx is bounded, with \\RX\\ < ||x||.
Conversely, using (5.4.1.1)
||x|| = sup |/(x)| = sup |/(x)|HI= sup ||(/©w)(x)||<||Bs||
ll/ll<i ll/llll>«ll<i ll/©HI<i

5.6 Composition Operators
151
Towards (5.6.2.2), Theorem 2.9.1 gives ||Lr|| < ||T|| and \\RT\\ < \\T\\.
Conversely,
||Lr||= sup ||TI7||> sup \\h®(Tx)\\ = sup ||r*||||fc||
\\U\\<1 \\hQx\\<l \\x\\\\h\\<l
= sup ||Tx|| = ||n|
IWI<1
and
||i2r||= sup \\VT\\> sup \\gTOw\\= sup \\w\\\\gT\\
\\v\\<i \\gQw\\<i IMIIWI<i
= sup \\gT\\ = ||Tt|| = ||T||
l|y||<i
using (5.4.1.2) at the end. ■
The reader will recall from the remarks following (2.9.1.3) that for
W = X and W = Y we already knew one or other of the equalities (5.6.2.2).
Extending Theorem 5.5.2 and Theorem 5.5.3, we have:
5.6.3 THEOREM If W ^ 0 and T G BL(X, Y) then there is implication
LT dense =>- T dense <$=^ RT one-one (5.6.3.1)
LT almost open => T almost open <=> RT bounded below (5.6.3.2)
RT dense ==> T one-one <=> LT one-one (5.6.3.3)
RT almost open => T bounded below <$=^ LT bounded below (5.6.3.4)
Proof: Towards (5.6.3.1) suppose LT is dense. Then for arbitrary y € Y
and h G W^ there is [Un) in BL{W,X) for which TUn ^ hey. If h ^ 0
then there is w G W with h(w) = 1, and now Txn —* y where xn = Unw,
showing that T is dense. Towards the second implication it is Theorem
3.2.1 that if T is dense then RT is one-one, while if T is not dense then by
(5.5.2.1) there is g G Y^ for which g ^ 0 = gT, and now if w ^ 0 in W then
V = g © w gives V ^ 0 = VT, which means that RT is not one-one.
Towards (5.6.3.2) suppose LT is almost open. Then there is k > 0 for
which, for arbitrary t/G7 and h G W^ there is (Un) in BL(W,X) for which
TUn -> h(Dy, with in addition ||Un\\ < k\\h®y\\ = A;||t/||||/i||. If, in particular,
||ty|| = 1 and h{w) = \\h\\ = 1, we have Txn —► y and ||xn|| < k\\y\\ where
xn = Unw, showing that T is almost open. Towards the second implication
it is Theorem 3.5.4 that if T is almost open then RT is bounded below.
Conversely, if RT is bounded below there is k > 0 for which || V|| < fc|| VT||
for each V G BL(Y,W), in particular for V = g © w with w G W and

152
5. Linear Functionals and Duality
g G Yf, so that ||t£;||||^|| < ib||t£;||||^T|| and hence ||^|| < k\\gT\\. Thus, T is
bounded below and hence (5.5.2.2) T is almost open.
Towards (5.6.3.3) suppose RT is dense: then if Tx = 0 we have, for
arbitrary / G X^ and w G W a sequence (Vn) in BL(Y,W) for which
f ®w = limn VnT and hence f(x)w = (/ © w)(x) = limn t^Tx = 0: thus
f{x) = 0 for each / G Xt, giving (Theorem 5.4.1) x = 0. Towards the
second implication, if T is one-one then by Theorem 3.2.1 LT is one-one.
Conversely, if T is not one-one there is x G X with i^0 = Tx, and now if
O^/iG^tandl7 = /i0xwe have TU = 0^U.
Towards (5.6.3.4) suppose RT is almost open: then there is k > 0
for which if / G X^ and w G W are arbitrary there is (Vn) in BL(Y,W)
for which / © w = limnVnr with ||Vn|| < fc||/©w|| = fcHI 11/11- Taking
h G W^ and w G W with \\h\\ = \\w\\ = h(w) = 1 gives gnT -> / with
||<7nll — ^11/IN where <7n = hVn: since / is arbitrary it follows that T^ is
almost open and hence (5.5.3.2) that T is bounded below. Towards the
second implication, if T is bounded below then LT is bounded below by
Theorem 3.3.4. Conversely, if T is not bounded below then there is (xn) in
X for which ||xn|| = 1 and ||Txn|| -> 0, and now if h G W1" with \\h\\ = 1
then Un = h © xn gives ||l/n|| = 1 and \\TUn\\ -> 0. ■
If X, y, Z, and W are normed spaces and if T G BL(X, Y) and 5 G
BL(jy, Z) are bounded operators then we may combine the left and right
multiplications with the row and column operators of (2.4.0.3) and (2.4.0.4),
writing
row(B5,Lr) : [V,U) -+VS + TU
(BL(Z,Y) x BL{W,X) —► BL{W,Y))
and
col(L5,*r) : W — (SW,WT)
(BL(Y,W) —► BL(y,Z) x BL(X,W))
5.6.4 THEOREM If T G BL(X,Y) and S G BL(W,Z), then there is
implication
i(w{Rs,LT) dense => co\(Ls,RT) one-one =>- 5 one-one or T dense
(5.6.4.1)
and
icw(Rs,LT) almost open => co\(Ls,RT) bdd below
=> 5 bdd below or T almost open

5.6 Composition Operators
153
Proof: If iow(Rs,LT) is dense, then for arbitrary R G BL(W, Y) there are
sequences [Un) and (Vn) in BL{W,X) and BL(Z, Y) for which \\R-VnS-
ri/n|| -> 0, so that for arbitrary W G BL(Y, W) we have
|| {WVn) {SW) + {WT) [UnW) - WRW\\ —► 0 as n —► oo (5.6.4.3)
Taking in particular R = h® y with h G W* and t/ G Y gives
|| (WVn) {SW) + (WT) (17nW) - {hW) 0 (Wy) || —♦ 0 as n —► oo
(5.6.4.4)
Thus, if SW = 0 = WT, then the left-hand side of (5.6.4.4j^vanishes for
arbitrary y G Y and arbitrary /i G W", which means that W = 0: for if
f^O then by definition there is y G Y for which Wt/ 7^ 0, and by Theorem
5.4.1 there is h G W^ for which hW ^ 0. This proves the first implication
of (5.6.4.1). Towards the second we claim that if co\{Ls,RT) is one-one
then there is implication, for arbitrary w G W, g eY and R G BL(W, Y),
gT = 0 = Sw => gRw = 0 (5.6.4.5)
for apply co\{Ls,RT) to the operator W = {gRw)g © w. From (5.6.4.5) it
then follows that either S is one-one or T is dense: for if Sw = 0 ^ w and
gT = 0 ^ g,v/e can manufacture # = /i©t/ for which <7#iu = <7(t/)/*(w) 7^ 0.
This finishes the proof of (5.6.4.1), and the arguments for (5.6.4.2) follow
the same pattern. If iovt{Rs,LT) is almost open, then there is k > 0
such that for arbitrary R G BL(W, Y) we have \\R - VnS - TUn\\ -> 0
with in addition \\Un\\ < k\\R\\ and ||Vn|| < &||-R||, so that for arbitrary
W G BL(Y,W) (5.6.4.3) holds with the same qualification. This time take
R = hnOyn with yn G Y and hn G W"l" chosen so that
l|y»ll = l=IIU and ||WVn|| > ±\\W\\
^ __ (5.6.4.6)
\\hnW\\ > $\\W\\
t/n exists by definition of ||W|| and hn by definition of ||Wt||, using (5.4.1.2).
But now
||W||2 < 4||WW||
< 4\ffivnsw\\ + 4||i?rtfnw||
<4*||«||||w||(||5W|| + ||wr||
This proves the first implication of (5.6.4.2). Towards the second take
W = (gRw)g © w to see that if co\{Ls,RT) is bounded below then there is

154
5. Linear Punctionals and Duality
k > 0 for which, for arbitrary w G W, g G Yf and R G BL(W, Y),
kfitn|<*||fi||(|W|||5ti;|| + |^r||||ti;||) (5.6.4.7)
This is turn forces either S to be bounded below or T to be almost open:
for, if not, then there are (wn) in W and (gn) in 7^ with ||wn|| = 1 =
||<7n|| and \\SwJ + ||^nT|| -+ 0, and then [hn) in W* and (t/J in Y with
fc»W = ! = QniVn) and IKII = 1 > H^nll But nOW Rn = K® Vn Sives
<7nii>n = 1 and ||5ti;n||||i2n||||^n|| + IKIHKUII^II - 0, contradicting
(5.6.4.7). ■
5.7 ENLARGEMENTS
Using the Hahn-Banach theorem, we can replace each of (3.4.5.2) and
(3.2.5.2) by the full analog of (3.3.5.2):
5.7.1 THEOREM If T G BL(X, Y) then there is implication
Q(r) dense => T almost open => Q(T) open (5.7.1.1)
Proof: If T G BL(X, Y) is not almost open then by the separation theorem
(5.5.2.2) the dual operator T^ is not bounded below, and hence there is
g = (gn) in Y* with
||*n|| = l and ||<7nr||-*0 (5.7.1.2)
and therefore also y = (yn) in Y for which
\\yj = l and \gn(yn)\ > ± (5.7.1.3)
Now we claim
x e U*) =► dist(y - Tx,c0(Y)) > \ (5.7.1.4)
Indeed if x G ^{X), then for each n G N we have
llv» - rxB|| = ||<7n||||yn - TxB|| > \gn(yn - Txn)\
> \9n(yn)\ - \\9nT\\\\xj > i - lixiLH^rii — \ (5-7-L5)
It follows that if y € l^Y) is given by (5.7.1.3) then
distfo.rp^X)) + c0(Y)) > i (5.7.1.6)
This gives the first implication of (5.7.1.1), and the second is part of
(3.4.5.2). ■

5.7 Enlargements 155
An alternative derivation of the first implication of (5.7.1.1) involves
certain linear functionals on the numerical space l^ = /^(N, K):
5.7.2 THEOREM If 0 G (Z^)1" satisfies
CoC^-^O) (5.7.2.1)
then for each normed space X there is a bounded linear mapping <t>x :
Q(Xt) ->Q(X)t given by the formula
<f>*x(f+c0(Xi))(x+c0(X)) = *(/.(*.)) for each x G lOQ(X),fe l^X*)
(5.7.2.2)
where, as in (2.9.4.5), /.(x.) G l^ is defined by setting
(/.(*.))» = /nW ^r each neM (5.7.2.3)
If T G BL(X, Y) is a bounded linear mapping then there is equality
Q(r)t o ft = ^x o Q(rt) (5.7.2.4)
Proof: If 0 G (Jqo)* is arbitrary then <f>(f.(x.)) G K is well defined for each
x G ^(-X^) and / G l^X^) and induces a bounded linear mapping
4>X{f) ■ * —■ *(/.(*.)) from /„,(*) to K (5.7.2.5)
for each / G /00(X"''). If in particular 0 satisfies the condition (5.7.2.1) then
<£((/. (x.)) =0 if either x G c0(X) or / G c0(Xf) (5.7.2.6)
Thus the mapping <££■ is well defined by (5.7.2.2). If also T G BL(X, Y)
then (5.7.2.4) follows from the associativity
<f>(g.(T"x).) = <f>((g o TN).(x.)) for each x G /«,(*)>0 € U^f) ■
(5.7.2.7)
If / G /00(X"'') is not in c0(Xt), so that limsupn ||/n|| > 0, then there
is x G Ioq{X) for which limsupn |/n(xn)| > 0, which means that
/.(*.) £c0 (5.7.2.8)
Now by (5.4.1.3) there is <j> G (/00)t for which (5.7.2.1) holds, satisfying
<£(/.(*.)) ^0 (5.7.2.9)
It now follows easily that if Q(T)t is one-one then so is Q(Tt):

156
5. Linear Punctionals and Duality
5.7.3 THEOREM If X and Y are normed spaces and T G BL(X,Y)
then
Q(r)f one-one => Q(Tf) one-one (5.7.3.1)
Proof: Suppose Q(Tf) : Q(Yf) -> Q(Xf) is not one-one: then there is
0£'oo(yt) for which
$ o TN G c0(Xf) and (? 0 c0{Y^) (5.7.3.2)
and hence also y G /^ (y) for which
0.(y.)gco (5.7.3.3)
As in (5.7.2.9) there is now <j> G (/00)t satisfying (5.7.2.1) for which
*((*(y.))^0 (5.7.3.4)
But now
Q(r)t(^(^ + c0(yt)) = OGQ(x)t
and (5.7.3.5)
O^0y(^ + co(^t))eQ(y)t
Thus Q(r)t is not one-one. ■
An alternative derivation of the first implication of (5.7.1.1) now
follows: by (5.5.2.1) and (5.7.3.1) we have
Q(r) dense => Q(T)f one-one => Q(Tf) one-one (5.7.3.6)
and then by (3.3.5.2) and (5.5.2.2) we have
Q(Tf) one-one => Tf bounded below => T almost open (5.7.3.7)
5.8 SEQUENCE AND FUNCTION SPACES
If X = K is the scalar field then obviously the dual space is given by
K: this is just (2.9.2.1). More generally for each ?iGN the dual of Kn is
isomorphic to Kn: this is (2.4.1.2). Generalizing still further, we can show
that the duals of the sequence spaces c0, lx and l2 are given by /1? l^, l2
respectively. For roughly the same work, we can show much more.
Suppose that H is a nonempty set, and W a normed space: then if
x : n -► W and y : Q -► W^ are arbitrary we shall define
VA(X) = Y,yt(xt) e K if Y,yt(xt) exists (5.8.0.1)
ten ten

5.8 Sequence and Function Spaces
157
For example if either x or y is "terminating" in the sense of (1.9.0.4) then
t/A (x) is certainly well defined.
5.8.1 THEOREM The mapping y -> t/A : Y -> X1" is well defined and
an isometric isomorphism if {X,Y) is (c0(Q,W), /1(Q,Wt)), or (/1(Q,W),
/oo(n,wt)),or(/2(n,w),/2(n,wt)).
Proof: If either x or t/ are "terminating" then
|y* (x)| < ]T \\*t\\\\Vt\\ < mindlxlLHyJl!, Hxll^ML, ||x||2||y||2) (5.8.1.1)
ten
Thus in all three cases the mapping y —► t/A : Y —► X^ is well defined and
linear, with bound < 1. We must show that it is in each case isometric,
and onto. For isometry
llvll < llvl (s-8-1-2)
we apply the mapping t/A to a shrewdly chosen family of elements x G X,
usually depending on y. For example if X = /1(Q,ty) take
x = 6tQq (5.8.1.3)
as in (1.8.0.6), with t G H and w eW: evidently
Hxll! = \\w\\ and t/A(x) = t/» (5.8.1.4)
If e > 0 is arbitrary and (GHwe can arrange
yt{w) = \\yt\\ and \\w\\ <l + e (5.8.1.5)
It now follows that for each t
WvtW = ytH = yA(*) < Ils/IIMI = Il3/Alllhll < llvAll(i + *)
giving (5.8.1.2) since e > 0 is arbitrary.
If instead X = c0(Q, W), or alternatively /^(n, W), take
x = ^2 sgnyt{zt)6t © zt (5.8.1.6)
tGK
with a finite subset K C ft and a mapping z : Q —► W with zt = 0 if t g K
chosen in such a way that w = zt satisfies (5.8.1.5) for each t G K. In
(5.8.1.6) we are writing
sgn(0) = 0 and sgn(re^) = eid if r > 0 and 6 G R (5.8.1.7)
so that sgn(re^) = e~id.

158
5. Linear Punctionals and Duality
Evidently x G c00(Q,W) C c0(Q,W) and satisfies
Iklloo < IklL < 1 + s and yA(x) = £ Vt(*t) = E Wl C5-8-1-8)
Allowing K -> n gives 112/111 < ||2/A||(1 + e) and hence again (5.8.1.2).
If instead X = l2{n,W) take
x=Y,yt(zt)St®zt (5.8.1.9)
with K and 2 as before, giving
/ \ 1/2
INl2<(EW2) (1 + &) and yAW = EW2 (5-8-1-10)
Once again (5.8.1.2) follows.
Finally we must see that in each case the mapping y —► t/A is onto.
Thus if / G X* is arbitrary we are looking for a sequence y = A/ G 7 for
which t/A = /. Since it is necessary that t/A(x) = f(x) for each x = 6t © W
from (5.8.1.3) we must take
[Af)t(w) = f(St © w) for each t G H and each w eW (5.8.1.11)
Thus, to finish the proof we must verify, in each case, that
AfeY and (A/)A=/ for each / G X1" (5.8.1.12)
The first inclusion follows from the proof of the inequality (5.8.1.2); towards
the second equality it is clear that / and (A/)A agree on the subspace
c00(Q,W) C X in each case, and hence we have only to confirm that in
each case c00(n,W) is a dense subspace, in the appropriate norm, of the
space X. m
Theorem 5.8.1 tells us something about the dual of the space ^(H, W)
for a nonempty set Q and a normed space W: there is isometric embedding
y — yA : 1,(0^) —♦ l^W)* (5.8.1.13)
given by the formula (5.8.0.1). Indeed it is only at the last ditch that we fail
to identify the dual of ^(r^W) here: the terminating sequences c00(Q, W)
are not in general a dense subspace of l^ (Q, W), and so / G l^ (H, W) t need
not be uniquely determined by the associated sequence
determine the dual of l^(fi, W) we begin by extending (5.8.1.13) to additive

5.8 Sequence and Function Spaces 159
set functions. Write
/xA(x) = I x dfj, = lim ^2 n(K)(x(tK)) for each /x G Mx{n9W^)
(5.8.1.14)
meaning that if e > 0 there is a partition Ke of Q for which
Uxdp- £ m(#)(*(**))
KeK
if K€ C /C and lKGX for each KgK.
< e (5.8.1.15)
5.8.2 THEOREM There is isometric isomorphism fi -► /xA : Mx (H, K) =
/oo(n,K)t.
Proof: When W and hence also W^ is the scalar field K then the "Riemann
sums" of (5.8.1.12) becomes a little simpler:
■ 0 under refinement of partition K
(5.8.2.1)
By the completeness of the field K the limit exists whenever Wx]]^ and \\fi\\i
are both finite:
|MA(x)| < IklLIMIi (5-8.2.2)
If K is a finite partition of H and we take
x= ^2sgRfi{K)6K (5.8.2.3)
writing now 6K for the characteristic function of the set K, then
||x|L < 1 and M*(x) = £ \„(K)\ (5.8.2.4)
Thus
IHi < llMll (5.8.2.5)
Finally, suppose / is a bounded linear functional on the space ^(H, K) and
define
(Af)(K) = f(6K) for each K C Q (5.8.2.6)

160
5. Linear Functionals and Duality
Then evidently \i = A/ satisfies the conditions (1.8.2.2), and we have
(*/)*(*) = f{x) if x = £ *(**)«* (5.8.2.7)
The proof is completed by showing that such "simple functions" form a
dense subspace of 1^(0). In the real case K = R suppose x G ^(H) and
e > 0, and take
K€ = {Hj:j = l,2,...,n}
withify = {ten:cj+1 <x{t) <Cj}{j = l,2,...,n)
(5.8.2.8)
with
b — a . . .
c=a+ ——(j = 1,2,...,n)
where a < — |lxlloo5& > H^Hoo and n >
b — a
(5.8.2.9)
This will give
KeK
< e whenever K£ C K
(5.8.2.10)
The reader is invited to make up some suitable sets H ■ in the complex case
K = C. ■
5.9 THE SECOND DUAL
When we call X^ the "dual" of X there is a suggestion that X is the dual
of X^. It is not as simple as that. Recall the evaluation maps
Rx:f —► f{x) from X to K
of (2.9.2.3), with W = K.
(5.9.0.1)
5.9.1 THEOREM If X is a normed space then, under the evaluation map
x —► Rx, there is isometric isomorphism X = XA, where XA is a subspace
ofXtt = (xt)t.
Proof: This is just (5.4.1.1) ■
In general, the subspace XA C X^ need not be the whole space X^,
even if X is complete. For example if X = c0, then X^ = l^, with a

5.9 The Second Dual
161
correspondence zv <-> z given by the formula
zv(yA) = yA(z) for each y G lx and each z G l^ (5.9.1.1)
Then it is clear that if z G l^ there is implication
zv G (c0)A = {Rx: x G c0} «=> zGc0 (5.9.1.2)
Theorem 5.9.1 offers an alternative candidate for a "completion" X~
ofX:
X~ =clXA CXft (5.9.1.3)
It is much easier, using Theorem 4.3.4, to see that clXA is complete than
to see that the space c(X)/c0(X) of Theorem 4.5.3 is. On the other hand,
to show that the embedding J : x —> R from X to X~ is isometric uses the
Hahn-Banach Theorem 5.3.2.
If X and Y are normed spaces and T G BL(X, Y), then T^ = (Jt)t :
X^ —► yl"l" is, through the medium of Theorem 5.9.1, just an extension
of r.
The situation of (5.9.1.1), in which XA is not even a dense subspace
of X, shows that the "dual" of the bipolar result (5.4.1.3) is liable to fail:
indeed
(XA)0 = {0} C Xf and hence ((XA)0)° = Xft (5.9.1.4)
We can also show that the implication (5.6.2.1) of Theorem 5.6.2
cannot in general be reversed: for if X = lx, Y = c0 and T = J : lx —► c0
is the natural injection then, through the medium of the isomorphisms
of Theorem 5.8.1, T^ : Y^ —*■ X^ is represented by the natural injection
J' : lx —*■ c0. Since J1 is defined on the same set as J, and takes the same
values at each point, the distinction between J and J' is, to say the least,
elusive; but of course J has dense range, while J1 has not. Thus, T is
one-one while T^ is not dense.
Sometimes the space X does furnish the dual of its dual X^:
5.9.2 DEFINITION A normed space X is called reflexive if there is
equality
{Rx:xeX} = X^ (5.9.2.1)
For example the space K, and Kn for each n G N, are reflexive, as is
the space l2. Curiously enough the condition
X = Xft (5.9.2.2)
is not sufficient for (5.9.2.1), although the known examples are difficult to

162
5. Linear Functionals and Duality-
understand. If X is reflexive then so are its closed subspaces Z C X and
their quotients X/Z:
5.9.3 THEOREM If X is a Banach space and Z C X is a closed sub-
space, then
X reflexive <£=> Z reflexive and X/Z reflexive (5.9.3.1)
Proof: Two applications of (5.6.4.3) and (5.6.4.4) give isomorphism
{X/Z)" S X^/Z00 and Z^ S Z00 (5.9.3.2)
in which the correspondences are those of (2.3.2.2). If we write
Y = X W = Z°° and T(x) = Rx e Xft for each x G X
(5.9.3.3)
then T(Z) C W and hence Theorem 3.11.1 applies. Further all three
mappings T, Tz, and T/Z are isometric, in particular one-one. Thus, (3.11.1.4)
gives implication
X/Z reflexive, Z reflexive => X reflexive => X/Z reflexive (5.9.3.4)
while (3.11.2.4) gives implication
X reflexive =>■ Z reflexive ■ (5.9.3.5)
A space X is reflexive if and only if its dual is:
5.9.4 THEOREM If X is a Banach space then
X reflexive <£=*> X* reflexive (5.9.4.1)
whether or not X is reflexive,
(Xf)A is a complemented subspace of (Xf)ft (5.9.4.2)
Proof: Towards (5.9.4.1) we have
X reflexive =>- X^ reflexive =>- X" reflexive =>- X reflexive (5.9.4.3)
the first implication is obvious after a moment's reflection, the second
follows from the first, while the third is (5.9.3.5). Towards (5.9.4.2) write Rx
for the evaluation map form X to X", and observe
(J2x)t • Rxt = Ixt (5.9.4.4)

5.10 An Uncomplemented Subspace
163
is the identity on X\ and hence Rx^ • (RxV : X^ ~~*" X^ 1S idempotent,
with range (Xf)A. ■
5.10 AN UNCOMPLEMENTED SUBSPACE
We have not yet had evidence that there is such a thing as an
uncomplemented subspace. In fact there is one under our nose: c0 is an
uncomplemented subspace of l^. We begin with an auxiliary result:
5.10.1 THEOREM If fn G (J^)1" for each n G N then there is implication
oo oo
«o £ D fn » => f) f-^O) g c0 (5.10.1.1)
n=l n=l
Proof: We shall construct a family (xa)A€r in l^ and prove that if c0 C
n^=i/rT1(°) then for at least one A G R the function xA is in n^/"1^),
but not in c0. We begin with an enumeration of the rationals, in the sense
of a mapping q : N —► R which is one-one, and onto the subset of rationals
Q C R, and next associate with each point A a sequence (An) of distinct
rationals converging to A: now we set
Kx = {neN:qne {Am:m G N}} for each A G R (5.10.1.2)
and finally take xA to be the characteristic function of the set Kx:
xx{n) = lifneKx and xx{n) = 0 if n G N \ Kx (5.10.1.3)
Observe that each Kx is infinite, while the intersection of any pair of distinct
Kx is finite. Thus for each A,/x G R
dist(xA,c0) = limsupxA(ra) = 1 and A^/i =>■ xAxM G c00 C c0
n
(5.10.1.4)
Suppose now g G [1^)^ and-write
Q(9)={\eR:g(xx)?0} = U~=1Qn(g)
where Qn{g) = {A G R: \g{xx)\ > 1/n} (5.10.1.5)
We claim
c0 G <7_1(0) =>- cardinal Qn{g) < ||<7|| for each nGN =>- Q(^) countable
(5.10.1.6)
Indeed if H is an arbitrary (finite) subset of Qn{g) and
ZH=Y, sgn<7(*A)xA = Y, (|0(xa)I/<7(*a))*A (5.10.1.7)

164
5. Linear Functionals and Duality
and also z'H G l^ is defined by setting
4W = (^> «i*e*-e*J-m (51018)
[ 0 else
so that z'H (n) = 0 unless n is in one and only one of the sets {Kx)XeH,
then we have
z'wW = $hM unless n G K\ n if „ with A ^ a
^ A M (5.10.1.9)
=>*2r-**e0 '(0)
and now
Ikirlloo ^ ! and 9{zh) = 9(zh) = \9(zh)\ > ~ cardinal (H)
(5.10.1.10)
giving (5.10.1.6). From (5.10.1.6) the implication (5.10.1.1) follows: for if
c0 C n^f-1^) then U™=1Q(fn) is countable and hence
oo
3AGR\ [J Q(/J (5.10.1.11)
n=l
■^oo f-l(
But for such A we have xx G H^L1/n 1(0), by (5.10.1.5), and xx & c0 by
(5.10.1.4). ■
Theorem 5.10.1 means that c0 cannot be a complemented subspace
of/„:
5.10.2 THEOREM c0 is not complemented in Z^, and hence is not a
dual space.
Proof: Suppose there were E = E2 G BL^,^) for which ^(Z^) = c0,
then we would have
oo
co=fl/n1(0) (5.10.2.1)
n=l
where for each nGN
fn{x) = {{I - E)x)n for each x e l^ (5.10.2.2)
But this contradicts (5.10.1.1). Thus, c0 is not complemented, and hence
by (5.9.4.2) cannot be the dual of any Banach space X. ■

5.11 Extreme Points 165
5.11 EXTREME POINTS
The "extreme points" of a convex set are like the vertices of a triangle:
5.11.1 DEFINITION If K = cvx(K) is a nonempty convex subset of a
real linear space X then a face of K is a nonempty convex subset H C K
with a convex complement in K:
H G Face(X) <=> 0 ^ H = cvx{H) C X and X \ H = cvx{K \ H)
(5.11.1.1)
An extreme point of K is a face consisting of a single point:
Extreme(X) = {x e X: {x} G Face(X)} (5.11.1.2)
Here we are writing
cvx{K) = lY^tjXjine N,*G [0,l]n,^^- = l,xeKn\ (5.11.1.3)
S=i 3=1 '
for the set of "convex combinations" of elements of K. Evidently
H G Face(K),#' G Face(if) =>► H' G Face(K) (5.11.1.4)
and
{#y:i G J} C Face(K), f] H) ^ 0 => f] Hj G Face(K) (5.11.1.5)
If a convex combination of elements of K is contained in a face, then so
must some of the vectors: if H G Face(if) and t G [0, l]n with 2y=i *y = *
then
n
x G iT\ 5^*yxy Gff,^0=^ xt- G if (5.11.1.6)
Linear functional cut faces off convex sets:
5.11.2 THEOREM If K C X is a nonempty compact convex subset of a
normed space X and if / G X*, then
k = sup {Re f{x):x e K} =^ Kf = {x e K: Re /(x) = A;} G Face(X)
(5.11.2.1)

166
5. Linear Functionals and Duality
Proof: By compactness Kj is nonempty, closed, and hence compact, and
it is easy to see that K* is also convex. If 0 < t < 1 and x0,x1 G K satisfy
(1 — t)x0 + tx1 G Kj, then
(1 - t) Re f(x0) + t Re /(xj = Re /((l - *)x0 + txx) = k
and since Re/(xy) < k this forces Re/(x0) = Re/(x1) = k and hence
x0,x1eKf. m
The Krein-Milman theorem guarantees that compact convex sets have
lots of extreme points:
5.11.3 THEOREM If K C X is a nonempty compact convex subset of a
normed space X, then
K = cl(cvx(Extreme(i:C))) (5.11.3.1)
Proof: We begin, using Zorn's lemma and the Hahn-Banach theorem, by
showing
Extreme^) ^ 0 (5.11.3.2)
Certainly the set of closed faces of K, partially ordered by downward
inclusion, satisfies Zorn's condition (1.11.1.2), using (5.11.1.5) and compactness.
There is, therefore, H G Face(if) which is minimal with respect to set
inclusion. If such a minimal face H is not an extreme point, containing
distinct points x0, x1? then by the Hahn-Banach Theorem 5.3.2 (for real space)
there will be (Theorem 5.4.1) /GXf for which Re/(x0) ^ Re/^). But
since both x0, xx cannot be in the set Hf of (5.1.1.2.1) this would contradict
the minimality of H.
We have proved (5.11.3.2): for (5.11.3.1) we will need the separation
theorem. Indeed if there exists x0 G K \ cl(cvx(Extreme(K))), then by
Theorem 5.5.1 there is / G X1" and k1 G R for which
Re/(x0) > A;'Re/(x) for each x G cl(cvx(Extreme(K))) (5.11.3.3)
Evidently the set Kf of (5.11.2.1) is disjoint from Extreme(if), but by
(5.11.3.2) there are extreme points in Kf, and by (5.11.1.4) these must lie
in Extreme(if). This contradiction means that x0 cannot exist, and proves
(5.11.3.1). ■
5.12 DIFFERENTIAL CALCULUS
The differential calculus for mappings from the real numbers into a normed
space is partly elementary, partly relies on the Hahn-Banach theorem, and

5.12 Differential Calculus
167
partly needs the space X to be complete. The derivative is just the limit
of the obvious difference quotient:
5.12.1 DEFINITION If X is a normed space and x : J -> X is denned
on an interval JCR and 5 G J is an interior point then the derivative of
x at 5 is the limit
(*®) =«'W = IimfWzfW (5.12.L1)
V * )t=. K ' ^ t-s K
If x'(s) exists for each 5 G J, we shall say that x is differentiable on J.
More generally, if 5 is an end-point of J, we may speak of a "left-hand"
or "right-hand" derivative of x at s. The elementary rules for differentiating
conbinations of functions need neither completeness nor the Hahn-Banach
theorem:
5.12.2 THEOREM If x : J -> X is differentiable at 5 G J, then x is
continuous at 5. If also y : J —► X and T : J ^ BL(X,Y) are differentiable
at 5, and ^ : J' —> J is differentiable at 5' G J' with <£(s') = s, then
3(ax + 0y)'{s) = ax'(s) + /?t/(s) for each a,/? G K (5.12.2.1)
3 (^^)t_3 = r'M*M + T(sW(s) (5.12.2.2)
3(x o 4>y(s') = <f>'{s')x'{<t>{s')) = ^'(« VM (5.12.2.3)
Proof: Examine, respectively, the difference quotients
ax(t) + 0y(t) - ax(s) - 0y(s) = ^x(t) - x(s) + ^yffl-yfc) g
Z 5 Z 5 Z 5
T(tHt)-T(s)x{s) = T{t)-T[s) + T{s) «(«)-«(«)
£ — 5 £ — 5 t — S
remembering here that x(t) is a continuous function of t at £ = 5, and
*' - 5' t'- s' <j>{t') - <j>{s')
where, as is customary when X = R, we interpret
(5.12.2.6)
x{*m~-x$)]) = AHs')] ifm = Hs']and' * ' "
(5.12.2.7)

168
5. Linear Functionals and Duality
For example the derivative of a constant is 0: if a, b G X then
— (ta + b) ) = a for each s G R (5.12.2.8)
dt J t=a
This gives an interesting special case of (5.12.2.2): if T G BL(X, Y) then
4(T(«(*)))=r (£:(*)) (5.12.2.9)
Two other cases are of interest: if T(t) is scalar multiplication by <j>{t) then
-^(t)x(t)) = <f>'(t)x(t) + 4>(t)x'(t) (5.12.2.10)
If instead T : J -> BL(X, Y) and S : J ->■ BL(F, Z) are differentiate, then
4-AS{t)T{t)) = S'{t)T(t) + S(t)T'(t) (5.12.2.11)
at
Finally, if x : J —► A and y : J —> A are different iable for a normed algebra
A then
-^(s(t)y(O) = *'(')</(') + *(')</'(') (5.12.2.12)
If T : J —> BL(X, Y) is different iable at 5 G if, and also invertible near
5 G J and if we know that r(£)_1 is a different iable function of £ at t = s,
then it is clear from (5.12.2.11) what the derivative of T^)-1 must be. We
can of course prove more:
5.12.3 THEOREM If T : J -> BL(X,Y) is different iable at 5 G J,
and T(t) is invertible for each t G J sufficiently close to 5, then T_1 is
different iable at 5, with
(^§-)t_3 = -rW^wrw-1 (5.12.3.1)
Proof: By the first part of Theorem 5.12.2 together with Theorem 3.1.4,
the inverse r(£)_1 is a continuous function of t. Now examine the difference
quotient
^ = .rMr(,rI . (5.12.3.2)

5.12 Differential Calculus
169
When X and Y are complete, so that BL_1(X, Y) is an open set, then
the invertibility of T(t) for t sufficiently close to 5 G J follows automatically
from the invertibility of T(s). If BL-1 (X, Y) is not an open set then we can
sustain a version of Theorem 5.12.3 in which the differentiability of r(£)_1
is analogous to the "left" or "right"-hand differentiablility above.
As we observed in (5.12.2.8), the derivative of a constant is 0. For the
converse we need the Hahn-Banach theorem:
5.12.4 THEOREM If x : J -> X is differentiate on an interval J G R
and [a,P] C J then
\\x{p)-x{a)\\<(p-a) sup ||x'(*)|| < (/? - <*)sup ||*'(-)|| (5.12.4.1)
Ot<t<0 J
In particular there is implication
x'(.) = 0 on J =>■ x(-) constant onJ (5.12.4.2)
Proof: If / G X^ = BLR(X, R) is an arbitrary (real) linear functional on
X then by (5.12.2.9) the numerical function / o x is differentiable on J,
with derivative (/ o x)' = / o x', and by the numerical mean value theorem
there is inequality
|/(*(/?)) - /(x(a))| < (/? - a) sup |/(x'(t))| < (/? - o)||/|| sup ||x'(.)||
ot<t<(3 [a,/?]
(5.12.4.3)
This together with (5.4.1.1) gives (5.12.4.1), and hence (5.12.4.2). ■
If the normed space X is complete, then every continuous mapping
x : J —► X can be represented as a derivative:
5.12.5 THEOREM If x : J —► X is a continuous mapping from an
interval JCR into a Banach space X, then for each 5 G J and each a G X
there is a unique mapping y : J —► X for which
t/(5) = a and t/(*) = x(t) for each * G J (5.12.5.1)
Proof: The uniqueness of y : J —► X satisfying (5.12.5.1) follows from
(5.12.4.2). The existence of y is through the medium of the integral
(4.2.2.5), interpreted as a limit of Riemann sums: for each t G J
y(t) =a+ [ x{t') dt' = a + lim I £(tj. - t'^Jxpj) : t' —+ [s,t] I
(5.12.5.2)

170
5. Linear Functionals and Duality
where t1 = (*o,*i,... 9tfn) is a partition of [s,t] and t'j G W_i,*y] for each
j. The continuity of x ensures that the Riemann sums are "Cauchy" on
the space of partitions of the subinterval [5, t] C J, and Theorem 4.2.2 then
ensures that the limit exists in X. To see that the derivative of y is x we
examine the difference quotient
— f x(t') dt'-x(s)\\ = — [(x(t')-x(s)) dt'\\ <sup||x(.)-x(5)||
~SJ II 11*-s J II [«,*]
3 3
(5.12.5.3)
and the right-hand side tends to 0 as t — s tends to 0, while of course the
equality and the inequality follow from the corresponding assertions for
Riemann sums. ■
If x : J —► X is continuous then generally
I x(t) dt\\ < f \\x(t)\\ dt<{/3- a) sup ||x(.)|| if [a,/?] C J
\\J II J [a,/?]
(5.12.5.4)
and
P
/ x(t) dt e (p-a)c\cvx{x{t):a<t < (3} (5.12.5.5)
When x = y' is continuous and is the derivative of y then also, whether or
not X is complete,
P
/ y'(i) dt = y{$) - y{a) for each [a,/?] C J (5.12.5.6)
We conclude by a discussion of power series. We shall write z : R —► R
for the function defined by setting
and occasionally write
z(t) = t for each iGR
dx .
x'(t) = —(t) for each t G J
dz
(5.12.5.7)
(5.12.5.8)
when x : J —► X is differentiate.

5.12 Differential Calculus 171
5.12.6 THEOREM If X is complete and k > 0, and if x = (xn) is a
sequence in X for which
oo
E KH*B < °° (5-12-6-1)
n=0
then the function
oo
^2 zTlxn = ] " M[ —► oo (5.12.6.2)
n=0
is bounded and continuous, with derivative
, oo oo
— E znxn = £>+ l)znxn+1 e C^G - k,k[,X) (5.12.6.3)
n=0 n=0
Proof: If (5.12.6.1) holds then the sequence
n
(£*%)n€N € 0(^(1 ~k,k[,X)) (5.12.6.4)
r=0
is Cauchy in the space C^Q — k,k[,X), which is complete by Theorem
4.3.3. This gives the first part of the theorem. Towards the second, the
linear operator
J0 : CTO(] - k,k\,X) —> CU] - k,k[,X) (5.12.6.5)
defined by setting
3
(l0y) (s) = / y(0 * for each«e ] - M[,y e £«>(] - *, *[,x)
0
(5.12.6.6)
is well-defined, linear, and bounded. It follows that, if — k < s < k,
t oo oo 1 °° 1
/ E <xdt = E ^Tsn+1^ = E -•"*-! (5-12-6-7)
q n=0 n=0 n=l
and hence, by (5.12.5.3),
E *"*» = Tz E ^2"x»-i (5.12.6.8)
n=0 n=l
To deduce (5.12.6.3) we go back to (5.12.6.1), and claim that if (5.12.6.1)

172
5. Linear Functionals and Duality
holds then
oo
0 < k' < k => J2 nWxn\\{k')n < °° (5.12.6.9)
n=0
since eventually n{k')n < kn. We have therefore proved the set
membership claim at the end of (5.12.6.3). This means that (5.12.6.8) holds with
((n + l)xn+1) in place of (xn), which gives the equality at the beginning of
(5.12.6.3). ■

6
Finite Dimensional
Spaces and Compactness
If bounded linear operators between normed spaces are to be thought of as
a generalization of finite dimensional linear algebra, the reader is entitled
to wonder why "boundedness" and "continuity" seem to be missing from
matrix theory.
6.1 LINEAR DEPENDENCE AND INDEPENDENCE
Suppose X is a linear space over the field K. Then a family x = {xt)ten
in X is said to be linearly independent if there is implication, for arbitrary
finite tfCH and arbitrary 5 = {st)ten, in K,
^2 stxt = ° e x => st = ° for each t e n' (6.1.0.1)
ten'
Otherwise it is called linearly dependent. The family {xt)ten is said to be
linearly generating, or spanning, if for each y G X there is finite Q' G H and
ist)ten' m K for which
y=Yl stxt (6.1.0.2)
ten'
If x = [xt) is both linearly independent and linearly generating then it is
called a Hamel basis for X. If y = (t/JieA 1S an extension of x = {xt)ten, in
the sense that Q C A, and xt = yt for each t G H, then there is implication
y linearly independent =£► x linearly independent (6.1.0.3)
and
x linearly generating =$■ y linearly generating (6.1.0.4)
173

174
6. Finite Dimensional Spaces and Compactness
Also, if x = {xt)ten is linearly dependent, and Q has more than one element,
then for at least one point t G H,
xte ^2 Kxt' (6.1.0.5)
tjtt'en
When Q is a subset of X and xt—t for each iGfl, then the family (xt)ten
reduces to the set Q. Partially ordered by inclusion, it is easy to check that
the linearly independent subsets of X satisfy Zorn's condition (1.11.1.2),
and also that a linearly independent subset of X is maximal if and only if
it is linearly generating. Thus every linearly independent subset of X can
be extended to a Hamel basis.
If Q = {1,2,..., 7i} is finite then, making an amalgam of the notations
(2.4.0.4) and (2.9.2.2), we shall write, for each x G Xn,
row(Lx)(s) = s1x1 + 52x2 H h snxn G X for each 5 G Kn (6.1.0.6)
Now evidently
x linearly independent <£=$> row(Lx) one-one (6.1.0.7)
and
x linearly generating <£=$> row(Lx) onto (6.1.0.8)
If there is n G N for which X has a basis x G Xn then X is said to be finite
dimensional. It is familiar that the number n is uniquely determined:
6.1.1 THEOREM If x G Xn is linearly independent and y G Xm is
linearly generating then
n<m (6.1.1.1)
and there is a basis z G Xp for which
{x:j = l,2,...,n} C{2.:j = l,2,...,p}
(6.1.1.2)
C {Xj:j = 1,2,... ,n} U {y-.j = 1,2,... ,m}
Proof: The argument is known as the Steinitz replacement process. We
begin with a deletion process: if y G Xm is a sequence for which
m > 2 and y is linearly dependent (6.1.1.3)
then we will write 6{y) G Xm_1 for the sequence obtained by deleting the
first term y • that is a linear combination of its predecessors. If yx =0 we

6.1 Linear Dependence and Independence
175
will delete y1. Formally,
*(v) = (Vi,"-,*y,-,Vm) (6-1.1.4)
where
j = 1 if yx = 0 and j = min I k: yk G ^ Kt/- I ify1^0 (6.1.1.5)
Evidently
m > 2 and y linearly generating =>- 6{y) linearly generating (6.1.1.6)
By repeating the operation 6 we either reach a single element 0 or a basis for
X, assuming we start with a generating sequence. Thus in a sense (6.1.1.2)
holds if n = 0. If x G Xn is linearly independent and y G Xm is linearly
generating, then the sequence
(xn,yi)...)3/m)eXm+1 (6.1.1.7)
is both linearly generating and linearly dependent.
^n,y) = {xn,y\,...,ylm)eXm (6.1.1.8)
is still linearly generating, and of course the deletion process cannot affect
the first term xn. Similarly
'(*»-i>*(*»>v)) = (*»-i> *».»!> •••.»«) e*m (6.1.1.9)
is linearly generating, and neither of its first two terms can be affected by
deletion. Eventually we reach a sequence
(x1,x2,...,xn,...,y^)GXm (6.1.1.10)
which is linearly generating, and may or may not be linearly independent;
the notation should not disguise the possibility that there are no t/'s in the
sequence (6.1.1.10). The inequality (6.1.1.1) is clear from (6.1.1.10). Of
course the proof of (6.1.1.1) consists in the fact that if x G Xn is linearly
independent then there can be no obstacle to the process of passing from
(6.1.1.7) to (6.1.1.8). If the sequence (6.1.1.10) is linearly independent then
we may take it to be z G Xp for (6.1.1.2); if not then we apply 6 repeatedly
until we reach a linearly independent z G Xp. ■
If x G Xn and y G Xm are both bases for X then (6.1.1.1) tells us
n < m < n => m = n (6.1.1.11)

176
6. Finite Dimensional Spaces and Compactness
We are hence justified in calling it the dimension of X. If X = 0 we shall
count it as finite dimensional, with dimension 0.
If / G L(X, K)n is a linearly independent sequence of linear functionals
on X then we can find a sort of "dual" sequence in X:
6.1.2 THEOREM If / = (/l5/2,. • • ,/J is linearly independent in
L(X,K) then
there is x G Xn for which /t(x) = 6- for each i,j G {1,2,... ,n}
(6.1.2.1)
and
n n
ifge L(X, K) satisfies f) /r1 (0) C ^(O) then g G J^ Kfj t6-1-2-2)
3=1 3=1
Proof: We use induction on n. The trick is to prove both at once. We
begin by showing that (6.1.2.2) follows from (6.1.2.1): for if
E = E1+E2 + --- + En with Ej = fj © xj {j = 1,2,... ,n) (6.1.2.3)
using the notation (2.9.2.4), then
n
E = E2 and jE?_1(0) = f) /"^O) (6.1.2.4)
3 = 1
Note that EiEj = S^Ej for each ij G {1,2,... ,n}. Now if T : X -> y is
arbitrary then
n n
T_1(0) D f| /ri(o) = (/- £)(X) ^=> T = T£ = £ fj ©Txy (6.1.2.5)
Applying this to T = g G L(X,K) then gives (6.1.2.2).
To see that (6.1.2.1) holds if n = 1 suppose / = fx ^ 0, so that there
is x[ G X for which fi{x[) ^ 0, and then take xx = x^/f^x^).
If finally (6.1.2.1) and (6.1.2.2) hold for n = k, we must show that
(6.1.2.1) holds with n = k + 1. Thus suppose that / = (/i,/25- • • ifk>fk+i)
is linearly independent in L(X, K), and apply (6.1.2.l)n=fc to find (x^x^,
..., x'k) G X* for which
fi(x'j) = *»y for each *>i ^ (M, • • • >*} (6.1.2.6)
The linear independence of /, together with (6.1.2.2)n=fc, implies that
/fc+i(0) does no^ C°ntain the intersection of the remaining /T1(0): thus

6.1 Linear Dependence and Independence
177
there is xjfc+1 G X for which
4+1 (4+i) ^ 0 = /y(4+1) for each j G {1,2,..., A;} (6.1.2.7)
Now, for (6.1.2.1), define [x1,x2,...,xk,xk+1) by taking
xj = x'j + six'k+iU = 1,2,...,A;) and **+i = **+i4+i (6.1.2.8)
where the scalars Sj,sk+i are chosen in such a way that
4+iK) + ^4+i(4+i)=0 (j = l,2,...,fc)
and (6.1.2.9)
5fc+i/fc+i(xfc+i) =1 ■
For a sort of dual to Theorem 6.1.2, say that the subspace Y C L(X, K)
separates points of X iff, for each x, x' G X,
t/(x') = t/(x) for each y eY => x' = x (6.1.2.10)
For example, if Y = X^ then (6.1.2.10) is part of Theorem 5.4.1, and if
yci^, then necessary and sufficient for (6.1.2.10) is
Y0 = {0} (6.1.2.11)
6.1.3 THEOREM If Y G L(X, K) is a subspace which separates points
of X and if x G Xn is linearly independent then there is y G Yn for which
Viixj) = Sij for each *>y G {l,2,...,n} (6.1.3.1)
Proof: If 7i = 1 suppose xx ^ 0 and find ^67 for which 21(x1) 7^ 0 and
put yx = z1/z1(x1). If Theorem 6.1.3 holds for n = k, and if y G Xfc+1 is
linearly independent, then there is z G Yk for which
*;(*;) = *»i for each ^J e {1,2,...,/;} (6.1.3.2)
Then by linear independence we have
k
and then, since Y separates points of X, an element zk+1 G Y for which
k
zk+i ixk+i ~ Yl ZJ (xk+i)xj) = 1 (6.1.3.4)

178 6. Finite Dimensional Spaces and Compactness
Now put
y*+i = **+i and y3-= z3-- z3ixk+i)zk+i (6 13 5)
for each j G {1,2,... ,k} m
By Theorem 5.4.1, (6.1.2.1) is a consequence of (6.1.3.1). Of course
(6.1.2.1) was proved without the use of the Hahn-Banach Theorem 5.3.1.
6.2 FINITE DIMENSIONAL SPACES
If X is a finite dimensional linear space then it can be normed, and all
possible norms give the same topology. We begin by applying one of our
"Riesz lemmas":
6.2.1 THEOREM If norms ||-|| and ||-||' on a linear space X both make
ycXa closed subspace, and if the topology of ||-|| is stronger on Y than
that of ||• ||', then for each x1 G X the same is true for the subspace Y-\-Kx1.
Proof: By Theorem 1.7.1 we are assuming that there is k > 0 for which
||t/||' < k\\y\\ for each y eY (6.2.1.1)
Then if 5 G K we have
lit/ + sXl\\' < \\y\\' + MUsJ' < fc||y|| + H(||sJ7II*iII)II*iII
<max(A;,||x1||7||x1||)(||t/|| + |5|||x1||)
and, provided xx £ Y = cl(Y) = cl'(Y) we have by (1.5.3.1) M > 0 for
which
||t/|| + H||x1||<M||t/ + 5x1||
Thus, if y G Y and 5 G K
\\y + Mill' < k'\\y + sxj with *' = Mmax(fc, ||*i||7ll*ill) (6.2.1.2)
Of course if xx G Y then (6.2.1.2) reduces to (6.2.1.1). We must see that
the larger subspace remains closed: thus suppose Y is ||»||-closed and that
l|y» + sn*i - A\ —♦ 0 with x G X,yn G y,and sn G K (6.2.1.3)
Then by (1.5.3.1), provided xx & Y,
\\yn-ym\\ + K-sm\\\xi\\ <M\\yn+snxi-ym-smxiII —^°
as m > n —> oo (6.2.1.4)

6.2 Finite Dimensional Spaces
179
Thus (t/n) and (sn) are Cauchy in Y and K, respectively. Since the scalar
field K is complete, it follows that (sn) converges, to s G K say: but this
forces (t/n) also to converge, with limit y G X. However, since Y is closed
in X we must have t/G7, and hence
x = y - sx1 G Y + Kxx (6.2.1.5)
Thus, if Y is closed in the ||-||-topology then so is Y + Kx1? and similarly
for the ||• ||'-topology. ■
The reader might like to verify that also
Y complete => Y + Kx1 complete (6.2.1.6)
The uniqueness of norm topology for finite dimensional spaces is now
clear:
6.2.2 THEOREM If X is finite dimensional, then there exists a norm
||*|| : X —*> [0, oo]. All norms define the same topology on X, and make it
complete.
Proof: If X = 0 then the result is clear, and rather trivial. If X is of
dimension ?iGl\l, with a basis e = (e1,e2,... ,en), then we obtain a norm
on X by setting
||51e1 + 52e2 + --- + 5nen|| = maxflsj, |s2|,- • •, \sn\) for each 5 G Kn
(6.2.2.1)
The reader can readily supply alternatives. In this norm the completeness
of X follows from Theorem 4.3.1. To prove that the topology is unique
we proceed by induction. If n = 1 then all possible norms are positive
scalar multiples of one another, making X isomorphic to K, and of course
complete. If the result is known for n = k, and if X is of dimension k + 1,
then we can write
X = Y + Kxx with yciof dimension k (6.2.2.2)
If ||• || and ||-||' are norms on X then by the inductive hypothesis they induce
the same topology on Y, which will necessarily be closed in X since (4.2.1.3)
it is complete. Now Theorem 6.2.1 (twice) says that ||-|| and ||-||' give the
same topology on X. m
If X is a finite dimensional normed space, then all its linear subspaces
are closed, and every linear mapping T : X —*Y into a normed space Y is
bounded:

180
6. Finite Dimensional Spaces and Compactness
6.2.3 THEOREM If T : X -> Y is a linear mapping between normed
spaces then
X finite dimensional => T bounded (6.2.3.1)
X finite dimensional and T one-one => T closed (6.2.3.2)
Y finite dimensional and T dense => T open (6.2.3.3)
Proof: If ||• || is a norm on X and T : X —► Y is linear, into a normed space
Y, then we obtain another norm ||»||' on X by setting
||x||' = ||x|| + ||Tx|| for each x G X (6.2.3.4)
and evidently the induced operator T : X' —> Y is bounded, with \\T\\' < 1,
where X' is the space obtained by imposing the norm ||-||' on the space
X. If X is finite dimensional, then by Theorem 6.2.2 the new norm ||»||'
gives the same topology as the norm ||»||. This proves (6.2.3.1). Towards
(6.2.3.2) consider the operator S : TX -► X/r_1(0) defined by
S(Tx) = x + T~1{0) e X/T^O for each x e X (6.2.3.5)
which is well defined and linear without restriction on T. If T is one-one
then S goes into X, and if X is finite dimensional then also TX C Y is finite
dimensional, so that by (6.2.3.1) the operator S is bounded, and hence the
operator
x —-► Tx : X —-► T{X) (6.2.3.6)
is left invertible, and hence closed in the sense of (3.3.1.2). An application
of (3.3.2.2) gives (6.2.3.2). If instead T is onto, then S goes from Y to
X/T~10, and if Y is finite dimensional then S is bounded, making the
operator
x + r_10 —> Tx : X/r_10 —> Y (6.2.3.7)
right invertible, hence open; now (6.2.3.3) follows from (3.4.2.1). Of course
if Y is finite dimensional and T is dense then T is onto. ■
The implication (6.2.3.3) is slightly clearer when T is assumed to be
bounded, and hence T-1(0) is closed in X. When T-1(0) is not closed then
X/r_1(0) is only a seminormed space. The continuity of the operator S is
still valid, since the argument for (6.2.3.1) extends to the case in which the
space Y carries only a seminoma.
In the notation of (3.2.1.4), (3.2.1.5), (3.3.1.4), and (3.4.1.3), we have
shown
X finite dimensional => rleft BL(X, Y) = 7rleft BL(X, Y) (6.2.3.8)

6.2 Finite Dimensional Spaces
181
and
Y finite dimensional => rright BL(X, Y) = 7rright BL(X, Y) (6.2.3.9)
It should now be clear why these ideas of "boundedness" and "continuity"
are missing from matrix theory. They happen automatically, and all the
"spatial" nonsingularities reduce either to one-oneness or ontoness. One
slightly unexpected consequence is that the set of bases for an ra-dimensional
space X is open in Xn:
6.2.4 THEOREM If X is a normed linear space and n G N then
{x G Xn:x is linearly independent} is open in Xn (6.2.4.1)
and
{x G Xn: x is linearly generating} is open in Xn (6.2.4.2)
Proof: Recall the topological isomorphism
x <—► row(Lx) : Xn ~ BL(Kn, X) = L(Kn, X) (6.2.4.3)
given by (6.1.0.6): then by (6.1.0.7) and (6.2.3.2)
x linearly independent <$=^ row(Lx) bounded below (6.2.4.4)
while by (6.1.0.8) and (6.2.3.3)
x linearly generating <$=^ row(Lx) almost open (6.2.4.5)
Now (6.2.4.1) and (6.2.4.2) follow from (3.3.3.1) and (3.4.3.1),
respectively. ■
Finite dimensional spaces are isomorphic if and only if they have equal
dimensions:
6.2.5 THEOREM If X and Y are finite dimensional spaces then
dim(X) < dim(Y) (6.2.5.1)
if and only if there are T G BL(X, Y) and S G BL(Y,X) for which
ST = I GBL(X,X) (6.2.5.2)
There is implication
X - Y <^ dim(X) = dim(Y) (6.2.5.3)

182 6. Finite Dimensional Spaces and Compactness
Proof: If x G Xn and y €Ym are bases, with n < m, define S and T by
setting
(n \ n I m \ n
E «y*y = E W and 5 E W = E *y*y (6-2-5-4)
y=i y y=i ^y=i y y=i
for each 5 G Km: then S and T are both bounded, by (6.2.3.1), and satisfy
(6.2.5.2), while
n = m=^TS = Ie BL(Y, Y) (6.2.5.5)
This proves that (6.2.5.1) => (6.2.5.2). Conversely, if (6.2.5.2) holds and
x G Xn is linearly independent then also Tx G Yn is linearly independent,
giving (6.2.5.1) by (6.1.1.1) from Theorem 6.1.1. ■
Linear operators between finite dimensional spaces are always regular:
6.2.6 THEOREM If X and Y are finite dimensional and T G BL(X, Y)
then T is regular, and
dim(X) = dim(y) <$=^ T decomposably regular (6.2.6.1)
Proof: If W C X is a subspace and w G Xp is a basis for W then by
(6.1.1.2) there is a basis x G Xn for which w^ = x^{j = 1,2,... ,p): now
the mapping
n p
E : ^2 sjxj —> E 5i™i G X for each 5 G Kn (6.2.6.2)
j=i 3=1
is a projection with range W, and of course bounded by (6.2.3.1). Thus,
for arbitrary T G BL(X, Y) both the range and the null space are
complemented, while the core is invertible (Theorem 6.2.3). This gives the first
part. For (6.2.6.1) observe
dim(X) = dimr_10 + dimX/r_10
and (6.2.6.3)
dim(Y) = dimTX+ dimY/TX
and
dimX/T^O = dim TY (6.2.6.4)
Now (6.2.6.1) follows from (3.8.6.1) and (6.2.5.3). ■

6.3 Operators of Finite Rank
183
The first part of Theorem 6.2.6 gives an improvement of (6.2.3.8) and
(6.2.3.9):
aleft BL(X, Y) = 7rleft BL(X, Y) (6.2.6.5)
and
aright BL(X, Y) = 7rright BL(X, Y) (6.2.6.6)
6.3 OPERATORS OF FINITE RANK
Perhaps, if we are trying to generalize matrix theory, we should look at
"finite dimensional" operators:
6.3.1 DEFINITION A linear operator from X to Y is said to be of finite
rank if
T(X) is finite dimensional (6.3.1.1)
If X and Y are normed spaces write
KL0(X,Y) = {Te BL(X,Y):T{X) is finite dimensional} (6.3.1.2)
for the bounded finite rank operators.
For example the "rank one" operators / © y of (2.9.2.4) are in
KL0(X,Y):
6.3.2 THEOREM If X and Y are normed spaces and T G KL0(X, Y)
then there are n G N, y G Yn and / G (X*)n for which
n
Proof: If T = 0 take n = 1, yx = 0 G Y and fx = 0 G Xf. If T ^ 0 take
t/ G Yn to be a basis for T(X), so that for arbitrary z G T(X) we can write
n
* = £*yO%y (6.3.2.2)
Evidently the coefficients g®{z) are uniquely determined by z [this is
(6.1.0.3) and (6.1.0.4)], and the resulting mappings g^ : T(X) —*■ K
necessarily linear. By (6.2.3.1) each g? is also bounded: now if we write, for each
iG{l,2,...,7i},

184
6. Finite Dimensional Spaces and Compactness
fj (x) = g°{Tx) for each x G X (6.3.2.3)
then /y G X1" for each j G {1,2,... n}, and (6.3.2.1) is satisfied. ■
The bounded finite rank operators form a "two-sided ideal" of
BL(X, Y) in the sense of (3.9.6.7):
6.3.3 THEOREM If X, Y, and Z are normed spaces then
KL0(X,Y) +KL(X,Y) C KL0(X,Y)
BL(Y,Z)KL0(X,Y) C KL0(X,Z)
and
KL0(X,Y)BL(Z,X) C KL0(Z,Y)
Proof: If T and T' are both of finite rank then
(r + r')(x) cr(i)+r'(i)
is necessarily of finite dimension, giving (6.3.3.1). If T = ^?=
KL0(X,Y)then
n
ST = Yl fj © Sy3- G KL0(X, Z) for each S G BL(Y, Z)
3=1
and
n
TS = Y,fjS® Vj e KLo(^> Y) f°r each 5 e BL(Z, X)
giving (6.3.3.2) and (6.3.3.3). ■
The finite rank operators in BL(X, Y) are regular in the sense of
Definition 3.8.1, with finite rank generalized inverses:
6.3.4 THEOREM If T G KL0(X, Y) then T is regular, with a regular
generalized inverse, and there is inclusion
rBL1(X, Y) + KL0(X, Y) C rBL1(X, Y) (6.3.4.1)
Proo/: If T = 0 then we can take T' = 0 to satisfy (3.8.1.1). If T ^ 0 take
y G Yn to be a basis for T(X) and form again the coefficient functional
g*j : T(X) —► K as in (6.3.2.2), which can now be regarded as bounded
linear functionals on T(X). By the Hahn-Banach Theorem 5.3.1 there is
g G (Y*)n for which each g^ is an extension of #°, and then / G (X*)n
(6.3.3.1)
(6.3.3.2)
(6.3.3.3)
(6.3.3.4)
i fj © Vj €
(6.3.3.5)
(6.3.3.6)

6.3 Operators of Finite Rank
185
given by /y = g?T = g3-T(j = l,2,...,n) for which (6.3.2.1) holds. We
claim that
/ G (Xf)n is linearly independent (6.3.4.2)
for otherwise the range of T would be of smaller dimension than the space
generated by y G Yn. Now by Theorem 6.1.2 there is x G Xn for which
(6.1.2.1) holds, and we take
n
T' = Y, 9j © xj € BL(Y, X) (6.3.4.3)
y=i
to satisfy (3.8.1.1). This proves the first part of Theorem 6.3.4, so that
KL0(X, Y) C rBL1(X, Y) (6.3.4.4)
now (6.3.4.1) follows from (3.9.6.9) and Theorem 6.3.3. ■
In particular, finite dimensional subspaces of normed spaces are
complemented:
6.3.5 THEOREM If Y and Z are closed subspaces of a normed space X
then
Y finite dimensional, Z complemented => Y + Z complemented
(6.3.5.1)
and
Y finite codimensional, Z complemented => Y n Z complemented
(6.3.5.2)
Proof: Towards (6.3.5.1) suppose first that Z = {0}: then if T = J :
Y —► X is the natural injection we have T G KL0(Y,X), regular by
Theorem 6.3.4, so that, in particular Y = T(X) is complemented. Now
for (6.3.5.1) suppose P = P2 G BL(X,X) with Z = P(X), and put
Y' = {I-P)Y. Evidently
Y'CY' + Z = Y + Z (6.3.5.3)
By what we have just proved, the finite dimensional space Y' is
complemented in Y' + Z. Now, if Q = Q2 :Y + Z->Y + Zisa continuous
projection with range Y' then
P + Q{I - P) : X —+ X (6.3.5.4)
is a continuous projection on X whose range is Y + Z.

186
6. Finite Dimensional Spaces and Compactness
Towards (6.3.5.2), suppose first that Z = X: then if T = K : X ->
X/Y is the quotient we have T G KL0(X, X/Y), regular by Theorem 6.3.4,
so that in particular Y = T_1(0) is complemented. Now for (6.3.5.2),
suppose P = P2 G BL(X,X) with Z = P{X), and observe that Y n Z has
finite codimension in Z. By what we have just proved there is a continuous
projection Q = Q2 : Z —► Z with range Y n Z: but now
QP :X —>X (6.3.5.5)
is a continuous projection on X whose range isYdZ. m
If T = ]£y=i /y © t/y G BL(X, Y) is of finite rank, with generalized
inverse T' = £*=1 </y © Xj G BL(Y, X) as in Theorem 6.3.4, then
n
Q = ^2gjeyjeBL(Y,Y) (6.3.5.6)
and
n
P = ^ /y © xy G BL(X,X) (6.3.5.7)
y=i
are continuous projections with the same range and null space as T,
respectively. We conclude by showing that a finite-rank operator is bounded if
and only if its null space is closed:
6.3.6 THEOREM If X and Y are normed spaces and T G L(X, Y) is of
finite rank, then
T G KL0(X, Y) <£=> T"1 (0) closed (6.3.6.1)
Proof: Forward implication is clear. If T ^ 0 is of finite rank, and t/G7n
is a basis for T(X) C Y, then we recover the linear functional g® on T{X)
of (6.3.2.2), and the linear functional /y on X of (6.3.2.3): now we claim
T-1 (0) closed => Fy bounded for each j" G {1,2,..., n} (6.3.6.2)
Indeed, for each j G {1,2,..., n} write
Wj = r_1(0) + ^ Kxt- (6.3.6.3)
where x G Xn is chosen so that
Txy = t/y for each j G {1,2,..., n} (6.3.6.4)

6.4 Fredholm Operators
187
Then by Theorem 6.2.1 each space Wj is closed if T 1(0) is closed. We
claim
W,. = /^(O) for each j G {1,2,... ,n} (6.3.6.5)
Certainly, T_1(0) C ff1^), and also x{ G fj~1{0) if i i1 j, giving inclusion.
Conversely, if x G X is in /~* (0) then
This proves (6.3.6.5). It now follows from Theorem 4.8.3 that each /• is
bounded, and hence also (6.3.2.1) T = Y%=i fj ® 2/y- ■
6.4 FREDHOLM OPERATORS
The "Fredholm operators" are in a sense at the opposite extreme from the
operators of finite rank:
6.4.1 DEFINITION If X and Y are normed spaces then T G BL{X,Y)
is essentially one-one iff
T_1(0) is finite dimensional (6.4.1.1)
is essentially dense iff
Y/c\(TX) is finite dimensional (6.4.1.2)
and is Fredholm, or spatially Fredholm, iff
T is essentially one-one, essentially dense, and proper (6.4.1.3)
We shall write
tt^ BL(X, Y)={T€ BL(X, Y): T is not essentially one-one} (6.4.1.4)
and
7r^ht BL(X, Y) = {Te BL(X, Y):T is not essentially dense} (6.4.1.5)
and
<7essBL(X,Y) ={Te BL(X,Y):T is not Fredholm} (6.4.1.6)
The reader will recall that an alternative description for the regular
operators of Definition 3.8.1 was "relatively Fredholm":

188
6. Finite Dimensional Spaces and Compactness
6.4.2 THEOREM Necessary and sufficient for T G BL(X, Y) to be Fred-
holm is that it be regular, not a left zero divisor modulo KL0(X, X) and
not a right zero divisor modulo KL0 (Y, Y).
Proof: We begin with an "essential" analogue of Theorem 3.2.2. We claim
T essentially one-one <£=*> L^}{0) C KL0(W,X)
for all W and LT = BL{W,T) (6.4.2.1)
and
T essentially dense <£=> R^(0) C KL0(Y,W)
for all W and RT = BL(T,W) (6.4.2.2)
Certainly if r_1(0) is finite dimensional and U G BL(W,X) then
TU = 0 => U{W) C r_1(0) finite dimensional => 17 G KL0(W,X)
(6.4.2.3)
while if instead Y/cl(TX) is finite dimensional and V G BL(Y, W) then
VT = 0 => ^-1(0) D cl(TX)
w " v ; (6.4.2.4)
=> dim^(y) = dimW/V^O < dimW/cl(TX) < oo
Conversely, if T_1(0) is not finite dimensional take W = T_1(0) and U =
J : T_1(0) —► X the natural injection, to find U G L^x(0) which is not of
finite rank, while if cl(TX) is not of finite codimension take W = Y j cl(TX)
and V = K : Y -> Y/c\(TX) the quotient map, to find V G R^{0) which
is not of finite rank. This proves (6.4.2.1) and (6.4.2.2). Now, if T is
Fredholm, then (6.4.2.1) with W = X says that T is not a left zero divisor
modKL0(X,X), and (6.4.2.2) with W = Y says that T is not a right
zero divisor modKL0(Y,Y). At the same time, by Theorem 6.3.5, both
T_1(0) and cl(TX) are complemented, so that if T is also proper then the
condition (3.8.2.1) holds, making T regular. Conversely, if T is regular,
with T = TT'T, then r_1(0) and cl(TX) are both complemented, with
projections P = T'T and Q = TT' satisfying (3.8.2.3). If at the same time
T is not a left zero divisor then J — P G KL0(X,X), giving (6.4.1.1), and if
T is not a right zero divisor then I — Q € KL0(Y, Y), giving (6.4.1.2); also
if T is regular then it must be proper. ■
Atkinson's theorem says that Fredholm operators are "essentially in-
vertible":

6.4 Fredholm Operators
189
6.4.3 THEOREM If X and Y are normed spaces then T G BL(X, Y) is
Fredholm iff
T is left invertible modulo KL0(X,X) (6.4.3.1)
and
T is right invertible modulo KL0(Y, Y) (6.4.3.2)
Proof: If T is Fredholm then by Theorem 6.4.2 it is regular, with a
generalized inverse T' G BL(Y,X) satisfying (3.8.1.1), and since T is essentially
one-one we have
I-T'T G Z^1 (°) ^ KLo(*>*) (6.4.3.3)
giving (6.4.3.1), and since T is essentially dense we have
I + TT' G ^(O) ^ KL0(Y,Y) (6.4.3.4)
giving (6.4.3.2). Conversely, if either (6.4.3.1) or (6.4.3.2) hold then
T - TT'T G KL0(X, Y) C rBLI(X, Y) (6.4.3.5)
using Theorem 6.3.4, so that T is regular by (3.8.3.1). At the same time
(6.4.3.1) and the implication (3.9.4.8) say that T is not a left zero divisor:
TU = 0 => U = (I - T'T)U G KL0(X,X) (6.4.3.6)
while (6.4.3.2) says that T is not a right zero divisor:
VT = 0=^V =V{I- TT') G KL0(Y,Y) ■ (6.4.3.7)
The product of Fredholm operators is Fredholm:
6.4.4 THEOREM If X, Y and Z are normed spaces and T G BL(X, Y),
S eBL{Y,Z) then
5, T Fredholm => ST Fredholm =^ S,T regular (6.4.4.1)
and
ST Fredholm => (S Fredholm <£=> T Fredholm) (6.4.4.2)
If T is regular with generalized inverse T' G BL(Y, X), then
T Fredholm => T' Fredholm (6.4.4.3)
If also XGBL(X,Y), then
T Fredholm, J + T'K Fredholm => T + K Fredholm (6.4.4.4)

190
6. Finite Dimensional Spaces and Compactness
and
T Fredholm, J + KT' Fredholm => T + K Fredholm (6.4.4.5)
In particular,
T Fredholm, K finite rank =» T + K Fredholm (6.4.4.6)
Proof: The first part of (6.4.4.1) follows from Theorem 3.9.3 together with
Atkinson's theorem, since the ideals
A = KL0{X,X) B = KL0{Y,Y) and D = KL0{Z,Z) (6.4.4.7)
satisfy the condition (3.9.3.3), enabling us to apply (3.9.3.4) and (3.9.3.5).
For the second part of (6.4.4.1) recall the argument of (6.4.3.5): if U is a
generalized inverse for the Fredholm operator ST, then
T - T{US)T e KL0(X, Y) C rBL1(X, Y)
and (6.4.4.8)
S - S(TU)S e KL0{Y,Z) C rBL1(Y,Z)
Now apply (3.8.3.1). For (6.4.4.2) we use Atkinson's theorem again . If U
is a generalized inverse for ST, then
J - UST e KL0(X,X) and J - STU e KL0{Z,Z) (6.4.4.9)
so that
T Fredholm <=> I - TUS e KL0(Y, Y) <£=> S Fredholm (6.4.4.10)
For (6.4.4.3) we recall from (6.4.3.3) and (6.4.3.4) that J - T'T and
j _ rprp/ are Q£ flnjte rank? and use Atkinson's theorem to deduce that T'
is Fredholm. Towards (6.4.4.4) and (6.4.4.5), we begin with (6.4.4.6). If K
is of finite rank then
I-T'{T + K) = {I- T'T) - T'K e KL0(X, X) (6.4.4.11)
and
I - [T + K)T' = {I- TT') - KT1 e KL0(Y,Y) (6.4.4.12)
so that T + K is Fredholm by Atkinson's theorem. Now for (6.4.4.4) we
argue
T'{T + K) = (I + T'K) -(I- T'T) e {I + T'K) + KL0(X, X) (6.4.4.13)

6.4 Fredholm Operators
191
using (6.4.3.3). Thus, by (6.4.4.6) the product T'{T + K) is Fredholm, and
then (6.4.4.2) and (6.4.4.3) together make T + K Fredholm. The argument
for (6.4.4.5) is the same. ■
If we specialize the perturbation slightly, we deduce slightly more:
6.4.5 THEOREM If T G BL(X, Y) is Fredholm, with generalized inverse
T' G BL(y,X), and if K G BL(X, Y) satisfies
J + T'K G BL_1(X,X) invertible (6.4.5.1)
then
&m{T + K)-l{ti) < dimT-^O) (6.4.5.2)
and
dim Y/cl(T + K)X < dimY/cl(TX) (6.4.5.3)
Proof: Towards (6.4.5.1), suppose
J + T'K one-one (6.4.5.4)
Then by (6.4.4.13) we have
(T + X)_1(0) n (/ - T'T)-1^) C (/ + T'K)-1^) = {0} (6.4.5.5)
so that (T + if)-:l(0) is contained in some complementary subspace for the
finite dimensional space (J — T'T)-1^), giving
dim(T + K)-\0) < dimX/(I - TT)"1^)
(6.4.5.6)
= dim^T)-1^) = dimr-^O)
This proves (6.4.5.2). Towards (6.4.5.3) recall from (3.1.3.1) that if (6.4.5.1)
holds then also i" + KT' G BL_1 {Y,Y) is invertible. If we merely assume
J + KT' onto (6.4.5.7)
then by the analogue of (6.4.4.13) we have
(T + K)X + {I - TT')X D {I + KT')X = X (6.4.5.8)
giving
dimY/{T+K)X < dim(/-rr')X = dimY/TT'X = dimY/TX (6.4.5.9)
Since of course both TX and (T + K)X are closed, we have proved
(6.4.5.3). ■

192
6. Finite Dimensional Spaces and Compactness
6.5 WEYL OPERATORS AND THE INDEX
The difference between the two finite dimensions which characterize Fred-
holm operators turns out to play a significant role:
6.5.1 DEFINITION If T G BL(X,Y) is a Fredholm operator between
normed spaces, then
index(T) = dimr_1(0) - dimY/cl(TX) (6.5.1.1)
We shall call T G BL(X, Y) a Weyl operator iff
T is Fredholm and index(T) = 0 (6.5.1.2)
We shall write
ue3SBL(X,Y) ={Te BL(X,Y) : T is not Weyl} (6.5.1.3)
We offer two characterizations of Weyl operators:
6.5.2 THEOREM If T G BL(X, Y) then the following are equivalent:
T is a Weyl operator (6.5.2.1)
T is Fredholm and decomposably regular (6.5.2.2)
T G BL_1(X,Y) +KL0(X,Y) (6.5.2.3)
Proof: If T G BL(X,Y) is Fredholm then by (6.2.5.3) the condition
(6.5.1.2) is equivalent to isomorphism
T"1 (0) ~ Y/ c\(TX) (6.5.2.4)
which for regular T is the decomposable regularity condition (3.8.6.1). Thus
the conditions (6.5.2.1) and (6.5.2.2) are equivalent. If (6.5.2.2) holds then
T has an invertible generalized inverse T' G BL(Y, X), which also satisfies
(6.4.3.3). Thus,
T = (T')"1 - (T')"1^ - T'T) G BL_1(X, Y) + KL0(X, Y) (6.5.2.5)
giving (6.5.2.3). Conversely, if (6.5.2.3) holds then we can write
T = S(I + K) with S G BL_1(X, Y) and K G KL0(X,X) (6.5.2.6)
and to deduce (6.5.2.1) we will show
I + Ke BL(X,X) is Weyl (6.5.2.7)

6.5 Weyl Operators and the Index
193
To see this we must decompose the space X into four subspaces, taking
X0 = K'1 (0) n c\{KX) = K'1 (0) n KX (6.5.2.8)
and then choosing closed subspaces Xx, X2 and X3 in such a way that
X0 + Xx = K{X) and X0nX1 = {0} (6.5.2.9)
and
X0 + X2 = K~10 and X0 n X2 = {0} (6.5.2.10)
while
(KX+K~10) + X3 = X and (XI + X^O) nX3 = {0} (6.5.2.11)
The existence of X1? X2 and X3, chosen in such a way that the induced
projections are bounded, follows from (6.3.5.1); in fact Theorem 6.3.6 shows
that the induced projections must always be bounded. Since each of X0,
Xx and X3 are finite dimensional it is clear that also
X2 = X0 + Xx + X3 is finite dimensional
If x E X is arbitrary we claim
{I + K)xeX'2^>xeX'2
This is clear from the matrix representation
(6.5.2.12)
(6.5.2:i3)
I + K-
^01
0 K(
03
0 I + Kn 0 K13
0 0/0
0 0 0/
rx°i
x2
Lx3J
—
rx°i
*2
Lx3J
(6.5.2.14)
In particular there is inclusion (/ + K)X2 C X2, and hence / + K induces
a linear operator—necessarily bounded—
U:X2
X'
(6.5.2.15)
To verify (6.5.2.7) we show
dim^+X)"^ = dimtf^O = dimX'2/UX'2 = dimX/{I+K)X (6.5.2.16)
The first equality follows from (6.5.2.13), which implies
(/ + K)-^ = U^O C X2 (6.5.2.17)

194
6. Finite Dimensional Spaces and Compactness
The second equality in (6.5.2.16) is (6.2.6.1), and finally, for the third, we
have
x = (i + k)x e w = {i + K)x2 e (i + k)x2 e w
with X2 = UK'2 0 W ■ (6.5.2.18)
The product of Weyl operators is Weyl:
6.5.3 THEOREM If X, Y and Z are normed spaces and T G BL(X, Y),
SGBL(Y,Z),then
5,T Weyl => ST Weyl => (5 Weyl <£=> T Weyl ) (6.5.3.1)
If T is regular with generalized inverse T' G BL(Y,X), then
T Weyl => T' Weyl (6.5.3.2)
If also XGBL(X,y), then
T Weyl, / + T'X Weyl => T + X Weyl (6.5.3.3)
T Weyl, / + XT' Weyl => T + K Weyl (6.5.3.4)
and in particular
T Weyl, X finite rank => T + X Weyl (6.5.3.5)
Proof: Most of this follows from the characterization (6.5.2.3). If T =
U + K and 5 = V + if with invertible U and V and finite rank K and H
then
ST = (^17) + [VK + HU + HK) G BL_1 (X, Z) + KL0(X, Z) (6.5.3.6)
giving the first implication of (6.5.3.1), while if instead T = U + K and
ST = V + H then
5 = (VU-^ + iH-SK^-1 GBL_1(y,Z) + KL0(y,Z) (6.5.3.7)
which proves half of the second implication. The other half is similar. For
(6.5.3.2) suppose T = U + K with invertible U and finite rank K; then
T + TT'T => T' - 17"1 G KL0(Y, X) (6.5.3.8)
The implication (6.5.3.5) is immediate from (6.5.2.3), and now the argument
for (6.5.3.3) and (6.5.3.4) is similar to that for (6.4.4.4) and (6.4.4.5). ■

6.5 Weyl Operators and the Index 195
The most surprising thing about the index is that it is a homomor-
phism:
6.5.4 THEOREM If X, Y, and Z are normed spaces and T G BL(X, Y),
S e BL{Y,Z) are Fredholm, then
index(ST) = index(S) + index(T) (6.5.4.1)
Proof: To see this we decompose the space Y into four subspaces, taking
Y0 = (TX) n 5_10 = cl(TX) n 5_10 (6.5.4.2)
and then choosing closed subspaces Y1? Y2 an(^ ^3 so that
Y0 + Y1= TX and Y0nY1= {0} (6.5.4.3)
and
Y0 + Y2 = S~10 and Y0 n Y2 = {0} (6.5.4.4)
while
(TX + 5-10)+Y3 = Y and (TX + 5_10) n Y3 = {0} (6.5.4.5)
The existence of Y1? Y2, and Y3, chosen in such a way that the induced
projections are continuous, follows from Theorem 6.3.5. We now claim
dim(5r)_10 = dimr_10 + dim Y0 (6.5.4.6)
since the operator TA = core(T) : X/T~10 —► TX induces isomorphism
{ST^O/T^O -Tin S-x0 = Y0. Also
dim Z/STX = dim Z/SY + dim Y3 (6.5.4.7)
since the operator SA = core(S) : Y/S~10 —► SY induces isomorphism
Y/{TX-\- 5_10) ^ SY/STX. At the same time it is clear from (6.5.4.3)
and (6.5.4.4) that
dimY0 + dimY2 = dim5_10 (6.5.4.8)
and
dim Y2 + dim Y3 = dim Y/TX (6.5.4.9)
If we now combine (6.5.4.J) for j = 6,7,8,9 we get (6.5.4.1). ■

196
6. Finite Dimensional Spaces and Compactness
The proof of Theorem 6.5.4 incorporates an alternative proof for the
first part of (6.4.4.1). If S and T are both Fredholm, then (6.5.4.6) and
(6.5.4.7) tell us that (5r)_1(0) is finite dimensional and that c\(ST)X D
ST(X) is of finite codimension. It remains to see that ST is also proper.
In fact, we can show that it is regular. If we write
'T0'
2\
0
_ 0
:X—►
'Y0~
Y1
Y2
Y3_
and
[0 S, 0 S3
Y1
then we can find S[ : Y1 —► Z and T[ : Yx —> X for which
S'1S1=Tiri = I:Y1-+Y1
giving
ST + S^ = S^T^iS^S^ = ST{T[S[)ST
(6.5.4.10)
(6.5.4.11)
(6.5.4.12)
With the help of the index theorem we can add more detail to the
perturbation theory of Fredholm operators, and rederive that for Weyl
operators:
6.5.5 THEOREM If X and Y are normed spaces and T G BL(X, Y) is
Fredholm, with generalized inverse T' G BL(Y,X), then
index(T') = -index(T) (6.5.5.1)
If also XGBL(X,Y), then
I+T'K Fredholm => index(r+iT) = index(T)+index[I+T'K) (6.5.5.2)
and
I+KT' Fredholm => index(T+iT) = index(r)+index(/+iTr/) (6.5.5.3)
In particular
K finite rank => index(T + K) = index(T) (6.5.5.4)
Proof: For (6.5.5.1) we use (6.5.4.1) and (3.8.1.1) to get
index(T) = index(T) + index(T') + index(T) (6.5.5.5)
For (6.5.5.4) we argue that if K is of finite rank then
T\T + K) = (I + T'K) -{I- T'T) is Weyl (6.5.5.6)

6.6 Compactness and Total Boundedness
197
so that T'(T + K) has index zero, giving
index(T + K) = -index(T') = index(T) (6.5.5.7)
From (6.5.5.4) we get (6.5.5.2) by arguing
index(T + K) = index(J + T'K) - index(T') (6.5.5.8)
The argument for (6.5.5.3) is the same. ■
We conclude with an extension of the characterization (6.5.2.3) to more
general Fredholm operators:
6.5.6 THEOREM If T G BL(X,Y) is Fredholm then there are S G
BL(X, Y) and S' G BL(Y,X) for which
T-Se KL0(X,Y) is finite rank (6.5.6.1)
with
S'S = 1 if index(T) < 0 (6.5.6.2)
and
SS' = 1 if index(T) > 0 (6.5.6.3)
Proof: Since the Fredholm operator T is regular we can find projections
P = T'T and Q = TT' satisfying (3.8.2.3), where T' G BL(Y,X)
satisfies (3.8.1.1), and by (6.2.5.2) we can find Sv : P_10 -► Q_10 and
S'v . q-i0 _► p-i0 for which either S'vSv = J'v or SvS'v = Jv,
according as which finite dimensional subspace has the greater dimension.
For (6.5.6.2) or (6.5.6.3) we now take
S = TP + SV{I-P) and S' = T'Q + S'w {I - Q) ■ (6.5.6.4)
When T is Weyl then both (6.5.6.2) and (6.5.6.3) hold, so that S
is invertible as in (6.5.2.3), and indeed coincides with the operator S of
(3.8.6.6).
6.6 COMPACTNESS AND TOTAL BOUNDEDNESS
We recall that a subset K of a topological space Q is compact iff every open
cover has a finite sub cover: equivalent ly
Ut G Nbd(*) for each t G K => K C |J Ut for some finite X'cn
teK1
(6.6.0.1)

198
6. Finite Dimensional Spaces and Compactness
If Q = X is a normed space and (6.6.0.1) holds for discs of equal radius
then K C X is said to be totally bounded:
if e > 0 then K C Disc(if'; e) = (J Disc(x ; e) for some finite K' C X
(6.6.0.2)
Compactness and total boundedness can be tested with sequences:
6.6.1 DEFINITION if C X is said to be sequentially compact iff
every sequence (xn) in K has a subsequence (x^) which converges in K
(6.6.1.1)
and is said to be sequentially precompact iff
every sequence (xn) in K has a Cauchy subsequence (x^) (6.6.1.2)
These three notions are related, together with completeness:
6.6.2 THEOREM If X is a normed space and K C X then
if totally bounded <£=£> if sequentially precompact
and
K complete and totally bounded <£=£> K sequentially compact (6.6.2.2)
Proof: If K C X is not totally bounded then there must be £ > 0 for
which there does not exist finite K' C.K such that K C Disc(if ;£). Then
inductively there is a sequence (xn) in K for which
n
xn+1 G if \ (J Disc(xy ; <5) for each n G N (6.6.2.3)
and evidently (xn) has no Cauchy subsequences. Conversely, if K is totally
bounded and (xn) is an arbitrary sequence in K, and e1 > 0 is arbitrary,
then there must be a subsequence (xjj of (xn) for which {x\:n € N} C
Disc(t/1 ; 6-l) for some yx G X. Repeating the argument with (xJJ in place
of (xn), inductively we get subsequences (xJJ1) of x for each m G N such
that
{x^: n G N} C Disc(ym ; em) and (^+1) is a subsequence of (x^1)
(6.6.2.4)
(6.6.2.1)

6.6 Compactness and Total Boundedness 199
If we do this with em —► 0 then the sequence (xJJ defined by setting
x'n = x* for each n G N (6.6.2.5)
is a Cauchy subsequence of (xn). This proves (6.6.2.1), and makes the
(standard) argument for (6.6.2.2) very easy: trivially, K sequentially
precompact and complete => compact => precompact (6.6.2.6)
while if (xn) is a Cauchy sequence in a sequentially compact subset K, and
therefore has a convergent subsequence, then by (4.1.2.3) the sequence (xn)
must itself converge. ■
For the record,
6.6.3 THEOREM If X and Y are normed spaces, with K C X and
HCY then
K,H totally bounded => K x H totally bounded (6.6.3.1)
and
K, H sequentially compact => K x H sequentially compact (6.6.3.2)
Also if <t>: K —> Y is continuous, then
K totally bounded => <£(if) totally bounded (6.6.3.3)
and
K sequentially compact => <j>{K) sequentially compact (6.6.3.4)
If Y = X and H C K then
K totally bounded => if totally bounded (6.6.3.5)
and
K sequentially compact => K closed and cl(if) sequentially compact
(6.6.3.6)
Proof: This is left to the reader. ■
For subsets of normed spaces, compactness and sequential compactness
are the same thing.

200
6. Finite Dimensional Spaces and Compactness
6.6.4 THEOREM If K C X for a normed space X, then
K sequentially compact => K compact (6.6.4.1)
Proof: We claim
K compact => K Bolzano-Weierstrass => K seq. compact (6.6.4.2)
and
K seq. compact => K Lebesgue compact => K compact (6.6.4.3)
Here the "Bolzano-Weierstrass property" says that any infinite subset H C
K has accumulation points in X:
acc(JJ) = 0 => H finite (6.6.4.4)
"Lebesgue compactness" mean total boundedness together with the
"Lebesgue covering property": if Ux G Nbd(x) for each x G K there must be
6 > 0 and a mapping <f>: K —> K for which
* Disc(x; g) C 170(x) for each xeK (6.6.4.5)
To verify the first implication of (6.6.5.2), note that if H C K has no
accumulation points then there is {Ux)xeK for which H d Ux C {x} at
each point. Clearly any subcover of K will have to use all the [Ux)xGH.
For the second implication, observe that if (xn) is a sequence in K with
the Bolzano-Weierstrass property then either ace {xn: nGN}^0or there
is y G X for which {n G N: xn = y} is infinite. The second implication of
(6.6.5.3) is very easy, and can be left to the reader. For the first we will have
to argue by contradiction. Indeed, if K does not have the Lebesgue covering
property then there is for each neighborhood family {Ux)xGK a sequence
(xn) in K for which none of the Disc(xn; l/n) lie in any of the U . If in spite
of this K is sequentially compact then there will be z = x^ = limn x^ for
some subsequence (^)(x(n)) of (xn). But now Uz must eventually contain
A topological space Q is called locally compact if
for each t G H and each U G Nbd(i)
there is compact V G Nbd(*) with V CU (6.6.4.6)
If Q = X is a normed space then by translation invariance
X locally compact <£=£> DiscjJC(0 ; 1) sequentially compact (6.6.4.7)

6.6 Compactness and Total Boundedness
201
6.6.5 THEOREM If X is a normed space, then
X locally compact <£=£> X finite dimensional (6.6.5.1)
Proof: If X is finite dimensional then
X ~ Kn for some n G N (6.6.5.2)
and if we give Kn the H-jj^ norm then (6.6.3.2) shows that Disc^O ; 1) C
Kn is sequentially compact. Conversely, we use the Riesz lemma
(Theorem 1.5.2), and part of Theorem 6.2.1 , to show that if X is infinite
dimensional then there is a sequence (xn) in X for which
/ » \ 1
for each n e N, ||xn|| = 1 and dist i -=1, Y^ Kiy > - (6.6.5.3)
V i=i J 2
Using (6.6.2.1), it is clear that Discx(0 ; 1) is not totally bounded. ■
The reader will observe that again we have proved more than we have
stated. He is invited to reflect on which, if any, normed spaces are actually
compact.
We conclude here with the famous Tychonoff product theorem, which
says that arbitrary products of compact spaces are compact.
6.6.6 THEOREM If (Hy)yGj is a family of compact topological spaces,
then
JJ Qy is compact (6.6.6.1)
J'€J
Proof: Suppose J is a set of subsets of Yljej ^y w^h the finite intersection
property, so that Q7' ^ 0 for each finite 7' C J: then with the help of
Zorn's lemma 1.11.2 we can find a family Q of subsets of ILgj fy> maximal
with respect to having the finite intersection property, for which Q D 7. If
we suppose that each member set of J is closed in the topology of Yljej Hy,
we should not assume that this is true of the sets in Q. For each j G J the
family
{GfG € 9} = {*3(G):G € 9} (6.6.6.2)
of subsets of Qy has the finite intersection property, as does the related
family
{cl(Gy):G e 9} = {clK-(G)): G € 9} (6.6.6.3)
By the compactness of H • it follows that the intersection of each family
(6.6.6.3) is nonempty. Thus (with another concealed application of Zorn's

202
6. Finite Dimensional Spaces and Compactness
lemma) we can find a point x G Ilyej fy f°r which
xj G D {*y (G): G G 5} for each j'eJ (6.6.6.4)
The compactness of the product will be established if we can show that this
point x lies in the intersection of the sets in J'. Certainly for each j G J
and each neighborhood U- of x ■ and each G G Q we have
^yn7ri(G)^0 (6.6.6.5)
It follows that for each j G J
for each Ge Q, 7rr1(Uj)nG^0 (6.6.6.6)
and hence, by the maximality of §,
^Jl{Uj) G Q for each j eJ (6.6.6.7)
This is true if each Uj is the projection of a basic neighborhood U of p in
nyeJny. Thus,
UnG^0 for each G G 7 (6.6.6.8)
Since each G G J is closed, we have proved x G nGeyCr ^ &- ■
6.7 ESSENTIAL ENLARGEMENT
If H is a nonempty set and X is a normed space, then the "compactness"
or otherwise of a mapping x : Q —► X is classified by its range x(Q) =
{xt:*GH}CX:
6.7.1 DEFINITION If Q is a nonempty set and X is a normed space,
then the totally bounded mappings from Q to X form the set
m(Q,X) = {x G xn:{xt:£ G H} is totally bounded in X} (6.7.1.1)
and the relatively compact mappings from Q to X form the set
m^HjX) = {x G Xn:cl{xt:£ G H} is compact in X) (6.7.1.2)
If x G /^(njX) is bounded then the measure of noncompactness of x is
given by
p(x) = inf {A; > 0: {xt : t G H} C Disc(X'; Jfe) for some finite K' C X}
(6.7.1.3)

6.7 Essential Enlargement
203
The totally bounded and the relatively compact mappings are linear
subspaces of ^(H, X), and the measure of noncompactness is a seminoma:
6.7.2 THEOREM If Q is a nonempty set and X is a normed space, then
m(Q,X) and m1(Q,X) are linear subspaces of 1^(0,X) for which
m^n.X) Cm(Q,X) =clm1(Q,X) Cl^faX) (6.7.2.1)
with
dist(x,m1(Q,X)) = dist(x,m(Q,X)) = p(x) for each x G l^V^X)
(6.7.2.2)
Proof: The inclusions (6.7.2.1) are clear, and to see that linear
combinations of relatively compact mappings are relatively compact, use (6.6.3.2)
and (6.6.3.4), while for totally bounded mappings use (6.6.3.1) and (6.6.3.3).
To see that m(Q,X) is closed in l^fi^X) suppose that ||xn — x^ —► 0,
with xn G m(Q,X) and x G Z^,X): then for each S > 0 and e > 0 there is
N G N and finite K' C K for which
n > N => ||xn - xH^ < 8 and x^(Q) C Disc(X'; e) (6.7.2.3)
But now
x(Q) C Disc(if' ; 6 + s) (6.7.2.4)
and since 6 and e were both arbitrary it follows x G m(Q,X). To see that
the closure of m1(Q,X) is m(Q,X) it will be sufficient to prove (6.7.2.2).
Toward this suppose 6 > dist(x,m(Q,X)) is arbitrary. There then is y G
m(Q,X) for which ||x H^ < 6, and for each e > 0 there is finite K' C X
for which t/(Q) C Disc(if; e), giving x(Q) C Disc{K',6 + s). By choice of
6 and e it follows
p(x) < dist(x,m(Q,X)) (6.7.2.5)
Conversely, if £ > p(x) is arbitrary then there is finite K' C X for which
x(Q) C Disc (if'; 6), and hence for each ?iGN there is yn G K' for which
||xn — t/n|| < £: now we have y G m1(Q,X) and ||x — t/H^ < S. By choice
of £ it follows
dist(x,m1 (Q,X)) < p(x) ■ (6.7.2.6)
The reader may like to see that the argument of Theorem 5.8.2
identifies the dual space of m(Q,X): under the correspondence (5.8.1.12)
m(n,X)f ^ M^Q,^) (6.7.2.7)
When n = N we write m(X) for m(Q,X), and m^X) for m1(Q,X).

