Banach
Algebra Techniques
in Operator Theory
RONALD G. DOITGLAS
Department of Mathematics
State University of New York at Stony Brook
Stony Brook, New York
1972
ACADEMIC PRESS New York and London

Copyright © 1972, by Academic Press, Inc.
all rights reserved
no part of this book may be reproduced in any form,
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other means, without written permission from
the publishers.
ACADEMIC PRESS, INC.
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United Kingdom Edition published by
ACADEMIC PRESS, INC. (LONDON) LTD.
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Library of Congress Catalog Card Number: 78-187253
AMS (MOS) 1970 Subject Classifications 47B05,47B15, 47B30,
47B35, 46L05,46J15,46-02,47-02
PRINTED IN THE UNITED STATES OF AMERICA

To my family
Nan, Mike, Kevin, Kristin

Contents
Preface xi
Acknowledgments xiii
Symbols and Notation xv
1. Banach Spaces
The Banach Space of Continuous Functions I
Abstract Banach Spaces 2
The Conjugate Space of Continuous Linear Functionals 5
Examples of Banach Spaces: c0, I1, and f 7
Weak Topologies on Banach Spaces 8
The Alaoglu Theorem 10
The Hahn-Banach Theorem 12
The Conjugate Space of C([0,1]) 18
The Open Mapping Theorem 23
The Lebesgue Spaces: Ll and £°° 23
The Hardy Spaces: Hl and Ha 26
Notes 27
Exercises 27
2. Banach Algebras
The Banach Algebra of Continuous Functions 32
Abstract Banach Algebras 34
Abstract Index in a Banach Algebra 36
The Space of Multiplicative Linear Functions 39
The Gelfand Transform 40
vii

viii Contents
The Gelfand-Mazur Theorem 42
The Gelfand Theorem for Commutative Banach Algebras 44
The Spectra] Radius Formula 45
The Stone-Weierstrass Theorem 46
The Generalized Stone-Weierstrass Theorem 50
The Disk Algebra 51
The Algebra of Functions with Absolutely Convergent Fourier Series 54
The Algebra of Bounded Measurable Functions 56
Notes 58
Exercises 58
3. Geometry of Hilbert Space
Inner Product Spaces 63
The Cauchy-Schwarz Inequality 64
The Pythagorean Theorem 66
Hilbert Spaces 66
Examples of Hilbert Spaces: C", P, L2, and H2 66
The Riesz Representation Theorem 72
The Existence of Orthonormal Bases 75
The Dimension of Hilbert Spaces 76
Notes 77
Exercises 78
4. Operators on Hilbert Space and C*-Algebras
The Adjoint Operator 81
Normal and Self-adjoint Operators 84
Projections and Subspaces 86
Multiplication Operators and Maximal Abelian Algebras 87
The Bilateral Shift Operator 89
C*-Algebras 91
The Gelfand-Naimark Theorem 92
The Spectral Theorem 93
The Functional Calculus 93
The Square Root of Positive Operators 95
The Unilateral Shift Operator 96
The Polar Decomposition 97
Weak and Strong Operator Topologies 100
W*-Algebras 101
Isomorphisms of Lm-Spaces 104
Normal Operators with Cyclic Vectors 105
Maximal Abelian H/*-Algebras 108
*-Homomorphisms of C*-Algebras 110
The Extended Functional Calculus 112
The Fuglede Theorem 114
Notes 115
Exercises 115

Contents ix
5. Compact Operators, Fredholm Operators, and
Index Theory
The Ideals of Finite Rank and Compact Operators 121
Approximation of Compact Operators 124
Examples of Compact Operators: Integral Operators 125
The Calkin Algebra and Fredholm Operators 127
Atkinson's Theorem 128
The Index of Fredholm Operators 130
The Fredholm Alternative 131
Volterra Integral Operators 132
Connectedness of the Unitary Group in a ff*-Algebra 134
Characterization of Index 138
Quotient C*-Algebras 139
Representations of the C*-Algebra of Compact Operators 142
Notes 144
Exercises 145
6. The Hardy Spaces
The Hardy Spaces: H\ H2, and Ha 149
Reducing Subspaces of Unitary Operators 151
Beurling's Theorem 153
The F. and M. Riesz Theorem 154
The Maximal Ideal Space of Hm 155
The Inner-Outer Factorization of Functions in H1 158
The Modulus of Outer Functions 159
The Conjugates of H1 and La>IH0<° 161
The Closedness of Hw + C 163
Approximation by Quotients of Inner Functions 163
The Gleason-Whitney Theorem 164
Subalgebras between HK and Lm 164
Abstract Harmonic Extensions 166
The Maximal Ideal Space of Ha + C 168
The Invertibility of Functions in H^ + C 170
Notes 171
Exercises 172
7. Toeplitz Operators
Toeplitz Operaiors 177
The Spectral Inclusion Theorem 179
The Symbol Map 179
The Spectrum of Self-adjoint Toeplitz Operators 183
The Spectrum of Analytic Toeplitz Operators 183
The C*-Algebra Generated by the Unilateral Shift 184
The Invertibility of Toeplitz Operators with Continuous Symbol 186

x Contents
The Invertibility of Unimodular Toeplitz Operators and Prediction Theory 187
The Spectrum of Toeplitz Operators with Symbol in Hm + C 189
The Connectedness of the Essential Spectrum 194
Localization to the Center of a C*-Algebra 196
Locality of Fredholmness for Toeplitz Operators 199
Notes 200
Exercises 203
References 208
Index 213

Preface
Operator theory is a diverse area of mathematics which derives its impetus
and motivation from several sources. It began as did practically all of modern
analysis with the study of integral equations at the end of the last century.
It now includes the study of operators and collections of operators arising in
various branches of physics and mechanics as well as other parts of math-
mathematics and indeed is sufficiently well developed to have a logic of its own.
The appearance of several monographs on recent studies in operator theory
testifies both to its vigor and breadth.
The intention of this book is to discuss certain advanced topics in operator
theory and to provide the necessary background for them assuming only
the standard senior-first year graduate courses in general topology, measure
theory, and algebra. There is no attempt at completeness and many
"elementary" topics are either omitted or mentioned only in the problems.
The intention is rather to obtain the main results as quickly as possible.
The book begins with a chapter presenting the basic results in the theory
of Banach spaces along with many relevant examples. The second chapter
concerns the elementary theory of commutative Banach algebras since these
techniques are essential for the approach to operator theory presented in the
later chapters. Then after a short chapter on the geometry of Hilbert space,
the study of operator theory begins in earnest. In the fourth chapter operators
on Hilbert space are studied and a rather sophisticated version of the spectral
theorem is obtained. The notion of a C*-algebra is introduced and used
throughout the last half of this chapter. The study of compact operators and
Fredholm operators is taken up in the fifth chapter along with certain ancillary
xi

xii Preface
results concerning ideals in C*-algebras. The approach here is a bit un-
unorthodox but is suggested by modern developments.
The last two chapters are of a slightly different character and present a
systematic development including recent research of the theory of Toeplitz
operators. This latter class of operators has attracted the attention of several
mathematicians recently and occurs in several rather diverse contexts.
In the sixth chapter certain topics from the theory of Hardy spaces are
developed. The selection is dictated by needs of the last chapter and proofs
are based on the techniques obtained earlier in the book. The study of Toeplitz
operators is taken up in the seventh chapter. Most of what is known in the
scalar case is presented including Widom's result on the connectedness of
the spectrum.
At the end of each chapter there are source notes which suggest additional
reading along with giving some comments on who proved what and when.
Although a reasonable attempt has been made in the latter chapters at citing
the appropriate source for important results, omissions have undoubtedly
occurred. Moreover, the absence of a reference should not be construed to
mean the result is due to the author.
In addition, following each chapter is a large number of problems of
varying difficulty. The purposes of these are many: to allow the reader to test
his understanding; to indicate certain extensions of the theory which are now
accessible; to alert the reader to certain important and related results of
which he should be aware along with a hint or a reference for the proof; and
to point out certain questions for which the answer is not known. These latter
questions are indicated by a double asterisk; a single asterisk indicates a
difficult problem.

Acknoivledginents
This book began as a set of lecture notes for a course given at the University
of Michigan in Spring, 1968 and again at SUNY at Stony Brook in the
academic year, 1969-1970.
I am indebted to many people in the writing of this book. Most of all I
would like to thank Bruce Abrahamse who prepared the original notes and
who has been a constant source of suggestions and constructive criticism
ever since. Also I would like to thank Stuart Clary and S. Pattanayak for
writing portions of later versions and for their many suggestions. In addition,
I would like to thank many friends and colleagues and, in particular, Paul
Halmos, Carl Pearcy, Pasquale Porcelli, Donald Sarason, and Allen Shields
with whom I have learned many of the things presented in this book. Special
thanks are due to Berrien Moore III who read and criticized the entire manu-
manuscript and to Joyce Lemen, Dorothy Lentz, and Carole Alberghine for typing
the various versions of this manuscript. Lastly, I would like to thank the
National Science Foundation and the Alfred E. Sloan Foundation for various
support during the writing of this book.
xiii

Symbols and Notation
A
AutO
C
C"
C(X)
Xn
co(Z+)
Gx
D
exp
SO
Si,
<$
I*
h
H1
Ho1
51
61
1
66
1
26
7
93
52
36
127
130
173
34
40
182
26
161
H2
H"
Hi
Ho™
j
ker
/Hz+)
/HZ)
/2(Z+)
/2(Z)
la>(l+)
L1
I2
IF
L°
Qf )
£C( )
#SO
A.
70
149
161
26
27
) 163
130
83
7
54
67
89
7
23
68
23
23
21
121
121
36

Symbols and Notation
u
r()
«()
ran
P()
p.O
*()
z
T
4
41
57
83
41
191
41
4
26
Afj, 39 Tv 177
Af,, 87 Tx 126
Mro 155 2() 179
M{X) 20
[/ 89
P 177 U+ 96
^ 51 l/<+B) 136
^+ 51
PC 176 w() 115
Pu« 86 **"() 115
3CT 101
QC 176
IIU
*
()i
II lip
©
±
®
1
6,82,119
10
24
29
30,71
31,62,79,118
64
108

Banach Spaces
1.1 We begin by introducing the most representative example of a Banach
space. Let A" be a compact Hausdorff space and let C(X) denote the set of
continuous complex-valued functions on X. For fx and f2 in C(X) and X
a complex number, we define:
With these operations C(X) is a commutative algebra with identity over the
complex field C.
Each function/in C(X) is bounded, since it follows from the fact that/is
continuous and X is compact that the range of /is a compact subset of C.
Thus the least upper bound of |/| is finite; we call this number the norm of
/and denote it by
The following properties of the norm are easily verified:
A) U/ll K = 0 if and only if /=0;
B) ||A/|U |A|||/||
C)
D)

2 1 Banach Spaces
We define a metric p on C(X) by p(f,g) = ||/-#|L- The properties of
a metric, namely,
A) P(f,g) = 0 if and only if/= #,
B) P</,<7) = P(g,f), and
follow immediately from properties (l)-C) of the norm. It is easily seen that
convergence with respect to the metric p is just uniform convergence. An
important property of this metric is that C(X) is complete with respect to it.
1.2 Proposition If A"is a compact Hausdorff space, then C(X) is a complete
metric space.
Proof If {/„}"= ] is a Cauchy sequence, then
l/.(*)-/„(*)! < II/.-/JL = PifnJJ
for each x in X. Hence, {fn(x)}™= t is a Cauchy sequence of complex numbers
for each x in X, so we may define /(x) = lim,,.* „,/,(*). We need to show
that / is in C(X) and that lim,,.^ H/-/JL = 0. To that end, given s > 0,
choose N such that n,m^ N implies \\fn—fm\\a, < £- For x0 in X there exists
a neighborhood 1/ of x0 such that |/jv(^o)~/jv(x)I < £ for x in U. Therefore,
l/(*o)-/«l < lim \fn(x0)~fN(x0)\ + |/w(*0)-/w(x)|
< 3e
which implies / is continuous. Further, for n > N and x in X, we have
\L(x)-f(x)\ = |/n(x)- lim/m(x)| = lim |/n(x) -/m(x)|
m-»oo m-»oo
< limsup||/n -/mL sg £.
m-»co
Thus, lirn^^^ \\fa-f\\m = 0 and hence C(X) is complete. ■
We next define the notion of Banach space which abstracts the salient
properties of the preceding example. We shall see later in this chapter that
every Banach space is isomorphic to a subspace of some C(X).
1.3 Definition A Banach space is a complex linear space SC with a norm
|| || satisfying