204
6. Finite Dimensional Spaces and Compactness
6.7.3 THEOREM If X is a normed space, then
m(X) = {iG /^(X): every subsequence of x has a Cauchy subsequence}
(6.7.3.1)
and
mAX) ={iG I (X): every subsequence of x
(6.7.3.2)
has a convergent subsequence}
Hence also
c{X) C m(X) and cx(X) C mx(X) (6.7.3.3)
If T G BL(X, Y) is a bounded linear operator between normed spaces, then
TNm(X) C m{Y) and rV^X) C m^Y) (6.7.3.4)
Proo/: Equality (6.7.3.1) is given by (6.6.2.1) for K = {xn:n G N}, and
equality (6.7.3.2) by (6.6.2.2) for K = cl{xn:n G N}. The first inclusion of
(6.7.3.4) follows from (6.6.3.3), and the second from (6.6.3.4). ■
More generally, Theorem 6.7.3 is valid for a nonempty set Q with the
homology of finite subsets. When Q = N we can use m{X) in place of
c0(X) in the definition of an enlargement.
6.7.4 DEFINITION If X is a normed space, then the essential
enlargement of X is
P(X)=UX)/m(X) (6-7.4.1)
and the semi-essential enlargement of X is
P1(X)=l00(X)/m1(X) (6.7.4.2)
If T G BL(X, Y) is a bounded operator, then the essential and semi-essential
enlargements of T are the operators
P(T) : P(X) —+ P(Y) and PX(T) : PX(X) —+ P^Y) (6.7.4.3)
induced by T: for each iG/^ (X)
P(T)(x + m(X)) =TNx + m(Y) (6.7.4.4)

6.7 Essential Enlargement
205
Unless it coincides with ~P(X), the space PX(X) carries a seminorm
which is not a norm. It would have been nice to express the norm ||P(T)||
in terms of the quotient norm
||r||e3S = dist(r,KL(x,y)) (6.7.4.5)
6.7.5 THEOREM IfreBL(X,Y),then
l|P(T)ll<P1i;»<2||P(T)|| (6.7.5.1)
where
||r||,ess = p{rx:||x||<l} (6.7.5.2)
Proof: It is clear that
||P(r)|| = sup p(TNx) < sup j>(TK) (6.7.5.3)
p(x)<l pffl<l
and also, by taking K = {x G X : \\x\\ < 1}, that
\\T\\'ess< sup j>(TK) (6.7.5.4)
P(K)<1
We claim that
\T\\'e33= sup p(«0<2||P(T)|| (6.7.5.5)
pW<i
Half of the first equality in (6.7.5.5) is given by (6.7.5.4): conversely if
j>(K) < 1 and e > 0 and 6 > 0 then there are finite K'e C X and HS CY
for which
K C Disc(i^; 1 + e) and {Tx: \\x\\ < 1} C Disc(if<5; 6) (6.7.5.6)
It follows that for each x G K'€ there is inclusion T Disc(x; 1+e) C Disc(if5 +
Tx;^(l + e)) and hence
T(K) CTDisc{K'e;l + s)cmsc{H6 + TK'c;6{l + e)) (6.7.5.7)
The finiteness of the set HS + TK'e and the choice of 8 and e finishes
the proof of equality in (6.7.5.5). For the second inequality suppose that
0 < S < p(K) < 00 and find x G l^X) for which
{xn: n G N} C K and p(x) > \d (6.7.5.8)

206
6. Finite Dimensional Spaces and Compactness
Simply choose xx G K arbitrarily and then arrange
n
xn+1 e K \ (J Disc(xy; 6) for each n G N ■ (6.7.5.9)
3 = 1
6.8 COMPACT OPERATORS
A more comprehensive definition of "finite dimensional" operators exploits
compactness or total boundedness.
6.8.1 DEFINITION A linear operator T e L{X, Y) is said to be compact
iff
cl {Tx: \\x\\ < 1} is compact in Y (6.8.1.1)
and is said to be totally bounded iff
{Tx: \\x\\ < 1} is totally bounded (6.8.1.2)
We write
KL^Y) = {Te L{X,Y):T is compact} (6.8.1.3)
and
KL(X,y) = {Te L(X,Y):T is totally bounded} (6.8.1.4)
The compact and the totally bounded operators form "two-sided
ideals" of BL(X,y) in the sense of (3.8.3.12):
6.8.2 THEOREM If X and Y are normed spaces, then
KL0(X,y) C KL^X^Y) C KL{X,Y) = clKL(X,y) C BL{X,Y)
(6.8.2.1)
There is inclusion
KL(X,y) + KL(X,y) C KL(X,y)
and (6.8.2.2)
KL^y) + KL1(X,y) C KL^X^Y)
and if Z is another normed space
BL(y,Z)KL(X,y) C KL{X,Z)
and (6.8.2.3)
BL(y,Z)KL1(X,y) C KL^XtZ)

6.8 Compact Operators
207
and
KL(X,Y)BL(Z,X) C KL(Z,Y)
and (6.8.2.4)
KL^YJBL^X) CKL^Y)
Proof: Much of this can be derived from Theorem 6.7.2, if we observe that
the totally bounded and the compact operators are those whose restrictions
to Q = DiscjJC(0 ; 1) are in m(Q,Y) and m^tl^Y) respectively. It is clear
that if T G L(X,Y) is totally bounded then it is also bounded, and that
if T is compact then it is totally bounded. If T G KL0(X, Y) is of finite
rank, then by Theorem 6.6.5 the subspace T(X) is locally compact, and
also complete, and hence T(X) n Discy (0 ; k) has compact closure for each
k > 0, which makes T compact. To see that KL(X, Y) is closed in BL(X, Y)
we repeat the argument which proves that m(Q,X) is closed in l^Q^X)
from Theorem 6.7.2. This proves (6.8.2.1). Towards (6.8.2.2), if S and T
are in KL(X,Y) and s,tGK then
{{sT + tS)x: \\x\\ < 1} C 5 {Tx: \\x\\ < 1} + t {Sx: \\x\\ < 1} (6.8.2.5)
is totally bounded, and if in particular S and T are both compact then
c\{[sT + tS)x\\\x\\ < l}C5cl{Tx:||x|| < l} + *cl{Sx: ||x|| < 1} (6.8.2.6)
is compact. Also if U G BL(Z,X) and V G BL(Y,Z) and T G KL(X,Y)
then
{TUz: \\z\\ < 1} C {Tx: \\x\\ < \\U\\} = \\U\\ {Tx: \\x\\ < 1} (6.8.2.7)
is a subset of a totally bounded set, and of a compact set if T G KLX (X, Y),
while
{VTx: \\x\\ < 1} = V {Tx: \\x\\ < 1} (6.8.2.8)
is the continuous image of a totally bounded or of a "relatively compact"
subset. ■
Compactness and total boundedness of operators can be tested with
sequences:
6.8.3 THEOREM If X and Y are normed spaces and T G BL(X,Y),
then
T G KL(X, Y) <£=> T^l^X) C m(Y) <£=> P(T) = 0 (6.8.3.1)

208
6. Finite Dimensional Spaces and Compactness
and
T G KL^X, Y) <£=> T^l^X) C m^Y) <£=> P^T) = 0 (6.8.3.2)
Proof: The first implication of (6.8.3.1) is obtained by applying (6.6.2.1)
with K = {Tx: \\x\\ < l}, while the second comes from the definition of
P(T). The first implication of (6.8.3.2) comes from (6.6.2.1) with K =
cl{Tx: ||x|| < 1}, and the second from the definition of P^T). ■
Finite dimensionality can be tested with the essential enlargements:
6.8.4 THEOREM If X is a normed space, then
/ G KL(X, X) <£=> X finite dimensional <£=> I e KLX(X, X) (6.8.4.1)
and
PX(X) = 0 <£=> X finite dimensional <£=> P(X) = 0 (6.8.4.2)
Proof: If X is finite dimensional, then by Theorem 6.6.5 the subset Disc(0;
1) is compact, giving / G KL1(X,X). If X is not finite dimensional, then
again, by Theorem 6.6.5 the subset Disc(0; 1) is not totally bounded, giving
/ £ KL(X,X). This proves (6.8.4.1), and hence also (6.8.4.2). ■
We conclude with the observation that the only regular compact
operators are of finite rank:
6.8.5 THEOREM If X and Y are normed spaces, then
KL(X, Y) n BT(X, Y) C KL0(X, Y) (6.8.5.1)
Proof: If T G 'BL'(X, Y) is regular, with generalized inverse T" G
BL(Y,X),then
TT'y = Iy for each y G T{X) (6.8.5.2)
and hence if T G KL(X,Y) is totally bounded then the restriction of the
identity / to the range T{X) is totally bounded. By (6.8.4.1) T{X) is finite
dimensional. ■
6.9 SEMI-FREDHOLM OPERATORS
A "semi-Fredholm" operator has about two thirds of the Fredholm
property:

6.9 Semi-Eredholm Operators
209
6.9.1 DEFINITION If X and Y are normed spaces, then T G BL(X, Y)
is upper semi-Fredholm iff
T is essentially one-one and proper (6.9.1.1)
and is lower semi-Fredholm iff
T is essentially dense and proper (6.9.1.2)
We shall write
cr+sBL(X,y) = {T e BL(X,Y):T is not upper semi-Fredholm}
(6.9.1.3)
and
cr"sBL(X,y) ={Te BL(X,Y):T is not lower semi-Fredholm} (6.9.1.4)
Evidently
T Fredholm <£=£> T upper semi-Fredholm and essentially dense (6.9.1.5)
and
T Fredholm <£=$> T lower semi-Fredholm and essentially one-one
(6.9.1.6)
We can use the semi-essential enlargement functor Px of (6.7.4.2) to test
for upper and lower semi-Fredholmness:
6.9.2 THEOREM If X and Y are normed spaces and T G BL(X,Y),
then
Px(r) one-one <£=> T upper semi-Fredholm (6.9.2.1)
Proof: Forward implication is divided into three parts. We claim that
PX(T) one-one => T essentially one-one (6.9.2.2)
PX(T) one-one => T relatively open (6.9.2.3)
PX(T) one-one and T relatively open => T proper (6.9.2.4)
The argument for (6.9.2.2) is the same as for Theorem 6.6.4: if T_1(0) is
infinite dimensional then by the Riesz lemma, Theorem 1.5.2, and part of
Theorem 6.2.1, there is a sequence x = (xn) in X satisfying the conditions

210
6. Finite Dimensional Spaces and Compactness
(6.6.4.3), so that
xel^X) and x^m{X)Dm1{X)
and (6.9.2.5)
Tx = 0 G c0{Y) C m^Y) C m{Y)
which means that neither PX(T) nor P(T) is one-one. For (6.9.2.3) suppose
that T is not relatively open: then there is x = (xn) in X for which
||xn|| = 1 and dist(xn,T_1(0)) > \ and ||Txn|| —> 0
(6.9.2.6)
We claim
16/^(1) and Tx G c0{Y) C m^Y) and xgm^X)
(6.9.2.7)
Indeed the first two observations are immediate. If, however, x G m1(X)
then there is a subsequence x' of x converging to an element x^ G X: but
now
T(x'oo)=0 and dist^r^O)) > i (6.9.2.8)
a contradiction.
Toward (6.9.2.4), suppose that T is relatively open, so that there is
k > 0 for which
y e T(X) =>y€ {Tx: \\x\\ < k\\y\\} (6.9.2.9)
Then if y G cl(TX) there is x' = (x'J in X for which ||t/n-rx'J| -► 0, and by
(6.9.2.9) there is x = (xn) in X for which Txn = Tx'n with ||xn|| < A:||rx^||.
Since the sequence Tx' is bounded, it follows that x G /^(X), and then
also the sequence Tx = Tx' G c^Y) C 77^ (Y). If, therefore, P^T) is also
supposed to be one-one, then we must have x G m1(X) and hence there
must be a subsequence x" of x converging to an element x'^ G X. Now we
have
t/ = lirnTx'^ = Tx'4 G T(X) (6.9.2.10)
n
Since t/ G cl(TX) was arbitrary it follows that T has closed range, which
together with being relatively open makes it proper by (3.5.6.1).
For backward implication in (6.9.2.1) suppose that T is upper semi-
Fredholm: then since T_1(0) is finite dimensional there is by (6.3.5.1) P =
P2 G BL(X,X) with P_1(0) = r_1(0), and since T is relatively open there
is k > 0 for which, if x G X,
\\Px\\ < ||P||dist(x,p-1(0)) = ||P||dist(x,T-1(0))
<A;||P||||rx|| = A;||P||||rPx|| (6.9.2.11)

6.9 Semi-Fredholm Operators
211
Now if x G ^(-X^) is such that Tx G tu1(Y) then for each subsequence x'
of x there is a subsequence x" of x' for which Tx" has a limit t/^ G Y, and
since T{X) is closed there must be x'^ G X for which y£ = Tx'^ = TPx^.
From (6.9.2.11) it follows that Px'^ -► Px'^. Since P"1^) = (7-P)(X) is
finite dimensional there must be a subsequence x'" of x" for which (I-P)x'"
converges to an element x'£ = {I - P)x'£ in X. But now x'" = Px'" +
(/ — P)x!n converges to Px^ + x'^ G X. Since x'" is a subsequence of x',
which was an arbitrary subsequence of x, we have shown that x G m1(X),
and hence that PX(T) is one-one. ■
Lower semi-Fredholmness can very nearly be tested with Px:
6.9.3 THEOREM If X and Y are normed spaces and T G BL(X, Y),
then
PX(T) onto <£=£> T essentially dense and relatively almost open (6.9.3.1)
and
Px(r) one-one onto => T Fredholm => Px(r) invertible (6.9.3.2)
Proo/: Towards (6.9.3.1) suppose T is not essentially dense, so that
Y/cl(TX) is infinite dimensional. Then by the Riesz lemma
(Theorem 1.5.2) and part of Theorem 6.2.1 there is y = (t/n) in Y for which,
for each n G N,
n
\\yn\\ < 2 and dist(t/n+1,cl(TX) + £ Kt/y) > 1 (6.9.3.3)
3 = 1
We claim
y G UY) and y £ T^X) + m(Y) D T^l^Y) = m^Y)
(6.9.3.4)
Certainly y is bounded, while if there were x = (xn) in l^X) for which
t/ — Tx had a Cauchy subsequence then there would be n and m > ra for
which
||r*n-r*m-yn+ym||<i (6.9.3.5)
contradicting (6.9.3.3). Thus if T is not essentially dense then neither
PX(T) nor P(T) is onto. If instead T is not relatively almost open, then
there is g = (gn) in Y* for which
\\gj = 1 and distfon, (T)"1^)) > I and ||^T|| —♦ 0
(6.9.3.6)

212
6. Finite Dimensional Spaces and Compactness
Evidently
g^rn^) =m{Y^) (6.9.3.7)
for if g' = (g^) is a subsequence of g for which g'n —*■ g^ G^ then
dist(^, (T)"1^)) > I and g^T = 0 (6.9.3.8)
a contradiction. By the completeness of Y\ m^Y^) = m(Y^) and since
</ £ m(yt)
0 < P(^) = inf {6 > 0:{gn:ne N} C Bisc(H ;6) for some finite H C Ff}
(6.9.3.9)
If now 0 < £ < P(<7), then there is a subsequence g' = [g!)n of g for which,
if ne N,
n
0n+i£ |J Disc(^- ; *) (6.9.3.10)
3 = 1
and then there is {y'n m)n^m in F for which
» # ™ =► IKmll = 1 and K(y;>m) - ^«J| > |* (6.9.3.11)
Now claim that if x'n m G X for each n ^ m then
sup K.JI < oo =► P {j£ - rx'nim: n?m}>\6 (6.9.3.12)
Indeed if
e > 0 and {t/^m - Tx'nrn\ n ^ m) C Disc(if ; e) for some finite ifC7
(6.9.3.13)
then
9'n(y'n,m)-9'm(y'n,m)£ {J I>isc((g'n - g'm)(y") ; s)
y"€K
+ (g'n-g'm)(Tx'ntm) ifn^m. (6.9.3.14)
and using the last part of (6.9.3.7) there is {N£,)£l>0 in N for which, if
e'>0,
n>m>Ne,^g'n(y'nim)-g'm(y'nim)e \J D\sc(K,e + e') (6.9.3.15)
y"GK
From (6.9.3.13) and (6.9.3.11) it follows
\6 < e' for each e' > 0 => ±<5 < e (6.9.3.16)

6.9 Semi-Fredholm Operators
213
giving (6.9.3.12). If we now arrange the pairs (n,m) with n ^ m in a
sequence and write
y = (y„) = (y'n,m) (6.9.3.17)
then (6.9.3.12) can be rewritten
* € foot*) => IIS/ - Tx||g > \6 (6.9.3.18)
so that in particular the coset y + m1(Y) is not in the range of the operator
Pi(T).
We have now proved forward implication in (6.9.3.1). Conversely, if
T is essentially dense and relatively almost open then by (6.3.5.2) there is
Q = Q2 e BL(Y, Y) with Q{Y) = cl(rX), and then k > 0 for which
y = Qy => y e el{rx: ||x|| < %||} (6.9.3.19)
Now if y G ^(Y) is arbitrary we claim there is x = (xn) in X for which
iG^fl) and y-Txem^Y) (6.9.3.20)
Certainly for each n 6 N we can find xn 6 X for which
||Qyn -TxJ| < I||yn|| and ||xj| < *||yn|| (6.9.3.21)
Thus
xG^fl) and Tx-Qyec0{Y) (6.9.3.22)
It follows
y-Tx = y-Qy + Qy-Txe ^(Q"1 (0)) + c0(Y)
= ^i(Q_1(0)) + c0{Y) C mi(Y) (6.9.3.23)
using the fact that Q_1(0) is a finite dimensional subspace of Y.
Toward the implication (6.9.3.2) it is now clear from (6.9.3.1) and
(6.9.2.1) that PX(T) one-one and onto implies T upper semi-Fredholm and
essentially dense, therefore Fredholm by (6.9.1.5). Conversely, if T is Fred-
holm, with generalized inverse T', which necessairly satisfies (6.4.3.1) and
(6.4.3.2), then
P1(T')P1(T)-P1(J)=0 and P^rjP^f) - P^J) = 0
(6.9.3.24)
so that PX(T') is a two-sided inverse for PX(T). ■
Of course both implications in (6.9.3.2) are two-way, since the last
condition implies the first. We should not, however, forget that PX(T) is

214
6. Finite Dimensional Spaces and Compactness
in general defined on seminormed spaces rather than on normed spaces.
An immediate consequence of Theorems 6.9.2 and 6.9.3 is the extension of
Atkinson's Theorem 6.4.6 and the perturbation Theorem 6.4.3 from finite
rank to compact operators:
6.9.4 THEOREM If X and Y are normed spaces and T G BL(X,Y)
satisfies
T left invertible modulo KL^X.X) (6.9.4.1)
and
T right invertible modulo KLX{Y,Y) (6.9.4.2)
then T is Fredholm. Also if K G BL(X, Y), then
T Fredholm, K compact => T + K Fredholm (6.9.4.3)
Proof: If (6.9.4.1) and (6.9.4.2) hold then there is V G BL(Y, X) for which
/ - T'T and / - TT' are both compact, giving (6.9.3.24), so that T is
Fredholm by (6.9.3.3). Also if T is Fredholm and K is compact then P^T)
is invertible and P^jK") = 0, so that VX{T + K) = P^T) is invertible, and
hence by (6.9.3.3) T + K is Fredholm. ■
6.10 ALMOST FREDHOLM OPERATORS
The "almost semi-Fredholm operators" are slightly more general than the
semi-Fredholm operators:
6.10.1 DEFINITION If X and Y are normed spaces, then T G BL(X, Y)
is almost upper semi-Fredholm iff
T is relatively open and essentially one-one (6.10.1.1)
is almost lower semi-Fredholm iff
T is relatively almost open and essentially dense (6.10.1.2)
and is almost Fredholm iff
T is almost upper and almost lower semi-Fredholm (6.10.1.3)
If T is almost Fredholm we shall define index(T) as in (6.5.1.1). We shall
write
3+,BL(X,Y) = {T€BL(X,Y):
T is not almost upper semi-Fredholm} (6.10.1.4)

6.10 Almost Fredholm Operators
215
&-sBL(X,Y)={TeBL(X,Y):
T is not almost lower semi-Fredholm} (6.10.1.5)
and
<ressBL(X,y) ={Te BL{X,Y):T is not almost Fredholm} (6.10.1.6)
Evidently
upper semi-Fredholm => almost upper semi-Fredholm
=> essentially one-one (6.10.1.7)
and
lower semi-Fredholm =>• almost lower semi-Fredholm
=> essentially dense (6.10.1.8)
The almost Fredholm operators form an open set, and the index is
continuous:
6.10.2 THEOREM If X and Y are normed spaces and T G BL(X,Y),
then there is ST > 0 for which, if T' G BL(X,Y) satisfies ||r' - T|| < ST,
then
T almost upper semi-Fredholm => Tf almost upper semi-Fredholm
(6.10.2.1)
and
T almost lower semi-Fredholm => Tf almost lower semi-Fredholm
(6.10.2.2)
and
T almost Fredholm => T' almost Fredholm, with index (T') = index(r)
(6.10.2.3)
Proof: Suppose T is almost upper semi-Fredholm. Then since T_1(0) is
finite dimensional, there is by (6.3.5.1) P = P2 e BL(X,X) for which
P_1(0) = r_1(0), and since T is relatively open there is k > 0 for which,
if x e X,
\\Px\\ < ||P||dist(x,p-1(0)) = ||P||dist(x,r-1(0))
<A:||P||||rx|| = A:||P||||rPx|| (6.10.2.4)

216
6. Finite Dimensional Spaces and Compactness
If now T' e BL(X,y) satisfies
Jfc||P||||r'-r|| <1 (6.10.2.5)
then, for arbitrary x G X,
k\\P\\\\T'x\\ > k\\P\\\\Tx\\ - k\\P\\(T' - T)x\\ > \\Px\\ - \\Tf - T\\\\x\\
(6.10.2.6)
so that in particular
x = Px =* fc||P||||T'x|| > (1 - *||P||||r' - r||)||x|| (6.10.2.7)
Thus the restriction of T' to P(X) C X is bounded below, and in particular
one-one:
(r')_1(o) n p(x) = {o} (6.10.2.8)
It follows that
dimtr')"1^) < dimX/P(X) = dimP-^O) = diml^O) < oo
(6.10.2.9)
and hence that V is essentially one-one. By (6.3.5.1) both (T')"1^) and
the sum (r')_1(0) + P(X) are complemented: thus we obtain P' = (P')2 G
BL(X,X) with
(P')-!(0) = (r')_1(0) and P'(X) = P{X) +W (6.10.2.10)
where W is a closed subspace complementary to T_1(0) + P(X). We claim
that the induced mapping T : X/P/_1(0) —»• Y is bounded below, for by
(6.10.2.7) the condition (3.3.6.2) of Theorem 3.3.6 is satisfied with (r')A
in place of T and W = (P(X) + P/-1(0))/P/"1(0), while the restriction
of (T')A to W is one-one since already (T')A is one-one. Since W is finite
dimensional (6.2.3.2) says that the condition (3.3.6.1) of Theorem 3.3.6 also
holds, and now Theorem 3.3.6 gives the result. This proves the implication
(6.10.2.1).
If instead T is almost lower semi-Fredholm, then since cl(TX) is of
finite codimension there is by (6.3.5.2) Q = Q2 e BL(y,y) for which
Q(Y) = cl(TX), and since T is relatively almost open there is k > 0 for
which, if y G Y,
y = Qy => y e el {Tx: \\x\\ < k\\y\\} (6.10.2.11)
Now if T' e BL(y,y) satisfies k\\T' - T\\ < 1, then by the argument of

6.10 Almost Fredholm Operators
217
Theorem 3.4.3 we have
y = Qy => y e cl {T'x: \\x\\ < k'\\y\\} with *' = (1_fe||*,_r|[)
(6.10.2.12)
so that in particular
Q{Y) C cl(r'X) (6.10.2.13)
It follows that
dimY/cl(r'X) < dimQ-^O) = dimY/Q(Y) = dimYcl(rX) < oo
(6.10.2.14)
and hence that T' is essentially dense. By (6.3.5.2), c\(T'X) and c\(T'X) D
Q_1(0) are complemented, giving us Q' = (Q')2 G BL(Y, Y) with
Q'{Y) = cl(r'X) and Q_1(0) = (Q')"1!0) + z' (6.10.2.15)
where Z1 is a closed complement for cl(T'X) fl Q_1(0). We claim that
the mapping T' is relatively almost open; for by (6.10.2.13) the condition
(3.4.6.4) is satisfied with {T')A : X -> Q'{Y) in place of T : X -> Y
and Z = Q{Y), while the induced mapping (r')A~ : X -> 0;(y)/0(y) is
dense since already (T')A is dense. Since Q'(Y)/Q(Y) is finite dimensional
(6.2.3.3) says that the condition (3.4.6.3) of Theorem 3.4.6 also holds, and
hence by Theorem 3.4.6 the mapping (T')A is almost open, giving the result.
This proves the implication (6.10.2.2).
If T is almost Fredholm and if V G BL(X, Y) satisfies both (6.10.2.5)
and the condition following (6.10.2.12), then all four projections P, Q, P'
and Q' are present, and we can write
X=P{X)®W®{P')-10 and Y = Q{Y)eZe{Qf)~10 (6.10.2.16)
We claim that the operator
(r')A : P{X) © W —> T'(PX) © T'{W) = Q'tQ^O) (6.10.2.17)
induced by T' is one-one, so that
index(r') = dim(r')_10 - (dimQ"1^) - dimr'(^))
= dim(r')_10 + dimW - dimQ"1^)
= dimr"1(0) -dimQ-^O)
= index(r) ■

218
6. Finite Dimensional Spaces and Compactness
On the face of it we have again proved more than we have stated. The
index is not only continuous, but actually locally constant. Since, however,
normed spaces are connected, and locally connected, this is the only way
that a mapping into the integers could have been continuous.
To test for almost Fredholmness it is more appropriate to use the
enlargement functor P instead of the semi-enlargement P x:
6.10.3 THEOREM If X and Y are normed spaces and T G BL(X, Y),
then
T almost upper semi-Fredholm => P(T) bounded below (6.10.3.1)
and
T almost lower semi-Fredhom => P(T) open (6.10.3.2)
and hence
T almost Fredholm => P(r) invertible (6.10.3.3)
Proof: If T is almost upper semi-Fredholm, then P(T) is certainly one-one:
for if P = P2 G BL(X,X) with P_10 = r_10 and k > 0 satisfies (6.10.2.4),
and if x G l^X) with TNx G m(Y), then for each subsequence x' of x
there is a subsequence x" of x' for which TNx" G c(Y) is Cauchy, so that
by (4.1.3.2) Px" G c(X) is also Cauchy. At the same time, the sequence
(/ — P)x" is bounded and in the finite dimensional space (I — P)X =
T_10, which is locally compact by Theorem 6.6.4. Therefore, there is a
subsequence x,n of x" for which (/ — P)x,n G m^X) C m(X), and hence
x'" = Px'" + (J - P)x'" G m{X) (6.10.3.4)
To see that the operator P(T) is bounded below we show that for each
dist(x, m(X)) = dist(Px, m{X)) < 2k dist(rx, m{Y)) (6.10.3.5)
For if 6 > dist(Tx, m(Y)) is arbitrary, then by (6.7.2.2) there is a finite set
H C Y for which {Txn:n G N} C Disc(H ; <5), hence a sequence y = (yn)
in/^(Y) with
{yn: n G N} C H and ||y - Tx^ < 6 (6.10.3.6)
and finally a finite subset K C {xn:7i G 6A^} and a sequence x' G ^oo(^)
for which
{x'n:ne N} C K and |lTx' - y]]^ < 6 (6.10.3.7)

6.10 Almost Fredholm Operators
219
Thus
{Txn: n G N} C Disc(r(K) ; 26) (6.10.3.8)
and hence
{Pxn: n G N} C Disc(P(K) ; 2k6) (6.10.3.9)
By (6.7.2.2) again we have
dist(x,m(X)) = dist(Px,m(X)) < 2kS (6.10.3.10)
which by the choice of 6 gives (6.10.3.5).
If instead T is almost lower semi-Fredholm then there is Q = Q2 G
BL(y,y) with Q{Y) = d{TX), and k > 0 for which (6.10.2.12) holds. Now
if y G Joo(y) is arbitrary we claim that there is x = (xn) in X for which
xel^X) and y-Txem{Y) and H*^ < kM^
(6.10.3.11)
Certainly for each n G N we can find xn G X for which
||Txn - Qyn|| < - and ||xn|| < k\\yj (6.10.3.12)
n
which ensures that x = (xn) is bounded and with norm dominated by Hyll^.
At the same time the sequence Qy — Tx is in c0(Y) C m^Y), while the
sequence y — Qy has its terms in the finite dimensional subspace Q_1(0),
therefore is in m1(Y): thus ^
y-Tx = y-Qy+Qy-Tx G m1{Y) + c0{Y) C m^Y) C m{Y) (6.10.3.13)
We have proved (6.10.3.1) and (6.10.3.2), and hence also (6.10.3.3). ■
As we have already proved in the argument for (6.9.2.1),
P(T) one-one =>■ T essentially one-one (6.10.3.14)
we claim that also
P(r) dense =^ T essentially dense (6.10.3.15)
For if T is not essentially dense, so that y/cl(TX) is infinite dimensional,
and if y = (yn) is a sequence satisfying the conditions (6.9.2.18), then we
claim
p(y - TNx) > \ for each x G /^(X) (6.10.3.16)
where p is the measure of noncompactness (6.7.1.3).

220
6. Finite Dimensional Spaces and Compactness
Theorem 6.9.3 and the continuity of the index enable us to improve
(6.4.4.6.) and (6.5.5.4):
6.10.4 THEOREM If X and Y are normed spaces and T G BL(X, Y),
KeBL{X,Y) then
T Fredholm, K compact
=> T + K Fredholm with index(r + K) = index(r) (6.10.4.1)
Proof: By (6.9.3.2)
T + tK is Fredholm for each t e K (6.10.4.2)
and by (6.10.2.3)
index(T + tK) is a continuous function of t ■ (6.10.4.3)
6.11 COMPLETENESS
If X and Y are Banach spaces, then the theory of compact and of Fredholm
operators is considerably simplified:
6.11.1 THEOREM If X and Y are Banach spaces and T G BL(X, Y),
then
T totally bounded => T compact (6.11.1.1)
T almost upper semi-Fredholm => T upper semi-Fredholm (6.11.1.2)
T almost lower semi-Fredholm => T lower semi-Fredholm (6.11.1.3)
and hence
T almost Fredholm => T Fredholm (6.11.1.4)
Proof: Implication (6.11.1.1) follows from Theorem 6.6.2, applied to K =
cl{Tx: ||x|| < 1}. Implications (6.11.1.2) and (6.11.1.3) follow from
Theorem 4.4.3 and Theorem 4.4.4 applied to the operator core(T) : T/X~l (0) —►
ci(rx). ■
If X and Y are complete, then Theorem 6.8.2 tells us that the compact
operators KL1(X,Y) = KL(X, Y) form a closed ideal of BL(X, Y), and
Theorem 6.10.2 tells us that the Fredholm operators form an open subset
of BL(X, Y). Alternatively, this can now be deduced from either (6.4.6.4)
or (6.4.6.5) of Theorem 6.4.6. For if K is so small that ||3r'|| ||JFC|| < 1 then
by the observation (4.4.5.15) both / + TK and / + KT1 will be invertible

6.11 Completeness
221
and hence Fredholm. At the same time, since I+T'K and I+KT' also have
index zero, (6.5.3.2) and (6.5.3.3) from Theorem 6.5.3 offer an alternative
proof that the index is continuous.
When X and Y are complete then both "essential enlargements"
coincide:
6.11.2 THEOREM If X and Y are complete and T G BL(X, Y), then
m{X) = mx(X) and m{Y) = m^Y) (6.11.2.1)
and
P(r) =Pi(r) (6.11.2.2)
Proof: Equality (6.11.2.1) follows by the same argument as (6.11.1.1), and
then gives (6.11.2.2). ■
If X and Y are complete, then Theorem 6.9.2 and Theorem 6.10.3
coalesce:
6.11.3 THEOREM If X and Y are Banach spaces and T G BL(X, Y),
then
P(T) one-one => T upper semi-Fredholm
=> P(r) bounded below (6.11.3.1)
and
P(r) one-one dense => T Fredholm => P(r) invertible (6.11.3.2)
Proof: For (6.11.3.1) use (6.9.2.1) and (6.10.3.1), and for (6.11.3.2) use
(6.10.3.15) together with (6.9.2.1). ■
Theorem 6.11.3 displays the upper semi-Fredholm operators, and the
Fredholm operators, as continuous counterimages of open subsets of
BL(P(X),P(Y)), and therefore as open subsets of BL(X,Y).
Alternatively, since by (4.4.5.9) the invertible operators form an open set, (6.4.4.4)
tells us that the Fredholm operators form an open set. We can also see
that the topological boundary of the Fredholm operators is disjoint from
the upper semi-Fredholm operators.
6.11.4 THEOREM If X and Y are complete and T G BL(X, Y) is upper
semi-Fredholm, then
T G closure {Fredholm operators} => T Fredholm (6.11.4.1)

222
6. Finite Dimensional Spaces and Compactness
Proof: If T is upper semi-Fredholm and \\T - Tn\\ -> 0 with Fredholm
(rj, then by (6.11.3.1) and (6.11.3.2), P{T) is bounded below and ||P(T) —
P(rn)|| -► 0 with dense P(TJ, so that by Theorem 3.5.1 P(r) is almost
open, and therefore (6.11.3.2) T is Fredholm. ■
When the spaces X and Y are complete there is a rather simpler
characterization of Fredholm operators:
6.11.5 THEOREM If X and Y are complete, then T G BL(X,Y) is
Fredholm iff
T_1(0) and Y/T{X) are finite dimensional (6.11.5.1)
Proof: Whether or not the spaces X and Y are complete, the condition
(6.11.5.1) is necessary for T to be Fredholm, since Fredholm operators have
closed range, and is sufficient for T to be both essentially one-one and
essentially dense. We claim that if X and Y are complete, then (6.11.5.1)
implies that T is proper. For Theorem 4.8.2 says that T(X) is closed, and
now Theorem 4.5.3 says that core(T) is invertible, so that T is proper. ■
Theorem 6.11.5 means that some of the arguments in our development
can be shortened if we confine ourselves to complete spaces. For example,
if we use the argument of Theorem 6.5.4 to prove that the product of
Fredholm operators is Fredholm, it is unnecessary to make the observation
(6.5.4.12), while if we want to prove that the Fredholm operators form an
open set we need only establish (6.10.2.9) and (6.10.2.14) from the argument
for Theorem 6.10.2.
6.12 DUALITY THEORY
The dual of a compact or a Fredholm operator has the same property:
6.12.1 THEOREM If X and Y are normed spaces and T G BL(X,y),
then
t e KL0(x,y) <^> rf g KL0(yt,xt) (6.12.1.1)
T Fredholm ^=> Tf Fredholm (6.12.1.2)
t e KL(x,y) <^> rf e KL(yt,xt) (6.12.1.3)
Proof: If T* G BL(yt,Xt) is not of finite rank then there is a linearly
independent sequence / = (fn) in T^(Y^) C X*, and if m G N is arbitrary
there is by (6.1.2.1) a sequence (x1,x2,... ,xm) is X for which
QiiTxj) = /t-(xy) = «.. for each ij G {1,2,...,m} (6.12.1.4)

6.12 Duality Theory
223
where g = (gn) in Y^ is chosen so that gnT = fn for each n. Since it is clear
from (6.12.1.4) that (Tx1,Tx2,... ,Txm) is linearly independent, it follows
from Theorem 6.1.1 that T(X) cannot be of dimension < m. Since m is
arbitrary it follows that T cannot be of finite rank. This proves forward
implication in (6.12.1.1). Conversely, if T* is of finite rank then so is the
second dual T^, by what we have just proved, and hence also T. Towards
(6.12.1.2) we claim
T essentially dense <£=> T' essentially one-one (6.12.1.5)
and
T* essentially dense =>• T essentially one-one (6.12.1.6)
For by (5.6.0.1) we have
T1" essentially one-one <!=> T(X)° finite dimensional (6.12.1.7)
and by (2.3.2.2) the dual space of y/cl(rX) is (TX)°: thus we may
apply (6.12.1.1) to the identity I : y/cl(rX) -+ y/cl(rX). This proves
(6.12.1.5), and hence also (6.12.1.6):
T* essentially dense => T'' essentially one-one
=> T essentially one-one (6.12.1.8)
If we recall (5.6.4.2) that T is proper if and only if T* is, then we have
shown
T lower semi-Fredholm <£=> T^ upper semi-Fredholm (6.12.1.9)
and forward implication in
T* lower semi-Fredholm ^=^ T upper semi-Fredholm (6.12.1.10)
Conversely, if T is upper semi-Fredholm, and P = P2 G BL(X,X) has
P_1(0) = r_1(0), then I- P is of finite rank and there is k > 0 for which
||Px|| < ||rx|| for each x G X (6.12.1.11)
By (6.12.1.1) the dual operator P*, also a projection, has finite dimensional
null space since it — P* is of finite rank, and we claim
Pf(Xf) CT^yt) (6.12.1.12)
Indeed, if / = fP e X1" we may define g° : TX -+ K by setting g°{Tx) =
f(x) for each x G X*, using (6.12.1.11) to see that g° is well defined,

224
6. Finite Dimensional Spaces and Compactness
and then bounded, and then extending it to g G Y1" by the Hahn-Banach
Theorem 5.3.2.
Toward (6.12.1.3) we claim that, for arbitrary T G BL(X, Y),
p {T*g: \\g\\ < l} < 4p {Tx: \\x\\ < 1} (6.12.1.13)
where p is given by (6.7.1.3). Indeed if 6 > p{Tx: ||x|| < 1} is arbitrary
then there is n G N and (x1?x2,... ,xn) G Xn for which {Tx: ||x|| < 1} C
Uy=1 Disc(Txy ; 26), and then since the mapping
W : g —ygTx from Y1" to Kn (6.12.1.14)
is of finite rank, there is for each 6 > 0 a finite set H C Y* for which
{Wg: \\g\\ < 1} C Disc(W(#) ; e) (6.12.1.15)
We claim
{T*g: \\g\\ < l} C Disc^jff) ; 4(5 + e) (6.12.1.16)
for if g G Y1" with ||^|| < 1 there is h e H for which max,- I^Tx^ - /iTx^l <
||W<7 — W/i||,andnow if ||x|| < 1 we can choose j for which ||Tx —Txy|| < 26,
so that
|<?rx - hTx\ < \gTx - gTxj\ + \gTxj - hTx^ + \hTXj - hTx\
< \\g\\ \\Tx - Txyll + e + ||/i|| ||rxy - Tx|| < 4(5 + e
The choice of 6 and e now gives (6.12.1.13). In particular it follows from
(6.12.1.13) that
T totally bounded => Tf totally bounded (6.12.1.17)
and hence also
Tf totally bounded =^ Tft totally bounded =^ T totally bounded ■
(6.12.1.18)
The implication (6.12.1.6) cannot in general be reversed. For example,
if T = J : lx —► c0, as discussed following (5.9.1.3), then T is one-one but
is not essentially dense. Using Theorem 5.6.1 and Theorem 5.6.2, the
reader may like to show
T almost lower semi-Fredholm <£=> T almost upper semi-Fredholm
(6.12.1.19)

6.12 Duality Theory
225
and
T almost upper semi-Fredholm <£=> T* almost lower semi-Fredholm
(6.12.1.20)
We can also supplement (6.12.1.2) by adding
T Fredholm => index(rf) = -index(r) (6.12.1.21)
In particular
T Weyl <^> Tf Weyl (6.12.1.22)
Theorem 6.12.1 and Theorem 6.11.3 combine to give the "perturbation
theory" for Fredholm and semi-Fredholm operators:
6.12.2 THEOREM If X, y and Z are Banach spaces and T G BL(X, Y),
S<EBL(y,Z),then
5, T upper semi-Fred =>• ST upper semi-Fred =>• T upper semi-Fred
(6.12.2.1)
and
5, T lower semi-Fred => ST lower semi-Fred => S lower semi-Fred
(6.12.2.2)
and if also K G BL(X,Y), then
T upper semi-Fred, K compact => T + K upper semi-Fred (6.12.2.3)
and
T lower semi-Fred, K compact => T + K lower semi-Fred (6.12.2.4)
Proof: (6.12.2.1) follows from (6.11.3.1) with either (3.2.3.1) or (3.3.2.1)
applied to P(r) andP(5), while (6.12.2.2) follows from (6.12.2.1) applied to
r1" and 5f, together with (6.12.1.9). For (6.12.2.3) use (6.11.3.1), and finally
(6.12.2.4) follows from (6.12.2.3) applied to T* and K^ using (6.12.1.9) and
(6.12.1.3). ■
For a converse to (6.12.2.3) and (6.12.2.4), we begin with an auxiliary
result:
6.12.3 THEOREM Suppose X and Y are normed spaces and T G
BL(X,y); then if T is not relatively open there are (xn) in X and (fn)

226
6. Finite Dimensional Spaces and Compactness
in X^ for which
oo
fi{*j) = Sij for each iJ and L Hrxnllll/nll < oo (6.12.3.1)
n=l
while if T is not relatively almost open there are (yn) in Y and (gn) inY^
with
oo
SiiVj) = S{j for each ij and ]£ 11^^1111^11 < oo (6.12.3.2)
n=l
Proof: Suppose (en) is an arbitrary sequence of positive numbers. Then
is T is not relatively open we shall find (xn) and (fn) for which
fiixj) = sij and ||rxy||||/y|| < Gj for each ij G N (6.12.3.3)
while if T is not relatively almost open we shall find (gn) and (yn) for which
fc(y;)=fy and II^IHI^-II^^ for each j'jGN (6.12.3.4)
Thus, if X)yLi ej < oo we will satisfy (6.12.3.1) and (6.12.3.2).
Towards (6.12.3.2) there is certainly xx G X for which HxJI = 1 and
IITxjH < el9 since core(T) is not bounded below, and then by the Hahn-
Banach Theorem 5.3.2 there is fx G X1" with H/J = 1 = /JxJ. If n G
N and sequences (x1,x2,... ,xn) in X and (/!,/25- • • ,fn) in X* satisfy
(6.12.3.3) for each i,j G {1,2,... ,?i}, we claim that there are xn+1 G X
and /n+1 G X1" for which (6.12.3.3) holds for ij G {1,2,... ,71,71 + l}. The
subspace
n
Wn=P[fi1W (6.12.3.5)
3 = 1
is of finite codimension in X, and by Theorem 3.11.1 T cannot be bounded
below on Wn. Thus, if e°+1 > 0 is arbitrary we can find xn+1 G Wn with
K+ill = 1 and ||r*n+1|| < e°+1, and then /°+1 € X* with ||/°+1|| = 1 =
fn+iixn+i)- Now put
OO
/n+1 = /n+1 " £ /n+1 K)/y (6.12.3.6)

6.12 Duality Theory
227
Evidently
fj{xn+\) =° = /n+i(x>) =° for eachjG {l,2,...,m}
and
(6.12.3.7)
ll/n+lll<l + Ell/ill
3 = 1
To extend (6.12.3.5) therefore it is only necessary to do all this with
(l + Ell/,llW°+i<^+i (6-12-3.8)
Toward (6.12.3.4) observe that the operator core(Tt) is not bounded
below, so that there is gx G Y^ for which ||^x || = 1 and ||^1T,|| < ^e1, and
then yx e Y with ||j/iII < 2 and 9i{Vi) = 1- If sequences (y1?y2,... ,yj
and (g1,g2, • • • ,<7n) are consistent with (6.12.3.4) then the restriction of T*
to the annihilator of 5Z?=i ^Vj cannot be bounded below, so that if e^+i
is arbitrary there will be <7n+1 in this annihilator for which ||<7n+1|| = 1 and
\\9n+iT\\ <e°+1, and then y°+1 6 F with ||y°+1|| < 2 and<7n+1(y°+1) = 1.
Now put
n
Vn+i = yj+i - E^(Ci)^ (6.12.3.9)
3 = 1
Evidently
9j{yn+i) = ° = dn+iiVn) for each J G {1,2,... ,n}
and
/ » \ (6.12.3.10)
ll»n+lll<2|l + X)Nl]
For consistency with (6.12.3.4) we need only do this with
2 1 + L INI K+1 < eB+1 ■ (6.12.3.11)

228
6. Finite Dimensional Spaces and Compactness
6.12.4 THEOREM If X and Y are Banach spaces, then T G BL(X, Y)
is upper semi-Fredholm iff
K compact =>• T — K essentially one-one (6.12.4.1)
and T G BL(X, Y) is lower semi-Fredholm iff
K compact =>• T — K essentially dense (6.12.4.2)
Proof: If T is upper semi-Fredholm then (6.12.4.1) is part of (6.12.2.3),
and if T is lower semi-Fredholm then (6.12.4.2) is part of (6.12.2.4). If
T is not upper semi-Fredholm then either it is not essentially one-one, in
which case (6.12.4.1) fails with K = 0, or else it is not proper, therefore
not relatively open, in which case (6.12.4.1) fails with
oo
K=Y,fn®T*n (6.12.4.3)
n=l
where (xn) and (/n) satisfy (6.12.3.1): for by the completeness of BL(X, Y)
and the closedness of KL(X,Y) = KL1(X,Y) the operator K is well
defined, and
xn e {T - K)_1(0) for each neN (6.12.4.4)
If instead T is not lower semi-Fredholm then either it is not essentially
dense, in which case (6.12.4.2) fails with K = 0, or else it is not proper,
therefore not relatively almost open, in which case (6.12.4.2) fails with
oo
ff=X>„T©yB (6.12.4.5)
n=l
where (gn) and (yn) satisfy (6.12.3.2). For again, K is well defined and
compact, with
cl((r - K)X) C g-l(Q) for each neN ■ (6.12.4.6)
Theorem 6.12.4 enables us to round off Theorem 6.11.3:
P(r) dense =^ T lower semi-Fredholm =^ P(r) open (6.12.4.7)
Indeed, the second implication is (6.10.3.2), while the first follows from
the second part of Theorem 6.12.4 if we apply (6.10.3.15) to T - K for all
compact K eBL{X,Y).

6.13 Composition Operators
229
6.13 COMPOSITION OPERATORS
We have seen in Theorem 6.12.1 that the dual of a finite rank or compact
operator has the same property. This does not extend to more general
compositions RT = BL(r,PY) and LT = BL(W,T):
6.13.1 THEOREM If 0 ^ T G BL(X, Y) and W is an infinite
dimensional normed space, then neither LT = BL(W,T) nor RT = BL(T,W) is
totally bounded.
Proof: If W is infinite dimensional then by the Riesz lemma 1.5.2 there is
(wn) in W for which, for each m,n G N,
|K|| = land^m^ \\wn - wm\\ > \ (6.13.1.1)
and if T ^ 0 there is / G X* for which ||/r|| = 1: now
||/ 0 wn\\ = ll/H and n + m => \\RT(f 0 wn) - RT(f 0 "m)ll > J||/||
(6.13.1.2)
Since W is also infinite dimensional there is [hn) in W for which, for each
m,n G N,
||fcB|| = 1 and n ± m =► ||fcn - fcj| > ± (6.13.1.3)
and since T ^ 0 there is x G X for which ||Tx|| = 1: now
\\hn 0 x|| = ||x|| andn^m^ ||LT(/in 0 x) - LT{hm 0 x)|| > ±||x|| ■
(6.13.1.4)
In spite of Theorem 6.13.1, total boundedness can still be transmitted
to composition operators. If T : X —> Y and T'X —> Y are linear mappings
between the same two spaces we shall write
T' A T : (I" - r)_10 —> Y (6.13.1.5)
for the common restriction of T and T' to the subspace {x G X: T'x = Tx}
on which they agree. We shall also recall || • ||gSS from (6.7.5.2):
6.13.2 THEOREM If X, Y, Z, W and E are normed spaces and if T G
BL(X,Y), 5 G BL(PY,Z), U : E -+ BL(Y,Z) and Y : £ -+ BL(X,PY) are
bounded linear operators, then
||(Br • V) A (Ls • V)||;„ < 4\\U\\\\T\\'ess + 2||V||||5||iM (6.13.2.1)
In particular
T,S totally bounded => (RT • U) A [Ls • Y) totally bounded (6.13.2.2)

230
6. Finite Dimensional Spaces and Compactness
Proof: Write
F={RT-U-LS. V)-1^) = {eeE: U{e)T = SV{e)} (6.13.2.3)
and
$ = (RT • U) A {Ls • V) : F —-► BL(X, Z) (6.13.2.4)
Choose arbitrarily 6 > ||T||gSS and e > \\S\\fe33: we then claim that
Unless <4||tf||5 + 2||V||£ (6.13.2.5)
Indeed, since ||T||gSS < 6, there is a finite subset K6 C Discx(0;l) for which
T Disc(0 ; 1) C Disc{T{K6) ; 26) (6.13.2.6)
and for each x' G K6 there is a finite subset He(X') C Disc^O ; 1) =
jPDisc^^ ; 1) for which
{SV(e)x': e G DiscF(0 ; 1)} C \\V\\SDiscw(0 ; 1)
C Discz{SVHe{x')x'; 2\\V\\e) (6.13.2.7)
where once again the factor of 2 comes from the effort to take the finite
subset of W to be of the form VHe(xf). Consider now the finite set
He)6= (J ff,(x')CDiscF(0;l) (6.13.2.8)
x'EK6
If e G Disc^(0 ; 1) and x G Discx(0 ; 1) are arbitrary, then there are by
(6.13.2.6) x' G K6 for which ||rx-rx;|| < 26, and by (6.13.2.7) e' G He{x')
for which ||5VP(c)x/-5VP(c/)x/|| < 2\\V\\e. Taking account of the definition
(6.13.2.3) of F we have
||(«(e) - «(c'))x|| < \\U(e)(Tx - Tx')\\ + \\S(V(e) - V(e'))x'\\
+ \\U(e')(Tx-Tx')\\
<2||J7||« + 2||Vr||e + 2||J7||«
Since x G Discx(0 ; 1) is arbitrary it follows
||$(e) - $(e')|| < 4\\U\\6 + 2\\V\\e (6.13.2.9)
which by the choice of 6 and e gives (6.13.2.1). ■
For example (6.12.1.13) follows from (6.13.2.1) if we take
W = Z = K; E = Y^; S = I; U = I; V = T1" (6.13.2.10)

6.13 Composition Operators
231
so that
F = E = yf and RT • U = Ls . V = Tf and ||S||'ess = 0 (6.13.2.11)
6.13.3 THEOREM If X, Y, Z and W are normed spaces and T G
BL(X,Y), S e BL{W,Z), then
ii^^Tiress<4ii5iiiir'iress+2iiriiii5iress (6.13.3.1)
In particular
t e KL(x,y), s e kl{w,z) => lsrt e kl{bl{y,w),bl{x,z))
(6.13.3.2)
If Y = X then also
l|£rA£r||Us<6|miUs (6.13.3.3)
In particular
T e KL(X,X) =>LTARTe KL{comm{T),BL{X,X)) (6.13.3.4)
Proof: For (6.13.3.1) apply (6.13.2.1) with
E = BL(Y, W) [7 = L5 V = RT (6.13.3.5)
so that
F = E = BL(Y, W) and £5 • *7 = LT • V = LSRT (6.13.3.6)
For (6.13.3.3) take
W = Z = Y = X E = BL{X,X) S = T U = V = I (6.13.3.7)
so that
F = comm(r) ={56 BL(X,X): ST = TS}
and (6.13.3.8)
RT • U = RT and Ls • V = Lr ■
We can slightly generalize Theorem 6.13.1 to give a converse to
(6.13.3.2):

232
6. Finite Dimensional Spaces and Compactness
6.13.4 THEOREM If X, Y, Z and W are normed spaces and T G
BL(X, Y), S G BL(W,Z), then there is implication
LSRT totally bounded => (S totally bounded or T = 0)
and (S = 0 or T totally bounded) (6.13.4.1)
Proof: If S ^ 0 and T is not totally bounded, so that by (6.12.1.3) T* is
not totally bounded, and then by (6.8.3.1) there is (gn) in Y* and w G W
for which
ye^^) and fonr)gm(Xt) and Sw^ 0 G Z
(6.13.4.2)
Since now
(gn0w)eloo{BL(Y,W)) and {S(gnOw)T) $m(BL(X,Z))
(6.13.4.3)
it is clear that LSRT is not totally bounded. If instead S is not totally
bounded and T ^ 0, then again by (6.8.3.1) there is (wn) in W and # G Y*
for which
KJe'ooW and (Su;n)gm(Z) and ^^06^
(6.13.4.4)
This time
(0©™JeUBL(Y,WO) and (% © wjr) £ m(BL(X,Z)) ■
(6.13.4.5)
Theorem 6.13.3, and in particular (6.13.3.2), says something about
the "essential invertibility" of the operators co\(Ls,RT) and row(JR5,LT)
of (5.6.3.6) and (5.6.3.5): we are using Definition 3.9.1 relative to the ideals
of totally bounded operators.
6.13.5 THEOREM If X, Y, Z and W are normed spaces and T G
BL(X,Y), S G BL{W,Z), then
S essentially left and T essentially right invertible (6.13.5.1)
implies
co\(Ls,RT) essentially left and tow(Rs,Lt) essentially right invertible
(6.13.5.2)
Proof: If (6.13.5.1) holds, then there are S' G BL{Z,W) and V G
BL(Y,X) for which
TT'-I=KeKL{Y,Y) and S'S-I = H G KL{W,W) (6.13.5.3)

6.13 Composition Operators
233
and hence, using (6.13.3.2)
= lhrk € KL(BL(y,Py),BL(y,Py)) (6.13.5.4)
and
yii>jnli>jnt — JtC>jn)[ljQljQt — ■^JTj = \-*JS S1 — l)\ TJ"'T,, — I)
= rhlk e KL(BL(Z,X),BL(Z,X)) (6.13.5.5)
Noting that each left and each right multiplication commute with one
another, this can be rewritten in the form
row((L5/L5 — I)RT,,RS,) co\(Ls,RT) + I = LHRK totally bounded
(6.13.5.6)
and
row(JR5, LT) co\{Rs,{LTLT, — I),LT) + I = Rh^k totally bounded ■
(6.13.5.7)
It is left to the reader to state and prove the analogue of Theorem 6.13.5
for "almost essential invertibility." In the other direction, we have the
essential analogue of Theorem 5.6.4:
6.13.6 THEOREM If X, Y, Z and W are normed spaces and T G
BL(X,Y), S e BL{W,Z) then
tow(Rs,Lt) essentially dense or co\(Ls,RT) essentially one-one
(6.13.6.1)
implies
(S essentially one-one or T dense) and (S one-one or T essentially dense)
(6.13.6.2)
and
iow(Rq,LT) almost lower semi-Fredholm
^ S T) (6.13.6.3)
or co\(Ls,RT) almost upper semi-Fredholm
implies
(S almost upper semi-Fredholm or T almost open)
and (S bounded below or T almost lower semi-Fredholm) (6.13.6.4)

234
6. Finite Dimensional Spaces and Compactness
Proof: If (wn) in S 1(0) and (gm) in (T*) 1(0) are linearly independent,
then
(gm ®wn) in col(L5,jRr)-1(0) is linearly independent (6.13.6.5)
and
{dm ® wn) m (row^S'^r)*)-1^) *s linearly independent (6.13.6.6)
where gm <g> wn : R —► ((7mJRiun) is the linear functional introduced in
(2.9.2.6) immediately following our first encounter with the finite rank
operator gm 0 wn in (2.9.2.4). This already proves (6.13.6.1). If we can find
either an infinite linearly independent sequence (wn) and a single nonzero
(gn) =5l5ora single nonzero (wn) = w1 and an infinite linearly
independent (<7m), then the infinite sequence (gm Qwn) will mean that co\(Ls,RT)
is not essentially one-one, while the infinite sequence (gm <g> wn) will mean
that iow(Rs,LT) is not essentially dense. If we only assume the failure of
the condition (6.13.6.4) then we must work harder, and construct sequences
{gn 0 wn) and (gn ® wn) with the help of Theorem 6.12.3 for which
\\Sn © "nil = \\9n ® Wn\\ = 1 for each "> (6.13.6.7)
and
\\9n ®Wn-9m® Wm\\ > 1 and K ®wn-9m® Wm\\ >1 if n + m
(6.13.6.8)
and
||col(L5,jRr)(0n 0 wn)\\ + ||[gn ® iyn)row(JR5,Lr)|| —»• 0 as n —► oo
(6.13.6.9)
There are of course two cases. If S is not almost upper semi-Fredholm then
we can find (wn) and (hn) for which, as in (6.12.3.1),
oo
IK|| = 1 and hn(wm) = 6nm and ^\\Swn
n=l
and
||ffj| = l and ||ffnr||—>0
incorporating the assumption that T is not almost open. Both (6.13.6.7)
and (6.13.6.8) are clear. Since \\hn\\ > 1, it follows that the sequence {Swn)
is absolutely summable and therefore converges to 0, giving (6.13.6.9). If
instead T is not lower semi-Fredholm and S not bounded below, then find
ll^nll < °°
(6.13.6.10)

6.13 Composition Operators 235
(wn), (gn) and (y) for which, as in (6.12.3.2),
oo
lkll = l and 9n(ym) = 6nm and £ ||ff„r||||yB|| < oo
n=1 (6.13.6.11)
and
|K|| = 1 and ||5u;B||—»0
The reader can again verify (6.13.6.7), (6.13.6.8), and (6.13.6.9). ■

7
Operator Algebra and Commutivity
When Y = X the space of operators BL(X, Y) is a normed algebra in the
sense of Definition 1.10.1. Much of the theory of BL(X, X) can be extended
at no extra cost to more general normed algebras A:
7.1 COMMUTANTS AND DOUBLE COMMUTANTS
If A is a normed linear algebra, or more generally a ring, then elements
a, 6 G A are said to commute iff
ba = ab (7.1.0.1)
7.1.1 DEFINITION If K is a subset of the ring A then its commutant
is the set
comm(K) = commA{K) = {a G A:ba = ab for each 6 G K} (7.1.1.1)
and its double commutant is the set
comm2(jFC) = comm^(jFC) = comm(comm(JK')) (7.1.1.2)
If a G A is an element we shall write
comm(c) = comm({a}) (7.1.1.3)
Commutants and double commutants have properties analagous to the
poles and polars of Theorem 5.2.2:
237

238
7. Operator Algebra and Commutivity
7.1.2 THEOREM If A is a normed linear algebra and K C A, then
comm.{K) is a closed subalgebra of A. If also K' C A there is implication
K Ccomm2{K) (7.1.2.1)
and
Ka'=> comm(K') C comm(K) (7.1.2.2)
Hence also
comm(comm2(K)) = comm(K) (7.1.2.3)
Proof: This is left to the reader. ■
Notice that if a G A, then
commA(a) = {La - flj"1^) (7.1.2.4)
is the null space of the linear operator La — Ra : A —> A of (2.10.0.1) and
(2.10.0.2). A subset K C A is called commutative if (7.1.0.1) holds for
every pair a, 6 G if. Necessary and sufficient is that
K C comm(K) (7.1.2.5)
or equivalently that
comm2(K) C comm(K) (7.1.2.6)
If K = A, then its commutant comm(A) is called the center of A. For
example, the algebras ^(ft) and (7^(17) are always commutative, while
BL(X, X) is about as noncommutative as it is possible to be:
7.1.3 THEOREM If fi is a topological space and A = C^fi), then
commA(A) = A (7.1.3.1)
If X is a normed space and A = BL(X,X), then
KL0(X,X) C K C A => commA(K) = K/ (7.1.3.2)
Proof: The implication (7.1.3.1) is clear. Towards (7.1.3.2) suppose T G
BL(X,X) commutes with KL0(X,X), so that for arbitrary x G X and
/ G X* we have
/ 0 (Tx) = T o (/ 0 x) = (/ 0 x)T = (/ o T) 0 x (7.1.3.3)
It follows
/(y)Tx = /(Ty)x for each y67 (7.1.3.4)

7.1 Commutants and Double Commutants
239
If, in particular, X ^ 0, then by Theorem 5.4.1 there are / G X* and y G X
for which f(y) = 1, and now (7.1.3.4) gives
T = kl with k = f{Ty) (7.1.3.5)
This proves (7.1.3.2) if X ^ 0, and the reader can check that it remains
valid when X = 0. ■
We recall the subgroup A-1 of invertible elements of A introduced
following (1.10.1.8). By the arguments of Theorem 3.1.2 and Theorem 3.1.3
we have implication, for a, 6 G A,
{a, 6} C A-1 <^=> {6a, ab} C A-1 (7.1.3.6)
and
1 - ba G A-1 <^=> 1 - a6 G A-1 (7.1.3.7)
and also
ab G A"1 =^(aG A-1 <^=> 6 G A-1) (7.1.3.8)
We recall also the Definition (1.10.1.7) of /(a) G A when a £ A and / :
K —► K is a polynomial. Alternatively this can be presented "inductively,"
by setting
z(a)=a and t(a) = t for each t G K (7.1.3.9)
and requiring that if g and h are polynomials then
(g + h){a) = g{a) + h(a) and (a • h){a) = g(a)h(a) (7.1.3.10)
Thus to prove a proposition about f(a) it is sufficient to verify it for the
constants t and the coordinate z, and to prove that it is transmitted to
sums and products. For example, we claim that f(a) G A is always in the
"double commutant" comm^(o):
7.1.4 THEOREM If A is a normed linear algebra and a G A, then there
is implication
a G A"1 => a"1 G comm^(a) (7.1.4.1)
If / : K —► K is a polynomial, then also
f{a) Gcomm^(a) (7.1.4.2)
and there is inclusion
/commA(a) C commA(a) C commA /(a) (7.1.4.3)

240
7. Operator Algebra and Commutivity
and
commA f(a) C commA(a) C commA /(a) (7.1.4.4)
Proof: Suppose a G A-1 is invertible; then if 6 G commA(c) we have
a~lb = a~l{ba)a~l = a~l{ab)a~l = 6a"1 (7.1.4.5)
This proves (7.1.4.1). Also, (7.1.4.2) is clear if the polynomial / is either
the coordinate z or one of the constants t G K, and if (7.1.4.2) holds when
f = g and when f = h then it must also hold when f = g + h and when
/ = g • h. By the remarks following (7.1.3.10) this proves (7.1.4.2) for all
polynomials /. Towards (7.1.4.3) observe that if a, 6 G A are arbitrary,
then
6 G commA(a) <£=> a G commA 6 (7.1.4.6)
The second inclusion of (7.1.4.3) now follows: by (7.1.4.2) and (7.1.4.6)
6 G commA(a) => /(a) G commA(6) => 6 G commA /(a) (7.1.4.7)
Now also the first inclusion follows: using (7.1.4.7)
6 G commA (a) => a G commA (6)
=> a G commA /(6) => /(6) G commA(a) (7.1.4.8)
Towards (7.1.4.4), the first inclusion follows from the second inclusion of
(7.1.4.3) together with (7.1.2.2), while the second follows from (7.1.4.2),
(7.1.2.6), and (7.1.2.2). ■
We conclude by noting, via Zorn's lemma, that every commutative
subset of a ring A is contained in some maximal abelian subset, which will
necessarily be a subring, and a closed subalgebra if A is a normed algebra.
The reader may like to verify that if K C A is a nonempty commutative
subset then
commA(jFC) = I) {B D K: B is a maximal abelian subalgebra of A}
(7.1.4.9)
and
comm^jK") = [j {B D K: B is a maximal abelian subalgebra of A}
(7.1.4.10)

7.1 Commutants and Double Commutants
241
7.2 MAXIMAL IDEALS AND THE RADICAL
We recall the "two-sided ideals" J C A of (1.10.1.5): more generally a left
ideal J C A is a linear subspace J for which
A-JCJ (7.2.0.1)
while a right ideal J C A satisfies instead
J -AC J (7.2.0.2)
A proper ideal J is of course one for which
J ^ A (7.2.0.3)
For example, if a G A, the set Aa = #A(A) is a left ideal, proper iff
a e aleft(A), while the set L'1^) = {b e A:ab = 0} is a left ideal, proper
iff a G 7rleft(A). A "maximal ideal" will mean a maximal proper ideal:
7.2.1 DEFINITION A maximal left ideal of the ring A is a proper left
ideal J C A for which there is implication, for arbitrary left ideals Jf C A,
J C J' => J = J' or J' = A (7.2.1.1)
Maximal right, and maximal two-sided ideals are defined in the same way.
We write
MLI{A) = {J C A: J is a maximal left ideal} (7.2.1.2)
and
MRI(A) = {J C A: J is a maximal right ideal} (7.2.1.3)
An easy application of Zorn's lemma shows that there are lots of
maximal left and maximal right ideals:
7.2.2 THEOREM If A is a ring and J C A is a proper left, right, or two-
sided ideal, then there is a maximal left, right, or two-sided ideal MCA
for which J C M.
Proof: We recall our convention that a ring A always contains an identity
1. Then it is clear that an ideal J C A is proper if, and only if, 1 ^ J.
Now we claim that, relative to set inclusion, the proper left ideals of A
satisfy Zorn's condition (1.11.1.2) since the union UAeAJA of any totally
ordered family («/A)AeA of proper left ideals is a proper left ideal. Thus
Theorem 7.2.2 for left ideals follows from Theorem 1.11.3. The argument
for right and for two-sided ideals is the same. ■

242
7. Operator Algebra and Commutivity
If A-1 is open (in particular, if A is complete), then maximal ideals
are always closed: for if M C A is a maximal left or maximal right ideal
then there is 6 > 0 for which, if x G A is arbitrary,
||x|| < 6 => 1 + x G A"1 => 1 + x £ M (7.2.2.1)
giving
1 £ cl(Af) DM=>MC cl(Af) ^ A => M = cl(Af) (7.2.2.2)
We can use maximal ideals to test for left or right invertibility in A.
7.2.3 THEOREM If A is a ring, then
<rleft(A) = (J MLI(A) and aright(A) = (J MRI(A) (7.2.3.1)
Also
.{aGA:l-AaC A"1} = {a G A: (1 - Aa) n aleft(A) =0} = f| MLI(A)
(7.2.3.2)
and
{aGAil-aACA-^laG A: (1 - aA) n aright(A) = 0} = |°| MRI(A)
(7.2.3.3)
Hence also
P| MLI(A) = P| MRI(A) is a two-sided ideal (7.2.3.4)
Proof: If a G A is in aleft(A), then Aa is a proper left ideal and so by
Theorem 7.2.2 there is M G MLI(A) for which a £ Aa C M. Conversely, if
a G M G MLI(A) then l^MD Aa, and hence, a G aleft(A). This proves
the first part of (7.2.3.1), and the argument for the second part is identical.
Towards (7.2.3.2) suppose a g PlMLI(A), so that a g M for some maximal
left ideal MCA; then M + Aa is a left ideal of A for which M g M + Aa.
By (7.2.1.2) it follows that M + Aa = A, and hence there is 6 G A for which
1-6-aGM Caleft(A) (7.2.3.5)
Thus, 1 — Aa is not disjoint from aleit(A). Conversely, if this is so, with
6 G A satisfying (7.2.3.5), then certainly a £ M, since otherwise
1 = (1 - 6a) + ba G M (7.2.3.6)
Thus, a ^ flMLI(A). This proves the second equality in (7.2.3.2). Towards
the first, it is clear that the first set in (7.2.3.2) is a subset of the second. We

7.2 Maximal Ideals and the Radical
243
have to reverse the inclusion. If, therefore, 1 — Aa is disjoint from aleft(A),
then for each 6 £ A there is b' £ A for which
6'(1 -ba) = l (7.2.3.7)
so that
b' = 1 - ca with c = -b'b (7.2.3.8)
and hence, by (7.2.3.7) there is also c' G A for which
c'bf = c'(l - ca) = 1 (7.2.3.9)
As is very familiar (Theorem 3.1.2), it follows that c' = 1 — ba = (6')"1
making 6' a two-sided inverse for 1 — ba G A-1. This finishes the proof of
(7.2.3.2), and (7.2.3.3) is exactly the same. Now (7.2.3.2) and (7.2.3.3)
together with (7.1.3.7) give the equality (7.2.3.4), and then force the common
set to be both a left and right ideal. ■
We shall write
n MLI(A) = n MRI(A) = Radical(A) (7.2.3.10)
and refer to the elements a £ Radical(A) as the radical elements of A. From
Theorem 7.2.3 we can extract four apparently different characterizations of
a radical element.
7.2.4 THEOREM If H is a topological space, then
Radical CTO(n) = {0} (7.2.4.1)
If X is a normed linear space, then
Radical BL(X, X) = {0} (7.2.4.2)
Proof: If 0 ^ a G C^H), then there must be t G H for which a(t) ^ 0,
and now
6 = l/a{t) GKC CTO(n) =J> (1 - 6a) {t) = 0 (7.2.4.3)
which means by (3.12.5.1) that 1 — ba is not invertible, proving (7.2.4.1).
If 0 ^ T e BL(X,X) then there must be x e X for which Tx ^ 0, and
then by Theorem 5.4.1 there is / G X1" for which f{Tx) = 1. Now in the
notation (2.9.2.4)
5 = /0x=^(J- ST){x) =0^x (7.2.4.4)

244
7. Operator Algebra and Commutivity
which means by (3.2.7.2) that I-ST is not invertible, proving (7.2.4.2). ■
If a G Radical(A) then it is clear from (7.2.3.2) that
A_1+aC A-1 (7.2.4.5)
Conversely, the reader may like to verify that (7.2.4.5) implies a G
Radical (A) provided
A~l + A~l = A (7.2.4.6)
Through the medium of the radical, we have a process of enlarging
two-sided ideals to their "inessential hulls":
7.2.5 DEFINITION If A is a ring and J is a two-sided ideal of A then
Hull(J) = {aeA:a + Je Radical(A/J)} (7.2.5.1)
Evidently, Hull (J) is a two-sided ideal of A, with
J C Hull(J) (7.2.5.2)
and
J C J' => Hull(J) C Hull(J') (7.2.5.3)
For example
Radical(A) = Hull({0» (7.2.5.4)
From (7.2.3.2) and (7.2.3.3) it is clear that each of the following conditions
is equivalent to having a G Hull (J):
for each 6 G A there is c G A for which 1 - c(l - 6a) G J (7.2.5.5)
for each 6 G A there is c G A for which 1 - (1 - ab)c G J (7.2.5.6)
It turns out that J and Hull (J) give rise to exactly the same
"essentially invertible" elements of A:
7.2.6 THEOREM If A is a ring and J C A is a two-sided ideal then, if
ae A,
a+Je ale{t{A/J) <^> a + Hull(J) G aleft(A/Hull(J)) (7.2.6.1)
and
a + Je aright(A/J) <^> a + Hull(J) G aright(A/Hull(J)) (7.2.6.2)

7.3 Regularity
245
Hence also
a = Je {A/J) <^> a + Hull(J) G (A/ Hull(J))"1 (7.2.6.3)
and
Hull(Hull(J)) = Hull(J) (7.2.6.4)
Proof: If a + J has a left inverse in A/ J then it is cbvious that a + Hull(J)
has a left inverse in A/ Hull (J). Conversely, if a + Hull (J) has a left inverse
in Aj Hull (J) then there is 6 G A for which
1-bae Hull(J) (7.2.6.5)
and then applying (7.2.5.5) with 1 — 6a in place of a and 1 in place of 6
gives c G A for which
1 - c(6a) = 1 - c(l - (1 - 6a)) G J (7.2.6.6)
But this means that cb + J is a left inverse for a + J in A/J. This proves
(7.2.6.1), and similarly (7.2.6.2), and hence also (7.2.6.3). Finally,
combining (7.2.6.3) with (7.2.3.2), there is implication
a + Je Radical(A/J) <^=> a + Hull(J) G Radical(A/ Hull(J)) (7.2.6.7)
giving (7.2.6.4). ■
If A = BL(X,X) and J = KL0(X, X) is the ideal of finite rank
operators, then the inessential hull of J includes the compact operators:
7.2.7 THEOREM If X is a normed space, then
KL^X,*) C HullKL0(X,X) (7.2.7.1)
Proof: If T G KL^XjX) is compact and S G BL(X, X) is arbitrary, then
ST G KLX(X,X) is also compact, and by Theorem 6.9.3 it follows that
/ - ST is Fredholm . By Atkinson's Theorem 6.4.3 it follows that I - ST
has an invertible coset relative to the ideal of finite rank operators. ■
7.3 REGULARITY
The "regularity" of Definition 3.8.1. has an analogue in the style of the
"algebraic invertibility" of Definition 3.10.1, in which the generalized
inverse T' G BL(y, X), which satisfies (3.8.1.1), is also supposed to lie in
M C BL(y,X), as in (3.10.1.1). We give the analogue here for an element
of a ring.

246 7. Operator Algebra and Commutivity
7.3.1 DEFINITION If A is a ring, then a G A is said to be regular, or
relatively Fredholm, iff
a G aAa (7.3.1.1)
and is said to be decomposably regular, or relatively Weyl, iff
aeaA~la (7.3.1.2)
We shall write
r£= {ae A: a is regular} (7.3.1.3)
Theorems 3.8.3 and 3.8.4 are valid for the regular elements of a ring.
7.3.2 THEOREM If A is a ring and a, a! G A, then
a - aa'a G 'A1 <^> a e^ (7.3.2.1)
and
1 - a'a G ^ -£=> 1 - ad G ^ (7.3.2.2)
If also 6, 6' G A with a = aa'a and 6 = 66'6, then
a& G ^ <^> a'a66' G 'A1 (7.3.2.3)
If a = aa'a and if d G A satisfies
1 - a'd G A"1 and (1 - aaf)d{l - a;d)_1(l - a'a) G 'A1 (7.3.2.4)
then
a-de'A! (7.3.2.5)
Proof: This is exactly the same as the proofs of Theorems 3.8.3 and
3.8.4. ■
If J C A is a two-sided ideal then, as for (3.9.6.9),
JCrA,=>rA,+ JCrA1 (7.3.2.6)
We can prove more.
7.3.3 THEOREM If A is a ring and if J C A is a two-sided ideal
satisfying
JC'A1 (7.3.3.1)

7.3 Regularity
247
then for each a G A there is implication
a + Je {A/J)^ =^aerA! (7.3.3.2)
Also
A_1 + Hull(J) C'A1 (7.3.3.3)
Proof: If a + J is regular in Aj J, then there is a' G A for which
a - aa'a G J (7.3.3.4)
so that (7.3.3.1) and (7.3.2.1) give a G "A1, proving (7.3.3.2). Towards
(7.3.3.3) we claim that if a G A then
a + Hull(J) £ aleft(A/ Hull(J)) =>aerA! (7.3.3.5)
and
a + Hull(J) £ aright(A/ Hull(J)) =^aerA! (7.3.3.6)
For example, if a+Hull (J) has a left inverse in A/ Hull (J), then by (7.2.6.1)
a + J has a left inverse in Aj J, and hence, by (7.3.3.2), a E^. Since every
element of A-1 + Hull(J) has a two-sided inverse in A/Hull(J) it is clear
that (7.3.3.3) follows from (7.3.3.5). ■
Theorem 7.3.3 and Theorem 7.2.7 confirm what we already knew in
(6.9.3.3) the sum of an invertible operator and a compact operator is
regular. The decomposably regular elements can be decomposed into products
of invertible and idempotent elements:
7.3.4 THEOREM If A is a ring and
A = {ae A: a2 = a} (7.3.4.1)
is the set of idempotents of A, then
{a G A: a is decomposably regular} = A~lA = AA~l (7.3.4.2)
A left or right invertible decomposably regular element is invertible:
[A-1 A) \ aleft(A) = {A-1 A) \ aright(A) = A"1 (7.3.4.3)
If A is a Banach algebra, there is equality
rA1 n cl(A_1) = A'1 A (7.3.4.4)

248
7. Operator Algebra and Commutivity
Proof: Suppose a = aa'a with invertible a'\ then p = a'a G A and
a = (a')~1p G A-1 A. Conversely, if a = cp with c G A-1 and p G A,
then c~la = c~1ac~1a, and hence a = ac~1aaA~1a. This proves the first
equlaity in (7.3.4.2), and similarly the second. Towards (7.3.4.3) suppose
a = cp has a left inverse a', with invertible c and idempotent p. Then also
p = p2 has a left inverse, forcing p = 1 and a = c G A-1. This proves about
a quarter of (7.3.4.3). We leave the rest to the reader. Towards (7.3.4.4)
if A is a normed algebra and a = cp G A~lA then \\a — an\\ —► 0 with
an = c(p + (l/n)(l — p)) G A-1. Conversely, with no restriction on A, we
can follow the argument for Theorem 3.8.7. If a = aa'a G 'A1 and if 6 G A
satisfies
6 G A-1 and 1 + (6 - a)af G A-1 (7.3.4.5)
then
p = a'a and c = (l + (6 - a)a')~lb (7.3.4.6)
gives
a = cp (7.3.4.7)
Evidently p is idempotent and c in invertible. If A is complete and a G
cl(A_1) we can satisfy (7.3.4.5) by taking
be A'1 and ||6 - a\\ \\a'\\ < 1 ■ (7.3.4.8)
It is clear from the proof that (7.3.4.4) extends to those normed
algebras A for which A-1 is open, as in Theorem 4.4.7.
For a rather more special kind of regular element, only available in
BL(X, Y) when Y = X, we make the following:
7.3.5 DEFINITION If A is a ring, then a G A is called simply polar iff
a G aBa with B = commA(a) (7.3.5.1)
For example, invertible elements and idempotents are simply polar, as are
commuting products of invertibles and idempotents. The simply polar
elements are decomposably regular. If a = aa'a with a'a = aa' = p, then
a" = a' + (1 - p) => a = aa"a and (a")"1 = a + (1 - p) (7.3.5.2)
When A = BL(X,X) for a normed space X then we can very nearly
characterize the simply polar elements spatially:

7.3 Regularity
249
7.3.6 THEOREM KTG BL(X, X) for a normed space X is simply polar,
then
T is regular and T{X) = T2{X) and T'1^) = T~2{0) (7.3.6.1)
Conversely, if (7.3.6.1) holds, then T is simply polar if either X is complete,
or T is Fredholm, T is of finite rank.
Proof: If T is simply polar then it is certainly regular, and if T' G comm(T)
satisfies (3.8.1.1) then
T{X) = TT'T{X) = T2T'{X) C T2{X) C T{X) (7.3.6.2)
and
r-^o) = (rr'r)-1^) = (r'r2)-1^) = r-^r'-^o) 2 r-*(o) d r-^o)
(7.3.6.3)
Conversely, if (7.3.6.2) and (7.3.6.3) hold, then
T{X) + r_1(0) = X and T{X) n r_1(0) = {0} (7.3.6.4)
and hence there is a linear projection P : X —> X for which
P{X) = T{X) and P_1(0) = r_1(0) (7.3.6.5)
Explicitly
if x e X then Px = Ty where Tx = T2y and y G X (7.3.6.6)
Further, by the argument for Theorem 3.8.2., it is clear that if T is proper
and P is continuous then there is T' G BL(X,X) for which
T'T = TT' = P (7.3.6.7)
which makes the operator T simply polar. Now if the space X is complete
then the projection P is continuous by Theorem 4.8.1, if T is Fredholm
then I — P is continuous by Theorem 6.3.6, and if T is finite rank then P
is continuous by Theorem 6.3.6. ■
To see that the projection P may fail to be continuous look at X = c00
and consider the operator
T = : X2 —- X2 (7.3.6.1
I 0 0 I V

250
7. Operator Algebra and Commutivity
where for each iGl
(Wx)n = {l/n)xn for each n G N
(7.3.6.9)
Evidently W is one-one and onto, but not bounded below: the linear inverse
W is unbounded. The operator T is regular in the sense of Definition 3.8.1,
with (3.8.1.1) satisfied by the bounded operator
T' =
0 -I'
-I 0
Also, in the ring of all linear operators on X2 the conditions of
Definition 7.3.5 are satisfied by the unbounded operator
rpff _
W o
o W
(7.3.6.10)
so that the conditions (7.3.6.2) and (7.3.6.3) both hold. The projection P
of (7.3.6.6) is, however, not bounded. Specifically,
P rprpll rpllrp
i w'
0 0
(7.3.6.11)
The product of regular operators need not be regular. For example, let
X = lp with p e {l,2,oo} or X = c0, and recall the forward and backward
shifts of (2.8.2.2) and (2.8.2.3). Define U and V on X by setting for each
xeX
and
Evidently
U(x1,x2,x3, ) — (0,x1,£2, )
V(X1,X25X3' • • V = lx2'X3'X45 • • V
(7.3.6.12)
(7.3.6.13)
(7.3.6.14)
VU = I ^ UV
Thus, both U and V are regular and neither of them are invert ible. If also
W{xl9x29x39...) = (xl5 \x2, \xz,...) (7.3.6.15)
Then W is one-one and dense but not invertible, and is also compact but
not of finite rank, and therefore by Theorem 6.8.5 not regular. Now if
A = BL(X2,X2) then
0 0
W 0
eA=>a2=0erA!a.nda^rA!
(7.3.6.16)

7.4 Quasinilpotent Elements
251
since a generalized inverse for a in A would give rise to a generalized inverse
for W in BL(X,X). If instead t G K is sufficiently large then
0 0
U 0
tl W
W tl
e^
and
a2^
(7.3.6.17)
Indeed,
0 V
0 0
is a generalized inverse for
"0 0"
u o.
, and
'tl W~
W tl_
IS
invertible if |£| > \\W\\, which means that a is regular, while
9
a2 =
"0 0"
.u o.
" 0
W
W
0 .
'tl W
W tl.
—
0 0
UWU 0
tl W
W tl
(7.3.6.18)
and since U is also Fredholm on X the operator UWU is again compact
and not of finite rank, hence again not regular.
7.4 QUASINILPOTENT ELEMENTS
A quasinilpotent is nearly as noninvertible as it can be:
7.4.1 DEFINITION If A is a normed algebra, then a G A is said to be
quasinilpotent if
\\nn\\lln
0
and is said to be nilpotent if
(7.4.1.1)
(7.4.1.2)
0e{an:ne N}
Quasinilpotents are topological zero divisors:
7.4.2 THEOREM If A ^ {0} and a e A, then
a quasinilpotent => a a left and a right topological zero divisor
(7.4.2.1)
and
a nilpotent => a a left and a right zero divisor (7.4.2.2)
Proof: If 0 ^ a G A is nilpotent then there is m G N for which
am+1 = 0 ^ am (7.4.2.3)
which means that
Lau = Rau = 0^u with u = am (7.4.2.4)

252 7. Operator Algebra and Commutivity
If a = 0 then we may take u = 1. This proves (7.4.2.2), and also gives
the idea for the proof of (7.4.2.1). If 0 ^ a is quasinilpotent, then for each
sufficiently small e > 0 there is m G N for which
||am||l/m > e > ||am+l||l/(m+l) (7.4.2.5)
which means that
||Lat*|| = \\Rau\\ < e||u|| with u = am ■ (7.4.2.6)
Sums and products of commuting quasinilpotents are quasinilpotent:
7.4.3 THEOREM IfcGA and if 6 G A is quasinilpotent and commutes
with a, then
ab is quasinilpotent (7.4.3.1)
a quasinilpotent => a + b quasinilpotent (7.4.3.2)
1 — ab is almost invertible (7.4.3.3)
a almost invertible => a — b almost invertible (7.4.3.4)
Proof: Towards (7.4.3.1) suppose that 6 is actually nilpotent; then
{ab)m = ambm =0 if bm = 0 (7.4.3.5)
so that ab is also nilpotent. More generally
||(a6)m||1/m < ||a||||6m||1/m < e||a|| if ||6m||1/m < e (7.4.3.6)
This gives (7.4.3.1). If a and 6 are both nilpotent then so is a + 6:
2m
am = bm = 0 =► (a + bfm = Yi ( ) ay62m_y
= bmY,(2m\a?bm-i+am ^ (2m)ay-162m-1=0
More generally, if 0 < e < min(||a||, ||6||) and if
n > N => \\an\\ < en and ||6n|| < en (7.4.3.8)

7.4 Quasinilpotent Elements
253
then
„ > 2JV =H|(a + 6)"|| < £ (") 11^11116^11
(N 2N-1 n \ , v
£+ £ +£ (") 11^1111^11(7.4.3.9)
y=o y=jv+i j-2NJ ^J '
< (e + e) max I ,
This proves (7.4.3.2). Towards (7.4.3.3) suppose a G A is nilpotent; then
am+1 = 0 => (1 - a)(l + a + --- + am) = l = (l+a + --- + am)(l - a)
(7.4.3.10)
Thus
a nilpotent => 1 - a G A"1 (7.4.3.11)
More generally
limsup||an||1/n < 1=> ||l-(l+a + --- + an)(l-a)|| —>0 (7.4.3.12)
n
In particular
a quasinilpotent => 1 — a almost invertible (7.4.3.13)
Now (7.4.3.3) follows from (7.4.3.13) together with (7.4.3.1). Towards
(7.4.3.4), if a is actually invertible then a + 6 = a(l + a-16) is almost
invertible by (7.4.3.3). More generally, suppose that a is almost left
invertible, with
|| 1 - a'na\\ —► 0 and sup ||a'J| < oo (7.4.3.14)
n
Then for each m,n G N
/ m \ m
!-<( ! + DO'*')(«"*) = l-<a+^(a'J^(l-a'na)^+(a'J^16^1
(7.4.3.15)
Now if k = supn ||aJJ| then
oo
1 + ]T A;r||6r|| = k' < oo (7.4.3.16)
r=l

254
7. Operator Algebra and Commutivity
Then if 6 > 0 and e > 0 are arbitrary there is N G N for which
n > N => ||1 - a'na\\ < 6 and ||6n|| < en (7.4.3.17)
Now
mn,> N =► ||1 - a'n f 1 + X^K)r6r ) (a - 6)|| < A;'<5 = jfcm+1em+1
(7.4.3.18)
Since <5 and e are entirely arbitrary it is clear that a — b is almost left
invertible. This proves the analogue of (7.4.3.4) for almost left invertibility.
The argument for almost right invertibility is identical, and hence (7.4.3.4)
follows. ■
When the algebra A is complete, then almost invertibility implies
invertibility, and hence, if the algebra is also commutative then the quasinil-
potents all lie in the radical:
7.4.4 THEOREM If A is a normed algebra, then the quasinilpotents lie
in the closure of the almost invertible elements, and hence, if 0 ^ A is
complete, then
a quasinilpotent => a G d(A-1) (7.4.4.1)
If also A is commutative, then
a quasinilpotent => a G Radical(A) (7.4.4.2)
Proof: If a G A is quasinilpotent and O^iGK, then t — a = t(l — (l/t)a)
is almost invertible by (7.4.3.3), and by allowing t to be arbitrarily close to
0 we find
a quasinilpotent => a G cl {6 G A: b almost invertible} (7.4.4.3)
By (4.4.5.12) and (4.4.5.13) it follows that the almost invertible elements
of A are in A-1 when A is complete, giving (7.4.4.1). Finally, if A is also
commutative, then (7.4.3.1), (7.4.3.13), and (7.2.3.2) give (7.4.4.2). ■
We might remark that (7.4.3.12) gives an improvement on (4.4.5.15)
for complete algebras A. The reader may like to verify that the following
condition is necessary and sufficient for a G A to be quasinilpotent:
||(*a)n||—>0 for each teK (7.4.4.4)

7.4 Quasinilpotent Elements
255
It then follows that if a G A is quasinilpotent, and m £Y\ and 61,62,...,
6m,c1,...,cmGA are arbitrary, then also
\\bm • • • &£6?ancnc?c£ • • • Cll —► ° ™ n —> °° (7.4.4.5)
At the other extreme from quasinilpotents and radical elements, we shall
call an element a G A conservative iff there is k > 0 for which
IKH > A^HI* for each n G N (7.4.4.6)
We conclude here with an introduction to Lomonosov's lemma:
7.4.5 THEOREM If X is a normed space and K G BL(X, X) is compact,
quasinilpotent and nonzero, then it has a nontrivial hyperinvariant closed
subspace: there is a closed subspace W C X with {0} ^ W ^ X for which,
ifTGBL(X,X),
TK = KT => T{W) C Py (7.4.5.1)
Proof: We shall show that if if is compact with ||jK"|| = 1 and has no
nontrivial hyperinvariant closed subspaces then it cannot be quasinilpotent.
Choosing x0 G X for which ||ifx0|| > 1, so that also ||x0|| > 1, write
U0 = Disc(x0; 1) and V0 = cl K{U0) (7.4.5.2)
Evidently U0 and V0 are both closed, neither of them contain 0 G X, and
V0 is compact in the full sense of (6.6.0.1). Suppose now 0 ^ x G X: then
Wx = cl {Tx: T G comm(K)} = X (7.4.5.3)
since evidently Wx ^ {0} satisfies (7.4.5.1). In particular x0 G Wz if
0 ^ x G X, which means
I\{0}CU {r-^int^o)):T G comm(K)} (7.4.5.4)
The right-hand side of (7.4.5.4) is therefore an open cover for the compact
set V0. By (6.6.0.1) there is a finite set of operators H C comm(jFC) for
which
VoCUlT-^intiUjy.TeH} (7.4.5.5)
Inductively we may pick a sequence (Tn) in BL(X, X) for which, for each
neN,
TneH and TnK • • • T2KTxKx^ G U0 (7.4.5.6)
Indeed there is I\ G # for which T^Xq G C/0, giving KT-^Kxq G V0, and

256
7. Operator Algebra and Commutivity
then T2eH for which ^(KT^Xq) G U0, and so on. If we write
m= inf llxll and M = max||r|| (7.4.5.7)
then, remembering that H C comm(jFC), we have
m < \\TnK... T^T.KxoW = \\KnTn ... T2Txx0\\
< ||Kn|||Mn|||x0|| for each n G N (7.4.5.8)
which means that
||#n||1/n > ^(iM)1^ —^ m as n —+ oo (7.4.5.9)
Since m > 0, K cannot be quasinilpotent. ■
7.5 POLAR AND QUASIPOLAR ELEMENTS
Begin by recalling the idempotents p = p2 G A of a normed linear algebra
as in (7.3.4.1): if p G A, then
pAp = {pap: a G A} = {a G A:a = ap = pa} (7.5.0.1)
is a normed linear algebra in its own right, and also a closed subspace of
A, but (unless p = 1) not a subalgebra in the sense of (1.10.1.4), since the
identity of pAp is p and not 1. The commutant of an idempotent is the
direct sum of two algebras:
commA(p) = pAp + (l — p)A(l — p)
and (7.5.0.2)
pAp fl (1 - p)A(l - p) = {0}
When a G A commutes with p G A we shall say that it is reduced by p. For
example, if P = P2 G BL(X,X) then Theorem 2.5.3 tells us that
comm(P) = {T G BL(X,X):T{PX) C PX and r(P_10) C P^O}
(7.5.0.3)
If p = p2 E A then we shall say that a G A is invertible relative to p iff
pap G (pAp)-1 (7.5.0.4)
with similar conventions for left and right invertibility, and for almost
invertibility. When a is reduced by p then invertibility in A can be tested in
pAp and (l — p)A(l — p):

7.5 Polar and Quasipolar Elements
257
7.5.1 THEOREM If a G A commutes with p G A, then
a e A"1 <t=> pap G [pAp)~l and (l - p)a(l - p) G ((1 - p)A(l - p))"1
(7.5.1.1)
Proof: Suppose ba = 1 with 6 G A: then
{pbp)(pap) = pbap = p (7.5.1.2)
so that pap has a left inverse in pAp, and similarly (l — p)a(l — p) in
(1 — p)A(l — p). Conversely, if
b'ap = p and 6"a(l - p) = (l - p) (7.5.1.3)
then
1 = p + (1 - p) = (b'p + 6"(1 - p))a (7.5.1.4)
so that a has a left inverse in A. This proves the analogue of (7.5.1.1) for
left inverse. The argument for right inverse in identical, and together they
give the argument for two-sided inverses. ■
When a commutes with p then the condition (7.5.0.4) is equivalent to
p G (Aa) n (aA) (7.5.1.5)
A "polar" element of a ring or algebra is an element a G A for which one can
find an idempotent p = p2 G commA (a) such that a is invertible relative to
p and nilpotent relative to 1 — p:
7.5.2 DEFINITION If A is a normed algebra then a G A is called almost
quasipolar if there is p G A for which
p = p2 and ap = pa (7.5.2.1)
with
p G cl(Aa) n cl(aA) (7.5.2.2)
and
||an(l-p)||1/n—>0 asn—> oo (7.5.2.3)
If in addition
p G (Aa) n (aA) (7.5.2.4)
then a is said to be quasipolar. If also
0e{an(l-p):ne N} (7.5.2.5)
then a is said to be polar.