Abstract Banach Spaces 3
A) H/ll = 0 if and only if/= 0,
B) ||A/|| = |A| Il/H for X in C and/in %, and
C) \\f+g\\^ 11/11 + y\ for/and^ in SC,
such that 3C is complete in the metric given by this norm.
1.4 Proposition Let 3C be a Banach space. The functions
a: SC x 9C -> SC defined c(/,^) =f+g,
s:Cxf-»f defined j(A,/) = Xf and
«:Z->R+ defined «(/) = Il/H
are continuous.
Proof Obvious. ■
1.5 Directed Sets and Nets The topology of a metric space can be described
in terms of the sequences in it that converge. For more general topological
spaces a notion of generalized sequence is necessary. In what follows it will
often be convenient to describe a topology in terms of its convergent general-
generalized sequences. Thus we proceed to review for the reader the notion of net.
A directed set A is a partially ordered set having the property that for each
pair a and /? in A there exists y in A such that y > a and y > /?. A net is a
function a -» Xa on a directed set. If the Xa all lie in a topological space X,
then the net is said to converge to X in X if for each neighborhood U of X
there exists av in y4 such that Xa is in U for a > a^. Two topologies on a space
X coincide if they have the same convergent nets. Lastly, a topology can be
denned on X by prescribing the convergent nets. For further information
concerning nets and subnets, the reader should consult [71].
We now consider the convergence of Cauchy nets in a Banach space.
1.6 Definition A net {fa}aeA in a Banach space X is said to be a Cauchy
net if for every £ > 0, there exists a0 in A such that a1;a2 5= ceo implies
ll/a,-/aJ<e-
1.7 Proposition In a Banach space each Cauchy net is convergent.
Proof Let {fa}aeA be a Cauchy net in the Banach space SC. Choose o.x
such that a > a, implies \\fa—/J| < 1. Having chosen {at>t=1 in A, choose
an+, ~$s an such that a^a,+ 1 implies
\\Ja~Jan+t\\ < . •

4 1 Banach Spaces
The sequence {/,„}"= 1 is clearly Cauchy and, since SC is complete, there
exists/in SC such that limn^00/an=/.
It remains to prove that \imaeAfa =f. Given e > 0, choose n such that
l/n < e/2 and \\fan—/|| < e/2. Then for cc > c^ we have
I/.-/II < WL-fJ + 11/*,-/I < l/n + e/2 < e. ■
We next consider a general notion of summability in a Banach space
which will be used in Chapter 3.
1.8 Definition Let {fa}aeA be a set of vectors in the Banach space SC. Let
& = {F<=A : F finite}. If we define jF, ^F2 for jF, <= F2, then & is a
directed set. For each Fin #", let#F = '£MeFfa. If the net {^F}Fe^ converges
to some g in SC, then the sum X!aexA is said to converge and we write
9 = Zaex/a-
1.9 Proposition If {fa}aeA is a set of vectors in the Banach space SC such
*at £aeX ||/J converges in U, then £aeX/a converges in SC.
Proof It suffices to show, in the notation of Definition 1.8, that the net
{gF}Fe& is Cauchy. Since £aeX ||/J converges, for e > 0, there exists jF0 in
#" such that F^ Fo implies
- Z Il/J < e-
Thus for F1}F2^ Fo we have
1 fa' 1 /J
aeFl\F2 aeF2\Fi
I II/JI+ I Il/J
< 1 Il/J - I Il/J < B.
aeFiUF2 aeF0
Therefore, {gF}FeSr is Cauchy and X!aex/« converges by definition. ■
We now state an elementary criterion for a normed linear space (that is,
a complex linear space with a norm satisfying (l)-C) of Definition 1.3) to be
complete and hence a Banach space. This will prove very useful in verifying
that various examples are Banach spaces.

The Conjugate Space of Continuous Linear Functional 5
1.10 Corollary A normed linear space SC is a Banach space if and only if
for every sequence {/,}"=! of vectors in SC the condition £;£,, ||/J < oo
implies the convergence of
Proof If SC is a Banach space, then the conclusion follows from the
preceding proposition. Therefore, assume that {#„}"= i is a Cauchy sequence
in a normed linear space SC in which the series hypothesis is valid. Then we
may choose a subsequence {#„*}£=! such that ££,, ||#Bt+1-#,J| < oo as
follows: Choose nx such that for i,j~^nx we have ||#j—g-{ < 1; having
chosen {«*}*-! choose mn+1 > «N such that/J > %+1 implies ||^,—#,-|| < 2~N.
If we set fk = g^-g^ for k > 1 and /! = gni, then £»„, ||/J < °°, and
the hypothesis implies that the series X£L! fk converges. It follows from the
definition of convergence that the sequence {#BJ£Li converges in SC and
hence so also does {gn}%L 1. Thus SC is complete and hence a Banach space. ■
In the study of linear spaces the notion of a linear functional is extremely
important. The collection of linear functionals defined on a given linear space
is itself a linear space and this duality is a powerful tool for studying either
space. In the study of Banach spaces the corresponding notion is that of a
continuous linear functional.
1.11 Definition Let 3C be a Banach space. A function <p from SC to C is a
bounded linear functional if:
A) (pihfi+hfi) = h(p<Ji) + h<P<J2) for/1,/2 »n SC and A1;A2 in C;
and
B) There exists M such that \q>(f)\ < M\\f\\ for every/in 3C.
1.12 Proposition Let <p be a linear functional on the Banach space SC. The
following statements are equivalent:
A) q> is bounded;
B) (p is continuous;
C) q> is continuous at 0.
Proof A) implies B). If {fa}aeA is a net m & converging to /, then
-/ll = 0. Hence,
= tim\q>(fa-f)\ < limM||/a-/|| = 0,
aeA aeA
which implies that the net {<p (/J}aeX converges to q>{f). Thus q> is continuous.
B) implies C). Obvious.

6 1 Banach Spaces
C) implies A). If cp is continuous at 0, then there exists 5 > 0 such that
Il/H < d implies \<p(f)\<\. Hence, for any nonzero g in X we have
W(ff)\ =
d
and thus <p is bounded. ■
<P
5
2\\gW
<?l
We next define a norm on the space of bounded linear functionals which
makes it into a Banach space.
1.13 Definition Let$** be the set of bounded linear functionals on the Banach
space 3C. For <p in SC*, let
Then SC* is said to be the conjugate or dual space of 9£.
1.14 Proposition The conjugate space 9C* is a Banach space.
Proof That 9C* is a linear space is obvious, as are properties A) and B)
for the norm. To prove C) we compute
Finally, we must show that SC* is complete. Thus, suppose {<pn}^ 1 is a Cauchy
sequence in 9C*. For each/in 3C we have \<pn(f)-<pm(f)\ < l^-^ml ll/ll so
that the sequence of complex numbers {<pn(f)}%L t is Cauchy for each/in SC.
Hence, we can define <p(f) = linin.,^ <pn(f). The linearity of <p follows from
the corresponding linearity of the functionals <pn. Further, if N is chosen so
that n, m ^ N implies \\<pn — <pm\\ < 1, then for /in SC we have
\<p(f)\
limsup ||<pn-<pN|| ll/ll + ||<pN

Examples of Banach Spaces: c0,11, and /°° 7
Thus <p is in 3C* and it remains only to show that lim,,_»oo \\<p — <)<>n|j = 0. Given
e > 0, choose N such that n,m^ N implies \(pn — <pm|| < e. Then for/in 3C
and m,n^ N, we have
Since lim^^ |fa>-<pm)(/)| =0, we have ||<p-<pj < £. Thus the sequence
{<pn}™= i converges to <p and 9C* is complete. ■
The reader should compare the preceding proof to that of Proposition 1.2.
We now want to consider some further examples of Banach spaces and
to compute their respective conjugate spaces.
1.15 Examples Let/°°(Z+) denote the collection of all bounded complex
functions on the nonnegative integers Z+. Define addition and multiplication
pointwise and set \f\x = sup{|/(n)| :neZ+}. It is not difficult to verify
that /°°(Z+) is a Banach space with respect to this norm, and this will be left
as an exercise. Further, the collection of all functions/in /°°(Z+) such that
lin^ _♦„/(«) = 0 is a closed subspace of /°°(Z+) and hence a Banach space;
we denote this space by co(Z+).
In addition, let llA+) denote the collection of all complex functions <p
on Z+ such that Y^=o |^>(«)| < oo. Define addition and scalar multiplication
pointwise and set \<p\i = Y.^=o \<p(n)\- Again we leave as an exercise the task
of showing that /X(Z!+) is a Banach space for this norm.
We consider now the problem of identifying conjugate spaces and we
begin with co(Z+). For cp in ll(I.+) we define the functional 0 on co(Z+)
such that $(/) = H^=oV>(n)f(n) for/in co(Z+); the latter sum converges,
since
\4>(f)\ =
I <P(n)f(n)
W(n)\ \f(ri)\
I W(n)\ = ll/llc
Moreover, since 0 is obviously linear, this latter inequality shows that 0 is
in co(Z+)* and that \<p\l > ||0||, where the latter is the norm of <p as an
element of co(Z+)*. Thus the map a(<p) = Q from /'(Z+) to co(I+)* is well
defined and is a contraction. We want to show that a is isometric and onto
To that end let L be an element of co(Z+)* and define the function <pL
on Z+ so that <pL(ri) = L(en) for n in Z+, where en is the element of co(Z+)

8 1 Banach Spaces
defined to be 1 at n and 0 otherwise. We want to show that @L = L, and that
ll^illi ^ \\L\\- F°r eacn N'mZ+ consider the element
n=0 I~V~WI
of co{1+), where 0/0 is taken to be 0. Then \fN\x < 1 and an easy com-
computation yields
* T(e) N N
\\L\\ > \L(fN)\ = 2 yi~L{en) = E \Uen)\ = £ \<pL(n)\;
hence ^ is in ll(J.+) and H^li < ||-L||. Thus the map /?(£) = ^i from
co(Z+)* to Z1 (Z+) is also well defined and contractive. Moreover, let L be in
co(Z+)* and g be in co(Z+); then
lim ||<7— E S(n)en\\oo = 0
iV->oo n=0
and hence we have
£(#) = lim < E ^(«)i(O[ = lim { E S(n)<pL(n)
N-+O0 («=0 I N~* OO (« = 0
E 0
«=0
Therefore, the composition ceo/? is the identity on co(Z+)*. Lastly, since
0 = 0 implies ^> = 0, we have that a is one-to-one. Thus a is an isometrical
isomorphism of ll{1+) onto co(Z+)*.
Consider now the problem of identifying the conjugate space of I1 (Z+).
For/in /°°(Z+) we can define an element/of /HZ")* as follows:/^) =
Y.n=of(n)(P(n)- We leave as an exercise the verification that this identifies
/HZ+)*as/°°(Z+).
1.16 We return now to considering abstract Banach spaces. If a sequence of
bounded linear functionals {<pn}™=0 in 3C* converges in norm to q>, then it
must also converge pointwise, that is, lim,,.^ <pn(f) = <p(f) for each/in SC.
The following example shows that the converse is not true.
For k in Z+ and/in l1^) define Lk(f) =f(k). Then Lk is in /HZ+)*
and HZ-jJ = 1 for each k. Moreover, lim^^^Lk(f) = 0 for each/in ll(J.+).
Thus, the sequence {Lk}k=0 converges "pointwise" to the zero functional 0
but ||Ifc-0|| = 1 for each k in Z+.

Weak Topologies on Banach Spaces 9
Thus, pointwise convergence in 3C* is, in general, weaker than norm
convergence; that is, it is easier for a sequence to converge pointwise than it
is for it to converge in the norm. Since the notion of pointwise convergence
is a natural one, we might expect it to be useful in the study of Banach spaces.
That is indeed correct and we shall define the topology of pointwise con-
convergence after recalling a few facts about weak topologies.
1.17 Weak Topologies Let X be a set, Y be a topological space, and #" be
a family of functions from X into Y. The weak topology on X induced by
& is the weakest or smallest topology &~ onX for which each function in #"
is continuous. Thus 3~ is the topology generated by the sets {/~ * (U) :/e #",
U open in Y}. Convergence of nets in this topology is completely characterized
by lim^x,, = x if and only if hmaeAf(xa) =f(x) for every/in J87. Thus &
is the topology of pointwise convergence.
If Fis Hausdorffand #" separates the points of X, then the weak topology
is Hausdorff.
1.18 Definition For each/in 3C let/ denote the function, on 3C* defined
f(<p) = <p(f). The w*-topology on 3C* is the weak topology on <£"* induced
by the family of functions {/:/e 2£}.
1.19 Proposition The w*-topology on 3C* is Hausdorff.
Proof If <Pi / q>2, then there exists / in 9C such that <px(f) / (Pzif)-
Hence,/(^J #/(^J) so that the functions {f\fe9£} separate, the points
of 3C*. The proposition now follows from the remark at the end of Section
1.17. ■
We point out that the w*-topology is not, in general, metrizable (see
Problem 1.13). Next we record the following easy proposition for reference.
1.20 Proposition A net {^a}aex in 3C* converges to <p in 3C* in the w*-
topology if and only if lim^,, <&,(/) = <p(f) for every/in $£.
The following shows that the w*-topology is determined on bounded
subsets of 3C* by a dense subset of 3C and this fact will be used in subsequent
chapters.
1.21 Proposition Suppose Jt is a dense subset of SC and {(pa}aeA ls a
uniformly bounded net in 9£* such that limaeA(pa(f) = q>{f) for/in M.

10 1 Banach Spaces
Then the net {(pJ^A converges to <p in the M>*-topology.
Proof Given g in 3C and e > 0, choose/in Jt such that ||/—g\ < s/3M,
where M = sup{||^>[|, \tpa\ : ae A}. If a0 is chosen such that cO cc0 implies
\(pa(f)—<p(f)\ < e/3, then for a > a0, we have
\<P.(g)-9a(f)\ + \<Pa(f)-<p(f)\ + to(f)-<p(s)\
Thus {(pJaEA converges to <p in the w*-topology. ■
1.22 Definition The unit ball of a Banach spaced is the set {/e X: \\f\\ < 1}
and is denoted
1.23 Theorem (Alaoglu) The unit ball (#■*), of the dual of a Banach space
is compact in the w*-topology.
Proof The proof is accomplished by identifying (^*), with a closed sub-
subset of a large product space the compactness of which follows from Tychonoff's
theorem (see [71]).
For each/in (^), let C/ denote a copy of the closed unit disk in C and
let P denote the product space X/em, C/. By Tychonoff's theorem P is
compact. Define A from (#"•), to P by A(<p) = <p | (#"),. Since A(<Pi) = A(<p2)
implies that the restrictions of <px and <p2 to the unit ball of 3C are identical,
it follows that A is one-to-one. Further, a net {^a}aeA in 3C* converges in the
w*-topology to a <p in SC* if and only if lim^,, q>a(f) = <p(f) for/in SC if
and only if lirn^^ <pa(f) = q>(J) for/in CT)l if and only if limaey4 W<P«)(f) =
A(<p)(f) for/in (^),. This latter statement is equivalent to lim^,, A(%) = A(^>)
in the topology of P. Thus, A is a homeomorphism between (^*)i and the
subset A [(£•*),] of P.
We complete the proof by showing that A [($■*)!] is closed in P. Suppose
{A(<pa)}0l£A is a net in A[($■*),] that converges in the product topology to
^ in P. If/ g, andf+g are in (^),, then
= limA(<pa)(f+g) = lim A(%)(/) +
aeA aeA aeA
Further, if/and Xf are in {Sf)l, then
= Mim<pa(f) =

The Ataoglu Theorem 11
Hence ^ determines an element $ of (!%*)! by the relationship
for/in 9C. Since ^(/) = \j/{f) for/in (#"),, we see not only that # is in (#"*),
but, in addition, A($) = ^. Thus A[(^*)J is a closed subset of P, and
therefore (^*)i is compact in the w*-topology. ■
The importance of the preceding theorem lies in the fact that compact
spaces possess many pleasant properties. We shall also use it to show that
every Banach space is isomorphic to a subspace of some C(X). Before doing
this we need to know something about how many continuous linear functionals
there are on a Banach space. This and more is contained in the Hahn-Banach
theorem. Although we are only interested in Banach spaces in this chapter,
it is more illuminating to state and prove the Hahn-Banach theorem in
slightly greater generality. To do this we need the following definition.
1.24 Definition Let $ be a real linear space and p be a real-valued function
defined on €. Then p is said to be a sublinear functional on & if p(J+g) <
p{f)+p{g) for/and g in & and p(Xf) = Xp(f) for/in S and positive X.
1.25 Theorem (Hahn-Banach) Let $ be a real linear space and let p be a
sublinear functional on S. Let J^ be a subspace of $ and <p be a real linear
functional on $F such that (p(f)^p(f) for/in J*. Then there exists a real
linear functional $ on § such that <&(/) = <p(f) for/in J^ and <&(#)</>(#)
for g in S.
Proof We may assume without loss of generality that SF / {0}. Take /
not in & and let ^ = {g+Xf: X e U, g e &}. We first extend <p to & and to
do this it suffices to define <!>(/) appropriately. We want <b{g+Xf) ^p(g+Xf)
for all X in U and g in &. Dividing by \X\ this can be written <b(J—h) <p (f—h)
and <$>(-/+h) <p(-f+h) for all h in SF or equivalently,
-/>(-/+ h) + <p(h) < <&(/) < P(f- h) + <p(h)
for all h in !F. Thus a value can be chosen for <&(/) such that the resultant
<t> on ^ has the required properties if and only if
sup{_p(-/+ h) + 9m < inf
However, for h and k in #", we have
- <p(k) = <p(h -k)^ p(h -k)^ p(f- k) + p(h -

12 1 Banach Spaces
so that
-P(h -f) + <p(h) < p(f- k)
Therefore, <p can be extended to O on ^ such that <&(h) ^p(h) for h in ^.
Our problem now is to somehow obtain a maximal extension of <p. To that
end let & denote the class of extensions of <p to larger subspaces satisfying the
required inequality. Hence an element of SP consists of a subspace ^ of $
which contains #" and a linear functional <f)9 on 9 which extends <p and
satisfies <£>g(g) < p(g) for g in ^. There is a natural partial order defined on
9, where (S?l5 <&,,) < (^2,«W if ^i c S?2 and 3>^(/) = <b9l(f) for/in »,.
To apply Zorn's lemma to the'class &, we must show that for every chain
{(&«>®0j}<teA in SP there is a maximal element in SP. (Recall that a chain is
just a linearly ordered set.) If {(&a,®&J}a£A is a chain in ^», let ^ = \JaeA^a
and define $ on ^ by <£(/) = ®<gJJ), where/is in ^a. It is easily verified that
^ is a subspace of <? which contains 3F; that <t> is well defined, linear, and
satisfies <&(/) ^p(f) for/in ^; and that E^,$^) < (^,<&) for each a in y4.
Thus the chain has a maximal element in & and Zorn's lemma implies that
& itself has a maximal element (^0,<t>0). If ^0 were not <?, then the argument
of the preceding paragraph would yield a strictly greater element in & which
would contradict the maximality of (%, <&^0). Thus &0 = g and we have the
desired extension of <p to <?. ■
The form of this result which we need in this chapter is the following.
1.26 Theorem (Hahn-Banach) Let Jt be a subspace of the Banach space SC.
If <p is a bounded linear functional on ^, then there exists <& in 3C* such that
= Hf) for/in .# and ||<Dj| = |M|.
Proof If we consider SC as the real linear space §£, then the norm is a
sublinear functional on St and ^ = Re <p is a real linear functional on the
real subspace Jl. It is evident that ||^|| < ||^||. Setting p{f) = \\y\\ ||/|| we
have i]/ (/)</?(/) and hence from the preceding theorem we obtain a real
linear functional ¥ on SC that extends ^ and satisfies ¥(/)< j|9)|| |/|| for
/in «". If we now define $ on SC by $(/) = xV(f)-ixi'(if), then we want to
show that <& is a bounded complex linear functional on SC that extends <p and
has norm |^>||.
For/and g in SC, we have
- W{tg)

The Hahn-Banach Theorem 13
Further, if A, and A2 are real and/is in SC, then $(*/) = ¥(*/) - /¥(-/) =
j<t>(/) and hence
) = a,
Thus, O is a complex linear functional on SC. Moreover, for /in Ji we have
Lastly, to prove ||$|| = ||*|| it suffices to show that ||$|| < ||*|| in view of
the fact that ||*F|| = ||^|| and $ is an extension of <p. For/in SC write $(/) =
rew. Then
|<D(/)| = r = e-i6®(J) = ®{e-lef) = ¥(«"»/)
< i«p(e-w/)i < \\n ii/ii,
so that <& has been shown to be an extension of q> to SC having the same
norm. ■
1.27 Corollary If/ is an element of a Banach space SC, then there exists q>
in SC* of unit norm so that <p(f) = |/||.
Proof We may assume// 0. Let M = {A/: A e C} and define ^ on Jt
by ^(A/) = A ||/1|. Then ||^|| = 1 and an extension of i]/ to SC given by the
Hahn-Banach theorem has the desired properties. ■
1.28 Corollary If <p(/) = 0 for each <p in SC*, then /= 0.
Proof Obvious. ■
We give two applications of the Hahn-Banach theorem. First we prove
a theorem of Banach showing that C(X) is a universal Banach space and then
we determine the conjugate space of the Banach space C([0,1]).
1.29 Theorem (Banach) Every Banach space SC is isometrically isomorphic
to a closed subspace of C{X) for some compact Hausdorff space X.
Proof Let X be {SC*)^ with the w*-topology and define P from SC to
C(X) by (fif)(<p) = <p(f). For/! and/2 in SC and kx and A2 in C, we have
= cpiX.f, + A2/2) = A, <p(/i) + A2<

14 1 Banach Spaces
and thus /? is a linear map. Further, for/in SC we have
= sup |/J(fl(<P)| = sup |<p(/)|< sup Ml 11/11 < li/ll,
and since by Corollary 1.27 there exists <p in {S^*)t such that <p(f) = \\f\\,
we have that ||/f (/)!!«, = ||/||. Thus /? is an isometrical isomorphism. ■
The preceding construction never yields an isomorphism of SC onto
C((%*)i) even if SC is C(Y) for some Y. If #" is separable, then topological
arguments can be used to show that X can be taken to be the closed unit
interval.
Although this theorem can be viewed as a structure theorem for Banach
spaces, the absence of a canonical X associated with each 3C vitiates its
usefulness.
1.30 We now consider the problem of identifying the conjugate space of
C([0,1]). By this we mean finding some concrete realization of the elements
of C([0,1])* analogous with the identification obtained in Section 1.15 of the
conjugate space of Z1 (Z+) as the space /°° (Z+) of bounded complex functions
on Z+. We shall identify C([0,1])* with a space of functions of bounded
variation on [0,1]. We shall comment on C(X)* for general X later in the
chapter. We begin by recalling a definition.
1.31 Definition If cp is a complex function on [0,1], then <p is said to be of
bounded variation if there exists M^O such that for every partition
0 = t0 < tx <■•• <tn< tn+l = 1, it is true that
t l<P('i+i) - <P(td\ < M.
i=0
The greatest lower bound of the set of all such M will be denoted by \\<p\\v.
An important property of a function of bounded variation is that it
possesses limits from both the right and the left at all points of [0,1].
1.32 Proposition A function of bounded variation possesses a limit from
the left and right at each point.
Proof Let <p be a function on [0,1] not having a limit from the left at
some t in @,1]; we shall show that q> is not of bounded variation on [0,1].
If <p does not have a limit from the left at t, then for some s > 0, it is true
that for each d > 0 there exist s and s' in [0,1] such that t—d<s<s' <t and
\(p(s) — <p(s')\ > e. Thus we can choose inductively sequences {jn}^=i and