258
7. Operator Algebra and Commutivity
For example T G BL(X,X) is quasipolar iff there is commuting P =
P2 G BL(X,X) for which T has an invert ible restriction to P(X) and a
quasinilpotent restriction to P_1(0); also <f> G C^Q) is quasipolar iff the
set <£_1(0) is open as well as closed, with inf^tw0 \<f>(t)\ > 0. In general, in-
vertible, idempotent, and quasinilpotent elements are all quasipolar, while
an almost invertible element is almost quasipolar. The simply polar
elements of Definition 7.3.5 are just the polar elements for which (7.5.2.5) can
be improved to
a = ap (7.5.2.6)
More generally, the reader can verify that
a polar <£=> an simply polar for some n G N (7.5.2.7)
If a G A is quasipolar then the idempotent p of Definition 7.5.2 is
unique, and lies in the double commutant of a:
7.5.3 THEOREM If A is a normed algebra and a G A is quasipolar, then
the idempotent p = a of Definition 7.5.2 is unique and lies in comm^(a),
and there is 6 G A with
ab = ba = p and b = bp = pb (7.5.3.1)
If (7.5.3.1) holds then b = ax is unique and lies in comm^(c)
Proof: We begin by verifying (7.5.3.1). From (7.5.2.4) there are u,v G A
with p = ua = av, giving p = {pup) (pap) = [pap)(pvp), and hence, pup =
pvp = b say. Now if also q = q2 satisfies the conditions (7.5.2.1), (7.5.2.4),
and (7.5.2.3), with ac = ca = q and cq = qc = c, then we have
p - pq = pn(l -q)= bnan{l - q) —>0 asn —>oo (7.5.3.2)
and
q-pq = (l- p)qn = (l - p)ancn —> 0 as n —> oo (7.5.3.3)
Thus, p — pq = q is unique. Also if w G commA(a) then
wp — pwp = (1 — p)wp = (1 — p)wpn = (1 — p)wanbn
(7.5.3.4)
= (1 — p)anwbn —> 0 as n —► oo
giving wp = pwp and similarly pwp = pw. The uniqueness of the element
b = ax satisfying (7.5.3.1) is just the uniqueness of inverse (3.1.2.2) for

7.5 Polar and Quasipolar Elements
259
{pap) x G pAp. Finally, if w G commA(o) then, using the fact that pw =
wp,
wb = wpb = bawp = bwap = bwp = bwp = bpw = bw ■ (7.5.3.5)
If a G A is almost quasipolar we shall call p = a the support of a, and
if a G A is polar then the element b = ax is known as the Drazin inverse of
a. The perturbation theory for polar and quasipolar elements is suggested
by the perturbation theory for quasinilpotents:
7.5.4 THEOREM If a G A is quasipolar and b G A commutes with a,
then
b quasipolar =>► ab quasipolar (7.5.4.1)
b quasinilpotent =>► a + b almost quasipolar (7.5.4.2)
b and 1 + axb almost invertible => a + b almost invertible (7.5.4.3)
If, in particular, a is polar and b commutes with a, then
b polar => ab polar (7.5.4.4)
b nilpotent =>► a + b polar (7.5.4.5)
b and 1 + axb invertible =>• a + b invertible (7.5.4.6)
Proof: If a and b are quasipolar and commute then by the doubly
commuting components of Theorem 7.5.3 the set
I a,6,a,6,ax,6x > is commutative (7.5.4.7)
In particular a and b commute, and hence
ab G A is idempotent (7.5.4.8)
To prove (7.5.4.1) we claim that ab is the support for ab. The condition
(7.5.2.1) follows from (7.5.4.7) and (7.5.4.8), then condition (7.5.2.4) follows
from
(6X ax)ab = ab = ab{bx ax) (7.5.4.9)
and finally the condition (7.5.2.3) comes from the observation that
ab{l - ab) = [aa)(6(1 - b)) + [bb) (a(l - a)) + (a(l - a)) (6(1 - 6)) (7.5.4.10)
is by (7.4.3.1) the sum of three commuting quasinilpotents, and therefore
quasinilpotent by (7.4.3.2). If, in particular, a, 6 are polar then a6(l — ab) is

260
7. Operator Algebra and Commutivity
nilpotent by (7.4.3.5) and (7.4.37), giving (7.5.4.4). To prove (7.5.4.2) note
that (a + b)a is almost invertible in aAa by (7.4.3.4), and that (a + 6)(l —d)
is quasinilpotent by (7.4.3.2). If, in particular, a is polar and b is nilpotent
then (a-\-b)a is invertible in aAa by (7.4.3.11) and (a+b)(l — a) is nilpotent
by (7.4.3.7), giving (7.5.4.5). To prove (7.5.4.3) we have
(1 + axb)a = ax(a + b)a (7.5.4.11)
so that (a + b)a is almost invertible in aAa by (3.7.3.1) or (3.7.3.2), while
(a+ 6)(1 — a) is almost invertible in (1 — a)A(l — a) by (7.4.3.4), and hence
a + b is almost invertible in A by the analogue of (7.5.1.1). If in particular
a, b and 1 + axb are invertible then (7.1.3.6), (7.4.3.11), and (7.5.1.1) give
(7.5.4.6). ■
In general we cannot expect implication:
ab = ba polar => a almost quasipolar (7.5.4.12)
since the left-hand side holds with b = 0 for arbitrary a G A. When A is
complete then the almost quasipolar elements are quasipolar, which offers
some simplification in Theorem 7.5.4:
7.5.5 THEOREM If A is a Banach algebra and a G A, then
a almost quasipolar => a quasipolar (7.5.5.1)
If a G A is quasipolar and b G A commutes with a then
b quasinilpotent =>► a + b quasipolar (7.5.5.2)
and
a quasipolar b and 1 + axb invertible => a + b invertible (7.5.5.3)
Proof: By (4.4.5.12) and (4.4.5.13) the almost invertible elements of the
algebra pAp are invertible in pAp whenever p G A is an idempotent in A,
so that the conditions (7.5.2.2) and (7.5.2.4) are equivalent when ap = pa.
This gives (7.5.5.1), which together with (7.5.4.2) gives (7.5.5.2). Finally,
(7.5.5.3) is clear from (7.5.4.3) together with (4.4.5.12) and (4.4.5.13). ■
The second part of the condition on the left-hand side of (7.5.4.3) is
satisfied if
II«X|HHI<1 (7-5.5.4)

7.6 Homomorphisms and Fredholm Theory 261
in particular by taking 0 ^ b G K we can improve (7.4.4.3) to
a quasipolar => a G cl {b G A: b almost invertible} (7.5.5.5)
so that if A is complete then
a quasipolar => a G cl(A_1) (7.5.5.6)
When a G A is quasipolar then its Drazin inverse is regular, since a is itself
a generalized inverse for a:
ax =ax.a-ax =^ ax G T (7.5.5.7)
As we saw in (7.3.6.16) however it is possible for a G A to be polar but not
regular.
7.6 HOMOMORPHISMS AND FREDHOLM THEORY
If T G HBL(A,B) is a homomorphism in the sense of (2.10.1.7), so that
T(l) = 1 and T(ab) = T(a)T(b), then there is inclusion
T(A_1) C B'1 (7.6.0.1)
By analogy with Atkinson's Theorem 6.4.3, we may regard the elements of
T-1(B~1) as another kind of "Fredholm" element:
7.6.1 DEFINITION If T G HBL(A,5) is a homomorphism of normed
algebras, then a G A will be called T-Fredholm iff
T{a) G B~l (7.6.1.1)
and almost T-Fredholm iff T(a) is almost invertible in B. Iff
aeA'1 +T~1{0) = {c + d:ceA-\T{d) = 0} (7.6.1.2)
then we shall call a T- Weyl .
If T G BL(A, B) is a homomorphism then
r_1(0) is a closed two-sided ideal of A (7.6.1.3)
proper iff B ^ {0}. Conversely, if J is a closed two-sided ideal then the
quotient
JA : A —> A/J (7.6.1.4)
is a homomorphism. Evidently the "Weyl" elements are unchanged if the
homomorphism T is replaced by the quotient JA with J = T_1(0), although

262
7. Operator Algebra and Commutivity
the T-Fredholm elements may be more general than the JA-Fredholm
elements: this is familiar when, for example, the homomorphism T is one-one,
or isometric.
If homomorphism T is induced by an ideal J which satisfies the
regularity condition (7.3.3.1) then much of the "spatial" Fredholm theory carries
over:
7.6.2 THEOREM If J C ^ is a regular two-sided ideal of a normed
algebra A, then a G A is JA-Fredholm iff
aG^ and L'1^) C J and R'1^) Q J (7.6.2.1)
Whether or not the ideal J is regular, if instead
1 + JCA-1! (7.6.2.2)
then a G A is JA-Weyl iff
JA{a) G {A/J)'1 and a G A'1 A (7.6.2.3)
Proof: The conditions (7.6.2.1) are as in the characterization
Theorem 6.4.2, and to prove the first part of Theorem 7.6.2 we have only to
repeat the argument for Atkinson's Theorem 6.4.3, using (7.3.3.2).
Towards (7.6.2.3) suppose a = cp is Fredholm with invertible c G A-1 and
idempotent pGi: then p = p2 is Fredholm, so
1-peJ (7.6.2.4)
and hence a = cp G c(l + J) C A~l + J. Conversely, if (7.6.2.2) holds, then
A-l + J<ZA~lA ■ (7.6.2.5)
Theorem 7.6.2 characterizes the Fredholm and the Weyl elements when
A = BL{X,X) and J = KL0(X,X) (7.6.2.6)
As we have said, we proved (7.6.2.1) in Atkinson's Theorem 6.4.3, while the
condition (7.6.2.2) was part of Theorem 6.5.2. The same characterizations
are valid when
A = BL{X,X) and J = KL1{X,X) (7.6.2.7)
using the fact (7.2.7.1) that the compact operators lie in the essential hull
of the finite rank operators. Indeed, Theorem 7.3.3 says that (7.6.2.1) is
valid whenever
J C Hull(J0) and J0 C ^ (7.6.2.8)

7.6 Homomorphisms and Fredholm Theory
263
which applies if J = KL1(X,X), while the condition (7.6.2.2) follows from
Theorem 6.10.4, which says that everything in 1 + J has index zero, and
Theorem 6.5.6, which tells us that Fredholm operators of index zero are
Weyl, and therefore decomposably regular as in Theorem 6.5.2.
If T = Jv :A-+A/J with J C 'A1, then we can weaken the conditions
for (7.3.2.5). If a = aa'a G *A! and if d G A satisfies either
T{l-afd)e{A/J)~1
and (7.6.2.9)
T((l - aa')d){T{l - a'd))-lT{l - a'a) G [A/Jf
T{l-daf) e{A/J)-1
and (7.6.2.10)
T(l - a'a){T{l - da!))-lT(d(l - a'a)) G (A/Jp
then (7.3.2.5) says that T(a - d) G (-A/J)r~1 and now (7.3.3.2) says that
a — d G 'A1.
The analogue of Theorem 6.4.4 extends to T-Fredholm elements:
7.6.3 THEOREM If T G HBL(A, B) is a homomorphism of normed
algebras and a,b G A then
a, b T-Fredholm ^=> ba, ab T-Fredholm (7.6.3.1)
and
ab T-Fredholm =>► (a T-Fredholm <=>> b T-Fredholm) (7.6.3.2)
and
1 - ab T-Fredholm => 1 - ba T-Fredholm (7.6.3.3)
If in particular a = aa'a G *A! then
a T-Fredholm => a' T-Fredholm (7.6.3.4)
and
a T-Fredholm,l + a'b T-Fredholm =>► a + b T-Fredholm (7.6.3.5)

264 7. Operator Algebra and Commutivity
Proof: The first three implications follow from (7.1.3.6), (7.1.3.8), and
(7.1.3.7) applied to the elements T(a), T{b) in B. For (7.6.3.4) observe
that
T{a) = T{a)T{a!)T{a) G B~l =>► T{a') G B~l (7.6.3.6)
Finally, for (7.6.3.5) argue
T{a')T{a + b) = T(l + a'b) G B~l
=^{T{a!)eB-1 =^T{a + b)eB~l) m (7.6.3.7)
The analogue of Theorem 6.5.3 extends to T-Weyl elements:
7.6.4 THEOREM If T G HBL(A, B) is a homomorphism of normed
algebras and a, b G A, then
a, b T-Weyl =>► ab T-Weyl =>► (a T-Weyl <^> b T-Weyl) (7.6.4.1)
If, in particular, a = aa'a G 'A1, then
a T-Weyl =>► a' T-Weyl (7.6.4.2)
and
a T-Weyl, 1 + a'b T-Weyl =>► a + b T-Weyl (7.6.4.3)
and
a T-Weyl, 1 + bo! T-Weyl =^ a + 6 T-Weyl (7.6.4.4)
Proof: If a = c + tt and 6 = d + v with invertible c,d and u,v in T_1(0)
then
a6 = cd + (ci; + ud + uv) G A"1 + T_1(0) (7.6.4.5)
giving the first implication of (7.6.4.1). If instead a = c + u and ab = d + v,
then
b = c~ld + c_1(t; - ub) G A'1 + T_1(0) (7.6.4.6)
giving half of the second implication of (7.6.4.1), and the other half is
similar. If a = aa'a = c + u with c G A~l and T(u) = 0 then
a! = c~l + c~\d - da'c - ca'd - da'd)c~l G A'1 + T"1^) (7.6.4.7)
giving (7.6.4.2). Finally, for (7.6.4.3) we have
a'{a + b) = {l + a'b)-{l-a'a) G (A"1 -hT"1^)) -hT"1^) = A^+T'1^)
(7.6.4.8)

7.7 Browder Operators
265
so that a + b is T-Weyl by (7.6.4.2) and the second part of (7.6.4.1). The
argument for (7.6.4.4) is the same. ■
There can be no analogue of the reverse implication in (7.6.3.1) for
Weyl operators. For example if X = lp and A = BL(X2,X2) with J0 =
KL0(X2,X2) and Jx = KL^X2,*2), and if U and V are the shifts of
(7.3.6.12) and (7.3.6.13), then
a =
U + I 0
0 V -I
(a+l)(a-l) e A~l + J0
and
(7.6.4.9)
a + 1 £ A'1 + JX a-lg A'1 + Jx
One way to see this is to compute the index: since
index(a + 1) = index(J7 + 21) + index(Vr) =0 + 1 = 1 (7.6.4.10)
and
index(a - 1) = index(J7) + index(F - 21) = -1 + 0 = -1 (7.6.4.11)
Neither a + 1 nor a — 1 can be Weyl operators, while Theorem 6.5.4 gives
index((a + l)(a - 1)) = index(a + 1) + index(a - 1) = 0 (7.6.4.12)
Alternatively, (7.3.6.14) says that U has a left inverse and V has a right
inverse, while neither of them is actually invertible. It follows that the same
is true of a—1 and a+1, respectively. Thus, by (3.8.6.8) and (3.8.6.9) neither
of them can be a Weyl operator. On the other hand, if
{U + 2I)U [U + 2J)(J - UV){V - 21) 1
0 V{V-2I) \
V(U + 2I)~l 0 1
{V - 2I)~1{I - UV){U + 2/)"1 (V - 2I)~lU\
c =
(7.6.4.13)
then the reader should verify that
c'c = 1 = cc' and (a + l) (a - 1) - c G J0 (7.6.4.14)
7.7 BROWDER OPERATORS
The Browder operators are even more special than the Weyl operators:

266
7. Operator Algebra and Commutivity
7.7.1 DEFINITION ' If X is a normed space then T G BL(X,X) is said
to be Browder, or spatially Browder, iff
T is Fredholm and polar (7.7.1.1)
By Theorem 7.3.6, a Fredholm operator T is polar if and only if there
is m G N for which
r"m(0) = r"2m(0) and Tm{X) = T2m{X) (7.7.1.2)
or equivalently
r"m(0) = T'171-1^) and Tm{X) = Tm+1{X) (7.7.1.3)
Using the Riesz lemma, Theorem 1.5.2, T = I + K is Browder if K is
compact.
7.7.2 THEOREM If X is a normed space and K G BL(X, X), then
K G KLX (X, X) => I + K Browder (7.7.2.1)
Proof: If the first part of (7.7.1.3) fails for T = / + K then we claim that
K cannot be totally bounded: for if
{0} £ T-^O) £ r~2(0) £ • • • (7.7.2.2)
then by Theorem 1.5.2 there is a sequence x = (xn) in X for which, for
each n G N,
||xn|| = 1 and xn G r-n_1(0) and dist(xn,r_n(0)) > \
(7.7.2.3)
Since T = I + K, it follows that
m?n=>\\Kxm-Kxn\\>± (7.7.2.4)
which by (6.8.3.1) means that K cannot be totally bounded, and hence
cannot be compact. If instead the second part of (7.7.1.3) fails, and if in
addition Tm(X) is closed for each m €N, then again T = I + K means K
is not totally bounded. For if
X ^ T{X) ^ T2 (X) ^ • • • (7.7.2.5)
then by Theorem 1.5.2 there is a sequence x = (xn) in X for which, for

7.7 Browder Operators
267
each n G N,
||xj| = 1 and xn G Tn(X) and dist(xn,T"+1(X)) > \
(7.7.2.6)
With T = I + K we again get (7.7.2.4). It is now impossible for K to be
compact, since on the one hand the compact operators are totally bounded,
while on the other hand, if K is compact then by (6.9.3.3) T = I + K is
Fredholm and hence Tm(X) is closed for each m EN. m
To derive (7.7.2.1) for an operator K G KL0(X, X) of finite rank we
can alternatively return to the proof of (6.5.2.7), with the decomposition
(6.5.2.12) of the space X and the operator U : Xf2 —*■ Xf2 induced by I + K
on the finite dimensional space X'2. It is clear at once that the operator U
must be polar, and the reader can verify that, for each m EN,
{I + K)-m{0) = (J + K)"m_1(0) <^=> U~m{0) = U-™-1^) (7.7.2.7)
and
[I + K)m{X) = (7 + K)m+1(X) ^ Um{Xf2) = Um+1{Xf2) (7.7.2.8)
To see the relationship between Weyl and Browder operators, and the
perturbation theory for Browder operators, it is convenient to look again at
the Fredholm theory associated with a homomorphism of normed algebras:
7.7.3 DEFINITION If T G HBL(A,5) is a homomorphism of normed
algebras then a G A is called T-Browder if
oGA'^r^O) = {c + d: c e A'1 ,d e T'1 {0),cd = dc} (7.7.3.1)
Here we are writing
H<±.K = {x + v: x G H, y G K, xy = yx} (7.7.3.2)
for a sort of "commuting sum" of the subsets H and K in a ring A.
Evidently if a G A there is implication
a invertible => a T-Browder => a T-Weyl => a T-Fredholm (7.7.3.3)
Conversely, a T-Fredholm element which is polar must also be T-Browder:
7.7.4 THEOREM If T : A -> B is a homomorphism of normed algebras
and a G A, then
a T-Fredholm and polar => a T-Browder (7.7.4.1)

268
7. Operator Algebra and Commutivity
Conversely, provided the homomorphism T satisfies the "Riesz condition,"
l + r_1(0) C {a G A: a polar} (7.7.4.2)
there is implication
a T-Browder => a T-Fredholm and polar (7.7.4.3)
Proof: If a G A is polar then also T(a) G5 is polar, with support T(a) G
B, and if, in particular, T(a) G B~l then by the uniqueness component of
Theorem 7.5.3 it follows T(p) = T(l), and hence, 1 - p G r_1(0). Now if
ax is the Drazin inverse of a and we write
c = ad+(l-d) c' = axd+(l-d) d=(a-l)(l-d) (7.7.4.4)
we have
a = c + d cd = dc c'c=l = cc' T(d) = 0 (7.7.4.5)
which means that a is T-Browder. Conversely, if a is T-Browder, with
c,d satisfying (7.7.4.5), then by the Riesz condition (7.7.4.2) the element
1 + c~ld is polar, and since cd = dc it also commutes with 1 + c_1d, so that
a = c(l + c~1d) is polar by (7.5.4.4). ■
It is clear that the Riesz condition (7.7.4.2) is necessary for the
implication (7.7.4.3). The Riesz condition plays a vital role in the perturbation
theory of T-Browder elements. We begin with an auxiliary result:
7.7.5 THEOREM If T G BL(A, B) is a homomorphism of normed
algebras and a, 6 G A commute with one another, then
ab T-Fredholm polar => a T-Browder, b T-Browder (7.7.5.1)
a T-Fredholm polar, T(b) = 0 => a + b T-Browder (7.7.5.2)
a, 1 + ax b T-Fredholm polar => a + b T-Browder (7.7.5.3)
Proof: For (7.7.5.1) we repeat the argument of (7.7.4.1). If ab = ba is
T-Fredholm and polar, with support (ab)' = p and Drazin inverse (ab)x,
then again 1 — p G T_1(0), and is we take
c = ab+{l-p) c' =(ab)xbp+{l-p) d={a-l){l-p) (7.7.5.4)
then the conditions (7.7.4.5) are satisfied, so that a is T-Browder. The
argument for b is the same: the reader is invited to write it down. For

7.7 Browder Operators
269
(7.7.5.2) we can write
a + b= c + d with c = aa+(l-a) and d= (a-l)(l-a) + b (7.7.5.5)
and observe that c is invertible as in (7.7.4.4), that T(d) = 0, and finally
that since a is in the double commutant of a we have cd = dc. For (7.7.5.3)
we have
a(l + axb)a = (a + b)a (7.7.5.6)
and hence also a(l + axb)p = (a + 6)p where p = d(l + ax6)', giving
a+6 = a(l + axb)p+{a+b)(l-p) = a(l+axb) + (-a(l+axb) + {a+b))(l-p)
(7.7.5.7)
which is T-Browder by (7.7.5.2). ■
When T_1(0) has the Riesz property, Theorem 7.7.5 translates into
the perturbation theory for T-Browder elements:
7.7.6 THEOREM If the homomorphism T G BL{A,B) has the Riesz
property (7.7.4.2) and if a, b G A commute with one another, then
ab T-Browder «=* a, b T-Browder (7.7.6.1)
a T-Browder, T(b) =0=>a + b T-Browder (7.7.6.2)
a, 1 + ax b T-Browder => a + b T-Browder (7.7.6.3)
a T-Browder, 6,1 + axb invertible => a + b invertible (7.7.6.4)
Proof: If a and 6 are T-Browder, then by (7.7.4.3) they are T-Fredholm
polar, so that ab is T-Fredholm polar by (7.5.4.3) and (7.6.3.1), hence T-
Browder by (7.7.4.1). Conversely, if ab is T-Fredholm polar by (7.7.4.3),
so that a and b are T-Browder by (7.7.5.1). This proves (7.7.6.1). Each
of (7.7.6.2) and (7.7.6.3) follows from (7.7.4.3) together with (7.7.5.2) or
(7.7.5.3). Finally (7.7.6.4) is just (7.5.4.6) with again (7.7.4.3). ■
Forward implication in (7.7.6.1) is the more significant in view of
the failure of the analogous result for general polar elements (7.5.4.8),
and also of the analogous result for Weyl operators (7.6.4.9). By
Theorem 7.7.2 each of the ideals KL0(X,X) and KL^XjX) has the Riesz
property (7.7.2.3), and hence the spatially Browder operators of
Definition 7.7.1 are the JA-Browder operators of Definition 7.7.3 for the
homomorphism JA : BL(X,X) -► BL(X,X)/J with J = KL0(X,X) and also
with J = KLX (X, X). It follows that the spatially Browder operators satisfy
Theorem 7.7.6.

270 7. Operator Algebra and Commutivity
If the algebra A is complete, then Theorem 7.7.4 and Theorem 7.7.5
have analogues in which "polar" is replaced throughout by "quasipolar".
The reader is invited to check the details. Whether or not the normed space
X is complete however, it follows from (6.8.5.1) that if T G BL(X, X) then
T Fredholm and quasipolar => T polar (7.7.6.5)
Indeed if P is the support projection of T then I — P is compact and
therefore of finite rank, and hence also T(I — P) is of finite rank. Now a
quasinilpotent operator of finite rank is necessarily nilpotent.
To see that there actually exist Weyl operators which are not Browder,
recall the operator (a + l)(a - 1) G A = BL(X2,X2) of (7.6.4.9): we saw
there that this is a Weyl operator and that neither a + 1 nor a — 1 is Weyl.
It follows that neither a+ 1 nor a — 1 can be Browder operators, and hence
by (7.7.6.1) that (a + l)(a — 1) cannot be a Browder operator. For another
example claim
a =
Here again U and V are the shifts of (7.3.6.12) and (7.3.6.13). Indeed if
o uv-n
o o J
(7.7.6.7)
then
a = c + d c'c = l = ccf deKL0{X2,X2) (7.7.6.8)
so that a is a Weyl operator, while if x = 6n+1 = (0,0,... ,0,1,0,...) G
X = lp, then Vnx ^ 0 = Vn+1x, and hence
(7.7.6.9)
Thus, the first part of (7.7.1.3) fails with T = a, which therefore cannot be
a Browder operator.
The condition that a and b commute cannot be dropped from (7.7.6.1),
for if T e BL(X,X) is Weyl and not Browder, then by (7.6.2.3)
T = SP = QS with invertible S and Fredholm idempotents P, Q
(7.7.6.10)
so that
S and P are Browder but SP is not Browder (7.7.6.11)
=^ a Weyl but not Browder (7.7.6.6)
U I-UV
O V
V 0"
I-UV U
d =
u
_0
0
vm
71
0
X
±
0
.0.
—
u
.0
0
vm
rt-r i

7.8 Ascent and Descent
271
and also
S~1T and TS'1 are Browder but T is not Browder (7.7.6.12)
7.8 ASCENT AND DESCENT
If T : X —► X is linear then the range and the null space of T are just the
leading terms of two monotonic sequences of subspaces.
{o} c r_1o c r~2o c • • • c r~no c r~n_1o c • • • (7.8.0.1)
and
X D T{X) 2T2{X)D'"2 Tn{X) D Tn+1{X) D • • • (7.8.0.2)
7.8.1 DEFINITION If T : X -> X is a linear operator then the hyper-
kernel of T is
oo
r-°°(o) = P| r-n(o) (7.8.1.1)
n=l
and the hyperrange of T is the subspace
oo
r°°(X) = P| Tn{X) (7.8.1.2)
n=l
We shall say that T has ascent < k iff
r-°°(0) =T~k{0) (7.8.1.3)
and that T has descent < k iff
r°°(X) =T/C(X) (7.8.1.4)
If k exists for which (7.8.1.3) holds, then T is said to be "of finite
ascent," in which case "the ascent of T" will be the smallest such value of
A;: similarly for "the descent of T."
7.8.2 THEOREM If T : X -► X is linear and k G N and k < kf G N,
then
ascent(r) < k ^ r~*(o) = r~*#(o) <^ r~*(o) = r-°°(o) (7.8.2.1)
and
descent(r) < k ^ Tk{X) = Tk'[X) ^ Tk{X) = T°°{X) (7.8.2.2)

272
7. Operator Algebra and Commutivity
If also 7i G N is arbitrary, then
Tk{X) n r_1(0) = {0} => ascent(r) < k => Tk{X) n r"n(0) = {0}
(7.8.2.3)
and
T'k0+T{X) =X=> descent(r) < k => T-k{0)+Tn{X) = X (7.8.2.4)
If T has finite ascent and finite descent then
ascent(r) = descent(r) (7.8.2.5)
Proof: For (7.8.2.1) and (7.8.2.2) we need only observe
r-*(o) = r-*-1^) =► r-*-1^) = r~*-2(o) (7.8.2.6)
and
Tk{X) = Tk+1{X) =► Tk+1{X) = Tk+2{X) (7.8.2.7)
For (7.8.2.3) and (7.8.2.4), suppose T : X -> Y and S : Y -> Z are linear.
We then claim
T{x) n 5_1(o) = {0} <^> r-^o) = (5r)_1(o) (7.8.2.8)
recalling the isomorphism between (5r)_1(0)/r_1(0) and T(X) n 5_1(0)
used in the proof of (6.5.4.6) as part of the index theorem, while
T{X) + S_1(0) = Y <^> S{Y) = {ST){X) (7.8.2.9)
recalling the ismorphism between S{Y)/(ST)(X) and Y/{T(X) + 5_1(0))
used for (6.5.4.7). Taking (T,Tk) in place of (S,T) in (7.8.2.8) and (7.8.2.9)
gives the first implications in (7.8.2.3) and (7.8.2.4), while for the second
implications we argue
(suj-^o) = (su2)-\o) = s-^o) =► (su^y^o) = s~\o)
(7.8.2.10)
and
(V.S^X) = (V2S)(X) = S(X) =* (^F25)(X) = S(X) (7.8.2.11)
Towards (7.8.2.5) we show that descent(T) < ascent(T) by proving the
implication
[VTU){W) = {VT){X) and [VT)'1^) = r_1(0) => {TU){W) = T{X)
(7.8.2.12)

7.8 Ascent and Descent
273
and that ascent(T) < descent(T) by proving the implication
(VTU)-lQ = (TU)-^ and [TU){W) = T{X) => (VT)"1*) = T"1^)
(7.8.2.13)
The details are left to the reader. ■
When X is a normed space and T G BL(X, X) then the subspaces
T~n(0) are all closed, but there is nothing to suggest that either r~°°(0)
or T°°(0) should be closed. They both have the "hyperinvariant" property
(7.4.5.1):
7.8.3 THEOREM If X is a normed space and T G BL(X, X), then
r-i(r-°°(o)) C r-°°(0) (7.8.3.1)
and
T essentially one-one => T{T°°{X)) = T°°{X) (7.8.3.2)
If S e BL(X,X) commutes with T then
S{T-°°{0)) C r-°°(0) and S{T°°{X)) C r°°(X) (7.8.3.3)
If S G BL(X, X) is invertible and commutes with T, then
[T - 5)-1(0) C r°°(X) and r-°°(0) C (T - S){X) (7.8.3.4)
Proof: For (7.8.3.1) observe that if n G N is arbitrary and x G X then
Tx g r-n(o) => x g r-n"1(o) c r-°°(o) (7.8.3.5)
For (7.8.3.2) suppose y G T°°(X). Then if y = Tx' the cosets T~1y =
x1 + r_1(0) are the same and evidently, for each n G N,
(x' + r_1(0)) n Tn{X) ^ 0 (7.8.3.6)
There are therefore sequences z = (zn) in X for which, for each n G N,
||*J|=1 zneT~1(0) x' + zn€Tn(X) (7.8.3.7)
If T is essentially one-one, so that T_1(0) is finite dimensional, then by
Theorem 6.6.4 the sequence (zn) has convergent subsequences zf = (zfn).
Now, if we take x = x' + limn z'n then
oo
x 6 fl (x' + r_1(0)) n T"(X) (7.8.3.8)
n=l

274 7. Operator Algebra and Commutivity
so that
y = Tx and xeT°°{X) (7.8.3.9)
The inclusions (7.8.3.3) are clear. For (7.8.3.4) note that if S is
invertible and commutes with T then for arbitrary n G N
x e [T - 5)_1(o) => x = 5_1rx = s~nTnx = Tns~nx g Tn{x)
(7.8.3.10)
giving the first part of (7.8.3.4), while if x G X then
rn+1x = 0=)>
x = (5-r)5"1(/+5-1r+-+5-nr)iG (r-s)(x) ■ (7.8.3.11)
The first part of (7.8.3.4) holds if instead S commutes with T and
T — S is onto and X/Tn(X) is finite dimensional for each n G N
(7.8.3.12)
For by (3.11.1.4), the induced mappings (T - S)A : X/Tn{X) -> X/rn(X)
are also onto, and by finite dimensionality also one-one, giving
{T - 5)-1(0) C Tn{X) = {T- S){Tn{X)) for each neN (7.8.3.13)
The second part of (7.8.3.4) holds if instead S commutes with T and
T — S is one-one and T~n(0) is finite dimensional for each n G N
(7.8.3.14)
For by (3.11.1.1) the induced mappings (T - S)A : T~n(0) -> r_n(0) are
also one-one, and by finite dimensionality also onto, giving
{T - 5)-1r-n(0) = r"n(0) C (T - S){X) for each n G N (7.8.3.15)
The hyperrange T°°(X) is used in the "punctured neighborhood
theorem":
7.8.4 THEOREM If T G BL(X,X) is Fredholm, with generalized
inverse T' G BL(X,X), and if S and S' in BL(X,X) are both invertible and
commute with T, and also satisfy
/ + T'S invertible and / + T'S' invertible (7.8.4.1)
then
dim(r - 5;)_1(0) = dim(r - 5)_1(0) < dimr-1^) (7.8.4.2)

7.8 Ascent and Descent
275
and
dimX/(r - S')X = d\mX/{T - S)X < dimX/TX (7.8.4.3)
Proof: If I + T'S is invertible then Theorem 6.4.5 gives the inequalities at
the end of (7.8.4.2) and (7.8.4.3). Writing *7A : r°°(X) -► r°°(X) for the
operator induced by U G comm(T), we claim
dimfr-S)-1^) = dim(r-5)A-1(0) = index(r-S) = index(r) (7.8.4.4)
The first equality comes from the first inclusion of (7.8.3.4), the second
equality comes from the fact that, by (7.8.3.2), (T — 5)A is onto, and the
third equality is Theorem 6.5.5. Since the right-hand side of (7.8.4.4) is
independent of S, equality follows in (7.8.4.2), and hence also in (7.8.4.3). ■
Theorem 7.8.4 derives its name from the consequence that, under
certain circumstances, every operator in a certain "punctured neighborhood"
of a Fredholm operator must be invertible:
7.8.5 THEOREM If X is a Banach space and A C BL(X,X) is a sub-
space, and if
T eAC comm(r) (7.8.5.1)
then there is implication
T Fredholm, Ted{A\ BL_1(X,X)) => T e iso{A \ BL_1(X,X))
(7.8.5.2)
Proof: We are writing
iso(K) = K\ acc(K) (7.8.5.3)
for the isolated points of K C H, where the accumulation points form the
set
acc(K) = {t e fi: U e Nbd(*) =>UnK\{t}^0} (7.8.5.4)
We claim
T Fredholm, T G cl(BL_1(X,X) n comm(r))
=> T £ acc(comm(r) \ BL_1(X,X))
Indeed, if T is Fredholm, then by Theorem 7.8.4 there is 6 > 0 for which,
whenever S and Sf are in the set
W{T,S) = Disc(0;<5) nBL_1(X,X) ncomm(r) (7.8.5.6)

276
7. Operator Algebra and Commutivity
there is equality
dim(r - 5'/)"1(0) = dim(r - 5)_1(0)
and (7.8.5.7)
dimX/(r - S')X = dimX/{T - S)X
If, in addition, T is in the closure of BL_1(X,X) n comm(T) then for
arbitrary 6' > 0 there is S' G W(T,Sf) for which T - Sf is invertible, so
that
dim(r - 5")_1(0) = dimX/(r - S')X = 0 (7.8.5.8)
Taking in particular 8' = 6 it follows from (7.8.5.7) that for arbitrary
SeW{T,6)
dim(r - S)-1 (0) = dimX/(r - S)X = 0 (7.8.5.9)
Since X is complete it follows from Theorem 4.6.3 that T — S is invertible.
This proves (7.8.5.5), and hence (7.8.5.2) with A = comm(T), and hence
(7.8.5.2) whenever A satisfies (7.8.5.1). ■
7.9 SEMI-BROWDER OPERATORS
A semi-Browder operator will satisfy about two-thirds of the conditions for
a Browder operator:
7.9.1 DEFINITION If X is a normed space then T e BL(X, X) will be
called upper semi-Browder if
T is upper semi-Fredholm of finite ascent (7.9.1.1)
and will be called lower semi-Browder if
T is lower semi-Fredholm of finite descent (7.9.1.2)
As for semi-Fredholm operators, we can only obtain the perturbation
theory for semi-Browder operators in complete spaces.
7.9.2 THEOREM If X is a Banach space and S, T e BL(X, X) commute
then
S,T upper semi-Browder ^=> ST upper semi-Browder (7.9.2.1)
S,T lower semi-Browder <£=> ST lower semi-Browder (7.9.2.2)
T upper semi-Browder, S compact
=> T + S upper semi-Browder (7.9.2.3)

7.9 Semi-Browder Operators
277
and
T lower semi-Browder, S compact
(7.9.2.4)
T + S lower semi-Browder
Proof: If S and T are upper semi-Browder then ST is upper
semi-Fredholm by (6.12.2.1), and also of finite ascent: for if r_n(0) = r-n_1(0) and
S-n(0) = S-n_1(0) then, remembering that ST + TS,
(ST)-n(0) = S-nT-n(0) = S-nT-n-1(0)
K J K J w w (7.9.2.5)
= r"n"15,"n(o) = r_n"15'"n"1(o) = (ST)-n~1(o)
Conversely, if ST = TS is upper semi-Fredholm then S and T are upper
semi-Fredholm by (6.12.2.1), and for some n G N,
(ST)-°°(0) = (ST)-n(0) is finite dimensional (7.9.2.6)
giving for arbitrary n G N
r~m(o) c T~ms~m(o)
v (7.9.2.7)
= (ST)-m(0) C (ST)-°°{0) finite dimensional
Thus T, and similarly 5, are also of finite ascent, finishing the proof of
(7.9.2.1). If instead S and T are lower semi-Browder then by (6.12.2.2) ST
is lower semi-Fredholm, and also of finite descent. For if Tn(X) = Tn+1(X)
andSn(X) =5n+1(X) then
{ST)n{X) = SnTn{X) = SnTn+1{X)
= Tn+1Sn{X) = Tn+1Sn+1{X) = [ST)n+1{X) (7.9.2.8)
Conversely, if ST = TS is lower semi-Fredholm then S and T are lower
semi-Fredholm by (6.12.2.2), and for some n G N
(ST)°°(X) = {ST)n{X) is closed of finite codimension (7.9.2.9)
giving for arbitrary m €N
Tm{X) D TmSm{X) = [ST)m{X) D {ST)°°{X) of finite codimension
(7.9.2.10)
Thus T, and similary S, are of finite descent, finishing the proof of (7.9.2.2).
Towards (7.9.2.3) it follows from (6.12.2.3) that if T is upper semi-
Browder and S is compact then T + S is upper semi-Fredholm, and hence
by (6.12.2.1) also (T + S)n for each n e N, so that in particular
(T + S)~n(0) is finite dimensional for each n e N (7.9.2.11)

278
7. Operator Algebra and Commutivity
If, in particular, T is bounded below and S commutes with T then by
(7.8.3.15)
r_1(r + S)-n{0) = {T + S)-n{0) C T{X) for each n G N (7.9.2.12)
If k > 0 is such that ||x|| < A;||Tx|| for each iGl then we claim
dist(x, (r + S)~n(0)) < ikdist(rx, (r + S)~n(0)) for each n G N
(7.9.2.13)
Indeed, if x G X and yn G (T + S)-n(0) are arbitrary then by (7.9.2.12)
there is zn G X for which yn = Tzn, and also by (7.9.2.12) it follows that
zne {T + S)-n{0): now
dist(s,(r + S)-n(0)) < \\x-zj < k\\T(x-zn)\\ =k\\Tx-yn\\ (7.9.2.14)
Since yn was arbitrary,. (7.9.2.13) follows. We now claim that T + S must
be of finite ascent. For if not then there is a sequence (xn) in X for which,
for each n G N,
||xj| = 1 xn G (T + 5) —x(0) dist(xn, (T + 5)-»(0)) > J
(7.9.2.15)
It follows that if n and m> n + 1 are arbitrary, then
*||Sxm - SxJ = k\\Sxm - (T + 5)xn + TxJ
> kdist{Txn,{T + S)~n{0)) > \ (7.9.2.16)
which contradicts the compactness of the operator S. This contradiction
proves (7.9.2.3) in the special case when T is bounded below. If more
generally T is upper semi-Browder, with T~p(0) = r~p_1(0) finite
dimensional, then the induced mapping TA : X/T~p(0) -+ X/T~p(0) is
bounded below and hence, by what we have just proved, the induced
mapping (r + S)A : X/T~p{0) -+ X/T~p(0) is upper semi-Browder, in
particular, of finite ascent. It follows that there is m G N for which n > m
implies
(r + S)~n{0) C(T + S)-nT~p{0) = {T + S)-mT-p{0) finite dimensional
(7.9.2.17)
which forces T + S to be of finite ascent, and finishes the proof of (7.9.2.3).
Towards (7.9.2.4) it follows from (6.12.2.4) that if T is lower semi-
Browder and S is compact then T + S is lower semi-Fredholm, so that
using (6.12.2.2)
(T + S)nX is closed of finite codimension for each neN (7.9.2.18)

7.9 Semi-Browder Operators
279
If, in particular, T is open and commutes with S then by (7.8.3.13)
r_1(0) C (T + S)n{X) = T{{T + S)n{X)) for each n G N (7.9.2.19)
If k > 0 is such that dist(x,r_1(0)) < A;||Tx|| for each x G X then we claim
dist(x, (r + s)n(x)) < A;dist(rx, (r + s)n(x))
for each n G N (7.9.2.20)
Indeed, if x G X and yn G (T + S)n(X) are arbitrary, then by (7.9.2.19)
there is zn G (T + S)nX for which yn = Tzn: now using again (7.9.2.19)
dist(x, (T + 5)n(X)) = dist(x - zn), (T + S)n(X))
< dist(x - z^T-'iO)) < k\\Tx - yn\\ (7.9.2.21)
Since yn was arbitrary, (7.9.2.20) follows. We now claim that T + S must
be of finite descent. For if not there is a sequence (xn) in X for which, for
each n G N,
KH = i xne(T + s)n(X) distfo.cr + sr+^ui (7.9.2.22)
It follows that if 7i and m > n + 1 are arbitrary, then
*||Sxm - SxJ = k\\Sxm - (T + 5)xn + TxJ|
> k dist(rxn, (r + 5')n+1 (X)) > \ (7.9.2.23)
which contradicts the compactness of S. This contradiction proves (7.9.2.4)
in the special case when T is open. If, more generally, T is lower semi-
Browder, with TP(X) = TP+1(X) closed and of finite codimension, then
the induced mapping TA : TP(X) —► TP(X) is open, and hence, by what
we have just proved, the induced mapping (T + S)A : TP(X) -+ TP(X) is
lower semi-Browder, in particular of finite descent. It follows that there is
m G N for which n > m implies
(r + S)n{X) D {T + S)nTp{X) = {T + S)mTp{X) of finite codimension
(7.9.2.24)
which forces T + S to be of finite descent, and finishes the proof of
(7.9.2.4). ■
For Fredholm operators of finite ascent or descent the perturbation
theory of Theorem 7.9.2 extends to incomplete spaces, since now we can
use (6.9.3.3) instead of Theorem 6.12.2. A semi-Browder operator which is
Weyl must be a Browder operator:

280 7. Operator Algebra and Commutivity
7.9.3 THEOREM If X is a normed space and T G BL(X, X) then
T Weyl and upper semi-Browder =>• T Browder (7.9.3.1)
and
T Weyl and lower semi-Browder =>► T Browder (7.9.3.2)
Also
T Fredholm of finite ascent =>► index(r) < 0 (7.9.3.3)
and
T Fredholm of finite descent =>► index(r) > 0 (7.9.3.4)
Proof: We use the Index Theorem 6.5.4. If T is Weyl then for arbitrary
n e N
dimr"n(0) - d\mX/Tn{X) = index(rn) = n index(r) = 0 (7.9.3.5)
so that if either dimT~n(0) or dimX/Tn(X) is a constant then so is the
other. This proves both (7.9.3.1) and (7.9.3.2). If T is Fredholm of finite
ascent p, then
n index(r) = index(rn) = dimT"p(0) - dimX/rnX whenever n > p
(7.9.3.6)
so that either T has finite descent q, in which case
n index(r) = dimr_p(0) - dimX/r9(X) independent of n > max(p, q)
(7.9.3.7)
which forces index(T) = 0, or T does not have finite ascent, in which case
n index(r) = dimr_p(0) - dimX/Tn(X) —> -oo as n —► oo
(7.9.3.8)
which forces index(T) < 0. This proves (7.9.3.3), and the argument for
(7.9.3.4) is exactly similar. ■
7.10 CONNECTEDNESS AND HOMOTOPY
A subset K of a topological space H is said to be connected if there are
no disconnections of K in H, where a disconnection of K in H is a pair
(G1, G2) of open subsets of H for which
KCGjUGj KHG1nG2 = 0 KnG1^0^KnG2 (7.10.0.1)
For example, the connected subsets of R are the intervals:
J connected in R <^=> (a < b =^ [a,b] C J) for each a,b e J (7.10.0.2)

7.10 Connectedness and Homotopy 281
If K and H are subsets of f2, then
K connected, K C H C cl(jRT) => jff connected (7.10.0.3)
and
if, jff connected, Kfl^^0=^KUff connected (7.10.0.4)
If K C n and jff C A, then
if connected in fi, if connected in A => K x H connected infix A
(7.10.0.5)
If <f> : H —► A is a mapping then
^ continuous on if, if connected in H =>• <£(if) connected in H
(7.10.0.6)
Both (7.10.0.4) and (7.10.0.5) have extensions to arbitrary families [K-) -€J:
in particular if t G K C H then
CompK(i) = (J {# C K:t E H connected in H} C K is connected in Q
(7.10.0.7)
We call the set (7.10.0.7) the connected component of t in if. Evidently if
5 and t are points of K then
either CompK(£) = CompK(s) or CompK(£)nCompK(s) =0 (7.10.0.8)
Thus, the set of connected sets {CompK(£):£ G K} forms a "partition" of
the set K.
If if C H and if t G H \ if then the complement of the component of t
in the complement of if will be known as the "connected hull" of if with
respect to t:
7.10.1 DEFINITION If if C fi and t G fi \ if then the connected hull of
if with respect to t is the set rjtK for which
n\tltK = CompQ\K{t) (7.10.1.1)
A hole in if with respect to t is a component of H \ if other than H \ rjtK.
Thus the connected hull of if is obtained by "filling in the holes."
7.10.2 THEOREM If if and H are subsets of fi \ {t} then
Tjt{TjtK) = VtK (7.10.2.1)

282
7. Operator Algebra and Commutivity
and
H C K => rjtH C 77tK (7.10.2.2)
If K is closed and H is locally connected then
rjtK = c\(rjtK) (7.10.2.3)
Proo/: (7.10.2.1) and (7.10.2.2) are clear from (7.10.1.1). If fi is locally
connected, in the sense that
t e n, U e Nbd(r) => * G V C Z7 for some connected 7 G Nbd(*)
(7.10.2.4)
and if s G fi \ t^K is arbitrary, then since fi \ if is a neighborhood of 5
there is a connected neighborhood V of 5 in H for which V C fi \ jK". Since
V and fi \ rjtK are connected and not disjoint then also V U (fi \ rjtK) is
connected. By the definition of r\tK it follows that V C fi \ rjtK. This
means that fi \ rjtK is open, giving (7.10.2.3). ■
There is an interaction between the connected hull and the topological
boundary:
7.10.3 THEOREM If H, K and L are subsets of U then
H C K and dK C jff U L => aiT C (Sjff) U L (7.10.3.1)
and if if and K are closed, then
dK CH and dH C K
=> a(K uF)c (ax) n (an) ca(KnF)cxn£ (7.10.3.2)
If fi is locally connected and H and if are closed subsets of fi \{t}, then
dKCHCK=^HCKCTjtH<^ drjtK C H C K (7.10.3.3)
and
drjtK CdK CK CTjtK = TjtdK (7.10.3.4)
Proof: If H C if and <9K C H U L, then
aK C (HuL)n(fi\int(K)) C (jffUL)n(n\int(jff)) C (dH)uL (7.10.3.5)
giving (7.10.3.1). If dK C H and K is closed then
d{KuH)CdKudHCHCKuH (7.10.3.6)

7.10 Connectedness and Homotopy
283
and hence by (7.10.3.1) dK C dH. Interchanging the roles of K and H
now gives the first inclusion of (7.10.3.2). At the same time (7.10.3.1) also
gives
dKCHnKCK=>dKC d{H n K) C K (7.10.3.7)
and interchanging the roles of K and H gives the second inclusion of
(7.10.3.2). Towards the first part of (7.10.3.3) suppose that L C fi is
connected: then
^nL = 0^(KaorKDLorKnL^0) (7.10.3.8)
since otherwise (int(jK"),fi \ cl(K)) would be a disconnection of L in H. If
in particular L = H \ 77t.ff with if C if C Q \ {£} then the left-hand side of
(7.10.3.8) holds, while neither the second nor the third of the alternatives on
the right-hand side is possible. This gives the first implication of (7.10.3.3).
Towards the second implication recall that by local connectedness each
component of H \ K is open, giving
{rjtK) \ K C int(i7ttf) (7.10.3.9)
if K = cl(jRT) C fi \ {t}, so that
d{rjtK) CK CrjtK (7.10.3.10)
By (7.10.2.1) and (7.10.2.2) it follows that if H C K C rjtH then rjtH =
rjtK, and applying (7.10.3.10) with H in place of K gives drjtK = drjtH C
H. This gives the second implication in (7.10.3.3). Finally (7.10.3.10) is
already part of (7.10.3.4), and gives the first inclusion by application of
(7.10.3.1) with [H,K,L) replaced by (K,rjtK,0): the rest then follows at
once. ■
As a supplement to (7.10.3.4) observe that if fi = X is a normed space
and if 5 G dK \ drjtK then there are sequences (sn) in X and [Hn) of
connected components of X \ K for which
snedHn and ||5-5J|—>0 (7.10.3.11)
Indeed, if we do not already have 5 G dH for some fixed component H of
X\K, then there will be [s'n) in X \ K for which \\s'n - s\\ -> 0, and for
sufficiently large n we must have sfn G Hn with connected Hn C X\K\
now if we look at the line segment {ts + (1 — t)sfn:0 < t < 1} we will find
at least one point sn G dHn.
The first inclusion of (7.10.3.3) is not in general reversible: for example

284
7. Operator Algebra and Commutivity
take H = C and
H = {seC:2 < \s\ <3} K = {s G C: 1 < \s\ < 3} and \t\ > 3
(7.10.3.12)
If X and Y are normed spaces then, in the notations (3.3.1.3), (3.3.1.4),
(3.7.1.3), and (3.7.1.4), it follows from (7.10.3.2) that
d{rleft BL(X, Y) U f right BL(X, Y)) C fleft BL(X, Y) n fright BL(X, y)
(7.10.3.13)
and
<9(<jleft BL(X, Y) U <rright BL(X, Y)) C aleft BL(X, y) n <Jright BL(X, y)
(7.10.3.14)
For by (3.5.3.3) and (5.5.6.1) the left-hand side of (7.10.3.2) holds with K =
<rleftBL(X,y) and H = frightBL(X,y), while by (3.7.5.1) and (3.7.5.2)
the same is true with K = <jleft BL{X,Y) and H = <jri*ht BL{X,Y). By
(7.10.3.1) it now follows that the left-hand side of (7.10.3.13) is contained
in the right-hand side of (7.10.3.12).
The analogues of (7.10.3.12) and (7.10.3.13) also hold when the space
of operators BL(X,y) is replaced by a normed algebra A. Since normed
spaces are easily seen to be locally connected, we can also derive results
about connected hulls:
7.10.4 THEOREM If A is a closed subalgebra of a normed algebra B,
then
<9(<7left(A) U <jright(A)) C rleit{B) n fright(5) n A (7.10.4.1)
and
<jleft(A) U <jright(A)) C T)i{Tleit{B) n fright(£) n A (7.10.4.2)
where we are taking the connected hull relative to the identity 1 G A C B.
Proof: For (7.10.4.1) we combine (3.10.7.1), (3.10.7.2) and (3.10.6.5), and
then (7.10.2.4) is the first implication of (7.10.3.3). ■
The "punctured neighborhood" Theorem 7.8.5 suggests applications of
a modified version of (7.10.3.3), holding "to within isolated points." If H
and K are subsets of H then
H CK => acc(H) C acc(K) (7.10.4.3)

7.10 Connectedness and Homotopy
285
and we have a decomposition of the boundary of K:
dK = <9(acc(K)) U iso(K) with <9(acc(K)) n iso(jRT) = 0 (7.10.4.4)
Thus also
acc(dK) = <9(acc(K)) and iso(di<:) = iso(jRT) (7.10.4.5)
7.10.5 THEOREM If fi is locally connected <md H C K C Q \ {t} are
closed subsets, then
t]tK = 77t(acc(K)) U iso(K) (7.10.5.1)
and
dK C H U iso{K) =>KCrjtHU iso{K) => drjtK CHU iso{K)
(7.10.5.2)
Proo/: Toward (7.10.51) it is clear that
K C 77t(acc(K)) U iso(K) C rjtK (7.10.5.3)
We claim that also, whenever L is the closure of a hole in K,
L C 77t(a77tL) C 77t(acc(K)) (7.10.5.4)
To prove (7.10.5.4) we show
dL n iso(K) C n \ a?7tL (7.10.5.5)
If s G iso(K), then by the local connectedness of H there is a connected
neighborhood U G Nbd(s) for which U D K \ {s} = 0, and if also 5 G dL
then LnU\{s} ^ 0. By (7.10.0.4) it follows that U \ {s} C int(L), and
hence that 5 £■ d77t£. This establishes (7.10.5.5), so that also
drjtL C acc(K) (7.10.5.6)
which with (7.10.4.3) gives (7.10.5.4). This completes the proof of
(7.10.5.1). Towards (7.10.5.2) suppose that H C K <md dK C H U iso{K):
then by (7.10.4.4)
<9(acc(K)) C H (7.10.5.7)
and hence by (7.10.3.2)
acc(K) C rjtH (7.10.5.8)

286
7. Operator Algebra and Commutivity
giving the first implication of (7.10.5.2). If instead K C rjtH U iso(K) then
also (7.10.5.8) holds, giving with (7.10.5.1)
rjtH C rjtK C rjtH U iso(K) (7.10.5.9)
and hence by (7.10.3.1)
d{t]tK) C d{rjtH) U iso(jRT) CffU iso(jRT) ■ (7.10.5.10)
For example, if X is a Banach space and A G BL(X, X) is a linear
subspace, and if T G A C BL(X,X) as in (7.8.5.1) then (7.8.5.2) gives
implication
TedK=>TeHU iso{K) (7.10.5.11)
where
K = {S G A: S is not invertible}
and (7.10.5.12)
H = {S e A: S is not Fredholm}
Since we may not have inclusion dK C H U iso(K) it does not immediately
follow that
T e K => T e rjtH U iso(K) (7.10.5.13)
If H and A are topological spaces, then continuous functions f,g G
C(H, A) are said to be homotopic if there is a mapping H : [0,1] x H —► A
for which
fT(O,.) = /(0 H(l,.)=^(«)
and (7.10.5.14)
if G C([0,1] x n, A) is continuous
When (7.10.5.4) holds we shall write
f~g and ge[f]„ (7.10.5.15)
and often speak of (^t)o<t<i = (-^(^0)o<t<i BS a homotopy between h0 =
f and hx= g. If / G C(H, A) is homotopic to a constant we shall say that /
is contractible. If the identity I: Q —> Q is homotopic to a constant we shall
call the space H contractible. For example, normed spaces, convex subsets
of normed spaces, and bounded linear mappings are all contractible. If
/ : H —► A and g : A —► M are continuous, then
/ contractible or g contractible => g o f contractible (7.10.5.16)

7.10 Connectedness and Homotopy
287
In particular, every continuous mapping in or out of a contractible space
is contractible. A space fi is said to be arcwise connected if every pair of
mappings into fi from the one-point space {0} are homotopic. Here the
homotopy (^t)o<i<i is referred to as an arc. Evidently if H is a topolgical
space
H contractible => fi arcwise connected => fi connected (7.10.5.17)
In partial converse, if fi is "locally arcwise connected," in the sense that
every neighborhood of every point contains an arcwise connected
neighborhood of that point, then
K C fi open and connected => K arcwise connected (7.10.5.18)
A topological space fi is called simply connected if every pair of continuous
mappings from the circle S C C to H are homotopic: evidently necessary
and sufficient is fi arcwise connected with the property that mapping from S
to H is contractible. It is significant, and perhaps not immediately obvious,
that the circle S is not simply connected. Towards a proof of this, introduce
mappings
ex : R —► S and lg:S\ {-1} —► R (7.10.5.19)
where
ex{t) = e27rit if t e R and lg{e27rit) = t if - \>k < t < \>k
(7.10.5.20)
If 4> : fi —► S is an arbitrary mapping, then a lift of <f> will be a mapping
<f>v : H —► R for which </> = ex o <f>w.
7.10.6 THEOREM If fi is a topological space and <f> e C(S, fi), then
<f> contractible <£=> <f> has a continuous extension 4>A : D —> Q (7.10.6.1)
If instead <f> G C(H,S), then
<f> contractible ^=^ <j> has a continuous lift <j>w : fi —> R (7.10.6.2)
Proof: Reverse implication in each case is clear from (7.10.5.16).
Conversely, for (7.10.6.1), suppose that {ht)0<t<1 is a homotopy in C(S,H):
then we claim
3/ig e c(D, n) => 3/ij e c(d, n) (7.10.6.3)

288
7. Operator Algebra and Commutivity
Specifically, define for each iGR and each r G [0,1],
h${rc2xit) = h*{2re2Kit) if 0 < r < \
and (7.10.6.4)
h^re2*") = h2r^{e2irii) if \ < r < 1
Intuitively, we have /^ : D -> A -> fi with A = (D x {0}) U (S x [0,1]), where
the embedding of D in A is achieved by pasting the interior of the disc across
the top of the open cylinder down the sides and across the bottom. This
gives the forward implication in (7.10.6.1) if we take h0 to be constant and
hx = <f>. For the forward implication in (7.10.6.2), suppose that {ht)0<t<1
is a homotopy in C(n,S): then claim
3/tg e C(fi, R) <=>> 3h\ e C(fi, R) (7.10.6.5)
By the compactness of [0,1] there is a partition (t-)^=1 with 0 = t0 <
t-L < - - - < tn = 1 for which supwGn \htj-(u) — /^.^(w)! < 2 for each
j = 1,2,..., n: if we now define
g (u) = 9tAy for each ueVt,j = l,2,...,n (7.10.6.6)
then ^y(H) C S\{—1} for eachj and h^w) = h0(u)g1(uj).. .gn{u) for each
cj; thus we can lift hx by taking
n
MM = MM + X^MW)) for each w G n (7.10.6.7)
Thus for forward implication in (7.10.6.2) take h0 to be constant and hx =
Theorem 7.10.6 enables us to define a winding number for continuous
functions on the circle:
winding number [4>) = ^(1) - ^(0) (7.10.6.8)
where
<£* = V>v is a continuous lift for ip = <f> o ex : R —► S (7.10.6.9)
Explicitly
c2«Mt) = <£(e27rt"<) for each t G R (7.10.6.10)

7.10 Connectedness and Homotopy
289
The winding number is well defined, and an integer, since if H is connected
then any two lifts for a continuous function </> : H —► S must differ by
a constant. The winding number picks out the contractible continuous
functions on the circle:
7.10.7 THEOREM If <fr : S -> S is continuous then the following are
equivalent:
<f> is contractible (7.10.7.1)
<j> has a continuous extension <f>A : D —► S (7.10.7.2)
4> has a continuous lift <f>v : S —► R (7.10.7.3)
winding number (cf>) = 0 (7.10.7.4)
The circle S is not contractible.
Proof: The equivalence of the first three conditions is Theorem 7.10.6. If
{ht)0<t<1 is a homotopy in C(S,S) then we claim
winding number (h0) = winding number (/ix) (7.10.7.5)
This is because the mapping t —► winding number (ht) is continuous, and
maps the connected space [0,1] into the integers Z. Since of course the
winding number of a constant is 0 we have proved that (7.10.7.1) implies
(7.10.7.4). Conversely, if (7.10.7.4) holds then so does (7.10.7.3): for we
may define <f>y by setting
P(e2irit) =<f>*{t) if 0 < t < 1 (7.10.7.6)
This completes the proof that the four conditions are equivalent. Since for
each n G Z it is clear that
winding number (zn) = n (7.10.7.7)
the identity / = z : S —> S fails the condition (7.10.7.4), so that by
(7.10.7.1) the circle is not contractible. ■
Intuitively, the winding number of <f> : S —► S counts the number of
times <f>(u) G S winds round the point 0 G C when the point wG S winds
once round 0 G C. If also ip : S —► S is continuous then
winding number [<f> • ip) = winding number (<f>) + winding number (tp)
(7.10.7.8)

290
7. Operator Algebra and Commutivity
thus acting as a group homomorphism from C(S,S) to Z, and also
winding number (ip o cj>) = winding number (<f>) • winding number (tp)
(7.10.7.9)
7.11 GENERALIZED EXPONENTIALS
If A is a normed algebra, then the invertible elements A~* form a subgroup
of the multiplicative semigroup A, and of course also a topological space.
The connected component of 1 in A-1 turns out to be a subgroup of A~x:
7.11.1 DEFINITION If A is a normed algebra we shall write
A'1 = CompA_! (1) C A'1 (7.11.1.1)
In the notation of Definition 7.10.1 we are writing
A\A^1=nl{A\A-1) (7.11.1.2)
The subset Aq1 is both a subgroup, and a relatively closed subset, of
A-1:
7.11.2 THEOREM If A is a normed algebra, then Aq1 is a subgroup of
A-1 £ A, and there is inclusion
a~xA^xa C Aq1 for each a G A'1 (7.11.2.1)
and equality
A'1 fl c\{Aq x) = Aq1 (7.11.2.2)
If A-1 is an open subset of A then also
Aq1 and A'^Aq1 are open subsets of A (7.11.2.3)
and
d{Ao x) C fleft(A) n fright(A) (7.11.2.4)
Proof: By definition, A^1 is a connected subset of A-1 containing 1, and
there is implication, using (7.10.0.4), for subsets K C A,
1EKCA-1 and K connected => K C Aq1 (7.11.2.5)
By the continuity of multiplication and inversion, together with (7.10.0.5)
and (7.10.0.6), the left-hand side of (7.11.2.5) is satisfied if K = (Aq x)(Aq x)
and if K = (Aq1)"1, so that by the right-hand side of (7.11.2.5) Aq1 is a
group. For the same reason each of the sets a~1A0~1a with a G A-1 satisfies

7.11 Generalized Exponentials
291
the left-hand side of (7.11.2.5), so that we obtain (7.11.2.1). By (7.10.0.3)
the left-hand side of (7.11.2.5) is also satisfied if K = A~x n c^Aq1), so
that we obtain (7.11.2.2). Finally if A~* is open in A, so that there is 6 > 0
for which
{1 - x: \\x\\ <6}C A'1 (7.11.2.6)
then the same argument (7.11.2.5) says that
{l-x:\\x\\<6}CA^ (7.11.2.7)
It follows at once that if a G A-1 is arbitrary then
aeAQ1=^ {a(l - x): \\x\\ < 6} C Aq1 (7.11.2.8)
and
cGA_1\ A'1 => {a(l - x): \\x\\ < 6} C A'1 \ Aq1 (7.11.2.9)
giving (7.11.2.3). Also Theorem 7.10.3 and Theorem 7.10.4, together with
Theorem 4.4.7, give
0(Ao x) = drj1{a]eit{A)uaTlsht{A)) C f]eit{A) nfright(A) ■ (7.11.2.10)
The condition (7.11.2.1) is what it means for Aq1 to be a normal
subgroup of A-1. If it holds then the sets of left and right cosets
{aA-^.aeA-1} = {A^aiaeA'1} = A~1/Aq1 (7.11.2.11)
coincide, and carry a natural group structure. The argument for (7.11.2.4)
can be made more explicit. We claim, if A~* is open in A,
c1(Aq *) \ (fleft(A) n fright(A)) C Aq1 (7.11.2.12)
Indeed, suppose a G c^A^"1), with (an) in A~l satisfying \\a — an\\ —► 0 as
n —► oo. Fixing 6 > 0 for which (7.11.2.6) holds, it is possible that there is
m G N for which
\\*m\\\\*-*m\\<* (7.11.2.13)
in which case by (7.11.2.7)
a^a =l-xmeAo1=^a = am{a^a) G A"1 (7.11.2.14)
If on the other hand there does not exist m G N for which (7.11.2.13) holds
then, for each nGN,
maxfllcOlJa-^ll) < (1 + l/tf)||o-on|||K1|| (7.11.2.15)

292
7. Operator Algebra and Commutivity
and, since \\a - an\\ -> 0, this means that a G r]eit(A) n fright(A).
If A is actually complete then we can express the elements of Aq1 in
terms of exponentials:
7.11.3 DEFINITION If A is a Banach algebra and a G A, then
oo n
exp(a) = ea = 1 + Y" ^- (7.11.3.1)
n=l
and
Exp(A) = {efllefl2---efln:n€ N,{al5a2,... ,an} C A} (7.11.3.2)
More generally, we can talk about exponentials of certain elements of
an incomplete algebra. If afc+1 = 0 then
ea = exp,(a) = l+£^- (7.11.3.3)
while if instead a2 = ta for some iGK then
ea = 1 + (c* - l)a (7.11.3.4)
In general, however we may as well work in the completion A~ of an
incomplete algebra A, and possibly inquire afterward whether or not an
exponential happens to lie in the original.
When A is complete then the sets Exp [A) and Aq1 coincide:
7.11.4 THEOREM If A is a Banach algebra and a, 6 G A, then
ab = ba => exp(a + 6) = exp(a) exp(6) (7.11.4.1)
and if x G A, then
||x|| <l=>l-x = ea with a = - ^ — (7.11.4.2)
n=l
There is equality
Aq1 = Exp(A) (7.11.4.3)

7.11 Generalized Exponentials
293
Proof: We offer two proofs of (7.11.4.1). If a and 6 commute in A then for
each n G N
exPn(a) exPnW - exPn(a + h)
= Y, { V?: r^ G {0,1,..., n} , n + 1 < r + 5 < 2n 1 (7.11.4.4)
and hence
||expn(a)expn(6) - expn(a + 6)|| < expn(||a||) expn(||6||)
- expn(||a|| + H&H) —+ 0 asn-^oo (7.11.4.5)
Alternatively, still assuming ab = 6a, (5.12.6.3) and (5.12.2.12) give
— (exp(*a + 6)) = aexp(*a + 6) (7.11.4.6)
at
and
— (exp(-ia) exp(*a + 6)) = 0 (7.11.4.7)
at
for all t G R. By (5.12.4.2) it follows that exp(—ta) exp(ta + 6) is
independent of t, giving again (7.11.4.1). Towards (7.11.4.2) define, if ||x|| < 1 and
-Kt\\x\\<l9
f{t) = exp(-</(*)) and g(t) = £ — (7.11.4.8)
By (5.12.6.3) both g and / are differentiate, with
g'(t) = x(l - tx)-1 and f'(t) = -g'(t)f(t) (7.11.4.9)
so that
^((l-*x)-7W)=0 (7.11.4.10)
By (5.12.4.2) it follows that /(l) = /(0)(l-x), which is (7.11.4.2). Towards
(7.11.4.3) it is clear that if a G A then
ea G A'1 with (ea)_1 = e~a (7.11.4.11)
and then, by applying the argument of (7.11.2.5) with K = {eta: 0 < t < l},
that
ea G Aq1 for each aeA (7.11.4.12)

294
7. Operator Algebra and Commutivity
This proves that the set Exp [A) is a subset of Aq1. With the aid of
(7.11.4.2) we claim that also
Exp(A) and A'1 \ Exp(A) are open in A (7.11.4.13)
Indeed if a G A-1 then
a G Exp(A) => {a(l - x): \\x\\ < 1} C Exp(A) (7.11.4.14)
and
cGA_1\ Exp(A) => {a(l - x): ||x|| < 1} C A"1 \ Exp(A) (7.11.4.15)
Now if equality (7.11.4.3) were to fail, then (Exp(A),A_1 \Exp(A)) would
give a disconnection of A^"1, which would contradict the fact that Aq1 is
connected. ■
Homomorphisms of normed algebra map generalized exponentials into
generalized exponentials, and sometimes onto:
7.11.5 THEOREM If A and B are normed algebras and T G HBL(A, B)
is a homomorphism then
T{Aq j) C Bq-1 (7.11.5.1)
If A and B are complete, and T is onto, then
T{AZx) = B0_1 (7.11.5.2)
and
r-1^1) C A'1 + r_1(0) (7.11.5.3)
If in addition A-1 is connected then (7.11.5.3) holds with equality.
Proof: For (7.11.5.1) apply (7.11.2.5) with Bq1 in place of Aq1, taking K
to be the set T{Aq1). If B is complete and 6 G i?^1 is arbitrary, then by
(7.11.4.3) there is ra G N and dl9 d2, • • •, dm in 5 for which
6 = edl ed2 • • • edm G Exp(E) (7.11.5.4)
If T is onto then there are c1,c2,..., cm in A for which
dy = T(cy) for each y = 1,2,..., m (7.11.5.5)
By the continuity of T we have
a = eCl eC2 • • • eCm => b = T{a) with a G Exp(A) C Aq x (7.11.5.6)

7.11 Generalized Exponentials
295
This proves (7.11.5.2), and hence also
T"1^1) = Aq1 + T-^O) (7.11.5.7)
from which (7.11.5.3) follows at once. Finally if A~* is connected then
A~l = Aq1, so that (7.11.5.7) becomes (7.11.5.3) with equality. ■
If T G BBL(A,B) is an onto homomorphism then, in the notation of
(7.6.1.4), T = JA where J = T_1(0) is obtained by quotienting out an
ideal of A: now (7.11.5.3) says that the elements of A which have cosets
which are generalized exponentials must all be "Weyl" elements relative to
the homomorphism. Conversely, an element 6 G B~x will be a generalized
exponential if and only if there is implication
r_1(6) cr^r^o)
whenever T G HBL(A,5) is a homomorphism onto (7.11.5.8)
For consider the algebra
A = {/ S C([0,l],B):/(0) C Kl} (7.11.5.9)
together with the homomorphism T : A —► B defined by setting
T(f) = /(l) for each / G A (7.11.5.10)
If 6 G B'1 can be put in the form 6 = T{f) then {f{t):t G [0,1]} is a
continuous arc in B~x connecting 6 to an element of the form tl with t G K,
which in turn is in the same component as 1 G -B-1, forcing 6 G Bq1.
The connected component of the identity in the algebra A = C^ft)
can be described in terms of homotopy:
7.11.6 THEOREM If Q is a separated normal space, then
Coo^o"1 = {<t>^ Coo(n)"1:sgn(^) is contractible} (7.11.6.1)
where sgn(^) G C(S,S) is defined by setting
sgn(^)(w) = </>(u)/\<l>{u)\ for each wGH^G (^(n)"1 (7.11.6.2)
There is abelian group isomorphism
cM(n, q-'/c^n, c)^1 = [cun, s) (7.11.6.3)

296
7. Operator Algebra and Commutivity
Proof: If a and 6 are in C^ (H) * then we claim
ab'1 e C^Q)-1 ^=> sgn(a) ~ sgn(6) (7.11.6.4)
and
sgn(a)o-1 G CM(n)o J (7.11.6.5)
Indeed, if a and 6 are in the same connected component of C^Q)-1, which
is an open subset of the normed space (^(H), which is locally arcwise
connected, then by (7.10.5.18) there is a homotopy {ht)0<t<1 in C^Q)_1
with h0 = a and h± = b, which induces a homotopy sgn(ht) between sgn(a)
and sgn(6), and gives forward implication in (7.11.6.1). Also the family
(a/|a|*)0<i<1 constitutes a homotopy between a and sgn(a), interpreted for
the moment as an invertible mapping from ft to C. This proves (7.11.6.5),
and gives backward implication in (7.11.6.1). Finally, to prove (7.11.6.3)
when the scalar field K is complex, we have only to confirm that every
homotopy equivalence class of mappings from Q to S corresponds to some
coset in C00(n)~1/C00(n)^"1: but since C(f2,S) can be interpreted as a
subset of (^(n)_1 this is clear. ■
The condition that C'00(r2)_1 is connected takes the form
[CUn,S) = {[l]J. (7.11.6.6)
which is in a sense dual to the condition that ft be simply connected:
[CUS,n) = {[lU (7.11.6.7)
The condition that the space H is connected can also be expressed in terms
of the exponential function. If the scalar field K is complex then
n connected 4=> exp_1(l) = 2ttZ1 C C^fi) (7.11.6.8)
More generally, if a G C'00(n) is arbitrary then exp(a) = 1 if and only if for
each connected component K of Q there is n G Z for which a = 27rin on K.
7.12 CONTINUOUS FUNCTIONS
If A = 0^(0.) = (^(n, K) and if H is a separated normal space then there
is a duality between closed subsets of f2 and closed ideals of A. For if CO
and JCiwe shall write
K° = {aeC^Ciy.KC a_1(0)} C A (7.12.0.1)
and
J0 = n{c-1(0):cGJ} CO (7.12.0.2)

7.12 Continuous Functions
297
The analogue of Theorem 5.2.2 is immediate:
KCK' CQ=> (K')° C K° and K C {K°)0 (7.12.0.3)
and
JCJ'U^ (J')0 C J0 and J0 C (J0)° (7.12.0.4)
Using Urysohn's lemma (Theorem 3.12.4), we can get sharper results
analogous to the consequences of the Hahn-Banach theorem:
7.12.1 THEOREM If Cl is a separated normal space and K C ft is a
closed subset, then
K = (K°)0 (7.12.1.1)
and there is isometric isomorphism
a + K «—» 0|K : C^nyK0 = C^tf) (7.12.1.2)
In particular
Coo(20 = {a|jr:o€Coo(n)} (7.12.1.3)
Proof: If a; G H \ jfiT, then by Urysohn's lemma (Theorem 3.12.4) there is
u G C(n,[0,1]) for which u(u) = 1 and u = 0 on K, which means that
u G K° is not annihilated by u: thus, u £ [K°)0. This reverses the second
inclusion of (7.12.0.3) and proves (7.12.1.1). Towards (7.12.1.2) it is clear
that the restrictiion mapping is well defined on the quotient C00(n)/JK'0,
linear and multiplicative, and bounded with bound < 1. To see that it is
isometric, suppose that e > 0 is arbitrary, and put
He = | u G n: |a(w) | > e + sup |a(.) 11 (7.12.1.4)
Then by Urysohn's lemma there is ue G C(f2, [0,1]) for which ue = 0 on K
and ue = 1 on He. Now be = uea satisfies
b£ G K° and \\a - b£\\ < e (7.12.1.5)
and it follows that the quotient norm in (7.12.1.2) is < \\a>\K\\ + s. Since e
is arbitrary, the isometry (7.12.1.2) follows. It remains to be proved that
the mapping from a + K° to a* is onto C^K), which we do by proving
(7.12.1.3). We consider the real and the complex cases separately, and
suppose first that 6 G C00(iiC) satisfies
sup6(.) = 1 and inf &(•) = -1 (7.12.1.6)
K K

298
7. Operator Algebra and Commutivity
(7.12.1.7)
iw2x»i (7.12.1.10)
Putting
Ko={u€K:b(u)<-±}
K$ = {u€K:b{U)>±}
there is by Urysohn's lemma w0GC(n,[-|,|j) for which
u0 = -| on Kq and u0 = | on Xj" (7.12.1.8)
Evidently
||u0|| < | and 116!|| < | where bx = b - u0 on K (7.12.1.9)
Inductively we now define sequences (un) in (^(H), (6n) in C00(JK'), and
(jfiT"), (if+) of closed subsets of H, taking for each n
K~ = {weK:bn(w) < -(!)(§)"}
1C = {«S If: 6» >(|)(|)"}
using Urysohn's lemma to find un with
«.6C(n,-(i)(i)*,(i)(i)") «n = ± (J) (|r on K±
(7.12.1.11)
and finally
*>n+i = K ~ un = b ~ K> + *i + * * * + tO on ^ (7.12.1.12)
Since ||un|| < (|)(|)n and C'00(r2) is complete (Theorem 4.3.3), we can now
define
oo
«=E«»eCj(l) (7.12.1.13)
n=0
Since ||6n|| < (|)(|)n -> 0, it is clear that a = b on K, with ||a|| < 1 =
||6||. This proves that 6 = a* whenever 6 G (^(H) = C^HjR) satisfies
(7.12.1.6). More generally, if c G C'00(r2, R) is arbitrary we can write
c= (az + 0)ob with/? + a = supc(-) and 0 - a = inf c(-) (7.12.1.14)
K K
where 6 satisfies (7.12.1.6). Now, if a G C^Q) is given by (7.12.1.13) then
the composed mapping (az+/3) oa is an extension of c to H. This completes
the proof of (7.12.1.3) for real scalars. Finally, for complex scalars, we need
only extend real and imaginary parts separately. ■

7.12 Continuous Functions
299
The extension theorem incorporated in Theorem 7.12.1 is known as
Tietze's theorem: it is interesting that Urysohn's lemma is an immediate
consequence. Indeed, if c\{K0) C in^l^) C ft then the mapping 6 : K —►
[0,1] obtained by setting
K = c\{K0) U (n \ (KJ) b = 1 on cl(lf0) 6 = 0 on n \ int^)
(7.12.1.15)
is continuous, and if u = a : ft —► [0, l] is a continuous extension of 6
then u satisfies (3.12.4.3). In the special case of a metric space there is a
direct derivation of Tietze's theorem which bypasses Urysohn's lemma: if
be C^HjR) satisfies
inf 6(.) = 1 and sup6(«) = 2 (7.12.1.16)
K K
we obtain a continuous extension a of 6 to ft if we define
a{u) = b{u) if u G K
and
, x . _ dist(u,uf)b(uf) ._ ^ % ,.,
a(w) = inf —J.\,/ V\ if a; G n \ X
v y w'gk dist(a;,i;r) x
(7.12.1.17)
The continuity of a must be established separately for points interior to K,
points outside K, and points of the topological boundary of K.
We might remark that we could have used Tietze's theorem in the
proof of (7.10.6.1) instead of the explicit construction we actually used. On
the other hand, Theorem 7.10.7 serves to warn us that we need not expect
an analogue of Tietze's theorem in which the unit interval is replaced by,
for example, the circle.
Theorem 7.12.1 tells us something about certain special ideals of
C^Q): it is clear from (7.12.0.3) and (7.12.0.4) that if K C Q then
J = K° => J = (J0)° (7.12.1.18)
and, if in particular, ft is normal and separated, then
J = K°=* CM(n)/J = CM) (7.12.1.19)
If, in particular, the space f2 is compact then this holds for all closed ideals:
7.12.2 THEOREM If n is compact Hausdorff and J C C^Cl) is an
ideal, then
d(^) = (J0)° (7.12.2.1)

300
7. Operator Algebra and Commutivity
Proof: We begin by proving, for arbitrary a G (^(H),
J0 C int a"1 (0) => a G J (7.12.2.2)
Indeed, for each u G H \ J0 there is 6W G J for which &w(u;) = 1, and then a
neighborhood Uu = {u/efh |&w(k/)| > ^} of a; in Cl. By compactness there
is a finite subset fl'CO\ inta_1(0) for which
n\a_1(0) cn\inta_1(0) C |J Uu (7.12.2.3)
wen'
If we now put
&=£U>W (7.12.2.4)
then
6 G J and |6(-)| > 1/4 on n \ inta_1(0) (7.12.2.5)
It follows that the reciprocal 1/6 is bounded and continuous on the closed
set H \ a_1(0), and therefore by Tietze's Theorem 7.12.1 has a continuous
extension to H. If c G 0^(0.) is such an extension then we have
a = abce acJ C J (7.12.2.6)
We now claim that if a G (^(H), then
JQ C a"1 (0) => a G cl( J) (7.12.2.7)
Indeed, for each e > 0 there is by Urysohn's lemma ue G C^Q) for which,
if u G H,
0 < u£(cj) < 1; |a(w)| < |c => ue(u) = 0; |a(w)| > e => u£(cj) = 1
(7.12.2.8)
If we now take be = ueA then
||a-6j| = ||a(l-ue)|| < e and J0 C a_1(0) C int&J^O) (7.12.2.9)
By (7.12.2.2) be G J for each e > 0, giving (7.12.2.7). ■
As a special case of (7.12.2.2) it follows
J0 = 0 => J = C^Q) (7.12.2.10)
We note that in this case the function 6 of (7.12.2.4) has a continuous
reciprocal on the whole of H, so that we do not need to use Tietze's theorem

7.12 Continuous Functions
301
to extend it. It is also clear exactly what are the maximal ideals of C^fi):
in the notation of Definition 7.2.1
n compact => MLIC^n) = MRIC^H) = Ut}° :teCl\ (7.12.2.11)
Urysohn's lemma, and in particular (7.12.1.1), tells us that the algebra
A = 0^(0.) separates points of the space H, in the sense that if a;,a/ G H,
then
u/ /. v =» a{u') ^ a(w) for some a G A (7.12.2.12)
The converse to this is called the Stone- Weierstrass theorem. In the complex
case we need to assume that the algebra A is self-adjoint, in the sense that
A = A*={a:aeA} (7.12.2.13)
of course every subset of C^HjR) is automatically self-adjoint. Before
stating the Stone-Weierstrass theorem, we note Dini's theorem:
7.12.3 THEOREM If Q is a compact Hausdorff space and if (un) and v
in (7^(0, R) satisfy
for each t G H,0 < un{i) < un+1{t) —> v{t) as n —> oo (7.12.3.1)
then
||v-un||—>0 as n—> oo (7.12.3.2)
Proof: IfoOandiGfi there are Uf G Nbd(*) and Nf G N for which
n>Nf=> v(t) - e < un{t) < v(t) (7.12.3.3)
and
6 G U* => max(|t;(«) - v{t)\9 \uNte{s) - uN.{T)\) < e
By compactness there is a finite subset (]£C(] for which Q =
Now if n > maxten N* and if t G H is arbitrary then there is
which t G U*,, and we have
< Ht) - v(t')\ + +(v(t') - uN, («')) + |uw. (0 - aw. (t)|
t' t' t'
<36T ■
(7.12.3.4)
*' G H£ for
(7.12.3.5)

302
7. Operator Algebra and Commutivity
Dini's theorem is a partial converse to Theorem 1.8.2, where C^ft) is
proved to be closed in 1^(0,). We need it to see that \z\ is a uniform limit
of polynomials on [0,1]. Indeed, if (pn) is defined inductively by setting
p0 = 0 and pn+1 =Pn + \{z- p2n) for each n = 0,1,2,...
(7.12.3.6)
then we have, by induction on ra,
0 < pn{t) < pn+1(*) <y/t if 0 < t < 1 (7.12.3.7)
so that by (7.12.3.2)
0< sup Vi-pn{t)—>0 asn=^oo (7.12.3.8)
0<t<l
The Stone-Weierstrass theorem is established separately for real and
for complex scalars. The theorem for real subalgebras proceeds though
the medium of the corresponding theorem for sublattices of the partially
ordered space C^H, R):
7.12.4 THEOREM If Q is a compact Hausdorff space of more than one
point and if A C C^Q) is a self-adjoint subalgebra which separates points
of H, then
cl(A) = CUO) (7.12.4.1)
Proof: We begin with the real case, and observe that if A separates points
of H then it has the following "two-point transitive" property:
if a,/? G R and s ^ t m Cl there is a G A with a(s) = a and a(t) = f3
(7.12.4.2)
Indeed if 6 G A satisfies 6(5) ^ b(t) we can take
v; b{t) - 6(5) H b{s) - b(t) K J
Next, we claim that the closure of a subalgebra is a sublattice:
{a, 6} C cl(A) => {max(a,6),min(a,6)} C cl(A) (7.12.4.4)
Indeed, in view of the identity
max(a,6) = |(a + 6+|a-6|) min(a,6) = \{a + b- \a-b\) (7.12.4.5)
we need only establish
a e A => \a\ e cl(A) (7.12.4.6)

7.12 Continuous Functions
303
But this is clear from (7.12.3.8), since with the polynomials (pn) of
(7.12.3.6) we have
a G A => \a\ = lim \\a\\pn (^^j G c\(A) (7.12.4.7)
The real case of the Stone-Weierstrass theorem will now follow if we can
show that (7.12.4.1) holds when A is a sublattice with the two-point
transitive property. If this is the case then for arbitrary a G C^Q), e > 0 there
are b3 t G A for each pair s, t of distinct points of Q for which b3 t(s) = a(s)
and b3jt(t) = a(t). Now
ult = {*' € n: &,jt(t') < a{t') +e}e Nbd(s) (7.12.4.8)
for each s G H and each t G f2\{s}. By compactness there are finite subsets
Hf of H for which {U*ts G H*} s an open cover for ft. Now put
&f = min bea t for each t G n (7.12.4.9)
Evidently &£ G A and satisfies 6f (t) = a(t) and 64(-) < a(-) + e. Also
Ul = {t' G n: bt(t') > a{t') -e}e Nbd(t) (7.12.4.10)
for each £ G f2. By compactness there is a finite subset Cle for which
{£//:£ G H£} is an open cover for Q. Now put
6* =max6f (7.12.4.11)
<Gne * v ;
Evidently be G A and satisfies
a(.) - e < &(•) < a(-) +e on n (7.12.4.12)
This means that ||a — 6e|| < e, and since & is arbitrary, that a G cl(A).
Since a G (7^(0) was arbitrary, we have proved that A C C^Q) is
dense provided it is a linear sublattice, containing the constants, which has
the two-point transitive property. All this applies to c\(A) if A C C^ft) is
a subalgebra which separates points of H. Thus, we conclude that c\(A) is
dense in, and hence equal to, C00(r2) = C^H, R). Finally, if K = C is the
complex field, suppose that A C C^Q) is a subalgebra which separates
points of CI: then we claim
Re(A) = {Re(a): a G A} C (^(H, R) separates points of n (7.12.4.13)
and also that
C^Ct, R) fl A = {a G A: a(H) CR}C (^(H, R) is a subalgebra
(7.12.4.14)

304
7. Operator Algebra and Commutivity
Indeed (7.12.4.14) is immediately obvious, while if 5 ^ t in Cl there is a G A
for which a(s) ^ a(t), and now
either Re(a(s)) ^ Re(a(*)) or Re(2a(s)) ^ Re(2a(*)) (7.12.4.15)
giving (7.12.4.13). If in addition the subalgebra A has the self-adjoint
property (7.12.2.12), then also
Re(A) = C^H, R) fl A (7.12.4.16)
For inclusion one way is trivial; if A is self-adjoint then
aeA=> Re(a) = \{a + a*) G (^(H, R) fl A (7.12.4.17)
From (7.12.4.13), (7.12.4.14), and (7.12.4.16) and the real case of
(7.12.4.1) it follows that
C^n.R) =clRe(A) =clC00(f2,R)nA (7.12.4.18)
which at once gives the complex case of (7.12.4.1). ■
7.13 LINEAR FUNCTIONALS AND STATES
If A is a real or a complex normed algebra, then its dual space is a
Banach space, but not in any obvious way another algebra. It does have
"two-sided A-module structure": if a G A and / G A^, we can define
(/ . a) (6) = f{a • 6) and (a • /) (6) = f(b • a) for each 6 G A
(7.13.0.1)
The resulting products satisfy conditions like (1.1.0.2) from the definition
of a linear space over the field K. We also get an analogue of the normed
algebra condition (1.10.0.1): For each a £ A and each f E A*
||a./||<||a||||/|| and ||/ • a\\ < \\a\\\\f\\ (7.13.0.2)
Certain subspaces MCA will have the status of left or right submodules,
in the sense that
A-MCM or M-ACM (7.13.0.3)
This is the case if for example M = J° in the sense of (5.2.0.1), where
J C A is a right or a left ideal of the algebra A. If a G A, then we recall the
bounded operators La : A -► A and Ra : A -► A of (2.10.0.1) and (2.10.0.2),
and how (Theorem 3.10.4) they can be used to decide whether or not the
element a is almost invertible; then by Theorem 5.5.2 the same information
can be expressed in terms of the dual operators (LaY : A* —> A* and
(iJjt^t^At:

7.13 Linear Functionals and States
305
7.13.1 THEOREM If A is a normed algebra and a G A then there is
implication
{Ra)^ one-one => a almost left invertible => [RaV bounded below
(7.13.1.1)
and
(La)t one-one => a almost right invertible => {La)^ bounded below
(7.13.1.2)
Also
a not a topological left zero divisor 4=> [La)^ almost open (7.13.1.3)
and
a not a topological right zero divisor 4=> (Ra)^ almost open (7.13.1.4)
Proof: The implication (7.13.1.1) is just the combination of (3.10.4.3)
with (5.5.2.1) and (5.5.2.2), while (7.13.1.2) is (3.10.4.4) with (5.5.2.1) and
(5.5.2.2). For (7.13.1.3) we note that a is a topological left zero divisor if
and only if La is not bounded below, and then apply (5.5.3.2). For (7.13.1.4)
we repeat the argument with Ra m
It might be thought that a more natural "dual" for a normed algebra
would be the subset of multiplicative linear functionals on A, the bounded
homomorphisms from A to K:
7.13.2 DEFINITION A state on the normed algebra A is a linear
functional f £ A* for which
11/11 < 1 = /(l) (7.13.2.1)
and a character is a state for which
f[a . 6) = f{a)f{b) for each a, 6 G A (7.13.2.2)
Of course, unless 1 = 0 and A = 0, the norm of a state must actually
equal 1, and the characters are just the bounded homomorphisms from A
to K in the sense of (2.10.1.7). The null space of a character is both a
closed two-sided ideal of A, and a "maximal proper subspace" of A, and
therefore certainly both a maximal left and a maximal right ideal in the
sense of Definition 7.2.1. We can improve the first part of Theorem 7.13.1,
and determine almost invertibility using states:

306
7. Operator Algebra and Commutivity
7.13.3 THEOREM If A is a normed algebra and a G A, then there is
equality
[J 0 (/ • »rHO) = *""(A) = {a € A: jrf ||/ • Ra\\ = o}
/GState(A) 6GA ^ V ' }
(7.13.3.1)
and
U fl (»• Z)"1 (0) = -rigLt(A) = {« € A: b£ ||/ . L.|| = o}
(7.13.3.2)
Proof: We begin by proving the first equality of (7.13.3.1). If a G A is in
the left-hand side then there is a state / G State(A) for which A-a C /_1 (0),
and since the null space of / is closed we have
l£/_1(0) Dcl(A.a) (7.13.3.3)
so that a is not almost left invertible. Alternatively, this part of the
argument is contained in (7.13.1.1). Conversely, if a is almost left invertible,
so that 1 §? c\(A • a), then by the Hahn-Banach Theorem (5.4.1.3) there is
/ G A for which /(l) ^ 0 and A-cC /_1(0). If we inspect the proof of
(5.4.1.3) it is clear that we can arrange more, getting
'« = 1 and ii'i"***^*1 (7-13-3-4)
deriving inequality at the end from (3.10.4.3):
||a - 1|| < 1 => 1 € cl(A .a)<=>a<£ d]e{t{A) (7.13.3.5)
This gives the first part of (7.13.3.1). If we combine this with (7.13.1.1) we
now have
|J (f.A)0Da^(A)
/GState(A)
2{aeA:MJ\f.Ra\\=o}
2f6AA)IKI=0} (7-13-3-6)
This forces equality throughout (7.13.3.6), and finishes the proof of
(7.13.3.1). The argument for (7.13.3.2) is identical. ■

7.13 Linear Functionals and States
307
We conclude with some characterizations of the characters of
Definition 7.13.2:
7.13.4 THEOREM If A is a normed algebra and / G A1" satisfies /(l) =
1 then the following are equivalent
for each a G A, /(a) = 0 => /(a2) = 0 (7.13.4.1)
for each a G A, /(a2) = /(a)2 (7.13.4.2)
for each a G A, /(a) = 0 => {f{ab):b G A} = {0} (7.13.4.3)
for each a,6 G A, f{ab) = /(a)/(6) (7.13.4.4)
Proo/: If /(l) = 1 then
for each a G A, /(a - /(a)l) = 0 (7.13.4.5)
Combining this with (7.13.4.1) gives
0=/((a-/(a))2) = /(a2)-/(a)2
which is (7.13.4.2). If this is assumed then for each a, 6 G A we have
f{ab + 6a) = 2/(a)/(6) and hence implication
/(a) = 0 => f{ab + 6a) = 0 => f{{ab + 6a)2) = 0 (7.13.4.6)
Since (a6 — 6a)2 = 2a(6a6) + 2(6a6)a — 2(a6 + 6a)2 we have also implication
/(a) = 0 => f{{ab - 6a)2 = 0 => f(ab - 6a) = 0 (7.13.4.7)
Adding (7.13.4.6) and (7.13.4.7) gives (7.13.4.3). If this is assumed then,
using (7.13.4.5) again,
0 = /((a - f(a))b) = f(ab - f{a)b) = f(ab) - f(a)f(b)
which is (7.13.4.4). If this is assumed then (7.13.4.1) is obvious. ■
When the algebra is complete then every multiplicative linear
functional is a character:
7.13.5 THEOREM If A is a Banach algebra then
HL(A, K) = HBL(A, K) C State(A) (7.13.5.1)

308
7. Operator Algebra and Commutivity
Proof: Suppose that / : A —► K is multiplicative and linear, with /(l) = 1.
We claim that there cannot exist x £ A for which
||*|| < 1 = f{x) (7.13.5.2)
Indeed, if (7.13.5.2) holds, then
OO
y = Y,xU => y = x(x + y) (7.13.5.3)
n=l
and now
/(y) = f{x) + fix)fiy) = 1 + /(y) (7.13.5.4)
Since (7.13.5.4) is impossible, (7.13.5.2) cannot happen, which means that
/ must be a state. ■

8
Inner Products and Orthogonality
There is a point of view which says that the only normed spaces of any
significance are those in which the norm is derived from an "inner product."
8.1 INNER PRODUCTS
An inner product on a linear space X over the field K is a certain kind of
mapping of pairs of vectors into scalars:
8.1.1 DEFINITION An inner product on the linear space X is a mapping
(;) : X x X —► K for which, if x, y, z G X and s, t G K,
(sx + ty\z) = s (x \z)+t (y ; z) (8.1.1.1)
(y ; x) = (x ; y)~ the complex conjugate of (x ; y) (8.1.1.2)
(x ; x) > 0 (8.1.1.3)
(x;x) =0=>x = 0 (8.1.1.4)
Note that if K is the real field R then the complex conjugate of (x ; y)
is just (x; y). From (8.1.1.1) and (8.1.1.2) it is clear that
(z ; sx + ty) = s(z;x) + i(z ; y) (8.1.1.5)
Thus, an inner product is linear in the first variable and "conjugate linear"
in the second variable, as defined in (5.1.2.8). If (;) is an inner product on
X we shall write
y*{x) = (x ; y) for each x,y G X (8.1.1.6)
309