The Hahn-Banach Theorem 15
{■?„'}"= 1 such that 0 < s, < j,' < ••• < sn < sn' < t and \<p(sn)-<p(sn')\ ^ s.
Now consider the partition t0 = 0; t2k+1 = sk for k = 0,1,..., AT— 1; ;2k =
sk' for A: = 1,2, ...,N; and <2n+i = 1- Then
2N N
k=0 h=1
which implies that cp is not of bounded variation on [0,1]. The proof that
<p has a limit from the right proceeds analogously. ■
Thus, if <p is a function of bounded variation on [0,1], the limit <p(t~)
of <p from the left and the limit <p(t+) of <p from the right are well defined for t
in [0,1]. (We set <p@~) = <p@) and <p(l+) = <p(l)-) Moreover, a function
of bounded variation can have at most countably many discontinuities.
1.33 Corollary If cp is a function of bounded variation on [0,1], then <p
has at most countably many discontinuities.
Proof Observe first that <p fails to be continuous at t in [0,1] if and only
if <p(t) / <p(t+) or <p(t) / <p(t~). Moreover, if t0, tu ..., tn are distinct points
of [0,1], then
N N
(=0 /=0
Thus for each e > 0 there exists at most finitely many points t in [0,1] such
that \(p(t)—(p(t+)\ + \(p(t)—(p(t~)\ > e. Hence the set of discontinuities of
<p is at most countable. ■
We next recall the definition of the Riemann-Stieltjes integral. For / in
C([0,1]) and <p of bounded variation on [0,1], we denote by §lfd(p, the
integral of/with respect to <p; that is, $lfd<pis the limit of sums of the form
HUof(ti)[.<p(ti+i)-<P(tl)'], where 0 = to<t1 < - < tn< tn+1 = 1 is a par-
partition of [0,1] and t{ is a point in the interval [tb ti+1~\. (The limit is taken
over partitions for which max1^£^ll|fj+1 —ft| tends to zero.) In the following
proposition we collect the facts about the Riemann-Stieltjes integral which
we will need.
1.34 Proposition If / is in C([0,1]) and q> is of bounded variation on
[0,1], then llfdq> exists. Moreover:
A) Jo(^i/i + ^2/2) d<p = A, JJ/j dcp + A2JJ/2 dq> for ft and f2 in
D» !])> ^1 and A2 in C, and <p of bounded variation on [0,1];

16 1 Banach Spaces
B) HfdQ.! q>x + k2q>2) = A, \\fdcpx + k2\\fdq>2 for / in C([0,1]), A,
and k2 in C, and cpx and q>2 of bounded variation on [0,1]; and
C) \\ofdtp\ < 11/11 m \\q>\\v for /in C([0,1]) and <p of bounded variation
on [0,1].
Proof Compare [65, p. 107]. ■
Now for q> of bounded variation on [0,1], let 0 be the function defined
by 0if) = lofdcp for/in C([0,1]). That 0 is an element of C([0,1])* follows
from the preceding proposition. However, if <p is a function of bounded
variation on [0,1], t0 is a point in [0,1), and we define the function ^ on
[0,1] such that ^(/) = <p@ for t / t0 and ^(t0) = <p(to~~), then an easy
computation shows that l\fd(p = \\fdty for / in C([0,1]). Thus if one is
interested only in the linear functional which a function of bounded variation
defines on C([0,1]), then <p and ^ are equivalent, or more precisely, 0 = fy.
In order to avoid identifying the conjugate space of C([0,1]) with equivalence
classes of functions of bounded variation, we choose a normalized repre-
representative from each class by requiring that the distinguished function be left
continuous on @,1).
1.35 Proposition Let q> be of bounded variation on [0,1] and i]/ be the
function defined ij/(t) = q>(t~) for t in @,1), ^@) = (?@), and ^A) = q>(\).
Then ^ is of bounded variation on [0,1], ||^|c < ||^>||c, and
[fdcp = P
Jo Jo
/# for /inC([0,l]).
Proof From Corollary 1.33 it follows that we can list {j,}j»i the points
of [0,1] at which <p is discontinuous from the left. Moreover, from the
definition of ^ we have ^/(t) = <p(i) for t # st for i > 1. Now let 0 = t0 <
h < ■- < tn < tn+i = 1 be a partition of [0,1] having the property that if
tt is in 5 = {st: i > 1}, then neither tv_x nor ti+1 is. To show that ^ is of
bounded variation and ||^||,, < \\<p\\v, it is sufficient to prove that
1 = 0
Fix e > 0. If tt is not in 5 or i = 0 or n+1, then set tt' = tt. If tx is in 5
and //0, n + l, choose f£' in (tt-utt) such that \<p(ti~)-q>(ti)\ <e/2n+2.
Then 0 = t0' < tx' <■■■ <tn' < t'n+1 = 1 is a partition of [0,1] and

The Hahn-Banach Theorem 17
Z liK*,+i) - iKOI = Z bd+i) - 9@1
i=O z=0
Z Ivft+i) - v«+i)l + Z k«+i)- vftO
1=0 1 = 0
Z
i=0
Since e is arbitrary, we have that ^ is of bounded variation and that
HiHI.<lbL.
To complete the proof for N an integer, let t]N be the function denned
tjN(t) = 0 for t not in {*„j2. • ••>%} and j/N(jj) = <?(.?()-^(jO for 1 ^ i < iV.
Then it is easy to show that ]imN^o0\\<p-(i]/+tiN)\\v = 0 and \\,fdt}n = 0
for/in C([0,1]). Thus, we have from Proposition 1.34 that
Jo Jo N-»co Jo Jo
/#.
Let BV[0,1] denote the space of all complex functions on [0,1] which
are of bounded variation on [0,1], which vanish at 0, and which are con-
continuous from the left on @,1). With respect to pointwise addition and scalar
multiplication, BV[0,1] is a linear space, and || ||,, defines a norm.
1.36 Theorem The space BV[0,1] is a Banach space.
Proof We shall make use of Corollary 1.10 to show that BV[0,1] is
complete and hence a Banach space. Suppose {<pn}%Li is a sequence of
functions in BV[0,1] such that ££=i H^X < oo. Since
for t in [0,1], it follows that ^Li q>n(t) converges absolutely and uniformly
to a function <p defined on [0,1]. It is immediate that ^>@) = 0 and that q>
is continuous from the left on @,1). It remains to show that <p is of bounded
variation and that Hrnf,.^ \\(p - ££= 19X = °-
If 0 = t0 < tt < ••• < tk < tk+1 = 1 is any partition of [0,1], then
Il9(*i+i)-9@l = Z
i=0 i=0
Z 9.
l
Z
n=l n=l

18 1 Banach Spaces
& CO
< E 5>.
00
< E
_ , _ , s E lb.ll..
h=1 Li = O J n=l
Therefore, <p is of bounded variation and hence in BV[0,1]. Moreover, since
the inequality
N
n=l
k «j
<E E
1 = 0 n=N
E 9«
,i+ift
E
=N+l
holds for every partition of [0,1], we see that
<P- E <Pn
n=l
E II?X
t, n=N+l
for every integer iV. Thus (p = X!^°=i <pn in the norm of BV\_0,1] and the proof
is complete. ■
Recall that for <p of bounded variation on [0,1 ], we let 0 be the linear
functional denned 0{f) = Hfd<p for/in C([0,1]).
1.37 Theorem (Riesz) The mapping <p -> 0 is an isometrical isomorphism
between BV[0,1] and C([0,1])*.
Proof The fact that $ is in C([0,1])* follows immediately from
Proposition 1.34 and we have, moreover, that \0\ < II^L- To complete the
proof we must, given an L in C([0,1])*, produce a function ^ in BV[0,1]
such that $ = L and ||^||,, < ||L||. To do this we first use the Hahn-Banach
theorem to extend L to a larger Banach space.
Let B[0,1] be the space of all bounded complex functions on [0,1].
It is by now routine to verify that B\_0,1] is a Banach space with respect to
pointwise addition and scalar multiplication, and the norm ||/||s =
sup {|/@1 : 0 < t < 1}. For E a subset of [0,1], let IE denote the characteristic
(or indicator) function on E, that is, IE(t) is 1 if t is in E and 0 otherwise. Then
for every E the function IE is in B\_0,1].
Since C([0,1]) is a subspace of B[0,1] and since L is a bounded linear
functional on C([0,1]), we can extend it (but not necessarily uniquely) to a
bounded linear functional L' on B [0,1] such that ||L'|| = ||L||. Moreover, L

The Conjugate Space of C([0,1]) 19
can be chosen to satisfy L'(^{0}) — 0' since we may first extend it in this manner
to the linear span of /{0} and C([0,1]) in view of the inequality
|L'(/+ A/{0})| = |L(/)| < |L||||/||oo < IILIIII/+^{O}I|S
which holds for/in C([0,1]) and X in C.
Now for 0 < t < 1 define (p(i) = £(/(o,f]X where @, i] is the half open
interval {s: 0 < s < i} and set <p@) = 0. We want first to show that <p is of
bounded variation and that |M|,,< ||L||.
Let 0 = t0 < tx < ■■■ < tn < tn+1 — 1 be a partition of [0,1] and set
k = [<K'*+i)-<K'*)]/l<K'*+i)-~<Pfe)l
if q>(tk+1) / <p0fc) and 0 otherwise. Then the function
n
is in B\_0,1] and ||/||s < 1. Moreover, we have
J;obfe+i)-?c*)i = fo
n
= z/.
= £(/)
and hence <p is of bounded variation and
We next want to show that L(g) = \\,g d<p for every function g in
C([0,1]). To that end, let g be in C([0,1]) and £ > 0; choose a partition
0 = t0 < tx <■■■ <tn< tn+1 = 1 such that
£
\g(s)-g(s')\ <
2\\L\\
for s and s' in each subinterval \jk, tk+1~\ and such that
o
-Y. 9(tk)(<p(tk+1)-<p(tk))
Then we have for/= 'Ll=og{tk)Jlf^1^a+g(fi)l@} the inequality
f1
Jo
- \ gd<p
Jo
k=0
Jo

20 1 Banach Spaces
Thus L(g) = llgdq> for g in C([0,1]).
Now the q> obtained need not be continuous from the left on @,1).
However, appealing to Proposition 1.35, we obtain \j/ in BV[0,1] such that
1 dq> = L(g)
for g in C([0,1]). Thus $ = L and combining the inequality obtained from
the first paragraph of the proof with the one just above, we obtain
||<H|r= ||L||. Thus BV[0,1] = C([0,1])*. ■
1.38 The Conjugate Space of C(X) If X is an arbitrary compact Hausdorff
space, then the notion of a function of bounded variation on X makes no
sense. Thus one must search for a different realization of the elements of
C(X)*. It can be shown with little difficulty that each countably additive
measure defined on the Borel sets of X gives rise to a bounded linear functional
on C(X). Moreover, just as in the preceding proof we can extend a bounded
linear functional on C(X) to the Banach space of bounded Borel functions
by the Hahn-Banach theorem and then obtain a Borel measure by evaluating
the extended functional at the indicator functions for Borel sets. This repre-
representation of a bounded linear functional as a Borel measure is not unique.
If one restricts attention to the regular Borel measures on X, then the pairing
is unique and one can identify C(X)* with the space M(X) of complex regular
Borel measures on X. We do not prove this in this book but refer the reader to
[65]. This result is usually called the Riesz-Markov representation theorem.
We shall need it at least for X a compact subset of the plane.
1.39 Quotient Spaces Let 3C be a Banach space and Jt be a closed sub-
space of SC. We want to show that there is a natural norm on the quotient
space 3C\Jt making it into a Banach space. Let 9C\Jt denote the linear space
of equivalence classes {[/] :/e SC), where [/] = {f+g : g e Jt}, and define
a norm on 9C\Jt by
IIL/DII = inf \\f+g\ = inf \\h\\.
Then ||[/]|| = 0 implies there exists a sequence {#„}"=, in Jt with
IimII_oo||/-|-0j = 0. Since Jt is closed, it follows that / is in Jt so that
[/] = [0]. Conversely, if [/I = [0], then/is in Jt and 0 «S \\f\\ < II/-/I =°-
Thus, IIL/IH = 0 if and only if [f] = [0]. Further, if/, and/2 are in SC and

The Conjugate Space of C([0,1]) 21
X is in C, then
= inf P/i+^H = |A| inf ||_
and
= IIL/i+/2]|| = inf |/,+/2+*||
= inf \\f,+g,+f2+go\\ < inf U,+gA+ inf
< urai +
Therefore, || ■ || is a norm on 9C\M and it remains only to prove that the space
is complete.
If {[/„]} ™=! is a Cauchy sequence in ^/^, then there exists a subsequence
{/^}»=1 such that || [/Bfc+,]-[/„ J|| < 1/2*. If we choose ^ in [/„_-/»*]
such that ||/ij| < 1/2*, then 2*°=i II^11 < J an^ hence the sequence {hk} is
absolutely summable. Therefore, h = '££=1hk exists by Proposition 1.9.
Since
1=1 1=1
we have lim^ „ [/nfc-/ni] = [h]. Therefore lim^^ [/nJ = [/;+/„,] and
is seen to be a Banach space.
We conclude by pointing out that the natural map /-> [/] from SC to
3C\Jt is a contraction and is an open map. For suppose /is in 3C, e > 0, and
K(J) = {ge%: \\f-g\\ < e}. If [h] is in
then there exists /i0 in \_h] such that ||/— h0 \\ < s. Hence, [fi] and, in fact,
all of Nc([f~\) is in the image of Nc(f) under the natural map. Therefore, the
natural map is open.
1.40 Definition Let SC and Q/ be Banach spaces. A linear transformation
Ffrom 3C to <& is said to be bounded if
f*o 11/11
The set of bounded linear transformations of SC to <& is denoted
with 2B£,9T) abbreviated 2(9T). A linear transformation is bounded if and
only if it is continuous.
1.41 Proposition The space 2.{9C, <&) is a Banach space.

22 1 Banach Spaces
Proof The only thing that needs proof is the completeness of
and that is left as an exercise. ■
Although an essential feature of a Banach space is that it is complete in
the metric induced by the norm, we have not yet made any real use of this
property. The importance of completeness is due mainly to the applicability
of the Baire category theorem. We now present one of the principal appli-
applications, namely the open mapping theorem. The equally important uniform
boundedness theorem will be given in the exercises.
1.42 Theorem If S£ and <W are Banach spaces and Tin S,{SC,<&) is one-to-
one and onto, then T~i exists and is bounded.
Proof The transformation T~1 is well denned and we must show it to
be bounded. For r > 0 let (ST)r = {f&9£\ \f\ < r}. To show that T'1 is
bounded it is sufficient to establish T~i{C/)xc:{9C)r for some r > 0 or
equivalently, that {®}\ c T(8£)N for some integer N.
Since Tis onto, we have \J ™= 1 T[(^n)] = ®/. Further, since <& is a complete
metric space, the Baire category theorem states that <& is not the countable
union of nowhere dense sets. Thus, for some N the closure clos{r[(^)N]}
of r[($\] contains a nonempty open set. It follows that there is an h in
(#)N and an e > 0 such that
Th + (<&), = {/e 9 : \\f-Ih\\ < e} <= clos{r[Cr)N]}.
Therefore, (©% <= -77!+clos{r[(#\]} <= clos{r[(#JN]} so that (<80, <=
clos{r[(^)r]}, where r = 2N/e. Except for the fact that this is the closure,
this is what we need to prove. Thus we want to remove the closure.
Let/be in {®)\. There exists gx in B£)r with ||/- Tgx \ < \. Since/- Tgx
is in 03O,/2, there exists g2 in (#)r/2 with \\f-Tgl-Tg2\<\. Since
/- Tgy - Tg2 is in («01/4, there exists g3 in {T)m with ||/- Tgx - Tg2 - Tg31|
<^. Continuing by induction, we obtain a sequence {#n}™=i such that
llo.ll < r/2-1 and I/-I?,, Tgt\\ < 1/2". Since
1^
the series Y^=iSn converges to an element g in (^Jr. Further,
Tg = r(lim f gk) = lim f Tgk =/.
\h-»k> k=l / n->oo k=l
Therefore, (W^ <= T[(SfJr] which completes the proof. ■

The Lebesgue Spaces: Ll and V 23
1.43 Corollary (Open Mapping Theorem) If SC and <W are Banach spaces
and Tis an onto operator in 2(SC, <&), then Tis an open map.
Proof Since T is continuous, the set M = {/e SC : Tf= 0} is a closed
subspace of SC. We want to define a transformation 5 (see accompanying
figure) from the quotient space SC\Jt to <& as follows: for [/] in #"/./$■ set
S\J]=Tg for g in [/]. Since gx and #2 in [/] imply that gx -g2 is in M,
we have Tj^, = Tg2 and hence 5 is well defined. Obviously, 5 is linear and
the inequality
|
fief/]
which holds for [/] in #"/^, shows that 5 is bounded. Moreover, if 5[/] = 0,
then 7/= 0, which implies that / is in Jl and [/] = [0]. Therefore, 5 is
one-to-one. Lastly, 5 is onto, since T is, and hence the preceding theorem
demonstrates that 5 is an open map. Since the natural homomorphism n
from SC to SC\J( is open and T= Sn, we obtain that Tis open. ■
We conclude this chapter with some classical examples of Banach spaces
due to Lebesgue and Hardy. (It is assumed in what follows that the reader is
familiar with standard measure theory.)
1.44 The Lebesgue Spaces Let n be a probability measure on a <r-algebra
£P of subsets of a set X. Let if1 denote the linear space of integrable complex
functions on X with pointwise addition and scalar multiplication, and let
Jf be the subspace of null functions. Hence, a measurable function / on X
is in Sel if Jx |/| d\i < 00 and is in JV if Jx |/| du = 0. We let L1 denote the
quotient space SC1/^ with the norm || [JQ || j = Jx |/| d\x. That this satisfies
the properties of a norm (that is, (l)-C) of Section 1.1) is easy; the com-
completeness is only slightly more difficult.
Let {£/„]}£, be a sequence in L1 such that ££., \\lfJU < M< 00.
Choose representatives/, from each [/„]; then the sequence {Z»=il/.|}w=i
is an increasing sequence of nonnegative measurable functions having the
property that the integrals
7x:

24 1 Banach Spaces
are uniformly bounded. Thus, it follows from Fatou's lemma that the function
h = IT=i |/»l is integrable. Therefore, the sequence {£%=lfn}%=1 converges
almost everywhere to an integrable function k in SC1. Finally, we have that
1 JX
I /. - I /.
n-1
CO f*
E I/.I
= AM-1 JX
E IIC/JII.,
- Thus, L1 is a Banach space.
denote the collection of functions / in if1 which
and hence 2^=i [/J =
For 1 <p < oo let
satisfy Jx |/|p dn< od and set .xT" = J^ n if". Then it can be shown that
ifp is a linear subspace of if1 and that the quotient space IP =
is a Banach space for the norm
The details of this will be carried out for the case p — 2 in Chapter 3; we
refer the reader to [65] for details concerning the other cases.
Now let if °° denote the subspace of ^C1 consisting of the essentially
bounded functions, that is, the functions/for which the set
{xeX:\f(x)\ > M}
has measure zero for M sufficiently large, and let ||/|| m denote the smallest
such M. If we set Jf °° = Jf n i£ °°, then we can easily show that for / in
if °° we have \f\x = 0 if and only if/is in Jf °°. Thus || H^ defines a norm
on the quotient space L00 = if a>/Jr °°. To show that L00 is a Banach space
we need only verify completeness and we do this using Corollary 1.10. Let
{ [fnl}n= i be a sequence of elements of L °° such that £n°i 1 \\ [/„] || „ < M < oo.
Choose representatives/, for each [/„] such that |/J is bounded everywhere
by || [/] 1 oo ■ Then for x in [0,1] we have
I I/.WI < I
l I
Therefore the function h(x) =
bounded since
M.
is well defined, measurable, and
\h(x)\ =
E /„(*)
Thus, h is in if °° and we omit the verification that limN_
= 0. Hence, L00 is a Banach space.

The Lebesgue Spaces: Ll and U* 25
Although the elements of an Lp space are actually equivalence classes of
functions, one normally treats them as functions. Thus when we write / in
L1, we mean /is a function in if1 and/denotes the equivalence class in L1
containing/ Hereafter we adopt this abuse of notation.
We conclude by showing that (L1)* can be identified as L00. This result
should be compared with that of Example 1.15. We indicate a different proof
of this result not using the Radon-Nikodym theorem in Problem 3.22.
For <p in L00 we let 0 denote the linear functional denned as
= f
Jx
for /inL1.
1.45 Theorem The map <p -> 0 is an isometrical isomorphism of L00 onto
(L1)*.
Proof If <p is in L00, then for/in L1 we have |(<p/)(x)| < ML !/(*)!
for almost every x in X. Thus ftp is integrable, 0 is well denned and linear,
and
[fq>dn < ML f
Jx Jx
Therefore, 0 is in (L1)* and Ml < IML-
Now let L be an element of (L1)*. For E a measurable subset of X the
indicator function IE is in L1 and \\lE\\i = \xIEdp = p(E). If we set
X(E) = L(/£), then it is easily verified that A is a finitely additive set function
and that |A(£)| < ju(£)||L||. Moreover, if {En}™=1 is a nested sequence of
measurable sets such that f)™=1 En = 0, then
I lim A(£L)| < lim \A(EJ\ < IILII lim »(£.) = 0.
Therefore, A is a complex measure on X dominated by ju. Hence by the
Radon-Nikodym theorem there exists an integrable function <p on X such
that X(E) = \xIE<pdp for all measurable E. It remains to prove that q> is
essentially bounded by ||L|| and that L(f) = lxf<p dp. for /in I).
For JV an integer, set
x6X:||L||+i< \<p(x)\ ^
Then EN is measurable and IEN<p is bounded. If /= £?=i c,/£i is a simple
step function, then it is easy to see that L(f) = \xfq> dp.. Moreover, a
simple approximation argument shows that if/is in L1 and supported on

26 1 Banach Spaces
EN, then again L(f) = \xf<P dp. Let g be the function defined to be
<p(x)l\q>(x)\ if x is in En and <p(x) / 0 and 0 otherwise. Then g is in L1, is
supported on EN, and ||^|| ^ = ti(EN). Therefore, we have
H(EN)\\L\\ > \L(g)\ =
which implies p(EN) = 0. Hence, we obtain ju(Un=i £n) = 0, which implies
<p is essentially bounded and ML ^ Wl- Moreover, the above argument
can be used to show that
L(f) = f fq> dp for all/in I},
Jx
which completes the proof. ■
We now consider the Banach spaces first studied by Hardy. Although
these spaces can be viewed as subspaces of the IF spaces, this point of view
is quite different from that of Hardy, who considered them as spaces of
analytic functions on the unit disk. Moreover, although we study these spaces
in some detail in later chapters, here we do little more than give the definition
and make a few elementary observations concerning them.
1.46 The Hardy Spaces If T denotes the unit circle in the complex plane
and p is the Lebesgue measure on T normalized so that p(J) = 1, then we
can define the Lebesgue spaces L?(J) with respect to p. The Hardy space Hp
will be defined as a closed subspace of LP(T). As in the previous section we
consider only the cases p = 1 or co.
For n in Z let %n denote the function on T defined #n(z) = z". If we define
H1 = |/6 £(¥): ^ j^fxn dt = O for n = 1,2,3,
then H1 is obviously a linear subspace of L1 (T). Moreover, since the set
is the kernel of a bounded linear functional on I}(T), we see that H1 is a
closed subspace of L1 (T) and hence a Banach space.
For precisely the same reasons, the set
Ik Jo
= L e LX(J) : ± Cyxn * = 0 for n = 1,2,3, •
I Ik J

Exercises 27
is a closed subspace of L°°(T). Moreover, in this case
is the zero set or kernel of the w*-continuous function
1 C2n
%n((p) = — I q>%n at
In Jo
and hence is w*-closed. Therefore, Hx is a w*-closed subspace of L°°(T).
If we let Ho°° denote the closed subspace
then the conjugate space of H1 can be shown to be naturally isometrically
isomorphic to Lm(J)IHox. We do not prove this here but consider this
question in Chapter 6.
Notes
The basic theory of Banach spaces is covered in considerable detail in
most textbooks on functional analysis. Accounts are contained in Bourbaki
[7], Goffman and Pedrick [44], Naimark [80], Riesz and Sz.-Nagy [92],
Rudin [95], and Yoshida [117]. The reader may also find it of interest to
consult Banach [5].
Exercises
Assume in the following that X is a compact Hausdorff space and that 9C is
a Banach space.
7. / Show that the space C(X) is finite dimensional if and only if X is finite.
1.2 Show that every linear functional on 3C is continuous if and only if
3C is finite dimensional.
1.3 If Jt is a normed linear space, then there exists a unique (up to iso-
isomorphism) Banach space SC containing Jl such that clos Jt = 9C.
1.4 Complete the proof begun in Section 1.15 that I1 (Z+)* = / °°(Z+).
7.5 Determine whether each of the following spaces is separable in the
norm topology: co(Z+), l\Z+), /°°(Z+), and /°°(Z+)*.