310
8. Inner Products and Orthogonality
Evidently the mapping y* is a linear functional on X, and the mapping
y —► y* : X —► LK{X, K) is conjugate linear in the sense of (5.1.2.8).
An inner product induces a norm: if x G X we shall write
||s|| = <x;s>1/2 = V^> (8.1.1.7)
Of course this definition is made possible by (8.1.1.3). We claim that ||-||
really is a norm:
8.1.2 THEOREM If (;) is an inner product on the linear space X then
||-|| is a norm, with the additional quadratic property that
||x + y||2 + ||x-y||2 = 2W2 + 2||y||2 (8.1.2.1)
for each x,y G X. Conversely if ||-|| is a norm on the linear space X which
satisfies the quadratic property then there is an inner product (;), given
by the formula
4Re(x;y) = ||x + y||2 - ||z-</||2 for each x,y G X (8.1.2.2)
from which the norm ||*|| can be derived by the formula (8.1.1.7).
Proof: If (;) is an inner product and ||-|| is given by (8.1.1.7), then the
conditions (1.1.1.1), (1.1.1.2), and (1.1.1.3) of Definition 1.1.1 are evident.
We must prove the triangle inequality (1.1.1.4). Explicitly, we must prove
that if x, y G X, then
||x + y||2<W2 + 2||x||||y|| + ||y||2 (8.1.2.3)
Since
Ik + y||2 = {x + y;x + y) = ||x||2 + (x ; y) + (y ; x) + \\y\\2
= ||x||2 + 2Re(x;y) + ||y||2
it is evidently sufficient to prove the Schwarz inequality
|<*;y>|<NI||y|| (8.1.2.4)
Necessary and sufficient is that (8.1.2.4) should hold when \\y\\ = 1: in this
case the intuitive idea of the argument is conveyed by the following diagram
inR2:

8.1 Inner Products
311
ToTT^I
The diagram of course is not necessary for the proof:
if ||y|| = 1 then ||x||2 - | (x ; y> |2 = \\x -(x;y) y\\2 > 0 (8.1.2.5)
We have, therefore, shown that an inner product space does indeed
carry a norm, and it is now an easy exercise to verify that the norm ||-||
must satisfy the quadratic property (8.1.2.1), and that also the (real part
of the) inner product can be expressed in terms of the norm by means of
the formula (8.1.2.2). If, in particular, X is a complex space, and if we
know the inner product to be linear in its first variable, then the argument
of Theorem 5.1.2 shows that
(x ; y) = Re (x ; y) - i Re (ix ; y) (8.1.2.6)
Thus in the complex case (8.1.2.2) implies
4 (x ; y) = \\x + y\\2 - \\x - y\\2 + i\\x + iy\\2 - i\\x - iy\\2 (8.1.2.7)
Suppose finally that ||-|| is a norm on the space X which satisfies,
in addition to the conditions of Definition 1.1.1, the quadratic property
(8.1.2.1), and then consider the mapping Re(;) : X X X —► R given by
the formula (8.1.2.2). It is clear that the conditions (8.1.1.3) and (8.1.1.4)
of Definition 8.1.1 are satisfied (with Re(;) in place of (;)), and also
that the relationship (8.1.1.7) holds. Towards (8.1.1.1), it is also clear
that
Re (tx ; ty) = \t\2 Re (x ; y) for each t G K
We claim that also, for each x,y,z G X,
(8.1.2.8)
Re (x + y ; z) = Re (x ; z) + Re (y ; z) (8.1.2.9)

312
8. Inner Products and Orthogonality
for we have
4Re(x;y)+4Re(y;z} = \\x + z\\2-\\x-zf + \\y + z\\2-\\y-z\\2
= (||x + 2||2 + ||y + 2||2)-(||x-2||2 + ||y-^||2)
= l(\\x + y + 2z\\2 + \\x-y\\2)
-!(||z + y-2*||2 + ||z-y||2)
using (8.1.2.1)
= L\\x + y + 2yf-±\\x + y-2z\\2
= 2||I(x + y)+2||2-2||I(x + y)-2||2
= 8Re(±(x + y);*)
In particular, taking y = 0,
4Re(x;z)+0 = 8Re(±x;z) (8.1.2.10)
and then putting this (with x + y in place of x) back in the previous
expression gives (8.1.2.10). Together (8.1.2.10) and (8.1.2.11) give (8.1.1.1)
for arbitrary rational s and t. The proof of (8.1.1.1) for real s and t now
follows by the continuity of the mappings
x-+\\x + y\\* -\\x-y\\2 :X -+R
for each y G X. Explicitly, if s G R and \s — sn\ —► 0 with rational (sn)
then
4Re(6x;y)-4Re(6nx;y) = ||6X+y||2-||6x-y||2-||6nx+y||2 + ||6nx-y||2
and
Ulac + y||2 - ||snx + y\\2\ = \\\sx + y|| - ||«nx + y|||
x(||«x + y|| + ||«nx + y||)
< |||5x + y|| - ||snx + y|||
x (2||«x + 2/|| + Hlac + 2/|| - ||«nx + y|||)
<|6-6n|||x||(2||6x + y|| + |6-6n|||x||)
using (1.2.0.11)
—► 0 as n —► oo ■
The fact that a quadratic norm can be derived from an inner product is
of almost no practical use to us. The fact that the inner product from which
such a norm can be derived can be expressed in terms of the norm is much

8.2 Orthogonality
313
more useful. The formulas (8.1.2.2) and (8.1.2.8) are sometimes termed
"polarization." The quadratic property (8.1.2.1) has a nice interpretation in
R2: it says that the sum of the squares on the diagonals of a parallellogram
is equal to the sum of the squares of the sides.
We met an early example of an inner product space in (1.6.0.3): if
Y = Z = K, then ||(y,*)||2 = (|y|2 + H2)1/2 = (yj/+^)1/2 = «V,*) 5 (v,*)>
where
<(y, z); (y', *')> = yy' + ^' if y, *,v\^eK (8.1.2.11)
evidently satisfying the conditions of Definition 8.1.1. More generally the
norm ||-||2 introduced in (1.9.0.3) also comes from an inner product, defined
by setting
(x ; y) = linJ ]T x(t)y(t)~: finite O0Cf] (8.1.2.12)
[ten0 J
The reader will remember that we gave a foretaste of the proof of the
Schwarz inequality (8.1.2.4) in Theorem 1.6.1.
8.2 ORTHOGONALITY
If X is an inner product space, then vectors x, y in X for which
<s;y> = 0 (8.2.0.1)
will be called orthogonal or perpendicular. We shall also write
x JL y (8.2.0.2)
If K C X is a nonempty subset we shall write
KL = {y eX:{x\y)=0 for each y G K} (8.2.0.3)
and refer to K1- as the orthogonal complement of K. The orthogonal
complement has properties similar to those of the annihilator K° of a sub-
space K:
8.2.1 THEOREM If X is an inner product space and i,t/Gl, then
y ±x<=>x±.y (8.2.1.1)
and
x JL x =► x = 0=^ x±.y (8.2.1.2)

314
8. Inner Products and Orthogonality
If K and H are nonempty subsets of X, then
K1- = [K')L (8.2.1.3)
is a closed subspace of X, where K' is the closed subspace generated by K,
and there is implication
K<ZH=>HL <ZKL (8.2.1.4)
K C KLL (8.2.1.5)
and
Kfl^CjO} (8.2.1.6)
Finally
{0}± = X and X± = {0} (8.2.1.7)
Proof: The first two implications follow from the Definition (8.2.0.2), and
the arguments for the rest are the same as the arguments of
Theorem 5.2.2. ■
We could also have included in Theorem 8.2.1 the observation that
K1- = KLLL (8.2.1.8)
analogous to (5.2.2.3). Also, if H and K are nonempty subsets, or sub-
spaces, then
[K U H)L = {K + H)L =KLf\HL (8.2.1.9)
One consequence of orthogonality is "Pythagoras' theorem":
xJLy^||x + y||2 = ||x||2 + |M|2 (8.2.1.10)
The reader will remember this from (8.1.2.6). Generalizing (8.2.0.2), we
shall refer to a system [xAjeJ of vectors in X as orthogonal iff
,-94i=>(z,.;xy) = 0 (8.2.1.11)
and as orthonormal, iff in addition
i = j=>(xi;xi) = l (8.2.1.12)
giving therefore a pairwise orthogonal system of unit vectors. An orthonor-
mal system of vectors is always linearly independent:

8.2 Orthogonality
315
8.2.2 THEOREM If X is an inner product space and {xj)jej is a system
of vectors in X, then
ixj)jeJ orthonormal =$■ [xj)jeJ linearly independent (8.2.2.1)
If (sn)nGN is a linearly independent sequence in X, then there is an or-
thonormal sequence (yn)nGN which is linearly equivalent, in the sense
n n
]T Kyy = ]T Kxj for each n e N (8.2.2.2)
y=i y=i
Proof: To say that [xj)jeJ is linearly independent is to say (6.1.0.1) that
(xy)y£Jo *s ^nearly independent for each finite subset J0 C J. If (xy)jGJ is
orthonormal and J0 C J is finite and (*y)yej0 is arbitrary in K, then
]T *jxj = ° => *fc = X] (*jxj ; xk) = \Y1 lixi ' xfc / = <° '» xk) = °
jGJo j'GJo \jGJo /
for each k £ J0, which proves (8.2.2.1). Toward (8.2.2.2), we construct two
sequences [zn) and (yn), starting with
* = 2i = ii?ii (8-2-2-3)
llxill
noting that x1 ^ 0 since (xn) is linearly independent, and then
z2 = x2- (x2 ; yx) yx and y2 = y-^-
ll22ll
noting that z2 ^ 0, since otherwise (x1,x2) would be linearly dependent.
Inductively, if (t/^yj,... , yn) is orthonormal and linearly equivalent to
(x1,x2,...,xn), define
n
*n+1 = xn+1 - £ (xn+1 ; y ■) y ■ and yn+1 = -^±1- (8.2.2.4)
y = 1 IFn+lll
Evidently zn+1 7^ 0, since otherwise xn+1 would be a linear combination of
(x1,x2,...,xn), and then Hy^J = 1 and
(yn+i 5 ty) = ° for each i = 1,2,..., n
Finally, the two systems (x1?x2,... ,xn,xn+1) and (yi,y2, • • • 52/n>2/n+i) are
linearly equivalent. ■

316
8. Inner Products and Orthogonality
The construction of (yn) is called the Gram-Schmidt process. As well
as "orthonormalizing" a linearly independent sequence (xn), it also serves
to detect linear dependence: if (x1?x2,... ,xn) is linearly independent and
(x1,x2,..., xn, £n+1) is linearly dependent, then (y1? y2? • • • ? 2/n) W1^- be or~
thonormal and linearly equivalent to (x1? x2,..., xn) and zn+1 = 0.
If the Gram-Schmidt process is just a generalization of the argument
(8.1.2.6) used in the proof of the Schwarz inequality, then it should come
as no surprise that there is a corresponding generalization of the Schwarz
inequality, known as (the finite) BesseVs inequality.
8.2.3 THEOREM If (ey)yGJ is a finite orthonormal system in an inner
product space X, then for arbitrary x £ X there is inequality
£l(*;<v>2<NI
(8.2.3.1)
with implication
Ml2 = E I <* s e;> I2 <=►x e E Kei (8-2-3-2)
and membership
(8.2.3.3)
Proof: By induction on the cardinal number of the set J the argument for
the Schwarz inequality and the Gram-Schmidt process gives
Ml8 = E !<*!«/> I' +
x-^(x;ey)ey
(8.2.3.4)
To see it directly just expand the second term on the right-hand side,
using the orthonormality of the sequence (e,-)yGj. This gives the inequality
(8.2.3.1), and also implication one way in (8.2.3.2). If there is equality in
(8.2.3.1), then
jeJ jeJ
The inclusion (8.2.3.3) is also clear, if we just write down the inner product
of the right-hand side with each ek. Finally, to finish the proof of (8.2.3.2),

8.3 The Nearest Point Theorem
317
suppose that x = YljeJ *jej ^ ^jeJ ^ej' an(* °^serve
tk = (x ; ek) for each k £ J
and hence
||s||2 = <s;s> = £|(s;ey>|2 .
jeJ
8.3 THE NEAREST POINT THEOREM
Certain subsets of an inner product space have "nearest points" to every
other point of the space:
8.3.1 THEOREM If K C X is a nonempty complete convex subset of
an inner product space X, then for each x £ X there is a unique point
y = EKx £ X for which
yeK and ||x - y\\ = dist(x, K) (8.3.1.1)
The mapping EK is idempotent and continuous.
Proof: This follows by a combination of the convexity of the set K and
the quadratic property of the norm. By definition of dist(x, K) there must
be a sequence (yn) in X for which
yneK and ||x -yn\\ —► dist(x,K) (8.3.1.2)
Rather unexpectedly, we claim
(yn) is Cauchy in X (8.3.1.3)
For each m and n in N we have
o<l|y„-ym||2 = ll(yn-s)-(ym-s)ll2
= 2||yn - x||2 + 2||ym - x||2 - ||yB + ym - 2x||2
= 2||yB - x||2 + 2||ym - x||2 - 4|| \{yn + yj - x||2
< 2||yB - x\\2 + 2\\ym - x\\2 - 4 dist(x,iT)2
since by convexity \{yn + ym) € K, and now this last expression tends to
0 as m,n —> oo. Thus, (yn) is Cauchy in X, and lies in the complete subset
K, and therefore has a limit y € K; evidently
||i — y\\ = lim||a; — yn\\ = dist(i,iif)

318
8. Inner Products and Orthogonality
This proves the existence of y G X satisfying (8.3.1.1). If also yf satisfies
(8.3.1.1), then
0 < \\y' - y\\2 = 2\\y' - x\\2 + 2\\y - x\\2 - 4||±(y' + y) - x\\2 < 0
so that y' = y. This proves the uniqueness of y G X satisfying (8.3.1.1),
and entitles us to speak of the mapping EK : x —► y. To see that EK is
continuous we follow the same kind of argument. If y = EKx and yf =
EKx\ then
0 < \\yf - y\\2 = 2||y' - s'||2 + 2\\y - s'||2 - 4|| \{y + y') - x'f
< 2dist(x/,JfiT)2 + 2\\y - x'||2 - 4dist(x/,JfiT)2
<2(||y-x|| + ||x-x'||)2-2dist(x',ii:)2
< 2| dist(x,K)2 - dist(x',K)2\ + 2\\x - x'\\ dist(x, K) + ||x - x'\\2
—> 0 as x1 —► x
remembering that dist(x, K) is a continuous function of x. ■
When the complete subset K is also a linear subspace then we find
that the mapping EK is also linear, and bounded:
8.3.2 THEOREM If K is a complete subspace of an inner product space
X then EK is linear, with H-Ej^ll < 1. There is equality
K + KL = X and K n KL = {0} (8.3.2.1)
and
KLL = K (8.3.2.2)
Proof: If x G X is arbitrary and y = ^x, then we claim
x-yeK1- (8.3.2.3)
For if y' G if and t EK are arbitrary, then y — tyf e K, so that
||x-y||2 = dist(x,y)2<||x-y + V||2
giving
||*y'||2 + 2 Re(* (x - y ; y')) > 0 (8.3.2.4)
If, in particular,
t = k(y';x-y) with A; G R

8.3 The Nearest Point Theorem
319
we get
I (x - y; y')2 (\\y'\\2k2 + 2fc) > o (8.3.2.5)
Taking -6 < k < 0 for sufficiently small 6 > 0 forces ||y'||2fc2 + 2k < 0,
which means that the only way to salvage (8.3.2.5) is to have
\(x-y,y')\2 = 0 (8.3.2.6)
Since y' £ K is arbitrary, this gives (8.3.2.3).
If we combine (8.3.2.3) with (8.2.1.6) we get (8.3.2.1). Thus (8.3.1.1)
can be rewritten in the form
yeK and x-yeKL (8.3.2.7)
This shows that EK = n^ coincides with the linear mapping of (2.5.1.10),
"the projection of X upon K in the direction of K-1" To see its bound we
recall the Pythagoras theorem (8.2.1.10): if y = EKx, then
\\EKx\\2 = ||y||2 < ||y||2 + ||* - y||2 = ||z||2 (8.3.2.8)
This has proved everything except (8.3.2.2), which is now easy. If x G KJ-L
is arbitrary and y = EKx, then x and y are both orthogonal to x — y, and
hence x — y JL x — y, giving x = y G K. ■
Equation (8.3.2.1) tells us that K C X is supplemented, in the sense
of (1.6.1.5). Of course, the continuity of EK already tells us the stronger
fact that K is complemented. We usually refer to EK as "the orthogonal
projection on X upon K." Since the only time we ever want to know that
EK is continuous is in the linear case, the slightly tedious argument at
the end of Theorem 8.3.1 can be ignored. Unless K = {0} we recall from
(2.5.1.10) that the norm of a projection on K is never less than 1. Thus
K±{0}^\\EK\\ = l (8.3.2.9)
We are now in a position to also substantiate some of the remarks following
the proof of Theorem 1.5.3: if K C X is a complete subspace of an inner
product space, then the "Riesz lemma" result (1.5.2.1) can be sharpened
to hold with t = 1.
We conclude here with a sort of converse to (8.2.1.3):
8.3.3 THEOREM If K, if, and G are subspaces of an inner product
space X, then there is implication
HCK + G^HnK-LcG (8.3.3.1)

320
8. Inner Products and Orthogonality
If, in particular, K is complete, then also
KCH^HCK+iHHK-L) (8.3.3.2)
Proof: For (8.3.3.1) we argue
H fl K1- C (K + G) fl K1- = G fl if-1 C G
For (8.3.3.2) we use (8.3.2.1) to write, for each y G H,
y = x + z with x E K and 2 G KL
Now if if C H we have
y-i = 2G(if + F)nxi = J?nifi
giving
yex+{Hf\KL)<ZK + {Hf\KL) ■
8.4 COMPLETENESS
An inner product space which is complete is called a Hilbert space. Since
(4.2.1.4) closed subsets of complete spaces are complete, Theorems 8.3.1
and 8.3.2 became slightly simpler:
8.4.1 THEOREM If K C X is a nonempty closed convex subset of a
Hilbert space, then there is a mapping EK : X —► X which is continuous
and satisfies
EK{x) e K and dist(x,K) = ||x - ^(x)ll f°r eacn x e X
(8.4.1.1)
If, in particular, if is a closed subspace of X, then K is complemented and
EK = tt^ e BL(X, X) (8.4.1.2)
Proof: This is just a summary of Theorems 8.3.1 and 8.3.2. ■
When X is a Hilbert space, then Bessel's inequality extends to
arbitrary orthonormal systems, and leads to an alternative construction of the
orthogonal projections EK:

8.4 Completeness
321
8.4.2 THEOREM If {ej)jej is an orthonormal system in a Hilbert space
X, then for arbitrary iGX there is inequality
]TKs;ey>|2<||s||2 (8.4.2.1)
with implication
ll*U2 = £ I <* 5 «y> I2 <=> * € cl I £ Key I (8.4.2.2)
yej V€J y
The generalized sequence [Yljej (x i ey) ey: finite J0 C J) converges, with
*"£<*; <V> <V € {ey:y € J}"1 = [ £ Key (8-4-2-3)
j€J \jeJ J
Proof: Recognizing the left-hand side of (8.4.2.1) as the supremum over
finite subsets J0 C J makes (8.4.2.1) an immediate consequence of (8.2.3.1).
This in turn ensures that the numerical sequence E-Gj | (x; e;) |2)
(finite J0 C J) is Cauchy in the sense of Definition 4.1.1. It next follows
easily that the vector-valued sequence {Yl3ej0 (x 5 ey) ey) is Cauchy in X.
If X is complete, then Theorem 4.2.2 ensures that this Cauchy generalized
sequence converges in X. From (8.2.3.4) it follows
IWI2 = £ I <* 5 <V> I2 + »*-£<*; «y> «ylla (8-4-2.4)
From (8.4.2.4) we get implication one way in (8.4.2.2), exactly as in
Theorem 8.2.3, and also membership (8.4.2.3). Finally if x is in the closed
subspace generated by the (e-) then the generalized sequence
(x — YljeJo (x *' ey) ey) converges to 0 as J0 —> J, so that (8.4.2.4) gives
equality in (8.4.2.1). ■
Everything we can say about orthonormal "systems" applies to
orthonormal sets: we just take the index set J to be a subset of the space
X. Under the partial order of inclusion, orthonormal subsets of an inner
product space X clearly satisfy Zorn's condition (1.11.1.2). Thus, by Zorn's
lemma (1.11.2)
orthonormal subsets of X are contained in maximal orthonormal sets
(8.4.2.5)
Theorem 8.4.2 gives us the nicest possible characterization of maximal sets:

322
8. Inner Products and Orthogonality
8.4.3 THEOREM Necessary and sufficient for the orthonormal set J C
X in the Hilbert space X to be maximal is that
X = cl^Ke (8.4.3.1)
eeJ
Proof: If (8.4.3.1) fails then we can find x G X for which
||x|| = 1 and xeJL (8.4.3.2)
For take x = x1/\\x1\\ where xx = XQ—YleeJ (xo > e) e w^n xo & c^YljeJ ^e*
Evidently the set J U {x} is orthonormal and properly contains J.
Conversely, if this is so, then equality must fail in (8.4.2.1), and hence by
(8.4.2.2) x is not in the closed subspace generated by J. ■
Theorem 8.4.3 leads us to call maximal orthonormal subsets of Hilbert
space orthonormal bases. To obtain an alternative expression for the
orthogonal projection EK on a closed space if C X we have only to find an
orthonormal basis {^j)j^.j for K. Then we have
ek{x) = YL (x ' ej) ei for each x e X (8.4.3.3)
Of course many different bases for K will give the same final answer.
The complemented subspace property means that different kinds of
nonsingularity coincide in a Hilbert space:
8.4.4 THEOREM If T G BL(X, Y) is a bounded linear mapping between
Hilbert spaces, then
T bounded below =► T left invertible (8.4.4.1)
and
T almost open =► T right invertible (8.4.4.2)
More generally
T proper => T regular (8.4.4.3)
Proof: The last part (8.4.4.3) follows from Theorem 3.8.2 and
Theorem 8.4.1. Now this together with (3.3.7.3), (4.4.4.1), and (3.8.3.12)
gives (8.4.4.1), and together with (3.4.7.2), (4.4.4.2), and (3.8.3.13) gives
(8.4.4.2). ■

8.5 Duality
323
In the notation of (3.3.1.3), (3.4.1.4), (3.6.1.3), and (3.6.1.4) this is
rleft BL(X, Y) = aleft BL(X, Y) (8.4.4.4)
and
f rieht BL(X, Y) = aright BL(X, Y) (8.4.4.5)
8.5 DUALITY
A Hilbert space furnishes its own dual:
8.5.1 THEOREM If X is a Hilbert space there is isometric conjugate-
linear isomorphism
y <—* y* : X = X^ (8.5.1.1)
given by the formula y* (x) = (x ; y)
Proof: The formula is (8.1.1.6), which associates with each vector y £ X
the (possibly discontinuous) linear functional y* : X —► K; now the Schwarz
inequality (8.1.2.4) shows that y* is always continuous, with
||y*|| < ||y|| for each y eY (8.5.1.2)
and then equality is obtained in (8.5.1.2) by applying y* to the vector x = y:
l|y||2 = v*(v)<l|y||||yl (8.5.1.3)
The mapping y —► y* : X —► XMs therefore conjugate linear and isometric,
but possibly not onto. If, however, 0 ^ f £ X^, we can find y0 £ X with
/(y0) = l and y0 e /^(O)1- (8.5.1.4)
This uses Theorem 8.3.2. We now claim
If x £ X is arbitrary we can write
x = {x - f{x)y0) + f(x)y0 (8.5.1.6)
and find
y*(x) = {x- f(x)y0 ; y0/lkl|2>+/(*) <% 5 Vo/IIVoll2> = 0+/(x) (8.5.1.7)
since y0 is orthogonal to x — /(«)y0 ^ /_1(^)- ■

324
8. Inner Products and Orthogonality
Under the correspondence y —► y* the orthogonal complement of a
subspace Y C X is mapped onto the annihilator Y° of the same subspace.
y* (x) = 0 for each x eY <=» y JL x for each iG7 (8.5.1.8)
The resemblance between Theorems 8.2.1 and 5.2.2 is therefore no accident.
The Hahn-Banach Theorem 5.3.1 is obvious for a Hilbert space: if f0
for a subspace Y C X, then by Theorem 8.5.1 we have / = y* for some
vector y G cl(Y), where for the moment we interpret y* as the restriction to
Y of the mapping x —► (x ; y). Clearly, however, the same vector y defines
a linear functional y* on the whole of X.
If T G BL(X, Y) is a bounded linear mapping between Hilbert spaces
then Theorem 8.5.1 shows that the dual mapping T^ : Y* —► X* can be
transferred to the spaces Y and X:
8.5.2 DEFINITION If X and Y are Hilbert spaces, then the adjoint of
T G BL(X, Y) is the mapping T* : Y -+ X given by
(r*y)* = y* • r = rf(y*) for each y G Y (8.5.2.1)
Equivalent ly,
(x ; T*y) = (Tx ; y) for each xeX,yeY (8.5.2.2)
Note that the inner products on each side of this may be defined in different
spaces. The adjoint has the same sort of properties as the dual; abstractly
the purely algebraic properties are those of an "involution."
8.5.3 THEOREM If X and Y are Hilbert spaces, then the mapping
T —► T* : BL(X, Y) —► BL(Y, X) is conjugate linear and isometric. There
is equality
(r*)*=r (8.5.3.1)
and
{sT + tS)* = sT* + tS* (8.5.3.2)
for each s,t G K and each S,T G BL(X, Y). If also U G BL(Y,Z), then
{UTY =T*U* (8.5.3.3)
Also
||r|| = ||T|| (8.5.3.4)

8.6 Positive Operators
325
and
||r*r|| = ilHlimi = llr||2 (8.5.3.5)
Proof: Out of all this only (8.5.3.5) need detain the reader more than a
moment: we shall give a proof of (8.5.3.5) in Theorem 8.6.3 below. ■
The identity 7": X —► X is its own adjoint:
I* = 1 (8.5.3.6)
The invertibility or singularity of T G BL(X, Y) is determined by that
of its adjoint. From (5.5.1.4) and (8.4.2.1), or directly, it is clear that
r_1(0) = [T*Y)L C X (8.5.3.7)
and
cl(rX) = [T*)-l{0)L C Y (8.5.3.8)
This gives about a third of the next result.
8.5.4 THEOREM If T G BL(X, Y) between Hilbert spaces, then
T dense 4=> T* one-one (8.5.4.1)
and
T almost open 4=> T* bounded below (8.5.4.2)
Proof: This is Theorem 5.5.2 and Definition 8.5.2. ■
In the notation of (3.2.1.4), (3.2.1.5), (3.3.1.3), and (3.4.1.4) this is
7rrightBL(X,Y)* =7rleftBL(Y,X) (8.5.4.3)
and
fright BL(X, Y)* = fleft BL(Y,X) (8.5.4.4)
8.6 POSITIVE OPERATORS
As well as its obvious similarity to the dual operator T^, the adjoint T* of
T G BL(X, Y) has some of the flavor of a generalized inverse for T. We
begin by specializing to the situation in which Y = X: here the adjoint is
a bit like complex conjugation.

326
8. Inner Products and Orthogonality
8.6.1 DEFINITION If X is a Hilbert space we shall say that T G
BL(X, X) is hermitian, or self-adjoint, iff
T* = T (8.6.1.1)
and that T G BL(X, Y) is positive iff in addition
(Tx ; x) > 0 for each x G X (8.6.1.2)
For example, the identity / : X —► X is positive, and real scalar
multiples of i" are self-adjoint. When the field K is the complex field C then
(8.6.1.1) is a consequence of (8.6.1.2):
8.6.2 THEOREM If X is a complex Hilbert space and T G BL(X,X),
then
T = 0 4=> (Tx ; x) = 0 for each x G X (8.6.2.1)
and
T = T* 4=> (Tx ; x) G R for each x G X (8.6.2.2)
Proof: Whether K is real or complex, the following polarization identity
holds for arbitrary T G BL(X,X):
4Re (Tx ; y) = (T(x + y) ; x + y) - (T{x -y);x-y) for each x,y G X
(8.6.2.3)
The reader need only expand the right-hand side. In the complex case it
follows, as in (8.1.2.7), that
4 (Tx ; y) = (T{x + y) ; x + y) - (T{x -y)]x-y)
+ * (T(x + iy) ; x + *y) - * (T{x - iy) ; x - *y) (8.6.2.4)
This makes (8.6.2.1) immediately obvious, and (8.6.2.2) not very hard to
see. ■
Theorem 8.6.2 fails in R2: the mapping T : (s,t) —> (—t,s) of rotation
through ^7r satisfies the right-hand side of (8.6.2.1) but not the left-hand
side of (8.6.2.2).
If T G BL(X, Y) then the operators T*T : X -> X and TT* : Y -> Y
are both "positive:"
8.6.3 THEOREM If T G BL(X,Y), then T*T G BL(X,X) is positive,
and if also S G BL(X, X) is positive, then for each x £ X
||r*||2<||(rr + s)z||H (8.6.3.1)

8.6 Positive Operators
327
and
||5x||2 < \\S\\(Sx;x) (8.6.3.2)
and
||(||5||-5)x||2<||5||(||5||||x||2-(5x;x)) (8.6.3.3)
Proof: For (8.6.3.1) we observe
||rx||2 = (Tx ; Tx) = (T*Tx ; x) < (T*Tx ; x) + (Sx ; x)
= ({T*T + S)x ; x) < \\(T*T + 5)x||||x|| (8.6.3.4)
Taking the supremum over ||x|| < 1 gives
||r||2 < ||r*r + 5|| (8.6.3.5)
Specializing to S = 0 proves the formula (8.5.3.5) from Theorem 8.5.3.
For (8.6.3.2) we argue, using (8.5.3.5),
||r*rx||2 < ||r*||2||rx||2 = ||rf (Tx; Tx) = \\t*t\\ (t*tx ; x) (8.6.3.6)
From (8.6.3.2) it is easy to see (8.6.3.3): the reader need only expand the
left-hand side. ■
Theorem 8.6.3 will enable us to express the invertibility of T in terms
ofrTandrr*:
8.6.4 THEOREM If T G BL(X, Y) and if S e BL(X, X) is positive, then
T bounded below => T*T + S invertible (8.6.4.1)
and
X ^ {0} =► T*T - \\T\\2I <£ BL_1(X, X) (8.6.4.2)
Proof: If T is bounded below, then (8.6.3.1) says that T*T + S is also,
which means by (8.4.4.1) that it is left invertible and hence, being self-
adjoint, invertible. For (8.6.4.2) find a sequence (xn) of unit vectors in X
for which ||Txn|| —> \\T\\ and substitute them into the right-hand side of
(8.6.3.3) with S = T*T - \\T\\2I. m
Theorem 8.6.4 has implications for self-adjoint operators: if S = S*,
then
S one-one 4=> S dense 4=> S one-one and dense (8.6.4.3)

328
8. Inner Products and Orthogonality
and
S bounded below 4=> S almost open 4=> S invertible (8.6.4.4)
8.6.5 THEOREM If T e BL(X,Y) is bounded linear between Hilbert
spaces, then
T one-one 4=> T*T one-one and dense (8.6.5.1)
and
T dense 4=> TT* one-one and dense (8.6.5.2)
Also
T bounded below 4=> T*T invertible (8.6.5.3)
and
T almost open 4=> TT* invertible (8.6.5.4)
Proof: From (8.6.3.1) it follows that, if S e BL(X,X) is positive,
(r*r + s,)-1(o) cr_1(o) c (r*r)_1(o) (8.6.5.5)
Specializing to S = 0 gives equality, and hence proves (8.6.5.1). Taking
adjoints in (8.6.5.1) gives (8.6.5.2). Taking S = 0 in (8.6.4.1) gives (8.6.5.3),
and taking adjoints in (8.6.5.3) gives (8.6.5.4). ■
We have a little more to say about self-adjoint operators:
8.6.6 THEOREM If S e BL(X, X) for a Hilbert space X, then
S = S* 4=> ||S||J - S positive (8.6.6.1)
If K = C is the complex field, then
5 = 5* =► {S + il, S - il} C BL"x(X, X) (8.6.6.2)
Proof: If ||5||/ — 5 is positive then, in particular, it is self-adjoint, which
forces S* = S. Conversely, if 5* = 5, then ||5||/—S is also self-adjoint, and
then also positive because {Sx;x) < ||5||||x||2. For (8.6.6.2) we observe,
using S* = 5,
(5 + iI)(S -U) = S*S + I={S- iI){S + il) (8.6.6.3)
and apply (8.6.4.1). ■

8.7 Regularity
329
8.7 REGULARITY
When an operator between Hilbert spaces is regular in the sense of
Definition 3.8.1, then we can arrange that the induced projections P and Q are
orthogonal:
8.7.1 THEOREM If X and Y are Hilbert spaces and T G BL(X,Y),
then
T e rBL1(X, Y) 4=> T* e W:{Y, X) 4=> T*T e rBL1(X, X) (8.7.1.1)
If T e rBL1(XJ Y) is regular there is unique S e BL(Y, X) for which
T = TST S = STS [ST)* = ST {TS)*=TS (8.7.1.2)
Proof: The first part of (8.7.1.1) is immediate:
T = TST => T* = T*S*T* (8.7.1.3)
To reverse the implication (8.7.1.3), substitute T* for T. To see that the
last part of (8.7.1.1) implies the first, we claim
rT = {T*T)U{T*T) =^T = T{UT*)T (8.7.1.4)
To see this, argue that the first part of (8.7.1.4) implies
{T{UT*)T-T)*{T{UT*)T-T) = T*T-T*T-T*T + T*T = 0 (8.7.1.5)
which with (8.5.3.5) gives the right-hand side of (8.7.1.4). Conversely, to
reverse this implication, assume (8.7.1.2), which gives
{T*T){SS*){T*T) = T*{TS){TS)*)T = T*{TS)2T = T*{TST) = T*T
(8.7.1.6)
This finishes the proof of (8.7.1.1), provided we can prove (8.7.1.2). To
satisfy (8.7.1.2), find projections P = P2 e BL(X,X) and Q = Q2 e
BL(Y, Y), using Theorem 8.3.2, with
P* = p P-1(0) = r_1(0) Q* = Q Q{Y) = c\{TX) (8.7.1.7)
and then define S = V as in (3.8.2.8) from the proof of Theorem 3.8.2.
The uniqueness of S follows from the uniqueness of P and Q: to see that
P and Q are both unique suppose that orthogonal projections E and F on
X have the same range and argue
F = FE = F*E* = (EF)* = E* = E m (8.7.1.8)

330
8. Inner Products and Orthogonality
If T £ BL(X,Y) is regular, then T*T is decomposably regular: more
generally, if S = S* e BL(X,X), then
5-^0) = (5*)"1(0) S X/c\{SX) (8.7.1.9)
The analogue of Theorem 8.7.1 holds with "Fredholm" in place of
"regular." We prove more:
8.7.2 THEOREM If X and Y are Hilbert spaces and T G BL(X,Y),
then
T upper semi-Fredholm 4=> T* lower semi-Fredholm (8.7.2.1)
and
T upper semi-Fredholm 4=> r*T Fredholm 4=> r*T Weyl (8.7.2.2)
If T e BL(X,y) is Fredholm, then
index(r*) = -index(r) (8.7.2.3)
Proof: The implication (8.7.2.1) follows from (8.7.1.1) and (8.5.3.7), while
the first implication of (8.7.2.2) follows from (8.7.1.1) and (8.6.5.1). Then
the second implication of (8.7.2.2) follows from (8.7.1.9). Finally, (8.7.2.3)
follows from (8.5.3.7) and (8.5.3.8). ■
Part of the argument for (8.7.2.1) tells us that
T essentially one-one 4=> T* essentially dense (8.7.2.4)
When we compare Definition 6.4.1 and Definition 3.9.1, and recall
Theorem 6.4.2, we have equivalence
T essentially one-one 4=> T not a left zero divisor mod KL0(X, X)
(8.7.2.5)
and
T essentially dense 4=> T not a right zero divisor mod KL0 (Y, Y)
(8.7.2.6)
When X and Y are Hilbert spaces, and we replace the finite rank
operators KL0(X,X) and KL0(Y,Y) by the compact operators KL(X,X) and
KL(y, y), then we get left and right Fredholm operators:

8.7 Regularity
331
8.7.3 THEOREM If X and Y are Hilbert spaces and T G BL(X,Y),
then
T not a left zero divisor mod KL(X,X) 4=> T left Fredholm (8.7.3.1)
and
T not a right zero divisor mod KL(Y, Y) 4=> T right Fredholm
(8.7.3.2)
Proo/: If T G BL(X, Y) is left Fredholm, with I-T'Te KL(X, X), then
it is upper semi-Fredholm, and by Theorem 6.11.3 we have P(T) one-one.
This gives implication, for arbitrary U G BL(X,X),
TU G KL(X, Y) => P(r)P(l7) = P(ri7) = 0
^P([/)=0^I7G KL(X, X) (8.7.3.3)
This proves implication one way in (8.7.3.1); the corresponding implication
of (8.7.3.2) folllows in the same way. For the converse we recall
Theorem 6.12.4: if T G BL(X, Y) is not upper semi-Fredholm, then there is
K G KL(X, X) for which (T — jfiT)-1(0) is infinite dimensional. Now, with
U = U2 a projection whose range is (T — jfiT)-1(0) we have
U & KL(X, X) and TU = KU G KL(X, Y) (8.7.3.4)
This reverses the implication of (8.7.3.1), and similar argument does the
same for (8.7.3.2). ■
When X and Y are Hilbert spaces we can simplify somewhat the proof
of Theorem 6.12.4. If T{X) ^ cl(rX), then neither J"1^)1- nor T(X) can
be finite dimensional (Theorem 6.2.3). If we find orthonormal sequences
(en) in T-^O) and (/J in T(X), then
oo
*=E-<0ren£KL(X,Y)
n=1 (8.7.3.5)
and
{em:men}<Z{T-K)-1{0)
and
K=JT±KTQfn€KL(X,Y) and {fm:m e N} C {T-K){X)L
(8.7.3.6)

332
8. Inner Products and Orthogonality
8.8 HILBERT ALGEBRA
If A is a linear algebra over K, then an involution on A is a mapping
* : A —> A such that, for each a, 6 £ A and each s,t £ K,
(*a+*&)* = sa*+f&* (8.8.0.1)
{a*)*=a (8.8.0.2)
(a&)* = b*a* (8.8.0.3)
1* = 1 (8.8.0.4)
8.8.1 DEFINITION A Hilbert algebra A is a Banach algebra with an
involution * for which
||a||2 = ||a*a|| for each a G A (8.8.1.1)
The condition (8.8.1.1) is also called "the B* condition," and more
usual terminology is to refer to our "Hilbert algebra" as either a "£*-
algebra" or a "C*-algebra." Indeed, it is more usual to apply the term
"Hilbert algebra" to the rather special situation of a Banach algebra which
is also a Hilbert space. Of course the most important example of
Definition 8.8.1 is the algebra of operators on a Hilbert space:
8.8.2 THEOREM If X is a Hilbert space, then
BL(X,X) is a Hilbert algebra (8.8.2.1)
If H is a topological space, then
(7^(0) is a Hilbert algebra (8.8.2.2)
Proof: If * is denned on BL(X,X) by (8.5.2.1) then (8.8.2.1) is
Theorem 8.5.3. If * is defined on C'00(r2) by setting
/*(*) = /(*)" for each t G H, for each / G C^Q) (8.8.2.3)
then (8.8.2.2) is clear, as may be checked by the reader. ■
It is easily checked that if (8.8.1.1) holds then also
||a*|| = ||a|| for each a £ A (8.8.2.4)

8.8 Hilbert Algebra 333
8.8.3 DEFINITION If A is a Hilbert algebra, then a G A is called normal
iff
a*a = aa* (8.8.3.1)
If, in particular,
a* =a (8.8.3.2)
then a is called hermitian, or self-adjoint. If instead
a*a = 1 = aa* (8.8.3.3)
then a is called unitary.
We shall define
h^(a) = \{a + a*) for each ae A (8.8.3.4)
Evidently /i+ : A —► A is bounded and real linear on A, and idempotent.
The set h+{A) of self-adjoint elements of A is a real subspace, and if A is
commutative a subalgebra:
if a = a* and 6 = 6* then (a& = (a&)* 4=> a6 = 6a) (8.8.3.5)
In the complex case we can say more:
8.8.4 THEOREM If A is a complex Hilbert algebra, then
A = ht(A) +ih+{A) (8.8.4.1)
and
K{A) Ccl(A_1) (8.8.4.2)
Proof: To verify (8.8.4.1) we have only to observe, if a G A is arbitrary,
a = 6 + ic with 6 = \{a + a*) and c = -\i{a - a*) (8.8.4.3)
Towards (8.8.4.2), we claim that if a = a* G A, then
0^eR=>c + tte A-1 (8.8.4.4)
To see this suppose r > ||a||2/2£, so that
(r + t)2>\\a\\2+r2 = \\a-ir\\2

334 8. Inner Products and Orthogonality
using the B* condition (8.8.1.1), giving
a + it = i{r + t)(l- i{r + i)~l{a - *>)) G A~l
using the geometric series argument of (4.4.5.15). ■
When A = BL(X,X) and t = 1 or t = -1 then (8.8.4.4) reduces to
(8.6.6.2) and (8.8.4.2) combines with (7.3.4.4) from Theorem 7.3.4 to give
implication
a = a* erA!=^ae A'1 A (8.8.4.5)
We saw a special case of this in (8.7.1.9).
We conclude here with a characterization of normal operators:
8.8.5 THEOREM If A is a Hilbert algebra and a G A is normal, then
||a*6|| = ||a6|| for each be A (8.8.5.1)
If X is a Hilbert space, then necessary and sufficient for T G BL(X, X) to
be normal is that
||r*x|| = ||rx|| for each x G X (8.8.5.2)
Proof: If a G A is normal and 6 G A is arbitrary, then
||a*6||2 = ||(a*6)*(a*6)|| = ||&*aa*6|| = \\b*a*ab\\ = ||(a&)*a&|| = ||a&||2
which is (8.8.5.1). If T G BL(X,X) is normal and iGl, then
||r*x||2 = (TT*x ; x) = (T*Tx ; x) = ||rx||2
giving (8.8.5.2). For the converse, use (8.6.2.1). ■
Condition (8.8.5.2) implies that (8.7.1.9) holds when S is normal: thus
also
T normal regular => T decomposably regular (8.8.5.3)
and
T normal Fredholm =► T Weyl (8.8.5.4)
8.9 ENLARGEMENTS
The enlargements of a Hilbert space should really be Hilbert spaces; for the
enlargement Q based on the natural numbers N this is almost never so:
X ^ 0 =► Q(X) not a Hilbert space (8.9.0.1)

8.9 Enlargements
335
Indeed, if 0 ^ e G X define x and y in l^X) by setting
x = (e,0,e,0,...) and y = (0,e,0,e,...) (8.9.0.2)
Then, in violation of (8.1.2.1),
II* + V\\2 + II* " yf = 2||e||2 * 4||e||2 = 2||x||2 + 2||y||2 (8.9.0.3)
A modification in the construction of Q does give rise to Hilbert spaces:
the idea is to replace the bounded structure on N used in Definition 1.9.2
by one which is "maximal" in a Zorn's lemma sense. Rather than do that,
we can achieve the same effect in a different way:
8.9.1 DEFINITION A Banach limit A : l^ -> K is a linear functional
for which
and
A(x*) = A(x)~
A(x) < limsupxn
for each iG^
if x = x* e l^
(8.9.1.1)
(8.9.1.2)
It is not hard to verify that these conditions imply
liminf xn < A(x) < limsupxn (8.9.1.3)
n n
xn < yn{n £N)^ A(x)n < A(y)n(n € N) (8.9.1.4)
y» = Vi(^N)^A(i) = A(y) (8.9.1.5)
Thus a "Banach limit" is a possible extension of the limit of a convergent
sequence to all bounded sequences. That such things exist at all follows
from the proof of the Hahn-Banach Theorem 5.3.2: the argument for
Theorem 5.3.1 extends to the situation in which limsup replaces a multiple of
the norm.
If X is a Hilbert space and if A is a Banach limit on l^ in the sense of
Definition 8.9.1, then we can define a Hilbert space enlargement QA{X) by
imposing the following "semi-inner product" on the normed space l^X):
(x;y)A = A(x. ; y.) for each x,y el^ (8.9.1.6)
Exactly as in (5.7.2.2), we are evaluating the functional A at the sequence
(x. ; y.) el ao for which
((*.; y.»n = <*»; y»> f°r each * e N (8.9.1.7)

336
8. Inner Products and Orthogonality
1 II
The vector space l^X) thus inherits a seminorm x —► (x ; x)A7 : evidently
(x ; x)A = 0 -<=> x e c0{X) (8.9.1.8)
It follows that the quotient space Q{X) = loo(X)/c0(X) is actually normed
by A, although not necessarily complete. To construct QA(X) we now need
only complete Q{X) with respect to this norm, as in Theorem 4.5.3.
If X and Y are Hiibert spaces and T £ BL(X, Y) is a bounded operator
then it is clear how we define QA(T) : QA(X) —► QA{Y). In fact, we can
regard QA(T) as the unique extension of Q(T) as defined in (2.7.2.3) from
Q{X) to QA(X). It is clear that all the conclusions of Theorem 2.7.3 hold
with QA in place of Qn. In addition,
QaCO = QA(r) (8.9.1.9)
For the record, (3.3.5.2) from Theorem 3.3.5 and (5.7.1.1) from
Theorem 5.7.1 hold with QA in place of Q:
8.9.2 THEOREM If T e BL(X, Y) is a bounded linear operator between
Hiibert spaces and if A : l^ —► K is a Banach limit, then there is implication
QA(T) one-one => T bounded below => QA(T) bounded below
(8.9.2.1)
and
QA(r) dense => T almost open => QA(T) open (8.9.2.2)
Hence also
T invertible <£=> QA(T) invertible (8.9.2.3)
Proof: For (8.9.1.2) repeat the proof of (3.3.5.2), and for (8.9.2.2) apply
(8.9.2.1) with T* in place of T. m
In practice while it is satisfying that the "enlargement" of a Hiibert
space should be another Hiibert space, we will find the original nonquadrat-
ic version from Definition 1.9.2 with ft = N does everything we need. The
same comments apply to the "essential enlargements" T*{X) and PX(X)
of Definition 6.7.4. The reader is invited to see whether he can invent a
modified version of P(X) which is always a Hiibert space, and for which
he can prove the analogue of Theorem 6.11.3. If he succeeds, he is then
invited to see whether his version has been able to tell him anything he has
not already learned from P(X).

9
Liouville's Theorem
and Spectral Theory
The "spectrum" of a linear operator generalizes the set of eigenvalues of a
matrix, and the range of a numerical function.
9.1 LIOUVILLE'S THEOREM
The differential calculus for mappings from the complex field into a normed
space is superficially the same as the real calculus of Definition 5.12.1:
9.1.1 DEFINITION If X is a complex normed space and / : Q -► X
is defined on a neighborhood ft C C of the complex number s, then the
derivative of / at 5 is the limit
If f'{s) exists for each sGflwe shall say that / is differentiable on H. If /
is differentiable on Q and if the mapping /' : Q —► X is continuous on H,
then / is said to be holomorphic on H.
If z : C —> C denotes the complex coordinate we shall also write
If mappings x : C —> R and y : C —► R are defined by the identity
z = x + iy (9.1.1.3)
337

338
9. Liouville's Theorem and Spectral Theory
then we shall define the partial derivatives of / at 5 by setting
f£M = lim|Mz/W:yW = y{s),t _> s| (9.1.1.4)
and
g(S) = lim |MzM:xW = ,(,),, _ SJ (9.1.1.5)
Thus if <7(u) = /(u + i/3) and /i(u) = /(a + zu) we have
|£(a + »7?) = </'(a) and |^(a + »7?) = &'(/?) (9.1.1.6)
where g' and /i' are defined as in (5.12.1.1). It is at once clear that if the
complex derivative (df/ dz) (s) exists then so do both the real derivatives
{df /dx)(s) and (df /dy)(s): they are also related by the Cauchy-Riemann
equation
The converse holds, but not for single points:
9.1.2 THEOREM If n G C is open and if / : n -> X maps n into a
Banach space X and has continuous partial derivatives df/dx and df/dy
for which
-L(s)=i-L(s) foreachsGH (9.1.2.1)
oy ox
then / is holomorphic on H.
Proof: Suppose s0 G H and suppose K = {t G C: \t — s0\ < R}: then we
claim
I—ol<*=>/W = ^/^7 (9-1.2.2)
dK
and hence
~&5) = 2^ / {I-**)* for each 5 with |5"5°' < * t9-1-2-3)
ax

9.1 Liouville's Theorem
339
Here we interpret
J gdz= I g(s0 + Reie)i Reie
dK -7T
do
(9.1.2.4)
whenever g is continuous from dK to X, using (4.2.2.5) and the
completeness of X for the integral. To prove (9.1.2.2) we use a special case of what
is known as Green's theorem. For certain compact subsets K' of f2 we have
equality
/ gdx + hdy= / ( -^ J dx A dy (9.1.2.5)
dK'
whenever dg/dy and dh/dx are continuous on K'. For example, (9.1.2.5)
is rather clear if K' is a rectangle {s + it: a_ < s < a+,/?_ <t< /?+}, if
we interpret the right-hand side as either of two repeated integrals, and the
left-hand side as the sum of four integrals which correspond to following
an anticlockwise path round the boundary. To establish (9.1.2.2) we need
(9.1.2.5) for sets of the form
K'5 = {t e C: \t - s0\ < R and \t - s\ > 6}
(9.1.2.6)
Here the boundary dKf5 is made up of the circle dK of (9.1.2.4), taken
"positively" or anticlockwise, together with another circle r5, of center 5
and radius 6, this time described "negatively" or clockwise.
If we apply (9.1.2.5) with Kf = Kf6 and ig = h = i[z — s)-1/ then we
find, for sufficiently small 6 > 0,
/^/^+/((i-£)<->-'/)^»<-»)
dk r6 K'
To finish the proof of (9.1.2.2) the reader should verify that
and that
v
fdz
-27rif(s)\
as S
(9.1.2.8)
(9.1.2.9)

340
9. Liouville's Theorem and Spectral Theory
The right-hand side of (9.1.2.2) is an instance of what is known as
"Cauchy's integral formula": the argument of Theorem 9.1.2 also extends
this to more general compact subsets K C Q. One consequence is the
maximum modulus principle and Liouville's theorem:
9.1.3 THEOREM If n C C is open and if / : n -> X is continuously
differentiate on f2, then, whenever K C ft is a compact disc,
sup ||/(5)|| < sup ||/(,)|| (9.1.3.1)
seK sedK
If, in particular, / is continusouly differentiable on C, then
/ bounded => f constant (9.1.3.2)
and
||/(5)|| —► 0 as \s\ —> 00 =► / = 0 (9.1.3.3)
Proof: The inequality (hence equality) (9.1.3.1) follows immediately from
Cauchy's integral formula (9.1.2.2). For (9.1.3.2) we differentiate the Cau-
chy formula, as in (9.1.2.3), and argue that if this holds for circles dK
of arbitrarily large radius then (df/dz)(s) = 0 for each s G C. Finally
(9.1.3.3) follows from (9.1.3.2), because first the left-hand side implies that
/ is bounded, therefore by (9.1.3.2) constant, and then the left-hand side
forces that constant to be zero. ■
The most famous consequence of Liouville's theorem is the
"fundamental theorem of algebra":
9.1.4 THEOREM If p : C —► C is a nonconstant polynomial, then p has
at least one zero in C; hence, if c G Cn there is s G Cn for which
zn + c^-1 +... + Cn = (z-s1)(z-s2)---{z- sn) (9.1.4.1)
Proof: If p is a nonvanishing polynomial on C, then l/p is continuously
differentiate on C, and we observe that
p nonconstant =
p(s)
0 as \s\ —► 00 (9.1.4.2)
By (9.1.3.3) it follows that if p is not constant, then l/p = 0, which is
impossible. Next (9.1.4.1) follows by induction on n: if p is the left-hand
side of (9.1.4.1) and p(s1) = 0, then
p=(z-s1)(zn-1+c'1zn-2 + ... + c'n_1) m (9.1.4.3)

9.2 The Spectrum
341
An alternative derviation of Theorem 9.1.4 is achieved by examining
the "winding number" of the mappings </>: S —> S induced by a nonvanish-
ing polynomial p : C —► C, in the sense of (7.10.6.8).
We remark that we have chosen to repeat for vector-valued functions
a simplified version of the very standard theory of "functions of a complex
variable." The reader is invited to see how, using the Hahn-Banach
theorem, he can extend results valid for numerical functions <f> : Q —> C, for an
alternative derivation of Theorems 9.1.2 and 9.1.3 in normed spaces.
9.2 THE SPECTRUM
The "spectrum" of a linear operator, or more generally a Banach algebra
element, is closely analogous to the range of a continuous function. While
the theory really works properly for complete algebras, we can as usual
say quite a lot in incomplete algebras. We will however look only at the
complex case.
9.2.1 DEFINITION If A is a complex normed algebra, then the spectrum
of an element a G A is the set
°a(°) = {seC:a-s <£ A'1} (9.2.1.1)
for which a — s is not invertible in A. The left and the right spectrum are
the sets
^A* M = {s e C:l ^ A{a - s)} = {s e C:a - s e cle{t{A)} (9.2.1.2)
and
<^ight(a) = {s G C: 1 g (a - s)A} = {s G C:a - 6 G aright(A)} (9.2.1.3)
The almost spectrum of a is given by
^W=^'(«)^fW (9.2.1.4)
where
^AftW = {* € C: 1 g cl(A(a - s))} = {s G C: a - 6 G aleft(A)} (9.2.1.5)
and
£PJght(a) = {5 G C: 1 g cl((a - *)A)} = {s G C:a - 6 G aright(^)}
(9.2.1.6)
There are some obvious inclusions among these sets:

342
9. Liouville's Theorem and Spectral Theory
9.2.2 THEOREM If A is a complex normed algebra and a G A, then
*T («) C *!f M ^ht(a) C a^"(a) (9.2.2.1)
and
*a(«) C aA(a) = o^(a) U o*#\a) (9.2.2.2)
If A is complete, then the spectrum and the almost spectrum are the same:
<^ft(a) = a]fft(a) and 5A*ht{a) = <£lght(a) (9.2.2.3)
Proof: The inclusions (9.2.2.1) are contained in (3.10.6.2) from
Theorem 3.10.6. This gives the first inclusion of (9.2.2.2), and the second equality
is (3.1.2.2). Finally, (9.2.2.3) follows from (4.4.5.12) and (4.4.5.13). ■
The almost spectrum is closed and bounded, and nonempty unless
A = 0:
9.2.3 THEOREM If A is a complex normed algebra and a £ A, then
5A(a) = c\dA(a) C {s e C: \s\ < \\a\\} (9.2.3.1)
and
cA(a) =0=^1 = O=>A = {O} (9.2.3.2)
Proof: By (3.10.6.1) both <7leftU) and crl&ht(A) are closed subsets of A,
and hence <7|fft(a) C C and aTAe (a) C C are the counterimages of closed
sets by the continuous mapping a — z : C —► A. This proves the first part
of (9.2.3.1); for the second suppose that \s\ > \\a\\ and observe that
a fd\n II / cl\ , II II , / <i\ ||
(9.2.3.3)
To establish (9.2.3.2) we begin by assuming that A is complete, so that
the almost spectrum and the spectrum coincide, and note that the resolvent
function
(a - z)-1 : C \ oA{a) —► A (9.2.3.4)
is holomorphic:
±{a-z)-l = {a-z)-2 (9.2.3.5)
Evidently also
IKa-s)"1!!—>0 as \s\—► oo (9.2.3.6)

9.2 The Spectrum
343
Thus, we may use (9.1.3.3) to argue
aA(a) = 0 => (a - s)_1 = 0 =► 1 = (a - s){a - s)_1 = 0 (9.2.3.7)
To prove (9.2.3.2) we pass from A to its completion A, and observe, using
(4.5.6.3) and (4.5.6.4) for example, that
^(a)CaA(a)CaA(a) ■ (9.2.3.8)
Of course Theorem 4.5.7 tells us
ax(a) = aA(o) (9.2.3.9)
The spectrum and the almost spectrum of elements of a normed algebra
behave very properly when under attack by polynomials:
9.2.4 THEOREM If a e A for a normed algebra A and if p : C -> C is a
polynomial, then there is equality
*aP(<0 =P°a(°) (9.2.4.1)
and
*aP(«0 =P*a(«0 (9.2.4.2)
Proo/: The left-hand side of (9.2.4.1) and of (9.2.4.2) is defined using
(1.10.1.7), while the right-hand side is of the form {p(t):t G K} for K =
(jj{a) C C. If p is not constant and t £ C then by (9.1.4.1) we can write,
with 5 e Cn and 0 ^ c0 G C:
p - t = c0(2 - sJiz -82)---{z- sn) (9.2.4.3)
and hence also (1.10.1.7)
p(a) - t = c0(a - 5x)(a - s2) • • • (a - sn) (9.2.4.4)
By (7.1.3.6) and induction on n it follows that
t e oAp{a) *=> {6l552,... ,5n} n cA(a) ^0<=>te pcA{a) (9.2.4.5)
The analogue for oA is exactly similar, using Theorem 3.10.6.
If the polynomial p is constant then (unless A = 0) (9.2.4.1) and
(9.2.4.2) follow from (9.2.3.2); if A = 0, then of course both sides are
always empty. ■

344
9. Liouville's Theorem and Spectral Theory
Theorem 9.2.4 holds separately for left and for right spectra, although
we have not yet proved that these are always nonempty. There is also an
extension to rational functions:
9.2.5 THEOREM If a G A for a normed algebra A and if p and q are
polynomials, with q ^ 0 not constantly zero, then
q-\0) n aA{a) =0^aA (?) (a) = ^aA{a) (9.2.5.1)
and
q-1(O)naA(a)=0^aA(^ya) = P-aA{a) (9.2.5.2)
Proof: Here of course
-(a) = qM^pU) = p{a)q{a)-1 (9.2.5.3)
Q
We rely on (9.2.4.1) to see that it is well defined and on Theorem 7.1.4
to see that the order of the factors is immaterial. For each iGCwe now
write, assuming that p/q is not constant,
I t= Llll = ±c (* _ Sl){z - S)... (* _ 8) (9.2.5.4)
q q q
and hence
-(a) " * = <l{a)~lc0(a - sx){a - s2) • • • (a - sn) (9.2.5.5)
Now the argument of Theorem 9.2.4 applies. ■
The reader is invited to reflect upon why we cannot replace oA by aA
in the left-hand side of (9.2.5.2).
We conclude with a "semicontinuity" property for the almost spectrum:
9.2.6 THEOREM If (an) and a in the complex normed algebra A, and
(sn) and 5 in C, satisfy
IK "all —>° and K"5I—>° (9.2.6.1)
then there is implication
sn e cA(an) for each n G N =► 5 G cA{a) (9.2.6.2)

9.3 The Spectral Boundary
345
Proof: We prove this separately for left and right spectra. If 5 is not in
c^|ft(a), so that a — s £ cleit(A), then since an — sn converges to a — s we
eventually find an - sn & cle{t{A). m
The "upper semicontinuity" of a mapping u> : A —► P (f2) from A into
the subsets of a topological space Q is defined to mean that if a G A and
w(a) C int(V) then there is U G Nbd(a) for which u(U) C V.
9.3 THE SPECTRAL BOUNDARY
We begin by looking at the "fine structure" of the spectrum:
9.3.1 DEFINITION If a G A for a normed algebra A we define the left
and right approximate eigenvalues of a to be the sets
fleft
(a) = j« e C: ^nf x \\ax\\ = o| = {s € C:a - , e f'^A)} (9.3.1.1)
and
^"W = {seC: ^fajf x ||xa|| = o| = {, G C:a - * G f^*(A)}
(9.3.1.2)
The left and the n'flrfa eigenvalues of A form the sets
*!?*(«) = {* S C:L7i.(0) ^ {0}} = {S € C:a - 3 <= 7rleft(^)} (9.3.1.3)
and
^8ht(a) = {s G C:12-i,(0) ^ {0}} = {s G C:a- 6 G ^ht(A)}
(9.3.1.4)
The exponential spectrum of a forms the set
^aM = {^G C:a-6g Aq1} (9.3.1.5)
Here Aq1 is the connected component of 1 in A-1 as in Definition 7.11.1.
Theorem 9.2.2 extends to include the sets of Definition 9.3.1:
9.3.2 THEOREM If a G A for a normed algebra A, then
tJFM <= ***(«) C a™ (a) C ^(a) (9.3.2.1)
and
TA8ht(«) C ?right(a) C 5*"(a) C «#**(«) (9.3.2.2)

346
9. Liouville's Theorem and Spectral Theory
and
n^(a) U <^ht(a) = «™*{a) U a™ (a) = cA{a) C eA{a) (9.3.2.3)
Proof: For (9.3.2.1) and (9.3.2.2) we need only refer to (3.10.6.2) and
(3.10.6.3) from Theorem 3.10.6, and for the first two equalities in (9.3.2.3)
we refer to (3.10.6.4). The last inclusion of (9.3.2.3) simply reflects the
inclusion Aq1 C A~l. ■
The approximate eigenvalues, and if A is complete the exponential
spectrum, form closed subsets:
9.3.3 THEOREM If a e A for a normed algebra A, then f||ft(a) and
f^g (a) are closed and bounded in C, and satisfy
^(a)Crf(a)nff(«) (9.3.3.1)
which is nonempty if A ^ 0. If A is complete, then also eA{a) is closed and
bounded, with
deA(a) CdaA{a) (9.3.3.2)
Proof: It is clear from (3.10.6.1) that f J|ft(a) and f^ght(a) are both closed,
being continuous counterimages of closed subsets of A. It is also obvious
from Theorem 9.2.3 and Theorem 9.3.2 that they are both bounded. From
(3.10.6.5) it follows that
dclXft{a) C TTJlsht{a) C 6™ht{a) (9.3.3.3)
and
^fWCff(a)C5f(a) (9.3.3.4)
Now (9.3.3.1) follows from (3.10.6.7). Since (9.2.3.2) says that cA(a) is
nonempty if A ^ 0, and of course bounded, and since C is connected,
(7.10.3.8) implies that daA{a) is nonempty if A ^ 0.
Towards (9.3.3.2), it is clear from (7.11.2.3) that eA{a) is closed in C
if A-1 is open in A, and since {s G C: \s\ > \\a\\} is disjoint from oA{a) and
connected in C there is inclusion
eA{a) C {s e C: \s\ < \\a\\} (9.3.3.5)
Finally, (7.11.2.4) gives inclusion
3«a(«) C f!T(«) n ^ght(a) C eA(«) (9.3.3.6)
so that (7.10.3.1) gives (9.3.3.2). ■

9.3 The Spectral Boundary
347
The exponential spectrum is contained in the connected hull of the
spectrum: whether or not A is complete there is inclusion
£a(«) C n°A{°) (9-3-3.7)
Here we are using a modified version of Definition 7.10.1: if K C C is
bounded write
nK = riooK = 'ntK if \t\ > sup \s\ (9.3.3.8)
seK
If a G A is arbitrary, then (3.10.6.4) from Theorem 3.10.4 gives the
first two equalities of (9.3.2.3) and
oA{°) = *r («) U ^ght(a) = <^ht(a) U ^(a) (9.3.3.9)
and
^)=-fWu5f(«) (9.3-3.10)
where r^ft (a) consists of those 5 G C for which the operator La — si fails to
be "closed" in the sense of Definition 3.3.1. If in particular A = BL(X, X)
for a normed space X, then in addition
aA{T) = TlXn{T) U r*ght(r) for each T G A (9.3.3.11)
noting that rAlg (T) consists of 5 G C for which T — si is not open and
using (3.3.7.1).
The spectral mapping theorem for polynomials extends to the
approximate eigenvalues, and almost to the eigenvalues:
9.3.4 THEOREM If a G A for a normed algebra A and if p is a
polynomial, then there is equality
vp{a) = pLj{a) for each « G {a^ft,a^ght,f^ft,f7ht} (9.3.4.1)
There is also inclusion
P^ia) C 7r^eftp(a) and p^lght(a) C 7r^lghtp(a) (9.3.4.2)
with equality for nonconstant polynomials, and inclusion
eAP{°) ^P^W (9.3.4.3)

348
9. Liouville's Theorem and Spectral Theory
Proof: If 6 and c are in A there is implication
{6, c} C A \ w(A) 4=> {6c, cb}CA\ w(A) (9.3.4.4)
for each of the w of (9.3.4.1). This together with the factorization of
(9.2.4.3) proves (9.3.4.1) for nonconstant p. If p is constant then
(unless A = 0) equality (9.3.4.1) follows from the fact that a;(a) ^ 0, and of
course if A = 0 then both sides are always empty. The same argument
proves (9.3.4.2), since (9.3.4.4) holds with w = 7rleft and with w = 7rrisht.
Finally, (9.3.4.3) follows from the one way implication
{6, cjCA"1^ {6c, c6} C Aq1 m (9.3.4.5)
Theorem 9.3.4 also extends to rational functions p/q for which the
zeros of the denominator q are disjoint from the spectrum 0A(a), or for
(9.3.4.3) the exponential spectrum eA(a). We need only adapt the proof of
Theorem 9.2.5.
9.4 SUBALGEBRAS AND QUOTIENTS
If T : A —► B is a homomorphism of normed algebras, then (7.6.0.1) tells
us that the image T(a) of an invertible element a £ A must be invertible
in E.
9.4.1 THEOREM If T : A —► E is a homomorphism of normed algebras
and a £ A, then
"bW^xW (9-4.1.1)
If T is continuous, then also
a^{t{Ta)CalXn{a) and af'(Ta) C ^ght(a) (9.4.1.2)
and
eB{Ta) C eA(o) (9.4.1.3)
If instead T is one-one, then, in the other direction,
ir^c) C n%{t{Ta) and ^"(To) C 7r£ght(a) (9.4.1.4)
If T is bounded below and continuous, then
f!?ftM C ?£ft(ra) and f*ght(a) C f£ght(Ta) (9.4.1.5)

9.4 Subalgebras and Quotients
349
Finally, if A and B are complete and T is continuous and onto, then
*B(r<0= p| eA(a + d) (9.4.1.6)
Td=0
Proof: For (9.4.1.1) apply (7.6.0.1) to elements a - s with s G C. When
T is continuous observe
T{A\cleit{A)) CB\cleit{B) and T{A\cT'lsht{A)) C5\aright(B)
(9.4.1.7)
giving (9.4.1.2). Also, (7.11.5.1) says that T maps Aq1 into Bq1, giving
(9.4.1.3). For (9.4.1.4) we claim, if T is one-one, that
T-^BXn^iB)) CA\7rleit{A)
and (9.4.1.8)
T~1{B \ 7rright(J5)) C A \ 7rright(A)
If, for example, a G A and Ta is not in 7rleft (B) then, for each x G A,
ax = 0 =► r(a)r(x) = T(ax) = 0 =► T(x) = 0 =► x = 0 (9.4.1.9)
Similarly, for (9.4.1.5), if ||-|| < ik||r(-)|| on A and ||-|| < h\\{Ta) • || on B
then
\\x\\ <ik||rx|| </iA;||(ra)(rx)||
= /iA;||r(ax)|| < /iA;||r||||ax|| for each x G A (9.4.1.10)
Finally, (9.4.1.3) gives inclusion one way in (9.4.1.6). Conversely, (7.11.5.2)
says that T maps Aq1 onto Bq1 so that, if 5 §? eB{Ta), there is a' G Aq1
for which Ta — s = Taf, but now
rf = a'-a + 5=l>rfG r_1(0) and 5 £ eA(a + d) ■ (9.4.1.11)
Of course, (9.4.1.1) holds separately for left and right spectra. The
contrast between the inclusions of (9.4.1.1) and (9.4.1.2) and those of (9.4.1.5),
together with (9.3.3.1), gives rise to a certain "spectral permanence":
9.4.2 THEOREM If 1 G A C B for a normed algebra B and a G A, then
^A(a)Cafl(a)CaA(a) (9.4.2.1)
If A is complete and J is a closed two-sided ideal of A, then
°a/A" + J)^r\cAa + d)^ *I°a/j{* + J) (9.4.2.2)

350
9. Liouville's Theorem and Spectral Theory
Proof: For (9.4.2.1) we argue, using (9.3.3.1), (9.4.1.2), and (9.4.1.5),
**a(«) £ rlf{a) C f*f (a) C aB(a) C aA{a) (9.4.2.3)
The first inclusion of (9.4.2.2) is just (9.4.1.1), while for the second we argue,
using in succession (9.3.2.3), (9.4.1.6), and either (9.3.3.7) or (9.3.3.2) with
(7.10.3.3),
D °A{a + d) C f| eA{a + d) = eA/J{a + J) C WA/J{a+ J) m (9.4.2.4)
deJ dej
Example (7.10.3.12) cautions us against concluding from (9.4.2.2) that
d( f| "a(* + «*)) C aA/J(a + J) (9.4.2.5)
To see that this is so we will have to work a little harder.
If T : A —► B is a homomorphism of normed algebras then we shall
refer to oB{Ta) as the T-Fredholm spectrum of a, with similar
terminology for the various subsets of the spectrum introduced in Definition 9.3.1.
Clearly, anything that can be said about the spectrum of a can also be
said about the spectrum of Ta, which is the Fredholm spectrum of a. The
"Weyl spectrum" of a is defined in terms of the "Weyl elements" of A of
Definition 7.6.1:
9.4.3 DEFINITION If T : A -> B is a homomorphism of normed
algebras then the T- Weyl spectrum of a G A is the set
uT(a) = {seC:a-s<£A-1+T-1(0)}= f]cA{a + d) (9.4.3.1)
Td=Q
and the almost T- Weyl spectrum is the set
uT(a) = P| cA(a + d) (9.4.3.2)
Td=0
We can make analogous definitions for left and right spectra, eigenvalues
and approximate eigenvalues. The almost Weyl spectrum is closed and
bounded, and satisfies half the spectral mapping theorem:
9.4.4 THEOREM If T : A —► B is a homomorphism of normed algebras
and a G A, then uT(a) is closed in C and satisfies
cB{Ta) C wT(a) C aA{a) and oB{Ta) C CT(a) C cA{a) (9.4.4.1)

9.4 Subalgebras and Quotients
351
with
uT(a) C uT(a) (9.4.4.2)
There is equality in (9.4.4.2) if A is complete. If p is a polynomial
0JAp(a) C puT(a) and uTp(a) C puT(a) (9.4.4.3)
Proof: It is clear that uT(a) is the intersection of closed subsets of C and
hence closed. The first inclusion of (9.4.4.1) follows from
A'1 C A'1 + r_1(0) C r-1^"1) (9.4.4.4)
and the second from the analogue of (9.4.4.4) for "almost invertible"
elements of A and B. Since every invertible element is almost invertible the
inclusion (9.4.4.2) follows, together with the converse for complete A.
Finally, (7.6.4.1) says that the product of Weyl elements is Weyl, so that the
first inclusion of (9.4.4.3) is the same as the proof of (9.3.4.3), and then the
second inclusion is similar. ■
The element
r 11 + T n 1 n
EA = BL{X2,X2)
U + I 0
0 V -I
of (7.6.4.9) provides an example of the failure of equality in (9.4.4.3), taking
T : A —> B = A/J to be the Calkin homomorphism and the polynomial
p = [z + l)(z — 1). The same example can also be used to show that
equality can fail in (9.3.4.3): we need to know that the invertible group
A-1 is actually connected when X is a Hilbert space, and then observe
that
A'1 = Aq1 =► uT{a) = eB{Ta) (9.4.4.5)
We are ready to finish the job begun in Theorem 9.4.2:
9.4.5 THEOREM If T : A —► B is a homomorphism of Banach algebras
for which T{A) is closed in B, then for each a G A
duT{a) C cB{Ta) C uT{a) (9.4.5.1)
Proof: If J = T_1(0) we can factorize
B^A = B^-A/J^-A (9.4.5.2)
and observe that U is bounded below and V is onto. By Theorem 9.4.2

352
9. Liouville's Theorem and Spectral Theory
(9.4.5.1) holds with U in place of T, and the weaker
oB{Ta) C uT(a) C rjaB{Ta) (9.4.5.3)
holds separately for U and for V. It follows that also (9.4.5.3) holds for
T = UV:
u>T(a) C rjaA(a + J) C rjrj(7B(Ta) = rjaB{Ta)
From (9.4.5.3) and Theorem 7.10.3 it follows that
drjuT{a) C aB(Ta) C uT(a) (9.4.5.4)
Thus suppose that 5 G duT(a): if there is a hole if in uT(a) for which
5 G dH, then, taking arbitrarily t £ H, apply the mapping / = (2 — £)_1
to an element of the form a + d with d G T_1(0) and argue
f{s) G /(&#) = drjfojT(a) = drjujTf(a + d) C cBTf{a + d) = fcBT{a)
(9.4.5.5)
Here the first equality is because the mapping / = (2 — £)-1 sends the
hole H onto the unbounded component of the image of ujT(a), the second
equality is because the one-way spectral mapping theorem (9.4.4.3) extends
to rational functions, the third inclusion is (9.4.5.4) applied to f{a + d), and
the final equality is (9.2.5.1) applied to Ta G B. Acting on (9.4.5.5) with
the polynomial z — t now gives inclusion 5 G oB[Ta). To finish the proof we
must recall (7.10.3.11), which says that if 5 G dK\dr]K, then we can write
5 = limn sn with sn G dHn for a sequence [Hn) of holes in K. Applying
this with K = uT(a) gives, by what we have just proved,
5 = lim sn G cl cB [To) = oB [Ta) m
9.5 THE SPECTRAL RADIUS
The spectral radius measures how far away from 0 traces of the spectrum
can be found:
9.5.1 DEFINITION If A is a normed algebra and a £ A has nonempty
bounded spectrum then the spectral radius of a is the number
|a|a=sup{|«|:«GaA(a)} (9.5.1.1)
We can extend the idea by declaring \a\a = 00 if the spectrum of a
is unbounded, and \a\a = —00 if the spectrum is empty: of course,
Theorem 9.2.3 says that this cannot happen if A ^ 0. We can also extend
Definition 9.5.1 to |a|w for each of the subsets uA C aA of Definitions 9.2.1

9.5 The Spectral Radius
353
and 9.3.1, and to the Weyl spectrum of Definition 9.4.3. The crucial
property of the spectral radius is its relation to the norm:
9.5.2 THEOREM If a e A for a Banach algebra A ^ 0, then
limsup||an||1/n < \a\a < inf ||an||1/n (9.5.2.1)
n->oo neti
Proof: From (9.2.3.1) and (9.2.4.1) there is inclusion, for each n G N,
{sn:secA{a)} =cA{an) C {t G C: \t\ < \\an\\} (9.5.2.2)
giving the second inequality of (9.5.2.1). To establish the first we claim
\a\a <1=> \\an\\ —> 0 => limsup ||an||1/n < 1 (9.5.2.3)
n—KX>
To establish this we show that, if \a\a < 1,
2m = i [z - a)"1 dz = 2m lim (l - an)_1 (9.5.2.4)
J n—► oo
1*1 = 1
where the notation indicates that the integral is taken round the circle
S = {\z\ = 1} = {s £ C: |s| = 1} once in a counterclockwise direction.
To establish (9.5.2.4), recall that, with un = ex]>(2m/n), we have
zn-l = (z- wB) (* - «2) -(*- «») (9-5-2-5)
and hence
(9.5.2.6)
One other observation is that
lim n(un - 1) = — exp(27r*t)t=0 = 2m (9.5.2.7)
n—►oo dt
We are ready to prove the second equality of (9.5.2.4), assuming only that
the spectrum oA{a) is disjoint from the circle S: by taking "Riemann sums"
it follows that
i (z- a)"1 dz = lim XK"1 - a)"1 K - <~')
J n—►oo * ■*
n
= nU™ E(X - w»y°)-1(w» - !) = '^"K - 1)(1 - «n)_1
n—►oo r -* n—►oo
J = l

354
9. Liouville's Theorem and Spectral Theory
The second equality of (9.5.2.4) now follows from (9.5.2.7).
For the first equality of (9.5.2.4) we need to know that the spectrum
oA{a) is confined to {\z\ < 1} = {s G C: |s| < 1} the interior of S: then
Green's theorem (9.1.2.5) gives, as in the proof of Theorem 9.1.2,
R
> 1 => <b [z — a) 1 dz = <b [z — a) l dz —> 2ni as R
oo
1*1=*
For the passage to the limit at the end argue that if R > \\a\\ then
<f> [z — a)~x dz — 2ni = f {{z — a)~l — z~x) dz\\
(9.5.2.8)
\z\=R
\z\=R
I
\z\=R
z [z — a) adz
<
Ml
(9.5.2.9)
R-\\a\\
With the proof of (9.5.2.4) we have established (9.5.2.3), and if we
apply (9.5.2.3) with (l/k)a in place of a, taking k = la^ + e > la^ and
allowing e —► 0, then we will obtain the first inequality of (9.5.2.1). ■
Theorem 9.5.2 gives an immediate characterization of the quasinil-
potent elements of Definition 7.4.1, at least when the algebra is complete.
9.5.3 THEOREM If A ^ 0 is a Banach algebra and a G A, then
a quasinilpotent 4=> oA{a) = {0} (9.5.3.1)
In particular, the radical elements of A are quasinilpotent:
a e Radical(A) => ||an||1/n —► 0 as n —► oo (9.5.3.2)
Proof: The first equivalence follows at once from (9.5.2.1). For (9.5.3.2)
combine (7.2.3.10) with (7.2.3.2) and (7.2.3.3) from Theorem 7.2.3 to see
that
Radical(A) = {aeA:l-AaC A'1} = {aeA:l-aAC A'1}
(9.5.3.3)
Now compare this with the implication
aA(a) = {0} 4=> 1-CaCA
-l
(9.5.3.4)
and apply (9.5.3.1).

9.5 The Spectral Radius
355
Theorem 9.5.2 of course guarantees that the sequence (||an||1/n)
always converges; it very nearly but not quite says that it is monotonically
decreasing. The argument used to prove Theorem 7.4.3 also shows that if
ab = ba, then
l« + ft|a<l«L + H<r
and
M„ < l«U*l<r (9-5-3.6)
If A is commutative, then (7.4.4.2) and (9.5.3.1) can be combined to
reverse the implication in (9.5.3.2).
Theorem 9.5.2 has an unexpected significance: the two quantities
related in (9.5.2.1) have different invariance properties. Thus, the limit
limn ||an||1/n obviously depends on the norm ||*||, but is completely
independent of the algebra A: conversely, the supremum \a\a depends on the
algebra A but is independent of the norm ||-|j. For equality (9.5.2.1) it is
necessary that the norm satisfies the multiplicative condition (1.10.1.1) and
that the algebra is complete: both these conditions will be preserved if we
only look at topologically equivalent algebra norms on A. It is now not
surprising that there is another expression for the spectral radius:
9.5.4 THEOREM If a e A for a Banach algebra A, then
\a\a = inf {p{a):p an equivalent norm on A} (9.5.4.1)
Proof: The left-hand side is by (9.5.2.1) less than or equal to the right-
hand side. For the reverse inequality first specialize to the algebra BL(X, X)
of bounded operators on a Banach space. If T G BL(X, X) and e > 0, then
there is n G N for which
||T»+i||i/(»+i) < i^ + £ = k (9.5.4.2)
Now put
||x||' = fcn||x|| + A^Hrxll + • • • + ||rnx|| for each x G X (9.5.4.3)
and
\\S\\'= sup \\Sx\\' for each S£BL(X,X)
Nl;<i
Evidently
fcn||x|| < llxH' < {kn + kn-x\\T\\ + • • • + ||rn||)||x|| for each x e X
(9.5.4.5)
(9.5.3.5)
(9.5.4.4)

356
9. Liouville's Theorem and Spectral Theory
Thus by (1.7.1.2) ||-||' is an equivalent norm on X, and hence also ||-||' is an
equivalent norm on BL(X,X). Taking S = T in particular gives
in,<||T||'<*=|T|„ + e (9.5.4.6)
This proves Theorem 9.5.4 when A = BL(X,X), and hence in general, if
we apply what we have just proved to X = A and T = La £ BL{A, A), m
9.6 GELFAND'S THEOREM
A homomorphism T : A —> B will be said to have the Gelfand property if
there is implication, for arbitrary a £ A,
Ta e B'1 =^aeA-1 (9.6.0.1)
In the terminology of Definition 7.6.1, all the T-Fredholm elements of A
must be invertible. A simple example is the quotient map associated with
the radical: Theorem 7.2.6 says, in particular, that if J = Radical(A)
a + Je {A/J)'1 =^aeA-1 (9.6.0.2)
The Gelfand theorem provides each commutative normed algebra A with a
Gelfand homomorphism T : A —> B = C^Q) for a topological space H.
We begin with an auxiliary result, which tells us that the only normed
division algebra is the complex field. This is known as the Gelfand-Mazur
lemma:
9.6.1 THEOREM If A ^ 0 is a normed division algebra, in the sense
that
A = A'1 U {0} (9.6.1.1)
then A reduces to the complex field:
A = CI (9.6.1.2)
The same is true if more generally there is k > 0 for which
||a||||6|| < fc||a&|| for each a,6 G A (9.6.1.3)
Proof: From (9.6.1.1) there is implication
5 e cA{a) => a = s e CI (9.6.1.4)
and (9.2.3.2) guarantees that oA{a) is not empty. If we assume only (9.6.1.3)
we have implication
5 e rlX{t{a) => a = s e CI (9.6.1.5)
and (9.3.3.1) guarantees that fj|ft(a) is nonempty. ■

9.5 The Spectral Radius
357
The topological space H which furnishes the Gelfand homomorphism
is manufactured from the "characters" of Definition 7.13.2:
9.6.2 DEFINITION If A is a complex normed algebra we shall write
A* = {/ G Af: Il/H < 1 = /(l) and f(ab) = f{a)f(b) for each a, 6 G A}
(9.6.2.1)
furnished with the topology of pointwise convergence on A, namely, U C A*
is a neighborhood of / G A* iff there is e > 0 and finite K C A for which
ig G Ax:max\g(a) - f(a)\ < e\ C U (9.6.2.2)
The reader should note that unless A is finite dimensional this is not
the same as the topology induced by the norm of A*. To see that Q = A* is
big enough to produce a Gelfand homomorphism we combine Theorem 7.2.3
with the Gelfand-Mazur lemma:
9.6.3 THEOREM If A is a complex Banach algebra and n = A* is its
character space, then the homomorphism T : A —► C^Q) defined by
setting
(Ta)(f) = f{a) for each / G H, for each aeA (9.6.3.1)
is well defined, of bound < 1, with the Gelfand property.
aeA-x^=^Tae ^(H)"1 (9.6.3.2)
Proof: To see that T is well defined we need to know that Ta is
continuous on fh but this is clear from (9.6.2.2). The boundedness and the
homomorphic properties of T are clear from the corresponding properties
(9.6.2.1) of the characters, and then forward implication in (9.6.3.2) is given
by (7.6.0.1). Conversely, we claim that if a G A then
a <£ A"1 => 0 G {/(a): / G Ax} (9.6.3.3)
Indeed, if a §? A-1 then by (7.2.3.1) there is a maximal ideal J C A for
which a £ J. By (7.2.2.2) J = cl(J) is closed, and now A/J is a normed
division algebra. If we define / : A —► C by setting
°A/j{b + J) = {/{*>)} ^ each 6 G A (9.6.3.4)
then it is clear that / G A* is a character for which f(a) =0. ■
An equivalent definition of the character / induced by the maximal

358
9. Liouville's Theorem and Spectral Theory
ideal J in (9.6.3.4) is to say that
6-/(6)16 J for each be A (9.6.3.5)
In (9.6.3.3) we have apparently proved more than we have stated: we do
not need to know that the continuous function Ta : Cl —> C has a bounded
reciprocal l/(Ta) : Q —> C, merely that it never vanishes. In fact, the
character space f2 = A* is compact, so that every continuous function is
bounded:
9.6.4 THEOREM If A is a complex normed algebra, then A* is compact.
Proof: If we impose the topology described in (9.6.2.2) on the whole of
the unit disc {/ G A*: ||/|| < l}, then the character space A* is a closed
subset. To see that the unit disc is compact in this "weak*topology" we
see how it can be embedded in the product space
n{N^Ni}=n{sec:H^Hi} (9-6-4-1)
By Theorem 6.6.4 each of the factor spaces is compact, and by Tychenoff's
Theorem 6.6.6, so is their product. ■
Theorem 9.6.4 explains why we have abandoned the norm topology of
in our discussion of A*. The "weak* topology" which we are using is
exactly the topology induced by the embedding of A* in the product C ,
and is the weakest topology for which all the functions Ta are continuous.
There is a superficial analogy between the dual space X* of a normed
space X and the character set A* of a normed algebra A. In general, the
Gelfand homomorphism need be neither one-one nor isometric, in contrast
to the embedding of a normed space in its second dual:
9.6.5 THEOREM Necessary and sufficient for the Gelfand
homomorphism on A to be one-one is that A is semisimple, in the sense that
Radical(A) = {0} (9.6.5.1)
Necessary and sufficient for the Gelfand homomorphism on A to be
isometric is that A is conservative, in the sense that
||a2|| = ||a||2 for each a e A (9.6.5.2)
If A is conservative, then necessary and sufficient for the Gelfand
homomorphism to be onto is that its range is self-adjoint in C00(A*).