28 1 Banach Spaces
Definition An element/of the convex subset K of 3C is said to be an extreme
point of K if for no distinct pair/t and/2 in K is/ = iC/i +/2)-
1.6 Show that an element/of C(X) is an extreme point of the unit ball if
and only if |/(x)| = 1 for each x in X.
1.7 Show that the linear span of the extreme points of the unit ball of
C(X) is C(X).
1.8 Show that the smallest closed convex set containing the extreme points
of the unit ball of C([0,1]) is the unit ball but that the same is not true for
1.9 Show that the unit ball of co(Z+) has no extreme points. Determine
the extreme points of the unit ball of Z1 B+). What about the extreme points
of the unit ball of Ll ([0,1])?
1.10 If K is a bounded w*-closed convex subset of 9£*, then {<p(f): <pe K}
is a compact convex subset of C for each/in SC. Moreover, if Ao is an extreme
point of {(p(f0): <p e K}, then any extreme point of the set {<p e K: <p(fo) = Ao}
is an extreme point of K.
1.11 If K is a bounded w*-closed convex subset of SC*, then K contains an
extreme point. * (Hint: If {fa}a e A is a well-ordering of 3C, define nested subsets
{Ka}aEA such that
where Xa is an extreme point of the set {<p(fj: <pe OpK^Kp}. Show that
f]asAKa consists of a single point which is an extreme point of K.)
1.12 (Krein-MU'man) A bounded w*-closed convex subset of 9£* is the
w*-closed convex hull of its extreme points.*
1.13 Prove that the relative K'*-topology on the unit ball of 3C* is metrizable
if and only if 3C is separable.
1.14 Let Jf be a subspace of 9C, x be in 3C, and set
If d > 0, then show that there exists q> in SC* such that <p(y) = 0 for y in Jf,
cp(x) = 1, and |M| = l/d.
1.15 Show that if we define the function f(q>) = q>(f) for/in 6C and <p in
%*, then/is in 9C** and that the mapping/-*/is an isometrical isomorphism
of 3C into

Exercises 29
Definition A Banach space is said to be reflexive if the image of SC is all
1.16 Show that SC is reflexive for SC finite dimensional but that none of the
spaces co{Z+), /1B+), / °°B+), C([0,1]), and Jj([O, 1]) is reflexive.
1.17 Let SC and <& be Banach spaces. Define the 1-norm \\f@g\\ i = ||/|| +1|#||
and the co-norm H/©^^ = sup{||/||, \\g\\} on the algebraic direct sum
SC®1®. Show that SC © <& is a Banach space with respect to both norms and
that the conjugate space of SC®®} with the 1-norm is SC*®^* with the
co-norm.
1.18 Let SC and <& be Banach spaces and | || be a norm on SC © <& making it
into a Banach space such that the projections n1 : SC®<2/ -+SC and rt2 :
SC@®/-*®/ are continuous. Show that the identity map between SC®^
in the given norm and SC®^ with the 1-norm is a homeomorphism. Thus
the norm topology on SC © <W is independent of the norm chosen.
1.19 {Closed Graph Theorem) If T is a linear transformation from the
Banach space SC to the Banach space <W such that the graph {{f, Tf) :/e SC}
of T is a closed subspace of SC © C/, then T is bounded. (Hint: Consider the
map /-»</, 7/).)
7.20 {Uniform Boundedness Theorem) Let f be a Banach space and
{<Pn}r=i be a sequence in SC* such that sup {| <pn{f) | :«eZ+}< co for/in SC.
Show that sup{||(pn||: n e Z+} < co.* (Hint: Baire category.)
7.27 If SC is a Banach space and {<pn}^=i is a sequence in SC* such that
{<Pn{f)} £°=i is a Cauchy sequence for each /in #", then limn_c09>n exists in
the w*-topology. Moreover, the corresponding result for nets is false.
7.22 Show that if SC is a Banach space and <p is a (not necessarily continuous)
linear functional on SC, then there exists a net {<pa}aeA in ^* sucn that
limaE/1%(/) = <?>(/) for/in ST.
7.25 Let #" and <& be Banach spaces and T be a bounded linear trans-
transformation from SC onto <&. Show that if M = ker T, then #"/./*' is topologically
isomorphic to <&.
Definition If SC is a Banach space, then the collection of functions {q> e S£*}
defines a weak topology on SC called the w-topology.
1.24 Show that a subspace M of the Banach space SC is norm closed if and
only if it is w-closed. Show that the unit sphere in SC is w-closed if and only
if SC is finite dimensional.*

30 1 Banach Spaces
1.25 Show that if 9C is a Banach space, then SC is w*-dense in 9C**.
1.26 Show that a Banach space SC is reflexive if and only if the w- and w*-
topologies coincide on SC*.
1.27 Let SC and <2/ be Banach spaces and T be in £BC,<&). Show that if q>
is in <&* then /-> <p(Tf) defines an element ^ of #"*. Show that the map
T*(p = ^ is in 2.(<3/*, #"•). (The operator T* is called the adjoint of T.)
1.28 If SC and ^ are Banach spaces and T is in £(^, SO, then T is one-to-
one if and only if T* has dense range.
1.29 If SC and <W are Banach spaces and Tis in £(^, <¥), then Thas a closed
range if and only if T* has a closed range. (Hint: Consider first the case
when T is one-to-one and onto.)
Definition If Jt is a subspace of the Banach space SC, then the annihilator
JtL of Jt is defined as JtL = {q> e SC* : <p(x) =0 for x e Jt}.
1.30 If 9C is a Banach space and Jt is & closed subspace of S£, then ^* is
naturally isometrically isomorphic to SC*\JtL.
1.31 If SC is a Banach space and Jf is a subspace of #"*, then there exists a
subspace Jl of S£ such that ^x = ./T if and only if Jf is w*-closed.
7.52 If the restriction of a linear functional <p on the Banach space M(X) of
complex regular Borel measures on X to the unit ball of M(X) is continuous
in the relative w*-topology, then there exists a function / in C{X) such that
for fi in M(X).*
(Hint: Obtain/by evaluating <p at the point measure Sx at x and use the fact
that measures of the form Y2=i ai^x, f°r Zi=i Kl < * are w*-dense in the
unit ball of M(X).)
1.33 (Grothendieck) A linear functional <p in SC** is w*-continuous if and
only if the restriction of <p to {9C*)y is continuous in the relative w*-topology.*
(Hint: Embed SC in C{X), extend <p to M{X) via the homomorphism from
M{X) to M(X)/2CL = SC*, and show that the function / obtained from the
preceding problem is in %.)
1.34 (Krein-Smul'yari) If 9C is a Banach space and ^ is a subspace of SC*,
then ■/*■ is w*-closed if and only if Jt n (^*), is w*-closed.* (Hint: Show
that Jt is the intersection of the zero sets of a collection of w*-continuous
linear functionals on SC*.)

Exercises 31
1.35 (Banach) If SC is a separable Banach space and Ji is a subspace of
ar*, then ^ is w*-closed if and only if M is w*-sequentially complete.
/J6 Let SC and ^ be Banach spaces and 9C®^ be the algebraic tensor
a
product of SC and ^ as linear spaces over C. Show that if for w in
we define
=1 i=i
then || • ||„. is a norm on 9£%<3J. The completion of ^®^ is the projective
a a
tensor product of 9C and ^ and is denoted 9C % <3/.
1.37 Let 9C and ^ be Banach spaces and 9C®<W be the algebraic tensor
a
product of SC and ^ as linear spaces over C- Show that if for w in SC ® ^
a
we define
sup
; yl,...,yne<3/;
then
||{ is a norm on
. The completion of
is the inductive
tensor product of ar and <W and is denoted
1.38 For ^ and <& Banach spaces, show that the identity mapping extends
to a contractive transformation from 9C ® <3/ to SC ® <&.
1.39 For Z and Y compact Hausdorff spaces show that C(X)®C(Y) =
C(Ix7). (Hint: Show that it is sufficient in defining ||-1|; to take <p and \jj
to be extreme points of the unit ball of 9C* and <&*.)
1.40 For X and Y compact Hausdorff spaces show that
C(X)®C(Y) = C(Xy.Y)
if and only if X or Y is finite. (Hint: show that there are functions h(x,y) =
YS=ifiix)Si{y) for which \\h\\x = 1 but ||A||n is arbitrarily large.) Thus the
tensor product of two Banach spaces is not unique.

Batmch Algebras
2.1 In Chapter 1 we showed that C{X) is a Banach space and that every
Banach space is, in fact, isomorphic to a subspace of some C(X). In addition
to being a linear space, C(X) is also an algebra and multiplication is con-
continuous in the norm topology. In this chapter we study C{X) as a Banach
algebra and show that C{X) is a "universal" commutative Banach algebra
in a sense which we will later make precise. We shall indicate the usefulness
and power of this result in some examples.
2.2 Recall that in Section 1.1 we observed that C{X) is an algebra over C
with pointwise multiplication and that the supremum norm satisfies
WfffW oo < ll/ll oo IML for/and g in C{X). These properties make C{X) into
what we will call a Banach algebra.
In the study of Banach spaces the notion of bounded linear functional is
important. For Banach algebras and, in particular, for C{X) the important
idea is that of a multiplicative linear functional. (We do not assume the
functional to be continuous because we show later that such a functional is
necessarily continuous.) Except for the zero functional, which is obviously
both multiplicative and linear, every multiplicative linear functional <p satisfies
q>(l) = 1 since <p f£ 0 means there exists an / in C{X) with <p(f) ¥*■ 0, and
then the equation <p A) <p(J) = <p (/) implies <p(l) = 1. Thus we restrict our
attention to the set Mcm of complex multiplicative linear functionals <p on
C{X) which satisfy <p(l) = 1. For each x in X we define the complex function
32

The Banach Algebra of Continuous Functions 33
<px on C{X) such that <px(f) =/(x) for/in C{X). It is immediate that <px
is in Mcm, and we let ^ denote the mapping from X to Mcm denned
^(x) = <px. The following proposition shows that \jj maps onto Mcm.
2.3 Proposition The map ^ defines a homeomorphism from X onto Mcm,
where Mcm is given the relative w*-topology on C(X)*.
Proof Let q> be in MCm and set
We show first that there exists x0 in X such that /(x0) = 0 for each / in ft.
If that were not the case, then for each x in X, there would exist fx in ft such
that/^x) # 0. Since fx is continuous, there exists a neighborhood Ux of x
on which fx¥. Since A' is compact and {Ux}xeX is an open cover of X,
there exists UXl,..., UXn with X=\JZ=1UXn. If we set # = £?= ,£„/*„,
then <p(g) = YZ= i <p(fxj<p(fxj = 0, implying that g is in ft. But g # 0 on
X and hence is invertible in C{X). This in turn implies ^>A) = (p(g)-<p(l/g)
= 0, which is a contradiction. Thus there exists x0 in X such that /(x0) = 0
for/in ft.
If/is in C(X), then f-<p(f)-1 is in ft since <p(f-(p(f)-1) = <p(f)-<p(f)
= 0. Thus
/(*o) - <?(/) = (/-<?(/)• 0(*o) = 0,
since q> — <p(f)-1 is in ft and therefore <p = <pxo.
Since each <p in MC(jr) is bounded (in fact, of norm one), we can give
Mcm the relative w*-topology on C{X)* and consider the map \jj : X-> MC(Xy
If x and y are distinct points of X, then by Urysohn's lemma there exists/in
C{X) such that/(x) //(j). Thus
Hx)if) = <P,U) =f(x) #/(y) = <p,(f) = HyKf),
which implies that \jj is one-to-one.
To show that \jj is continuous, let {xa}aeA be a net in A' converging to x.
Then limaeAf(xa) =/(x) for / in C(Z) or equivalently HmaeA*I/(xa)(f) =
^(x)(/) for each / in C{X). Thus the net {^(xj},^ converges in the
w*-topology to ^(x) so that \jj is seen to be continuous. Since ^ is a one-to-one
continuous map from a compact space onto a Hausdorff space, it follows
that ^ is a homeomorphism. This completes the proof. ■
We next state the definition of Banach algebra and proceed to show that
the collection of multiplicative linear functionals on a general Banach algebra
can always be made into a compact Hausdorff space in a natural way.

34 2 Banach Algebras
2.4 Definition A Banach algebra 95 is an algebra over C with identity 1
which has a norm making it into a Banach space and satisfying || 11| = 1
and the inequality \\fg\\ < \\f\\ \\g\\ for/and g in 95.
We let X denote the element of 95 obtained upon multiplying the identity
by the complex number X.
The following fundamental proposition will be used to show that the
collection of invertible elements in 95 is an open set and that inversion is
continuous in the norm topology.
2.5 Proposition If/is in the Banach algebra 95 and
invertible and
■I „ i II _ ■»
-/|| < 1, then /is
Proof If we set t] = || 1 -/|| < 1, then for N > M we have
II N M
I(i-/)"- £(!-/)"
\\n=0 n-0
I a-j
N
n=M+l
and the sequence of partial sums {Hb=oA~/)"}n=o is seen to be a Cauchy
sequence. If g = XT=oO -/)", then
fg = [1 - A -/)] ( £ A -/)") = lim ([1 - A -/)] £ A -/)")
\n=0 / N-.m \ H=0 /
since lim^^ ||(l-/)N + 1|j = 0. Similarly, gf=l so that / is invertible with
/-1 = g. Further,
= lim
n=0
1
2.6 Definition For 95 a Banach algebra, let ^ denote the collection of
invertible elements in 93 and let <&x, respectively, <St denote the collection
of left, respectively, right invertible elements in 95 that are not invertible.
The following result will be of interest in this chapter only as it concerns
^ but we will need the results about ^, and ^r in Chapter 5 when we study
index theory.

Abstract Banach Algebras 35
2.7 Proposition If © is a Banach algebra, then each of the sets 9, <SX,
and ^r is open in 93.
Proof If / is in 9 and \\f-g\\ < l/H/"!, then 1 > ||/-»|| \\f-g\\ >
\\l-f~lg\\. Thus the preceding proposition implies that f~lg is in ^ and
hence g=f(f~1g) is in ^. Therefore ^ contains the open ball of radius
1/||/~*1| about each element of/ in 'S. Thus 'S is an open set in 93.
If/is in %, then there exists h in 93 such that hf= 1. If ||/-#|| < l/\\h\\,
then 1 > ||h|| ||/-#|| 5= \\hf—hg\\ = ||1—%||. Again the proposition implies
that k = hg is invertible and the identity (k~lh)g = 1 implies that g is left
invertible. Moreover, if g is invertible, then h=kg~l is invertible which in
turn implies that/is invertible. This contradiction shows that g is in ^i so that
^i is seen to contain the open ball of radius ll\h\ about/ Thus ^, is open.
The proof that <St is open proceeds in the same manner. ■
2.8 Corollary If 93 is a Banach algebra, then the map on 'S denned
f-*f~l is continuous. Thus, ^ is a topological group.
Proof If/is in <S, then the inequality \\f-g\\ < 1/2H/!! implies that
|| 1 -/" V|| < i and hence
y~li < \\ff-ln\\rl\\ = ucrvr
Thus the inequality
Wf-'-ff-'W = \\f-l(f~9)9-1\\^2\\f-l\\2\\f-g\\
shows that the map f-*f~l is continuous. ■
There is another group which is important in some problems.
2.9 Proposition Let 95 be a Banach algebra, ^ be the group of invertible
elements in 95, and % be the connected component in 'S which contains
the identity. Then % is an open and closed normal subgroup of'S, the cosets
of % are the components of'S, and ^/% is a discrete group.
Proof Since ^ is an open subset of a locally connected space, its com-
components are open and closed subsets of ^. Further, if/ and g are in %, then
f% is a connected subset of ^ which contains/? and/. Therefore, % (jf%
is connected and hence is contained in %. Thus fg is in % so that % is a
semigroup. Similarly, f~l%'u% is connected, hence contained in %,

36 2 Banach Algebras
and therefore % is a subgroup of <§. Lastly, if/ is in 9, then the conjugate
group/%/"* is a connected subset containing the identity and therefore
f%f~x = %. Thus, % is a normal subgroup of ^ and &l% is a group.
Further, since f&0 is an open and closed connected subset of & for each
/ in 9, the cosets of ^0 are the components of <§. Lastly, ^/^0 ls discrete
since ^0 is an open and closed subset of <§. ■
2.10 Definition If © is a Banach algebra, then the abstract index group
for S3, denoted A», is the discrete quotient group ^/^0. Moreover, the
abstract index is the natural homomorphism y from ^ to A<b .
We next consider the abstract index group for a Banach algebra in a little
more detail.
2.11 Definition If © is a Banach algebra, then the exponential map on SB,
denoted exp, is defined
The absolute convergence of this series is established just as in the scalar
case from whence follows the continuity of exp. If SB is not commutative,
then many of the familiar properties of the exponential function do not hold.
The following key formula is valid, however, with the additional hypothesis
of commutativity.
2.12 Lemma If © is a Banach algebra and/and g are elements of SB which
commute, then exp(/+#) = exp/ exp g.
Proof Multiply the series denning exp/and exp# and rearrange. ■
In a general Banacti algebra it is difficult to determine the elements in the
range of the exponential map, that is, the elements which have a "logarithm."
The following lemma gives a sufficient condition.
2.13 Lemma If © is a Banach algebra and/is an element of SB such that
||1-/||< 1, then/is in exp SB.
Proof If we set g = X^=1 (l/n)(l-/)", then the series converges ab-
absolutely and, as in the scalar case, substituting this series into the series
expansion for exp g yields exp g = /. ■

Abstract Index in a Banach Algebra 37
Although it is difficult to characterize exp© for an arbitrary Banach
algebra, the collection of finite products of elements in exp© is a familiar
object.
2.14 Theorem If© is a Banach algebra, then the collection of finite products
of elements in exp © is %.
Proof If/= expg, then/• exp(-g) = exp(g-g) = 1 = exp(-#)/ and
hence/is in ^. Moreover, the map <p from [0,1] to exp© defined <p (A) =
exp(A#) is an arc connecting 1 to/and hence /is in %. Thus exp© is con-
contained in %. Further, if #" denotes the collection of finite products of elements
of exp©, then #" is a subgroup contained in %. Moreover, by the previous
lemma #" contains an open set and hence & being a subgroup is an open set.
Lastly, since each of the left cosets of #" is an open set, it follows that #"
is an open and closed subset of %. Since S?o is connected we conclude that
% = #", which completes the proof. ■
The following corollary shows that the problem of identifying the elements
of a commutative Banach algebra which have a logarithm is much easier.
2.15 Corollary If © is a commutative Banach algebra, then exp © = %.
Proof ByLemma2.12if©iscommutative, then exp © is a subgroup. ■
Before continuing, we identify the abstract index group for C{X) with
a more familiar object from algebraic topology. This identification is actually
valid for arbitrary commutative Banach algebras but we will not pursue this
any further (see [40]).
2.16 Let X be a compact Hausdorff space and let ^ denote the invertible
elements of C{X). Hence a function/in C{X) is in 'S if and only if/(x) # 0
for all x in X, that is, 'S consists of the continuous functions from X to C* =
C\{0}. Since 'S is locally arcwise connected, a function/is in % if there exists
a continuous arc {/J*e[o,i] of functions in 'S such that/0 = I and/t =/.
If we define the function F from Ix [0,1] to C* such that F(x,X) =/*(*),
then F is continuous, F(x,0) = 1 and F(x, 1) =/(x) for x in X. Hence/ is
homotopic to the constant function 1. Conversely, if g is a function in ^
which is homotopic to 1, then g is in %. Similarly, two functions gx and g2
in ^ represent the same element of A = &/% if and only if gt is homotopic
to g2. Thus A is the group of homotopy classes of maps from Zto C*.

38 2 Banach Algebras
2.17 Definition If X is a compact Hausdorff space, then the first co-
homotopy group nl (X) of X is the group of homotopy classes of continuous
maps from X to the circle group T with pointwise multiplication.
2.18 Theorem If X is a compact Hausdorff space, then the abstract index
group A for C(X) and nl(X) are naturally isomorphic.
Proof We define the mapping Q> from nl(X) to A as follows: A con-
continuous function / from X to T determines first an element {/} of nl(X)
and second, viewed as an invertible function on X, determines a coset/+^0
of A. We define <D({/}) =f+%. To show, however, that <D is well defined
we need to observe that if g is a continuous function from X to T such that
{/} = {g}> then/is homotopic to g and hence f+% = g+%. Moreover,
since multiplication in both nl (X) and & is defined pointwise, the mapping
<D is obviously a homomorphism. It remains only to show that <D is one to-one
and onto.
To show <D is onto let / be an invertible element of C(X). Define the
function F from Xx[0,l] to C* such that F{x,t)=f(x)l\f{x)\t. Then F
is continuous, F(x,0) =f(x) for x in X, and g(x) = F(x, 1) has modulus
one for x in X. Hence, f+% = g+% so that 0>({g}) =f+% and therefore
<D is onto.
If/ and g are continuous functions from X to T such that $({/}) =
<D ({#}), then/is homotopic to g in the functions in &, that is, there exists a
continuous function G from Jx [0,1] to C* such that G(x,0) =/(x) and
G(x, 1) = g{x) for x in X. If, however, we define F{x, t) = G{x,t)l\G{x,t)\,
then F is continuous and establishes that/and g are homotopic in the class
of continuous functions from X to T. Thus {/} = {g} and therefore <D is
one-to-one, which completes the proof. H
The preceding result is usually stated in a slightly different way.
2.19 Corollary If X is a compact Hausdorff space, then A is naturally
isomorphic to the first Cech cohomology group Hl(X,Z) with integer
coefficients.
Proof It is proved in algebraic topology (see [67]) that nl(X) and
Hl {X, Z) are naturally isomorphic. H
These results enable us to determine the abstract index group for simple
commutative Banach algebras.

The Space of Multiplicative Linear Functions 39
2.20 Corollary The abstract index group of C(T) is isomorphic to Z.
Proof The first cohomotopy group of T is the same as the first homotopy
group of T and hence is Z. ■
We now return to the basic structure theory for Banach algebras.
2.21 Definition Let © be a Banach algebra. A complex linear functional
<p on 93 is said to be multiplicative if:
A) <p(fff) = <p(f)<p(ff) for/and g in 93; and
B) <p(l) = 1.
The set of all multiplicative linear functionals on 93 is denoted by M = M&.
We will show that the elements of M are bounded and that M is a w*-
compact subset of the unit ball of the conjugate space of 93. We show later
that M is nonempty if we further assume that 93 is commutative.
2.22 Proposition If 93 is a Banach algebra and <p is in M, then \cp\ = 1.
Proof Let ft = kercp = {/e 93 : (p(f) = 0}. Since (p(f-<p(f)-1) = 0, it
follows that every element in 93 can be written in the form A+/for some X
in C and/ in ft. Thus
M = suP = sup
IMI {% {fo
because ||1+A|| < 1 implies that h is invertible by Proposition 2.5, which
implies in turn that h is not in ft. Therefore ||^|| = 1 and the proof is
complete. ■
Whenever we deduce topological properties from algebraic hypotheses,
completeness is usually crucial; the use of completeness in the proof of
Theorem 1.42 was obvious. Less obvious is the role played by completeness
in the preceding proposition.
2.23 Proposition If 93 is a Banach algebra, then M is a w*-compact subset
of (93*),.
Proof Let {<pa}aEA be a net of multiplicative linear functionals in M
that converge in the H-*-topology on (93*), to a <p in (93*),. To show that
M is w*-compact it is sufficient in view of Theorem 1.23 to prove that <p is

40 2 Banach Algebras
multiplicative and ^A)=1. To this end we have <p(l) = \ima<EA(pa(l) =
lim^ A 1 = 1. Further, for/and g in 93, we have
<p(fg) = Yim<pa(fg) = Yaa.<pa(J)<pa(g)
aeA aeA
= lim<pa(f)lim<pa(g) = <p(f)-<p(g).
aGA aeA
Thus <p is in M and the proof is complete. ■
Thus M is a compact Hausdorff space in the relative w*-topology. Recall
that for each/ in 93 there is a w*-continuous function /: (©*)t -> C given
by f(<p) = <?>(/)• Since M is contained in (93*)i, then /| M is also continuous.
We formalize this in the following:
2.24 Definition For the Banach algebra S3, the Gelfand transform is the
function T : 93-* C(M) given by T(f) =/| M, that is, r(f)(<p) = <p(f) for
^> in M.
2.25 Elementary Properties of the Gelfand Transform If © is a Banach
algebra and T is the Gelfand transform on 93, then:
A) F is an algebra homomorphism; and
B) Hr/IL* Il/H for/in 93.
Proof The only nonobvious property needed to conclude that T is an.
algebra homomorphism is that T is multiplicative and that argument goes as
follows: For/and g in 93 we have
nfg)W) = v(fg) = <p{f)v(si) = nf)(<p)-ng)W) = [r(/)-r(s-)](H
and hence T is multiplicative. To show that T is a contractive mapping we
let/be in 93 and then
Thus r is a contractive algebra homomorphism and the proof is complete. ■
2.26 Before proceeding we want to make a few remarks about the Gelfand
transform. Note first that T sends all elements of the form fg—gf to 0. Thus,
if 93 is not commutative, then the subalgebra of C{M) that is the range of T
may fail to reflect the properties of 93. (In particular, we indicate in the prob-
problems at the end of this chapter an example of a Banach algebra for which
M is empty.) In the commutative case, however, M is not only not empty
but is sufficiently large that the invertibility of an element/in 93 is determined