9.5 The Spectral Radius
359
Proof: If T : A -> C^Cl) is the Gelfand homomorphism then (9.6.0.1)
says
*a(«) = {/(«): / € A*} = (Ta)(n) (9.6.5.3)
The spectrum of a is the range of its Gelfand transform Ta. Thus, using
(7.4.4.2) since A is commutative,
Ta = 0 4=> a quasinilpotent 4=> a G Radical(A) (9.6.5.4)
This proves the first part. Towards the second, it follows from (9.6.5.3)
that
|o|a =sup{|/(o)|:/ € A*} = ||Ta|L (9.6.5.5)
The spectral radius of a is the standard norm of the function. If, therefore,
T is isometric, so that ||a|| = \a\a, then (9.6.5.2) follows from the spectral
mapping theorem (9.2.4.1). Conversely, if (9.6.5.2) is assumed, then ||an|| =
||a||n whenever n = 2m is a power of two and hence
||a|| = ||an||1/n if n G 2N and aG A (9.6.5.6)
By (9.5.2.1) this implies ||a|| = \a\a, proving the second part. The last
part now follows at once from the complex case of the Stone-Weierstrass
Theorem 7.12.4. Note that T{A) "separates points" of Cl = A* by definition,
and is necessarily closed in C^Cl) if T is isometric. ■
The "commutative Gelfand-Naimark theorem" says that the last part
of Theorem 9.6.5 applies to the "Hilbert algebras" of Definition 8.8.1:
9.6.6 THEOREM If A is a commutative Hilbert algebra, then the
Gelfand homomorphism is an isometric *-isomorphism. If, in particular, A is
the closed algebra generated by a and a*, then
Ax =cA{a) C C (9.6.6.1)
Proof: Begin with the observation, using (8.8.4.4), that if a G A then
a = a* => cA{a) C R (9.6.6.2)
For if 5 and t ^ 0 are in R then a + 5 is also self-adjoint and therefore
a + 5 + it is invertible. It follows at once that if / G A* is a character then
a = a* => f(a) G R (9.6.6.3)
and hence, writing a = b + ic and a* = 6 — ic as in (8.8.4.3),
/(a*) = /(a)" for each a G A and / G Ax (9.6.6.4)

360
9. Liouville's Theorem and Spectral Theory
We have proved that the Gelfand homomorphism is also a *-homomor-
phism. It follows that the range of the Gelfand homomorphism is a self-
adjoint subalgebra of C00(A*). It remains only to verify the "conservative"
condition (9.6.5.2). But, since A is commutative, every element a G A is
"normal" in the sense of (8.8.3.1), and now (8.8.5.1) with the B* condition
(8.8.1.1) gives
||a2|| = ||a*a|| = ||a||2 (9.6.6.5)
Toward (9.6.6.1), it is clear that each element of A* is mapped into the
spectrum of a under evaluation at a. Conversely, if A is generated by a and
a* then each / G A* is determined on A by the number /(a). ■
Whether or not a Hilbert algebra A is commutative, if a G A is normal
in the sense of (8.8.3.1) then the closed subalgebra B generated by a and
a* will be a commutative Hilbert algebra, hence subject to Theorem 9.6.6.
In the case A = BL(X, X) this is sometimes called "the spectral theorem"
for the normal operator a: linear functional on C^B*), or equivalently
measures on B*, can be brought to bear on a through the medium of the
Gelfand homomorphism.
It is less obvious that the double commutant comm^(c) of a normal
element a, as defined in (7.1.1.2), is also "self-adjoint" and therefore a
commutative Hilbert algebra. This result is known as Fuglede's theorem:
9.6.7 THEOREM If a G A for a Hilbert algebra A there is implication
a*a = aa* =$■ a* G comm^(a) (9.6.7.1)
Proof: The argument, due to Rosenblum, is a cunning use of Liouville's
theorem: if a and 6 are in A there is implication
ab = ba =$■ ex-p(isa)b • exp(—isa) = b for each 5 G C (9.6.7.2)
while if a is normal then (7.11.4.1)
exp(zsa* + isa) = exp(isa*) exp(zsa) for each 5 G C (9.6.7.3)
Since sa* +sa is self-adjoint in the sense of (8.8.3.2), the exponential on the
left-hand side of (9.6.7.3) is unitary in the sense of (8.8.3.3). Thus, there is
k > 0 for which
||et5a* m b m t-isa* || < fc||6|| for ea(.h 3 e c (9.6.7.4)
Since the function txza* -b-e~lza* is evidently holomorphic on C, Liouville's

9.6 Gelfand's Theorem
361
theorem (9.1.3.2) says that it is constant. In particular,
i{a*b - ba*) = Y^b' e~iZa*)z=o = ° " (9.6.7.5)
We conclude with a characterization of the characters in the dual space
of a Banach algebra:
9.6.8 THEOREM If A ^ 0 is a complex Banach algebra, then
Ax = {/ e A^: /(l) = 1 and 0 £ /{A'1)} (9.6.8.1)
Proof: It is clear, since A ^ {0}, that the left-hand side of (9.6.8.1) is
included in the right. Conversely, we claim that if / is in the right-hand side
of (9.6.8.1), then the implication (7.13.4.1) holds: /(a) = 0 => /(a2) = 0.
To see this, suppose /(a) =0 and n £ N and write
f((z-a)n) = (z-Sl)(z-s2)---(z-sn) (9.6.8.2)
using the fundamental theorem of algebra (9.1.4.1). At the same time
f((z - a)n) =zn- nf{a)zn-1 + \n{n - l)f{a2)zn~2 + • • • + (-l)n/(an)
(9.6.8.3)
Thus, if /(a) = 0 then also Y^=i sj = 0> while ^j<A: sjsk =
\n{n — l)/(a2): it follows
0 = E sn = E 5; +2 E sJsk = E 5y + n(n ~ W) (9-6-8-4)
V J J J i<k i
and hence
n{n-l)\f{a2)\ < El5j|2 <ttmax|sy|2 (9.6.8.5)
3
Suppose now that the functional / lies in the right-hand side of (9.6.8.1),
so that /(l) = 1 and 0 £ /(A-1). By (9.6.8.2)
b=(a- s1)(a - s2) • • • (a - sn) => f(b) = 0 => 6 £ A-1 (9.6.8.6)
It follows, using (9.2.3.1) at the end,
{61,62,...,6n} CcA{a) =4>max|6-|2 < ||a||2 (9.6.8.7)
3

362
9. Liouville's Theorem and Spectral Theory
Thus, for each n G N
(n-l)|/(a2)|<||a||2 (9.6.8.8)
forcing /(a2) = 0. This proves the implication (7.13.4.1), and hence by
Theorem 7.13.4 that / is a character on A. m
An equivalent condition to membership of the right-hand side of
(9.6.8.1) is
f{a) G cA(a) for each a G A (9.6.8.9)
Combined with Theorem 9.6.5, Theorem 9.6.8 gives its own extension: if
A is arbitrary and if B is commutative and semisimple, then necessary and
sufficient for T G BL(A,B) to be a homomorphism is that
r(l) = 1 and r(A_1) C B~l (9.6.8.10)
9.7 THE FUNCTIONAL CALCULUS
If we extend the "Cauchy integral formula" of (9.1.2.2) from complex
numbers to Banach algebra elements a G A, then it is plausible that we will
be entitled to call the answer "/(a)". We begin with the observation that,
if T C C is a "piecewise smooth" image in C of the circle S, then we can
define a mapping
Wr:s —>—. f(z- s)-1 dz from C \ T to Z (9.7.0.1)
27TI J
r
which then divides the complement of T in C into subsets VTIT1(n)nGZ;
some of these may be empty, or possibly not connected: but no connected
subset of C \ T can intersect more than one of them, and always
C\riTCWf1(0) (9.7.0.2)
In particular, if T is "simple," in the sense that the piecewise smooth
mapping from S onto T can be made one-one, then
C \ T = Wf1 (1) U Wf\0) (9.7.0.3)
We shall sometimes speak of Wf1^) as the inside, and V^IT1(0) as the
outside, of the curve T, and then extend this terminology to the situation
in which T is the union of finitely many such curves. Of course the interior
of the set T, as defined in (1.2.0.8), is the empty set. For brevity we shall
extend the term "simple" to apply to finite unions of curves for which
(9.7.0.3) holds.
With this preamble, we offer the "Cauchy integral formula" as our
definition of the functional calculus:

9.7 The Functional Calculus
363
9.7.1 DEFINITION If a G A for a complex Banach algebra A and if
/ : 0 C C is holomorphic on an open set 0 C C for which oA{a) C O, then
we write
/W = i / f{z){z-a)-1dz=±Jf[z){z-a)-1dz (9.7.1.1)
<rA(a) r
where T C C\cA(a) is piecewise smooth with oA{o) C VTIT1(l).
In words, we integrate once anticlockwise around the spectrum of a.
We must at once check that f(a) is well defined:
9.7.2 THEOREM If a G A for a Banach algebra A and if /, g and h are
holomorphic on a neighborhood of oA{a)> then /(a) G A is well defined,
and satisfies
/(a) G comm^(a) (9.7.2.1)
/(a) = l if / = 1 (9.7.2.2)
/(a) = a if / = z (9.7.2.3)
/(a) = g{a) + &(a) if / = g + /i (9.7.2.4)
f(a)=g(a)h(a) if / = g • /i (9.7.2.5)
Proof: The element /(a) is supposed to exist and be independent of the
particular curve T winding +1 round the spectrum of a. The existence, as
a limit of Riemann sums, follows from Theorem 5.12.2; the independence of
particular T follows from Cauchy's theorem and Green's theorem (9.1.2.5).
To see that f(a) is in the double commutant of a, apply (5.12.2.9) to the
operators Lb and Rb associated with each 6 G commA(c):
(27r*)6/(a) = / f{z)b{z - a)'1 dz
<7A(a.)
f{z){z -a)-1bdz = (2iri)f(a)b
-f
o-A(a)
We essentially proved (9.7.2.2) as part of Theorem 9.5.2. If R > \a\a,
then
(27rz)l(a) = j> (z — a)~1dz= j> [z — a)-1 dz—> 2m as J?—> oo
<rA(a) \z\=R

364
9. Liouville's Theorem and Spectral Theory
This now Combines with Cauchy's theorem (9.1.2.5) to give (9.7.1.3):
(2iri)(z(a) — a) = <f> z[z — a)-1 dz — a = <f> [z — d)[z — a)-1 dz = 0
<rA{a) <Ja{o>)
For (9.7.2.4) we need only look at Riemann sums to see that the integral
of a sum is the sum of the integrals. For (9.7.2.5) we need to work just a
little bit harder. Taking advantage of the independence of f(a) from the
particular curve T round the spectrum of a, if g and h are each differentiable
in a neighborhood of oA{a) we can arrange that Tg and Tk are finite unions
of piecewise smooth curves in C \ oA{a) for which
'aWC^'W and WF/MUIVCW^I) (9.7.2.6)
Thus, also
rkC ^(0) (9.7.2.7)
In particular, T and Tk are disjoint sets. We may now compute
(27ri)2g(a)h{a) = / g{z)(z — a)-1 dz / h(w)(w — a)-1 dw
zevg werh
= / ( / g(z)h(w)(w - z)-1
zevg wevh
X ((z - a)-1 — (w — a)-1) da J dz
zevg wevh
+ /" /iW^-aj-M f f{z){z-w)-1dz\dw
wevh zevg
= / g{z)(z- a)~1(27ri)h(z) dz
zeTg
+ / h(w)(w-a)~1 -Odw
werh
= 2ni / g(z)h(z)(z - a)"1 dz + 0 = (27rz)2(^ • /i)(a)
We used (9.1.2.2) with h in place of / for the first of these integrals, and
Cauchy's theorem (9.1.2.5) for the second. ■

9.7 The Functional Calculus
365
Theorem 9.7.2 in particular shows that if / is a polynomial then f(a)
agrees with the formula of (1.10.1.7): we might remark that this can be
done by the argument for (9.7.2.3), without going to the trouble needed for
(9.7.2.5). It is also clear from Theorem 9.7.2 that if / is a rational function
(well defined on oA{a) of course) then f(a) agrees with the formula of
(9.2.5.3). We need only verify this when / = l/q is the reciprocal of a
polynomial, and in this case (9.7.2.5) says that
*(«)(£)(«)=!= (£)(«)«(«) (9-7.2.8)
There is a certain continuity about the mapping / —► /(a): our
statement is complicated by the fact that the space of all relevant functions / is
not a normed space.
9.7.3 THEOREM If a e A for a Banach algebra A and if K C C is a
compact set for which oA{a) C int(jK'), then there is k > 0 for which
||/(a)||<fcsup|/| (9.7.3.1)
K
whenever / : O —► C is holomorphic on an open set Q ~D K. In particular,
if / = c0 + Yl^Li cnzU ls defined on a disc {\z\ < R} with R > \a\a, then
oo
/(«) = co + J2 cn<*n (9.7.3.2)
n=l
Proof: If we fix a piecewise smooth curve T, lying in the interior of K and
winding +1 round oA{a)> we can argue
||/(a)|| < i- length(r) max \\{z - a)'1]] sup ||/|| (9.7.3.3)
Zn r k
giving (9.7.3.1). It follows that the mapping / —► f(a) is bounded and
linear on the subspace of C00(jfiT) consisting of those continuous functions
which are differentiable at each point of int(jfir). In particular, if / is a
power series then f(a) is the limit of pn(a) for the partial sums pn of /. ■
Necessary and sufficient for the power series / = c0 + Yl^Li cnzTl to
be defined on the disc {\z\ < R} is the familiar condition,
limsup|cn|1/n < -^ (9.7.3.4)
The spectral mapping Theorem 9.2.4 extends to holomorphic functions:

366
9. Liouville's Theorem and Spectral Theory
9.7.4 THEOREM If a G A for a Banach algebra A and if / : n -> C is
holomorphic on an open neighborhood of oA{a)^ then
*a/(«)=/*a(«) (9-7-4.1)
Proof: If 5 G H is arbitrary we can write
/ - f(s) = (Z-S)g:n—+C (9.7.4.2)
with holomorphic g : Q —► C. It is clear how to define g(t) if t ^ 5, and we
take (7(5) = f'(s). Alternatively, the reader can write out a Cauchy integral
for g(t). It follows that
/(a) - f(s) = (a - s)<7(a) = <7(a)(a - s) (9.7.4.3)
giving inclusion one way in (9.7.4.1).
MeftW^r/(«) and fa^\a)<Za^f{a) (9.7.4.4)
Conversely, if £ G C \ /crA(a), then (/ — i)~l = h is holomorphic on some
open neighborhood of oA{a). Since (/ — i)h = 1 on a third neighborhood
of aA (a) it follows that
Ma)(/(a) -0 = 1= (/M " 0Ma) (9.7.4.5)
which excludes £ from the spectrum of /(a). ■
It is satisfying to know that the functional calculus for f(a) is consistent
with the calculus for a:
9.7.5 THEOREM If a e A for a Banach algebra A and if / : Qf -> C
and <7 : f2g —► C are functions holomorphic on open sets containing o~A(a),
and o~Af(a), respectively, then there is equality
(ff °/)(<*) =ff (/(a)) (9.7.5.1)
Proof: By the spectral mapping theorem 9.7.4 there is inclusion
^(^/-'(n,) (9.7.5.2)
and now g o / is holomorphic on the open set
«* = r1^) nn;5 aA{a) (9.7.5.3)
We can now choose piecewise smooth curves T* and T , winding +1 round
cA{a) and 0-A/(a), respectively, in such a way that T* and its "inside" lies

9.7 The Functional Calculus
367
in Qf, while Tg and its inside lies in VIg. Also, by continuity there will be
an open set n'h in f2 *, containing T* and its inside, for which f(n'h) lies
"inside" Tg. Now for each point t £ Tg the function ht = (t — /)_1 is
differentiate on nj^. By (9.7.2.5)
ht(a) = {t- /(a))-1 for each t £ Tg (9.7.5.4)
Now
(2*i)g(f(a)) = J g{w)(w - /(a))"1 dw
= J ^H^7 J K(z)(z-°)~ldz\ dw
= j{z - a)"1 l±-Jg(w)(w - /(*)) dwdz
= J{z-a)-1[g-f){z)dz m
The functional calculus also commutes with homomorphisms T €
HBL(A,B): if a £ A and if / : Qf —► C is differentiate on a neighborhood
of aA (a) D aB (Ta), then
f(Ta) = T(f(a)) (9.7.5.5)
One rather familiar application of the functional calculus is a
characterization of the "quasipolar" elements of Definition 7.5.2:
9.7.6 THEOREM If a e A for a complex Banach algebra A, then
a quasipolar -<=>• 0 £ accaA(a) (9.7.6.1)
Proof: If a £ A is quasipolar, then (7.5.5.3) from Theorem 7.5.5 ensures
that a — s € A-1 for sufficiently small complex numbers 5^0. Conversely,
if 0 is not an accumulation point of the spectrum we can find an open
neighborhood f2 of oA (a) which contains a disc K of center 0 for which
K and fi \ K are open in C and cA{a) n K C {0} (9.7.6.2)

368
9. Liouville's Theorem and Spectral Theory
If we now define / : VI —► C by setting
f{s) = \ 9.7.6.3
[0 ifsGK
then / is holomorphic on f2 and satisfies the identity f(z)2 = f(z). Using
the functional calculus formula (9.7.1.1) gives an element
a = f(a) = f{a)2 GA (9.7.6.4)
which commutes with a, as required by (7.5.2,1) from Definition 7.5.2. If
we define g : f2 —> C by setting
f - ]fsen\K
g(s) =\s (9.7.6.5)
I 0 if 5 £ K
then g is also holomorphic on U and satisfies f(z) = zg[z). Again the
functional calculus formula (9.7.1.1) gives
a = ag(a) = g(a)a € a A D Aa (9.7.6.6)
as required by (7.5.2.4). Finally, the spectral mapping theorem 9.7.4 says
aA(a(l - /(a))) = {*(1 -s):sG cA{a)} = {0} (9.7.6.7)
which combines with (9.5.3.1) from Theorem 9.5.3 to give (7.5.2.3). ■
It is the idempotent 1 — a whose functional representation is most
familiar:
1 - a = 1 - f{a) = — l{z- a)'1 dz (9.7.6.8)
27T2 J
0
integrating counterclockwise once along the boundary dK of a disc K of
center 0 which contains no other point of <JA{a). More generally, if K C
<JA{a) is, relative to <7A(a), both open and closed, then
P = XK{°) = ^lf{z-°)~ldz (9.7.6.9)
K
is a projection commuting with a for which (unless K = 0)
K = cA(ap) (9.7.6.10)
We are now able to extend (7.11.6.8) to more general A:

9.8 Essential Spectra
369
9.7.7 THEOREM If a e A for a Banach algebra A, then
oo
exp(a) = 1 <=> a = ^ 2ninXn(a) (9.7.7.1)
n=—oo
for a finite family {xn{a))neZ °f pairwise commuting projections with
n ± m => XnWxm(a) = 0 and £x»(<0 = 1 (9.7.7.2)
n
Proo/: If p = p2 £ A, then
exp(27T2» = 1 + (e2™ - l)p = 1 (9.7.7.3)
and hence, if a is in the right-hand side of (9.7.7.1), then exp(a) is a finite
product of l's. Conversely, if a is in the left-hand side of (9.7.7.1) then,
using Theorem 9.7.4,
aA(a)C27rlZ (9.7.7.4)
In particular, <JA{a) has no accumulation points: thus, using (9.7.6.8),
oo
b= Yi 2«nx„(o)=>aA(o-6) = {0} (9.7.7.5)
n=—oo
We now have
||(a-6)n||1/n —>0 and exp(a-6) = l (9.7.7.6)
This gives
oo
(a-6)(l+d) = ^(a-6)n/n! = exp(a-6)-l = 0 (9.7.7.7)
n=l
with
n=2
Since evidently crA(d) = {0} it follows that 1 + d (= A-1 and hence that
a-6 = 0. ■
9.8 ESSENTIAL SPECTRA
If the "Weyl spectrum" relative to a homomorphism is derived from the
"Weyl elements," then the "Browder elements" give rise to a "Browder
spectrum":

370
9. Liouville's Theorem and Spectral Theory
9.8.1 DEFINITION If T : A -> B is a homomorphism of normed
algebras, then the T-Browder spectrum of a € A is the set
c4omm(a) = {s G C:a - 5 £ A"1^"1^)}
= H ^(a + d):T(d) = 0,ad = da} (9.8.1.1)
We can obviously make analogous definitions for left and right spectra,
almost spectra, and (approximate) eigenvalues. The Browder spectrum is
closed and bounded and, if the homomorphism satisfies the "Riesz"
condition (7.7.4.2), satisfies the spectral mapping theorem:
9.8.2 THEOREM If T : A -> B is a homomorphism of Banach algebras
and a£ A, then
<rBiTa) C wr(a) C c4omm(a) C aB(Ta) U accaA(a) C aA{a) (9.8.2.1)
and if / : f2 —► C is holomorphic on a neighborhood f2 of cA(a), then
, .comm t(„\ r~ t, .comm/j
(9.8.2.2)
C aBr(/(a)) U accaA/(a) C f(aB(Ta) U accaA(a))
If the homomorphism T : A —► B has the Riesz property, then
cB{Ta) U accaA(a) (9.8.2.3)
(a) = c4omm/(a) (9.8.2.4)
Proof: The inclusions (9.8.2.1) are obtained by applying (7.7.3.3), and
(7.7.4.1) from Theorem 7.7.4, to elements a — s with s € C The
inclusions (9.8.2.2) are mostly clear: the middle one comes from (7.7.5.1)
from Theorem 7.7.5. If T has the Riesz property (7.7.4.2), then (7.7.4.3)
from Theorem 7.7.4 gives (9.8.2.3), which combines with (9.8.2.2) to give
(9.8.2.4). ■
When A = BL(X,X) for a Banach space X and $ : BL(X,X) ->
BL(X, X)l KL(X, X) is the Calkin homomorphism then, changing notation,
we shall write
<7essCn = ^BL(X>X)/Kt(x>x)(r + KL(X.X)) (9.8.2.5)
"ess(r) = ««(T) (9.8.2.6)
»
and

9.8 Essential Spectra
371
and
^essmm(T) = t4°mmCn (9.8.2.7)
respectively, for the Fredholm, Weyl, and Browder spectra of T. By
Theorem 7.7.2 the homomorphism $ has the Riesz property (7.7.4.2), so that
Theorem 9.8.2 applies:
9.8.3 THEOREM If T € BL(X, X) for a complex Banach space X, then
°V) = ^essCH U 7rleft(r) U 7rrieht(r) (9.8.3.1)
and
o(T) = wess(r) u (7rleft(r) n 7rright(r)) (9.8.3.2)
and
»%T(T) = <ress(r) U acea(T) (9.8.3.3)
Proof: If 5 € C is not in the right-hand side of (9.8.3.1), then T — si is
Fredholm, and hence in particular (6.4.4.1) regular, as well as being both
one-one and dense, therefore (3.8.3.12), (3.8.3.13) invertible. If instead
5 £ C is not in the right-hand side of (9.8.3.2), then T - si is by (6.5.1.2)
Fredholm of index zero, so that if it is either one-one or dense then it must
be both one-one and dense. By (9.8.3.1) it follows that T — si is invertible.
This proves the first two equalities, and the third is (9.8.2.3). ■
We shall write
^00(r) = isoa(r)\Wess(r) (9.8.3.4)
for the Riesz points of the spectrum of T. Evidently 5 £ 7r00 (T) means
that 5 is isolated in cr(T), and an eigenvalue for which the eigenspace (T —
5/)_1(0) is finite dimensional, while the range (T — sI)(X) is closed and
has equal finite codimension.
Using the "punctured neighborhood" Theorem 7.8.5, if we remove the
connected hull of the essential spectrum from the spectrum, only Riesz
points will remain:
9.8.4 THEOREM If T £ BL(X, X) for a complex Banach space X, then
da(T)\ae33(T) Cisoa(T) (9.8.4.1)
and hence
<r(T) C r,aess(T) U xM(T) (9.8.4.2)

372
9. Liouville's Theorem and Spectral Theory
In particular,
T £ KL(X,X) => a{T) C {0} U 7r00(r) (9.8.4.3)
Proof: For (9.8.4.1) apply Theorem 7.8.5 to the operators T - si, taking
for the subspace A C BL(X,X) the set {T -tl:te C}. From (9.8.4.1) and
(7.10.5.2) from Theorem 7.10.5 it follows
<r(T)CVce33(T)Uisoc(T) (9.8.4.4)
and of course
^ess(r) U isoa(T) = r,v„(T) U (isoa(T) \ aeas(T)) (9.8.4.5)
By (9.8.3.3) we have
isoa(r) \ aess(r) = isoa(T) \ Wec°3mm(r) = ^oo(r) (9.8.4.6)
This proves (9.8.4.2), and then (9.8.4.3) follows at once, noting that if T is
compact then its essential spectrum {0} is the same as the connected hull
n{o} = {o}. m
It is not necessary to use the punctured neighborhood Theorem 7.8.5
to derive (9.8.4.3):
9.8.5 THEOREM If T € BL(X, X) for a normed space X and if
0 ^ e3 G (T - sl)-1^) for each sGK (9.8.5.1)
then {e8)8eK is linearly independent in X. Hence,
T G KL(X, X) => ace 7rleft (T) C {0} (9.8.5.2)
Proof: The linear independence of eigenvectors corresponding to distinct
eigenvalues reduces to the linear independence (6.1.0.1) of finite families of
vectors, which can be proved by induction. Thus, if (ey)j€j are nonzero
eigenvectors corresponding to complex numbers {sj)jeJ with either J = {1}
or J = {1,2,..., m}, then the analogue of (9.8.5.1) is evident when m = 1,
and follows for m = n + 1 from the case m = n: for if (e1? e2,..., en) is
linearly independent and if en+1 = t-^e-^ + t2e2 H + £nen, then
n
0 = (J - 6n+1/)en+1 = J^«J.(6n+1 - «,.)cy => *x = *2 = • • • = tn = 0
forcing en+1 = 0, a contradiction.

9.8 Essential Spectra
373
If, in contradiction to (9.8.5.2), there is an infinite sequence (sy)jGN
of eigenvalues with inf • \sA = k > 0, then there will be corresponding
eigenvectors (e •), necessarily linearly independent. We write
n
E^^T-Sjir^O) (9.8.5.3)
3 = 1
and note that this is an increasing sequence of closed subspaces, none of
which equals its successor. By the Riesz lemma (Theorem 1.5.2) there is a
sequence [xn) in X for which, for each n £ N,
" *n € En; \\xn\\ = 1; dist(sn+1,£n) > \ (9.8.5.4)
If we notice also that, for each n £ N,
(T-snI)En+1CEn
then it is easy to see that
n?m=>\\Tzn-Txm\\>$k (9.8.5.6)
Specifically, we observe that if n > m+ 1 then Txn — Txm — snxn € En_1.
This gives \\Txn - Txm\\ > dist^x^E^) > i|«„| > \k. m
We can offer an alternative argument which shows that, relative to the
finite rank ideal, a "Browder operator" must be quasipolar:
T e BL_1(X,X)€KL0(X,X) => 0 0 acca(r) (9.8.5.7)
Indeed, if T = S + K with finite rank K and invertible S there will be
6 > 0 for which S — si is invertible whenever \s\ < 6, and hence if x £ X
(J - sl)x = 0 => x = (5/ - 5)_1ii:x = ^(5/ - 5)_1x G ^(X)
if SK = KS
Since -K^X) is assumed to be finite dimensional, the linear independence of
eigenvectors (9.8.5.1) ensures that for only finitely many 5 G C with \s\ < 6
can the operator S — si not be one-one. Since it is also a Weyl operator it
will be invertible whenever it is one-one.
We are also able now to finish Lomonosov's theorem, which we started
in Theorem 7.4.5:
(9.8.5.5)

374
9. Liouville's Theorem and Spectral Theory
9.8.6 THEOREM If X is a Banach space and if K € KL(X,X) \ CI
is a nonscalar compact operator, then it has a nontrivial hyperinvariant
subspace.
Proof: If 5 £ 7rleft(iJT) is an eigenvalue of K, then
W = (K-sI)-1^) (9.8.6.1)
is (unless K is a scalar) a nontrivial hypervariant subspace for K:
TK = KT^ T({K - sl)-1^)) C(K- s/)"1^) (9.8.6.2)
If 7rleit(K) = 0, then (9.8.4.3) says that c{K) = {0}, Theorem 9.5.3 says
that K is quasinilpotent, and now Theorem 7.4.5 says that K has a non-
trivial hypervariant subspace. ■
We conclude by showing that, on Hilbert space, it is possible with a
single compact operator to strip the spectrum of a bounded operator down
to its essentials. The proof divides into two parts, one of which works on
Banach space:
9.8.7 THEOREM If T e BL(X,X) for a Banach space I and O 0
then there is K € KL(X,X) with \\K\\ < e for which
Cm(^^)^essm (9.8.7.1)
Proof: Let {sn:n £ N} be a dense subset of C \ u;ess(T) and put e:0 = e:
then by Theorem 6.5.2 there is Kx £ KL0(X,X) with
H^! || < \e0 andT + Kx- sj GBL_1(X,X) (9.8.7.2)
Note that if K° is given by Theorem 6.5.2 then all nonzero multiples of K°
differ from T by an invertible operator. By the openness of BL_1(X,X)
there is ex > 0 for which, for arbitrary S (E BL(X, X),
\\S\\ <e1 =>► (J + Kx - sj) + S GBL-\X,X) and e± < \e0 (9.8.7.3)
By Theorem 6.5.2 again there is K2 E KL0(X, X) with
\\K2\\<^e1 and T + Kx + K2 - s2I G BL_1(X,X) (9.8.7.4)
By (9.8.7.3)) we also have
T + Kx + K2 - sj e BL_1(X, X) (9.8.7.5)

9.8 Essential Spectra 375
Inductively we obtain sequences [en) > 0 and [Kn) in KL0(X,X) for which
||5||<en=^lr + f^iry-«jbI:* = ll2l...lnl
C BL_1(X,X) and en < \en_l (9.8.7.6)
and
ll*n+ill<Kand <r+^AJ.-«jbI:* = l,2,...,n+ll
CBL_1(X,X) (9.8.7.7)
Put
oo
K=Y,Kn (9-8.7.8)
n=l
Evidently the series converges, giving K £ KL(X,X) with \\K\\ < e\ also
{T + K- snI: n £ N} C BL_1(X,X) (9.8.7.9)
Since {sn: n £ N} is dense in C \ cjess(T) it follows that
a(T + K) \ ue33(T) = c(T + K) \ ue33(T + K) C da(r + K) (9.8.7.10)
and now Theorem 9.8.4 gives (9.8.7.1). ■
The second part of the proof assumes the condition (9.8.7.1):
9.8.8 THEOREM If T £ BL(X,X) for a Hilbert space X then there is
K£ KL(X,X) for which
c(T + K)=ue33(T) (9.8.8.1)
Proof: We begin by finding K £ KL(X, X) for which
a{T + K)C c^s°smm(r) (9.8.8.2)
Since no compact subset of C \ wl33mm(T) can contain more than finitely
many points of c{T) we can write
a{T) \ u-ec°smm(r) C K: B£N} (9.8.8.3)

376
9. Liouville's Theorem and Spectral Theory
Now put
oo
I-P=Y,Qn withQ0 = 0
n=1 (9.8.8.4)
and
Qn = Qn = Ql With Qn(X) = (T- SnI)-°°(0)
the orthogonal projection whose range is the closure of the sum of the
hyperkernels. Since each sn is at worst in n00(T) each of these is finite
dimensional. Now find (s'n) for which, for each n € N,
*'» G du~™(T) and \,'n - sn\ = dist(5„,W™(r)) (9.8.8.5)
Finally put
oo
K=J2«-Sn)(Qn-Qn-l) (9-8.8.6)
n=l
Since the projections Qn — Qn_i are mutually orthogonal the series
converges to a compact operator of bound < supn \s'n — sn\: we must verify
(9.8.8.2). Begin by noticing
a(T + K) \ uttr(T) C a(T + K) \ ue33(T)
= o(t + k)\ cjess(r + k)c 7rleft(r + K)n 7rright(r + k) (9.8.8.7)
Also, if tG C \ c^s°smm(r) then
(T + K- tl)*-1^) = {{T + K- t^X^ C P{X) C (ir)-1(0) (9.8.8-8)
Thus
aright(r+ii:)\^s0smm(r) =7rrisht(r)\^s°smm(r) cfc^N} (9.8.8.9)
But now
t = sn => (J - */)*_1(0) C Qn{X) C P(X)-1- (9.8.8.10)
Thus if tea(r + K)\ ^°smm(r) then the operator (T + K - tl)* must
be not one-one, and its null space must coincide with that of (T — tl)*. By
(9.8.8.8) and (9.8.8.10), however, this null space lies in the intersection of
P(X) and P(X)-1-, which is {0}. This contradiction proves (9.8.8.2).
For (9.8.8.1) use Theorem 9.8.7 to find K0 £ KL(X,X) for which
Cm? + K0) = cess(r) = wess(r + K0) (9.8.8.11)

9.8 Essential Spectra
377
and then apply (9.8.8.2) to find Kx £ KL(X,X) for which
a(T + K0 + Kx) C a;ecs°smm(r + K0)
Then (9.8.8.1) holds with K = K0 + K1. m
(9.8.8.12)
It would be nice to be able to report equality in (9.8.8.2) for a compact
operator K commuting with T. This, however, may not be possible even
when T is a Riesz operator, in the sense that
°ess(T) = {0}
(9.8.8.13)
so that by Theorem 9.8.4 also u;ess(r) = w|^mm(r) = {0}. If we take
X = Y xY with Y = l2 and put
T =
W U
0 -W
(9.8.8.14)
where U is the forward shift of (7.3.6.12) and W the compact weight of
(7.3.6.15), then
K =
W 0
0 -W
and
S =
0 U
0 0
(9.8.8.15)
give
Ke KL(X,X) and
0
and K + S = T (9.8.8.16)
In particular, T satisfies the Riesz condition (9.8.8.13). It is clear also,
since U is not compact, that T is not compact. Notice that W has lots of
one-dimensional eigenspaces:
(W - (l/n)/)"1^) = C6n for each n e N (9.8.8.17)
We observe that for arbitrary A € BL(/2,y
OO , v -1
^_1(0) 2 |J [W --IJ (0) => A = 0 (9.8.8.18)
since linear combinations of the Kronecker deltas 6n are dense in l2. We
then claim that T has a similar property: for arbitrary S € BL(X, X),
s~1(o)d |J [t-±i) (o)u |J fr+^/J (0)^5 = 0
n=l ^ ^ n=l ^ '
(9.8.8.19)

378 9. Liouville's Theorem and Spectral Theory
Indeed, for each nGNwe have
(0) = C
and
T +
i/f(0) =
(9.8.8.20)
n(n+l)6n+l
-(2n + l)*n
so that if S € BL(X, X) is quasinilpotent and commutes with T then the
operator induced by S on each of the one-dimensional eigenspaces (T —
(l/n)/)-1^) and (T + (l/n)/)-1^) is quasinilpotent and therefore 0.
It is now clear that (9.8.8.2) cannot hold with compact K commuting
with T: for if so then T + K is quasinilpotent and commutes with T,
therefore is 0, which would make T compact.
One last observation is the extent to which the upper and lower semi-
Fredholm spectra mimic left and right approximate eigenvalues:
9.8.9 THEOREM If X is a Banach space and T e BL(X, X) then
^ess(T)=at33(T)Ua-3(T)
and
aaess(r)cae+ss(r)na-s(r)
Proof: Theorem 6.11.3 and (6.12.4.7) say that
7rleftp(r) = ae+ss(r) = f leftp(r)
and
7rleftp(r) u 7rriehtp(r) = aess(r) = aP(r)
and
7rrightp(r) = ae-ss(r) = frightp(r)
Now (9.8.9.1) follows from (9.3.3.11) and (9.8.9.2) from (9.3.3.1). ■
9.9 HILBERT ALGEBRA
The alert reader may have noticed an omission in Definition 8.8.3: we
offered no analogue in a general "Hilbert algebra" of the "positive" operators
of Definition 8.6.1. We begin by making an observation:
(9.8.9.1)
(9.8.9.2)
(9.8.9.3)
(9.8.9.4)
(9.8.9.5)

9.9 Hilbert Algebra
379
9.9.1 THEOREM If a G A for a complex Hilbert algebra and if k > \a\a,
then the following conditions are equivalent:
cA{a) CR+ = [0,oo[ (9.9.1.1)
aA(o) C R and \k - a\a < k (9.9.1.2)
Proof: If (9.9.1.1) holds, the first part of (9.9.1.2) is clear, and by rather
a trivial part of the spectral mapping theorem 9.2.4 we have
cA{k — a) = k — cA{a) C [0,A;]
giving the second part of (9.9.1.2). Conversely, if (9.9.1.2) holds, then
k — crA(a) = oA[k — a) C [—k,k]
so that
aA(a) C k-[-k,k] = [0,2k] C [0,oo[ ■
We are now prepared to make
9.9.2 DEFINITION If A is a Hilbert algebra, then aGAis called positive
iff
a = a* and aA(a) C R+ (9.9.2.1)
We write
A+ = {a€ A:a positive} (9.9.2.2)
Since (9.6.6.1) self-adjoint elements have real spectra, a £ A will be
positive iff a = a* and
IIH -a|| < ||a|| (9.9.2.3)
The positive cone A+ of a Hilbert algebra A satisfies the conditions of
Definition 1.11.4:
9.9.3 THEOREM If A is a complex Hilbert algebra, then
A+ n (-A+) = {0} (9.9.3.1)
A+ + A+ C A+ (9.9.3.2)
R+A+ C A+ (9.9.3.3)
cl(A+) = A+ (9.9.3.4)

380
9. Liouville's Theorem and Spectral Theory
Proof: If a G A+ and —a (E A+, then cA{a) = {0} and hence \a\a = 0.
By (9.5.3.1) and the "conservative" property (9.6.6.4) it follows that a = 0,
giving (9.9.3.1). If a and b are in A+ then (9.9.2.3) gives
llll^ll H- 11*11 — C« H- *)|| < ||ll«ll — «H H- llll*H — *ll < H«H H- H*H
and now (9.9.1.2) with k = \\a\\ + ||6|| gives a + b € A+, hence (9.9.3.2).
For (9.9.3.3) we use (9.9.2.1) together with another trivial instance of the
spectral mapping theorem 9.2.4, and for 9.9.3.4 we combine the definition
(9.9.2.1) with the "upper semicontinuity" result Theorem 9.2.6. ■
It is rather easy for a positive element to be invertible:
9.9.4 THEOREM If a e A+ and b £ A+ in a Hilbert algebra A, then
aU'^a + ^r1 (9.9.4.1)
and
max(||a||,||6||)<||a + 6||<||a|| + ||6|| (9.9.4.2)
Proof: Using the notation (1.11.4.2) for the partial order induced by A+,
which is transitive by (9.9.3.2), we can argue that if a £ A+ D A~l then
k = inf aA(a) => 0 < k < a < a + b => 0 < k < inf aA(a + b) (9.9.4.3)
which excludes 0 from crA(a + b) and gives (9.9.4.1). For (9.9.4.2) we argue
0 < a < a + b < \\a + 6|| => ||a|| < \\a + 6||
using (9.9.1.2) with k = \\a + 6|| at the end. This gives one quarter of
(9.9.4.2). Interchanging a and b gives another quarter, and the remaining
half is just a restatement of the triangle inequality (1.1.1.4). ■
If a = a* is self-adjoint in the Hilbert algebra A, and B is the closed
subalgebra of A generated by a and a*, then (9.6.6.1) and (9.4.2.1) give
B**aB(a)=aA(a) (9.9.4.4)
since (9.6.6.2) says that crB(a) C R coincides with crA(a). Thus the
commutative Gelfand-Naimark Theorem 9.6.6, applied to B, can be turned
inside out to yield a massive extension of the functional calculus (9.7.0.1):
if / € C^aM) ^s arbitrary, then f(a) € A is defined by setting
<£(/(a)) = /(4(a)) for each <j> G Bx (9.9.4.5)

9.9 Hilbert Algebra
381
If in particular a £ A+ is positive, so that cA{a) C [0,oo[, then (9.9.4.5)
applies to the square root function z1!2 : [0,00[ —► [0,oo[. This gives us
another characterization of the positive elements of A:
9.9.5 THEOREM If A is a complex Hilbert algebra and a £ A+ is
positive then there is a unique positive element a1/2 £ A for which
a1/2 e A+ and (a1/2)2 = a (9.9.5.1)
Proof: To show that a1/2 exists apply (9.9.4.5) with / = 21/2. To see that
it is unique suppose that b £ A+ and b2 = a: since a1/2 £ comm^(a) it
follows that 6a1/2 = a1/26, and now the double commutant D of {a1/2,6}
in A is a commutative Hilbert subalgebra of A. Evidently a1/2 and b have
the same Gelfand transforms in C(jD*), and therefore coincide. ■
Of course it is pretty clear that the square of a self-adjoint element is
always positive: we can prove more.
9.9.6 THEOREM If a £ A for a Hilbert algebra A, then
a*a£A+ (9.9.6.1)
Proof: We begin, rather mysteriously, with an auxiliary result: for
arbitrary a £ A there is implication
-a* a U+=^a = 0 (9.9.6.2)
To see this write a = b + ic and a* = b — ic, so that b and c are self-adjoint,
and
aa* = 262 + 2c2 - a*a (9.9.6.3)
Since b and c are hermitian, and hence have real spectra, the spectral
mapping theorem 9.2.4 says that b2 and c2 are both positive in the sense
of Definition 9.9.1. If -a*a £ A+ then (9.9.6.3) and (9.9.3.2) imply that
aa* £ A+, and now (7.1.3.7) says that also a*a £ A+. An application of
(9.9.3.1), with aa* in place of a, finishes the proof of (9.9.6.2). Toward
(9.9.6.1) we now write
a*a = b-c with 6,c £ A+ and be = cb = 0 (9.9.6.4)
In terms of the functional calculus we may take
b= \(a*a+{{a*a)2y!2) and c = \{-a*a+ {{a*a)2)^2) (9.9.6.5)

382
9. Liouville's Theorem and Spectral Theory
We now have
(ac)*ac = c*a*ac = c{b — c)c = — c3 £ —A+
which by (9.9.6.2) gives ac = 0 and hence c3 = 0 and hence c = 0, so that
finally a = b £ A+. ■
Theorems 9.9.6 and 9.9.4 can be used to extend the spectral
permanence from self-adjoint elements to arbitrary elements:
9.9.7 THEOREM If a £ A for a Hilbert algebra A, then
*Aft(a) C {s £ C: (a - s)*{a - s) <£ A"1} C ^{a)
and
<^lght(a) C {s £ C: (a - s){a - s)* <£ A'1} C f^ght(a) (9.9.7.2)
Proof: If a£ A and a*a£ A~l then certainly a is left invertible and hence
not a topological left zero divisor. Conversely, if a is not a topological left
zero divisor, and if k > 0 satisfies ||a6|| > k\\b\\ for each & G i, then,
recalling the B* condition (8.8.1.1),
||6||||a*a6|| > \\b*a*ab\\ = \\ab\\2 > k2\\b\\2 for each b £ A
so that a* a is also not a topological left zero divisor in A. Since a* a is
self-adjoint, therefore (9.6.6.2) has real spectrum, which therefore coincides
with its boundary, (9.3.3.1) tells us that if a*a is not a topological left zero
divisor then it is invertible. Applying this argument to a — s with s £ C
finishes the proof of (9.9.7.1), and applying (9.9.7.1) with a* in place of a
gives (9.9.7.2). ■
Rather late in the day, we are able to prove that the quotient of a
Hilbert algebra by a closed two-sided ideal is a Hilbert algebra:
9.9.8 THEOREM If A is a complex Hilbert algebra and if J C A is a
closed two-sided ideal of A, then
J* =J (9.9.8.1)
and
inf \\a*a - d\\ = ( inf \\a - d\\ J for each a £ A (9.9.8.2)
(9.9.7.1)

9.10 States and Representations 383
Proof: If b € A+ is arbitrary and c = 6(1 + 6)_1 then, using (9.9.4.5),
c£A+ and ||c|| < 1 and ||1 - c\\ < 1 (9.9.8.3)
If K C J is a finite set, with nK elements, write
bK = ^2 x*x and CK = bK^ + hK)~1 (9.9.8.4)
xeK
Evidently these are elements of J for which, in the sense of (1.9.0.5),
\\d-dcK\\ —► 0 as K —► J for each d G J (9.9.8.5)
This means that if d € J then d* G cl(J) = J, giving (9.9.8.1). Also if
a £ A is arbitrary then, in the same sense,
\\a - ack\\ —v dist(a, J) as K —► J (9.9.8.6)
This now transfers the B* condition (8.8.1.1) from A to A/ J. m
In particular, the Calkin algebra is a "Hilbert algebra": if X is a Hilbert
space then for each T (= BL(X, X)
dist(r*r, kl(x, x)) = dist(r, kl(x, x))2 (9.9.8.7)
9.10 STATES AND REPRESENTATIONS
The "states" of Definition 7.13.2 can be used to define another kind of
spectrum:
9.10.1 DEFINITION If A is a normed algebra, then the numerical range
of an element a £ A is the set
V{a) = VA(a) = {/(a): H/ll = 1 = /(l)} = {/(a):/ € State(A)}
(9.10.1.1)
and the pure numerical range is the set
a* {a) = {/(a): / G Extreme (State (A))} (9.10.1.2)
The numerical range is closed and convex, and includes the almost
spectrum:

384
9. Liouville's Theorem and Spectral Theory
9.10.2 THEOREM If a £ A for a normed algebra A, then
*a(«) £ Va{°) = cl(cvx(aj(a))) C {|*| < ||a||} (9.10.2.1)
and for arbitrary s,t £ C there is equality
VA(sa + t)= sVA(a) +1 . (9.10.2.2)
If also b £ A there is inclusion
VA(a + b)C VA(a) + VB(a) (9.10.2.3)
If T £ HBL(A, B) is a bounded homomorphism, then
T relatively open => VB(Ta) = f] VA(a + d) (9.10.2.4)
Td=0
Proof: Since State (A) is closed in the "weak* topology" of pointwise
convergence on A induced by the closed unit disc in A*, which by the proof
of Theorem 9.6.4 is compact, it is clear that VA(a) C C is the continuous
image of a compact set, and therefore closed. It is obvious that VA(a) is
contained in the disc {\z\ < \\a\\} C C; to see that it is convex observe that
if f,g £ State(A) and 0 < t < 1 then
\\(l-t)f+tg\\ < [l-t+t)l = {l-t)f{l)+tg{l) < ||(l-«)/+^|| (9.10.2.5)
From (5.11.3.1) it is clear that the numerical range is contained in the
closed convex hull of the pure numerical range. To see why it includes
the almost spectrum suppose 5 £ ^a{°) ls arbitrary, and define a linear
functional /0 : Ca + C —> C by setting
f0(ra + t) = rs + t for each r,t £ C (9.10.2.6)
By (9.2.3.1) from Theorem 9.2.3 this is well defined and satisfies the
condition (7.13.2.1), and by the Hahn-Banach Theorem 5.3.2 can be extended
to / £ State(A). Equality (9.10.2.2) and inclusion (9.10.2.3) can be left
to the reader. With no restriction on T £ KBL(A,B), if 5 = g{Ta) with
g £ State(B) then f = gT £ State(A), giving
VB(Ta)CVA(a) (9.10.2.7)
and, hence, forward inclusion in (9.10.2.4). Conversely, if 5 £ C is in the
right-hand side of (9.10.2.4) we can define g0 : CTa + C —> C by setting
g0{rTa + t) =rs + t for each r,t £ C (9.10.2.8)

9.10 States and Representations
385
Since T is relatively open, g0 is continuous, and of course well defined if 5 is
in the right-hand side of (9.10.2.4). We can now extend by Theorem 5.3.2
to0GState(5). ■
One small consequence of (9.10.2.4) is that the numerical range of an
element is independent of the algebra: if a £ A C B then (9.10.2.4) gives
VB(a) = VA(a). We shall write
Extreme(State(A)) = A* (9.10.2.9)
In a Hilbert algebra, the numerical range of an element determines
whether it is positive, self-adjoint or zero.
9.10.3 THEOREM If a, b € A for a Hilbert algebra A and / e State(A),
then
a£A+=> /(a) £ R+ (9.10.3.1)
f(a*)=f(a)- (9.10.3.2)
|/(6*a)\2 < f(a*a)f(b*b) (9.10.3.3)
Proof: For (9.10.3.1) suppose that f(a) = s + it with s,t £ R. We must
show that t = 0 < s. If k > 0 is sufficiently small then o~A(l — ka) C [0,1],
giving || 1 — A;a|| = |1 — ka\a < 1 and hence
|1 - ks\ < |1 - k(s + it)\ = |/(1 - ka)\ < 1
so that 5 >0. Also 6 = a +5 +Ogives ||62|| = ||6*6|| = ||(a + s)2 + A;2*2|| <
||a + s||2 + A;2*2 so that
(A;2 + 2k + l)*2 = |/(6)|2 < \\a + 6||2 + A;2*2 for all k > 0
Allowing k —> oo forces t = 0. This gives (9.10.3.1), and hence, by the
argument of (9.9.6.4) the numerical range of a self-adjoint element is always
real. Writing a = b + ic and a* = b — ic now gives (9.10.3.2). Finally,
(9.10.3.1) and (9.9.6.1) show that
f{a*a) > 0 if a € A and / € State(A) (9.10.3.4)
Now (9.10.3.3) follows by exactly the same argument used to prove the
Schwarz inequality (8.1.2.4). ■
Theorem 9.10.3, together with a certain amount of "pulling oneself up
by the bootstraps," now gives

386
9. Liouville's Theorem and Spectral Theory
9.10.4 THEOREM If a £ A for a complex Hilbert algebra A, then
of (a) = {0} <=> VA{a) = {0} <=> a = 0 (9.10.4.1)
a*[a) C R <=> VA(a) C R <=> a = a* £ h+(A) (9.10.4.2)
of (a) = R+ <=> VA(a) CR+^fla+ (9.10.4.3)
Proof: Theorem 9.10.3 suggests that at least
a = a* and VA{a) = {0} =>► a = 0 (9.10.4.4)
but this is clear, since (9.10.2.1) and (9.6.6.5) give
°a (<0 C VA(a) and ||a|| = \a\a (9.10.4.5)
To generate the second part of (9.10.4.1) from this write a = b + ic with
a* = b — ic, and recall from (9.10.3.2) that f(b) and /(c) are real for all
/ £ State(A): thus if VA(a) = {0} then for arbitrary / £ State(A) we have
0 = f(a) = f(b) + if(c) => /(&) = /(c) = 0 (9.10.4.6)
Now (9.10.4.4) gives b = c = 0 and hence a = b + ic = 0. This proves the
second part of (9.10.4.1), which in turn gives the second part of (9.10.4.2):
for if VA(a) C R then for arbitrary / £ State(A) we have, using (9.10.3.2),
f(a - a*) = /(a) - /(a*) = /(a) - /(a)" = 0 (9.10.4.7)
giving a = a* £ h^{A) by (9.10.4.1). This proves the second part of
(9.10.4.2), which then combines with Definition 9.9.2 to give the second
part of (9.10.4.3). Finally the first part of each part of Theorem 9.10.4
follows from the middle of (9.10.2.1) from Theorem 9.10.2. ■
From (9.10.4.5) it is also clear that
a normal in A => ||a|| C i\s\:s £ <jf{a)\ C {\s\:s £ VA(a)} (9.10.4.8)
Theorem 9.10.4 says, among other things, that a £ A is "positive" iff
/(a) > 0 for every state f £ A*; dually a linear functional f £ A* with
/(l) = 1 will be a state iff /(a) > 0 for every positive element a £ A:
9.10.5 THEOREM If A is a complex Hilbert algebra and / £ A^, then
/(A+)CR+«=>||/||=/(1) (9.10.5.1)

9.10 States and Representations
387
Proof: Backward implication is (9.10.4.3). Conversely, if the left-hand
side of (9.10.5.1) holds, then (9.9.6.4) gives inclusion f(h+(A)) C R. Also,
if a (E h+(A) is arbitrary, then ||a|| — a and ||a|| + a are both in A+, so that
a = a*=>-||a||/(l)</(a)<||a||/(l)
Applying this to the element a*a for arbitrary a £ A, and using (9.10.3.3),
gives
|/(a)|2 = |/(l*a)|2 < /(a*a)/(l * l) < ||/||||a||2/(l) ■
Linear functional f £ A* which satisfy either side of (9.10.5.1) make
up the "positive" elements of the dual space A*:
(A^)+ = {fe A:f{A+) C R+} (9.10.5.2)
The reader should verify the analogue of Theorem 9.9.3, showing that this
satisfies the conditions of Definition 1.11.4.
If A is a normed alegbra then, rather trivially, the mapping
a —► La : A —► BL(A, A) (9.10.5.3)
shows that A can be regarded as a closed subalgebra of the bounded
operators BL(X, X) on a normed space X. Rather less trivially, the "noncom-
mutative Gelfand Neumark theorem" says that if A is a Hilbert algebra
then we can do this with a Hilbert space:
9.10.6 THEOREM If A is a complex Hilbert algebra, then there exists
an isometric *-homomorphism
tta# : A — BL(tfA#,tfA#) (9.10.6.1)
from A into the bounded operators on a Hilbert space HA#.
Proof: This is acheived by the pure states on A. We begin by showing
how each / € State (A) induces a cyclic representation
ivf:A —► BL(Hf,Hf) (9.10.6.2)
from A into a Hilbert space H^ in the sense of a *-homomorphism of bound
< 1 for which there is a "cyclic vector"
£f e Hf for which Hf = cl {irf(a)Zf: a € A} (9.10.6.3)
Specifically, if we write [taking advantage of (9.10.3.3)]
Af = {a£ A: f(Aa) = {0}} = {a £ A: f(a* a) = 0} (9.10.6.4)

388
9. Liouville's Theorem and Spectral Theory
for the "left kernel" of /, then we can take
Hf = (A/A,)" (9.10.6.5)
to be the completion of the quotient space A/A, with respect to the norm
induced by the inner product defined by setting, for each a, b £ A,
(a + A, ; b + A,) = f(b*a) (9.10.6.6)
Evidently the coset 1 + A, of the identity 1 £ A is the cyclic vector £,.
If we now take a nonempty subset K C State (A) then we can combine
the cyclic representations 7r, associated with / £ K in the crudest possible
way, by setting
HK = l2(Hf)feK = \x £ n Hf' £ H*/H2 < °° [ (9-10'6-7)
[ feK feK J
and then defining
*K = *2(T/)/ejc : A —> BL(HK,HK) (9.10.6.8)
in the obvious way:
{nK{a)x)f = irf(a)(xf) for each a£A,f £ K,x£ HK (9.10.6.9)
Evidently nK is a *-homomorphism of bound < 1 whenever K C State (A)
is nonempty: to make it isometric it will be sufficient, using (9.6.6.5), to
ensure that it is faithful in the sense of being one-one. Now (9.10.4.8) shows
that nK is faithful whenever
Extreme(State(A)) = A* C K ■ (9.10.6.10)
When the Hilbert algebra is commutative then the pure states A#
coincide with the characters A*:
9.10.7 THEOREM If A is a Hilbert algebra there is inclusion
A* C A* (9.10.7.1)
with equality whenever A is commutative.

9.10 States and Representations
389
Proof: If / £ A* is a character and if
/ = (l-t)/o+«/i (9.10.7.2)
with 0 < t < 1 and states /0 and fx then certainly
A+ n r1 (o) c /o"1 (o) n /f1 (o) (9.10.7.3)
and, hence, using first the decomposition (9.9.6.4) of a hermitian element
as the difference of positive elements and then the decomposition (8.8.4.3)
of an arbitrary element as a linear combination of hermitian elements,
/"'(O) C f-'iO) and r^O) C /^(O) (9.10.7.4)
Note that we need the explicit formula (9.9.6.5) for the "positive" and
"negative" parts of a hermitian element here. From (9.10.7.4) and (6.1.2.2)
it follows that /0 and f1 are both scalar multiples of /, and hence that
fo = fi = /• Conversely, if A is commutative and / is a pure state we
claim that if a € A there is implication
0<a<l=>foLa = f(a)f (9.10.7.5)
Indeed, if 0 < f{a) < 1 we have (9.10.7.2) with
< = /(*) /iW"VoLa /0 = (l-/(a))-1(/-/oLa) (9.10.7.6)
giving the right-hand side of (9.10.7.5) by the pure condition. If instead
/(a) = 0 then for arbitrary b = b* £ h+(A) there is k £ R for which —k <
b < k, which means (using the commutivity of A) that — ka < ab < ka, and
hence, that —kf(a) < f(ab) < kf(a). Thus / o La annihilates hermitian
elements and is therefore 0, giving again (9.10.7.5). If instead f(a) = 1
then the same argument shows that foLa = f. Since every element of A is
a linear combination of elements a with 0 < a < 1 it follows that f°LA = f
for all a, which means that / is a character. ■

10
Comparison of Operators
and Exactness
The various kinds of invertibility and singularity have "relative" analogues,
in which one operator is compared to another. If we mix both left and
right comparisons and then specialize, we come down to concepts of
"exactness."
10.1 MAJORIZATION AND FACTORIZATION
If T £ L(X,Y) and S (E L(X, Z) are linear operators between normed
spaces with the same domain space X then there seem to be five basic
comparisons between S and T:
10.1.1 DEFINITION S e L(X,Z) is called a left multiple of T e
BL(X,y) if there is U £ BL(Y, Z) for which
S = UT (10.1.1.1)
and an almost left multiple ofTe BL(X,Y) if there is (Un) in BL(Y,Z)
for which
||5 - UnT\\ —► 0 and sup \\Un\\ —► oo (10.1.1.2)
n
and an approximate left multiple of T if there is (Un) in BL(Y, Z) for which
\\S-UnT\\^0 (10.1.1.3)
We shall say that S is majorized by T if there is k > 0 for which
\\Sx\\ < k\\Tx\\ for each x £ X (10.1.1.4)
391

392
10. Comparison of Operators and Exactness
and that S is determined by T if there is inclusion
r_1(0) C5_1(0) (10.1.1.5)
For example, S (E L(X, Z) is bounded if and only if it satisfies (10.1.1.4)
with Y = X and T = J; if instead S = I then (10.1.1.1) means that
T is left invertible, (10.1.1.2) and, using Theorem 3.7.2, (10.1.1.3), mean
that T is almost left invertible, (10.1.1.4) means that T is bounded below,
and (10.1.1.5) means that T is one-one. It is not clear how to extend the
"closed" condition (3.3.1.2) from i" to more general S. We recall from
(6.9.2.11) how (10.1.1.3) holds with S = P when T is upper semi-Fredholm
and P = P2 € BL(X,X) satisfies P_1(0) = T_1(0). Each of the conditions
in Definition 10.1.1, with the exception of (10.1.1.3), is transitive:
10.1.2 THEOREM Each of the conditions (lO.l.l.j) with j ^ 3 is
transitive. If T G BL(X,y) and S G BL(X,Z) there is implication
(10.1.1.1) =>► (10.1.1.2) =>► (10.1.1.4)
| | (10-1-2.1)
(10.1.1.3) =>► (10.1.1.5)
If T is proper with complemented range there is implication
(10.1.1.5) =)> (10.1.1.1) (10.1.2.2)
Proof: If S = UT and R = VS then R = {VU)T, so that (10.1.1.1)
is transitive. If \\S - UnT\\ -> 0 and \\R - VnT\\ -> 0 then, provided
supn||Vn||<oo,
||JZ - VnUnT\\ < \\R - VnS\\ + (sup ||VJ|) ||S - UnT\\ —> 0 (10.1.2.3)
m
and, if in addition, supn \\Un\\ < oo, then also supn ||Vnl7n|| < oo. This
proves that (10.1.1.2) is transitive. If ||5(-)|| < *||T(-)|| and ||-R(-)|| <
h\\S{-)\\ then ||iE(.)|| < ^||r(-)||, so that (10.1.1.4) is transitive, and the
transitivity of (10.1.1.5) is obvious. Towards (10.1.2.1) it is clear that if
(10.1.1.1) holds, then so does (10.1.1.2) with Un = U for each n £ N. If
(10.1.1.2) holds, then (10.1.1.3) is obvious, and (10.1.1.4) also holds with
k = supn \\Un\\: for if x € X is arbitrary then \\Sx — UnTx\\ —► 0 and
||C/nTx|| < fc||Tx||. We leave the last two implications to the reader. Finally,
if (10.1.1.5) holds we can define a linear mapping U0 : TX —► Z by setting
U0(Tx) = Sx for each xeX (10.1.2.4)
Now if T is relatively open then U0 is bounded, and if T(X) is closed and

10.1 Majorization and Factorization
393
complemented, with Q = Q2 £ BL(Y,Y) satisfying Q(Y) = T(X)9 then we
can satisfy (10.1.1.1) by setting
U = U0Q ■ (10.1.2.5)
The argument for (10.1.2.2) also shows that if TX is closed and
complemented, then (10.1.1.4) =>» (10.1.1.1). The proof of the transitivity of
the relation (10.1.1.2) also shows that if S is almost left invertible and T
is approximately left invertible then the product ST is approximately left
invertible.
If instead T £ L(X, Y) and S € L(Z, Y) have the same codomain space
Y then there are seven basic comparisons between S and T:
10.1.3 DEFINITION S e L(Z,Y) is called a right multiple of T e
BL(X,y) if there is U € BL{Z,X) for which
S = TU (10.1.3.1)
and an almost right multiple of T if there is (Un) in BL(Z,X) for which
\\S-TUn\\ —>0 and sup \\Un\\ < oo (10.1.3.2)
n
and an approximate right multiple of T if there is (Un) in BL(Z, X) for
which
\\S-TUn\\ —>0 (10.1.3.3)
We shall say that S is comajorized by T if there is k > 0 for which
Sz e TDisc(0 ; k\\z\\) for each z e Z (10.1.3.4)
and almost comajorized by T if there is k > 0 for which
Sz £ clTDisc(0 ; A;||^||) for each zG Z (10.1.3.5)
We shall say that S is range-included in T if there is inclusion
S(Z) C T{X) (10.1.3.6)
and almost range-included in T if there is inclusion
S{Z) C cl(rX) (10.1.3.7)

394
10. Comparison of Operators and Exactness
For example, if Y = X and T = J, then each of (10.1.3.4) and (10.1.3.5)
reduce to the condition that S be bounded: if instead Y = Z and S = I,
then the conditions of Definition 10.1.3 reduce to T being, respectively,
right invertible, almost right invertible (twice, using Theorem 3.7.2), open,
almost open, onto, and dense. We recall from (6.9.3.18) how (10.1.3.5) holds
when T is lower semi-Fredholm and S = Q = Q2 with Q(Y) = cl(TX).
With the exception of (10.1.3.3), each of the conditions of Definition 10.1.3
is transitive:
10.1.4 THEOREM Each of the conditions (10.1.3J) with j ^ 3 is
transitive. If T G BL(X,y) and S G BL(Z,Y) there is implication
(10.1.3.1) => (10.1.3.4) => (10.1.3.6)
i i
(10.1.3.2) => (10.1.3.5)
i \ J
L
(10.1.3.3) > (10.1.3.7)
(10.1.4.1)
If T is proper with complemented null space there is implication
.(10.1.3.7) => (10.1.3.1) (10.1.4.2)
Proof: Using the corresponding arguments for Theorem 10.1.2 as a guide,
we leave the transitivity arguments to the reader. If (10.1.3.1) holds then
so does (10.1.3.2), with Un = U for each n € N, and so also does (10.1.3.4),
with k = \\U\\. If (10.1.3.2) holds then (10.1.3.3) is obvious, and (10.1.3.5)
also holds, with k = supn \\Un\\. We leave the rest of the implications of
(10.1.4.1) to the reader. Finally, if (10.1.3.6) holds we can define a linear
mapping U0 : Z —► X/T_1(0) by setting, for each z € Z and x £ X,
U0z = x + T-1(0) ifSz = Tx (10.1.4.3)
If T(X) = c\(TX) is closed then of course (10.1.3.7) implies (10.1.3.6). If
also T is relatively open then U0 is bounded, and finally if T_1(0) = P_1(0)
with P = P2 £ BL(X, X) we may replace U0 by U : Z -> X for which
Uz = Px ifSz = Tx m (10.1.4.4)
10.2 MIXED INTERPOLATION
If linear operators T G L(X,Y) and S G L(Y, Z) satisfy
S-1{0)=T{X) (10.2.0.1)

10.2 Mixed Interpolation
395
then the pair (S, T) is sometimes described as exact This is a rather special
case of the common generalization of the comparisons of Definitions 10.1.1
and 10.3.1:
10.2.1 DEFINITION If T £ L(X,y), S e L(W,Z) and R e L(W,Y),
then we shall call R a (left,right) multiple of (S,T) if there are U €
BL(W,X) and V e BL(Z,Y) for which
R = VS + TU (10.2.1.1)
and an almost (left,right) multiple of (S, T) if there are (Un) in BL(W,X)
and (VJ in BL(Z,Y) for which
\\R-VnS-TUn\\^0 and sup ||iy +sup \\Vn\\ < oo (10.2.1.2)
n n
and an approximate (left,right) multiple of (S,T) if there are (Un) in
BL{W,X) and (VJ in BL(Z,Y) for which
\\R-VnS-TUn\\-+0 (10.2.1.3)
We shall say that R is (left,right) majorized by (S, T) if there are k > 0
and h > 0 for which
iZiu G Disc(0 ; /i||5t(;||) + T Disc(0 ; Jb||u;||) for each w G W (10.2.1.4)
and almost (left,right) majorized by (S, T) if there are k > 0 and h > 0 for
which
iZiu G cl(Disc(0;fe||5u;||)+rDisc(0;ik||t(;||)) for each w £ W (10.2.1.5)
We shall say that R is (left,right) determined by (S,T) if
JR(S'-1(0)) CT(X) (10.2.1.6)
and almost (left,right) determined by (S,T) if
RiS'1 (0)) C cl(rX) (10.2.1.7)
The reader may notice that we have used just seven of a possible 35
combinations of the conditions of Definitions 10.1.1 and 10.1.3. If, for
example, 5 = 0 then Definition 10.2.1 reduces to Definition 10.1.3 with R
in place of 5; if instead T = 0 then this all reduces to Definition 10.1.1
with (R,S) in place of (5,T). If W = Y and R = I, and if ST = 0, then
(10.2.1.5) reproduces the exactness condition (10.2.0.1). Thus, if R = I
and ST = 0 we can think of each of the conditions of Definition 10.2.1 as

396
10. Comparison of Operators and Exactness
some kind of "topological exactness." For example the condition (10.2.1.5)
is satisfied if R = I and T = J : X —► Y is the natural embedding of a
subspace X C Y, with S = K : Y —► Z = Y/c\(X) the corresponding
quotient; in this situation the condition (10.2.1.1) says that X is closed and
complemented in Y.
The conditions of Definition 10.2.1 satisfy the same pattern of
implication as those of Definition 10.1.3:
10.2.2 THEOREM If T £ BL(X, Y), S £ BL(W, Z) and R £ BL(W, Y)
there is implication
(10.2.2.1)
(10.2.1.1) => (10.2.1.4) => (10.2.1.6)
i i
(10.2.1.2) => (10.2.1.5)
i \ J
(10.2.1.3) > (10.2.1.7)
If S and T are both regular there is implication
(10.2.1.7) => (10.2.1.1) (10.2.2.2)
Proof: If (10.2.1.1) holds, then so does (10.2.1.2), with Un = U and Vn =
V for each n £ N, and so also does (10.2.1.4), with k = \\U\\ and h = \\V\\.
If (10.2.1.2) holds, then (10.2.1.3) is obvious, and (10.2.1.5) also holds with
k = supn \\Un\\ and h = supn ||Vn||: for if w £ W is arbitrary then \\Rw —
VnSw - TUnw\\ -> 0 with VnSw £ Disc(0 ; h\\Sw\\) and TUnw £ T Disc(0 ;
k||w||). The remaining implications of (10.2.2.1) are left to the reader. For
(10.2.2.2) suppose T = TT'T and S = SS'S with T £ BL(Y,X) and
S' £BL(Z,W) and write
P = S'S and Q = TT' (10.2.2.3)
for the projections of (3.10.1.3): then if (10.2.1.7) holds we have
(/- Q)"1^) = Q(Y) = c\(TX) D R(S-l(0))
= R(P-1(0))=R((I-P)(W)
which, is equivalent to
(J - Q)R{I - P) = 0 (10.2.2.4)

10.2 Mixed Interpolation
397
Thus
R = QR- QRP + RP = (RSf - TT'RS')S + TlT'R)
V (10.2.2.5)
= [RS')S + T{T'R - T'RS'S)
giving us two candidates for a pair (V, 17) to satisfy (10.2.1.1). ■
The transitivity properties of the relations of Definition 10.2.1 are more
complicated, and fall into two sets. The first six are fairly obvious:
10.2.3 THEOREM Suppose T £ BL(X,y), S £ BL(W,Z) and R £
BL(W,Y): then if T £ BL(X',Y) and S' £ BL(W,Z') we have the
following implications:
R a (left,right) multiple of (S, T) and S a left multiple of S'
and T a right multiple of T' (10.2.3.1)
=^i2a (left,right) multiple of (S',T')
R an almost (left,right) multiple of (S, T) and S an almost
left multiple of S'and T an almost right multiple of Tf (10.2.3.2)
=> R an almost (left,right) multiple of (Sf,Tf)
R (left,right) majorized by (5, T)
and S majorized by Sf and T comajorized by T' (10.2.3.3)
=> R (left,right) majorized by (S',T')
R almost (left,right) majorized by (5, T)
and S majorized by Sf and T almost comajorized by Tf (10.2.3.4)
=> R almost (left,right) majorized by (Sf,Tf)
R (left,right) determined by (S,T)
and S determined by S' and T range-included in T' (10.2.3.5)
=> R (left,right) determined by (S',T')
R almost (left,right) determined by (S, T) and S determined by Sf
and T almost range-included in T' (10.2.3.6)
=> R almost (left,right) determined by (Sf,Tf)

398
10. Comparison of Operators and Exactness
Proof: lfR = VS + TU and S = V'S' and T = T'U' then R = [VV')S' +
T'{U'U), giving (10.2.3.1). If \\VnS + TUn - R\\ -> 0 and \\V^S' - S\\ -> 0
and ||T'i74 - T|| -> 0, and if supn ||V^|| < oo and supn ||C/J| < oo, then
\\VnV^S' + T'U'nUn - R\\ -> 0, giving (10.2.3.2). If for each w e W and
each xGX we have J)isc{Rw; h\\Sw\\) nTJ)isc{0; k\\w\\) ^ 0 and ||5u;|| <
h'\\S'w\\ and Tx £ {T'x'i \\x'\\ < k'\\x\\} then also D'isc(Rw ; hh'\\S'w\\) D
r'Disc(0 ; ikik'Hwll) ^ 0, giving (10.2.3.3). The derivation of (10.2.3.4) is
very similar, and then (10.2.3.5) and (10.2.3.6) are left to the reader. ■
Not quite so obviously,
10.2.4 THEOREM Suppose T £ BL(X,Y), S £ BL(W,Z) and R £
BL(W,Y): then if S' £ BL(Y,Y') and T £ BL(W',W) satisfy
S'T = 0 = ST' (10.2.4.1)
We have the following implications:
R a (left,right) multiple of (S,T)
=> S'R a left multiple of S (10.2.4.2)
and RT' a right multiple of T
R an almost (left,right) multiple of (5, T)
=> S'R an almost left multiple of S (10.2.4.3)
and RT' an almost right multiple of T
R (left,right) majorized by (S, T)
(10.2.4.4)
=> S'R majorized by S and RT' comajorized by T
R almost (left,right) majorized by (5, T)
=> S'R majorized by S (10.2.4.5)
and RT' almost comajorized by T
R (left,right) determined by (5, T)
(10.2.4.6)
=> S R determined by S and RT range-included in T
R almost (left,right) determined by (5, T)
=> S'R determined by S (10.2.4.7)
and RT' almost range-included in T

10.2 Mixed Interpolation
399
Proof: If R = VS + TU then (10.2.4.1) give S'R = (S'V)S and RT' =
r(UT'), which proves (10.2.4.2). If instead \\R - VnS - TUn\\ -► 0 then
(10.2.4.1) gives \\S'VnS-S'R\\ -► 0 and \\TUnTf-RT'\\ -► 0, which proves
(10.2.4.3), and indeed the analogue for approximate (left,right) multiples.
If for each w £ W we have Disc^w; ; k||Su;||) D TDisc(0 ; k\\w\\) ^ 0 so
that there is x £ X with \\Rw — Tx\\ < h\\Sw\\ and ||x|| < k\\w\\ then
(10.2.4.1) says that ||«S"iZiu|| = \\S'(Rw - Tx)\\ < h\\Sw\\ and also that
if w = T'w' then \\Rw - Tx\\ = \\RT'w' - Tx\\ < h\\ST'w'\\ = 0 with
||x|| < Jb||u;|| = ik||rV|| < fc||r'||||u/||. This proves (10.2.4.4), and the
argument for (10.2.4.5) is the same. We leave (10.2.4.6) and (10.2.4.7) to
the reader. ■
The implication (10.2.2.2) at the end of Theorem 10.2.2 can be
extended to more general S and T if we specialize R:
10.2.5 THEOREM If T £ BL(X,y), S £ BL(W,Z), and R £ BL(W,Y)
satisfy
S relatively almost open and R of finite rank (10.2.5.1)
then there is implication
(10.2.1.6) => (10.2.1.1) (10.2.5.2)
Proof: If R £ BL(W, Y) is of finite rank then by Theorem 6.3.2 we may
write
n+m
R=YlkJ® yj (10.2.5.3)
y=i
where (t/y)?^^ is a basis for the finite dimensional space R(W) C Y, and
by Theorem 6.1.1 we can arrange that
n
J^ Cyy = R(W) n T{X) (10.2.5.4)
y=i
Thus there is (x;)y=1 in X for which
Txj = t/y for each j £ {l, 2,..., n} (10.2.5.5)
We claim, using condition (10.2.1.6) that iE(5_1(0)) C T(X), that
m
S-1^) C fl fc,-+m(0) (10.2.5.6)
t=l

400
10. Comparison of Operators and Exactness
because if Sw = 0 then R(w) £ T(X), which means that kj(w) = 0 if
j > n + 1. As in the proof of Theorem 5.5.3 this means that there is
(ky)*+£+i in Z* for which
kj = hjS for each j e {n + 1,..., n + m} (10.2.5.7)
Define tf : S(W) -► C by setting h^Sw) = kj(w) for each w £ W, using
the relatively almost open property of S to see that h% is well defined and
bounded, and then extend to h- by the Hahn-Banach Theorem 5.3.2. But
now
n n+m
U = Y,kjQxj,V = Y, hj® Vj =>VS + TU = R (10.2.5.8)
y=i y=n+i
giving (10.2.1.1). ■
Since the totally bounded operators KL(X, Y) form a "two-sided ideal"
in the sense of (3.9.6.7) it is clear that if (10.2.1.1) holds then
T £ KL(X, y), S £ KL(W, Z)=>R£ KL{W, Y) (10.2.5.9)
We can prove more:
10.2.6 THEOREM If T G BL(X,y), S G BL(W,Z), and R £ BL(W,Y)
satisfy
Rw G c\(D'isc(0 ;h\\Sw\\) + TDisc(0;A;||w;||) for each w G W (10.2.6.1)
then
\\R\\'~<4k\\T\\'m + 2h\\S\\'„ (10.2.6.2)
In particular
(10.2.1.5) =>► (10.2.5.9) (10.2.6.3)
Proof: By definition (6.13.1.6) there are for each e > 0 finite subsets Ke C
X and He C W such that
{Tx: ||s|| < 1} C Disc(T(Ke) ; 2||r||'ess + e)
and (10.2.6.4)
{Sw: H| < 1} C Disc(S(2T.); 2||5||US + e)
Now if w € W is arbitrary there is w' £ W for which
w' e \\w\\H£ and \\Sw - Sw'\\ < {2\\S\\'m + e)\\w\\ (10.2.6.5)

10.3 Exactness
401
and then for arbitrary 6 > 0 there is x £ X for which
\\R(w - w') - Tx\\ < h\\Sw - Sw'\\ + S\\w\\ and ||x|| < fc||tc; - w'\\
(10.2.6.6)
Finally there is xf (E X for which
x'€\\x\\Ke and \\Tx-Tx'\\ < (2\\T\\'eaa +e)\\x\\ (10.2.6.7)
Thus
\\Rw - Rw' - Tx'\\ < h\\Sw - Sw'\\ + \\Tx - Tx'\\ + 6\\w\\
< h(2\\S\\'ess + e)\\w\\ + (2||r||'ess + e)||x|| + 6\\w\\
<HnS\\'ess + s)\\w\\ + (2\\T\\'ess + s)k\\w-w'\\
+ 6\\w\\
< (2&||5||U + 4||T||U)lhll + {he + 2ke + 6)\\w\\
noting that \\wf\\ < \\w\\. This gives
{Rw:\\w\\ < 1} C Disc(L,„2fc||5||^, +4*11111^ + ^) (10.2.6.8)
with
Le, = R(He) + T(kKe) C W (10.2.6.9)
as soon as he + 2ke + 6 < e'. ■
10.3 EXACTNESS
When W = Y and R = I the conditions of Definition 10.2.1 can be thought
of as various kinds of exactness, particularly when ST = 0. We shall write
BL(X,y,Z) = {{S,T) GBL{Y,Z) x BL(X,Y):ST = 0} (10.3.0.1)
for the set of these chains from X to Y to Z: note that they form a closed
subset, not necessarily linear, of the product space BL(Y,Z) x BL(X, Y).
10.3.1 DEFINITION The chain (5, T) € BL(X, Y, Z) is called (hft,right)
invertible, or decomposably exact, if there are U £ BL(Y,X) and V €
BL(Z,Y) for which
VS + TU = I (10.3.1.1)
It is called almost invertible, or almost decomposably exact, if there are (Un)
in BL{Y,X) and (VJ in BL(Z,Y) for which
\\VnS + TUn-I\\^>0 and sup \\UJ +sup ||VJ| < oo (10.3.1.2)
n n

402 10. Comparison of Operators and Exactness
and is called approximately invertible, or approximately decomposably exact,
if there are (17J in BL{Y,X) and (VJ in BL{Z,Y) for which
\\VnS + TUn-I\\^>0 (10.3.1.3)
We shall call the chain (S, T) exact if there are k > 0 and h > 0 for which
for each y £ Y, y £ Disc(0 ; h\\Sy\\) + TDisc(0 ; fc||y||) (10.3.1.4)
and almost exact if there are k > 0 and h > 0 for which
for each y £ Y, y £ cl(Disc(0 ; h\\Sy\\) + TDisc(0 ; ife||y||)) (10.3.1.5)
We shall call the chain (S, T) linearly exact if
S_1(0) CT(X) (10.3.1.6)
and almost linearly exact if
S_1(0) Ccl(rX) (10.3.1.7)
As a matter of notation we shall write
BL_1(X,y,Z) = {{S,T) £ BL(X,Y,Z):(S,T) invertible} (10.3.1.8)
aieft,right BL^ y? ^ = BL^Xj y? Zj \ BL-i (x, Y, Z) (10.3.1.9)
aleft'rightBL(X,y,Z)
= {{S,T) £ BL{X,Y,Z): (S,T) not almost invertible}
fleft'rightBL(X,y,Z)
= {(S,T) £ BL(X,Y,Z): (S,T) not almost exact}
7rleft'rightBL(X,y,Z)
= {{S,T) £ BL(X,Y,Z): (S,T) not almost linearly exact}
Finally we shall write
(10.3.1.10)
(10.3.1.11)
(10.3.1.12)
rleft>rightBL(X,y,Z) = {(S,r) £ BL(X,Y,Z):(S,T) not strictly exact}
(10.3.1.13)
where we call (S,T) £ BL{X,Y,Z) strictly exact iff
(S,T) is exact and S(Y) is closed (10.3.1.14)
If the chain (S,T) is invertible in the sense of (10.3.1.1) then it is also

10.3 Exactness
403
regular in the sense that each of S and T are regular:
rBT(x, y, z) = bl(x, y, z) n fBTfy, z) x rBT(y, x)) (10.3.1.15)
In this case we can arrange that its "generalized inverse" is another chain.
Indeed if
ST = 0 and T = TT"T and S = SS"S
(10.3.1.16)
with T" = T"TT" and S" = S"SS"
and we take
(T', 5") = (T", (J - TT")S") or (T', S') = (T"(I - S"S),S")
(10.3.1.17)
then
T = TTfT and S = SS'S and r'S' = 0 (10.3.1.18)
Definition 10.3.1 extends to "chains" of length 3 and longer. If for
example, T G BL(X,Y), S £ BL(Y,Z) and R G BL(Z,W) satisfy
ST = 0 = RS (10.3.1.19)
we shall write
(R, S, T) £ BL(X, y, Z, W) (10.3.1.20)
and again speak of a chain. To extend Definition 10.3.1 we simply impose
corresponding conditions on each of the subchains (S,T) and (R, S). For
the first two conditions, the result can be improved:
10.3.2 THEOREM If (R, S, T) € BL(X, Y, Z, W) is invertible, then there
is a chain (T',S',Rf) e BL(W,Z,Y,X) for which
S'S + TT' = I = R'R + SSf (10.3.2.1)
If (R, S, T) € BL(X, y, Z, W) is almost invertible, then there are (T'n), (S'n),
and (R'n) for
\\S'nS + TT'n - I|| + \\R'nR + SS' - I\\ — 0
and (10.3.2.2)
8Up||^||+8Up|K||+8UP|K||<00

404
10. Comparison of Operators and Exactness
Proof: If ST = 0 = RS and
S'S + TT' = I = R"R + SS" (10.3.2.3)
then, multiplying on either side by S' or S",
(J - SS') (J - R"R) = 0 = (J - TT') (J - S"S) (10.3.2.4)
which gives (10.3.2.1) with
R'=(I- SS')R" (10.3.2.5)
If we assume only that (R,S,T) is almost invertible then there are (T£),
(5^), (54'), and (#£) furnishing "almost inverses" for (S,T) and (£,£),
respectively. Now the reader can confirm that (10.3.2.2) holds with
R'n = (I - SS')R% for each n £ N ■ (10.3.2.6)
If we observe how Theorem 10.3.2 is proved, it is clear how the result
extends to sequences (T1? T2,..., Tn) of operators in which 2y+1Ty = 0 for
each j.
Regularity is again the bridge between both ends of Definition 10.3.1:
10.3.3 THEOREM If (S,T) e BL(X,Y,Z) is invertible then it is also
regular, with an inverse chain (17, V) (E BL(Z, Y, X) for which
VS + TV = I and UV = 0 (10.3.3.1)
Necessary and sufficient for (S,T) £ BL(X, Y, Z) to be invertible is
(S,T) is regular and almost linearly exact (10.3.3.2)
Necessary and sufficient for T £ BL(X,Y) to be regular is that there are
Z and S £ BL(Y, Z) for which
ST = 0 and (S,T) is invertible (10.3.3.3)
Necessary and sufficient for S £ BL(Y, Z) to be regular is that there are X
and T £ BL(X,Y) for which (10.3.3.3) holds. Necessary and sufficient for
T £ BL(X,Y) to be decomposably regular is that there is S £ BL(Y,X)
for which
ST = 0 = TS and (S,T) and (T, S) are invertible (10.3.3.4)

10.3 Exactness
405
Proof: If (S, T) £ BL(X, Y, Z) satisfies the condition (10.3.1.1), then
multiplying on the left by 5, and multiplying on the right by T, give
S = SVS and T = TUT (10.3.3.5)
which tells us that S and T are both regular. If (10.3.1.1) and (10.3.3.5)
both hold then U' = UTU and V = VSV give
V'S + TU' = 1 and U' = U'TU' and V = V'SV' and U'V = 0
(10.3.3.6)
since U'V = U(TU)(VS)V and (TU)(VS) = {TU){I - TU) = 0.
This establishes (10.3.3.1), and shows that the first part of (10.3.3.2)
is necessary. The second part of (10.3.3.2) is necessary, from
Theorem 10.2.3, which also shows that the whole condition (10.3.3.2) is
sufficient. If T = TT'T is regular then the condition (10.3.3.3) holds
with Z = Y and S = I - TT'\ if instead S = SS'S is
regular then (10.3.3.3) holds with X = Y and T = I - S'S. Finally
if the condition (10.3.3.4) holds then there are U',V,U" and V" for
which
/ = u"T + SV" = V'S + TU'
Now, if we take U = U"TU' and V = V'SV", then we get, with P = UT,
R = T + V(I - P) and R' = R + PU,
P = P2 and RP = T and R'R = I = RR' (10.3.3.7)
By (3.8.5.2) this is the condition that T is decomposably regular:
conversely if (10.3.3.9) holds then so does (10.3.3.4) with S = {I -
P)R'. m
If (10.3.3.1) holds we shall call (U,V) £ BL{Z,Y,X) a (left,right)
inverse for the chain (5,T) e BL(X,Y,Z).
Almost invertible chains come in open sets; we begin with an auxiliary
result:
10.3.4 THEOREM Each of the following conditions is sufficient for J G
BL(y,y) to be a (left,right) multiple of {S + H,T + K) G BL(Y,Z) x
BL(X,y):
VS + TU = I and VSTU = V(S + H)(T + K)U
and (10.3.4.1)
H'{I + VH) =/=(/ + KU)T'

406
10. Comparison of Operators and Exactness
V(S + H) + (J + K)U e BL-\Y,Y)
and (10.3.4.2)
V(S + H){T + K)U = (T + K)UV(S + H)
V(S + H) + (T + K)U and U(T + K) are invertible
and (10.3.4.3)
V(S + H)(T + K) =0
^(5 + H) + (J + K)U and (5 + ^)7 are invertible
and (10.3.4.4)
(5 + ^)^ + ^)17 = 0
Proof: The reader can verify the identity
V(S + H)(I+KU) + {I+VH)(T+K)U-(I+VH){I+KU) = VS + TU-I
(10.3.4.5)
This means that the first two parts of (10.3.4.1) together imply
V(S + H)(I + KU) + (/ + V#)(r + K)U = (I + VH)(I + KU)
and this with the last part of (10.3.4.1) gives
(H'V) {S + H) + {T + K) [UK1) = I (10.3.4.6)
If instead (10.3.4.2) holds then, with R = V(S + H) + (T + K)U
R^ViS + H) + (T + K)UR~l = I (10.3.4.7)
If instead (10.3.4.3) holds, then
.R-1^^ + H) + (J + K)(U(T + K))-1 = I (10.3.4.8)
If instead (10.3.4.4) holds, then
((S + HW-^S + ^ + iT + K^R-1 =1 m (10.3.4.9)
The open set result now follows:
10.3.5 THEOREM If X, Y and Z are normed spaces, then
aleft>riehtBL(X,y,Z) is closed:
{(5,T) £ BL(X, y, Z): (5, T) is almost invertible} (10.3.5.1)
is open in BL(X, y, Z).

10.3 Exactness
407
Proof: If (S,T) G BL{X,Y,Z) satisfies (10.3.1.2) and if (S',T') = {S +
H, T + K) € BL(X, Y, Z) is sufficiently close to (S, T) that
sup||J+Vn#|| < 1 and sup \\I + KUn\\ < 1 (10.3.5.2)
where of course (Un) and (Vn) are taken from (10.3.1.2), then by the
argument of Theorem 3.7.2 there are (Hfn) and [K'n) for which
\\H'n(I + VnH) - I|| + ||(/ + KUn)K'n -1\\ — 0
and (10.3.5.3)
sup|K||+sup|K||<oo
n n
Using the identity (10.3.4.5) we now argue
\\H'nVn(S + H) + (T + K)UnK'n-I\\
< \\H'n(VnS + TUn - I)K'J + \\H'n(I + VnH)(I + KUn)K'n - I\\
+ \\H'nVn(S + H)(I-(I + KUn)K'J
+ \\(I-H'n(I + VnH)(T + K)UnK'J^0 Bsn^n
noting that for each n,
VnSTUn = 0 = Vn(S + H)(T + K)Un m
The fundamental examples of exact and almost exact chains come from
subspaces, closed or not, and their associated quotients. Recall the natural
injection Jw : W —► Y of (2.3.0.3) induced by a subspace W C Y, together
with the quotient mapping Kw : Y —► y/cl(W) induced by cl(W) as in
(2.3.0.4). Examples are the mappings ker(T) and coker(T) of (2.3.0.6) and
(2.3.0.7):
10.3.6 THEOREM If W C Y is a subspace, then
(KW,JW) is almost exact (10.3.6.1)
and if W is closed, then
(KW,JW) is exact (10.3.6.2)

408
10. Comparison of Operators and Exactness
If T e BL(X, Y) and S G BL(y, Z), then the following are equivalent:
J is almost (left,right) majorized by (coker(T),ker(5')) (10.3.6.3)
ST = 0 (10.3.6.4)
J is (left,right) majorized by (coker(r),ker(S)) (10.3.6.5)
Proof: If W C Y is a subspace then the Riesz lemma Theorem 1.5.1 says
that (10.3.1.5) holds with (S,T) = (KW,JW) and k = 2 and h = 1. If, in
particular, W is closed then the auxiliary result (1.5.1.7) says that (10.3.1.4)
holds with (S,T) = (Kw, Jw) and k = 2 and h = 1 + e > 1. Implications
(10.3.6.5) =>» (10.3.6.3) => (10.3.6.4) follow from Theorem 10.2.2. We
complete the proof by using the Riesz lemma result (1.5.1.7) to show that
(10.3.6.4) implies (10.3.6.5). Indeed if y € Y and e > 0 are arbitrary then
(1.5.1.7) gives w £ Y for which
Sw = 0 and ||y-w|| < (* + £) dist^S"1^))
(10.3.6.6)
and ||w;|| < 2\\y\\
The first part of this condition says
w = kei(S)w (10.3.6.7)
while if ST = 0 then the middle part of this condition gives
\\y - w\\ < (1 + e) dist(t/,cl(rX)) = (l + e)\\ cokei(T)y\\ ■ (10.3.6.8)
From Theorem 10.3.6 we are able to resolve exactness and almost
exactness of chains into components:
10.3.7 THEOREM Necessary and sufficient for (S,T) e BL(X,Y,Z) to
be almost exact is that there are k > 0 and h > 0 for which, for each t/^7,
dist(t/,rX) <h\\Sy\\ (10.3.7.1)
and
Sy = 0 => y G cl {Tx: \\x\\ < k\\y\\} (10.3.7.2)
Necessary and sufficient for (S, T) € BL(X, Y, Z) to be exact is that in
addition
Sy = 0 => y G {Tx: \\x\\ < k\\y\\} (10.3.7.3)

10.3 Exactness
409
Proof: If J is almost (left,right) majorized by (5, T) then, whether or not
ST = 0, both (10.3.7.1) and (10.3.7.2) follow at once, without change of
constants. If in particular J is (left,right) majorized by (S, T) then also the
condition (10.3.7.3) holds. Conversely, if ST = 0 then (10.3.6.3) says that
for each y GY there is (wn) in Y for which
Swn = 0 and limsup ||y + iun|| < dist(t/,Tx)
(10.3.7.4)
and KJ| < 2||t/||
By (10.3.7.2) there is (x J in X for which
||u;n-rsn||—0 and ||xn|| < A||u;n||
and now (10.3.7.1) says
limsup ||2/ — Txn\\ = limsup ||y — wn\\ < d\st(y,TX) < h\\Sy\\
n n
and (10.3.7.5)
Kll < *KJ| < 2*||y||
But this means that (5, T) satisfies the condition (10.3.1.5). If instead of
(10.3.6.3) we use (10.3.6.5) then for each y GY and arbitrary e > 0 there
is w £ Y for which
Sw = 0 and \\y - w\\ < (1 + e) dist(y,TX) and \\w\\ < 2\\y\\
(10.3.7.6)
If (10.3.7.3) is assumed then there isxGX for which
w = Tx and ||x|| < fc||tu||
Going back to (10.3.7.1) now gives
||y - Tx\\ = \\y - w\\ <(l + e) dist(y,TX) < h\\Sy\\
and (10.3.7.7)
||*|| <*HI<2*||y||
which means that (S,T) satisfies the condition (10.3.1.4). ■
The reader will of course have noticed that the "additional"
condition (10.3.7.3) actually includes the condition (10.3.7.2). The conditions
(10.3.7.1) and (10.3.7.2) say that coker(T) is majorized by S, in the sense
of (10.1.1.4), and that ker(S) is almost comajorized by T in the sense of
(10.1.3.5). The proof that these conditions together give (5, T) almost
exact is just the transitivity argument (10.2.3.4) with (coker(T),ker(5)) in

410
10. Comparison of Operators and Exactness
place of (S,T) and {S,T) in place of (Sf,Tf); the reverse implication is the
other transitivity argument (10.2.4.5) with T' = ker(S) and Sf = coker(T).
We have however preferred to prove Theorem 10.3.7 explicitly; we use it to
show that almost exact pairs come in open sets:
10.3.8 THEOREM If X, Y and Z are normed spaces, then
fleft>riehtBL(X,y,Z) is closed:
{{S, T) € BL(X, y, Z): {S, T) is almost exact} (10.3.8.1)
is an open subset of BL(X, Y, Z).
Proof: Suppose that k > 0 and h > 0 satisfy the condition (10.3.1.4) from
Definition 10.3.1: then if (S',T') € BL(X,Y,Z) satisfies
k\\T' - T\\ + h\\S' - S\\ + e = 6 < 1 with e > 0 (10.3.8.2)
we claim that (10.3.7.1) and (10.3.7.2) hold with (S',T') in place of (S9T)
and h/(l — 6) and k/(l — 6) in place of h and k. Indeed if y £ Y is arbitrary
then by (10.3.1.5) there is xx € X for which
llv-TsJ^fcHSyll + cllyll and \\Xl\\ < k\\y\\ (10.3.8.3)
and hence
\\y-T'x1\\<\\(T'-T)x1\\+h\\Sy\\+e\\y\\
< h\\S'y\\ + 11(2- - T)xx\\ + h\\(S' - S)y\\ + e\\y\\
so that, with yx = y — Tfxx,
y = yx+ Tfx1 and S'yx = S'y
and (10.3.8.4)
llyill^fcll^yll + tfHyll and KH < %||
Iteration gives sequences (t/n) in Y and (xfn) = [xx + x2 + • • • + xn) in X
for which
y = yn + T'{x'n) with S'yn = S'y (10.3.8.5)
and
||yn|| <h(l + 6 + --- + S^WS'yW + 6n\\y\\ < h'\\S'y\\ + Sn\\y\\ (10.3.8.6)

10.3 Exactness
411
and
\\x'n\\<hk(n + (n-l)6 + --- + 6n-1)\\S'y\\
+ *(1 + 6 + • • • + 6n~l)\\y\\ < nk"\\S'y\\ + k'\\y\\
(10.3.8.7)
with h' = h/(l-6),kf = k/(l-S) and k" = hk/{l-S). From (10.3.8.5) and
(10.3.8.6) it follows that (S",T") satisfies the condition (10.3.7.1), in which
no control is needed on the norm of xfn = x1+x2H hxn, and together with
(10.3.8.7) they show that (S",T') satisfies the condition (10.3.7.2), since our
lack of control of the norm of x'n is confined to a term which is multiplied
by ||5"t/||. By Theorem 10.3.7 the pair (S",T') is almost exact. ■
The openness just proved, and also that of Theorem 10.3.5, is relative
to the closed subset BL(X, Y, Z), and does not hold in the larger space
BL(Y,Z) x BL(X,y). For example let X = Y = Z = C2 and look at
1 0'
0 0
T =
0 0
0 1
(10.3.8.8)
Evidently (5, T) is an invertible chain in the sense of (10.3.1.1), but if
(10.3.8.9)
st =
1 t
0 0
Tt =
0 t~
0 1.
then ||5t -S\\ + \\Tt - T\\ -> 0 as t -> 0, while if 0 ^ t G C the pair (St, Tt)
is not even almost linearly exact in the sense of (10.3.1.7).
We conclude with a boundary result, generalizing Theorem 3.5.1:
10.3.9 THEOREM If {S,T) and {Sn,Tn) in BL(X,Y,Z) satisfy
S almost open and T bounded below (10.3.9.1)
and*
{Sn,Tn) almost exact (10.3.9.2)
and
\\S - Sn\\ + \\T - Tn\\ —► 0 as n — oo (10.3.9.3)
then
(5,r) is almost exact (10.3.9.4)

412
10. Comparison of Operators and Exactness
Proof: By (10.3.9.1) there are k > 0 and h > 0 for which if x € X and
z G Z then
||x|| < *||Tx|| and z G cl {Sy : \\y\\ < h\\z\\} (10.3.9.5)
and hence by Theorem 3.3.3 and Theorem 3.4.3
\\x\\ < kn\\Tnx\\ and *€cl{SBy:||y||<fcn||*ll} (10.3.9.6)
with
K = r^ ^77 and hn = —^— (10.3.9.7)
n l-*||Tn-r|| n 1-Sn y ]
assuming k\\Tn — T\\ < 1 and h\\Sn — S\\ < Sn < 1. In particular, taking
z = Sny, there is for arbitrary (en) > 0 a sequence (y'n) = [y — yn) for
which
PJJ < «»ll»ll with ||y - y'J < hJSny\\ (10.3.9.8)
By (10.3.9.2) there are (possibly unbounded) sequences (k£) > 0 and (h£) >
0 for which, for arbitary (e:*) > 0, there is a sequence (xfn) in X with
\\y'n - Tnx'J < hjSny'J + ejy'j with ||<|| < *B||yB||. (10.3.9.9)
Thus if y € Y is arbitrary we have
||y - Tx'J < \\y - y'J + \\y'n - Tnx'J + ||r„x'n - Tx'J
< h - y'J + KWSM + <WJ + \K - n\K\\
< (1 + eB)||y - y'J + fcBeB||y|| + <||y|| + ||r„ - T||fc||rx'J|
<(l + <)hJSny\\ + h^Jy\\+eJy\\
+ ||TB-r||*(||r«'B-y|| + ||y||)
giving
(1 - *||rB - T||) ||y - Tx'J < (1 + <)hjS\\ + <rB||y|| (10.3.9.10)
where
e'B = fc||TB - T|| + (1 + <)hjSn - 5|| + Ken + < (10.3.9.11)
The coefficient of ||y — Tx'n\\ is bounded below as n —► oo, the coefficient of
11 St/11 is bounded as n —► oo if we assume e^ —► 0, and finally we are free
to arrange that h^en —► 0, which makes e:" —► 0; thus the chain (5, T) is

10.3 Exactness
413
almost linearly exact in the sense of (10.3.1.7). At the same time
(1 - k\\Tn - T\\) K|| < (1 - k\\Tn - T\\) (k\\TJn - y\\ + ||*||y||) < *Z||y||
(10.3.9.12)
giving (10.3.9.4). ■
Theorem 10.3.9 generalizes Theorem 3.5.1, and hence furnishes a
generalization of Theorem 3.5.2: if a chain (S, T) £ BL(X, Y, Z) is in the
topological boundary (among chains) of the set of almost open chains
then either S € BL(Y,Z) fails to be almost open or T £ BL(X,Y) fails
to be bounded below. We have been unable to settle a "dual" result:
if chains (S,T) and [Sn,Tn) satisfy the conditions (10.3.9.4), (10.3.9.5)
and
Sn almost open and Tn bounded below (10.3.9.13)
does it follow that (S,T) satisfies the condition (10.3.9.1)? If z £ Z
is arbitrary then the first part of (10.3.9.13) gives (yn) in Y for which
\\z — Snyn\\ —► 0, and then the condition (10.3.9.4) gives k > 0 and h > 0
and a sequence (xn) in X for which
||yn - TxJ < h\\Syn\\ + hz\\ with ||»„|| < *||y„|| (10.3.9.14)
n
Now
\\z - Syn\\ < \\z - Snyn\\ + \\Sn - S\\\\yn - TxJ + \\(Sn - S)TxJ
<\\"-SnyJ + \\Sn-S\\(h\\SyJ + ^\\z\\j
+ \\{Sn-S)TxJ
giving
(l-fc||SB-S||)||*-SyB||
/ i\ (10.3.9.15)
< W* ~ SnyJ + \\Sn - S\\ [h + -J \\z\\ + \\(Sn - S)TxJ
If we knew that [Txn) were bounded this would show that S is dense;
without this information we are unable to proceed, and the reader is
invited to see if he can succeed where we have failed. For example if X, Y,
and Z are Hilbert spaces then we will see below that the result follows from
Theorem 3.5.1.
The alert reader will possibly have noticed that specialization of
Theorem 10.3.9 to the case 5 = 0 does not yield quite the full strength of
Theorem 3.5.1: for if S = 0 = Sn then our assumption (10.3.9.2) says that

414
10. Comparison of Operators and Exactness
each Tn is almost open, while for Theorem 3.5.1 we need only assume that
Tn is dense. Inspection of the proof of Theorem 10.3.9 however shows that
the constants (k£) play no part in the argument, so that we are actually
working with an assumption weaker than (10.3.9.2). We cannot similarly
weaken the assumptions on (Sn), as we see by specializing instead to T = 0
and recalling Theorem 4.7.4.
10.4 COMPOSITION OPERATORS AND DUALITY
The various relationships of Definition 10.2.1 are transmitted to
composition operators, and also some of them have alternative expressions in terms
of composition operators. For example the condition (10.2.1.1) says that
R C U{E) with U = iow{Rs,LT) and E = BL(Z,Y) x BL{W,X)
(10.4.0.1)
while the condition (10.2.1.3) says that
R C c\(U(E)) with U = iow{Rs,LT) and E = BL(Z,Y) x BL(X,Y)
(10.4.0.2)
The reader is invited to offer a formulation in these terms of the important
condition (10.2.1.2). If (10.2.1.1) holds then it also holds for induced left
and right compositions:
LR = LVLS + LTLV and RR = RVRT + RSRV (10.4.0.3)
where we may take LR = BL(E,R) for arbitrary E and then define the
other compositions consistently. Similarly (10.2.1.2) is transmitted: if
(5, T) satisfy it then also
\\LVnLs + LTLUn-LR\\^0 and \\RUnRT + RsRVn - RR\\ — 0
(10.4.0.4)
and of course the sequences {LUn), (-^yn)» i^un) an(^ G^vn) are bounded.
The condition (10.2.1.5) has an interesting expression in terms of
composition operators:
10.4.1 THEOREM If T € BL(X,Y) and S £ BL(Y,Z) and R £
BL(W, Z) the following are equivalent:
R is almost (left,right) majorized by (S,T) (10.4.1.1)
for arbitrary U G BL(E,W) and V G BL(y,F),
\\VRU\\<h\\V\\\\SU\\+k\\VT\\\\U\\ (l0-4-L2)
for arbitrary w € W and g € Y^, \gRw\ < /i||^||||5t(;|| + A;||^r||||w;||
(10.4.1.3)

10.4 Composition Operators and Duality
415
Proof: If (10.4.1.1) holds and E and U £ BL(£, W) are arbitrary, then for
each u £ E we have (xn) in X with supn ||xn|| < fc||C/u|| and limn ||i2!7u —
Txn\\ < h\\SUu\\, so that for arbitrary F and V £ BL(Y,F) there is
inequality
\\VRu\\ lim\\V(RUu - Txn)\\ + sup ||7r*B||
n n
< \\v\\h\\suu\\ + \\vt\\ sup kii < fe||F||||rc;«|| + *||7T||||CTu|I
n
Taking sup||u|i<:l gives (10.4.1.2), and hence by specializing to E = F = K,
(10.4.1.3). Finally, the implication (10.4.1.3) =^ (10.4.1.1) is the separation
Theorem 5.5.1: if (10.4.1.1) does not hold there is w £ W for which
Rw <£ c\(K) = (K°)0 with K = Disc(0 ; h\\Sw\\) + TDisc(0 ; k\\w\\)
giving g £ Y^ for which (10.4.1.3) fails. ■
The analogue of Theorem 10.4.1 holds for the condition (10.2.1.7):
10.4.2 THEOREM If T £ BL(X,y), S £ BL(W,Z), and R £ BL(W,Y),
then the following are equivalent:
R is almost (left,right) determined by {S,T); (10.4.2.1)
for arbitrary U £ BL(£,W) and V £ BL(Y,F),
SU = 0 = VT^ VRU = 0
for arbitrary w £W <md g £Y\Sw = 0 = gT ^>> gRw = 0 (10.4.2.3)
Proof: If (10.4.2.1) holds, so that i?^-1^)) C cl(TX), suppose that
SU = 0 = VT: then for arbitrary u £ E we have SUu = 0, and
hence, RUu £ cl(TX), giving VRUu = 0. This gives (10.4.2.2) and hence
(10.4.2.3). Finally, if (10.4.2.1) does not hold then there is y £ R{S~1{0)) C
cl(TX), so that y = Rw with Sw = 0 and also y £ cl(TX): now by the
Hahn-Banach theorem (5.4.1.3) there is g £ Y^ for which gT = 0 ^ g{y).
Evidently g and y together violate (10.4.2.3). ■
If in Theorem 10.4.1 and Theorem 10.4.2 we specialize by taking 5 = 0
then we have in particular
R almost comajorized by T <£=$► R* majorized by T* (10.4.2.4)
and
R almost range-included in T <£=> R^ determined by Tt (10.4.2.5)

416
10. Comparison of Operators and Exactness
The dual of (10.4.2.5) is only partially valid, while the dual of (10.4.2.4) is
even stronger:
10.4.3 THEOREM If T e BL{X,Y), S € BL{W,Z), R € BL{W,Y) the
following are equivalent:
R is almost (left,right) majorized by (T],S]) (10.4.3.1)
R is almost (left,right) majorized by (5,T) (10.4.3.2)
R* is (left,right) majorized by {T\S^) (10.4.3.3)
Proof: If (10.4.3.1) holds then by Theorem 10.4.1 we have
for arbitrary gGY^wG W^9 \wR^g\ < h\\wS*\\\\g\\ + *||r^||||o7||
(10.4.3.4)
Specializing to w = w (= W C W^ gives (10.4.1.3), and hence again by
Theorem 10.4.1 we get (10.4.1.1), which is (10.4.3.2). If (10.4.3.2) holds
then for arbitrary w £ W there are y GY and (xn) in X for which
\\Rw-Txn-y\\ —► 0 with \\y\\ < h\\Sw\\ and ||xj| < Jb||u;|| (10.4.3.5)
This enables us to define, for each g £ Y"l", a mapping /0 : SW —► K by
setting
f0{Sw) = g{y) if {w,y, (xj) satisfy (10.4.3.5) (10.4.3.6)
Since \g(y)\ < ||^||||y|| < fc||<7||||Siu||, ft is clear that /0 is well defined,
evidently linear, and then bounded with ||/0|| < k\\g\\. By the Hahn-Banach
Theorem 5.3.1 we can extend /0 to / (E X^ with ||/|| = ||/0||: now
\fSw - gRw\ = limllffTzJ < ||0T||*|HI (10.4.3.7)
n
and taking supy^n^ gives (10.4.3.3). Finally the implication (10.4.3.3) =>•
(10.4.3.1) is obvious. ■
Taking T = 0 in Theorem 10.4.3 gives
Ri almost comajorized by 5' =>• S majorized by R
«+ ..,,„+ (10.4.3.8)
=> R] comajorized by S]
10.4.4 THEOREM If T £ BL(X,Y), S € BL(W,Z), and R € BL(W,Y)
then there is implication
R] almost (left,right) determined by (T^S^)
(10.4.4.1)
=> R almost (left,right) determined by (S,T)

10.4 Composition Operators and Duality
417
Proof: If the first part of (10.4.4.1) holds then by Theorem 10.4.2 there is
implication
for arbitrary g eY^w€ W^,wS^ = 0 = T*g => wR^g = 0 (10.4.4.2)
Specializing to C7 = w gives condition (10.4.2.3), and hence
again by Theorem 10.4.2 the condition (10.4.2.1), which is the second part
of (10.4.4.1). ■
Taking T = 0 in Theorem 10.4.4 gives
Ri almost range-included in S* =>• R determined by S (10.4.4.3)
Even when R = I the implication (10.4.4.3) is not reversible, as we saw
following Theorem 5.4.1.
Theorem 10.4.2 and Theorem 10.4.3 help us to a result complementary
to Theorem 5.6.4:
10.4.5 THEOREM If T € BL(X, Y) and S e BL{Z, W) then
(iovt(—RT,Ls),co\(Ls,RT)) almost linearly exact
=>• (S or T one-one and S or T dense)
(10.4.5.1)
and
(iow(—RT,Ls),co\(Ls,RT)) almost exact
=>• (S or T bounded below and S or T almost open)
Proof: We are looking at the diagram
(10.4.5.2)
BL(X,W)
[ —Rt Ls
BL(y,W)l [Rt\
BL(X,
BL(Y,Z): (10.4.5.3)
the reader should note that there are two different operators Ls and two
different operators RT. If the right-hand side of (10.4.5.1) fails we claim
that there are x £ X, f eX\yeY,g eYi, zeZ,heZi,w eW and
k € Wt for which
[k <g) y h <g) x]
lirp
- [ -RT Ls}
g ®w
fQz
t^ [fc® y h®x]
= 0
g ®w
fez
(10.4.5.4)

418
10. Comparison of Operators and Exactness
for if neither S nor T are one-one we may take
Tx = 0 ^ f(x) and Sz = 0 ^ /i(^) and (y, g,w,k) = (0,0,0,0)
(10.4.5.5)
and if instead neither S nor T are dense we may take
gT = 0^g(y) and kS = 0 ^ k(w) and (x, /, 2, h) = (0,0,0,0)
(10.4.5.6)
By Theorem 10.4.2 the condition (10.4.5.4) gives the failure of the left-
hand side of (10.4.5.1), and hence proves the implication. The argument
for (10.4.5.2) is the same, working with sequences of vectors and functionals
and using Theorem 10.4.3. Details are left to the reader. ■
Decomposable exactness for composition operators, as in (10.4.0.3) but
without that specific "(left,right) inverse," is not at first sight sufficient for
the condition (10.2.1.1). For a flow of information in that direction we have
a generalization of Theorem 3.6.5:
10.4.6 THEOREM If [R,S,T) G BL(X,Y,Z,W) and if (R,S) G
BL(y, Z,W) is (left,right) invertible then there is implication
(LS,LT) linearly exact =>• (S,T) (left,right) invertible
(10.4.6.1)
r [ljQ)ljrp] exact
If instead (5, T) £ BL(X, Y, Z) is (left,right) invertible then there is
implication
(RS,RR) linearly exact => (R,S) (left,right) invertible
=> (RS,RR) exact
Proof: The second implication in (10.4.6.1) is part of (10.4.0.3), with no
restriction on (R, S). Conversely if RfR + SSf = i" then
S(I - S'S) = {I- SSf)S = R'RS = 0 (10.4.6.3)
and if (LS,LT) is linearly exact this gives
I-S'S £ Lsx (0) C LT BL(y, X)
so that
I-S'S = TT' for some T' € BL(Y, X) (10.4.6.4)
This proves (10.4.6.1), and the argument of (10.4.6.2) is exactly the
same. ■

10.5 Enlargement and Completion
419
The analogue of Theorem 10.4.6 holds for almost exactness,
generalizing Theorem 3.7.2: if (R, S) is almost (left,right) invertible then
(LS,LT) almost linearly exact
=>• (S,T) almost (left,right) invertible (10.4.6.5)
=>• (LS,LT) almost exact
while if instead (5, T) is almost (left,right) invertible then
(RS,RR) almost linearly exact
=>• (R,S) almost (left,right) invertible (10.4.6.6)
=>• (RS,RR) almost exact
The details are left to the reader. Both Theorem 10.4.6 and its
analogue have immediate extensions to more general chains (Tn,... ,T2,T1)
and (0,rn,...,7^,0).
10.5 ENLARGEMENT AND COMPLETION
The enlargement process can be used to test for majorization and co-
majorization:
10.5.1 THEOREM If T £ BL(X,Y), S £ BL(W,Z) and R £ BL(W,Y)
then the following are equivalent:
Q(R) is almost (left,right) majorized by (Q(S),Q(T)) (10.5.1.1)
R is almost (left,right) majorized by (5,T) (10.5.1.2)
Q(R) is (left,right) majorized by (Q(5),Q(r)) (10.5.1.3)
Proof: If (10.5.1.2) holds, so that for each w £ W and each e > 0 there is
x£X with
H-Rw - Tx\\ < h\\Sw\\ + e||tc;|| and ||x|| < fc||tc;||
then for each w = (wn) £ l^iW) there is x = (xn) in X with, for each
n£ N,
\\Rwn - TxJ < h\\SwJ + illtt,^ and ||xj| < *||«,n||
n
Evidently x £ /^(X), and also
limsup \\Rwn — Txn\\ < /ilimsup ||^w;n||
n n
with limsup ||xn|| < klimsup \\wn\\
n n

420
10. Comparison of Operators and Exactness
which is (10.5.1.3). Trivially (10.5.1.3) =>> (10.5.1.1). Finally, if (10.5.1.1)
holds then for each w € W and each e > 0 there is xe £ l^ (X) for which
limsup \\Rw - Txen\\ < h\\Sw\\ + e||u;|| and limsup \\xen\\ < k\\w\\
n n
Thus, if k' > k there is x' — x£N for which
||-Rt£7 - Tx'e\\ < h\\Sw\\ + 2e|H| and \\x'e\\ < k'\\w\\
giving (10.5.1.2). ■
When R = I and ST = 0 then we can improve:
10.5.2 THEOREM If (S, T) € BL(X, Y, Z) then the following are
equivalent:
((Q(S),Q(T)) is almost linearly exact (10.5.2.1)
(S,T) is almost exact (10.5.2.2)
((Q(S),Q(r)) is exact (10.5.2.3)
Proof: If (10.5.2.3) holds then so by Theorem 10.5.1 does (10.5.2.3), which
trivially implies (10.5.2.1). To see that (10.5.2.1) implies (10.5.2.2) we use
the subspace-quotient characterization Theorem 10.3.2. If (10.5.2.2) does
not hold then one of the two conditions (I0.3.2.y) must fail. If (10.2.3.1)
fails then there is (yn) in Y for which
||yj| = l and dist(yn,TX) > \ and ||SyJ| — 0
(10.5.2.4)
and then by the Hahn-Banach theorem (gn) in Y"l" for which
ll*»ll = lff»(y»)l = l ^d ||flBr||=0 (10.5.2.5)
Evidently
y € l^Y) and Sy € c0(Y) and (x € l^X) => dist(y - Tx,c0(Y)) > |)
(10.5.2.6)
If instead (10.2.3.2) fails, then using the Hahn-Banach separation theorem,
there is (gn) in Y for which
IbJIs-Ho) = 1 and \\snT\\^0 (10.5.2.7)
and then (yn) in Y for which
||y„|| = l and Syn = 0 and \gn(yn)\ > \ (10.5.2.8)

10.5 Enlargement and Completion
421
Now for arbitrary x € Ioq{X) we argue
l|y» - TxJ = ||ffB|lllv» - TxJ > \gn(yn - Txn)\
>lfl„(y„)l-llff»r||KH>|-
so that
limsup ||yn - TxJ > limsup (§ - ||*|U|ffBT||)
n n
Thus again we get (10.5.2.6), which violates (10.5.2.1). ■
We cannot extend Theorem 10.5.2 to more general R, as in Theorem
10.5.1. If for example X = Y = Z = l2 and T = S2 where S = W : (x J ->
(xjn) is the operator of (7.3.6.15), then (10.1.1.5) holds with (Q(S), Q(T))
in place of {S,T), but (S,T) does not satisfy (10.1.1.4):
||5x||2 < ||x||||rx|| for each xGX (10.5.2.9)
10.5.3 THEOREM If T € BL(X, Y) and S € BL(X, Z), and if either S
is bounded below or T is relatively open, then
Q(S) determined by Q(T) <=> S majorized by T (10.5.3.1)
Proof: Backward implication follows from Theorem 10.5.1 with (5, T,0)
in place of (R, S,T). Conversely, if S is not majorized by T, then there is
a sequence (xn) in X for which
\\Sxn\\ = 1 > n||rxj| for each n £ N (10.5.3.2)
Provided we know that (xn) is in l^X) this will show that Q(S) is not
determined by Q(T). If S is bounded below this is certainly the case:
||xn|| < k\\Sxn\\ = k for each n £ N (10.5.3.3)
If instead T is relatively open then we can arrange to replace (xn) by
(xn + un) in /^(X), but we have to argue by contradiction. Indeed, if T is
relatively open then there is k > 0 for which
dist(x,r_1(0)) > ik||rx|| for each xGX (10.5.3.4)
and hence, if (10.5.3.2) holds there is a sequence (un) in X for which
Tun = 0 and ||xn + un|| —► 0 (10.5.3.5)
Evidently the sequence x + u = (xn + un) is in l^X). If we now assume
that Q(S) is determined by Q(T), so that also S is determined by T, then
IMUIff»r||
_ 1.
— 2

422
10. Comparison of Operators and Exactness
we also have
5(tt„)=0 and ||5(xB + ttB)|| = ||5(*B)|| = l (10.5.3.6)
which means that (10.5.3.2) holds with x + u in place of x. Since x +
u £ /^(X), this contradicts the assumption that Q(S) is determined by
Q(r). ■
Argument similar to Theorem 10.5.1 shows that
Q(S) almost range-included in Q(T) =>• S almost range-included in T
(10.5.3.7)
If the spaces X, Y, and Z are complete, then exactness and almost
exactness for (S,T) £ BL(X,Y,Z) coincide:
10.5.4 THEOREM If X, Y, and Z are Banach spaces and (S,T) £
BL(X, Y, Z), then the following are equivalent:
(S,T) is almost exact (10.5.4.1)
5_1(0) = TX and S(Y) = c\(SY) (10.5.4.2)
(S,T) is exact (10.5.4.3)
Proof: If (5,r) £ BL(X,Y,Z) is almost exact then by Theorem 10.3.7
the conditions (10.3.7.1) and (10.3.7.2) both hold. By Theorem 4.4.3 and
Theorem 4.4.4, applied to the operators S and T : X —► 5_1(0), the
condition (10.5.4.2) holds. By the open mapping Theorem 4.6.2 the condition
(10.5.4.2) gives (10.3.7.1) and (10.3.7.3), which by Theorem 10.3.7 again
gives (10.5.4.3). Trivially (10.5.4.3) implies (10.5.4.1). ■
Baire's theorem gives us an interesting generalization of the open
mapping theorem:
10.5.5 THEOREM If T £ BL(X, Y) and S £ BL(Z, Y) and if Z is
complete, then
S range-included in T => S almost comajorized by T (10.5.5.1)
and
S range-included in T =)> 5t majorized by Tf (10.5.5.2)

10.5 Enlargement and Completion
423
Proof: If S is range-included in T, then
oo oo
Z = S~1 [j {Tx: \\x\\ <n}= [j 5_1rDisc(0 ; n) (10.5.5.3)
n=l n=l
and if Z is complete then by Baire's Theorem 4.6.1 there is n £ N with
0 ^ int cl S~XT Disc(0 ; n) = int S"1 cl T Disc(0 ; n) (10.5.5.4)
and hence
0 ^ int cl T Disc(0 ; 1) = (l/n) int S'1 c\T Disc(0 ; n) (10.5.5.5)
As usual (1.4.1.1), this means that 0 £ int 5_1 cl{Tx: ||x|| < l}, giving the
condition (10.1.3.4). This proves (10.5.5.1), and hence also (10.5.5.2), by
the duality result (10.4.2.4). ■
It would be nice to extend Theorem 10.5.5 to the situation of Definition
10.2.1. The best we can do is to make the observation that if T G BL(X, Y),
S e BL(W,Z), and R e BL(W,Y), and if the space W is complete, then
(10.5.5.1)
R{S-1{0)) CT(I) =>5-1(0) <Z{weW:Rwec\{Tx:\\x\\ <k\\w\\}}
(10.5.5.6)
The implication (10.5.5.1) is not in general reversible. If X = c0, Y = Z1?
Z = K, and if S and T are defined by setting
(Tx)n = n-*xn
and (10.5.5.7)
(St)n = tn~2 for each n € N, x € c0, t € K
then the right-hand side of (10.5.5.2) holds but the left-hand side does not.
If we combine Theorem 10.5.5 with Theorem 10.2.6 we find that if
T e BL(X, Y) and S £ BL(Z, Y) for Banach spaces X, Y, and Z then
S range-included in T compact => S compact (10.5.5.8)
It would have been nice to report implication (10.3.1.2) =>• (10.3.1.1)
for complete spaces; the best we have been able to do is the following:
10.5.6 THEOREM If X, Y, and Z are complete and T £ BL(X,Y),
S £BL(Y,Z) satisfy
\\VnS + TUn - I|| — 0 with sup ||tfj| + sup ||Vn|| < oo (10.5.6.1)

424
10. Comparison of Operators and Exactness
then each of the following is sufficient for J to be a (left,right) multiple of
(5,r):
VnSTUn = TUnVnS for each n £ N (10.5.6.2)
\\UnT -1\\ —> 0 and VnST = 0 (10.5.6.3)
\\SVn — JT|| —^ 0 and STUn = 0 (10.5.6.4)
If the spaces are complete, then BL_1(X, Y, Z) is an open subset of
BL(X,Y,Z).
Proof: Assuming only that the space Y is complete, (10.5.6.1) says that for
sufficiently large n € N the operator VnS+TUn is invertible. Now (10.3.4.2)
from Theorem 10.3.4 says that (10.5.6.2) makes i" a (left,right) multiple of
(S,T). If also the space X is complete, so that eventually UnT — I is also
invertible, then the sufficiency of (10.5.6.3) is given by (10.3.4.3). If instead
the space Z is also complete, then the sufficiency of (10.5.6.4) is given by
(10.3.3.4). Finally, if the space Y is complete and (S,T) G BL(X,Y,Z) is
invertible, with VS + TU = J, then by taking (S", T') = (S + H, T + K) so
close to (S, T) that i" + VH and i" + KU are both invertible we find using
(10.3.4.1) that (S",T') is also invertible. ■
It would as we have suggested have been more satisfying to prove, in
complete spaces, that almost invertible chains are invertible, and hence to
derive the last part of Theorem 10.5.6 from Theorem 10.3.5. An application
of (10.5.6.2) shows that if the space Y is complete then it is sufficient for
the chain (S,T) to be invertible that there are [Un) in BL(Y, X) and [Vn)
mBL(Z,Y) for which
\\VnS + TUn-I\\^>0
and (10.5.6.5)
sup||iy +sup||Vn|| < oo and UnVn = 0
n n
In words we require that (S,T) have an "almost inverse" consisting of
chains (Un, Vn). If we are unable to modify the argument for (10.3.2.1) to
show that (10.5.6.5) holds for every almost invertible chain we might be
tempted to replace the condition (10.3.1.2) by the condition (10.5.6.5) as
the definition of almost invertibility: it would then be important to show
that (10.5.6.5) defines an open subset of BL(X, Y, Z). The reader is invited
to see if he can show this by modifying the proof of Theorem 10.3.4: in the
notation of that proof he should try to arrange that
UnK'nH'nVn = 0 for each n £ N (10.5.6.6)

10.6 Essential Exactness
425
The pair of operators (S,T) of (10.3.8.8) shows that BL"1 (X,Y,Z)
need not be an open subset of the product space BL(Y,X) x BL(X, Y)
even for complete spaces.
10.6 ESSENTIAL EXACTNESS
The first three conditions of Definition 10.2.1 have obvious "essential"
analogues in the spirit of Definition 3.9.1, and in particular relative to the
finite rank operators as in Atkinson's Theorem 6.4.3. An "essential"
analogue of the last condition of Definition 10.2.1 also generalizes the "zero
divisor" condition of (3.9.4.7) and (3.9.4.8), and in particular the essential
one-oneness and denseness conditions of Definition 6.4.1:
10.6.1 DEFINITION We will call the pair (S,T) £ BL{X,Y,Z)
essentially (left,right) invertible, or spatially (left,right) Fredholm, if there are
T' £ BL(y,X) and S' £ BL(Z,Y) for which
S'S + TT' - I £ KL0(y, Y) (10.6.1.1)
and essentially (left,right) almost invertible if there are (T£) and (Sfn) for
which
\\S'nS + TT'n -I\\ —> 0 and sup \\T^\\ + sup \\S'n\\ < oo (10.6.1.2)
n n
We shall call the pair (S,T) £ BL(X, Y, Z) essentially (left,right) one-one,
or weakly (left,right) Fredholm, if
5"1(0)/cl(rX) is finite dimensional (10.6.1.3)
Once again, regularity is the bridge between the two extremes:
10.6.2 THEOREM If T £ BL(X, Y) and S £ BL(Y, Z) then
ST £ KL0(X,Z) <=► (S_1(0) +cl(rX))/5'-1(0) finite dimensional
(10.6.2.1)
and each of the following conditions are equivalent:
UT = 0 = SV ^UV £ KL0(X', Z') (10.6.2.2)
(S-^O) + cl(rX))/cl(rX) finite dimensional (10.6.2.3)
UT £ KL0(X,Z') and SV £ KL0(X',Z) =>► UV £ KL0(X',Zf) (10.6.2.4)
Necessary and sufficient for a chain (S,T) £ BL(X, Y, Z) to be (left,right)

426
10. Comparison of Operators and Exactness
Fredholm is
(S,T) is regular and weakly (left,right) Fredholm (10.6.2.5)
Proof: For (10.6.2.1) observe that the mapping SA : Y/S'1^) -► S(Y)
induces isomorphism
(S-^O) +cl(rX))/5-1(0) S E with ST{X) CEC c\ST{X)
(10.6.2.6)
Towards the next part, note that both sides of (10.6.2.1) are equivalent to
cl(rX) C S_1(0) + W for some finite dimensional W C Y (10.6.2.7)
If (10.6.2.3) holds and if U £ BL(Y,Zf) and V £ BL(X',Y) are such that
SV and UT are of finite rank, then there are finite-dimensional subspaces
E, F, and G of Y for which
cl(VX') C S_1(0) + E C cl(rX) + G + £ C J7"1 (0) + F + G + E
which means by (10.6.2.7) that UV is of finite rank. Conversely, if (10.6.2.3)
fails then (10.6.2.2) is violated by U = coker(T) and V = ker(S). This
proves the three conditions equivalent. Towards (10.6.2.5), it is clear that
a (left,right) Fredholm pair (S,T) is weakly Fredholm, whether or not it is
a chain, and if also ST = 0 then the operators T - TT'T and S - SS'S are
both of finite rank, hence regular by Theorem 6.3.4, which makes T and S
regular by (3.8.3.1). Conversely, if (S,T) is regular and weakly (left,right)
Fredholm, then we can apply (10.6.2.2) with U = 1- TT' and V = I - S'S
to see that (S,T) satisfies (10.6.1.1):
T = TT'T,S = SS'S => (I-TT')(I-S'S) £ KL0(Y,Y) m (10.6.2.8)
The definition of Fredholmness extends to triples (R, S, T) and longer
chains in the same way as the definition of invertibility, following
Definition 10.3.1. The reader can show the analogue of Theorem 10.3.2, that if
(R,S,T) is a Fredholm chain then there is another chain (T',S',Rf)
providing essential inverses for (S,T) and (R, S). At the same time it is clear
that we can always take the "essential inverse" of a Fredholm chain to be
another chain. If, in addition, the essential inverse is itself "invertible" then
we shall speak of a "Weyl" chain:
10.6.3 DEFINITION The chain (5, T) £ BL(X, Y, Z) will be called
spatially (ltft,right) Weyl if there is a chain (T',5") £ BL(Z,Y,X) for which
S'S + TT' - I £ KL0(y,y) and / £ SfBL(Y,Z) + BL(X,y)r'
(10.6.3.1)

10.6 Essential Exactness
427
If (S, T) is spatially Fredholm we shall write
Euler(5,r) = dim5_1(0)/cl(rX) (10.6.3.2)
If more generally (Tn,..., ^jTq) is spatially Fredholm we shall write
n
Eu\ei(Tn,...,T1,T0) = X>l)P_1 Buler^,^) (10.6.3.3)
P=i
We leave the reader to see whether he can deduce (10.6.3.1) from the
corresponding assumption using a pair (T", Sf) without the assumption that
T'S' = 0. To extend the definition of "Weyl" to longer chains we make
the obvious extension of (10.6.3.1), making explicitly the assumption that
the "invertible essential inverse" is another chain. The most important
examples of (10.6.3.3) are chains of the form (0, Tm,..., T1? 0), but we shall
also encounter chains of the form (S,T,S) with T £ BL(X, Y) and S £
BL(y,X).
When the spaces are complete, Weyl and Fredholm chains come in
open sets:
10.6.4 THEOREM If chains (5, T) and (S + H, T + K) in BL(X, Y, Z)
and (r',5") £ BL(Z,y,X) satisfy
S'S + TT'-I <EKL0(Y,Y) and {/ + SfH,I + KT'} C BL'1 {Y,Y)
(10.6.4.1)
then (S + H,T + K) is (left,right) Fredholm. If in addition there are
T" £ BL(X, Y) and S" £ BL(Y, Z) for which, with K' = (/ + KT)-1 and
H'=(I + S'H)-\
S's" + T"T' = I
and (10.6.4.2)
{/ = t"t'(k' -/),/ + {Hf - i)s's"} c bl"1 (y,y)
then (S + H, T + K) is (left,right) Weyl.
Proof: If we just assume that there are Hf and K' in BL (Y,Y) for which
{(/ + KT')K' - /, H'(I + S'H) - 1} C BL"1 (y, y) (10.6.4.3)
then the proof of Theorem 10.3.4, and in particular the identity (10.3.4.5)
with (T',5") in place of {U,V), shows that
{H'S')(S + H) + (T + K)(T'Kf) -l£ KL0(y,y) (10.6.4.4)

428
10. Comparison of Operators and Exactness
This shows that (S + H,T + K) is Fredholm. To see that it is Weyl we
ought to replace the pair {T'K',H'S') by a chain, and show that that
chain is invertible. Also, the only way we can hope to show such a chain
to be invertible is to keep it close to the original chain {T',S'), and use
Theorem 10.3.4 again. In fact the assumption (10.6.4.1) already tells us
that {T'K',H'S') is a chain: for if J + S'H and J + KT' have two-sided
inverses H' and K' then (3.1.3.1) says that J + HS' and J + T'K are also
invertible, with
H" = {I + HS')-1 =1- HH'S'
and (10.6.4.5)
K" = (I + T'K)-1 =1- T'K'K
It now follows that also
T'K' = K"T' and H'S' = S'H" (10.6.4.6)
For example,
T' = T'K'{I + KT') => T'{I - K') = T'K'KT'
=^T'K' = [I-T'K'K)T'
This at once makes [T'K', H'S') a chain:
{T'K'){H'S') = {K"T'){S'H") = K"{T'S')H" = 0 (10.6.4.7)
This together with (10.6.4.2) shows that the condition (10.3.4.1) is satisfied
with (r',5") in place of (S,T) and {T'K',H'S') in place of {S + H,T +
K), and now Theorem 10.3.4 says that {T'K',H'S') is invertible, so that
(10.6.4.4) says that {S + H, T + K) is Weyl. ■
The openness follows when the space Y is complete: the condition
(10.6.4.1) can be achieved by taking ||r'||||ii:|| < 1 and ||S"||||#|| < 1,
and then the condition (10.6.4.2) by taking K and H if necessary even
smaller. The reader should observe that ||i^' — i"|| —► 0 as \\K\\ —► 0, and
that ||jy; - J|| -> 0 as ||#|| -> 0: he is invited to find bounds for ||ii:||
and ||#|| which will ensure \\I + ifr'HII-fr'H < 1 and \\I + S"#||||#'|| < 1,
respectively.
Theorem 10.3.4 and Theorem 10.6.4 both extend to triples {R,S,T)
and longer chains. If for example
S'S + TT' = I = R'R + SS' (10.6.4.8)

10.6 Essential Exactness
429
and
S'STT' = S'(S + H){T + K)T'
and (10.6.4.9)
R'RSS' = R(R + L)(S + H)S'
and
H'(I+S'H) =1= (I+KT')Kf and L'(I+R'L) = 1= {I+HSf)H"
(10.6.4.10)
then Theorem 10.3.4 tells us that
(H'S')(S + H) + (T + K)(T'K') = 1= (L'R')(R + L) + (S + H)(S'Hf)
(10.6.4.11)
and since (10.6.4.9) and again (3.1.3.1) tell us that both I+S'H and I+HSf
have two-sided inverses we have, as in (10.6.4.6),
S'H" = H'Sf (10.6.4.12)
As for single operators, the index of a Fredholm chain (0, S, T, 0)
vanishes when the chain is Weyl:
10.6.5 THEOREM If (5, T) e BL(X, Y, Z) is a regular chain, then
necessary and sufficient for (0, S, T, 0) to be Fredholm is that
r_1(0) and 5"1(0)/cl(rX) and Z/c\(SY) are finite dimensional
(10.6.5.1)
Necessary and sufficient for (0, S, T, 0) to be Weyl is that in addition
Euler(0, S, T, 0) = 0 (10.6.5.2)
Proof: The first part is clear from (10.6.2.5). If (0,5, T,0) is Fredholm
then there is an essential inverse (0, T", Sf, 0) for (0, S, T, 0) for which
T'S' = 0 and T = TT'T and S = SS'S (10.6.5.3)
Now write
T'T = Pr, TT' = Qr, S'S = Ps and SS' = Qs (10.6.5.4)
As in (3.8.2.2) these are idempotents, and since PSQT = 0 we recall from
Theorem 2.5.4 that (J — QT)(I — Ps) is another idempotent, with
VP) = QAY) + ((J - QtW ~ PS))(Y) (10.6.5.5)

430
10. Comparison of Operators and Exactness
Thus, we have isomorphisms
Pf1(0)=T-1(0),((I-QT)(I-Ps))(Y) = S-\0)/c\(TX),
Q^{0) = Zlc\{SY)
Necessary and sufficient for Euler(5, T) = 0 is therefore
((/- QT){I- PS){Y) - Pf!(()) x Qs\0) = (I-PT)(X) x (I-Qs)(Z)
(10.6.5.7)
which is equivalent, by the definition of the cartesian product, to the
existence ofUG BL(X,y), V £ BL(y,Z), U' £ BL(y,X), and V G BL(Z,Y)
for which
VU = 0 = U'V
and (10.6.5.8)
U,U = I-PT,V'V + UU' = (I-QT)(I-PS),VV' = I-QS
If (10.6.5.8) holds, then we obtain an essential inverse (0, T',5',0) with an
inverse (0,S"',r",0) by taking
Tf = U\ Sf = V\ T" = T + U and S" = S + V (10.6.5.9)
Conversely, if we already have Sf, T', S" and T" then we can solve (10.6.5.9)
to get U, V, U' and V" satisfying (10.6.5.8). ■
The alert reader may noticed that the operators T' and S' of (10.6.5.9)
need not be the same as the operators Tf,Sf of (10.6.5.3): the same
projections PT and QT can be induced by more than one generalized
inverse r'.
Definition 10.6.1 does not include the common generalization of
upper and lower Fredholmness, and exactness, or of almost upper and lower
Fredholmness and almost exactness: we settle for an uneasy compromise.
10.6.6 DEFINITION We shall call a chain (5,T) £ BL(X,Y,Z) nearly
essentially exact if
5"1(0)/cl(rX) is finite dimensional (10.6.6.1)
and
S is relatively open (10.6.6.2)
and
S has closed range (10.6.6.3)

10.6 Essential Exactness
431
and
T is relatively almost open (10.6.6.4)
Near essential exactness can be tested with the essential enlargement
functors of Definition 6.7.4:
10.6.7 THEOREM If (5, T) £ BL(X, Y, Z) then
(P^S^P^r) linearly exact (10.6.7.1)
implies
(5, T) nearly essentially exact (10.6.7.2)
which implies
(P(5),P(r)) exact (10.6.7.3)
If X, y, and Z are complete then all three conditions are equivalent.
Proof: The first implication is a synthesis of the arguments of
Theorems 6.9.2 and 6.9.3, while the second follows the arguments of
Theorem 6.10.3. If (10.6.6.1) fails, so that the quotient 5_1(0)/ cl(TX) is infinite
dimensional, then by the Riesz lemma (Theorem 1.5.2) there is y = (yn) in
Y for which
Syn = 0 and ||yn|| = 1
and
(10.6.7.4)
disth/n+1,cl(rX) + ^Ct/y >I
which, means
y € JTO(y) and Sy € e0{Z) C m^Z)
and (10.6.7.5)
y^Tl00(X)+m1(Y)
so that
S + m1(7)€PI(S)-1(0)\P1(r)P1(X)CP1(7) (10.6.7.6)
violating (10.6.7.1). If instead (10.6.6.2) fails then there is y = (yn) in Y

432
10. Comparison of Operators and Exactness
for which
||yn|| = l and distfo^ST^O)) > \ and ||SyJ| — 0
(10.6.7.7)
We claim that this also gives us (10.6.7.5). The first two parts are clear: If
the last part fails then there is x € l^X) for which y — Tx has a convergent
subsequence, so that there is (xf, y') < (x, y) for which y'n — Tx'n —► y'^ € Y:
but now
Sy'^ = 0 and distfy^Sr *(())) > \ (10.6.7.8)
This contradiction verifies the last part of (10.6.7.5). To verify (10.6.6.3)
we assume both (10.6.7.1) and (10.6.6.2): If z € c\(S(Y)) then by (10.6.6.2)
there is yn in Y for which
\\z - Syn\\ —> 0 with ||yj| < h\\z\\ (10.6.7.9)
which means that y (E l^Y) and Sy (E c1(Z) C m1(Z). But now by
(10.6.7.1) there is x € l^X) for which y—Tx € m1 (Y), giving subsequences
(x',y') < (x,y) for which y'n-Tx'n ^y^GY; since \\z-S{y'n-Tx'n)\\ -> 0
it follows that z = Sy'^ G S(Y). If (10.6.6.4) fails then there is g = (gn) in
y t for which
\\9n\\ = Klls-i(0) = 1
and (10.6.7.10)
dist^trt)"1^))^! and llfcTH—.0
We claim
g <£ m^Y*) = m{Y^) (10.6.7.11)
Equality at the end follows from the completeness of Y^: if in violation of
the first part g^ £ Y^ were the limit of a convergent subsequence g' < g
then we would have g'JT = 0 and dist(^, (rt)-1(0)) > \. Now if
0 < 6 < p(^) = inf {k > 0 : {gn : n £ N}
(10.6.7.12)
C Disc(# ; k) with finite H CY^}
then there is gf ■< g for which, for each n (E N,
S'n+i * U Disc(^ ->6) (10.6.7.13)

10.6 Essential Exactness
433
and then {yfn}m)n^m in Y for which n^m implies
||y;>m|| = l and \g'n(y'n,m) ~ ?m«m)l > 5 (10.6.7.14)
We claim
sup ||x'Bim|| < oo =► p {y'n>m - Tx'n>m : n ? m} > \S (10.6.7.15)
Indeed if e > 0 and {yfn m — Tx'n m : n ^ m} C Disc(iJT ; e) with finite
KCY then
^(<J-^(<J£ U Disc((^-^)(t/");-) + (^-^)(Tx'njm)CC
(10.6.7.16)
and if
n>m>Nel^g'n(y'n)rn)-g'n(y'n)Je [j Disc(K;e + e') (10.6.7.17)
y"€K
then it follows that ^£ < e + e:' and hence, letting e1 —► 0, that j8 < e. If
we now arrange the pairs {(n,m) : n ^ m} in a sequence and write
y = (yn) = (y'n,m)n*m (10.6.7.18)
Then (10.6.7.15) can be rewritten
x € l^X) => p(y - Tx) > \6 (10.6.7.19)
which means that (10.6.7.6) holds again.
This finishes the proof of the first implication (10.6.7.1) =^ (10.6.7.2):
toward the second observe that if (10.6.7.2) holds then there are k > 0,
h > 0 and Q0 = Q20 £ BL(S,-1(0),S'-1(0)) for which
(?o(5'"1(0)) = cl(TX) C S_1(0) C Y (10.6.7.20)
and
y £ cl(rX) => y £ cl {Tx : \\x\\ < k\\y\\} (10.6.7.21)
and
yG7=^ dist^S-1 (0)) < h\\Sy\\ (10.6.7.22)
If now y £ loo{Y) then by the anti-Riesz lemma (Theorem 1.5.1) there is

434
10. Comparison of Operators and Exactness
w = (wn) in Y for which
and (10.6.7.23)
\\yn - wj < distfo^ST^O)) + (l/»)||yj| and ||u;n|| < 2||yn||
and then by (10.6.7.20) and (10.6.7.21) there is x = (xn) in X for which
\\Q0wn - TxJ < (±) \\wn\\ and ||xj| < k\\Q0wn\\ (10.6.7.24)
Evidently w € l^Y) and x € /^(X); also
Q0w -TxG c0(Y) C m^Y) C m{Y) (10.6.7.25)
and, since by (10.6.6.1) Qq1^) is finite dimensional,
w-Q0w€ l^Qo l{0)) = m^Qo'(0)) C m^Y) C m{Y) (10.6.7.26)
thus
P1(y-rx)=P1(y-u;) and P(y - Tx) = P(y - w) (10.6.7.27)
To see that (10.6.7.3) holds we have to verify that the chain (P(5),P(T))
satisfies (10.3.1.4); but we have from (10.6.7.23) and (10.6.7.24)
limsup||yn-Txn|| = limsup||yn-u;n|| <felimsup||5yn||+0 (10.6.7.28)
n n n
and
limsup ||xj| < 2lfe||Q0|| limsup ||yn|| ■ (10.6.7.29)
n n
Theorems 6.9.2 and 6.9.3 suggest that we ought to be able to reverse
the implication (10.6.7.1) =>• (10.6.7.2) for incomplete spaces X, Y, and
Z. To do this we would have to be able to construct the sequence w of
(10.6.7.23) in such a way that
y-wGm^Y) (10.6.7.30)
The reader can certainly do this if 5_1(0) is complemented in Y, taking
w = Py with P = P2 G BL(Y, Y) for which P"1^) = S"1^).
When X, Y, and Z are complete then "near essential exactness" looks
much simpler, and also coincides with obvious notions of "essential
exactness" and "almost essential exactness":

10.7 Algebraic Exactness
435
10.6.8 THEOREM If X, Y, and Z are Banach spaces then necessary
and sufficient for (S,T) £ BL(X,Y,Z) to be nearly essentially exact is that
dim5'-1(0)/r(X) < oo and S is proper (10.6.8.1)
Proof: If (S,T) is nearly essentially exact then S is relatively open by
(10.6.6.2) and therefore has closed range by Theorem 4.4.3, and hence is
proper by Theorem 4.6.4, while T is relatively almost open by (10.6.6.4),
therefore relatively open by Theorem 4.4.4, therefore has closed range
by Theorem 4.4.3, which together with (10.6.6.1) gives the first part of
(10.6.8.1). Conversely if (10.6.8.1) holds then (10.6.6.1) is immediate, and
since S is proper so is (10.6.6.2). The first part of (10.6.8.1) together with
Theorem 4.8.2 (and Theorem 6.2.2) says that T(X) is closed, therefore
proper by Theorem 4.6.4, and hence relatively almost open as needed for
(10.6.6.4) ■
10.7 ALGEBRAIC EXACTNESS
The first three conditions of Definition 10.2.1 have obvious "algebraic"
analogues in the spirit of Definition 3.10.1; we will confine ourselves to looking
at the analogue of Definition 10.3.1 in a normed algebra:
10.7.1 DEFINITION A pair of elements (6, a) £ A2 in a normed algebra
A will be called (left,right) invertible if
1 £ Ab + aA (10.7.1.1)
and almost (left,right) invertible if there are (afn) and (bfn) in A for which
||&'n6 + aa'n-l|| —>0 and sup ||a'J| +sup ||6'n|| < oo (10.7.1.2)
n n
We shall call the pair (6, a) £ A2 (left,right) bounded below if there are
k > 0 and h > 0 for which, for arbitrary u,v £ A,
\\uv\\ < &||u||||6v|| + fc||ua||||v|| (10.7.1.3)
and (left,right) one-one if there is implication, for arbitrary u,v £ A,
bv = 0 = ua => uv = 0 (10.7.1.4)
Once again we shall call a pair of elements (6, a) for which
ba = 0 (10.7.1.5)
a chain.

436
10. Comparison of Operators and Exactness
10.7.2 THEOREM If (6, a) £ A2, there is implication
(10.7.1.1) => (10.7.1.2) => (10.7.1.3) => (10.7.1.4) (10.7.2.1)
Necessary and sufficient for (6, a) (E A2 to be (left,right) invertible is that
(6, a) is regular and (left,right) one-one (10.7.2.2)
Proof: The first and third implications of (10.7.2.1) are both obvious. To
establish the second suppose that (10.7.1.2) holds, and see that for arbitrary
u,v€Avte have
|M| < ||«(6'n6 + aa'n - 1)„|| + ||«||K|||NI + IMIKIIIMI
for each n (E N
Allowing n —► oo gives (10.7.1.3) with k = supn \\a'n\\ and h = supn \\b'n\\.
To prove (10.7.2.2) simply repeat the argument for (10.3.3.2) from Theorem
10.3.3. ■
It is left to the reader to see how the rest of Theorem 10.3.3 holds for
pairs of normed algebra elements, and how to extend the first two conditions
of Definition 10.7.1 to triples (c,6, a) and longer chains. For the record, we
shall write out the proof that, if the algebra is complete, invertible chains
come in open sets:
10.7.3 THEOREM If A is a normed algebra, then each of the following
conditions is sufficient for the pair (6 + h, a + k) (= A2 to be (left,right)
invertible:
b'b + aa' = 1 and b'baa! = b'{b + h)(a + k)af
and (10.7.3.1)
ti{l + b'h) = 1 = (l + ka!)k'
b'(b + h) + (a + k)a' GA'1 and b'(b + h)(a+k)a! = {a + k)a'b'{b + h)
(10.7.3.2)
{&'(6 + h) + (a + k)a',a'[a + k)} C A'1 and b'{b + h)(a + k)=0
(10.7.3.3)
{b'(b + h) + (a + k)a', (6 + h)b'} C A~l and (6 + h) (a + fc)a' = 0
(10.7.3.4)

10.7 Algebraic Exactness
437
Proof: Repeat the argument for Theorem 10.3.4: for example the second
part of the condition (10.7.3.1) gives the identity
b'(b + h)(l + kaf) + (1 + b'h)(a + k)a' - (1 + b'h)(l + ka')
= b'b + aa'-l ■ (10.7.3.5)
Theorem 10.7.3 tells us that almost invertible chains come in open sets:
10.7.4 THEOREM If A is a normed algebra, then the set
{(6,a) £ A2:ba = 0 and (6,a) is almost invertible} (10.7.4.1)
is an open subset of the closed set of chains in A. If the algebra A is
complete then also the set
{(6, a) e A2: ba = 0 and 1 e Ab + aA] (10.7.4.2)
is open in the set of chains.
Proof: For the first part repeat the argument for Theorem 10.3.5, which
uses the identity (10.7.3.5); for the second part repeat the argument for the
last part of Theorem 10.5.6. ■
Even in the algebra of 2 x 2 matrices the openness of Theorem 10.7.4 is
only relative to the closed set of chains: we recall again the pair of operators
of (10.3.8.8).
In a normed algebra A we would expect to find that the chains (6, a)
which satisfy the boundedness below condition (10.7.1.3) form another open
set: we encourage the reader to see if he can succeed where we have failed.
If, in particular, A = BL(X, X) then this is the case, because then the
bounded-below chains coincide with the pairs of operators which are almost
exact in the sense of (10.3.1.5):
10.7.5 THEOREM If A = BL(X,X) for a normed space X and (6, a) €
A2 is a chain, then
(6, a) (left,right) bounded below -<=>• (6, a) almost exact (10.7.5.1)
and
(6, a) (left,right) one-one -<==> (6, a) almost linearly exact (10.7.5.2)

438
10. Comparison of Operators and Exactness
Proof: If (6, a) is almost exact, then by Theorem 10.4.1 the condition
(10.4.1.2) holds, in particular with the spaces E and F both equal to the
space X = Y = Z, which says that (6, a) is (left,right) bounded below.
Conversely, if this is so then we may further restrict the condition (10.4.1.2)
to apply only to operators U and V of rank one: but this gives the linear
functional condition (10.4.1.3), and hence by Theorem 10.4.1 the almost
exactness condition (10.4.1.1). This proves both implications of (10.7.5.1),
and the argument for (10.7.5.2) is the same, using Theorem 10.4.2 instead
of Theorem 10.4.1. ■
If we apply Definition 10.7.1 in the seminormed algebra BL(X, X)/
KL0(X, X), then the first and last conditions applied to a pair (6, a) = (S +
KL0(X,X),T + KL0(X,X)) reproduce the Fredholm and weak Fredholm
conditions of Definition 10.6.1 for the operators (S,T). This is obvious for
the first condition. To see it for the last condition we use Theorem 10.6.2
and argue as in Theorem 10.7.5.
In the other direction, the exactness condition latent in Theorem 10.6.5
can be reproduced in the form (10.7.1.1):
10.7.6 THEOREM If (S, T) e BL(X, Y, Z), then the following are
equivalent:
(0,5, T,0) is (left,right) invertible
"0 0 0"
T 0 0
.0 S 0.
?
"0 0 0"
T 0 0
.0 S 0.
is (left,right) invertible
(10.7.6.1)
(10.7.6.2)
Proof: If (0,5, T,0) is (left,right) invertible, then, using Theorem 10.3.2,
there is (r',S") £ BL(Z,Y,X) for which
T'T = I and S'S + TT' = I and SS' = I (10.7.6.3)
This gives
OS' 0
0 0 T'
0 0 0
"0 0 0"
T 0 0
.0 S 0.
+
"0
T
.0
o o-
0 0
s o.
0 S'
0 0
0 0
o ■
r
0 .
=
-/ o o-
0/0
.0 0 /.
(10.7.6.
4)

10.7 Algebraic Exactness
439
and hence
'I 0 0"
0/0
.0 0 I.
/
GBL
V
'X-
Y
_Z.
?
■x-
Y
_Z_
\ 1
;
0 0 0
TOO
0 S 0
+
0 0 0"
TOO
0 S 0.
(
BL
V
'X-
Y
_Z_
?
-xi
Y
_Z_
(10.7.6.5)
Conversely, if (10.7.6.5) holds, then there is U = (U{j) and V = (V{j) for
which
0 0 0
TOO
L0 S 0J
+
"0 0 0"
T 0 0
l0 S 0.
U =
-I 0 0"
0/0
.0 0 /.
and hence
W
"0
T
.0
o o-
0 0
s o.
+
0
T
0
0 0"
0 0
s o.
w =
"I 0 0"
0/0
.0 0 /.
with W = U
0 0 0
TOO
0 S OJ
But now (10.7.6.3) holds with
and
T' = W23 and S' = W12 (10.7.6.6)
Similar representations exist for the exactness of chains of the form
(0,Tn,...,T2,T1,0) withr^BLp^,*,.) (10.7.6.7)
(0,an,...,a2,a1,0) with aj £ A ■ (10.7.6.8)
Theorem 10.4.6 has an analogue for normed algebra elements:
10.7.7 THEOREM If (c,6,a) € A3 is a chain and if (c,6) e A2 is
(left,right) invertible then there is implication
(Lb,La) linearly exact => (b,a) (left,right) invertible =>• (Lb,La) exact.
(10.7.7.1)

440
10. Comparison of Operators and Exactness
If instead (6, a) £ A2 is (left,right) invertible then there is implication
(Rb,Rc) linearly exact =>• (c,6) (left,right) invertible =>• (Rh,Rc) exact.
(10.7.7.2)
Proof: Repeat word for word the proof of Theorem 10.4.6 ■
The analogue of Theorem 10.7.7 also holds for almost exactness:
analogous to (10.4.6.5), if (c,6) £ A2 is almost (left,right) invertible then
(Lb,La) almost linearly exact
=>• (6, a) almost (left,right) invertible (10.7.7.3)
=>• (Lb,La) almost exact
while, analogous to (10.4.6.6), if instead (6, a) £ A2 is almost (left,right)
invertible then
(Rb,Rc) almost linearly exact
=>• (c,6) almost (left,right) invertible (10.7.7.4)
=>• (Rb,Rc) almost exact
Like Theorem 10.4.6, Theorem 10.7.7 and its "almost" analogue extend to
longer chains (an,... , c^,*^) and (0,an,... ,a2,a1,0).
The Fredholm and Weyl theory of a homomorphism of normed
algebras extends to pairs of elements, giving another perspective on "essential
exactness."
10.7.8 DEFINITION If T £ BBL(A,B) is a homomorphism of normed
algebras then a chain (6, a) £ A2 will be called (left,right) T-Fredholm if
l£BT(b)+T(a)B (10.7.8.1)
weakly (left,right) T-Fredholm if there is implication, for arbitrary u,v £ B,
uT(a) = 0 = T(b)v =)> uv = 0 (10.7.8.2)
and (leftjright) T-Weyl if there are a', 6', a", b" in A for which
b'b + aaf -1£T~1(0) and a'bf = 0 and b'b" + a" a' = 1
(10.7.8.3)
This gives us back Definitions 10.6.1 and 10.6.3 if we take A =
BL(X,X) and B = BL(X,X)/KL0(X,X), with T the "finite Calkin
functor." For certain such homomorphisms, we recover the relationship between
regularity, Fredholmness, and weak Fredholmness:

10.8 Hilbert Spaces
441
10.7.9 THEOREM If T £ HBL(A, B) satisfies
T-^OJC^ and T(A) = B (10.7.9.1)
then it is necessary and sufficient for a chain (6, a) € A2 to be (left,right)
T-Fredholm that
(6, a) is regular and weakly T-Fredholm (10.7.9.2)
Proof: This is a rewrite of the proof of (10.6.2.5) from Theorem 10.6.2. ■
Theorem 10.6.4 and Theorem 10.7.3 have a common generalization
which gives conditions sufficient for a pair (6 + h, a + k) to be T-Fredholm,
or T-Weyl: we leave it to the reader to state and prove.
10.8 HILBERT SPACES
For Hilbert spaces, most of the conditions of Definition 10.1.1 and of
Definition 10.1.3 are the same:
10.8.1 THEOREM Suppose that X, Y, and Z are Hilbert spaces, and
that T £ BL(X,y) and S G BL(X,Z): then there is implication
S majorized by T =>► S a left multiple of T (10.8.1.1)
If instead T £ BL(X,Y) and S £ BL(Z,Y), there is implication
S almost comajorized by T =>• S a right multiple of T (10.8.1.2)
and
S range-included in T =)> S a right multiple of T (10.8.1.3)
Proof: For (10.8.1.1) we need only assume that Y is a Hilbert space and
that Z is complete, and follow the argument at the end of Theorem 10.1.2. If
the condition (10.1.1.4) holds, then the mapping U0 : TX -► Z of (10.1.2.4)
is well defined, bounded, and linear; if the space Z is complete then there
is a bounded extension U1 of U0 to the closure cl(TX), and finally, if Y is
a Hilbert space there is Q = Q2 € BL(Y,Y) with Q(Y) = cl(TX). If we
now define
U = UlQ:Y—> Z (10.8.1.4)
analagously to (10.1.2.5) then we will satisfy the condition (10.1.1.1). For
(10.8.1.2) we use instead the argument at the end of Theorem 10.1.4: if
the condition (10.1.3.5) is satisfied then the operators U of (10.1.4.4) will

442
10. Comparison of Operators and Exactness
satisfy the condition (10.1.3.1). Finally, since Hilbert spaces are complete,
the condition (10.8.1.3) is derived from (10.8.1.2) using (10.5.5.1). ■
For example Theorem 10.8.1 and Theorem 9.9.5 combine to give the
"polar decomposition" of a bounded operator between Hilbert spaces: If
TGBL{X,Y) then
T = sgn(r) . \T\ (10.8.1.5)
with
|r| = (r*r)1/2 e bl(x,x) (10.8.1.6)
and
sgn(r) = sgn(r)sgn(r)*sgn(r) e BL(X,y) (10.8.1.7)
Here the positive operator \T\ is given by (9.9.5.1) with a = T*T and
A = BL(X,X): since \T\ = \T\* and \T\2 = T*T we have
|||r|||2 = (|r|2x;x) = {T*Tx;x) = \\Tx\\2 for each x £ X (10.8.1.8)
so that T is majorized by \T\ in the sense of Definition 10.1.1. We shall
define sgn(T) to be the operator U obtained in the analogue of (10.8.1.4)
with T and \T\ in place of S and T, taking Q to be the orthogonal projection
on the range of \T\.
An operator U for which U = UU*U is called partial isomttry.
It would have been nice to extend the argument of Theorem 10.8.1 to
the conditions of Definition 10.2.1: the reader is invited to try his hand.
In the particular case of the exactness of Definition 10.3.1 we have the
equivalence: we first pass through an auxiliary result.
10.8.2 THEOREM If X, Y, and Z are Hilbert spaces and (S,T) e
BL(X, Y, Z), then the following are equivalent:
T is one-one and S is dense
T*T and SS* are one-one and dense
col(5,r*) is dense
row(5*,T) is one-one
Each of the following are also equivalent:
T is left invertible and S is right invertible
T*T and SS* are invertible
col(5,T*) is right invertible
row(S*,T) is left invertible
(10.
(10.
(10.
(10.
(10.
(10.
(10.
(10.
8.2.1)
8.2.2)
8.2.3)
8.2.4)
8.2.5)
8.2.6)
8.2.7)
8.2.8)

10.8 Hilbert Spaces
443
Proof: The equivalence of (10.8.2.1) and (10.8.2.2) follows from (8.6.5.1)
and (8.6.5.2) from Theorem 8.6.5. In particular, (8.6.5.5) gives equality
r_1(o) = (r*r)_1(o) (10.8.2.9)
which establishes half of the equivalence (8.6.5.1), the other (8.6.5.2)
following by taking adjoints. If we apply (8.6.5.1) with row(S*,T) in place of
T, and apply (8.6.5.1) with col(5, T*) in place of T, and observe
col(5,r*)row(5*,r) =
S '
T*
IS*
L
ri =
J
'SS*
0
0
T*T
(10.8.2.10)
then we obtain the equivalence of each of (10.8.2.4) and (10.8.2.3) with
(10.8.2.1).
For the second part the equivalence of (10.8.2.5) and (10.8.2.6) is given
by (8.6.5.3) and (8.6.5.4) from Theorem 8.6.5, and then the equivalence of
(10.8.2.5) with each of (10.8.2.8) and (10.8.2.7) follows from (10.8.2.10). ■
More generally,
10.8.3 THEOREM If X, Y and Z are Hilbert spaces and (S,T) €
BL(X, y, Z) then the following are equivalent:
(S,T) is almost linearly exact (10.8.3.1)
S*S + TT* is one-one and dense (10.8.3.2)
col(5,r*) is one-one (10.8.3.3)
row(S"\T) is dense (10.8.3.4)
Each of the following is also equivalent:
(S,T) is (left,right) invertible (10.8.3.5)
S*S + TT* is invertible (10.8.3.6)
col(S,r*) is left invertible (10.8.3.7)
row(5*,r) is right invertible (10.8.3.8)
Proof: Whether or not the operators T € BL(X,Y) and S € BL(Y,Z)
form a chain, there is implication
s-1(o) c d(rx) ^=> sr^o) _l (r*)-1^) => s-\o) n (r*)-1^) = {0}
(10.8.3.9)

444
10. Comparison of Operators and Exactness
while
{ST = 0 and 5"x(0) n (T*)"1^) = {0}) =► 5"x(0) _L (T*)"1^)
(10.8.3.10)
This follows from (8.3.3.2) from Theorem 8.3.3 by taking K = c\{TX) and
H = 5_1(0). Thus if ST = 0 and the condition (10.8.3.1) holds then so
does the condition (10.8.3.3), which is therefore equivalent to (10.8.3.1).
The equivalence of (10.8.3.3) and (10.8.3.4) is just (8.5.4.1). Finally, the
equivalence of (10.8.3.3) and (10.8.3.2) is an application of (8.6.5.1) with
col(5,T*) in place of T, noting that
row(S*,T)col(S,r*) = [S* T]
S
= S*S + TT* (10.8.3.11)
Towards the second part suppose that (5, T) is (left,right) invertible,
with VS + TU = J: then it is also (left,right) bounded below in the sense of
(10.3.1.4) and hence satisfies the condition (10.4.1.3). With k = \\U\\ and
h = ||V|| we have for arbitrary y,w £ Y
Ky;u»>|<fc|HH|5y|| + fc||y||||r*u»|| (10.8.3.12)
and hence, taking w = y and then dividing across by ||y||,
\\y\\ < h\\Sy\\ + &||r*y|| for each y eY (10.8.3.13)
Thus if (10.8.3.5) holds then the operator col(5,T*) is bounded below, and
hence by (8.4.4.1) is left invertible, giving (10.8.3.7). The equivalence of
(10.8.3.7) and (10.8.3.8) is (8.5.4.2) together with Theorem 8.4.4: we finish
the whole argument by showing that (10.8.3.6) => (10.8.3.5). This is given
by the argument of Theorem 10.3.4: if ST = 0 then also T*S* = 0 and
hence
(5*5)(TT*) = (TT*)(5*5) (10.8.3.14)
S*S + TT* =ReBL-\Y,Y) (10.8.3.15)
If, in addition,
is invertible then
(jR_15*)5 + r(r*jR_1) = / (10.8.3.16)
which is the condition (10.8.3.5). ■
Part of the second part of Theorem 10.8.3 gives the promised
implication (10.3.1.4) => (10.3.1.1) for Hilbert spaces, and hence of course also
(10.3.1.2) => (10.3.1.1). The essence of Theorem 10.8.2 follows from
Theorem 10.8.3 by taking separately 5 = 0 and T = 0. As part of the proof

10.8 Hilbert Spaces
445
of Theorem 10.8.2 we noted in (10.8.2.9) that T £ BL{X,Y) is determined
by T*T e BL(X,X) in the sense of (10.1.1.5). The alert reader may have
wondered whether for the second part of the argument we could have shown
that T was majorized by T*T in the sense of (10.1.1.4): but this fails for
the operator T = W : (xn) —► {xn/n) of (7.3.6.14), defined on the sequence
space S = Y = l2: for if Sn = (0,0,..., 0,1,0,...) is the "Kronecker delta"
then, for each nGN,
ll*»ll = 1 ^d IITSJI = n"1
and (10.8.3.17)
\\T*TSn\\=n-i = n-l\\TSJ
Together with Theorem 3.5.1, these last two results give us a boundary
result dual to Theorem 10.3.9:
10.8.4 THEOREM If X, Y and Z are Hilbert spaces and if chains
(5,T) e BL(X,y,Z) and (5n,Tn) G BL{X,Y,Z) satisfy
(S,T) is (left,right) invertible (10.8.4.1)
and
Sn is dense and Tn is one-one (10.8.4.2)
and
||5n - 5|| + \\Tn - T\\ —> 0 as n —> oo (10.8.4.3)
then
S is dense and T is one-one (10.8.4.4)
Proof: By Theorem 10.8.3 the operator col(S,T*) is left invertible,
therefore bounded below, while by Theorem 10.8.2 each of the operators
col(Sn,T^) is dense, and also || col(5n,T^) - col(S,T*)|| -> 0. By
Theorem 3.5.1 the operator col(5,T*) is also dense, which by Theorem 10.8.2
again gives (10.8.4.4). ■
The second part of Theorem 10.8.3 extends to the "Hilbert algebras"
of Definition 8.8.1:
10.8.5 THEOREM If (6, a) G A2 is a chain in a Hilbert algebra A then
the following are equivalent:
(6, a) is (left,right) bounded below (10.8.5.1)
6*6 + aa* is not a topological zero divisor (10.8.5.2)

446 10. Comparison of Operators and Exactness
6*6 + aa* is invertible (10.8.5.3)
(6, a) is (left,right) invertible (10.8.5.4)
Proof: By Theorem 9.9.4 and Theorem 9.9.6 there is inequality, for
arbitrary a, 6 £ A,
max(||a||2, ||6||2) < ||6*6 + aa* || (10.8.5.5)
If now (6, a) £ A2 is (left,right) bounded below in the sense of (10.7.1.3)
then there are k > 0 and h > 0 for which, for arbitrary u £ A,
H|2 = ||u*u|| < A||ul||fru|| + *N|||u*a|| = \\u\\(h\\bu\\+k\\a*u\\)
(10.8.5.6)
which with the elementary (1.6.1.1) and the Hilbert algebra condition
(8.8.1.1) gives
ll^ll < (A:2 -H ^2)1/2([[&^||2 -H H^^H2)1/2
= (k2 + h2)1/2{\\u*b*bu\\ + \\u*aa*u\\)1/2
and hence, using (10.8.4.5),
INI2 < (A:2 + /i2)(||u*6*6u|| + ||u*aa*u||) < 2{k2 + h2)\\u*{b*b + aa*)u||
(10.8.5.8)
and hence finally,
INI2 < 2{k2 + /i2)||u||||(6*6 + aa*)u\\ for each u £ A (10.8.5.9)
Dividing across by ||u|| gives (10.8.5.2), and hence by Theorem 9.9.7 also
(10.8.5.3). Using for the first time the assumption ba = 0 and following the
argument for the last part of Theorem 10.8.3 gives the condition (10.8.5.4).
If c = 6*6 + aa* is invertible and (6*6)(aa*) = (aa*)(6*6) then
(c-x6*)6 + a{a*c-x) = 1 (10.8.5.10)
The final, obvious, implication (10.8.5.4) => (10.8.5.1) is part of Theorem
10.7.2. ■
Part of the argument for Theorem 10.8.5 shows that
(6, a) (left,right) one-one =*> 6*6 + aa* not a zero divisor (10.8.5.11)
Indeed whether or not ba = 0 there is implication, for arbitrary u £ A,
(6*6 + aa*)u = 0 =^ u*(6*6 + aa*)u = 0=^bu = u*a = 0 (10.8.5.12)

10.8 HUbert Spaces
447
The reader is invited to see whether he can reverse the implication
(10.8.5.12) when ba = 0, or indeed extend the implication (10.8.5.1) =>
(10.8.5.4) to the situation in which this is not assumed. This implication
also gives a "spectral permanence" for chains (6, a) £ A2 in a Hilbert
algebra A: if A C B is isometrically embedded in a larger normed algebra B,
then any chain (6, a) which is (left,right) invertible in B will be (left,right)
invertible in A, and any chain which fails to be (left,right) bounded below
in B will also fail in A.
Theorem 10.8.5 applies, in particular, in the Calkin algebra A =
BL(X, X)/KL(X, X) on a Hlbert space X. If we argue similarly for the
seminormed algebra BL(X, X)/ KL0(X, X) obtained by quotienting out the
finite rank operators, and then relax the requirement that all three spaces
be the same, we reach the following result.
10.8.6 THEOREM If (5, T) <E BL(X, Y, Z) for Hilbert spaces X, Y and
Z then the following are equivalent:
(S,T) is weakly (left,right) Fredholm (10.8.6.1)
S*S + TT* is essentially one-one and essentially dense (10.8.6.2)
col(5,T*) is essentially one-one (10.8.6.3)
row(5*,T) is essentially dense (10.8.6.4)
Each of the following is also equivalent:
(5,T) is (left,right) Fredholm (10.8.6.5)
S*S + TT* is Fredholm (10.8.6.6)
col(S,T*) is left Fredholm (10.8.6.7)
row(S*,T) is right Fredholm (10.8.6.8)
Proof: This is left to the reader. ■
We can add to the list in the second part of Theorem 10.8.6 if we
observe
(S,T) (left,right) Fredholm <£=> (5,T) nearly essentially exact (10.8.6.9)
To see this combine (10.6.2.5) with Theorem 10.6.8. We can also compute
the Euler number of certain kinds of Fredholm chains in terms of the index
of a related operator:

448
10. Comparison of Operators and Exactness
10.8.7 THEOREM If (S,T) <E BL{X,Y,X) and (T,S) <E BL{Y,X,Y)
then
(S, T, S) (left,right) Fredholm <^=> T + S* Fredholm (10.8.7.1)
with
Euler(S, T, S) = index(T + S*) (10.8.7.2)
in particular
(S, T, 5) (left,right) Weyl <=> T + 5* Weyl (10.8.7.3)
Proof: The first part follows from (10.8.6.6) when we observe
S*S + TT* = (5*+T)(T*+5) and SS*+T*T = (S + T*)(T + S*)
(10.8.7.4)
Towards (10.8.7.2) the argument of Theorem 10.8.3 says that if ST = 0
then
5"1(o)/ci(rx) ^5"1(o)n(r*)-1(o) = (5*5 + rr*)"1(o) (10.8.7.5)
and hence,
5"1(0)/cl(rX) © (T* + 5)_1(0) = r"1(0)/cl(5y) © (T + 5*)_1(0)
(10.8.7.6)
When (S,T) and (T, S) are Fredholm we can add dimensions and get
(10.8.7.2), and in particular (10.8.7.3) ■
10.9 SKEW EXACTNESS
Suppose T £ BL(X,y) and S £ BL(Y,Z): then each of the conditions
T{X) H S"1 (0) = {0} (10.9.0.1)
and
T(X) + S"1 (0) = Y (10.9.0.2)
is in some sense "orthogonal" to the exactness condition (10.2.0.1). We
begin by recalling a relationship between these conditions and the index
Theorem 6.5.4:
10.9.1 THEOREM If T <E BL(X, Y) and S <E BL(y, Z) there is
implication
T{x) n s_1(o) = {o} <=> r_1(o) = (5r)_1(o) (10.9.1.1)

10.9 Skew Exactness
449
and
T{X) + 5_1(0) = Y <=> S{Y) = {S{T{X)) (10.9.1.2)
Proof: The mapping TA : X/T~x{0) -> T(X) is one-one and onto, and
induces the isomorphism used in the proof of (6.5.4.6):
(sr)-1(o)/r-1(o) = [tx) n 5_1(o) (10.9.1.3)
since if x <E X then Tx <E S_1(0) & {ST)x = 0. The mapping 5A :
y/5_1(0) —> 5(y) is also one-one and onto, and induces the isomorphism
used in the proof of (6.5.4.7):
Y/{TX + 5_1(0)) = S{Y)/{ST{X)) (10.9.1.4)
since if y <E Y then Sy <E S(TX) &yeTX+ S_1(0). ■
There is an interaction between exactness and skew exactness which
generalizes our results on subspaces and quotients, Theorem 3.11.1 and
Theorem 3.11.2, as well as our "ascent and descent" Theorem 7.8.2:
10.9.2 THEOREM If U e BL{W,X), T e BL(X,Y) and V <E BL(y,Z),
then there is implication
J7_10 = (rm_10 and (VT)_10 C U(W)
V J ,~ , , (10.9.2.1)
=► T_10 = 0 =J> U-x0 = (Ttf) 0
[TU)W D V-x0 and VY = [VT)X =^TX = Y =^VY = [VT)X
(10.9.2.2)
[TU)W D V'-^O) and T~l{ti) = 0 =► [VT)'1^) C tf(W) (10.9.2.3)
{VT)-1^) C UW and T(X) = y =► (T^)(W) D y_1(0) (10.9.2.4)
Also
[VTU){W) = (VT)(X) and (VT)-1^) = r_1(0) =► (TJ7)ty = TX
(10.9.2.5)
and
(VTUy'iO) = (TU)-1^) and (TU)W = TX q q g
=>• (vr)_1o = r_1o

450
10. Comparison of Operators and Exactness
Proof: If U-x0 = (TU)-1*) and (VT)_10 C UW suppose Tx = 0, so
that also VTx = 0 giving x = Uw with w e (TU)-x0 = U~x0, which
makes x = 0. This is the first implication of (10.9.2.1), and the second is
clear from (10.9.1.1). If y e Y is arbitrary and VY = VTX then there
isiGl for which y - Tx <E 7_10, and if also TU{W) D 7_10 then
there is w e W for which y = Tx + TUw = T{x + Uw) G TX. This is
the first implication of (10.9.2.2), and the second is clear from (10.9.1.2).
If x e X satisfies VTx = 0 and V_10 C (TU)W then there is w <E W
for which Tx = TUw, and if also T is one-one then x = Uw: this proves
(10.9.2.3). If Vy = 0 and TX = Y then y = Tx with x £ (7r)_1(0),
and if also (7r)_10 C UW then x = Uw, giving y <E [TU)W. This
proves (10.9.2.4). To prove (10.9.2.5) argue y = Tx => Vy = VTUw with
T(x - Uw) G^O, hence T(x - Uw) = 0. Finally, for (10.9.2.6) argue
VTx = 0 and Tx = TUw => VTUw = 0=>Tx = TUw = 0. ■
Theorem 10.9.2 relates the weakest possible kind of "exactness" for the
pairs {VT,U) and {V,TU):
10.9.3 THEOREM If U <E BL(W,X), T G BL(X,Y) and V <E BL(y,Z),
then there is implication
U majorized by TU and (VT, U) almost exact
=> T bounded below ==> U majorized by TU
(V,TU) almost exact and V almost comajorized by VT
=> T almost open ==> V almost comajorized by VT
(V, TU) almost exact and T bounded below
=> (VT, U) almost exact
(VT, U) almost exact and T almost open
=r> (V,TU) almost exact
Also
VT almost comajorized by VTU and T majorized by VT
=r> T almost comajorized by TU
and
TU majorized by VTU and T almost comajorized by TU
=^ T majorized by VT
(10.9.3.1)
(10.9.3.2)
(10.9.3.3)
(10.9.3.4)
(10.9.3.5)
(10.9.3.6)

10.9 Skew Exactness
451
Proof: We closely follow the pattern of argument for Theorem 10.9.2,
substituting those parts of Theorem 3.11.1 and Theorem 3.11.2 which involve
boundedness below and almost openness for those which involve one-one-
and ontoness. Thus for (10.9.3.1) we suppose that there are k > 0 for which
||J7(.)|| < k\\TU{.)\\ on W and &',&" > 0 for which each x £ X has [wn)
in W for which limsupn ||x - Uwn\\ < k'\\VTx\\ with supn ||u;n|| < k"\\x\\.
Now for arbitrary iGlwe have
||s||<||*-CTWn|| + ||tf«,B||
<\\x-UwJ+k\\T{Uwn-x)\\+k\\Tx\\
<(l + *||T||)||s-CT«;J+*||T*||
and hence, taking limsupn,
||*|| < (1 + *||r||)A'||VTs|| + fc||Tx|| < (k + (1 + *||T||)A'||V|| Wyr)||T*||
This is the first implication of (10.9.3.1), and the second is obvious.
Towards (10.9.3.2) we suppose there are k,k',k" > 0 for which limsupn ||y —
TUwJ < k'\\Vy\\ with supn||Wn|| < fc||y|| and \\Vy - TxJ -» 0 with
supn ||zj| < fc"||y||. Then if y e Y is arbitrary and \\V(y - Txn)\\ -»• 0
there is Km) in W with limsupm ||(y - Txn) - wnm\\ < k'\\V{y - Txn)\\
and supm ||Wnm|| < A||y - TxJ < fc(l + ||T||fc")||y||. This gives
limsup||y - T{xn + Uwnm)\\ < limsupfc'||y(y - Txn)|| = 0
n,m n
and
K-^»«,ii<*"+*(i+imi*")
This proves the first implication of (10.9.3.2), and the second is
obvious. For (10.9.3.3) we have for arbitrary x £ X a sequence {wn) in
W with limsupn||x - Uwn\\ < A;limsupn \\Tx - TUwn\\ < kk'\\VTx\\
with supn||w;n|| < ||Tx|| < A;"||T||||x||. For (10.9.3.4) we have for
arbitrary y £ Y a sequence (xn) with ||y — Txm|| —> 0 and supm ||xm|| <
A^'lls/H, and then sequences {wmn) for which limsupn \\xm — Uwmn\\ <
k'\\VTxm\\ with supn \\wnm\\ < k\\xm\\9 giving limsupmjrl \\y - TUwmn\\ <
limsupm \\y - Txm\\ + ||T|| limsupmjrl \\xm - Uwmn\\ < ||T||A;'||V<,|| using
again ||y — Txm|| —> 0. Towards (10.9.3.5) we have for arbitrary x £ X a
sequence (wn) with ||VTx - VTUwn\\ -> 0 and supn ||u;n|| < k'\\x\\9 while
also ||T(.)|| < A;||7r(.)||. Thus, \\Tx-TUwn\\ < k\\VTx - VTUwn\\ -> 0.
Finally, for (10.9.3.6) we have for arbitrary x £ X a sequence (wn) with
supn ||Wn|| < fc'||x|| and \\Tx - TUwJ -» 0, while \\TU(-)\\ < k\\VTU{-)\\.
Thus, ||Tx|| < ||r*-r^wB|| + ||Tl7u;n|| < \\Tx- TUwJ + k\\VTUwn\\ <
\\Tx - TUwJ + jfc||Vri7u;n - VTx\\ + k\\VTx\\, and taking limits gives the
result. ■

452
10. Comparison of Operators and Exactness
Naturally, there is an analogue for the "decomposable exactness"
condition (10.3.1.1):
10.9.4 THEOREM If U <E BL(W,X), T £ BL(X,y), and V <E BL{Y,Z),
then there is implication
U = T'TU and I = V'VT + UU' =^ I = T"T =► U = T"TU (10.9.4.1)
I=V'V + TUU" and V = VTT" =► J = TT' =► V = VTT' (10.9.4.2)
I=V'V + TUU" and I = T'T => I = V'VT + UU' (10.9.4.3)
J = V'VT + UU" and I = TT' =^ I = V"V + TUU' (10.9.4.4)
Also
VT = VTUU' and T = V'VT =^T = TUU" (10.9.4.5)
and
TU = V'VTU and T = TUU' =J> T = V'VT (10.9.4.6)
Proof: If U = T'TU and I = V'VT + UU' then
0 = (/ - T'T)UU' = {I- T'T){I - V'VT)
giving (10.9.4.1) with T" = T'+V'V-T'TV'V. If instead J = V'V+TUU"
and V = VTT", then
0 = V'V{I - TT") = {I- TUU") {I - TT")
giving (10.9.4.2) with V = T" + UU" - UU"TT". For (10.9.4.3) we argue
I = T'T = T'{V'V + TUU")T = [T'V')VT + U{U"T)
and for (10.9.4.4) we argue
I = TT" = T(V'VT + UU")T" = [TV')V = TU{U"T")
Finally, if VT = VTUU' then V'VT = V'VTUU', giving (10.9.4.5) with
U" = U, and if TU = V'VTU then TUU' = V'VTUU', giving (10.9.4.6)
with V" = V. ■
It is left to the reader to show that the analogue of Theorem 10.9.4
holds with almost exactness and almost left or right multiples in place of
exactness and left or right multiples.
We conclude with another kind of analogue of the result which says
that if a product of two operators is left or right invertible then so is one
factor:

10.9 Skew Exactness
453
10.9.5 THEOREM If T e BL{X,Y), S e BL(Y,Z), R e BL(Z,W),
U e BL(y,y), and V e BL{Z,Z) satisfy
vs = su
then there is implication
(jR,5) and (S, T) nonsingular =
R 0
V S
S 0
-U T
for each of the nonsingularities of Definition 10.3.1.
Proof: If
R'R + SS" = I = S'S + TT'
then
B!
0
-S'VR' S'
R 0
V S
+
S 0
-U T
S" 0
T'US" T'
(10.9.5.1)
nonsingular
(10.9.5.2)
(10.9.5.3)
/ 0"
0 /.
(10.9.5.4)
The only nontrivial detail is the bottom left hand corner, which in the
left-hand side is given by
-S'VR'R + S'V - US" + TT'US" = S'V{I - R'R) -{I- TT')US"
= s'vss" - s'sus"
= S'(VS -SU)S"
This establishes (10.9.5.2) for the (left,right) invertibility of (10.3.1.1), and
the argument for the almost (left,right) invertibility of (10.3.1.2) is almost
identical. The same pattern of argument establishes (10.9.5.2) for the
exactness of (10.3.1.4): if for each z £ Z there are y' and x' for which
\\z-Sy"\\<h'\\Rz\\ with ||y'|| < fc'W
and
lly-Ts'll^fcUSyll with ||x'|| < *||y||
then there is also x" e X with ||x"|| < k\\y + Uy'\\ and
\\(y + Uy')-Tx"\\<h\\S(y + Uy')\\
= h\\Sy + VSy'\\ < h\\Sy + Vz\\ + h\\V\\\\Z - Sy'\\
(10.9.5.5)
(10.9.5.6)
(10.9.5.7)
Together (10.9.5.5) and (10.9.5.7) show that the right-hand side of
(10.9.5.2) holds for exactness in the sense of (10.3.1.4). The argument

454
10. Comparison of Operators and Exactness
for almost exactness in the sense of (10.3.1.5) is the same, and left to the
reader. Finally, if
jR_1(0) C S(Y) and 5_1(0) C T{X)
then for arbitrary
(10.9.5.8)
' z
.y.
e
R 0"
y s.
-i
"0"
.0.
we have, since Rz = 0, y' £ Y for which
z = Sy' and Sy = -Vz = -VSy' = -SUy'
Since this means S(y + Uyf) = 0 there must be x' £ X with
z = Sy' and y + Uy = Tx'
which means that
' z
.y.
=
' S 0'
-u r.
[y'l
This establishes (10.9.5.3) for linear exactness in the sense of (10.3.1.6), and
the argument for almost linear exactness can now be left to the reader. ■
Conversely,
10.9.6 THEOREM If T e BL{X,Y), S e BL{Y,Z), R <E BL{Z,W),
u e BL(y,y), v e bl{z,z) then
R 0
V S
S 0
-U T
nonsingular
=^ (col(jR,7),5) and (5,row(T,C7)) nonsingular
for each of the nonsingularities of Definition 10.3.1.
Proof: If
(10.9.6.1)
R' W
V S'
R 0
V S
+
S 0
-U T
s" w"
-U' T'
then
[jR; W]
+ SS" = I = S'S + [-U T]
I 0
0 /
r
(10.9.6.2)
(10.9.6.3)

10.9 Skew Exactness
455
which proves (10.9.6.1) for (left,right) invertibility, and similarly for almost
(left,right) invertibility. If instead the left-hand side of (10.9.6.1) holds for
the exactness of (10.3.1.4) then there are h > 0 and k > 0 for which, for
arbitrary z £ Z and y GY, there are yf GY and x' £ X such that
S 0] \y'
-U T
<h
R 0
V S
with
y]
<k
(10.9.6.4)
Taking the cartesian product norm to be the maximum of the norms on
the factors, and specializing in turn to y = 0 and to z = 0,
||s-Sy'|| < max{\\z-Sy'\\,\\Tx'-Uy'\\) < hmBx{\\Rz\\9\\Vz\\)
withmax(||y/||l||a/||)<*H
and
\\y + Uy' - Tx'\\ < max(||S</'||, \\y + Uy' - Tx'\\) < h\\Sy\\
with maxdli/'lUlx'l!) <A;||y||
(10.9.6.5)
(10.9.6.6)
This proves (10.9.6.1) for exactness, and similarly for almost exactness.
Finally, if
~R 0"
-i
0'
.0
c
" S 0'
-u r.
Y~
then
and
jR_1(o) n v-^o) c 5(^_1(rx)) c s{Y)
S~x(0) C T{X) + ^(5_1(0)) C T{X) + U{Y)
(10.9.6.7)
(10.9.6.8)
(10.9.6.9)
This proves (10.9.6.1) for linear exactness, and similarly for almost linear
exactness. ■
If, in particular,
RS = 0 = ST and VS = SU
then there are well-defined operators
colA(jR, V) : Z/c\(SY) —-> W x Z/c\(SY)
(10.9.6.10)
(10.9.6.11)

456
10. Comparison of Operators and Exactness
and
rowA(T,U) : X x 5_1(0) —-> S_1(0) (10.9.6.12)
Then (10.9.6.5) and (10.9.6.8) say that colA(jR,V) is bounded below and
one-one, respectively, while (10.9.6.6) and (10.9.6.9) say that rowA(T,J7) is
open and onto, respectively.

11
Multiparameter Spectral Theory
The various kinds of "spectra" of a normed algebra element have extensions
to n-tuples, and more general systems, of elements: for systems of linear
operators there is a more subtle concept of joint spectrum available.
11.1 LEFT AND RIGHT SPECTRA
If a = (al9 a2,..., an) £ An is an ra-tuple of elements in a normed algebra
A we shall write
n n
row(Lj : b —► ]T ay6y and row(jR0) : b —► ]T 6yay (11.1.0.1)
y=i y=i
for the "row" operators induced from An to A, and
col(L0) : c —> (axc, a2c,..., anc)
and (11.1.0.2)
col(jRj :c—► (ca^caj^.^caj
for the "column" operators induced from A to An.
11.1.1 DEFINITION If a e An in a normed algebra A, then the /e/*
spectrum of a with respect to A is the set
o*J*{a)= «eC»:lj!£x(«,.-,,.)
I ,tl J (11.1.1.1)
= {s 6 Cn:row(2E0_.) e rri*ht BL^",^)}
457

458
11. Multiparameter Spectral Theory
and the right spectrum is the set
oTleht{a)=\SeCn:lzJ2(a}.-Si)A\
{ ,=i J (11.1.1.2)
= {se Cn:row(L„_,) e r"*ht BL(An, A)}
The almost left spectrum of a with respect to A is the set
ar(a) = |«eC":l*cl£>(o,.-«,.)|
I i=i J (11.1.1.3)
= {« 6 Cn:row(i?a_3) e ?right BL{An, A)}
and the almost right spectrum is the set
af (a)= 5GCn^clEK--^
[ y=i J (11.1.1.4)
= {se Cn:row(La_J £ fright BL(An,A)}
Theorem 3.10.4 extends to ra-tuples a £ An, so that the row
operator #a_5 is onto iff it is open, and is dense iff it is almost
open: this has been incorporated in Definition 11.1.1. It is clear
that the right and almost right spectrum of a £ An are the same
as the left and the almost left spectrum of a with respect to the
algebra derived from A by "reversal of products." We make
similar extension to ra-tuples of eigenvalues and approximate
eigenvalues:
11.1.2 DEFINITION If a £ An in a normed algebra A, then the left
approximate eigenvalues of a with respect to A form the set
f^\a) = \seCn: mf T\\(aj-Sj)x\\=o\
{ M^ J (11.1.2.1)
-left i
= {sE Cn:co\(La_s) 6 fIrftBL(A», A)}

11.1 Left and Right Spectra
459
and the right approximate eigenvalues form the set
f^(a) = ls€Cn: inf x £ ||x(a,- - *,-)ll = 0
y=i ) (11.1.2.2)
= {se Cn:col(Ra_s) 6 fMtBL(An,A)}
The left eigenvalues of a with respect to A form the set
O) = 5 e C": f] (La, - ,yI)-i(0) ^ {0}
I ;=i J (n.i.2.3)
= {5 e C»:col(L0_.) 6 *uitBL(An,A)}
and the rtyfa eigenvalues form the set
^ghtW = {*€ C»: H (*., - V)_1(°) * W [
I y=i J (n.i.2.4)
= {s 6 Cn:col(J?0_s) e ?rleftBL(^n,A)}
We have the obvious inclusions:
11.1.3 THEOREM If a e An for a normed algebra A, then
*!?» £ f^W C 5f(«) C a™(a) (11.1.3.1)
and
right.
-right
w»«"(a) c f?<"(a) c ^rw c ,r»
right /
(11.1.3.2)
Proof: Only the middle inclusion needs any attention, and (11.1.3.2) will
follow from (11.1.3.1) by "reversal of products." By considering ra-tuples
a — s £ An it is sufficient to test membership of s = 0 £ Cn in each set.
Thus, if 0 £ <7l£{t{a) there is (6m) in An for which £y=1 ||6mj-ay - 1|| -> 0
with supmmaxy ||6my|| = k1 < 00, and hence if 0 < 6 < 1 we have for
arbitrary x £ A
x <
1-Y^bmiaj\X
+ supmax||6my||^||o,-x||
m }
y=i
(11.1.3.3)
<*MI+*'£>y*||

460
11. Multiparameter Spectral Theory
as soon as ||1 — X)y=i &my ay II < ^> giving for each x E A
||s|| < k JT \\ajX\\ with k = ^ (11.1.3.4)
y=i
We are of course following the argument of Theorem 3.10.5. This proves
(11.1.3.1), and similarly (11.1.3.2). ■
Joint spectra behave under homomorphism in the same way as ordinary
spectra:
11.1.4 THEOREM If T : A —► B is a homomorphism of normed algebras
and aE An, then
<7Bft(Ta) C alX{t{a) and a£ght(Ta) C a*ght(a) (11.1.4.1)
ETG HBL(A, B) is a continuous homomorphism, then also
a^{t{Ta) C alX{t{a) and agght(Ta) C a^ght(a) (11.1.4.2)
If instead T is one-one, then in the other direction,
^W £ TBft(ro) and ^ght(a) C 7r£ght(Ta) (11.1.4.3)
If T is bounded below and continuous, then also
rlXU{a) C fjg^Ta) and f*ght(a) C r*eht{Ta) (11.1.4.4)
Proof: We consider only the "left" inclusions, leaving "right" inclusions
to reversal of products, and test only membership of 6 = 0 E Cn. Ifa =
(ax, a2,..., an]) e An then of course Ta = [Tax, Ta2,..., Tan) e Bn. Thus,
suppose for (11.1.4.1), that 0 £ <jj|ft(a), so that 1 = £y=1 6yay with 6 £ An,
giving 1 = Y%=i T{bj)T{a>j) € B, which means that 0 £ aJfft(Ta). If, more
generally, 0 £ a^ft(a), with || £y=1 6myay-l|| -> 0 and supm maxy ||6my|| <
oo, then provided T is bounded we have || Ey=i T(&my)r(ay) ~ 1|| -+ °>
giving 0 £ ofjgftpra), and proving (11.1.4.2). If 0 G ^{a) and if ajx = ° ^
x then, provided T is one-one, T(aj)T(x) = 0 ^ Tx, giving 0 £ 7r^ft(Ta)
and proving (11.1.4.3). Finally, if 0 £ ^(Ta) and if T is bounded and
bounded below, then for arbitrary iGAwe argue
INI < *||T*|| < khJ2 ll(Tay)(T*)|| = khJ2\\T[a,-x)\\ < kh\\T\\ £ H*ll
y=i y=i y=i
giving 0 £ f!|ft(a) and proving (11.1.4.4). ■

11.2 Polynomials
461
11.2 POLYNOMIALS
To generalize to ra-tuples the spectral mapping theorem for polynomials,
we begin by looking at "noncommutative" polynomials. As a matter of
notation we shall write
Polyn = Alg{zuz2,...,zn) (11.2.0.1)
for the free algebra with identity generated by n independent elements
zl9z2, • • • ,zn. As usual we feel free to interpret the complex field C as a
subalgebra of Polyn, the constant polynomials. The remainder theorem
holds, and is proved by induction:
11.2.1 THEOREM If / <E Polyn and 5 <E Cn there are /', /" <E (Polyn)n
for which
/ - /W = £ fr (*/ " 5>) = D*y" 5y)'" t11-2-1-1)
y=i j=i
Proof: If either / = c0 is a constant or / = z- is a coordinate, then
(11.2.1.1) is clear. We claim that (11.2.1.1) holds if either / = g + h or
/ = g • h for polynomials g and h which satisfy (11.2.1.1). Certainly, if
/ = g + h we can satisfy (11.2.1.1) by taking f' = g' + ti and /" = g" + h"',
where g', g" and h', h" come from (11.2.1.1) for g and h, respectively: to
see how to extend (11.2.1.1) to a product / = g • h observe
g- h - g{s)h{s) = h(s){g - g(s)) + g{h - h{s)) = [g - g{s))h +{h- h{s))g{s)
noting that the scalars g(s) and h(s) commute with h — h(s) and g — g{s)
respectively. We can now leave it to the reader to write down /' and /" in
terms of g', h', g", and h". ■
If A is a linear algebra and / £ Polynj ^nen we write / : An —►
A for the obvious mapping. Formally it is defined inductively, setting
f(a) =seC if / = 6 G C (11.2.1.2)
f(a) = ajeA if / = Zj <E Polyn (11.2.1.3)
f(a) = g{a) + h(a) if / = g + /i (11.2.1.4)
/(a) = ^(a) . /i(a) if / = g • h (11.2.1.5)
If, for example, A = C is the complex field then the induced mapping /
is just a "polynomial in n complex variables." This in turn determines

462
11. Multiparameter Spectral Theory
/ : An —► A if A is commutative. More generally, if / = (/l9 /2,..., /m) £
Poly™ = (PolyJm we shall write f : An -+ Am for the mapping a ->
(/1(a),/2(a),...,/m(a)), in which each /^(a) is defined inductively as
above.
With these interpretations, we immediately obtain about half of the
spectral mapping theorem:
11.2.2 THEOREM If a £ A2 for a normed algebra A and / £ Poly™,
then
/aJf(a)CaJf/M and /^ght(a) C <£*h7(a) (11.2.2.1)
/^(aJCal^/H and /^ght(a) C a*ght/(a) (11.2.2.2)
/ffWCfr/W and /ff'(a) £ ?f V(«) (11.2.2.3)
Meft(a)C^ft/W and KJght(a) C ^ght/(a) (11.2.2.4)
Proof: Suppose 5 G Cn is such that f(s) is not in <7^ft/(a): then by the
remainder theorem, applied to each component fk, and the definition of
left spectrum, we have
m m n
i = £ MAM - AW) = Eb*E A»K - «,•)
fc=l fc=l j'=l
7 = 1 \A:=1 /
y=l \fc=l
which means that 5 is not in the left spectrum of a. This proves the first
part of (11.2.2.1), and the second follows by "reversal of products." The
argument for (11.2.2.2) is very similar, but messier: we leave it to the
reader. If 5 £ Cn is in 7r^ft(a) then there is x £ A with x ^ 0 = (a;- — Sj)x
for each j, giving
n
(/*(*) - fM)x = Y, f'kj(a)(aj - sj)x = ° for each k
3 = 1
so that f(s) is in 7r^ft/(a). This proves (11.2.2.4): we leave (11.2.2.3) to
the reader. ■
The almost left and right spectra are closed and bounded, and upper
semicont inuous:

11.2 Polynomials
463
11.2.3 THEOREM If a <E An then for each u <E {a^ft,a
left -right -left -right
A '"A
oj(a) = clo;(a) C {s £ Cn: \sA < \\a-\\ for each j}
If U C Cn is a neighborhood of u>{a) there is 8 > 0 for which
(11.2.3.1)
• u(a') C U
Proof: As a special case of Theorem 11.2.2 there is inclusion
oj(a) C (jj{ax) x oj(a2) x • • • x c<;(an)
(11.2.3.2)
(11.2.3.3)
Simply take / = z- for each j. An application of the first part of
Theorem 9.2.3 now proves the second part of (11.2.3.1) for each of aj|ft and
<5^ght, and hence also for f^ft and f^ght. For the first part of (11.2.3.1) we
need only observe that in each case uj(a) is the continuous counterimage of a
subset of BL{An,A) or BL(A, An) which is closed, either by (3.4.3.1) or by
(3.3.3.1). The same argument gives the upper semicontinuity (11.2.3.2). ■
Unlike the single element case Theorem 9.2.3, joint spectra are liable
to be empty. If, for example, A = BL(X,X) with X = C2 and a = (a1,a2)
with
a, =
0 1
0 0
0 0
1 0
(11.2.3.4)
then it follows from (11.2.3.3) that a||ft(a) and <j^lght(a) are contained in
the set {(0,0)}, while the fact that
2 , 2
1 0
0 1
(11.2.3.5)
excludes this point. This same example shows that we cannot expect
Theorem 9.2.4 to extend to arbitrary ra-tuples: indeed, if / = z\ + z\ and
a = (a1,a2) is as in (11.2.3.4) then for all possible w we have
0 = /(0) = fu(a) ± uf{a) = u(l) = {1}
Alternatively, whenever u(a) =0 take / to be a constant.
The one-way spectral mapping theorem extends to noncommutative
rational functions, interpreted as products of polynomials and their
reciprocals. If, for example, / £ Polyn is a polynomial and A is an algebra we
define 1// on part of An by setting
DomainA(l//) =/"1(r1)an
(11.2.3.6)

464 11. Multiparameter Spectral Theory
and
(l//)(o) = /(a)"1 if o € rl{A~l) (11.2.3.7)
11.2.4 THEOREM If A is a normed algebra, if a 6 An and / 6 Polyn,
then for each u, 6 {o^o^^o^o^^f™,^*™,^}
a € rl[A~l) =► w(») C C" n /-1(A_1) and (l//)w(o) C w(/(a)_1)
(11.2.4.1)
Proof: We have first a "remainder theorem": if a S An and s£Cn are
both in /_1(A-1) then
/(a)"1 - /(s)-1 = /(a)"1/(a)"1 (/(a) - /(,)) (11.2.4.2)
Adding in the remainder theorem for /, it follows as in Theorem 11.2.2 that
(l//)Ma) n /-'(A-1)) C ^(/(a)-1) (11.2.4.3)
It remains only to be shown that
w
(aJC/-1^"1) (11.2.4.4)
It is sufficient to show this for w = a^ft U aAlg : but this follows from
(11.2.2.1) and the assumption that /(a) £ A~l:
a £ /-'(A-1) => 0 £ oAf(a) D f(o^(a) U a?ht(a))
so that if 5 £ <^ft(a) u ^Aght(a) then /(5) is invertible. ■
Equality does hold in the one-way spectral mapping theorem for certain
special polynomials or rational functions:
11.2.5 THEOREM If a £ An and if / £ Poly™ satisfies either
m = n and gof = z = fogfoi some g £ Poly^ (11.2.5.1)
or
m > n and / = (z,$) for some g £ Poly™~n (11.2.5.2)
then
/w(a) =w/(a) (11.2.5.3)
for each a, £ {a™,o**\a™,a?ht,f^,f^^^A18"}-

11.2 Polynomials
465
Proof: For (11.2.5.1) we need only apply Theorem 11.2.2 twice:
"/(a) = /(*("/(a))) C fojg(f(a)) = fu[a) C uf(a)
Towards (11.2.5.2), observe that if a £ An and b £ Am are arbitrary then
w(a,6) C a;(a) x u(b) (11.2.5.4)
by two applications of Theorem 11.2.2 to the (m + 7i)-tuple (a, 6). Putting
m — 7i in place of m and g(a) in place of 6 in this and applying Theorem
11.2.2 again gives
{(5>0(5)):5 £ v(a)} C ct;(a,y(a)) C u;(a) x W0(a)
To complete the proof we must verify that
(M) <E w(a,y(a)) =► t = g{s) (11.2.5.5)
This also follows from Theorem 11.2.2: define h <E Poly™~n by setting
h(z,w) = w — g(z) and observe
t-g(s) = h{s,t) e hoj{a,g{a)) = w(0) = {0} ■
We shall refer to (11.2.5.4) as the subprojective property:
11.2.6 DEFINITION If w is a subprojective system of mappings from
An into subsets of Cn we shall write
w6=t(a) ={se Cn: (s, t) e a; (a, b)} (11.2.6.1)
and
ub=b(a) = (J ub=t(a) (11.2.6.2)
tecm
For example, if a;(a) is the range of a mapping a : Q —> Cn, then o;6_t(a)
is the range of its restriction to the level surface b~x(t) C H. The
"relative" spectrum ub=b(a) measures the failure of equality (11.2.5.3): for if w
satisfies the "one-way" Theorem 11.2.2 then (11.2.5.5) gives
"a=af{") = M<0 (11.2.6.3)
and also
WM ="/(a)=/(a)(a)
(11.2.6.4)

466
11. Multiparameter Spectral Theory
11.3 THE SPECTRAL MAPPING THEOREM
For commuting systems of elements, equality holds in Theorem 11.2.2 for
the almost spectrum and the approximate eigenvalues. Not surprisingly,
we need Liouville's Theorem 9.1.3. We begin with some very simple
observations:
11.3.1 THEOREM If U <E BL(X,X), T e BL{X,Y), and V <E BL(y,y)
are bounded between normed spaces then there is implication
col(T, U) one-one => U one-one on T-1(0) (11.3.1.1)
col(T, U) bounded below =>• U bounded below on T_1(0) (11.3.1.2)
col(T, U) closed =^ U closed on T"1 (0) (11.3.1.3)
row(T, V) dense =^ V dense to Y/ cl{TX) (11.3.1.4)
row(T, V) onto =J> V onto Y/ c\{TX) (11.3.1.5)
row(r,y) almost open =>• V almost open to Y/c\{TX) (11.3.1.6)
row(T, V) open =^ V open to Yj cl(TX) (11.3.1.7)
Proof: If col(T, U) is one-one from X to Y x X then for arbitrary x £ X
there is implication Tx = Ux = 0 => x = 0. Restricted to points x £ T_1(0)
this gives implication Ux = 0 => x = 0, which is (11.3.1.1). If col(T, U) is
bounded below on X then there are k,h > 0 for which, for each x £ X,
\\x\\ < k\\Tx\\+h\\Ux\\: restricted to T_1(0) this gives ||x|| < h\\Ux\\, which
is (11.3.1.2). If in addition there is implication ||Txn—2/|| + ||lfxn—z\\ —> 0 =>
{y>z) £ {(Tx, 17x):x £ X}, then restricted to T_1(0) there is implication
||[/xn-z|| -+0=> ze {Ux:x e T-^O)}, which is (11.3.1.3). If row(r,V) is
onto Y then it is clear that (the mapping induced by) V is onto Y/ cl(TX),
as required by (11.3.1.5): we leave (11.3.1.4), (11.3.1.6) and (11.3.1.7) to
the reader. ■
We combine the elementary observations of Theorem 11.3.1 with
Liouville's Theorem 9.2.3:
11.3.2 THEOREM If U <E BL(X,X), T <E BL(X,y), and V <E BL(y,y)
satisfy
^(r_1(o)) cr^o) (n.3.2.1)
and
V{T{X)) C cl(T(X)) (11.3.2.2)

11.3 The Spectral Mapping Theorem
467
then there is implication
T e 7rleft BL(X, Y) =* col(T, U-sI)e fleft BL(X, XxY) for some s <E C
(11.3.2.3)
and
TG7rrightBL(X,y)
(11.3.2.4)
=* row(T, V -sl)e fnght BL(X x Y, Y) for some s <E C
Proof: If T G 7rleft BL(X,y) so that (3.2.1.4) r_1(0) ^ {0} then
Theorem 9.2.3 says that the operator U : r_1(0) -+ r_1(0) has bounded
nonempty "almost spectrum" relative to the algebra BL(T~1(0),T~1(0)),
which therefore by (7.10.3.8) has nonempty topological boundary, which by
(9.3.3.1) lies in the set of approximate left eigenvalues of U. By (5.6.3.4) this
means that there is 5 £ C for which the operator U—sI is not bounded below
on r_1(0), and by (11.3.1.2) this means that col(T, U - si) is not bounded
below on X. This proves (11.3.2.1), and the argument for (11.3.2.2)
is very similar: if T G 7rrisht BL(X,Y) so that (3.2.1.5) cl(TX) ^ Y
then Theorem 9.2.3 says that the operator V : Y/c\(TX) -* y/cl(TX)
has bounded nonempty almost spectrum, which again by (7.10.3.8) has
nonempty topological boundary, which by (9.3.3.1) lies among the right
approximate eigenvalues of V on y/cl(TX). By (5.6.3.2) this means
that there is 5 G C for which V is not almost open on Y/c\(TX),
and by (11.3.1.6) this means that row(T, V — si) is not almost open to
y. ■
Note that (11.3.2.1) says that the operator TU is determined by T in
the sense of (10.1.1.5), while (11.3.2.2) says that the operator VT is almost
range-included in T in the sense of (10.1.3.7). Theorem 11.3.2 only begins
to bite when it is combined with the enlargement process:
11.3.3 THEOREM If U e BL(X,X), T e BL(X,y), and V <E BL(y,y)
satisfy
TU majorized by T (11.3.3.1)
and
VT almost comajorized by T (11.3.3.2)
then there is implication
T e fleft BL(X, y) =J> col(T, U-sI)e fleft BL(X, Y x X) for some seC
(11.3.3.3)

468
11. Multiparameter Spectral Theory
and
refrightBL(x,y)
. v (11.3.3.4)
=> row(T, V - si) £ f nght BL(X x Y, Y) for some seC
Proof: If (11.3.3.1) holds then Theorem 10.5.1 (and Theorem 10.1.2) says
that (11.3.2.1) holds with (Q(T),Q(17)) in place of (T,17), while if the
first part of (11.3.3.3) holds then Theorem 3.3.5 says that the first part
of (11.3.2.3) holds with Q(T) in place of T. By (11.3.2.3) there is 5 G C
for which col(Q(T), Q(U) - sQ(I)) is not bounded below, and by
Theorem 3.3.5 this means that (T, U — si) is not bounded below. This proves
(11.3.3.3), and the argument for (11.3.3.4) is very similar: if (11.3.3.2) holds,
then Theorem 10.5.2 (and Theorem 10.1.4) says that (11.3.2.2) holds with
(Q(T),Q(V)) in place of (T, V), while if the first part of (11.3.3.4) holds,
then Theorem 5.7.1 says that the first part of (11.3.2.4) holds with Q(T) in
place of T. By (11.3.2.4) there is 5 £ C for which row(Q(r),Q(y) -sQ(J))
is not almost open, and by Theorem 5.7.1, or alternatively Theorem 3.4.5,
this means that row(T, V — si) is not almost open. ■
11.3.4 THEOREM If a £ An is commutative and commutes with b £
Am, then for each w £ {a^ft,a^ght,f^ft,f^ght}, there is equality
"(*)="a=a(*) (11.3.4.1)
and hence also
w/(a) = /a; (a) (11.3.4.2)
if a £ An is commutative and / £ Poly™.
Proof: Equality (11.3.4.2) follows at once from (11.3.4.1) if we put b =
f(a) and use (11.2.6.3). It is sufficient, for (11.3.4.1), by induction on
n, to consider only the case of a single element a = ax commuting with
each element bk, and of course by "reversal of products" we need only
consider the almost left spectrum and the left approximate eigenvalues. To
establish (11.3.4.1) with n = 1 and w = aj|ft we use the second part of the
elementary Theorem 11.3.2: if t £ Cm is in a^ft(6) then T = iov/(Rb_t) £
7rrightBL(X,y) with X = Am and Y = A, remembering (11.1.1.3) from
Definition 11.1.1 and Theorem 3.10.4. If a = ax commutes with each bk
then V = Ra satisfies the condition (11.3.2.2). By (11.3.2.4) there is 5 £ C
for which (s,t) £ aj|ft(a, 6) = w(a, b) giving (11.3.4.1) in this case.
To derive (11.3.4.2) for left approximate eigenvalues needs the
enlargement process, funnelled here through Theorem 11.3.3. If££Cmin ^I|ft(6),
then by (11.1.2.1) col(L6_t) is in fleft BL(A, Am), and hence by (11.3.2.3)

11.3 The Spectral Mapping Theorem
469
there is s £ C for which col(L6_t,L0_a) is in fleft BL(A, Am+1), which
means by (11.1.2.1) that (s,t) is in fj|ft(a,6). ■
For an alternative derivation of (11.3.4.1) for the almost left spectrum
suppose t £ £j|ft(6) and write
m
N = cl^Mh~h) (11.3.4.3)
k=i
for the induced closed left ideal of A, together with
M = {c £ A: Nc C N} (11.3.4.4)
Then M is a closed subalgebra of A, containing the identity 1 and the ideal
iV, and N is actually a two-sided ideal of M:
NM C M (11.3.4.5)
Now if c £ M we claim, since N ^ M,
0 * d*M/N(c + n)Q f™}„{c + AT) C (<?!f)6=t(<) (11.3.4.6)
If, for example, c'{c — s) + X^fcLi ^5c(^A: — *fc) = *> then x = c'(c — 5)x +
Y^k=i ^kiPk — tk)x £ c'{c — s)x + N for arbitrary x £ M giving \\x + iV|| <
||c'j|||(c — s)x + N\\. If we assume only (s,t) £ aj|ft(c,6) the details are
messier, and left to the reader.
It is clear from the proof of Theorem 11.3.4 that equality (11.3.4.2)
can be attained for certain noncommutative systems a £ An. If n = 1
then it is sufficient for (11.3.4.1) with u = aleft that ax be in M for each
t £ Cm, where M is the algebra of (11.3.4.4); for more general n it is then
sufficient to impose in succession the corresponding conditions on each a -+1
with (6, al9..., a;) in place of 6. If a £ An and 6 £ Am are arbitrary and
£ £ Poly£+m then Theorem 11.2.2 says that for eight different w there is
inclusion
gu{a, b) C ug{a, b) (11.3.4.7)
while if (a, 6) £ An+m is commutative then Theorem 11.3.4 say that for
four different u>
ug(a,b) = gu(a,b) C g(u(a),u(b)) = g{u{a) x v{b)) (11.3.4.8)
Here we are writing u>(K,H) = u>(K x H): we shall also write
ug{K,b) = (J $(«,&) and ug(a,H) = (J w^(a,*) (11.3.4.9)

470
11. Multiparameter Spectral Theory
With this notation we can state and prove a result intermediate between
(11.3.4.7) and (11.3.4.8):
11.3.5 THEOREM Suppose a £ An, g £ Poly£+m and u £ {a^ft,
^ng t^ieft^ng ty, ^^ ^ a jg commutative and commutes with b there
is equality
ua=3g{a, b) = ua=3g{s,b) ^ each 5 € Cn (11.3.5.1)
and
ug{a,b) =va=ag{a,b) = (J wfl=^(6,6)Cwff(w(a),6) (11.3.5.2)
3Gw(a)
If instead 6 is commutative and commutes with a there is equality
Ub=t9ia>b) = Vb=t9{a> *) for each t e c™ (11.3.5.3)
and
ug{a,b) =ub=bg{a,b) = (J u>h=tg(a,t) C wy(a,w(6)) (11.3.5.4)
tGw(6)
Proof: If a £ An commutes with 6 £ Am then for each s £ Cn and r £ Cp
the (p + ra)-tuples
(<7(a, 6) — r, a — s) and (0(5,6) — r, a — s) (11.3.5.5)
generate the same left and right ideals in A, which establishes (11.3.5.1)
for the left and the right spectrum and hence also, taking closures, for the
almost left and the almost right spectrum. Passing to the "annihilators" of
these ideals gives also (11.3.5.1) for the left and the right point spectrum,
and hence, considering enlargements, for the left and right approximate
point spectrum. This now gives the second equality of (11.3.5.2); for the
first we must apply (11.3.4.1) with g(a,b) in place of 6. For (11.3.5.3) and
(11.3.5.4) we just interchange a and b in what just proved. ■
11.4 MANY VARIABLES
The spectra of Definition 11.1.1 and Definition 11.1.2 have extensions to
more general systems of elements:
11.4.1 DEFINITION If A is a normed algebra and a = (ay)jGJ £ A3 is
a mapping from the nonempty set J into A then for each
#. a /^-left „.right ;;left fright -left -right left right \ ill A 1 l\
Uje\aA i°A i°A *°A >TA >TA >*A >*A J (11.4.1.1)

11.4 Many Variables
471
we define
oj{a) = js G CJ: (sy)jGJ, G o((ay)yGJ,) for each finite J' C j} (11.4.1.2)
In each case the resulting set uj(a) can be represented by a definition
formally very like the definition for ra-tuples. The extension of Theorem 11.1.3,
Theorem 11.1.4, and Theorem 11.2.2 is mechanical:
11.4.2 THEOREM If a G AJ and u is in (11.3.1.1), then
oj{a) o(j>C u{a o <f>) (11.4.2.1)
for arbitrary mappings <f> : K —> J. There is inclusion
*r (a) C f^a) C a™(a) C a™ (a) (11.4.2.2)
and
**"(a) C f*ht(a) C a*"(a) C a^ht(a) (11.4.2.3)
If T G HBL(A, B) is a homomorphism, then
uB{Ta) C wA(a) for each a; G {aleft,aright,aleft,aright} (11.4.2.4)
If T is one-one, then also
^ W £ ^ft(Ta) and ^lght(a) C 7r£ght(Ta) (11.4.2.5)
if T is bounded below, then also
rlXft{a) C f^Ta) and f*ght(a) C T*ght(Ta) (11.4.2.6)
If / G Polyj is a system of polynomials then there is inclusion
fu(a) C w/(a) (11.4.2.7)
for each a; in (11.4.1.1).
Proof: This can be left to the reader. Note that each "polynomial" fk G
Polyj is in the algebra generated by the coordinates z^(j G J). ■
The almost left and right spectrum, and the left and right approximate
eigenvalues, of a G AJ form compact, but possibly empty, subsets of the
cartesian product space C . When J = H is either a topological space or
an algebraic system, and a G AJ is either continuous, or a homomorphism,
then this is inherited by every 5 G CJ in the left or right spectrum of a:

472
11. Multiparameter Spectral Theory
11.4.3 THEOREM If H is either a topological space, or a semigroup, or
a linear space, and if a £ An is either bounded, or continuous, or homo-
morphic, or linear, then so is each 5 G Cn in ^Aft(a) u °Ag (a)* ^ m
particular
n = A and at = t for each t £ ft (11.4.3.1)
then
alX{t{a) = a^ght(a) = A* = HBL(A, C) (11.4.3.2)
Proof: If £, £', and £" are in H and / £ Poly3 then st is in the spectrum of
at, st, —st is in the spectrum of at, —at and f{st,st,,st,,) is in the spectrum
of /(at,at/,at//), which means that
\st\< \\at\\ and |st, "5tl< ||at, - at\\
and (11.4.3.3)
|/(«t,«t/,«t//)|< ||/(at,at,,at„)||
The first of these ensures that 5 is bounded with a, the second that 5
is continuous with a, and the third that 5 satisfies any of the obvious
homomorphic or linearity properties of a. In particular if a is given by
(11.4.3.1) then this ensures that the left and the right spectrum of a are
both included in the Gelfand space A* of A: conversely (11.4.2.4) ensures
that the Gelfand space of A lies in both the left and the right spectrum
of a. ■
If B C A is a subset and at = t for each t £ B, then we are tempted
to write
w(a) = u{B) (11.4.3.4)
for each of the u> of Definition 11.4.1: when B = A this conflicts with some
of our earlier notations. Thus we shall write in the situation of (11.4.3.1)
w(a) = a; (A) = A* (11.4.3.5)
Commuting systems of elements a £ A3 have the "spectral mapping
property," in the sense that for arbitrary K and arbitrary / £ Polyj
/w(a) = w/(a) (11.4.3.6)
for the almost left and right spectrum w, and for the left and right
approximate eigenvalues w:
11.4.4 THEOREM If a £ AJ is commutative and commutes with 6 £

11.4 Many Variables
473
AK, then for each w 6 {af.^.ff.f J1*}
«(*)=««=«(*) (11.4.4.1)
In particular, if a £ A*7 is commutative and / £ Poly^f
oj(f(a)) = fu[a)) (11.4.4.2)
Proof: This is left to the reader. ■
The reader should also be able to write out an extended version of
Theorem 11.3.5. Together with (11.4.3.2), Theorem 11.4.4 gives an alternative
derivation of Gelfand's Theorem 9.6.3:
11.4.5 THEOREM If A is a Banach algebra the following are equivalent:
A has the left spectral mapping property (11.4.5.1)
A has the right spectral mapping property (11.4.5.2)
A/Radical(A) is commutative (11.4.5.3)
Proof: If a £ AJ is arbitrary, then (7.2.5.4) and Theorem 7.2.6 say that
^eftW=^/tRad(A)(« + Rad(A))
and (11.4.5.4)
^8ht(«)=^^d(A)(« + Rad(A))
and then if (11.4.5.3) holds Theorem 11.4.4 gives the spectral mapping
property for the system a + Rad(A) £ (A/Rad(A))J. This means that
a £ A3 has the spectral mapping property, in particular, if it is the system
given by (11.4.3.1). This means that both of the conditions (11.4.5.1) and
(11.4.5.2) follow from (11.4.5.3). Conversely, if say (11.4.5.1) holds then for
arbitrary a, 6, c £ A we have
<j^eft(c(6a - aft)) = {0} (11.4.5.5)
and hence, recalling the connected hull (9.3.3.8) of Definition 7.10.1,
aA{c{ba - ab)) C 7yaleft((c(6a - ab)) = r\ {0} = {0} (11.4.5.6)
Thus, if a, b £ A then
c £ A =J> 1 - c{ba - ab) £ A'1 (11.4.5.7)

474
11. Multiparameter Spectral Theory
By Theorem 7.2.3 and (7.2.3.10) this means that ba - ab £ Radical(A),
which gives the condition (11.4.5.3). If instead we start with the assumption
(11.4.5.2) the argument is the same. ■
We conclude with a converse to the "upper semicontinuity" result
Theorem 9.2.6:
11.4.6 THEOREM If a^ £ A and a £ AN satisfy
llaoo — an\\ —* 0 as n —► oo and {a^} U {an:n £ N} is commutative
(11.4.6.1)
then, for each a, £ {d™,e«*\?f,f«**}9
"(O = { lim sn:s £ c0 n J] "("») \ (11.4.6.2)
I nGN J
Proof: Theorem 9.2.6 says that the right-hand side of (11.4.6.2) is included
in the left. Conversely, if t £ ^(a^) is arbitrary then Theorem 11.4.4 says
that there is a sequence 5 £ C for which
(M)^(a,aJ (11.4.6.3)
By Theorem 11.4.3 it follows that \sn — t\ —> 0 as n —> oo, and by part of
Theorem 11.4.2 we must have sn £ oj(an) for each n £ N. ■
11.5 THE SILOV BOUNDARY
We recall from Theorem 9.1.3 that if / : H —> C is holomorphic on an open
set H containing a disc K C C, then it attains its maximum modulus on
the circle dK:
11.5.1 DEFINITION If E C C(H) is a linear subspace which separates
points of the compact Hausdorff space H, then the closed boundaries of E
in H are the sets
AE(Q) = Ix = cl(lf) C n:sup |/| = sup |/| for each f e e\ (11.5.1.1)
I k n J
The main result about boundaries is easily stated:
11.5.2 THEOREM If E C C(H) is a linear subspace of C(H) which
separates points of the compact Hausdorff space H, then
nAs(n) £/>F(n)
(11.5.2.1)

11.5 The Silov Boundary
475
Proof: We actually prove that
nAs(n) 2 dE(n) e AE(n) (11.5.2.2)
where
dE(n) = {t £ H: for each U £ Nbd(*)
there is K £ A^H) with K\U £ A^H)}
For the second part of this suppose that W = int(PV) is an arbitrary open
neighborhood of dE(Q), so that for each t £ H \ W there is Ut £ Nbd(£) for
which
AE{n)\utcAE{n) (11.5.2.4)
in the sense that K \ Ut is a boundary whenever K is a boundary: then
by compactness there is a finite subset H C H \ PV for which n \ W C
utGH^t = UyLify- But now
m
n\71GAB(n)=>(n\71)\72GAB(n)=>---=>n\|j7yGAB(n)
and hence
m
ten y=i
Since this holds for arbitrary neighborhoods W of d#(n) it follows that
dE(Q) is also a closed boundary, as required by the second part of (11.5.2.2).
For the first part of (11.5.2.2) suppose that t £ n is not in the intersection
flA£7(n), so that there is Kt £ A^ft) for which t g Kt, and then for each
point 5 £ Kt a function gs £ E for which gs(t) = 0 and gs(s) = 2. By
compactness there is a finite set K[ C Kt for which Kt C UsGKi {\gs\ > 1}:
now we claim that if we put
Ut = H {1^1 < !> (11.5.2.5)
then (11.5.2.4) will be satisfied. Certainly Ut £ Nbd(*) and
17t n ift = 0 (11.5.2.6)
If (11.5.2.4) is not satisfied there will be K £ A^H) for which K\Ut <£
A£7(H), and then / £ E for which sup^^ |/| < 1 = supn |/|. Now for

476
11. Multiparameter Spectral Theory
sufficiently large nGNwe have
maxsup |/n<75| < sup |/| maxsup|<75| < 1 (11.5.2.7)
sGK't q \K\Ut J sGK't q
Since Kt G A^H) there is t' e Kt for which \f{t')\ = 1, which with
(11.5.2.7) means that \g3{t')\ < 1 for each 5 £ K'v Thus>
t'eKtn ut (n.5.2.8)
The contradiction between (11.5.2.8) and (11.5.2.6) flows from the attempt
to deny (11.5.2.4), which is therefore proved. This finishes (11.5.2.2), which
gives (11.5.2.1). ■
From (11.5.2.2) it is clear that the set dE{Q) must actually equal the
intersection C\AE{Q), and now we can also simplify (11.5.2.3):
dE(Q) = {teQ: for each U <E Nbd(*), H \ U <£ AE{Q)} (11.5.2.9)
Certainly the right-hand side of (11.5.2.9) is a subset of the left: for the
opposite inclusion use (11.5.2.1) to see that the intersection of two boundaries
is always a boundary, which gives implication
n \ u c A^(n) =j> As(n) \ u c A^(n) (11.5.2.10)
When H = A* is the maximal ideal space of a commutative Banach
algebra A and E = AA is the image of A under the Gelfand transform then
the Silov boundary looks rather like the approximate eigenvalues:
11.5.3 DEFINITION If A is a commutative Banach algebra and a £ AJ
then the Silov boundary of a is the set
&a(°) = \ s £ CJ: inf > \(a- — sAu\„ = 0 for each finite Jr C J }
\ '"'-^ife ' "J
(11.5.3.1)
In particular, the Silov boundary of A is the set
8(A) = \<t> e Ax: inf ]T \au\a = 0 for each finite K C ^_1(0) >
(11.5.3.2)

11.5 The Silov Boundary
477
As we are suggesting, the Silov boundary of A coincides with the set
(11.5.2.3):
11.5.4 THEOREM If A is a commutative Banach algebra, then the Silov
boundary of A is the intersection of the closed boundaries of AA in A^\
B{A) = 3AA (A*) = nAAA (A*) (11.5.4.1)
Proof: Suppose <j> £ H = A* is in the set dAA (H) and use the formula
(11.5.2.9): then if a £ An satisfies <£(ay) = 0 for each j and if 0 < e < 1
there is U £ Nbd(<£) for which
n
tf Gtf =>£|tf(ay)|<e (11.5.4.2)
y=i
and then by (11.5.2.9) there is u £ A for which
\u\a = 1 and sup \ip(u)\ < ef with Y" \aAa e1 < e (11.5.4.3)
Use the condition that n \ U is not a boundary, normalize to make {u^ = 1
and then raise it to a sufficiently high power: we claim
n n
|ttU = l and ^2\aj-u\a= sup^2\tp(aj-u)\<e (11.5.4.4)
y=i ^Atj=i
for look at hKtt)l£y=i May)l with ^> G 17 and with ^ £ ft \ 17. This
shows that the intersection of the closed boundaries is a subset of the Silov
boundary: conversely, if <j> £ A* and U £ Nbd(<£), then by the definition of
the topology of A* there is s £ ]0,1[ and a £ An for which
I 1> £ A*: X] W*y) " *(ay)| < * > Q U (11.5.4.5)
and if ip £ 5(A) there is by (11.5.3.2) an element u £ A for which lu^ = 1
and Ey=i l(ay — ^(ay))ulo- < is* We cla™ that the right-hand side of
(11.5.2.9) holds with <f> in place of t:
iPeA*\U=^> \ip{u)\ < \ < 1 = \u\a (11.5.4.6)

478 11. Multiparameter Spectral Theory
Indeed, iitp&U then £y=1 |^(ay) - ^(ay)| > e by (11.5.4.5), while
y=i
n n
= £ WK - *(a/))*)l ^ E IK - <t>^M* < is ■
y=i y=i
The Silov boundary of a system of elements is a subset of the
approximate eigenvalues:
11.5.5 THEOREM If a £ An for a commutative Banach algebra A then
^WCr~rW=rTht(a) (11-5.5.1)
Hence also
3(A)Cf™(A)=f***(A) (H-5-5.2)
Proof: If 0 is not in f J|ft (a) then there is k > 0 for which
n
E Ha/ttll ^ *HttH for each u e A (11.5.5.3)
y=i
Consider the algebra
B=\f: £ 6^(1)4(1)-<W:(OinA,f; £ ||6j<ool
(^m=0|r/|=m m=0 |i/|=m J
(11.5.5.4)
normed as indicated, where we write
\i/\ = 1/(1) + i/(2) H \-i/(n) for each multi-index v (11.5.5.5)
Evidently A is a closed subalgebra of B: look at the element
n
c = Eay^ GjB (11.5.5.6)
y=i
and claim that, for each uGA,
||cu|| > fc||u|| and ||cr+1u|| > k\\cru\\ for each r eH (11.5.5.7)

11.5 The Silov Boundary 479
Indeed, writing
a? = a^(1)4(2)- ••<(") (11.5.5.8)
we have, using "multinomial coefficients,"
|i/|=r ^ ' |i/|=r ^ '
and then
||c'+1«|l = II £>,•*,■£ Q°"«*l
y=i |„|=r v y
= EEQ IK(1) • • • <(y)+1 • • • <w«r(1) • • • <w+1 • • • <(n) ii
y=i H=r v y
= e(:)eik^h
\»\=r V y
using (11.5.5.3). From (11.5.5.9) it follows, replacing uby ur, that
||(cu)r|| = ||crur|| > kr\\ur\\ for each u <E A and r <E N (11.5.5.10)
and hence, using Theorem 9.5.2, and (9.5.3.5),
n n
/J l^y^U > | 2_J CLjZjU^ > k\u\a for each uGA (11.5.5.11)
y=i y=i
This proves (11.5.5.1): for (11.5.5.2) we can argue
f Jjft(A) = f^ght(A) e AAA (A*) is a closed boundary for AA (11.5.5.12)
using (9.4.2.1) from Theorem 9.4.2 and (7.10.3.3) from Theorem 7.10.3 to
see that
\c\a = sup \s\ = sup \s\ for each c £ A ■ (11.5.5.13)

480
11. Multiparameter Spectral Theory
If A is commutative and a £ AJ is arbitrary, then there is inclusion
*a(<0 £ fA(a) C f| aB(a) C aA(a) (11.5.5.14)
ACB
For a single element cGA there is equality in the middle of (11.5.5.14):
11.5.6 THEOREM If a £ A for a commutative Banach algebra A, then
*a(*)= fl 'bW (11.5.6.1)
ACB
If jK" C A is a finite subset disjoint from the topological zero divisors, then
there is a commutative Banach algebra B for which
ACB and K C B'1 (11.5.6.2)
Proof: Suppose 0 £ ^(a), so that there is A: > 0 for which
||ac|| > k\\c\\ for each c £ A (11.5.6.3)
If t > 0 we shall write
D{t) = \c0 + Yl cJzJ': cj e A> Hcoll + L Hc; II*"' < °° \ (H-5.6.4)
normed as indicated, with the multiplication one would expect of "power
series with coefficients in A," and then put
B(t) = D{t)/J{t) where J{t) = clD(t)(l - az)J(t) (11.5.6.5)
quotienting out the closed ideal generated by 1 — az. Evidently the mapping
T : b —> b + J(t) from A to B(t) (11.5.6.6)
is a homomorphism, and of course B{t) is a commutative Banach algebra.
Also, the image T{a) = a + J{t) has an inverse in B{t), given by the coset
z + J{t). We claim that for sufficiently large t the embedding T of A in B{t)
is isometric. To see this observe that if b £ A and c0 + Sylj CjZ3 £ £)(£)
then
b-
(l-az)(c0 + J^c3-A
v y=i y
= n + (** -1)
(11.5.6.7)

11.6 Composition Operators
481
This is because
b-{l-az)lc0 + Y, C3zJ 1 = b ~ co + D(acj-i " ci)zJ
and hence
\b — (1 — az
)(co + X/y;
v y=i
ll*-*oll + X^K-i-'yll*y
y=i
oo
> (11*11 - Ikoll) + £(%y-ill - Ml)* = 11*11 + £(*' " ^V
y=i y=i
We have proved (11.5.6.2) for the set K = {a} consisting of a single element:
just ensure that kt > 1 and take B = B(t). For (11.5.6.1) apply what we
have just proved to elements a — s for arbitrary sGC, and for (11.5.6.2)
take a to be the product of all the elements in K. m
The reader should compare this characterization of the topological zero
divisors as "permanently noninvertible" with the characterization of
generalized exponentials obtained in (7.11.5.8).
11.6 COMPOSITION OPERATORS
It is instructive to compare the spectra of a system a £ An of normed
algebra elements with the corresponding spectra of the systems La £ Dn
and Ra £ Dn of left and right multiplications in the algebra D = BL(A, A),
in particular when A is itself an algebra of operators:
11.6.1 THEOREM If A is a normed algebra and a £ An then, with
D = BL{A,A),
and
Also
and
**?(*) = ^(La) and ^ght(a) = *™{Ra)
?!f M = W.) and ?^\a) = f™(Ra)
aF(a)=f**\Ra)=*«*t(Ra)
(11.6.1.1)
(11.6.1.2)
(11.6.1.3)

482
11. Multiparameter Spectral Theory
and
<#"(«) = r^\Ra) and a^\a) = r^(La) (11.6.1.4)
Proof: This is just a restatement of Definition 11.1.2 and Definition 11.1.1:
in (11.6.1.3) we are also using the second part of the remark immediately
following Definition 11.1.1. ■
If we had extended Definition 11.1.2 to establish notation for the case
in which the operators row(La) and row(jRa) were onto, as distinct from
open, then we would have been able to similarly augment (11.6.1.4). We can
also say something about systems (La, Rb) £ Dn+m associated with (a, 6) £
An+m, or more generally about systems (La,Rb) associated with (a,6) £
An x Bm in the situation discussed immediately following Theorem 3.10.4.
If A and B are normed algebras then we shall call the normed space M a
normed (left A,right B)-module if there are mappings
L:A—>BL(M,M) and R:B —> BL(M,M) (11.6.1.5)
linear and of bound < 1, which satisfy, for arbitrary a, a' £ A and 6, bf £ B,
La'a = LafLa, Rb,b = RbRbi, L± = I = R±,
and (11.6.1.6)
LaRb = RbLa
For example, this is the case if X and Y are normed spaces and
A = BL{X,X) B = BL{Y,Y) and M = BL{Y,X) (11.6.1.7)
As the reader will recall from (3.10.4.10) the general case can be
interpreted by regarding A, B, and M as subspaces of BL(X, X), BL(y, Y), and
BL(y, X). The other special case is of course
A = B = M is a normed algebra (11.6.1.8)
Once again (3.10.4.10) suggests that this is actually quite general.
11.6.2 THEOREM If A and B are normed algebras and M a normed
(left A,right £)-module, and if a £ An and b £ Bm are arbitrary, then,
with £> = BL(M,M),
S™{La,Rb)ce™{a)xeW[b)
and (11.6.2.1)
af(Lfl)^)Caf(a)xaf(&)

11.6 Composition Operators
483
and
<^(i.^)Cfff(a)xa3'ht(i)
and (11.6.2.2)
Proof: This is essentially Theorem 11.1.4 and Theorem 11.2.2. If, for
example, (5, t) <E Cn+m is not in a^ft (a) x a£ght (6) then either £y=i aj (ay -
5y) = 1 G A for some a' £ An or ^2™=1{bk — tk)b'k = 1 E 5 for some
6' 6 Bm, giving E;=1 W«, - *y J) + Er=i f*(*fc - '*') =^P with
either (17, V) = {La,,0) or (17, V) = (0,jR6/), which means that {s,t) is not
in 0pft(La,jR6). This proves the first part of (11.6.2.2), and all the rest is
left to the reader. ■
Specializing again to the case (11.6.1.8) and combining (11.6.1.3) with
(11.6.2.1), and taking either b = 0, or a = 0 after first interchanging a and
6, gives
and (11.6.2.3)
and
and (11.6.2.4)
If instead we specialize to the case (11.6.1.7) then we can prove more:
11.6.3 THEOREM If X and Y are normed spaces and A = BL(X,X),
B = BL(y,y), M = BL(y,X), and N = BL{X,Y) then for arbitrary
a e An and b <E Bn we have, with D = BL(Af,Af) and E = BL(iV,iV),
tJFW x *?"(*) Q *D{t(La,Rb) C ir?ht(iiolL6) (11.6.3.1)
and
f!2«(a) x frjght(6) C fj5»(L0,J?6) C f^ht(i?0,L6) (11.6.3.2)
Proof: This is Theorem 5.6.4: if 5 G Cn and t £ Cm are arbitrary, set
S = a — s and T = b — t and then use (5.6.4.1) to argue that row(jR5,Lr)

484
11. Multiparameter Spectral Theory-
dense => co\{Ls,RT) one-one and that col(L5,jRr) one-one => (S one-one
or T dense), giving (11.6.3.1). The same argument, using (5.6.4.2), gives
(11.6.3.2). ■
Interchanging a and b gives of course
^ght(a) x ^eft(6) C *%\Ra,Lh) C *i&\La,Rh) (11.6.3.3)
and
f^ht(a) x ff\b) C rlf(Ra,Lb) C f^ht(La,Rb) (11.6.3.4)
Theorem 11.6.3 extends to more general algebras and modules A, B,
M, and N as in (3.10.4.10): for (11.6.3.1) it is sufficient that there is
implication, for arbitrary u and t; in M,
uwv = 0 for each w e N =^ u = 0 e M oi v = 0 e M (11.6.3.5)
while for (11.6.3.2) it is sufficient that there are k > 0 and h > 0 for which
fc||tt||||tyt;|| + /i||twy||||t;|| < ||myt;|| for each u,v G M and w £ N
(11.6.3.6)
For example, suppose that (11.6.3.5) holds and that a £ A and b £ B are
such that the operator row(jRa,L6) : iV2 —> N is dense: then for arbitrary
w G N there are (aJJ and (6JJ in iV for which ||ty — o!na — bb'n|| —> 0, and
hence if u £ M is arbitrary
||uwu — ua'nau — ubb'nu||—► 0 as n—► oo (11.6.3.7)
From (11.6.3.7) it follows that
au = 0 = ub =^ uwu = 0 (11.6.3.8)
This together with (11.6.3.5) says that u = 0, and hence that the operator
col(La, Rb) : M —> M2 is one-one. This is the second inclusion of (11.6.3.1),
or at least a special case; for the first inclusion argue that if co\(La,Rb) is
one-one then for arbitrary u, v in M and w £ N we have
au = vb = 0 =r> a(uiut;) = (uiut;)6 = 0 ==> uiuv = 0 (11.6.3.9)
This together with (11.6.3.5) says that either u = 0 or t; = 0, and hence
that either La or Rb is one-one.
It is clear that there is implication (11.6.3.6) => (11.6.3.5). If A =
BL(X,X) and B = BL(Y,Y), then it is sufficient for (11.6.3.6) that N

11.6 Composition Operators
485
contain all the operators of finite rank:
KL0(X,y) C JV (11.6.3.10)
The reader can verify this by inspecting the proof of Theorem 5.6.4. If,
for example, N is equal to the finite rank operators, or their closure in the
norm of BL(X, Y), then the compatibility conditions of (3.10.4.10) will be
satisfied if either M = BL(Y,X) or M = KL0(y,X) or its closure.
Theorem 11.6.3 and the special case of Theorem 11.6.2 have an
essential analog:
11.6.4 THEOREM If X and Y are normed spaces and S £ BL(X,X)n,
T £ BL(y, Y)m are arbitrary systems of operators, then
^(Ls,RT)^st\Rs,LT) C (aJ2(5)xari«ht(T))u(aM(5)xa2ht(r)
(11.6.4.1)
(ir™(S)xirri«ht^
(11.6.4.2)
(*e+ss(*) x fri«ht(T)) U (f™(S) x a-s(T)) C a+.(L5,iir) n *+.(**, Lr)
(11.6.4.3)
Proof: If (s,t) £ Cn+m is not in the right-hand side of (11.6.4.1) then
there are three cases to consider: either S — si is almost left invertible, or
T — tl is almost right invertible, or both S — si is almost essentially left
invertible and T — tl is almost essentially right invertible. In the first case
Ls — si is almost left invertible and Rs — si almost right invertible, while
in the second case RT — tl is almost left invertible and LT — tl almost
right invertible. Thus, in each of the first two cases (Ls — si, RT — tl) is
almost left and [Rs — si, LT — tl) almost right invertible. For the third
case we apply the "almost" analog of Theorem 6.13.5, and note that the
analog of the condition (6.13.5.1) is satisfied with col(5 — si) : X —> Xn
and col(T - tl) : Y -► Ym in place of S : W -► Z and T : X -► Y:
then the analog of the condition (6.13.5.2) says that (Ls — sI,RT — tl) is
almost essentially left invertible and (Rs — si, LT — tl) almost essentially
right invertible. In all three cases therefore (s,t) £ cn+m is excluded
from the left-hand side of (11.6.4.1). For (11.6.4.2) and (11.6.4.3) we go to
Theorem 6.13.6: if (*,*) £ Cn+m is not in the right-hand side of (11.6.4.2),
or of (11.6.4.3), then condition (6.13.6.1), or condition (6.13.6.3), is satisfied
with S — si and T — tl in place of S and T, so that we deduce that also
condition (6.13.6.2), or condition (6.13.6.4), is satisfied, which excludes
{s,t) £ Cn+m from the left-hand side of (11.6.4.2), or of (11.6.4.3). ■
The obstacle, in incomplete spaces, to a corresponding result for left

486
11. Multiparameter Spectral Theory
and right essential spectra, as distinct from "almost" left and right essential
spectra, is that we ought to work with actually compact operators rather
than totally bounded operators in that context: unfortunately, we have only
established (6.13.3.2) for totally bounded operators. Of course when we
restrict attention to complete spaces these distinctions can be ignored. For
Hilbert spaces, and for Banach spaces and single operators, Theorem 11.6.2
and Theorem 11.6.3 combine effectively, as do both parts of Theorem 11.6.4:
11.6.5 THEOREM Suppose X and Y are Banach spaces and S £
BL(X,X)n, T e BL(y,y)m: then if X and Y are Hilbert spaces there
is equality
aleit{Ls,RT) = aright(jR5,Lr) = aleft(5) x aright(T) (11.6.5.1)
aright(L5,jRr) = cleit{Rs,LT) = aright(S) x aleft(T) (11.6.5.2)
= (<£?(*) x *right(T)) U (aleft(5) x aHf *(T))
aesfht(^5^r) = aleJsiRSiLT) (11 6 5 4l
= Wjf'P) x ^CH) U (aright(S) x aJ^(T))
If instead n = m = 1, so that S = S1e BL(X,X) and T = Txe BL(Y,Y)
are single operators, then there is inclusion
d(a(S) x a(T)) C <j(Ls,RT) C <j(S) x a{T) (11.6.5.5)
d{a{S) x a{T)) C a{Rs,LT) C a{S) x a{T) (11.6.5.6)
d((oe33(S)xo(T))u{o{S)xoe33{T)))
C ce33(Ls,RT) C (aess(5) x a(T)) U (a(S) x aess(T))
d((<Je33(S)xv(T))u(<j(S)x<je33(T)))
C ce33(Rs,LT) C (aess(5) x o(T)) U (a(5) x aess(T))
(11.6.5.7)
(11.6.5.8)
Proo/: Towards (11.6.5.1), Theorem 8.4.4 applied with col(5 - si) and
with col(T — tl) in place of T gives
fleft(5) x fright(T) = aleft(S) x <7right(T) (11.6.5.9)
which combines with (11.6.2.2) and (11.6.3.2) to give (11.6.5.1), and now
(11.6.5.2) follows from (11.6.5.1) if we interchange S and T. Towards
(11.6.5.3), Theorem 8.7.3 gives
*e+ss(S) x a-s(T) = aJ£(S) x a*f*(T) (11.6.5.10)

11.6 Composition Operators
487
which combines with (11.6.4.1) and (11.6.4.3) to give (11.6.5.3), and now
(11.6.5.4) follows by interchange of S and T. Towards (11.6.5.5) begin by
noting
d(K xH) = {(dK) xH)U{K x (dH)) (11.6.5.11)
in particular, with K = a{S) and H = c{T), and then recall Theorem 9.3.3,
and (9.3.3.11), which say
da{S) C fleft(5) H fright(S) and da{T) C rle{t{T) n fright(T)
(11.6.5.12)
and
G(S) = fleft(5) U fright(S) and c{T) = rleit{T) U frisht(T)
(11.6.5.13)
The first part of (11.6.5.5) now follows: using (11.6.5.11) and (11.6.3.2)
d{a{S) x a(T)) C (fleft(5) x fr'lght(T)) U (fright(5) x fleft(T))
C fleft(L5^r) Ufright(L5,.Rr) C c(Ls,RT)
The first part of (11.6.5.6) follows by interchange of S and T, and in each
case the second part is clear. The argument for (11.6.5.7) and (11.6.5.8)
follows the same pattern, and will mostly be left to the reader: we
supplement (11.6.5.11) with (7.10.3.6), which says that the boundary of the union
is contained in the union of the boundaries, and we replace (11.6.5.12) and
(11.6.5.13) with, using Theorem 9.8.7,
^ess(S) C ae+ss(S) H a-3(S) and aess(T) C ae+ss(T) n ^(T)
(11.6.5.14)
and
*ess(S) = *e+ss(S) U ^(5) and <7ess(T) = ae+ss(T) U c~3(T) m
(11.6.5.15)
One byproduct of the proof of Theorem 11.6.5 says that if X and Y
are Hilbert spaces and S £ BL(X,X)n, T G BL(y,y)m then
aleft {Ls, jRr) = rleft [Ls, jRt) and aright (Ls, jRr) = fright (Ls, jRr)
(11.6.5.16)
and
°™(I<sM = °L(Ls>Rt) and ***ht(Ls,RT) = g-„[LS9Rt)
(11.6.5.17)
Theorem 11.6.5 also gives us some more spectral mapping theorems:
11.6.6 THEOREM If S e BL(X,X)n and T £ BL(y,y)m for Hilbert

488 11. Multiparameter Spectral Theory-
spaces X, Y and g £ Poly£+m then
g(oie{t(S),cr>*ht(T)) C aie{tg(Ls,RT)
and (11.6.6.1)
ffK^(5),aleft(T)) C <Tri^(Ls,i?r)
g(a^(S),a"^(T)) U ^""(S^a^T)) C aJ^(Ls,*r) (H-6.6.2)
and
<,(<f'(^.^"(T)) U ?(arieht(S),<£"(r)) C ae^htff(Ls, J?r) (11.6.6.3)
If in particular S £ BL(X, X)n is commutative, then
aIeft<7(LSli2r)=ari^<7(aleft(5),r)
and (11.6.6.4)
a^tyLs.lEj.) = alett<7(ari^(5),r)
<£?*(£*.%) = ^r',ght<7(^(5),r) U<£f'^(S)^) (11.6.6.5)
and
aeriihtff(Ls,i?r) = aIeftff(o2f *(5),T) U a^g(a"^(S),T) (11.6.6.6)
If instead T £ BL(y,y)m is commutative, then
cMtg(Ls,RT)=aUttg(S,ar'^(T))
and (11.6.6.7)
ari^ff(Ls, RT) = a^g^o^T))
a^g(Ls,RT) = ^(S.a^CT)) U o^g(Sto^*(T)) (11.6.6.8)
and
<£f*g(Ls,RT) = a^g{S,alett{T)) U ^htff(S,a^(T)) (11.6.6.9)
If both S and T are commuting systems of operators, then
cMtg(Ls,RT) = g(aM\S),c"^{T))
and (11.6.6.10)
a^htg(Ls,RT) = g(a"^(S),aleii(T))
c£MLs,Rt) = g(^{S),a^(T))Ug(cleti(S),c^(T)) (11.6.6.11)

11.6 Composition Operators
489
and
<7l^g(Ls,RT) = <7(aeriiht(5),aleft(r))U<7(a^ht(5),^t(T)) (11.6.6.12)
Proof: This is mostly what we would have expected from the
combination of Theorem 11.6.5 with the spectral mapping theorems (11.3.4.7),
(11.3.4.8), and Theorem 11.3.5: in particular we are using the notation
(11.3.4.9). For (11.6.6.1) we use (11.3.4.7) together with (11.6.5.1) and
(11.6.5.2), and for (11.6.6.10) we use (11.3.4.8) together with (11.6.5.1) and
(11.6.5.2). For (11.6.6.2) and (11.6.6.3), and for (11.6.6.11) and (11.6.6.12),
we do the same, substituting (11.6.5.3) and (11.6.5.4) for (11.6.5.1) and
(11.6.5.2). Towards (11.6.6.4) we use Theorem 11.3.5 to see
cleitg(Ls,RT) = |J al^=3l9{sI,RT) = (J o^R^t)
se<rle{t(S) 5Go-left(5)
(11.6.6.13)
and
c"^g(Ls,RT)= (J ot?=si9{sI,RT)
right (H.6.6.14)
~ U aLs=sIRg(sI,T)
5Go-right(5)
and then use (11.6.5.1) and (11.6.5.2), together with (11.6.1.3), to see that
s e aleft(S) =► c^=sIRg{sIiT) = vle{tRg{3l,T) = ari«ht*(a/,T)
(11.6.6.15)
and
(11.6.6.16)
This proves (11.6.6.4), and the argument for (11.6.6.7) is exactly the same,
The arguments for the corresponding essential results follow the same
pattern, and are left to the reader. ■
To exploit the other half of Theorem 11.6.5 we need an auxiliary result
about polynomials.
11.6.7 THEOREM If K C Cn and H C C are compact and g <E Polyn+1,
then
g(K xH)= g{d(K x H)) (11.6.7.1)

490
11. Multiparameter Spectral Theory
Proof: We have
g(d{K x H)) = g{dK x H) U g{K x dH) (11.6.7.2)
and
g(K xH) = g{K x dH) U g(K x int(JET)) (11.6.7.3)
We claim
g{K x int(tf)) C g{dK x int(JET)) (11.6.7.4)
To see this suppose r £ C is arbitrary and define iVr : if —> C by setting
JVr(s) =#{te mt(H):g{s,t) - r = 0} for each sGX (11.6.7.5)
the number of zeros of the polynomial ga — r = g(s, •) — r in the bounded
open set H C C. Obviously, Nr(s) £ N U {0} is always an integer; it is also
a continuous function of 5, in view of its integral representation
N^ = hjWr~r *»<**•€* (H.6.7.6)
Thus if iVr vanishes on dK it must vanish on K, and hence if r £ C is not in
the right-hand side of (11.6.7.4) it must also be excluded from the left. ■
Using Theorem 11.6.7, the other half of Theorem 11.6.5 gives spectral
mapping theorems for single operators on Banach spaces:
11.6.8 THEOREM If X and Y are Banach spaces, if 5 £ BL(y, Y) and
T £ BL(X,X) are single operators, and if g £ Poly2 is a polynomial in two
variables, then
<jg{Ls,RT) = g{c{S),c{T)) (11.6.8.1)
and
*e3a9(Ls,RT) = g(oess{S),o(T))U9(*(S),c7ea3(T)) (11.6.8.2)
Proof: By (11.6.5.5) there is inclusion
gd(c(S) x g(T)) C gc(Ls,RT) C g{c(S) x c(T)) (11.6.8.3)
and (11.6.7.1) means that there is equality throughout. By (11.3.4.8) this
applies to the spectrum of g(Ls,RT), giving (11.6.8.1). The argument for
(11.6.8.2) is exactly the same, using (11.6.5.7) in place of (11.6.5.5). ■
As a simple application of Theorem 11.6.8, look at sums and products:

11.6 Composition Operators
491
11.6.9 THEOREM If X and Y are nonzero Banach spaces and S e
BL(X,X), T e BL(Y, Y), then
Ls - RT invertible <=> o{S) n o{T) = 0 (11.6.9.1)
and
LSRT invertible <£=> 0 £ <j(S) U o{T) (11.6.9.2)
Also
Ls - RT Fredholm ^=> a{S) n aess(T) = aess(5) n c{T) = 0 (11.6.9.3)
and
LSRT Fredholm <=► 0 £ a(5)aess(T) U aess(5)a(T) (11.6.9.4)
Hence, if neither X nor Y is finite dimensional, then
LSRT Fredholm <==> LSRT invertible (11.6.9.5)
Proof: If, more generally, M is a normed (left A, right B)-module and
a e A, b e B, then with D = BL(M,M)
aD(La - Rb) C aD(Lj - aD{Rh) C aA(a) - aB(6) (11.6.9.6)
and
aD(Ltti?6) C BD(La)5D(Rh) C <?A(a)aB(6) (11.6.9.7)
so that
aA(a) fl aB(6) = 0 =>• La - jR6 almost invertible (11.6.9.8)
and
0 £ oA{a) U <7B(6) =>► LaRb almost invertible (11.6.9.9)
When A = BL(X,X), B = BL(Y,Y), and M = BL(Y,X) and X, Y are
complete then all the inclusions of (11.6.9.6) and (11.6.9.7) become equality,
according to (11.6.8.1) from Theorem 11.6.8. This proves (11.6.9.1) and
(11.6.9.2). For (11.6.9.3) and (11.6.9.4) we use instead (11.6.8.2). Finally,
if neither 0e33(S) nor ^essl^O *s emPty, the right-hand side of (11.6.9.4)
takes the form
0 £ a(5) U aesa(T) U aess(S) U o(T) = o(S) U o{T) u

492
11. Multiparameter Spectral Theory
The implications (11.6.9.6) and (11.6.9.7) hold for left and right spectra
separately: in particular
a^ft(a) H a£ght(6) = 0 =► La - Rb one-one
and
bright.
*a (a) n 5B V>) = ®=^La-Rb onto
(11.6.9.10)
(11.6.9.11)
When A = B = M these two conditions have an unexpected consequence:
we shall write
D=L-Rn
for each a £ A
(11.6.9.12)
11.6.10 THEOREM If A is a normed algebra and a, b <E A satisfy
rlZ{t{a) H f*ght(6) = 0 (11.6.10.1)
then (a, 6) and (a + 6, a6) have the same commutator, in the sense that
D^{0) n JV(o) = D~lb(o) n p^(o) (11.6.10.2)
Proof: With no restriction on a and 6 there is inclusion
D^io) n 1^(0) c P-^(O) n ^(o) (11.6.10.3)
One way to see this is to observe
r t t 1 r n i
(11.6.10.4)
(11.6.10.5)
[Dab\
=
' I I '
Rb K.
.Db.
Formally taking the "adjugate" or "cofactor" matrix gives
L„-Rh
0
L„-R
b J
-R,
^a+b
Dab
This gives implication
La - Rb one-one =► D~lb(0) n D^(0) C D~X(0) n P^ (°) (11.6.10.6)
Finally, the argument for (11.6.9.6), together with (11.6.1.2), gives
fleft(L0 - Rb) C fIrft(L„) - T™\Rb) = f ™(a) - fr/ht(6) (11.6.10.7)
Thus the condition (11.6.10.1) ensures that La—Rb is bounded below, hence
one-one. ■

11.7 Tensor Products
493
11.7 TENSOR PRODUCTS
Just as the cartesian product, or direct sum, of Theorem 1.6.1 enables
certain kinds of two-variable functions to be represented as linear operators,
so the "tensor product" enables "bilinear" mappings, such as the algebra
multiplication (1.10.0.1) or the inner product of Definition 8.1.1, to be
represented as linear mappings:
11.7.1 DEFINITION If X and Y are linear spaces over the field K then
their tensor product is the linear space
X®Y = X®KY = c00{X xY/ (ST1 (0)) (11.7.1.1)
where (S)_1(0) is the linear subspace of c00(X <g> Y) generated by the set
U {{**+*',* ~ S*,y ~ 6x',y>6x,y+y> ~ Sx,y ~ 6x,y" ? l
6sx,y ~ SSx,y>tx,sy ~ s6xiy}'^^ eX,y,y' EY^sE K}
If X and Y are normed spaces then a crossnorm on X <8> Y is a norm for
which
||rc <8> 2/|| = ||x||||y|| for each x <E X,y <E Y (11.7.1.3)
A uniform crossnorm on X <8> Y is a crossnorm for which
j=i
£/(*yMyy) <ll/lllkll E*i®yy
J=l
(11.7.1.4)
for each / G X\g <E Y\ ^iy®yyGX®y
We shall call the normed space Z a tensor product of X and Y if it has a
dense subspace isomorphic to the tensor product X <8> Y with a crossnorm,
and a uniform tensor product if the crossnorm is uniform.
Here c00{X xY) = c00(X x Y, K) is the space of finite or terminating
functions c00(H) of (1.9.0.4), where H is X x Y with the bornology of finite
subsets, and Sxy is the Kronecker delta of the point (x,y) £ X x Y. We
are writing
E xi ® Vi = I £ *»y.w + ®_1(°) e coo(^ x n/ ®_1 (0) (H.7.1.5)
for the coset of a typical element of c00{X x Y). Later on, in a further

494
11. Multiparameter Spectral Theory
abuse of notation, we may write X <S> Y to represent "a tensor product
of X and Y" in the sense of the spaces Z at the end of the definition:
in particular if X and Y are both complete then we will be able to take
Z = X <S> Y also complete. If X = A and Y = B are linear algebras
then the tensor product X <8> Y = A <8> B becomes an algebra if we
define
(a <g> b)(af <g> 6') = (aaf) <g> (66') (11.7.1.6)
and extend to more general tensors by distributivity. We shall see
below that this is a "good definition." If A and B are normed algebras we
will usually look only at crossnorms on the product A <8> B that are
compatible with this multiplication. If instead X is a right module and Y a
left module over the linear algebra A then we can form the further
quotient
X(g)AY = (X(E)Y)/(g)^1(0)
where
^a1!0) = \J2xjaj ®yj-xj®a>jyj:neN,xexn,ye Yn,ae An \
(11.7.1.8)
Alternatively we may define ^^(O) as a subset of c00(X x Y) obtained
by adding to the set (g)-1^) of (11.7.1.2) all the elements Sxay - Sxay.
Naturally we write
!>;®Ayy= ll>i®»i \+®A1(0)e{X®Y)/®^1(0) (11.7.1.9)
It can be quite difficult to decide exactly when an element of X <8> Y is
zero:
11.7.2 THEOREM If X and Y are linear spaces and 0 ^ YJjLi x'j ® v'j €
X <8> Y then there are n £ N and linearly independent x £ Xn,2/ £ Yn for
which
m n
]T xj ® 2/y = Y, xj ®Vj£X®Y (11.7.2.1)
y=i j=i
If subspaces if C L(X, K) and jK" C L(Y, K) separate points of X and Y,
respectively, then for arbitrary n £ N, x £ Xn, and y £ Yn, the following
(11.7.1.7)

11.7 Tensor Products
495
are equivalent:
n
]T x. (g) y. = o £ X <g> Y (11.7.2.2)
y=l
n
Y, f(xj)yj = °£Y for each / € J5T (11.7.2.3)
n
]T y (yy)xy = 0GI for each (/GX (11.7.2.4)
j = i
Proof: If xj = X)y^t 5yxy *s a linear combination of the remaining x'j then
m
E xy ® *>y = E xy ® fry + 8M) (11.7.2.5)
y=i y**
and if at the same time the sequence {y'j)JL1 is linearly independent then
so is the sequence [y'j + Sy2/()y^i- Whether or not the sequence (y'J)JL1
is linearly independent we can repeat the same process with the sequence
(xf) ^t-, arguing that if it is still linearly dependent then the tensor
YJjLi x'j ® y'j can be written without the help of either x\ or x\ for some
other index i'. Since not all the x'j can be 0 (remember YlT=i xj ® Vj ^ 0)
this process will eventually yield a linearly independent subsequence x"
of (xf)Jl_1. If the corresponding sequence y" which has evolved is also
linearly independent then we have established (11.7.2.1); if not we now
repeat the process with the sequence {yn), noting that we can never lose the
linear independence of the sequence (x") (we made this comment
immediately following (11.7.2.5)). This finishes the proof of (11.7.2.1). Toward
the equivalence of the next three conditions observe that (11.7.2.2) can be
rephrased
XX^.e®-1^)
y=i
which makes the left-hand sides of (11.7.2.3) and (11.7.2.4) vanish for
arbitrary / £ L(X, K) and g £ L(y, K). In turn, this observation guarantees
that the left-hand sides of (11.7.2.3) and (11.7.2.4) are well defined as
functions of the tensor Y%Li xj ®Vj €X®Y. We have proved that (11.7.2.2)
is sufficient for each of the other two conditions; conversely if (11.7.2.2)
does not hold then we are entitled by (11.7.2.1) to assume that x £ Xn and
y £ Yn are linearly independent, and by what we have just noted above

496
11. Multiparameter Spectral Theory
to use this x £ Xn and y £ Yn in the left-hand sides of (11.7.2.3) and
(11.7.2.4). In particular,
xx^0^yi
and hence by the separating assumptions (6.1.2.10) there are / £ H and
g £ K for which
/K)^o^ff(yi)
But now the linear independence of y £ Yn means that for this / the
left-hand side of (11.7.2.3) does not vanish, and the linear independence
of x £ Xn means that for this g the left-hand side of (11.7.2.4) does not
vanish. ■
When X and Y are normed spaces then Theorem 5.4.1 tells us that
we can take H = X^ and K = Y^: more generally, we shall usually always
work with H C X* and K C yt.
If X and y are normed spaces then an example of a crossnorm on
X <8> y is the quotient norm on c00(X x Y)/ <8>_1 (0) induced by the norm
of ^(IxF):
5>y ® J = inf J £ ||*;-||||^||:£*y ® 2/y = £ xj- ® ^ V
y=i Hi [y=i j=i y=i J
(11.7.2.6)
The reader should verify that this is a crossnorm, a uniform crossnorm, and
that it lives up to its title, the greatest crossnorm for X<8> Y. At the other
extreme, the least uniform crossnorm is given by
Ex;®2/;
j=i
= sup
£/(*yMy,)
y=i
<i,IWI<i
(11.7.2.7)
The reader should verify that this also lives up to its name, and also that its
value is unchanged if the supremum is restricted to functional / £ H C X^
and g £ K C yt belonging to subspaces H and K which are "norm-
determining" in an obvious sense.
Various specific spaces can be recognized as tensor products in the
sense of the last part of Definition 11.7.1:
11.7.3 THEOREM If X and Y are normed spaces then the finite rank
operators
KL0(X,y) C BL(X,y) is a tensor product of X and Y (11.7.3.1)

11.7 Tensor Products
497
with
y=i
E/y®yy
y=i
and the linear functional
7f®IC BL(X, y)f is a tensor product of Yf and X
with
X>y®*y
y=i
X>y®*y
(11.7.3.2)
(11.7.3.3)
(11.7.3.4)
If also W and Z are normed spaces then the elementary operators
(11.7.3.5)
°£bl(x,y) CBL(BL(^,Z),BL(X,y))
is a tensor product of BL(P7, Z) and BL(X,y)
(11.7.3.6)
and, provided W ® X has a uniform crossnorm,
BL(W, Z) <g> BL(X, y) C BL(W <g> X, Z <g> y)
is a tensor product of BL(VF, Z) and BL(X, y)
If H is a nonempty set then for each u £ {c00,c0,c1,c,/1,/2}J
o;(n,X) is a uniform tensor product of oj{Q) and X (11.7.3.7)
and
m(H,X) is a uniform tensor product of ^(H) and X (11.7.3.8)
If H is a topological space then
C^HjX) n m(H,X) is a uniform tensor product of C^Q) and X
(11.7.3.9)
Proof: In each case it is obvious that there is an embedding of the tensor
product, as a set, in the space concerned: to see that it is one-one we apply
Theorem 11.7.3 with suitably chosen subspaces H and K, and it can then
be left to the reader to see that the induced norm is a crossnorm, and that
the image of the tensor product is a dense subspace. ■
The norm that best reflects the character of the tensor product is the
greatest crossnorm of (11.7.2.6):

498
11. Multiparameter Spectral Theory
11.7.4 THEOREM If X, Y, and Z are normed spaces then there is
isomorphism
x 0! (y ®x z) = [x ®x y) ®x z (11.7.4.1)
and
x <£>! (y 0! z) = (x <£>! y) 0! (x ® x z) (11.7.4.2)
and
BL(X ® x y, Z) = BL(X, BL(y, Z)) = BL(y, BL(X, Z) (11.7.4.3)
Proof: We may safely leave the first two parts of this to the reader, and
recall that the second part of (11.7.4.3) was given by Theorem 2.9.3. Toward
the first part of (11.7.4.3) we have a well-defined correspondence
0 <—► $ : BL(X <£>! y, Z) = BL(X,BL(y, Z))
given by the formulas
(11.7.4.4)
0 HC xj® yj) = ^2 *(xy)yy for each X] xy ® ty G x ® Y
\j=i J y=i y=i
(11.7.4.5)
and
$(x)(y) = 0(x<g> y) for each x G X,y G y (11.7.4.6)
If $ is in the right-hand side of (11.7.4.4) and 0 is given by (11.7.4.5) then
ehr^.ay,.
U=1
< 11*11 X>ylll|yyl
j=i
and hence, considering all possible synonyms for Yll=i xj ®Vj € X ®Y,
(11.7.4.7)
©lX>y®yy
< 11*11
I>y®yy
y=i
This proves that ||0|| < ||$||. If instead 0 is in the left-hand side of
(11.7.4.4) and $ is given by (11.7.4.6) then
llft^t^Hlieiillxsyil^neiiHiiyii
and hence, considering all vectors y £ Y of norm < 1,
||$(*)||<||0||||*|| (11.7.4.8)
This proves ||$|| < ||0||. ■

11.7 Tensor Products
499
As a special case of (11.7.4.3) we have isometric isomorphism
[X <£>! y)f = BL(X, Yf) £* BL(y, Xf) (11.7.4.9)
Thus, using Theorem 5.9.1, we have the option of realizing the tensor
product X <8> Y as either a subspace of the dual of BL(X,yt) or a subspace of
the dual of BL(y, X*). In this sense the linear functional f <2>y of (2.9.2.6)
are closer in spirit to our "tensors" than the finite rank operators / © y of
(2.9.2.5). The first two parts of Theorem 11.7.4 remain valid if the greatest
crossnorm is replaced by a general crossnorm: if on either side of (11.7.4.1)
or (11.7.4.2) we change the greatest crossnorm or the direct sum norm then
it becomes an interesting calculation to see what norms are induced on the
other side. If we try to replace the greatest crossnorm in the left-hand
side of (11.7.4.3) then we must replace the spaces of operators on the other
two sides by certain subspaces. For example, if we use the least uniform
crossnorm and then specialize to Z = K we will get a tensor product of X
and Y on the right, consisting of those operators from X to Y^ which can
be approximated in "greatest crossnorm" by finite rank operators.
Another way to reduce the right-hand sides of (11.7.4.3) is to replace
the tensor product X ®>x Y by its quotient X <8>A Y, if X is a normed
right and Y is a normed left module over an algebra A. Either the same
argument as in the proof of (11.7.4.3), or an immediate deduction from the
conclusion (11.7.4.3), gives
BL{X ®AY,Z) ~BLA{X,BL{Y,Z)) = BLA{Y,BL{X,Z)) (11.7.4.10)
Here we are writing, if $ <E BL{X,BL{Y,Z)) and # <E BL{Y,BL{X,Z)),
* e Bl/(X,BL(y,Z)) «=► «(xa)(y) = *(*)(ay) (n.7.4.11)
for each x E X,y £Y,a £ A
and
$ 6 BLA(Y,BL(X,Z)) ^ *(ay)(x) = *(y)(™) (n ? 4 n)
for each x E X,y EY,a E A
and of course using the greatest crossnorm of X <8> Y to define the norm of
X <8>A Y. If we remark that
X®AAS*X and A <g>A Y = Y (11.7.4.13)
then we notice (generalizing Theorem 5.1.2!) the special cases
BLA(y,Af) = yf and BLA(X,Af) ^ Xf (11.7.4.14)

500
11. Multiparameter Spectral Theory
In particular,
BLA{A\Af) = Aft and BLA(Af, Af) s Aft (11.7.4.15)
The reader should notice carefully that (11.7.4.15) represents the second
dual A^ of a normed algebra as a normed algebra, but in two different
ways, depending on whether the first dual is regarded as a left module or
as a right module over the algebra.
For tensor products of normed algebras, we have analogs of
Theorem 11.6.3 and Theorem 11.6.2:
11.7.5 THEOREM If A and B are normed algebras and A®B is a tensor
product of A and B then for arbitrary a £ An and b £ Bm there is inclusion
*Ah(°) x *l§h(b) C x™B(a ® 1,1 ® 6)
****(*) x n^(b) C ^g* (a ® 1,1 ® 6) (1L7-5'1)
flf(a) x f^'(6) C f$B(a ® 1,1 ® 6)
^ght(«) x f^ht(6) C f&h* (a ® 1,1 ® 6) (11'7-5-2)
*!&»(« ® 1,1 ® 6) C <#"((») x a£ft(6)
<tf |b(« ® 1,1 ® 6) C a7ht(a) x a£ght(6) (U-7-5-3)
If, in particular A <g> 5 is uniform then there is equality
*aSb(* 3 1,1 3 6) = ^eft(a) x a^ft(6)
*&"(* ® 1,1 ® 6) = *?ht(a) x a£ght(&) (11-7-5>4)
Proof: If 5 £ *rjfft(a) and t £ 7r^ft(6) then there are 0 ^ u £ A and
0 7^ t; £ B for which (a;- — Sj)u = 0 = {bk — tk)v for each j and A:, giving
((ay - sy) <g> l)(u <g> v) = (1 <g> (6fc - tk))(u ®v)=0®0^u®v (11.7.5.5)
for each j and A:. This proves the first part of (11.7.5.1), and the second
is exactly the same (formally, derive it from the first by "reversal of
products"). The argument for (11.7.5.2) is almost identical: if 5 £ fj|ft(a) and
t £ f!Jft(6) then there are sequences {ur) and (vr) in A and B for which
IM = 1 = \K\\ and ||(ay - 5j>r|| + \\{bk - tk)vr\\ - 0 for each j and k:
but now, using the crossnorm property for A <8> B,
and (11.7.5.6)
|| ((aj - 5y) ® 1) (ur ® t,r)|| + ||(1® {bk -tk)){ur®vr) II—0

11.7 Tensor Products
501
This proves the first part of (11.7.5.2), and similarly the second. Toward
(11.7.5.3) suppose that either £^=1aJ(ay-sy) = 1 £ A or ^=1b'k{bk-tk) =
leB: then
n m
]T cy((ay - Sj) (g) 1) + ]T ^(1 (g) (6fc - **)) = l<g>l£A<g>5 (11.7.5.7)
j = l fc=l
with either (c,d) = (a' (g) 1,0 (g) 0) or (c,d) = (0®0,1® 6'). This proves
the first part of (11.7.5.3), and hence also the second. Exactly similar
argument shows that the analogue of (11.7.5.3) holds for the almost
spectrum: whether or not the crossnorm is uniform there is inclusion one
way in (11.7.5.4). For equality in the uniform case suppose that (s,£) £
aj|ft(a) xa£ght(6),sothat
n m
l£M = c\y^A{aj-sj) and 1 £ N = cl ]T B{bk-tk) (11.7.5.8)
y=i fc=i
Arguing as in Theorem 7.13.3 there are by the Hahn-Banach theorem <f> £
A*, tp £ £t for which
<£(1) = 1 = ^(i) and 4>{M) = {0} = 1>(N) (11.7.5.9)
By the assumption that the tensor product is uniform there is a well-defined
bounded linear functional <f> <g> %j) : A <g> B —> K for which
/ p \ p p
[4> ® ^) X] ^ ® 6*" = ^ ^(a<)^(6i) for each Y,ai®bieA®B
\i=l J i=l i=l
(11.7.5.10)
If the finite rank tensors are only a dense subspace, <j> ® ip extends to the
whole of "A <g> B" by continuity. Since
[<t> <g> i>) (1 ® 1) = 1 and (<£ (g) ip) (M <g> 5 + A <g> TV) = {0} (11.7.5.11)
it follows that the closed left ideal of A <g> B generated by (a — s <g> 1,1 <g> b — t)
does not contain the identity 1®1, which means that (s,t) £ <7j|£B(a<g>l, 1(g)
6). This proves the first part of (11.7.5.4), and hence also the second. ■
The analog of Theorem 11.6.3 holds for normed algebras and uniform
tensor products:
11.7.6 THEOREM If A and B are normed algebras and D = A <g> B is a
uniform tensor product of A and B then for arbitrary a £ An, b £ Bm and

502
11. Multiparameter Spectral Theory
9 e Poly£+m
ff(*!f M.^'ifW) C d^g(a ® 1,1 ® 6)
<7(ar/hta),arjght(6)) C a^a ® 1,1 ® 6) (11-7'<U)
If in particular a £ An is a commuting system then
<?£ftff(a ® 1,1 ® 6) = a'jftff(^ft(a),6)
<?£ght«7(a ® 1,1 ® 6) = o%shtg(o«eht(a),b)) i11-7*-2)
If instead b £ Bm is a commuting system then
a^t<7(a®l,l®6)=^ft<7(a,aIJft(6))
c^tya ® 1,1 ® 6) = ar/htff(a,agght(&)) (n-7-6-3)
If both cGAn and b G Bm are commutative then
agftff(o®l,l®6)=ff(a5ft(a),ajgft(6))
ar^htff(a® 1,1®6) =ff(^ht(a),aright(6)) (U-7'6-4)
If in particular A, 5, and 2} are complete and m = n = p = 1 then
<7D<7(a<g> 1,1 <g> 6) = 0(aA(a),aB(6)) (11.7.6.5)
Proof: This is the argument of Theorem 11.6.6, using (11.7.5.4) in place
of (11.6.5.16); for the last part we use the argument of Theorem 11.6.8. ■
The second or third part of Theorem 11.7.6 gives an amusing expression
for the spectrum of an "operator matrix." If A is a complex linear algebra
we shall write
Aqq = A®cCqq = A*2 (11.7.6.6)
for the algebra of q x q matrices over A, with obvious addition and
multiplication, using (11.7.4.2) to recognize this as a tensor product: if A is a
normed algebra then any one of the usual product norms on Aq will be a
uniform crossnorm. If c = (a{) £ A we shall write
det(c) = Y, {sgn(7r)ai7r(i)^(2) * * * V(g) : * e ^eim{^ 2,..., g)} G A
(11.7.6.7)
for the "determinant" of c: of course its significance is greatly diminished
by the fact that we have had to make a specific choice in the order of the
factors in each of the products. Here sgn(7r) is either +1 or —1 depending
on whether the permutation is "even" or "odd." If in particular the matrix
c has a commuting sequence a = (an,a12,...,alg,...,a ) of entries then

11.7 Tensor Products
503
the ordering of the factors in the determinant no longer matters, and we
can go on to define a "cofactor" or "adjugate" matrix adj(c) with
cadj(c) = adj(c)c = det(c)J (11.7.6.8)
There is a multiplicative property for the determinant: if c' = {a'-) is
another matrix and if the whole system (a, a') £ A2q is commutative, then
det(cc') = det(c) det(c') = det(c') det(c) = det(c'c) <E A (11.7.6.9)
and
adj(cc') = adj(c') adj(c) <E Aq* (11.7.6.10)
11.7.7 THEOREM If c = (a{j) <E D = Aqq is a q x q matrix over the
normed algebra A with a commuting sequence of entries a = {an,a 12,
a^ft(c) = {reC:0e a^ det(c - r)} (11.7.7.1)
and
a£ght(c) = jrGC:0G ar/ht det(c - r)} (11.7.7.2)
and
aD{c) ={reC:0e<JA det(c - r)} (11.7.7.3)
Proof: Writing B = Cqq let
b={blub12,...,blq,...,bqq)eB* (11.7.7.4)
obvious ba
which
be the obvious basis for the vector space B = C , namely, the one for
<? q
c = (a{j) eD^>c = g{a, b) = ]T ]T a{j ®bijeA®B = D (11.7.7.5)
i=iy=i
By (11.6.7.2) we have
(11.7.7.6)

504
11. Multiparameter Spectral Theory
and
^6htW = ^g[a«*\*),b) = U |^ght EE'oV* € ^ht(a) 1
(11.7.7.7)
By "classical" determinant theory we have, for each 5 £ Cn,
<#**(*. *) = a?hVa,6) = ^M) = *?"*(*, 6)
= {r€ C: det($(s,6) - r) = 0}
The first four expressions coincide by (3.10.4.1), (6.2.6.5), and (6.2.6.1),
because B is a finite dimensional linear algebra, and then implication
g(s,b)-re B'1 <£=> det{g{s, b) - r) ^ 0 (11.7.7.9)
follows from (11.7.6.8) and (11.7.6.9). If we now apply Theorem 11.3.4 with
the polynomial / = det(g(z, b) — r) then we find implication
0 £ aj|ft det(c - r) <<==> det{g{s,b) - r) = 0 for some 5 £ <J!j;ft(a)
(11.7.7.10)
and
0 £ <5^ght det(c - r) «=> &et(g(s, b) - r) = 0 for some 5 £ <5^ght(a)
(11.7.7.11)
From (11.7.7.6), (11.7.7.8), and (11.7.7.10) we get (11.7.7.1), similarly
(11.7.7.2), and hence, if the algebra is complete, (11.7.7.3). Whether or
not A is complete, it is clear from (11.7.6.8) that if det(c — r) £ A~l then
c — r £ £>_1; also the reverse implication would flow from (11.7.6.9) if we
knew that the entries of a two-sided inverse (c — r)-1 had to commute with
one another and with the entries of c — r. To see that this is so observe
that for arbitrary c = (ai;) £ D and arbitrary u £ A there is implication
c{u <8> J) = (u ® I)c =>■ a{u = ua- for each i,j £ {1,2,..., q} ■
(11.7.7.12)
Superficially similar to Theorem 11.7.7 is the spectrum of an upper
triangular matrix, with no commutivity restriction on the entries:
c = (aiy) £ D => aD{c) = (J dA{ajj) (11.7.7.13)
where D is the algebra of upper triangular q x q matrices over a normed
algebra A. To see this observe that D = A <8> B for the algebra B of upper
triangular qxq matrices over the complex field C, and that for an arbitrary

11.7 Tensor Products
505
system b £ Bm
«B(6)={6ii.622,-,6mm} (11.7.7.14)
for the left and the right, and the almost left and right, spectra w: thus
also
ujBg{b) = gujB{b) if b £ Bm, g £ Poly^ (11.7.7.15)
The reader should now mimic the argument for Theorem 11.7.7,
interchanging the roles of A and B. The spectrum of a continuous vector-valued
function is what it ought to be: If / : H —► A is totally bounded and continuous
from a normal HausdorfF space H to a normed algebra A then
D = ^(fl, A) => oD(f) = closure (J 5J(t) (11.7.7.16)
ten
We prove more:
11.7.8 THEOREM If D is a uniform tensor product of normed algebras
A and B, where A is complete and B is commutative, then for each c £ D
there is equality
S%%)= U *aV(c) and **"(<:) = |J otf'VM
(11.7.8.1)
where
n n
^A(c) = lim ^2 ^{bj)aj :^L,an® bj —* c for each ^ e B%> c ^ ^
y=i y=i
(11.7.8.2)
Proof: The uniformity of the tensor product ensures that ipA is well
defined, bounded, and linear on the set A <8> B C £), and the completeness of
A means that it extends uniquely to D: now we have
il> £ Bx =► <PA £ HBL(£>, A) (11.7.8.3)
For equality (11.7.8.1) we need the extension of Theorem 11.7.6 to infinite
systems: If ip £ B* and r £ £jfft^A(c) then by the extended version of
(11.7.5.4) we have
tysr) £a£ft(l<g>£,^A(c)<g>l) CCflxC (11.7.8.4)
and hence by the extended version of Theorem 11.3.5
tys r) £ a£ft (1 <g> B, c) C CB x C (11.7.8.5)

506
11. Multiparameter Spectral Theory
In particular it follows that r £ <?pft(c). Conversely if this is assumed then
by Theorem 11.4.4 there is a character ip e B* for which (11.7.8.5) holds,
and hence again by (11.7.5.4) also (11.7.8.4); now Theorem 11.7.6 again
gives r £ <j^ft^A(c). This proves the first part of (11.7.8.1), and hence also
the second ■
To deduce (11.7.7.16) form Theorem 11.7.8, at least for compact Haus-
dorff spaces, simply take B = C^Q). For Allan's Theorem take B to be
the algebra
Holo^) = {f eC(K):
(11.7.8.6)
/ has a holomorphic extension to some open H D K}
associated with a compact set K C C. Note that Ho1o(jK") is a subalgebra of
C{K) but not in general closed in the usual norm; however, using a Cauchy
integration,
g £ closure Holo(ii:) =^ g G Holo(interior(X)) (11.7.8.7)
11.7.9 THEOREM Suppose that A is a Banach algebra, that K C C is
compact, and that H C interior[K): then for each / £ Holo(iJT, A) there is
implication
/(*) £ Aj;Jt for each t £ K =► / £ Holo(H,A)fJt (11.7.9.1)
and
/(<) € AS*ht for each t £ K =► / £ Holo(H, A)^ht (11.7.9.2)
Proof: If we take
£> = closure Holo(jRT, A) C ^(jRT, A) = C(jK", A) (11.7.9.3)
then Z? is a tensor product of A and B = Ho1o(jK"): for if / £ Ho1o(jK", A)
then
t£K^m = ±.ff{z){z_t)-idz
K
(11.7.9.4)

11.7 Tensor Products
507
giving
n
/ = lim^6iOay
(11.7.9.5)
with v ;
bJ = (SJ ~ *)"' and aJ = ^M5; " 5;-i)
By Theorem 11.7.8 we have
/(*) £ A^Jt for each t e K =^ f <£ cle{t(D) =>> / £ D^k (11.7.9.6)
and
/(*) € ^ht for each t € K => f <£ ar'*ht(D) =► / € I>^ht (11.7.9.7)
using Theorem 4.4.5 and the completeness of D at the end. But now if g £
D is either a left or a right inverse for / then (11.7.8.7) extends to vector-
valued functions to tell us that the restriction of g to H is holomorphic ■
Theorem 10.9.5 has an analogue for tensor products:
11.7.10 THEOREM Suppose T £ BL{X,Y), 5 £ BL{Y,Z), R £
BL(Z,W), U £ BL(y,y), and 7 <E BL(Z,Z) satisfy 75 = Stf, with
(jR, 5) not almost exact and (5, T) not almost exact (11.7.10.1)
and
V not bounded below or U not almost open (11.7.10.2)
then
jR® J 000'
7(8)7 5(g)/
5(g)/ 0<g>0l\ , , ,
is not almost exact (11.7.10.3)
-I®U T®l\J K J
Proof: We prove something quite different. If (R,S) and {S,T) are not
(left,right) one-one then there are z e Z, h E Z^, y E Y, and g £ yt for
which
Rz = 0 = hS and Sy = 0 = gT and /i(z) ^ 0 ^ $(</)
(11.7.10.4)
If 7 is not one-one or C7 is not onto then there are z' £ Z, hf E Z^, yr E Y,

508
11. Multiparameter Spectral Theory
and g' £ Y^ for which
Vz' = 0 = g'U and (/i'(z') # 0 = q' or 9'W) ± 0 = z')
(11.7.10.5)
In either case
R®I 0<g>0
/(g)^ 5(g)/
z <g> z' 0 <g> 0
L 0 (g) 0 2/0 2/'
_ r/Kgj/i' o® o
~ [ 0(g)0 g®g'\
0(g)0 0(g)0'
0(g)0 0<g>0.
1
5(g)/ 0<g>0"
_-J<g>i7
r<g>/_
(11.7.10.6)
and
/i<g> /i' 0 (g> 0
0(g)0 y
</'j
z® z'
0(g)0
0(g)0 "
y®yf
±
"0 0'
.° °.
(11.7.10.7)
so that the left-hand side of (11.7.10.3) is not (left,right) one-one.
To convert this argument to a proof of the theorem as stated we now
systematically replace vectors and linear functionals by bounded sequences,
and equality to zero by convergence to zero. The details are left to the
reader. ■
The reader might also like to note the following extension of
Theorem 11.7.5: If a e An, a' e An\ b e Bm, and 6' G £m' then there are
inclusions
^ett'right(a,a') x ^ft'right(6,6') C ^^ight((a,a') ® 1,1 ® (6,6'))
(11.7.10.8)
flf™ht(a,a') x f£ft'r*ht(6,6') C f^ht((a,a') ® 1,1 ® (6,6'))
(11.7.10.9)
<^t£ght(K a') ® 1,1 ® (6,6')) C <^ft >right (a, a') x a™'***(6,6')
and equality
(11.7.10.10)
(11.7.10.11)
11.8 QUASICOMMUTING SYSTEMS
The spectral mapping theorem extends to certain noncommuting systems
of normed algebra elements. We begin by looking at "derivations":
11.8.1 DEFINITION A linear operator D e L{A, A) on an algebra A is

11.8 Quasicommuting Systems 509
called a derivation if
D{ab) = a(Db) + (Da)b for each a,b <E A (11.8.1.1)
We shall write
DBL(A, A) ={De BL(A, A): D is a derivation} (11.8.1.2)
for the bounded derivations on a normed algebra A.
Evidently DBL(A, A) is a closed linear subspace of BL(A, A), and is
also closed under the formation of "commutators":
D,D' e DBL(A, A) =^ D'D - DD' <E DBL(A, A). (11.8.1.3)
Each element a £ A gives rise to the "inner derivation" of (11.6.9.12):
aeA^La-Rae DBL(A, A). (11.8.1.4)
The ordinary product of derivations is not usually a derivation. We have
however Leibniz9 rule: if D £ DL(A, A) and a,b G A then by induction
L>»(a6) = J2 [) [Dn~ra){Drb) for each n <E N (11.8.1.5)
r=0 ^ '
The Kleinecke-Sirokov theorem says that bounded derivations breed quasi-
nilpotents:
11.8.2 THEOREM If D e DBL(A, A) and a <E A satisfy either
D2a = 0 (11.8.2.1)
or
a(Da) = (Da)a (11.8.2.2)
then
||(£>a)n||1/n—>0 asn—> oo (11.8.2.3)
Proof: If D <E DBL(A, A) and x G AN satisfy
D2(xn) = 0 for each n <E N (11.8.2.4)
then we claim
Z)n(x1x2 • • • xn) = (n!)(Z)x1)(Z)x2) • • • (Dxn) for each n <E N
(11.8.2.5)

510
11. Multiparameter Spectral Theory
and
Dn+1{x1x2 • • • xn) = 0 for each n <E N (11.8.2.6)
To see this inductively assume both (11.8.2.5) and (11.8.2.6) for n = k and
apply (11.8.1.5) with n = k + 1, a = x1x2- — xk and b = xk+1 to derive
(11.8.2.5) for n = k + 1, and hence also (11.8.2.6) for n = k + 1. Taking
xn = a for each n and assuming (11.8.2.1) thus gives
||(Pa)"|| < ||i?|r||a|r/»! (11.8.2.7)
and hence (11.8.2.3), since (n!)_1/n ->0asn->oo. For the other
implication write Aa for the inner derivation on BL(A, A) obtained by substituting
La for a in (11.8.1.4) and observe
Aa{D) = -LDa (11.8.2.8)
Thus if (11.8.2.2) holds then
A2a(D) = 0 (11.8.2.9)
We may now use this in the same way as we have just used (11.8.2.1) to
see that
\\(Da)n\\ = \\(-LDan < l|A0|H|2>ir/»! < 2»||a|r||2?|r/n! ■
(11.8.2.10)
For example, either component of Theorem 11.8.2 gives implication
Da = 1 =^ 1 quasinilpotent =^ A = {0} (11.8.2.11)
The second part says that if 0 ^ 5 £ C then
Da = sa =r> a quasinilpotent (11.8.2.12)
Here we can say more: since
Da = sa=^ D{an) = nsan (11.8.2.13)
it is clear from Theorem 9.2.3 and Theorem 9.3.2 that a is actually nilpo-
tent:
Da = sa, n\s\ > \\D\\ =^ an = 0 (11.8.2.14)
The second part of Theorem 11.8.2 also tells us that bounded derivations
on commutative normed algebras are essentially trivial:
A commutative, D e DBL(A, A) =^ D{A) C Radical(A) (11.8.2.15)

11.8 Quasicommuting Systems
511
In particular
A = C^fl) =► DBL{A,A) = {0} (11.8.2.16)
From either part of Theorem 11.8.2 it is clear that if a, 6 £ A then
ba — ab £ comm(a) => ba — ab quasinilpotent (11.8.2.17)
and
ba — ab £ comm(6) =>• 6a — a6 quasinilpotent (11.8.2.18)
When both conditions hold at once then (a, 6) £ A2 is called a "quasi-
commuting pair":
11.8.3 DEFINITION b £ Am is said to quasi-commute with a £ An if
6a - a6 £ Anm commutes with (a, 6) £ An+m (11.8.3.1)
in the sense that each bka^ — afik commutes with each a^, and with each
bk,. If this holds with b = a then a £ An is called a quasi-commuting
ra-tuple.
Here we are interpreting the "commutator" 6a — a6 as a rectangular
matrix of elements of A, rather in the spirit of Theorem 11.4.3; alternatively
we can write it out (in one of two ways) as an (ram)-tuple. To prove spectral
mapping theorems for quasi-commuting systems, we need another auxiliary
concept:
11.8.4 DEFINITION The system 6 £ Am is said to be left invariant
under a £ An if
( 2^ Abk ay C closure 2J Abk J for each j £ {1,2,..., n}
(11.8.4.1)
right invariant under a £ An if
ay I 2~^ 6^ C closure I Y^ 6^ for each j £ {1,2,..., n}
(11.8.4.2)
and completely invariant under a G An if
b — t is left and right invariant under a £ An for each t £ Cm (11.8.4.3)

512 11. Multiparameter Spectral Theory
For example, if a £ An commutes with b £ Am then (11.8.4.3) holds; if
t £ Cm is not in <jleft(6) then b — t e Am is left invariant under arbitrary
a £ An. Quasi-commutivity induces this kind of invariance:
11.8.5 THEOREM Suppose a £ An and b £ Am: If ba - ab commutes
with a then
(6,6a — ab) is completely invariant under a (11.8.5.1)
If ba — ab commutes with (a, 6) then for each system of polynomials / £
(/(6),6a — ab) is completely invariant under a (11.8.5.2)
If ba — ab commutes with (a, 6) and a is commutative then for arbitrary
g 6 Poly*+m
(g(a,6),6a — a6) is completely invariant under a (11.8.5.3)
Proof: Begin with an observation about derivations: if D £ T)L{A,A) and
/ £ Polyn then there are systems of polynomials /',/" £ Poly£ for which
there is implication, for arbitrary a £ An,
n n
Da £ comm(a)'1 =► Df(a) = £ /;.(a)(D^) = ^(^O/y'W
y=i y=i
(11.8.5.4)
This is trivial if / is either a constant or a coordinate and trivially inherited
by sums: if (11.8.5.4) holds with f = g and with f = h then it must also
hold with f = g + h. To see that this is also inherited by products observe
(assuming that (11.8.5.4) holds with f = g and f = h)
n
D{g{a)h{a)) = {Dg{a))h{a) + g{a)Dh{a) = ]T(0y(a)/i(a) + g{a)h'j{a))Daj
3 = 1
noting that Daj commutes with h(a) for each j. Thus if / = gh then
/y = 9'jh + gh'jy and similarly /j' = g'Jh + gh'j, for each j £ {l, 2,..., n}.
Towards (11.8.5.1) we must prove that for each t £ Cm and r £ Cmn
(6 — t, ba — ab — r) is left invariant under a (11.8.5.5)
By (11.10.2.17) this is trivial if r ^ 0 £ Cmn, and if r = 0 and N is the

11.8 Quasicomnruting Systems
513
closed left ideal generated in A by (6 — t,ba — ah) we have for each j\k,i
(11.8.5.6J
{bka{ - a{bk)aj = a^b^ - a{bk) £ N
so that Naj C N for each j. The argument for right invariance is of course
the same. Towards (11.8.5.2) we use (11.8.5.4) to see that, since ba — ab
commutes with 6,
m
f{b)aj - ajf{b) = ]T f'k{b){bkaj - ajbk) for each j £ {1,2,... ,n}
whenever / £ Polym is a polynomial in m variables: now if instead / £
Poly^ and r £ Cp take N to be the closed left ideal generated by (/(6) —
r, ba — ab) and see that
m
{fk{t>)-rk)*j = a,-(/*(6)-rfc)+E/**(6)(6*0>-°y6*) e * for each -?>fc
1=1
(11.8.5.7)
which together with the second part of (11.8.5.6) gives again Na ■ C JV for
each j. Finally, for (11.8.5.3), apply (11.8.5.2) with (a,6) in place of b to
see that
((<7(a,6),6a — a&,aa — aa) is completely invariant under a (11.8.5.8)
and then specialize to a commuting system a £ An. ■
The almost left and right spectra, and the approximate eigenvalues, of
a quasi-commuting system have the "projective property":
11.8.6 THEOREM If a £ An and b £ Am are such that ba - ab is
commutative and commutes with b then there is equality, for each u £
{aleft,arisht,fleft,frisht},
"(*)="6a-a6=o(*) (11.8.6.1)
If instead a is quasi-commutative and commutes with ba — ab then there is
equality
<"»«-«»=o(6) =<".=. (*) (11.8.6.2)
Hence if a is quasi-commutative and quasi-commutes with 6 then
W(6)=a;0=0(6) (11.8.6.3)

514
11. Multiparameter Spectral Theory
Proof: If c = ba- ab £ Anm then (11.3.4.1) from Theorem 11.3.4 and
(11.8.2.8) give
u[b) = uc=c(b) = uc=0(b) (11.8.6.4)
for each of the four sets u: this is (11.8.6.1). Towards (11.8.6.2) we begin
with the observation, using (11.2.6.3) and (11.2.6.4), that with no
restriction on (a,6) £ An+m there is equality uj(a,b) = wba_ab=0(a,b) and hence
inclusion
<"«=«(*) C wto_ot_0(6) (11.8.6.5)
To reverse the inclusion we may use induction on ra, and indeed it is
sufficient to prove (11.8.6.2), and hence (11.8.6.3), when n = 1. If therefore
a = ax commutes with ba—ab £ Am then by (11.8.5.1) the system (6, ba—ab)
is completely invariant under a £ A; but now we may repeat the proof of
the first part of Theorem 11.3.4. The complete invariance guarantees that
with either T = row(jR6_t|6o_o6) or T = iov/(Lb_tM_ab), the conditions
of either Theorem 11.3.2 or Theorem 11.3.3 are satisfied with V = Ra and
U = La: by the argument of the first part of Theorem 11.3.4 it follows that
u(b,ba - ab) = w0=0(6,6a - ab) (11.8.6.6)
for each of the four sets oj. This proves (11.8.6.2) when n = 1, and hence for
more general n: to make the induction observe that for each j = 1,2,..., n—
1 the element aJ+1 commutes with each commutator daj+1 — aJ+1rf, where
d = (6, al9... ,a;). From (11.8.6.1) and (11.8.6.2) we then immediately get
(11.8.6.3). ■
Theorem 11.8.6 is only the analogue of the first part of Theorem 11.3.4;
unlike the commutative case the analogue of the second part is not here a
trivial consequence. The point is that if a £ An is quasi-commutative and
/ £ Poly^1 is a system of polynomials then it is possible that a does not
quasi-commute with b = f(a). In spite of this, with a bit of extra work,
the spectral mapping theorem does follow:
11.8.7 THEOREM If a £ An is quasi-commutative and quasi-commutes
with b £ Am then for each g £ Poly£+m and each w £ {aleft,aright,fleft,
-righty there is equality
ug{a,b) = ua=ag{a,b) = [j ua=3g{s,b) C ug{u{a),b) (11.8.7.1)
aGw(fl)
In particular if a £ An is quasi-commutative and / £ Poly^1 there is equality
ojf(a) = foj(a)
(11.8.7.2)

11.8 Quasicommuting Systems
515
Proof: If the commutator ba — ab is commutative and commutes with
(a,6), hence also with g(a,6), then it is an application of (11.3.4.1) from
Theorem 11.3.4 that
c = ba- ab =>■ u>g(a, b) = u>c=cg(a, b) = wc=0g{a, b) (11.8.7.3)
We claim that also
u(g(a,b),ba - ab) = va=a{g{a,b),ba - ab) (11.8.7.4)
and hence
"ba-ab=0<l{a> b) C Ua=a<l{a> b) (11.8.7.5)
This again is induction on ra, and it is sufficient to prove it in the case n = 1.
In the case n = 1 the system a £ An is of course commutative, and since
ba — ab commutes with (a, 6) then by (11.8.5.3) the system (g(a, 6),6a — ab)
is completely invariant under a, but now (11.8.7.4) follows by the same
argument as (11.3.4.1) from Theorem 11.3.4. This gives (11.8.7.5), which
together with (11.8.7.3) gives the first part of (11.8.7.1), and hence of course
(11.8.7.2). For the rest of (11.8.7.1) we observe that
ua=8g{", *>) = "a=sd{s, *>) ^r each s £ Cn (11.8.7.6)
Indeed if a — s £ An is left invariant under b £ Am then, with N =
closure2y=i A(ay - s;) and M = {d £ A : Nd C N},
g{a,b) £ Mp and g{a,b) - g{s,b) £ Np (11.8.7.7)
It is sufficient to prove this for p = 1, and this is established by verifying
it if g(a,b) = /(6), if g(a,b) = a;-, and if g is the sum or product of
polynomials for which it holds. Thus if a £ An is completely invariant
under b £ Am then (11.8.7.6) holds for the almost left and right spectrum,
since (g(a,b) — r,a — s) always generate the same closed left and right
ideals; by taking annihilators of ideals and considering enlargements we
also obtain (11.8.7.6) for approximate eigenvalues. Under the assumptions
of Theorem 11.8.7 we have by (11.8.5.1) that
(a, ba — ab) is completely invariant under b (11.8.7.8)
and hence
c = ba - ab =>• ^(ajC)=(5j0)y(a,6) = w(0|C)=( ,0)g{s,b) for each 5 £ Cn
(11.8.7.9)
applying (11.8.7.6) with (a, ba — ab) in place of a. But now (11.8.7.3)
conveys f11.8.7.9) ^ fU.8.7.6), and finishes the proof of (11.8.7.1) ■

516
11. Multiparameter Spectral Theory
The conditions (11.8.6.1) and (11.8.6.2) are neither necessary nor
separately sufficient for the conclusions of Theorem 11.8.6. For example, in
the algebra A of 3 x 3 matrices, take
U =
Then
and
ro
o
-0
1 0"
0 1
0 0.
V =
10 0
0 0 0
0 0 -1
and
W =
VU-UV = U but a{V) ± au=u{V)
11 0 -
0 0-1
0 0 -1.
(11.8.7.10)
(11.8.7.11)
W = VU + UV + V but <j{W) ^ <jv=v [W) (11.8.7.12)
Thus if a = U and b = V then (11.8.6.2) holds but not (11.8.6.1); if instead
a = V and b = U and g(a, b) =ba + ab + a = W then (11.8.6.1) holds but
not (11.8.7.1), and with 5 = 0 not (11.8.7.6). If instead A is the algebra of
upper triangular p x p matrices then equality (11.8.7.2) holds for arbitrary
cGAn and / £ Poly™. It is clear from their proofs that Theorem 11.8.6 and
Theorem 11.8.7 sit at the beginning of a chain of increasingly uninteresting
generalizations:
11.8.8 THEOREM If a £ An and b £ Am then each of the following
conditions is sufficient for equality u>(b) = o:a=a(b) :
D[a is quasi-commutative and commutes with D[+1a for all
r £ N U {0} and D^a commutes with b for some p £ N U {0}
{Dab, D^b,..., Drab) is commutative for each r eM and a is
quasi-commutative and commutes with D%b for some
p £ N U {0}
a is quasi-commutative and Dba = tb for some t £ Cn
(11.8.8.1)
(11.8.8.2)
(11.8.8.3)
Proof: The sufficiency of (11.8.8.1) is derived from Theorem 11.8.6 by
repeatedly using the conclusion (11.8.6.3) in the derivation of (11.8.6.1). For
the sufficiency of (11.8.8.2) observe that the system c = (b,Dab,Dlb,...,
D*b) is completely invariant under the system a, giving wc=Q{b) C w0=0(6).
Finally (11.8.8.3) is clearly sufficient if t = 0, and in any case there is
inclusion
uj(a) C ujb=0(a) Uuj(a + t)
(11.8.8.4)
Thus by the boundedness of u>(a) there is equality u>(a) = u>b=0(a) and
hence, since u>(b) = {0} and uj(a) ^ 0, equality uj(b) = c<;a_a(6) again. ■

11.9 The Taylor Spectrum
517
11.9 THE TAYLOR SPECTRUM
The "left" and "right" spectra of Definition 11.1.1 do not quite add up
to an adequate "spectrum" for a system of elements a £ An: we need
some "middle" terms. For commuting systems of linear operators, the right
way to do this has been discovered by Joseph Taylor: we shall tentatively
make an extension to noncommuting systems in normed algebras. For
an ra-tuple a £ An we need as an auxiliary the "exterior algebra on n
generators"
An = A(dz) = Alg{dz1,dz2,...,dzn) (11.9.0.1)
Whimsically perhaps we shall identify these generators with "differentials"
induced by the coordinates z = (z1,z2,...,zn). Exterior multiplication is
associative and anticommutative:
dz- A dz{ = —dz{ A dzj for each i,j and dzx A dz2 A • • • A dzn ^ 0 (11.9.0.2)
so that k{dz) is a linear space of dimension 2n, which is the direct sum
of 7i + 1 subspaces kv{dz) of "homogeneous forms of degree p," of
dimension (p). For each j the mapping Ay of multiplication by dz^ sends
Ap~1(dz) into KP{dz) for each p = 1,2,...,ra, and sends An(dz) into
0.
11.9.1 DEFINITION If X is a linear space we write
An{X) =A{X;dz) =X®k{dz) = I x0 + ]T ]T xj ® dz-xj <E X \
{ p=1 IjI=p J
(11.9.1.1)
where
J = Ui>J2> • • • >Jp) => \J\ = P and dzj = dzj\ A dzj2 A • • • A dzjp (11.9.1.2)
If x £ Xn is an ra-tuple of elements we shall write
n
Ax(x) = Y,xj ® rf*y € A£(X) (11.9.1.3)
If T £ L(X, X)n is an n-tuple of linear operators we shall write
n
A(T) = E Ti ® Ai e L(AnP0> An(X)) (11.9.1.4)

518
11. Multiparameter Spectral Theory
for the Koszul complex of T, or equivalently the induced sequence
(11.9.1.5)
An(T)
A»(X)< A^X)
A2(T) , AX(T)
For example if n = 2 the sequence (11.8.1.5) can be represented by the
operator matrices
[ -T2 Tx ]
[';]
(11.9.1.6)
while the operator (11.9.1.4) is represented by the matrix
0
0
0
0
0
T2
0
0
0
Ti
0"
0
0
0
:
'X"
X
X
_x_
—
"XI
x
X
_x_
(11.9.1.7)
Both we and the reader will find it helpful to examine all theorems and
definitions for the Taylor spectrum separately in this case. More generally
we can express the Koszul complex of an (n + l)-tuple (T, S) in terms of
the Koszul complex of the ra-tuple as a sort of lower triangular operator
matrix:
A(T,S) =
A(T) 0
5V A(T)
An(X)
LAn(X)
An(X)
LAn(X)
(11.9.1.8)
where the matrix 5V is a 2n x 2n diagonal matrix whose diagonal entries
are ±5: to see this represent the vectors x £ An+1(X) = A(dz, dw) in a
column in which all the terms involving dw are kept to the last: the reader
should experiment with the case n = 2.
The commutativity of the operator n-tuple A(T) is equivalent to the
chain condition (10.3.0.1) for the pair (A(T),A(T)), or its extension
(10.3.1.20) for the sequence (11.9.1.5), which of course is tacitly assumed
when we call it the Koszul "complex:"
11.9.2 THEOREM If T e L(X,X)n is an n-tuple of linear operators
then the following are equivalent:
T is commutative
Ap(r)Ap_1(r) = 0 for each p <E {1,2,... ,n}
A(T)2 = 0
(11.9.2.1)
(11.9.2.2)
(11.9.2.3)

11.9 The Taylor Spectrum
519
Proof: The equivalence of the first two conditions may safely be left to the
reader: for the equivalence of the second two we compute
Ap(T)(Ap-1(T)(xJ- <g> dzj = Y, (TiTi' ~ Ti'Ti)x3 ® dzi A dzi' A dzj
l<i<i'<n
(11.9.2.4)
for each j = (j\,j2, • • • )Jp-i) an(i eacn x £ AP_1(X). ■
If X = A is a linear algebra and a G An then A1 (a) = Y^=i ai ® dzi
can be treated as an element of the linear algebra An(A) = A <g> A(dz): the
argument of Theorem 11.8.2 now gives
a commutative <=> A1 (a)2 = 0 (11.9.2.5)
More generally we interpret Ap(a) for each p £ {0,1,..., n} as certain kinds
of rectangular matrix over the algebra A.
A system T £ BL(X, X)n of bounded operators on a normed space can
be classified as "Taylor nonsingular" if the pair of operators (A(T), A(T))
is nonsingular in the corresponding sense from Definition 10.3.1:
11.9.3 DEFINITION The n-tuple T £ BL(X,X)n will be called Taylor
invertible if there are U and V in BL(An(X), A.n(X)) for which
VA(T)+A(T)U = I (11.9.3.1)
and called almost Taylor invertible if there are {Um) and (Vm) in
BL(An(X),An(X)) with
||VmA(T) + A(T)Um - I|| — 0 and sup ||tfj| + sup ||VJ| < oo
m m
(11.9.3.2)
The n-tuple T £ BL(X,X)n will be called Taylor nonsingular if there are
k > 0, h > 0 for which
for each x £ An(X),x £ Disc(0;/i||A(T)x||)+A(T) Disc(0;A;||x||) (11.9.3.3)
and will be called almost Taylor nonsingular if there are k > 0, h > 0 for
which
for each x £ An(X),x £ cl(Disc(0 ; /i||A(T)x||) + A(T)(Disc(0 ; A;||x||))
(11.9.3.4)
The ra-tuple T £ BL(X, X) will be called linearly Taylor nonsingular if
A(T)-1(0)CA(T)An(X) (11.9.3.5)

520
11. Multiparameter Spectral Theory
and will be called almost linearly Taylor nonsingular if
ACT)"1^) C clA(T)An(X) (11.9.3.6)
If T £ BL(X,X)n is Taylor invertible then (11.9.3.1) holds for
operators U and V on An(X) of a special form: there are T',T" in BL(X,X)n
for which
A?{T")lL{T) + A{T)A^{Tf) = I (11.9.3.7)
where we define, for arbitrary S £ BL{X,X)n,
n
At(S) = ]TSy<g>A; (11.9.3.8)
i=i
Here Ay is the Hilbert space adjoint of the operator Ay induced by the
obvious inner product on the space A.(dz):
n
(a ; P) = a0P0 + J2J2 "jPjdzj (11.9.3.9)
p=i |j|=p
This is because, when U, V, and A(T) are represented as operator matrices,
many of the terms of U and V do not participate in the action, and when
these are replaced by zeros we find that (11.9.3.1) is reduced to (11.9.3.7).
One simple observation is clear from (11.9.3.7): if T € BL(X,X)n has
a left (equivalently a right) inverse S £ BL(X, X)n for which
{Tjij e {1,2,...,n}} U {Sjij e {1,2,...,n}} is commutative
(11.9.3.10)
then T is Taylor invertible: if (11.9.3.10) holds then
Y^ SjTj. = I=> Af(5)A(T) + A(T)Af(5) = I (11.9.3.11)
j
We can make the same comments about almost Taylor invertibility.
Relative to the closed set of commuting n-tuples, almost invertible and
almost nonsingular ra-tuples form open sets:
11.9.4 THEOREM If X is a normed space then
{T £ BL(X, X)n:T commutative and almost Taylor invertible}
(11.9.4.1)

11.9 The Taylor Spectrum
521
and
{T G BL(X,X)n:T commutative and almost Taylor nonsingular}
(11.9.4.2)
are open subsets of {T G BL(X,X)n:T commutative}.
Proof: Observe that the mapping T —► (A(T), A(T)) is continuous and
takes commuting n-tuples T G BL(X,X)n into chains (A(T), A(T)) G
BL(An(X),An(X),An(X)), and then apply Theorem 10.3.5 and
Theorem 10.3.8. ■
When the normed space X is complete, several of the Taylor nonsin-
gularities coalesce:
11.9.5 THEOREM If T G BL(X,X)n is a commuting n-tuple on a Ba-
nach space X then the following are equivalent:
T is Taylor nonsingular (11.9.5.1)
T is almost Taylor nonsingular (11.9.5.2)
T is linearly Taylor nonsingular (11.9.5.3)
If X is complete then also
{T G BL(X,X)n: T is commutative and Taylor invertible} (11.9.5.4)
is an open subset of {T G BL(X,X)n : T commutative}.
Proof: The first part is by application of Theorem 10.5.4 to the chain
(A(T), A(T)), noting that in this case the second part of condition (10.5.4.2)
follows from the first. The last part is the end of Theorem 10.5.6. ■
Once again we are unable to show that, for complete X and commuting
TGBL(X,X)n,
T almost Taylor invertible => T Taylor invertible (11.9.5.5)
Under passage to enlargement, almost Taylor nonsingularity takes several
forms:
11.9.6 THEOREM If X is a normed space and T G BL(X,X)n is com-

522
11. Multiparameter Spectral Theory
mutative then the following are equivalent:
Q(T) is almost linearly Taylor nonsingular (11.9.6.1)
T is almost Taylor nonsingular (11.9.6.2)
Q(T) is Taylor nonsingular (11.9.6.3)
Proof: Apply Theorem 10.5.2 to the chain (A(T), A(T)). ■
Taylor nonsingularity is preserved under duality:
11.9.7 THEOREM If T e BL(X,X)n is arbitrary then the following are
equivalent:
T* is almost Taylor nonsingular (11.9.7.1)
T is almost Taylor nonsingular (11.9.7.2)
Tf is Taylor nonsingular (11.9.7.3)
Proof: Specialize Theorem 10.4.3 to R = I and apply it to (A(T), A(T)). ■
Different definitions of "Taylor nonsingularity" give rise to different
kinds of "Taylor spectrum":
11.9.8 DEFINITION If X is a normed space and T <E BL(X,X)n is
arbitrary then the Taylor invertible spectrum of T is the set
<jTaylor(r) = {seCn:T- si is not Taylor invertible} (11.9.8.1)
The Taylor almost invertible spectrum is the set
(7Taylor(T) ={seCn:T- si is not almost Taylor invertible} (11.9.8.2)
The Taylor singular spectrum is the set
rTaylor(T) ={seCn:T- si is not Taylor nonsingular} (11.9.8.3)
and the Taylor almost singular spectrum is the set
fTaylor(T) ={seCn:T -si is not almost Taylor nonsingular}
(11.9.8.4)
For commuting systems of operators in incomplete normed spaces the
almost invertible and the almost singular spectrum are compact sets:

11.9 The Taylor Spectrum
523
11.9.9 THEOREM If T G BL(X,X)n is a commuting system on a
normed space X then
n
aTaylor(T) = claTaylor(r) C JJ v{T) (11.9.9.1)
and
fTaylor(T) = clfTaylor(T) C claTaylor(r) (11.9.9.2)
Proof: Both the almost singular spectrum and the almost invertible
spectrum are continuous counterimages, by the mapping T — zl : Cn —►
{S G BL{X,X)n: S commutative}, of closed subsets of {S G BL(X,X)n: S
commutative}, namely, the complements of the open subsets (11.9.4.2) and
(11.9.4.1). From (11.9.3.10) it follows that
n
aTaylor(r) Q ^(r) Q JJ ^ (11.9.9.3)
3 = 1
whenever A is a commutative subalgebra of BL(X, X) containing {T;-:
j = 1,2,...,n}, and from the "almost" analog of (11.9.3.10) we have, in
the same circumstances,
n
^Taylor(r) Q -^ Q JJ ~(r) (11.9.9.4)
y=i
Finally, to finish (11.9.9.2), we have
rTaylor(T) C aTaylor(T) and rT*yloT{T) C aTaylor(T) ■
(11.9.9.5)
The almost singular spectrum is nonempty, and subject to the spectral
mapping theorem for polynomials:
11.9.10 THEOREM If T G BL(X,X)n and S G BL(X,X)m form a
commuting system (T,S) G BL(X,X)n+m on a normed space X then
fTaylor^ = f£^>r(T) (11.9.10.1)
and if / G Poly™ then
fTaylor/(T) = frT^loT(T) (11.9.10.2)
Proof: By induction on m it is sufficient to prove (11.9.10.1) when m = 1.
To do this suppose that 5 G Cn and t G C satisfy \s,t) G fTaylor(T,5), so

524
11. Multiparameter Spectral Theory
that
/rA(T-sJ) 0
U(S-tOv A(T-sJ)
is not almost exact
and hence by Theorem 10.9.5,
(A(T - si), A(T - si)) is not almost exact (11.9.10.4)
Thus the right-hand side of (11.9.10.1) is included in the left. Conversely
if 5 e Cn satisfies (11.9.10.4) then by Theorem 10.3.7 either
A(T - sI)A : An(X)/cl A(T - sI)An{X) —► An{X) is not bounded below
(11.9.10.5)
or
A(T - sI)A : An{X) —> A(T - 5/)_1(0) is not almost open (11.9.10.6)
By Theorem 11.3.3 there is t <E C for which (11.3.3.1) either
col(A(T - si), (S - tI)v)A is not bounded below (11.9.10.7)
or
row(A(T - si), (5 - tI)v)A is not almost open (11.9.10.8)
By Theorem 10.9.6, in particular (10.9.6.5) and the analog of (10.9.6.6)
for almost exactness, each of (11.9.10.7) and (11.9.10.8) imply (11.9.10.3).
This proves (11.9.10.1) when m = 1, and hence for arbitrary m. Towards
(11.9.10.2) suppose that A C BL(X, X) is a commutative subalgebra
containing {T:j = 1,2,...,n}: then by (11.9.3.11) there is inclusion
fTaylor(r) Q ^Taylor^ Q ^ft(T) = (J^T) (11.9.10.9)
We claim that this gives equality
fTaylor(T) = {4>{T):4> e fTaylor(A)} (11.9.10.10)
where, as in (11.4.3.4),
fTaylor(A) = {<t>e A*: for each m <E N and S <E Am,<f>{S) G fTaylor(5)}
(11.9.10.11)
Indeed fTaylor(A) is a compact subset of A*, and since by (11.9.10.1) the
family
{{4> e A*:4>{S) e fTaylor(5)})nGN)5GA, (11.9.10.12)
A(T - si) 0
[(S-tI)v A(T-sI)
(11.9.10.3)

11.9 The Taylor Spectrum
525
has the "finite intersection property," fTaylor(A) is nonempty. The right-
hand side of (11.9.10.10) is obviously contained in the left; conversely if
s e f Taylor (T) then the family Qf sets
({</> e A*:*{T) = s and <j>{S) G rTaylor(5)})mGN5GAm (11.9.10.13)
has the finite intersection property, and if <f> £ A* is taken from its
intersection then 4>{T) = s. Equality (11.9.10.10) now gives (11.9.10.2): if
/ e Poly;? then
fTaylor/(r) = {^/(T): ^ ^Taylor (r)|
= {f<t>{T):<t> e fTaylor(T)} = /fTaylor(r) m
It is clear from Theorem 11.9.10 and the inclusion (11.9.9.5) that
the almost invertible spectrum aT*yloT(T) of a commuting n-tuple T G
BL(X,X)n is nonempty. To see that it has the spectral mapping theorem
for polynomials we can use Theorem 10.4.5 and its analogues (10.4.6.5) and
(10.4.6.6):
11.9.11 THEOREM If T e BL(X,X)n is a commuting n-tuple on a
normed space X then
aTaylor(r) = rTaylor(Lr) = rTaylor(i?T) (11.9.11.1)
and
<7Taylor(r) = fTaylor(Lr) = fTaylor(£r) (11.9.11.2)
Proof: For (11.9.11.1) apply the extended version of (10.4.6.1) and
(10.4.6.2) to the Koszul complex (11.9.1.5), augmented with zeros at
either end. For (11.9.11.2) use (10.4.6.5) and (10.4.6.6) ■
The Taylor spectrum of a tensor product n-tuple is the cartesian
product of the spectra of the factor operators:
11.9.12 THEOREM If T e BL(X, X)n is a commuting system of
operators on a normed space X then
n n
II ?(*)) £ fTaylor(T®) C aTaylor(T®) C H a(ry) (11.9.12.1)
y=i y=i
where we write
r® = (rf,rf,...,r®) withrf = /®---®rf ®---®/ (11.9.12.2)

526
11. Multiparameter Spectral Theory
If the space X is complete then
n
fTaylor^®) = JJ ^j (11.9.12.3)
Proof: The middle inclusion is just (11.9.9.5), and the final inclusion is
(11.9.9.1) together with (11.7.5.4) from Theorem 11.7.5. Towards the first
inclusion we claim that, if S G BL(X, X) is such that (T, S) G BL(X, X)n+1
is commutative then
fTaylor(T) x t{S) C tTaylor(T® <g> /, 7® <g> 5) (11.9.12.4)
To see this apply Theorem 11.7.10 with (A(T) - si, A(T) - si, A(T) - 5/,
(5 - tl)v, -(5 - *J)V) in place of {R,S,T,U,V). The first inclusion of
(11.9.12.1) is now (11.9.12.4) and induction. For (11.9.12.3) observe that if
X is complete then
f(Tj)=a(Tj)=a(Tj)
for each j, giving equality throughout (11.9.12.1). ■
11.10 ALGEBRAIC AND ESSENTIAL SPECTRA
The various kinds of "Taylor spectra" for an n-tuple a G An of normed
algebra elements are obtained by applying the conditions of Definition 10.7.1
to the Koszul complex chains (A (a — 5), A (a — s)) G An(A)2, where A (a — s)
is given by (11.9.1.3) with x = a — s:
11.10.1 DEFINITION If a G An for a normed algebra A then the Taylor
invertible spectrum of a with respect to A is the set
is not (left, right)invertible in An(A)2}
(11.10.1.1)
and the Taylor almost invertible spectrum of a with respect to A is the set
(11.10.1.2)
a^aylor(a) = {s G Cn : (A(a - s), A(a - s))
is not almost (left, right) invertible in An(A)2}
The Taylor almost singular spectrum of a with respect to A is the set
f^aylor(a) ={seCn. (A(a _ fi) A(a _ fi))
A v v J v (11.10.1.3)
is not (left, right) bounded below in An(A) }

11.10 Algebraic and Essential Spectra
527
The first two concepts are expressible in terms of the operators La G
BL(A,A)n, or the operators Ra <E BL(A,A)n:
11.10.2 THEOREM If A is a normed algebra and a G An is a commuting
n-tuple then
<r£aylor(a) = TT^lor{La) = rT^loT{Ra) (11.10.2.1)
and
c£*ylor(a) = fTaylor(La) = fTaylor(£a) (11.10.2.2)
Proof: For (11.10.2.1) apply the extended versions of (10.7.7.1) and
(10.7.7.2) to the Koszul complex
(0, An(a - 5),..., A2(a - s),A1{a - s),0) (11.10.2.3)
while for (11.10.2.2) use (10.7.7.3) and (10.7.7.4) ■
From the second part of Theorem 11.10.2 we obtain the compactness
and nonemptiness, and the spectral mapping theorem, for the "Taylor
almost invertible spectrum" of a commuting system of normed algebra
elements: for example, using (11.9.10.2),
/<^aylorM = A~Taylor(£J = fTaylo7(£J = fTaylor(£/(a)) = *Aayl°7(a)
(11.10.2.4)
In the special case A = BL(X, X) Definition 11.10.1 reproduces
Definition 11.9.8:
11.10.3 THEOREM If A = BL(X, X) and T e An then
aTaylor(r) = aTaylor(r) (11.10.3.1)
and
^Taylor (r) = ~Taylor(r) (11.10.3.2)
and
f Taylor^ = fTaylor^ (11.10.3.3)
Proof: Equality (11.10.3.1) is given by the observation (11.9.3.2), and
equality (11.10.3.2) by the analogue of (11.9.3.7) for almost Taylor invert-
ibility. For equality (11.10.3.3) we specialise Theorem 10.4.1 to R = I ■

528
11. Multiparameter Spectral Theory
We have no analogue of Theorem 11.10.2 for f Aay or in more general
A: the reader will remember from our comments immediately preceding
Theorem 10.7.5 that we have not even proved that fAayor(T) is a closed
set.
The obvious way to define an "essential Taylor spectrum" would be
to use Definition 11.10.1 with the Calkin algebra A = B/J, where B =
BL(X,X) and either J = KL(X,X) or J = clKL0(X,X); we will prefer
to be more "spatial" and use the "essential exactness" of Definition 10.6.1
and Definition 10.6.6:
11.10.4 DEFINTION The Taylor-Fredholmspectrum of T G BL(X, X)n
is the set
aJsasylor(T) = {seCn : (A(T - si), A(T - si)) is essentially invertible}
(11.10.4.1)
and the Taylor almost Fredholm spectrum is the set
^ylor(r) ={^Cn: (A(T - si), A(T - si))
ess v ; i v v ;> v ;; (11.10.4.2)
is essentially almost invertible}
The essential Taylor spectrum is the set
fJsaylor(r) ={seCn : (A(T - si),A(T - si)) is nearly essentially exact}
(11.10.4.3)
and the weak Taylor spectrum is the set
*l7iot(T) = {s€Cn: (A(T - si), A(T - si)) (n.10.4.4)
is essentially (left, right)one-one}
Of course the first two sets coincide for complete spaces, and there is
the obvious inclusion between them in general; also when X is complete
then Theorem 10.6.4 says that
T commutative => aJsasylor(T) = aJsaylor(T) is compact in Cn
(11.10.4.5)
For complete spaces the "near essential exactness" of Definition 10.6.6 used
in the makeup of the essential Taylor spectrum coincides with "essential"
and "almost essential" exactness; also by Theorem 10.6.7
T commutative => r^loT{T) = rTaylorp(r) (n.10.4.6)
so that the compactness and nonemptiness, and the spectral mapping
theorem, for the essential Taylor spectrum follow from the corresponding
theorems for the Taylor spectrum applied to the essential enlargement.

11.10 Algebraic and Essential Spectra
529
We can also define an index for a commuting Taylor-Fredholm system
of operators, and a Taylor- Weyl spectrum; we need to work with the more
complicated version of the Koszul complex:
11.10.5 DEFINITION The Taylor-Weyl spectrum of T G BL(X,X)n is
the set
"S^'Cn = {^C": (0,A»(r-«I),...,A1(T-«J),0) ^ g
is not spatially Weyl}
If T G BL(X, X)n is a commuting Taylor-Fredholm system then the Taylor
index of T is the number
index(T) = Euler(0, An(T),..., A^T^O) (11.10.5.2)
Theorem 10.6.4 says that, for complete spaces, the Weyl spectrum of
a commuting n-tuple is a compact set; Theorem 10.6.5, extended to these
longer chains, says that
T Weyl <=> T Fredholm of index zero (11.10.5.3)
The openness of the set of Weyl n-tuples in the set of all commuting n-
tuples is consistent with the continuity of the index: we prove this only for
operators on Hilbert space. We begin with the representation of the Taylor
spectrum of a system of Hilbert algebra elements as the ordinary spectrum
of a single linear operator:
11.10.6 THEOREM If a G An is a commuting system in a Hilbert
algebra A then the following are equivalent:
a G An is Taylor invertible with respect to A (11.10.6.1)
A (a) + A (a)* is invertible with respect to An(A) (11.10.6.2)
□(A) = A(a)*A(a) + A(a)A(a)* is invertible with respect to An(A)
(11.10.6.3)
a G An is Taylor nonsingular with respect to A (11.10.6.4)
Proof: The equivalence of (11.10.6.1), (11.10.6.3) and (11.10.6.4) is by
application of Theorem 10.8.5 to the chain (A(a), A(a)) G An(A)2, noting that
An(A) = A ® An is also a Hilbert algebra. The equivalence of (11.10.6.2)
and (11.10.6.3) follows from
□(a) = (A(a) + A(a)*)2 (11.10.6.5)

530
11. Multiparameter Spectral Theory
which holds because (A(a),A(a)) is a chain. We sometimes call D(a) the
"Laplacian" of cG An.
A special case of Theorem 11.10.6 is A = BL(X,X) for a Hilbert space
X: more generally if A C B for Hilbert algebras A and B then
Theorem 11.10.6 gives us the "spectral permanence" of the Taylor spectrum: if
a £ An C. Bn is commutative then
rJaylor(a) = r£aylor(a) = a£aylor(a) = a^aylor(a) (11.10.6.6)
The reader should note that Theorem 11.10.6 does not say
rTaylor(r) = ^-^
What it does say is that
rTaylor(T) ={seCn:0e cU{T - si)} (11.10.6.7)
There is a similar description of the "Taylor-Fredholm" spectrum:
'eTssyl°rCn = {s€Cn:0e <7essD(r - si)} (11.10.6.8)
This is just part of the more comprehensive statement that all possible
definitions of "Taylor-Fredholm" coincide on Hilbert space:
11.10.7 THEOREM If T e BL(X,X)n is a commuting system on a
Hilbert space X then the following are equivalent:
T is Taylor-Fredholm (11.10.7.1)
T is essentially Taylor nonsingular (11.10.7.2)
T + KL(X, X) is Taylor invertible in BL(X, X)/ KL(X, X) (11.10.7.3)
T + KL(X,X) is Taylor nonsingular in BL(X,X)/KL(X,X) (11.10.7.4)
P(T) is Taylor nonsingular (11.10.7.5)
P(T) is Taylor invertible (11.10.7.6)
A(T) + A(T)* is Fredholm (11.10.7.7)
D(T) = A(T)*A(T) + A(r)A(T)* is Fredholm (11.10.7.8)
Proof: Implication (11.10.7.1) =J> (11.10.7.3) =J> (11.10.7.6) =J> (11.10.7.5)
is elementary, and the equivalence (11.10.7.2) <» (11.10.7.5) is (11.10.4.6).
Equivalence (11.10.7.3) <» (11.10.7.4) <» (11.10.7.7) <» (11.10.7.8) is an
application of Theorem 11.10.6 with A = BL(X,X)/KL(X,X), and finally
(10.6.8.9) gives (11.10.7.2) <» (11.10.7.1) ■

11.11 Functional Calculus
531
The Weyl spectrum of a commuting n-tuple T G BL(X,X)n cannot
be related to the normal operator D(T), in view of (8.8.5.4). Instead of
assembling the Koszul complex of (11.9.1.5) into the single operator A(T)
of (11.9.1.4), we put together its "odd" and "even" parts, writing
A(T) =
0 Aodd(T)
Aeven(T) 0
A°ddP0
LArnP0J
where
Arn(*).
(11.10.7.9)
Arw = © KM
p even
and
and A°dd(X) = 0 A'PO
p odd
(11.10.7.10)
Aeven(T) = 0 Ap{T) and Aodd(T) = 0 AP{T) (11.10.7.11)
p odd
p even
where the operators AP(T) : AP_1(X) -► A£(X) are as in Definition 11.9.1.
Evidently, according to Definition 11.10.5 and Definition 10.6.3
index(T) = Euler(Aodd(T), Aeven(T), Aodd(T))
(11.10.7.12)
11.10.8 THEOREM If T e BL(X, X)n is a commuting Taylor-Fredholm
n-tuple on a Hilbert space X then
index(T) = index(Aodd(r) + Aeven(T)*)
Proof: This is Theorem 10.8.7 ■
(11.10.8.1)
It is now clear, at least on Hilbert space, that the index is a
continuous, therefore locally constant, function of commuting Fredholm n-tuples of
operators. To extend the continuity of the index to Banach spaces seems to
be harder: the boundary result Theorem 10.3.9 seems however to contain
the essence of the argument.
11.11 FUNCTIONAL CALCULUS
If a G An is a commuting system of Banach algebra elements and / : H —► C
is a function "holomorphic" on an open set H C Cn then we can sometimes
define an element f{a) G C. If for example the algebra A is commutative
and the set H contains the spectrum cA{a) C Cn then the construction of
f{a) is called the Silov-Arens-Calderon-Waelbroeck functional calculus; if
instead A = BL(X, X) is the algebra of operators on a Banach space X and
the set H contains the Taylor spectrum rTaylor(a) then the construction of

532
11. Multiparameter Spectral Theory
f(a) is called the Taylor functional calculus. In the special case of a Hilbert
space X we have rTaylor(a) = aTaylor(a), and a simplification of Taylor's
calculus due to Vasilescu. In this section we attempt a description of the
first and the third of these: we begin with a look at "polynomially convex"
subsets of Cn.
11.11.1 DEFINITION The polynomially convex hull of a compact set
K C Cn is
polycvx(fQ = js e Cn : \p{s)\ < max |p| for each p G Polyn j (11.11.1.1)
and its rationally convex hull is
ratiocvx(fr) = p| {p~lp{K) : p G Polyn} (11.11.1.2)
We shall call a set K polynomially convex iff
K = polycvx(jK"),
and rationally convex iff
K = ratiocvx(fr) (11.11.1.4)
It is clear that
K C ratiocvx(fr) C polycvx(iC) (11.11.1.5)
and that the mappings polycvx and ratiocvx are increasing and idempotent
on the set of compact subsets of C; also the rationally convex hull of K is
the same as the set obtained from (11.11.1.1) by replacing polynomials with
rational functions whose denominators do not vanish on K. Something very
similar is going on in Definition 11.5.1 leading up to the "Silov boundary".
When n = 1 the polynomially and rationally convex hulls simplify:
11.11.2 THEOREM If K C C is a nonempty compact set then
ratiocvx(fr) = K (11.11.2.1)
and
polycvx(fr) = rjK. (11.11.2.2)
Proof: Equality (11.11.2.1) is obtained by looking at the particular
polynomial p = z. Towards (11.11.2.2), in which rj represents the modified
"connected hull" of (9.3.3.8), inclusion one way comes from the extended
(11.11.1.3)

11.11 Functional Calculus
533
version of the maximum modulus principle Theorem 9.1.3: if H is a hole
in K and s G H then for arbitrary polynomials p we have
\p{s)\ < sup \p\ < sup \p\ (11.11.2.3)
dH K
Conversely suppose s G C \ rjK and suppose that (pn) is a sequence of
polynomials for which
sup \pn - (z - s)"11 —> 0 as n —> oo (11.11.2.4)
K
Then for sufficiently large n G N we will have
|pn(s)|>sup|pj (11.11.2.5)
To see that such a sequence of polynomials exists we need the Hahn-Banach
theorem: if X is the normed space of continuous complex-valued functions
on rjK and Y is the linear supspace consisting of the polynomials, and if
(j> G Y° C X^ is arbitrary in the annihilator of Y, then the mapping
5 —+ <j>{{z - s)'1) : C \ K —+ C (11.11.2.6)
is holomorphic, and vanishes for sufficiently large \s\:
oo
4>((z - s)-1) = J2 H^s-"-1) = 0 (11.11.2.7)
n=0
By (5.4.1.3) from Theorem 5.4.1 the function (z — s)-1 is in the closure of
Y m
If v = (i/x, i/2,..., vn) G R^ we shall write
A(i/) = {seCn : |sy| < i/y for each jG {l,2,...,n}} (11.11.2.8)
for "the closed poly disc of multiradius v"\ more generally a polynomial
polyhedron is a set of the form
(s.p)-^!/,!*) = {seCn: {s,p{s)) G A(i/,M) C Cn+m} (11.11.2.9)
induced by multi-indices v G R^, /x G R^ and a polynomial p G Poly^1.
As a matter of notation we shall sometimes write (z, w) for the identity on
Cn+m = Cn x Cm, and temporarily regard z as the projection of Cn+m
onto Cn obtained by keeping the first n coordinates. Polynomial polyhedra
are polynomially convex, and can be used to approximate more general
polynomially compact sets:

534 11. Multiparameter Spectral Theory
11.11.3 THEOREM If K = polycvx(ff) is compact and polynomially
convex and if
K C Q = int(H) C A{v) (11.11.3.1)
then there are ra E N and \i G Rm and p G Poly™ for which
K C (x^p^A^fi) C H (11.11.3.2)
Proof: For each s G A(j/) \ H there is a polynomial q3 G Polyn for which
\q3\{s)>l>sup\q3\
K
and then a neighborhood U3 of 5 in Cn for which
\q3{t)\ > 1 for each te U3
and then by compactness a finite subset H of A(i/) \ H for which
AW\nc (J u.
Now take m to be the cardinal number of the set H and write
\xk = 1 for each k G {1,2,...,m} and p G (p1?p2,... ,pm) = {qa)aeH
(11.11.3.3)
The first inclusion of (11.11.3.2) is clear: to verify the second we consider
two cases. If 5 G A{u) \ H then 5 G Ut for some t G H giving \qt{$)\ > 1;
if 5 G Cn \ A{v) then |2--(s)| > Vj for some j G {1,2,..., n}. In either case
(s,p(s)) ^ A(i/,m). ■
If we marginally increase the multi-radii u and \i we can replace
(11.11.3.2) by
K C int(2,p)_1A(i/,M) Q (2,p)_1A(i/,m) C Q (11.11.3.4)
In one rather special situation the joint spectrum is polynomially convex:
11.11.4 THEOREM If a G An is a commuting n-tuple for which
A = cl(alg(a)) (11.11.4.1)
then
At = cA{a) = polycvxaA(a) (11.11.4.2)

11.11 Functional Calculus
535
Proof: With no assumptions about either A or a G An the mapping
aA : <f> —► <f>{a) from A to Cn (11.11.4.3)
is continuous and takes A* into the joint spectrum cA{a)\ if a is
commutative and generates A then this mapping is also one-one, and onto the
spectrum, therefore topologically a homeomorphism. For the second part
of (11.11.4.2) suppose that s G Cn is in the polynomially convex hull of
crA(a): then for arbitrary p G Polyn we have
\p(s)\ < max |p| = max |p| = max|p(a)A| < ||p(a)||
crA(a) aA(A*) A*
The mapping
p{a) — p{s) : Alg(a) —♦ C (11.11.4.4)
is therefore a well-defined bounded linear homomorphism on a dense sub-
algebra of A: thus there exists <f> G A* for which
p{s) = <t>{p{a)) for each p G Polyn (11.11.4.5)
In particular, taking p = z,
s = 4>{a) G <JA(a) ■
We observed the isomorphism at the beginning of (11.11.4.2) in
Theorem 9.6.6, for the special case of a Hilbert algebra. The next move, known
as the "Arens-Calderon trick," partially reduces the joint spectrum of
arbitrary commuting a G An to the situation of Theorem 11.11.4:
11.11.5 THEOREM If A is commutative and a G An and if
cA{a) C n = int(n) C Cn (11.11.5.1)
then there is b G Am for which
B = cl(alg(a,6)) => cB(a) C H (11.11.5.2)
Proof: If 5 G Cn is not in cA{a) there is c3 G An for which Y^=i C3jiaj ~
5;) = 1, and, taking C3 = cl(alg(a,c3)), an open set U3 containing s for
which
teu3=>tgcCs{a)
Taking i/- = ||a.|| for each j G {1,2,... ,n}, so that cA{a) C A(i/), the sets
(U3) with 5 G A(i/) \ H form an open cover for the compact set A(i/) \ H,

536
11. Multiparameter Spectral Theory
so that there is a finite set H C A(v) \ Q for which
A(i/)\HC (J U3 (11.11.5.3)
seh
If we now put
* = (0.6* and 5 = cl(*lgM)) = cl (alS ( U Cs) ) (H.ll-5.4)
then (11.11.5.2) holds:
cA{a) C aB(a) C (J aC-(a) CO I (11.11.5.5)
sEH
We are ready to start talking about the functional calculus. If H C Cn
is open we shall call the function / : H —► C holomorphic if the partial
derivatives df/dxj and df/dy^ exist continuously throughout H and satisfy
the "Cauchy-Riemann conditions" of (9.1.2.1):
df/dyj = idf/dx- throughout H for each j'g{1,2,...,n} (11.11.5.6)
If iC C Cn is an arbitrary set, in particular compact, we shall write
Ro\o(K) = {feC(K):
K J X K J (11.11.5.7)
/ has a holomorphic extension to some open H 2 K}
Thus HoIo(jK') is a subalgebra of C(K), not necessarily closed. When n > 1
the space HoIo(jK') sometimes coincides with Ho\o(K') for some larger Kf:
this is known as Hartog's phenomenon. A specific example is given by
2f = A(lll)\intA(i|i) #' = A(1,1)
We can extend / <E Holo(iC) to / G Holo(fr') by setting
/(*i.*2) = (2^)"1 / ^I'O^i if (*i,0 e K\K' (11.11.5.8)
7 ^i — *i
|*i 1=3/4
For commutative algebras, the functional calculus is defined in two
stages: the first is very easy, essentially reducing to Definition 9.7.1:
11.11.6 DEFINITION If a e An for a commutative Banach algebra A
and if
/ e Holo A(i/) with cA{a) C int A(i/) (11.11.6.1)

11.11 Functional Calculus
537
then
/(a) = (2*t)-B J (■■■( J f(z){Z1-a1)-1..-
\zn\=vn \z\\=v\
fo.o.o.
{zn-an)~ldzl
The reader can easily extend Theorem 9.7.2 to this context: the mapping
/ —► f{a) is a homomomorphism from Holo A (j/) into A which sends each
coordinate function Zj to the corresponding a-. The full statement is more
complicated, and relies not only on the Arens-Calderon trick and related
matters, but also on the Oka extension theorem, which can be summarized
by saying
HoloK^p)"^!/,/*)) = (Holo A(i/,/x)) ° iz>P) (0.0.0.2)
In words, whenever / is holomorphic near a polynomial polyhedron
(^p)-1 A(j/,aO C Cn there is a function g holomorphic near the polydisc
A(i/,/x) £ cn+m with /(*) = 9{Z>P{Z))- We offer no Proof of oka's tneo"
rem, except to remark that it ultimately derives from Stokes' theorem and
the Cauchy integral formula.
0.0.1 DEFINITION If a e An for a commutative Banach algebra A and
if
/GHoloaA(a) (0.0.1.1)
then
f{a) = G{a,b,p{a,b)) (0.0.1.2)
where
b G Am and p G Poly^+m and polycvxaA(a, b) C mi{z,w,p)~l A(i/,/j,, A)
(0.0.1.3)
and
G e Holo A(i/,/x,A) and f(z) = G{z,w,p(z,w)) (0.0.1.4)
To see that this is a good definition we suppose that / is holomorphic
on an open set H D ^aM* an<^ use ^rs^ the Arens-Calderon trick Theo-
dz„

538
11. Multiparameter Spectral Theory
rem 11.11.5 to find b G Am for which (a, 6) G An+m is a commuting
system with cB{a) C H, where B is generated by (a,6). By Theorem 11.11.4
the set aB(a, 6) C Cn+m is polynomially convex, therefore contains the
polynomially convex hull of <rA{a,b). By (11.11.3.4) we can find a
polynomial polyhedron {z,w,p)~1A{i/,fi,A) C Cm containing crA(a,b) in its
interior, and by Oka's theorem (11.11.6.3), applied in Cn+m, we can find
G G Holo A(i/,/j,, A) satisfying (11.11.7.4): then of course we use
Definition 11.11.6 in (11.11.7.2).
For a commuting system a G An in a noncommutative algebra A we
can now write
f{a) G A if / G HoloaD(a) with {ay} CDC commA(D) C A (0.0.1.5)
Thus /(a) is defined whenever / is holomorphic near the spectrum of a
with respect to any commutative subalgebra of A containing the a;-. For the
largest family of such holomorphic functions one should make the spectrum
crD{a) as small as possible, and hence the algebra D as large as possible:
at the same time the algebra D should in some sense be determined by
the system a G An. It would be logical to work with a maximal abelian
algebra D containing a;-: but apparently different maximal abelian algebras
are liable to induce different spectra. The most reasonable candidate would
seem to be the double commutant of Definition 7.1.1:
D = comm^(a) (0.0.1.6)
When A = BL(X, X) is the algebra of operators on a Banach space then
Joseph Taylor has discovered how to define f(a) for functions / G
HolorTaylor(a): we make no attempt to reproduce his construction, which
is based on the "Cauchy-Weil integral." If we specialize to a Hilbert space
X then Vasilescu has descovered a simpler formula, based on the "Mar-
tinelli kernel": we must extend An(X) to differential forms in dz- and dz -,
and write
f{a) = / {2m)-nf{z)M{z - a) A dzx A dz2 A • • • A dzn (0.0.1.7)
E
for a surface E surrounding rTaylor(a), where
M{z-a) = {L{a-z)+k{a-zY)-l{dz{k{a-z)+k{a-zY)-1)n-1 (0.0.1.8)

11.11 Functional Calculus
539
It would be attractive to try to extend such a formula to functions holo-
morphic near crTaylor(a) on a Banach space X: an extension of Allan's
Theorem 11.7.9 to produce "holomorphic(left,right)inverses" might be helpful.

Notes, Comments, and Exercises
CHAPTER 1
1.1.1 EXERCISE: Prove that (1.1.1.5) =J> (1.1.1.3).
1.1.3 COMMENT: These conditions characterize "topological vector
spaces"; more generally if {Qj)j^.j is a family of seminorms on the linear
space X and we declare "U G Nbd(x)" to mean that there is finite J0 C J
and e > 0 for which
\ y G X: max qAy — x) < e\ QU
^ jeJo J >
then we satisfy the conditions (1.3.1). Necessary and sufficient for the
"separated" condition Nbd(x) = {x} is CijeJ^i0) = W •
1.5.1 COMMENT: If 0 < t < 1 then it may not be possible to find (yn)
in Y with \\yn -x0\\ - dist(x0,y) and ||yj| < ||x0|| + *dist(x0,y): this is
unpublished work of Edward Bach.
1.5.3 REFERENCE: Bonsall (1967b).
1.8.0 COMMENT: Minkowski's inequality is the triangle inequality for
II*||p (1 < P < o°)- it extends to mappings
"NIP=(/l*lp<^
l/p
541

542
Notes, Comments, and Exercises
on the space of V-measurable" functions x : H —► K associated with a
"positive measure" \i\ S —► [0,oo], defined and "countably additive" on a
"a-ring" % C P(Q) of subsets of the set H. The normed space induced
by the quasi-seminorm ||-|| is written LJfi) or Lp{fi). More generally,
"Young's inequality" leads to "Orlicz spaces."
1.8.2 EXAMPLE: C^{[a,b}) = {x e C{[a,b}) : 3x' <E C{[a,b}), the
space of continuously differentiable functions on [a, b] (necessary and
sufficient is uniform convergence of the difference quotient), with norm llxlloo^ —
INIco+Mco-
1.8.2 EXERCISE: Write down the definition of C^{[a,b}) .
1.9.0 COMMENT: Analogs of c0 and c1 on a topological space H are
the spaces C0(H) = C(H) fl c0(H) and C^H) = C(H) fl c^H), where the
spaces c0(H) and c1(H) are induced by the bornology of "relatively
compact" subsets of H: thus C00(Q) = C(H) flc00(n) comprises the continuous
functions "of compact support."
1.9.1.9 REFERENCE: This observation is due to Wolff.
1.9.2 COMMENT: If H is derived from N by substituting a "maximal"
bornology for the bornology of finite subsets then Qn{X) is called an
"ultraproduct"—such a thing is more of a precision instrument than our
crude but more elementary "enlargement": it can be used to "standardize
nonstandard analysis."
1.10.0 EXAMPLE: If H is a group then ^(H) is a normed algebra with
(a • b){u) = (a * b){u) = ^ a{s)b{t)
st=u
("convolution"). The mapping t —► St is a semigroup homomorphism from
the group H to the algebra /1(H).
CHAPTER 2
2.1.0 EXERCISE: Prove (2.1.0.3).
2.1.1 EXERCISE: Prove that each of the following is continuous, and
determine its bound:

Notes, Comments, and Exercises
543
(a) T = U: (i1,i2,i3,...)->(0,i1,i2,...)froml1to/1,froml2tol2
or from l^ to Z^;
(b) T = V: (x1,x2,x3,...) -► (x2,x3,x4,...) from lp to lp {p = l,2,oo);
(c) T = K: C([a,6]) -► C([a,6]) where (iCx)(0 = f*=ak{s,t)ds for
each £ G [a,6] and each x G C([a,6]), where k : [a,6] x [a,6] —► K is
continuous;
(d) T = S: C([a,6]) -► C([a,6]) where {Sx){t) = f*=ax{s)ds for each
(e) T = R: lp —»• /p where (ifo)^ = (xx + x2 + • • • + xn)/n for each
nGN,xG/p.
2.3.3 COMMENT: Our slightly aggressive concepts of "kernel" and "co-
kernel" come from the Cambridge notes of C. T. C. Wall on homological
algebra (1962-3).
2.5.4 REFERENCE: Kato (1966).
2.8.0 EXERCISE: C^{[a,b}) = K x C([a,b}) under the correspondence
y <-► (t,x) given by the formulas
{t,x) = {y{a),x') and y = t + x
CHAPTER 3
3.1.2 COMMENT: T"1 is the unique U for which UT = Ix, IY = TV.
3.1.4 COMMENT: BL_1(X,y) is a topological space in its own right:
U e Nbd(T) means U = V fl BL_1(X,y) for some neighborhood V of T
in the normed space BL(X, Y)\ it is not necessary that BL_1(X,y) should
be an open subset of BL(X, Y).
3.1.4 EXAMPLE (W. H. Ruckle): If X = Y = c00 is the space of
"terminating" sequences and T = I : X —► X then T is invertible but
J — (l/n)U —► T where U(x1,x2,x3,...) = (0,x1,x2,.. .)> and I — (l/n)U
is not onto, since S1 = (1,0,0...) is not in its range. Conclusion: The set
of invertible operators need not be an open set in BL(X, Y).
3.2.2 COMMENT: Compare this "homological" characterization of one-
one-ness, denseness with the "Wilson cloud chamber," in which unknown

544 Notes, Comments, and Exercises
particles are described by being bombarded with other equally unknown
particles.
3.3.2 EXERCISE: I - ST bounded below => I-TS bounded below
I-ST closed => I-TS closed
COMMENT: Usually "closed" means either "closed range" or
"closed graph": here we are thinking of "bounded below" as "almost
closed."
3.3.5 REFERENCE: Davis/Rosenthal (1974), Choi/Davis (1974).
3.4.2 EXERCISE: I - ST (almost)open => I-TS (almost)open.
3.4.2 EXERCISE: Define the "almost closure" of the range of T to be
cl(T,X) = {limTxn : Tx e c{Y) and x <E /^PO}
and verify
TXCc\{T,X) Ccl(TX)
and
cl(T,X) = {ye Y:q(y) G Q(T)Q(X)}.
3.4.2 EXERCISE: Call T e BL{X,Y) "almost onto" iff cl(T,X) = Y,
so that
T onto => T almost onto =^ T dense
and
T almost open => T almost onto => T dense
verify
5, T almost onto => ST almost onto => S almost onto
3.4.5 EXERCISE: Prove Q(T) dense =J> T dense.
3.5.4.1 COMMENT: The missing (3.4.5.8) is that Q(T) dense =J> T
dense.
3.7.3 EXERCISE: I - ST left (right) invertible => I - TS left (right) in-
vertible.

Notes, Comments, and Exercises
545
3.8.1 REFERENCE: Caradus (1974, 1977, 1978); Taylor/Lay (1980).
3.8.7 REFERENCE: The origin of this is an (incorrect) result of Treese/
Kelley (1977) which was corrected by Gonzalez and generalized by Harte
(1987a): the further simplification presented here is due to Gerard Murphy.
3.9.3 EXERCISE: I - ST left (right) almost invertible => I - TS left
(right) almost invertible.
3.11.1, 3.11.2 REFERENCE: These are inspired by the generalization of
(3.11.3) due to J. L. Taylor (1970a).
3.12.3.2 REFERENCE: The concept of "relief" comes from Mandelkern
(1983), who is however working in the context of "constructive analysis."
3.12.3.2 EXERCISE: If [Uj)jeJ is a finite open cover of a normal Haus-
dorff space then there is {uj)jej in C(H, [0, l]) for which
Y,uj = 1 and cl(n \ uJl(°)) ^ Uj
(The set cl(H \ uj1^)) is called the "support" of u;-; the family (u) is
called a "partition of unity" for H, "subordinate to" the cover {Uj)jeJ.)
CHAPTER 4
4.1.4 REFERENCE: This is an unpublished argument of Edward Bach.
4.2.3.11 This idea is developed and exploited by G. J. O. Jameson
(1974).
4.3.3 EXERCISE: C([0,l]) is not complete with respect to IHIx, ||-||2;
lx is not complete with respect to ||-||2, ||*||oo
C(1)([0,1]) is complete with respect to ||.||W but not y^
4.4.6 REFERENCE: This argument is taken from H. G. Heuser (1982).
4.4.7 REFERENCE: Fuster/Marquina (1984).

546
Notes, Comments, and Exercises
4.5.1 REFERENCE: This is an unpublished argument of Edward Bach.
4.6.1 REFERENCE: R. P. Boas (1960).
4.7.4 REFERENCE: A. Wilansky (1977).
4.8.4 COMMENT: Much of the application of functional analysis to
differential equations is achieved by extending the theory of bounded operators
to "densely defined operators with closed graph."
CHAPTER 5
5.3.2 EXERCISE: The Hahn-Banach theorem extends to seminorms,
and in the real case to "sublinear functional," which may take negative
values: for example, the functional limsup: l^ —► R.
5.3.2 EXERCISE (cf. Edwards (1964)): A linear functional on l^ which
is dominated by limsup is necessarily monotonic and translation invariant,
and is necessarily an extension of lim : c1 —► R: such things are called
"Banach limits."
5.3.2 EXERCISE (cf. Edwards (1964)): Use the Hahn-Banach theorem
to prove a "monotonic Hahn-Banach theorem," extending monotonic linear
functional from subspaces of partially ordered linear spaces, in a
monotonic linear fashion, and to prove an "invariant Hahn-Banach theorem,"
extending linear functional which satisfy f{x) = f{Tx) for certain families
of operators T, in an invariant fashion.
5.5.1.2 EXERCISE: A locally convex topological vector space (one in
which every neighborhood of 0 contains a convex neighborhood) has its
topology given by a family of seminorms as in exercises to (1.3.1)—use the
Minkowski functional of absolutely convex neighborhoods of 0.
5.5.1.3 REFERENCE: Taylor/Lay (1980).
5.5.6.2 REFERENCE: Wilansky (1978).
5.6.4 REFERENCE: Dash/Schechter (1970); Fialkow (1983, 1985); Car-
illo/Hernandez (1984).
5.7.1 REFERENCE: Albrecht/Mehta (1984).

Notes, Comments, and Exercises
547
5.7.2 REFERENCE: This idea is inspired by Martin Mathieu (Harte/
Mathieu (1986)).
5.8.2 EXERCISE: Do the complex case explicity
5.8.2 EXERCISE: If H is a normal topological space (e.g., [a, b]), identify
the quotient l^ (Q)^ / C ^(Q)0 with more conventional representations of
Coo(n)t.
EXERCISE: If H is a topological group then C^H)* is a normed
algebra, where we define convolution multiplication
{fi * v){x) = \i{y * x)
where
(i/*x){t) = v{xt)
where
xt{s) = x(ts)
This contains ^(H) as a subalgebra.
5.9.2 REFERENCE: The most famous example is due to R. C. James
(1951).
5.10.1 REFERENCE: Jameson (1974); other examples are in D. J.
Newman (1960), Rudin (1983).
CHAPTER 6
6.1.3 COMMENT: A quantitative version of this is stated without proof
in Banach (1932): the simple proof of this "lemma of Auerbach," due to
Ruston, is given by Jameson (1974).
6.4.2 REFERENCE: This argument is given by Taylor/Lay (1980).
6.4.3 REFERENCE: This proof is taken from Taylor/Lay (1980).
6.5.4 RLFERENCE: Variants of this argument are given by Taylor/Lay
(1980) and by Caradus/Pfaffenberger/Yood (1974); an alternative "one
diagram" ~>roof is h; Y*ng (1973).

548
Notes, Comments, and Exercises
6.7.2.7 EXERCISE: Verify that m(H,X)t ^ M^X*), and identify
(mC)(n,X)t, where (mC)(H,X) = C(H,X) fl m(H,X).
6.7.4 REFERENCE: This idea is due to John Buoni.
6.8.3 COMMENT: A Banach space X is said to have the "bounded
compact approximation property" if there is k > 0 for which, for each compact
subset HCX and each e > 0 there is K G KL(X, X) for which
sup \\Kx -x\\<e and \\K - I\\ < k
xEH
(Lebow/Schechter (1971), Lindenstrauss/Tzafriri (1979)). Necessary and
sufficient for X to have this property is that for arbitrary Banach spaces Y
the mapping
T + KL(X,y) —+P(T) :BL(X,y)/KL(X,y) —+BL(P(X),P(y))
is bounded below (Astala/Tylli (1983)).
6.9.2 COMMENT: A Banach space X is said to have the "subprojective
property" if each infinite dimensional closed subspace contains an infinite
dimensional closed subspace which is complemented in X (Lebow/Schechter
(1971)). If X is subprojective and Y is arbitrary and T G BL(X,Y) then
T essentially bounded below => T upper semi-Fredholm
with equivalence if in addition X has the bounded compact approximation
property (Astala/Tylli (1983)).
6.8.3, 6.9.2 REFERENCE: Caradus/Pfaffenberger/Yood (1974) do the
easy part of this with sets instead of sequences; proof of the hard part is due
to Wickstead (Buoni/Harte/Wickstead (1977), Harte/Wickstead (1977)).
6.11.5 EXERCISE: Use this to prove (6.11.3) again for complete spaces.
6.12.3 EXERCISE: Extend (6.12.3.1) to "almost upper semi-Fredholm"
and (6.12.3.2) to "almost lower semi-Fredholm."
6.12.4 REFERENCE: This comes from Lebow/Schechter (1971).
6.13.2 REFERENCE: This is the "quantitative" version of the
compactness condition for the operator (RToU)A (Ls o V), which is given by Bonsall
(1967b).

Notes, Comments, and Exercises
549
6.13.2 PROBLEM: Prove Theorem 6.13.2 directly with ||P(T)|| in place
of ||T||gSS. Is it possible to improve on the constants?
6.13.2 PROBLEM: Find a converse to Theorem 6.13.2—a quantitative
version of a generalization of Theorem 6.13.4?
6.13.5 PROBLEM: Find an analog for "almost essential invertibility."
6.13.5, 6.13.6 REFERENCE: Compare Carillo/Hernandez (1984).
CHAPTER 7
7.2.5 COMMENT: For commutative algebras, Hull(J) is usually
factored into the composition hull(kernel(J)).
7.3.2 PROBLEM: Is there implication 1-ab <E A'1 A =J> 1-ba <E A"1 A?
(Harte (1986))
7.3.2 PROBLEM: If A = BL(X,X) for a normed space X, write *~A =
{a e A.-a-^O) = X/c\(aX)}, so that A'1 A = ^fl A. Is A closed under
i i i i
multiplication? Conversely, is there implication ST, TS G A =>• S,T G A ?
If S,T,ST,TS e A, is there necessarily equality index(5T) = index(S) +
index(T)?
7.3.2 EXERCISE: If J : A = BL(X,X) -► B = A/J with J =
KLX(X,X) is the Calkin quotient then T"1^-^ U B^ht) C ^ and A"1 +
r-1(o) = AJnr-1(51;f1tu5Tg1ht).
7.3.4 REFERENCE: Harte (1987a); the argument here represents an
unpublished simplification due to Gerald Murphy. Compare also Rakocevik
(to appear).
7.3.6.8 REFERENCE: This is derived from an example of Jameson
(1974).
7.3.6.16,17 REFERENCE: These examples are presented by Caradus
(1977, 1984).
7.4.5 REFERENCE: This argument is due to Hilden, as presented by
Michaels (1977).

550
Notes, Comments, and Exercises
7.6.1 REFERENCE: This is done with J = socle(A) by
Barnes/Murphy/Smyth/West (1982), where socle(A) is the sum of the minimal left
ideals of A, provided that it is also the sum of the minimal right ideals.
Compare also Pearlman (1974), Rowell (1984), and O'Searcoid.
7.6.1 EXERCISE: If A is semisimple, prove that socle(A) C 'A1 contains
only "regular" elements, and decide whether or not they must all be "de-
composably regular."
7.6.4.9 REFERENCE: This example comes from Gramsch/Lay (1971)
and Berberian (1970).
7.6.4.9 COMMENT: If T e BL(X,X) is Weyl then it is possible to
decompose T = S + K with invariable S and compact K for which
(SK - KS)2 = 0? (Murphy/West (1981), Laffey/West (1982),
O'Searcoid).
7.7.4 PROBLEM: Must the ideal socle(A) have the "Riesz property"?
7.7.6.10 REFERENCE: This was noticed by Trevor West.
7.9.2 REFERENCE: Grabiner (1978).
7.10.0.2 EXERCISE: Prove that the connected subsets of R are the
intervals.
7.10.3 REFERENCE: Crimmins/Rosenthal (1967) attribute part of this
to Parrot; compare also Salinas (1972) and Harte/Wickstead (1981).
7.11.2 REFERENCE: J. L. Taylor (1975); cf. also Harte (1976).
7.13.4 REFERENCE: This account of Zelazko's theorem is due to Roit-
man/Sternberg (1981); cf. also Zelazko (1968) and Kahane/Zelazko (1968).
7.13.4 PROBLEM: Extend Theorem 7.13.4 to the "generalized
characters" of Allan (I967a,b).
CHAPTER 8
8.4.3 COMMENT: If (cy)yeJ and [fkikeK are orthonormal bases for a
Hilbert space X then the sets J and K must have the same cardinal number,

Notes, Comments, and Exercises
551
which we can call dim(X), the "dimension" of X. Necessary and sufficient
for Hilbert spaces X and Y to be isomorphic is dim(X) = dim(y).
8.5.1 EXERCISE: if K C X is a subspace then (If-1)* = K°.
8.8.1 COMMENT: The rest of the world calls these either "E*-algebras"
or "C"-algebras"; for some of the rest of the world a "Hilbert algebra" is
a Banach algebra which is also a Hilbert space, in the rather specialized
sense of Ambrose (1969).
8.8.5.3,4 COMMENT: The converse of this inoffensive observation is the
enormous work of Brown/Douglas/Fillmore (1977) (cf. Halmos (1983)).
8.9.1 EXERCISE: Verify (8.9.1.3)-(8.9.1.5).
8.9.1 REFERENCE: This is the original construction, due to Berberian
(1959), which leads to our "enlargement."
8.9.1 EXERCISE: If A = BL(X,X) for a Hilbert space X then, with a"
as in Problems 7.3.2,
lA = {ae A: dim a"1 (0) = dim(X/cl(aX))}
Hence if X is a "separable" Hilbert space—one with a countably infinite
basis—and a G A then
i i
a regular and not semi-Fredholm =>• a G A
CHAPTER 9
9.3.1 COMMENT: Compare Taylor/Lay (1980) and Goldberg (1966),
who give an older division of the spectrum cr{T) of T G BL(X,X) for a
Banach space X into the point spectrum 7rleft(T), the "residual spectrum"
fright (T) \ tfieftjr), and the "continuous spectrum" rrieht(T) \ 7rrieht(T).
9.3.3 EXERCISE: Determine a{T) for T = U and T = V, the forward
and backward shifts onX = /l5X = /2. What about the analogs on /X(Z)
or Z2(Z), the "bilateral" shifts?
9.3.4 EXERCISE: If T = P = P2 g {0,1} then a(T) = {0, l}.

552
Notes, Comments, and Exercises
9.3.4 EXERCISE: The spectral mapping theorem fw = wf for
polynomials / fails with w = da, w = r\o and u; = dr\o (Harte/Wickstead (1981)).
9.5.2 REFERENCE: Lorch (1962).
9.5.4 EXERCISE: If a G A'1 for a Banach algebra A then
sup{e>0:{a-*:*eC,|*| < e} C A'1} = lira (dist(an, A \ A_1)1/n :
cf. Makai/Zemanek (1983), Zemanek (1984, 1985).
9.6.5 EXERCISE: Show that it is sufficient if each element of A is
"conservative" in the sense of (7.4.4.6).
9.7 COMMENT: The Silov Idempotent Theorem says that if a{A) is not
connected then A g {0,1} (Silov (1955)).
9.7.3 EXERCISE: What do you understand by length(r)?
9.8.3 EXERCISE: If T = U*SU + K with unitary U {U* = U'1),
compact K, and normal S then <?es3{T) = u;ess(T).
9.8.7 REFERENCE: This was first established for Riesz operators by
West (1966b) and then extended by Stampfli (1974); generalizations are
due to Apostol (1976), Davidson/Herrero (1986), and O'Searcoid.
9.8.8.14 REFERENCE: This example is due to Gillespie/West (1968).
Chui/Smith/Ward (1976) show that a Riesz operator T on a Hilbert space
X can be decomposed T = S + K with compact K and quasinilpotent 5,
SK-KS.
COMMENT: Call T e BL(X,X) "diagonal" if X =
ciErecC71-*7)"1!0) with complemented null spaces (T-tl)-1^): then if
X is a Hilbert space there is implication T diagonal =^ T normal. The
converse is known as the Berg/Sikonia theorem (Berg (1971), Sikonia (1971)),
a precursor of the result of Brown/Douglas/Fillmore (1977).
9.9.6 COMMENT: It is the sequence of events leading up to this theorem
which is enshrined in the tentative-sounding terminology A*-, J5*-, C*-: cf.
Dixmier (1977), Kadison/Ringrose (1983), Goodearl (1982).
9.10.1 REFERENCE: Bonsall/Duncan (1971/73, 1973).

Notes, Comments, and Exercises
553
CHAPTER 10
10.1.1, 10.1.2 REFERENCE: Douglas (1966), Embry (1971).
10.2.1 REFERENCE: "Interpolations" were christened by Coburn/
Schechter (1968).
10.2.5 COMMENT: These two special cases were noticed by O'Searcoid.
EXERCISE: y G cl(T,X) ^ Ry almost co-majorized by T.
10.3.9 REFERENCE: Wrobel (1987) established this for complete
spaces; Cho/Takaguchi (1981) and Curto (1981) for Hilbert spaces.
10.3.9 EXERCISE: Specialize (10.3.9) to give a direct proof of (5.5.6).
10.4.3 REFERENCE: Dash (1973b); Dash/Schechter (1970).
10.4.6 REFERENCE: Slodkowsky (1977).
10.5.2.9 REFERENCE: This example is due to Wickstead (Harte
(1978)).
10.5.3 REFERENCE: Kitadani (1982), Harte (1978).
10.5.5 REFERENCE: Cf. Embry (1971).
10.5.5.6 COMMENT: The right hand here is another of the 35 possible
combinations of the conditions of Definitions 10.1.1 and 10.1.2 which could
have been added to the list of Definition 10.2.1.
10.5.5.7 REFERENCE: This example is similar to one given by Bouldin
(cf. Embry (1971)).
10.6.7 REFERENCE: Albrecht/Mehta (1984); Fainstein (I980b,c).
10.8.1 EXERCISE: If A = BL(X, X) for a Hilbert space X then A'1 =
Aq1 is connected (cf. Rudin (1973), Kuiper (1965)). This was used in
(9.4.4.5).

554
Notes, Comments, and Exercises
10.8.1 EXERCISE: If A = BL(X,X) for a Hilbert space X then A C
A~1A+ C cl(A_1) (since positive operators are Hermitian, therefore in
cUA"1)).
REFERENCE: Feldman/Kadison (1954), Bouldin (1982), Izumi-
ne (1979).
10.8.3 REFERENCE: Vasilescu (1977, 1978); Curto (1981).
10.9.5,6 REFERENCE: Slodkowsky (1977).
CHAPTER 11
11.1.1 REFERENCE: Coburn/Schechter (1968).
11.1.2 EXERCISE: Determine aleft(T), <7rieht(T), rleft(T), rrisht(T),
7rleft(T), 7rrisht(T), with T e BL(X,X)n given by (a) T = (ri5r2) = {U,V)
where U and V are the unilateral shifts on lx or on l2, (b) the corresponding
bilateral shifts, (c) if Z\ = S and T2 = P = P2 satisfies SP = PSP or
PS = PSP.
11.1.4 EXERCISE: If A and B are commutative Banach algebras and
TeBBL{A,B) then
T^{BX) = Ax <=> for each n G N,a <E An,aA{a) = cB{Ta)
(Corach/Suarez, 1987).
11.2.2 EXERCISE: Define the "commutant" and "double commutant"
spectra of a <E An as <^omm(a) = aB{a) and <^bc(a) = aD{a) with
B = commA(a), D = comm^(o), and verify that they are both compact,
possibly empty, with /<^bc(a) C a^bc/(a) for polynomials / G Poly™;
cf. Coburn/Schechter (1968), Dash (1973b), Dash/Schechter (1970), Harte
(1973c).
11.2.6 EXERCISE: If T G BL(X,X) commutes with P = P2 <E
BL(X,X) then 0>=1(T) is the spectrum of the restriction of T to its
invariant subspace P{X).
11.3.3 REFERENCE: Slodkowsky (1977) gives essentially this
argument; the idea of using the enlargement comes from Davis/Rosenthal (1974)
and Choi/Davis (1974).

Notes, Comments, and Exercises
555
11.3.4 REFERENCE: Originally proved for operators on Hilbert space
by Bunce (1971); extended to Banach algebras for the left and right
spectrum by Harte (I972a,b). The argument for approximate point spectra was
extended to operators on Banach spaces by Davis/Rosenthal (1974) and
Choi/Davis (1974) and to Banach algebras by Zelazko/Slodkowsky (1974).
11.3.4 EXERCISE: Give another derivation of (9.5.3.5) and (9.5.3.6).
11.4 REFERENCE: Clarke (1975), Curto (to appear), Zelazko (1979),
Slodkowsky (1977), Harte.
11.5 REFERENCE: Wermer (1971), Zelazko (1970), Arens (1961),
Harte (1975a).
EXERCISE: If a e A then daA{a) C dA{a) (Harte)
11.6.10 REFERENCE: Davis/Rosenthal (1970), Embry/Rosenblum
(1974), Harte (1974).
EXERCISE: Do the analog of (11.6.10) directly for 3x3 matrices
(Harte (1971)).
11.7 REFERENCE: Grothendieck (1955); Shchatten (1950); Brown/
Pearcy (1966); Ichinose (I978a,b).
11.7.7 REFERENCE: Halmos (1982); Harte (1985).
11.7.7.8 COMMENT: This remark uses (3.10.4.1), (6.2.6.5) and
(6.2.6.1).
11.7.7.13 COMMENT: The union of the cA (ayy) need not coincide with
the spectrum of the upper triangular matrix c with respect to the full matrix
algebra.
11.7.8 REFERENCE: Allan (I967a,b); this extends to m-tuples of
functions.
11.8.2 REFERENCE: It is the first part of this which is really the
"Kleinecke-Sirokov theorem" (Kleinecke (1957), Sirokov (1956)); the
second part is due to Singer/Wermer (1955). This derivation (sic) of the
Singer-Wermer result is due to Gerard Murphy.
11.8.2.7 EXERCISE: Prove that (n!)"1/n -► 0 as n -► oo.

556
Notes, Comments, and Exercises
11.8.3 EXERCISE: Extend (9.5.3.5) and (9.5.3.6) to quasicommuting
pairs.
11.8.6 PROBLEM: Find a pair T = {Tl9T2) of quasi-commuting upper
triangular 5x5 matrices and a system of polynomials / for which f{T) is
not quasi-commutative.
11.9.8 REFERENCE: J. L. Tayor (I970a,b); Vasilescu (1977, 1982);
Curto (to appear).
11.9.8 REFERENCE: Eschmeier calls our "Taylor invertible spectrum"
the "split spectrum."
11.9.8 REFERENCE: Hamet Isaev (to appear) relates the Taylor
spectrum to the "multi-parameter spectral theory" of Atkinson (1968, 1972,
1977), Browne (I972a,b; 1974a,b; 1977), and Binding (1980, 1981,
1982a,b,d).
11.9.9 EXERCISE: dfTaylor(T) C fle{t{T) U frisht(T) (Wrobel (1987),
Cho-Takaguchi (1981). Curto (1986, 1987)); this fails for triples (Curto
(1986)) and cannot be improved to replace the union of the left and right
spectra by their intersection (Wrobel (1987)).
11.9.10 EXERCISE: Prove directly /u;Taylor(T) C u;Taylor/(T) with u =
f,a and / e Poly^ (Harte (1981), Fainstain (1987)).
11.9.10 COMMENT: An alternative derivation of the spectral mapping
theorem for the Taylor spectrum uses Zelazko's Theorem 9.6.8 (Zelazko/
Slodkowski (1974)).
11.9.12 REFERENCE: Rynne (1987); Wrobel (1986b); Eschmeier
(1986, 1987).
11.10.1 REFERENCE: For noncommutative algebras consult J. L.
Taylor (1973), M. E. Taylor (1968), Anderson (1969), Albrecht (1982), Nelson
(1970).
11.10.3 EXERCISE: If
r = (r1,r2,r3,r4)
0 d/dz1
0 0
0 d/dz2
0 0

Notes, Comments, and Exercises
557
on the space X = C{V) x C^{V) with V = closure ((3D x 3D) \ (D x D))
then <7dbc(T) ± rTaylor(T) and /adbc(T) ^ adhcf{T). Cf. J. L. Taylor
(1970a), Slodkowsky/Zelazko (1974), Harte (1973c).
11.10.5 REFERENCE: If T G BL(X, X)n is "Taylor-Weyl" can we write
T = S + K with "invertible" S and compact K1
11.10.5 PROBLEM: What should one mean by "Taylor-Browder" ?
Compare Schechter/Snow (1975), Snow (1975), Buoni/Dash/Wadhwa
(1981), Curto/Dash (1987). In particular, do any of these definitions admit
"commuting decompositions" T = S + K with invertible 5, compact K1
11.10.5 PROBLEM: Prove that index(T) is a continuous function of
commuting Taylor-Fredholm systems T G BL{X,X)n: this is proved for
Hilbert spaces by Curto (1986, 1987) and for unbounded operators on Ba-
nach spaces (using "gap theory") by Vasilescu (1979) and by Albrecht/Vasi-
lescu (1974), but one might hope for argument as in Theorem 10.3.9 or
Wrobel (1987).
11.11.6.3 REFERENCE: Wermer (1971) gives an account of the Oka
extension theorem. For holomorphic functions consult Krantz (1982),
Henkin/Leiterer (1984), Hormander (1975), Gunning/Rossi (1965); the
functional calculus for commutative Banach algebras is expounded in
Wermer (1971), Bourbaki (1967), Curto (to appear). The Taylor calculus for
Hilbert spaces is given by Vasilescu (1982) and Curto (to appear).
EXERCISE: If T = {T^T2) is given by Tx = V <g> /, T2 =
V <g> (U - U2V) + J <g> U2V where U and V are the forward and backward
shifts then (0,0 G iso(aleft(T) U aT'^ht(T)) \ accaTaylor(T): thus the union
of the left and the right spectrum cannot possibly support a functional
calculus (Curto (1986)).

References
Albrecht, E. (1977). Generalized spectral operators. Proceedings Paderborn
Conference on Functional Analysis. North Holland Math. Stud. 27:
259-277.
Albrecht, E. (1979). On joint spectra. Studia Math. 64: 263-271.
Albrecht, E. (1982). Several Variables Spectral Theory in the
Non-commutative Case. Banach Center Publications Warsaw, Poland, vol. 8,
Spectral Theory, pp. 9-30.
Albrecht, E., and Frunza, S. (1976). Non-analytic functional calculi in
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Notation
K, R, C
ll-ll
dist(x,y)
Disc(x\S)
Int, CI
Nbd
dK = {c\K)\{mtK)
dist{x,K)
X/Y
Y®pZ
~
=
Hoc
Np
ip(n,x)
<f>0 x
coo(n,x)
c0(n,x)
cx(n,x)
QnPO
Q(X)
A-1
L(x,y) =
BL{X,Y).
LK(X,Y)
= BLK(X,y)
1
2
3
3
3
3
3
4
6
11
12
12
14
15
15
15
17
17
17
19
19
21
26
26
Ix
JY
ker(T)
Ky
coker(T)
core(T)
col(5,T)
row(5, T)
^U ^2
<, 7T2
Qn(r)
*Z
j-n
±jrp
Q(T)
<ln(*)
X*
jR/p
^x
*x
/0y
/®y
/.(*.)
^a
28
28
28
29
29
29
30
31
31
31
32
33
39
42
42
43
43
46
47
47
47
47
48
49, 457
581

582
Notation
K
HBL(A,J9)
BL+(X,y)
BL_1(X,y)
aBL(X,y)
7rleftBL(X,y)
7:T^htBL{X,Y)
fleftBL(X,y)
rleftBL(X,y)
TT{sht BL(X,y)
TT{shtBL{X,Y)
core0(T)
<7leftBL(X,r)
<7T{shtBL{X,Y)
aleftBL(X,y)
(7rightBL(X,y)
rBL1(X,y)
v)fBL{X,Y)
cf*htBL(X,Y)
RT/A
LT/B
o™BL{X,Y)
<4ghtBL(X,y)
(^t)m' (^t)m
C7left(A)
<7rieht(A)
7Tleft(A)
7TT{Sht(A)
T\v,z> T/W}Z
c{n,x)
X~, T~
graph(T)
xt
rt
BL£(X,y)
K°
H0
cvxK{K)
49, 457
50
51
54
54
57
57
61
61
65
65
72
73
74
76
76
80
86
86
88
88
92
92
94
95
95
95
95
98
108
125, 126
133
137
137
137
140, 296
140, 296
140
sgn
cvx{K)
Face(JC)
Extreme(jK')
KL0(X,y)
*Js?BL(x,y)
^fhtBL(X,y)
aessBL(X,y)
index(T) 192
wessBL(X,y)
m(H,X)
?7i1(n,x)
p(*)
P(X)
PiW
l|r||ess
ll^U'ess
KLx(X,y)
KL(X,y)
a+sBL(X,y)
a-sBL(X,y)
T'aT
comm(T)
comm(jK') = comm.A(K)
comm2 [K)
comm(o)
MLI(A)
MRI(A)
Radical(A)
Hull(J)
I1
A
a
ax
H<±K
r-°°(o)
T°°{X)
ascent (T)
157
165
165
165
183
187
187
187
,529
192
202
202
202
204
204
205
205
206
206
209
209
209
229
231
237
237
237
241
241
243
244
246
247
258
258
271
271
271

Notation
583
descent (T)
acc(jRT)
iso(jRT)
CcmpK(£)
Vt(K)
Ifl
\cun,n') =
{\fl-.f ec(n,n')}
winding number(<£)
A-1
exp(a)
Exp(A)
sgn(<£)
Re(A), Re(a)
State(A)
(x\y)
y*
y J- x
K±
r*
/i+(a)
*a(°)
*!?"(")
c*ht(«)
*!|ftM
a^ht(a)
W")
??%)
*!?'(*)
*2ght(<0
eA(°)
•7(10
u;r(a)
cDr(a)
W,
A*
WT
°ess(T)
271
275
275
281
281
286
295,296
288
290
292
292
295
303
305
309
309
313
313
324
333
341
341,457
341, 458
341, 458
341, 458
345,459
345,459
345, 459
345, 459
345
347
350
350
352
357
362
370
"ess^)
wr (a)
"eCs°smm(r)
*oo(T)
V(a)=VA(a)
«#(*)
A*
BL{X,Y,Z)
crleft'r''ehtBL(X,r,^)
BL_1 [X,Y,Z)
'W{X,Y,Z)
Euler(5,T)
Euler(Tn,...,r1,T0)
Polyn
Poiy^
w6=t(a)
w6=6(a)
g(K,b), g(a,H)
(A)=Al
AE(n)
dE(n)
dvA(a)
d*(A)
X®Y
X®AY
det(c)
\q
adj(c)
Holo(jRT)
DBL(A,A)
kn{X)=k{X',dz)
A\x)
A(T)
KM
At(5)
aTaylor(T)
aTaylor(r)
rTaylor/j.\
370
370
370
371
383
383
385
401
402
402
403
427
427
461
461
465
465
469
472
474
475
476
476
493
494
502
502
503
506,536
509
517
517
517
518
520
522
522
522

584
Notation
~Taylor(r)
Taylor/ •.
<^lor(a)
-Ta,lor(r)
fTay.or(r)
522
526
526
526
528
528
528
*eTssyl°r(r) 528
index(T) 529 (cf. p. 192)
D(a) 529
Aodd(T), Aeven(T) 531
polycvx(iC) 532
ratiocvx(iC) 532
A(*/) 533
[z.p)-1^) 533

Index
Absolutely convex hull, 140
Absolutely convex, 6
Accumulation points, 275
Adjoint, 137, 324
Adjugate matrix, 493, 503
Allan's theorem, 506
Almost (left,right) determined,
395
Almost (left,right) invertible,
401, 435
Almost (left,right) majorized,
395
Almost (left,right) multiple, 395
Almost comajorized, 393
Almost decomposably exact, 401
Almost exact, 402
Almost Fredholm, 214
Almost left invertible, 76
Almost left multiple, 391
Almost left spectrum, 532
Almost linearly exact, 402
Almost linearly Taylor nonsingu-
lar, 520
Almost lower semi-Fredholm, 214
Almost open, 65
Almost quasipolar, 257
Almost range-included, 393
Almost right invertible, 76
Almost right multiple, 393
Almost right spectrum, 532
Almost spectrum, 339
Almost Taylor invertible, 520
Almost Taylor nonsingular, 520
Almost upper semi-Fredholm,
213
Annihilator, 139
Approximate (left,right)
multiple, 395
Approximate left multiple, 391
Approximate right multiple, 393
Approximate (left,right)
invertible, 402
Approximately decomposably
exact, 402
Arcwise connected, 287
Arens-Calderon trick, 536
Ascent, 271
Atkinson's theorem, 188
Axiom of choice, 22
Back annihilator, 139-140
Backward shift, 44, 250
Baire's theorem, 127
Banach limit, 335
Banach space, 107
585

586
Index
Banach-Steinhaus theorem, 4,
135
Bessel's inequality theorem, 316,
321
Bolzano-Weierstrass property,
200
Bornological space, 17
Bornology, 17
Boundary operator, 71
Bounded, 26
Bounded below, 61
Bounded structure, 17
Browder, 266
Canonical factorization, 31
Cauchy integral formula, 363
Cauchy sequence, 108
Cauchy-Riemann equation, 338
Chain, 399, 435
Character, 305
Closed, 61
Closed boundaries, 474
Closed graph theorem, 133
Closure, 2
Comajorized, 393
Coarser, 13
Codomain, 29
Cofinal, 43
Cokernel, 29
Commutant, 237
Commutative Gelfand-Naimark
theorem, 381
Commute, 237
Compact, 197
Compact operator, 206
Complemented, 12
Complete, 110
Completely convex, 113
Completely invariant, 512
Completion, 125
Conjugate linear, 139
Conjugate, 137
Connected, 280
Connected component, 289
Connected hull, 281
Conservative, 255, 358
Continuous, 4, 25
Contractible, 286
Convergent sequence, 4, 17
Convex, 112
Convex combinations, 165
Crossnorm, 493
Cyclic representation, 387
Cyclic vector, 387
Decomposably exact, 401
Decomposably regular, 84, 246
Dense, 57
Dense proper subspace of, 130
Derivation, 509
Derivative, 167
Descent, 271
Determinant, 503
Determined, 392
Different iable, 167
Dimension, 175-176
Dini's theorem, 301
Directed, 21
Disconnection, 280
Double commutant, 237
Dual Space, 137
Enlargement, 19, 41
Epimorphism, 57
Equivalence of norms, 13
Essential enlargment, 204
Essential Taylor spectrum, 528
Essentially (left,right) almost
invertible, 425
Essentially (left,right) invertible,
425
Essentially (left,right) one-one,
425
Essentially dense, 187
Essentially one-one, 187
Euler number, 420, 426
Evaluation map, 47
Exact, 394, 402

Index
587
Exponential, 292
Exponential spectrum, 345
Exterior algebra, 517
Extreme point, 165
Face, 165
Faithful representation, 388
Finer, 13
Finite, 17
Finite ascent, 271
Finite descent, 271
Finite dimensional, 174
Finite rank operator, 183
Forward shift, 44, 250
Fredholm, 187
Fuglede's theorem, 360
Fundamental theorem of algebra,
340
Gelfand property, 356
Gelfand theorem, 357
Gelfand-Mazur lemma, 356
Generalized exponential, 292
Generalized inverse, 80
Greatest crossnorm, 496
Green's theorem, 339
Hahn-Banach theorem, 142
Hamel basis, 173
Hartog's phenomenon, 537
Hermitian, 326, 333
Hilbert algebra, 332
Hilbert space, 320
Hole, 281
Holomorphic, 337
Homomorphism, 49
Homotopic, 286
Hyperinvariant subspace, 255
Hyperkernel, 271
Hyperrange, 271
Ideal, 20
Idempotent, 247
Increasing, 50
Index, 192, 529
Index theorem, 194
Inessential hull, 244
Inner product, 309
Inside of a curve, 362
Interior, 2
Inverse, 20, 53
Invertible, 53
Invertible, relative to, 256
Isolated points, 275
Isometric, 12
Isomorphic, 12
Jordan property, 50
Kernel, 29
Kleinecke-Sirikov theorem, 510
Krein-Milman theorem, 166
Least uniform crossnorm, 497
Lebesgue covering property, 200
Left and right compositions, 46
Left and right multiplications, 46
Left approximate eigenvalues,
345, 458
Left eigenvalues, 345, 459
Left ideal, 241
Left invariant, 512
Left invertible, 73
Left invertible modulo, 86
Left M-invertible, 92
Left multiple, 391
(Left,right) bounded below, 435
(Left,right) determined, 395
(Left,right) invertible, 401
(Left,right) majorized, 395
(Left,right) multiple, 395
(Left,right) one-one, 435
(Left,right) T-Fredholm, 440
(Left,right) T-Weyl, 440
Left spectrum, 341, 457
Left topological zero divisor, 79,
95
Lift, 287

588
Index
Linear algebra, 19
Linear functional, 47
Linear space, 1
Linear subspace, 5
Linearly dependent, 173
Linearly exact, 402
Linearly generating, 173
Linearly independent, 173
Linearly Taylor nonsingular, 520
Liouville's theorem, 340
Locally compact, 200
Lomonosov's lemma, 255, 374
Lower semi-Fredholm, 209
Majorized, 391
Martinelli kernel, 539
Maximal, 22
Maximal ideal, 241
Maximum, 22
Maximum modulus principle,
340
Measure of noncompactness, 202
Minkowski functional, 145
Monomorphism, 57
Monotonic, 50
Natural injection, 28-29
Nearest point theorem, 317
Nearly essentially exact, 430
Neighborhood, 3
Nilpotent, 251
Noncommutative Gelfand-
Naimark theorem, 387
Noncommutative polynomials,
461
Nonsingular, 102
Norm, 2
Normal, 333
Normal space, 104
Normal subgroup, 291
Normed linear algebra, 20
Normed module, 482
Normed space, 2
Null, 17
Null space, 29
Numerical range, 383
Oka extension theorem, 537
One-one, 56
Onto, 56
Open, 65
Open mapping theorem, 128-129
Operator bound, 26
Orthogonal, 313
Orthogonal complement, 313
Orthonormal, 314
Outside of a curve, 362
Partial isometry,
Partial order, 21
Partially ordered normed space,
23
Perpendicular, 313
Polar, 257
Polar decomposition, 442
Polarization, 310
Pole, 140
Polynomial polyhedron, 534
Polynomially convex, 532
Polynomially convex hull, 532
Positive, 326
Precore, 73
Proper, 60
Proper ideal, 241
Pseudo-inverse, 80
Punctured neighborhood
theorem, 274
Pure numerical range, 383
Pythagoras' theorem, 314
Quadratic norm, 310
Quasi-commute, 571
Quasinilpotent, 251
Quasinorm, 2
Quasipolar, 257-258
Quotient, 6
Quotient map, 29
Radical, 243

[ndex
589
Radical element, 243
Range, 29
Range-included, 393
Rank one, 183
Rationally convex, 532
Rationally convex hull, 532
Reduced by, 256
Reflexive, 161
Regular, 80, 246
Relative spectrum, 465
Relatively almost open, 72
Relatively compact mapping, 202
Relatively Fredholm, 80, 246
Relatively open, 72
Relatively Weyl, 84, 246
Relief map, 105
Resolvent function, 342
Reversal of products, 21, 54
Riesz points, 371
Right approximate eigenvalues,
345, 461
Right eigenvalues, 345, 459
Right ideal, 241
Right invariant, 512
Right invertible, 73
Right invertible modulo, 86
Right M-invertible, 92
Right multiple, 393
Right spectrum, 341, 458
Right topological zerodivisor, 79,
95
Schwartz inequality, 310
Self-adjoint, 326, 333
Semi-essential enlargement, 204
Semisimple, 358
Seminorm, 2
Separated, 105
Separation theorem, 145
Sequentially compact, 198
Sequentially precompact, 198
Sesquilinear, 139
Silov boundary, 476
Simply connected, 287
Simply polar, 248
Spanning, 173
Spatially Browder, 266
Spatially Fredholm, 187
Spatially (left,right) Fredholm,
425
Spatially (left,right) Weyl, 426
Spectral mapping property, 472
Spectral mapping theorem, 343,
468
Spectral permanance theorem,
349
Spectral radius, 352
Spectrum, 341
Stampfli's theorem, 374, 375
State, 305
Steinitz replacement process
theorem, 174
Stone-Weierstrass theorem, 302
Strictly exact, 402
Strictly weaker than, 130
Stronger, 13
Subalgebra, 20
Submodule, 304
Subprojective property, 465
Supplemented, 11
T-Browder, 267
T—Browder spectrum, 370
T-Fredholm, 261
T—Fredholm spectrum, 350
T-Weyl, 261
T-Weyl spectrum, 350
Taylor almost Fredholm
spectrum, 528
Taylor almost invertible
spectrum, 523, 527
Taylor almost singular spectrum,
523
Taylor-Fredholm spectrum, 528
Taylor invertible, 520
Taylor invertible spectrum, 523,
527
Taylor nonsingular, 520

590
Index
Taylor singular spectrum, 523
Taylor-Weyl spectrum, 529
Tensor product, 493
Terminating, 17
Tietze's theorem, 297
Topological versus metric, 13
Topology, 3
Total order, 21
Totally bounded, 197
Totally bounded mapping, 202
Totally bounded operator, 206
Triangle inequality, 3
Tychonoff product theorem, 201
Uniform boundedness principle,
134
Uniform crossnorm, 494
Uniform tensor product, 493
Uniformly, continuous, 25
Unitary, 333
Upper bound, 21
Upper semicontinuity, 345
Upper semi-Fredholm, 209
Urysohn function, 105
Urysohn's lemma, 105
Usual axioms of set theory, 22
Vector space, 1
Weak Taylor spectrum, 528
Weak* topology, 384
Weaker, 13
Weakly (left,right) Fredholm,
425
Weakly (left,right) T-Fredholm,
440
Weyl operator, 192
Winding number, 288
Zelazko's theorem, 361
Zorn's condition, 23
Zorn's lemma, 22

about the book . . .
Focusing on the various kinds of "singularity" which prevent an operator from being
invertible, this introduction to functional analysis bases its presentation on the Open
Mapping theorem, the Hahn-Banach theorem, the Dual Space construction, the
Enlargement of normed space, and Liouville's theorem.
Invertibility and Singularity for Bounded Linear Operators makes these concepts easily
accessible to an elementary audience . . . develops the theory of open and almost open
operators between incomplete spaces... builds the "enlargement" of a normed space and
of a bounded operator . .. and sets up an elementary algebraic framework for Fredholm'
theory.
In addition, this new approach to elementary functional analysis extends from the
definition of a normed space to the fringe of modern multiparameter spectral theory. It
concludes with a discussion of the various kinds of "joint spectrum," including the
complicated ideas of Joseph Taylor.
Complete with exercises and a bibliography, this volume serves both as a text for
advanced undergraduate and graduate mathematics students in functional analysis courses
and as a monograph for mathematicians researching Fredholm theory, Banach algebras,
and multiparameter spectral theory.
about the author . . .
Robin Harte is Professor of Mathematics at Cork University, a college of the National
University of Ireland, where he has taught since 1968. Previously he taught at the
University College of Swansea in Wales, U.K., and he was a visiting professor at the
University of Iowa. His research interests include Fredholm and spectral theory for
bounded operators and Banach algebras. Dr. Harte was educated at Trinity College,
Dublin, and received the Ph.D. degree (1965) in mathematics from the University of
Cambridge in England.
Printed in the United States of America ISBN: 0—8247—7754—9
marcel dekker, inc./newyork • basel