The Gelfand Transform 41
by the invertibility of r/in C(M). This fact alone makes the Gelfand transform
a powerful tool for the study of commutative Banach algebras.
To establish this further property of the Gelfand transform in the com-
commutative case, we must first consider the basic facts of spectral theory. We
will not assume, in what follows, that 93 is commutative until this assumption
is actually needed.
2.27 Definition For 93 a Banach algebra and/ an element of 93 we define
the spectrum of/ to be the set
0<b(/) = {A e C :/— A is not invertible in SB},
and the resolvent set of/ to be the set
Further, the spectral radius of/ is defined
r<&(f) = sup{|A| :Ai
When no confusion will result we omit the subscript 93 and write only
o{f\ Pif), and r(f).
The following elementary proposition shows that <r(/) is compact. The
fact that a(J) is nonempty lies deeper and is the content of the next theorem.
2.28 Proposition If 93 is a Banach algebra and/is in 93, then <r(/) is compact
and r(f) < ||/||.
Proof If we define the function <p : C-*93 by <p (A) =/— A, then <p is
continuous and p(/) = q>~l(f&) is open since ^ is open. Thus the set <r(/)
is closed.
If |A| > Il/H, then
11/11
1 >
-H)
so that i -//A is invertible by Proposition 2.5. Thus/-A is invertible. There-
Therefore, A is in p(/), g(J) is bounded and hence compact, and r(f) < ||/||. ■
2.29 Theorem If 93 is a Banach algebra and/is in S3, then o(f) is non-
nonempty.
Proof Consider the function F:p(/)-»93 defined F(A) = (/-A).
We show that F is an analytic 93-valued function on p(f) that is bounded
at infinity and use the Liouville theorem to obtain a contradiction.

42 2 Bamch Algebras
First, since inversion is continuous we have for Ao in p (/) that
A —
In particular, for 9) in the conjugate space 93*, the function <p(F) is a complex
analytic function on p(/).
Further, for |A| > ||/|| we have, using Proposition 2.5, that 1-//A is
invert ible and
(-f
1
Thus it follows .that
lim ||F(A)|| = lim
l l
hm sup— -
= 0.
Therefore for <p in 93* we have lim^^^ <p (F(X)) = 0.
If we now assume that <r(/) is empty, then p(/) = C. Thus for <p in 93* it
follows that q>(F) is an entire function that vanishes at infinity. By Liouville's
theorem we have <p(F) = 0. In particular, since for a fixed A in C we have
(p(F{k)) = 0 for each <p in 93*, it follows from Corollary 1.28 that F{X) = 0.
This, however, is a contradiction, since F(X) is by definition an invertible
element of 93. Therefore a(f) is nonempty. ■
Note that although SB is not assumed to be commutative, the subalgebra
of 93 spanned by 1,/, and elements of form (/— X)~~l is commutative, and the
result really concerns only this subalgebra.
2.30 The following theorem is an immediate corollary to the preceding and
is crucial in establishing the desired properties of the Gelfand transform.
Recall that a division algebra is an algebra in which each nonzero element
is invertible.
2.31 Theorem (Gelfand-Mazur) If 93 is a Banach algebra which is a
division algebra, then there is a unique isometric isomorphism of 93 onto C.

The Gelfand-Mazur Theorem 43
Proof If/ is in 93, then a{f) is nonempty by the preceding theorem. If
Xs is in <r(/), then/-/^ is not invertible by definition. Since 93 is a division
algebra, then f—Xf = 0. Moreover, for A # Ar we have f-X = Xf—X which
is invertible. Thus a{f) consists of exactly one complex number Xf for each
/in SB. The map ^:93-*C defined ^(/) = A/ is obviously an isometric
isomorphism of 93 onto C. Moreover, if \jj' were any other, then $'{f) would
be in o(f) implying that xj/if) = \p'{f). This completes the proof. ■
2.32 Quotient Algebras We now consider the notion of a quotient algebra.
Let 93 be a Banach algebra and suppose that 5R is a closed two-sided ideal of
93. Since SOt is a closed subspace of 93 we can define a norm on 93/511 following
Section 1.39 making it into a Banach space. Further, since 5R is a two-sided
ideal in 93, we also know that 93/511 is an algebra. There remain two facts to
verify before we can assert that 93/5A is a Banach algebra.
First, we must show that ||[1]|| = 1, and this proof proceeds as follows:
||[l]|| = in£6jj|l-fli| = l, for if |1-#||<L, then g is invertible by
Proposition 2.5.
Secondly, for / and g in 93 we have
inf Hf-hJig-hJl < inf \\f-h.\\ inf \\g-h2\\
so that I [/][#] || < ||[/]| || [g-] ||. Thus 93/SK is a Banach algebra. Moreover,
the natural map/-* [/] is a contractive homomorphism.
2.33 Proposition If 93 is a commutative Banach algebra, then the set M of
multiplicative linear functionals on 93 is in one-to-one correspondence with
the set of maximal two-sided ideals in 93.
Proof Let cp be a multiplicative linear functional on 93 and let Si =
ker^> = {/e93 : <p(f) = 0}. The kernel R of a homomorphism is a proper
two-sided ideal and if/ is not in ft, then
Since A —f/<p(f)) is in ft, the linear span of/with ft contains the identity 1.
Thus an ideal containing both ft and/would have to be all of 93 so that ft
is seen to be a maximal two-sided ideal.

44 2 Banach Algebras
Suppose SOt is a maximal proper two-sided ideal in 93. Since each element
/of £01 is not invertible, then ||1 —/|| > 1 by Proposition 2.5. Thus 1 is not in
the closure of 511. Moreover, since the closure SH of SOt is obviously a two-
sided ideal and £01 c W £ 93, then SOI = W and SOI is closed. The quotient
algebra 93/SOl is a Banach algebra which because 951 is maximal and 93 is
commutative, is a division algebra. Thus by Theorem 2.31, there is a natural
isometrical isomorphism ^ of 93/SOl onto C. If n denotes the natural homo-
morphism of 93 onto 93/SOl, then the composition q> = \jjn is a nonzero
multiplicative linear functional on 93. Thus ip is in M and 951 = ker<p.
Lastly, we want to show that the correspondence <p <->■ ker <p is one-to-one.
If q>t and <p2 are in M with ker<pt = ker<p2 = SOI, then
Vdf) ~ <Pi(f) = if-Viif)) ~ (f-Viif))
is both in 9JI and a scalar multiple of the identity for each/in 93 and hence
must be 0. Therefore ker^ = ker<p2 implies q>^ = q>2 and this completes
the proof. ■
This last proposition is the only place in the preceding development
where the assumption that 93 is commutative is required.
Hereafter, we refer to M<g as the maximal ideal space for 93.
2.34 Proposition If 93 is a commutative Banach algebra and / is in 93,
then/is invertible in 93 if and only if F(/) is invertible in C(M).
Proof If/is invertible in 93, the H/) is the inverse of T(f). If/is
not invertible in SB, then S0lo = {gf: g e 93} is a proper ideal in © since 1
is not in 9Jl0. Since 93 is commutative, S9l0 is contained in some maximal
ideal S91. By the preceding proposition there exists q> in M such that ker tp = SOI.
Thus r(f)(q>) = (p(f) = 0so that T(f) is not invertible in C(M). U
We summarize the results for the commutative case.
2.35 Theorem (Gelfand) If 93 is a commutative Banach algebra, M is its
maximal ideal space, and T : 93 -» C(M) is the Gelfand transform, then:
A) Mis not empty;
B) r is an algebra homomorphism;
C) ||r/||co<||/|for/in93;and
D) /is invertible in 93 if and only if F(f) is invertible in C{M).
The crucial fact about statement D) is that it refers to T(/) being invertible
in C(M) rather than in the range of P.

The Spectral Radius Formula 45
We obtain two corollaries before proceeding to a result concerning the
spectral radius.
2.36 Corollary If 93 is a commutative Banach algebra and/is in 93, then
a(J) = ranger/and r(f) =
Proof If A is not in a(f), then f-X is invertible in 93 by definition. This
implies that T(f) — X is invertible in C{M), which in turn implies that
(Tf-X)(cp) # 0 for <p in M. Thus A7)(<p) # X for <p in M. If X is not in the
range of Tf, then Tf— X is invertible in C(M) and hence, by the preceding
theorem,/— X is invertible in 93. Therefore, X is not in a(f) and the proof is
complete. ■
If cp(z) = Y^=oan^ is an entire function with complex coefficients and
/is an element of the Banach algebra 93, then we let <p(f) denote the element
£„%«„/" of 93.
2.37 Corollary (Spectral Mapping Theorem) If 93 is a Banach algebra, / is
in 93, and q> is an entire function on C, then
Proof If cp{z) — Y^=oan^ is tne Taylor series expansion for <p, then
cp(f) = Y^=oanf" can be seen to converge to an element of 93. If 33O is tne
subalgebra of © generated by 1,/ and elements of the form (/—A) for X in
p(/) and (<p(/)—m) x for ju in p(<p(f)), then 930 is commutative and oVg(/)=
°Vbo(/) and ovn(<p(/)) = oVboOpC/))- Thus, we can assume that © is com-
commutative and use the Gelfand transform.
Using the preceding corollary we obtain
<r(<p(f)) = range T(^(/)) = range <p{Tf)
= (p(range T/) = <p{o{f)),
since T(<p{f)) = <p(I/) by continuity; thus the proof is complete.
We next prove a basic result due to Beurling and Gelfand relating the
spectral radius to the norm.
2.38 Theorem If 93 is a Banach algebra and / is in 93, then r^(J) =
lim II f\\1/n
11IUB-»00 11/ II

46 2 Banach Algebras
Proof If ©o denotes the closed subalgebra of 93 generated by the identity,
/, and {(/"-I)'1 \kepss,(/"), nel+}, then 93O is commutative and a^(/") =
o9{f") for all positive integers n. From the preceding corollary, we have
ffiBoGH = <%„(/)" and hence #■„(/)" = rB(/") < ||/"|; thus the inequality
rB(/) < liming \\r\\lln follows.
Next consider the analytic function
which converges for |A| > limsup,,.^ ||/"||1/n by Proposition 1.9. For
|A| > Il/H we have G(X) = (/-A) and therefore
(/•-A)G(A) = G(A)(/-A) = 1 for |A| > limsupH/T*1.
Thus
r.Cn > limsup||/T/n > liminf ||/"||1/n ^
n->co n-*co
from which the result follows. ■
2.39 Corollary If © is a commutative Banach algebra, then the Gelfand
transform is an isometry if and only if ||/2|| = ||/||2 for every/in 93.
Proof Since r{f) = ||rs/||„ for/in © by Corollary 2.36, we see that
Eg is an isometry if and only if r(f) = ||/|| for / in 93. Moreover, since
r(f2) = r(fJ by Corollary 2.37, the result now follows from the theorem. ■
We now study the self-adjoint subalgebras of C(X) for X a compact
Hausdorff space. We begin with the generalization due to Stone of the classical
theorem of Weierstrass on the density of polynomials. A subset 91 of C(X)
is-said to be self-adjoint if/in 91 implies/is in 91.
2.40 Theorem (Stone-Weierstrass) Let Ibea compact Hausdorff space.
If 91 is a closed self-adjoint subalgebra of C(X) which separates the points
of X and contains the constant function 1, then 91 = C{X).
Proof If 9Ir denotes the set of real functions in 91, then 9Ir is a closed
subalgebra of the real algebra Cr (X) of continuous functions on X which
separates points and contains the function 1. Moreover, the theorem reduces
to showing that 9Ir = Ct(X).
We begin by showing that/in 9Ir implies that |/| is in 9Ir. Recall that the

The Stone-Weierstrass Theorem 47
binomial series for the function <p(t) = A —1I/2 is H^°=o«h'". where o^ =
(— l)"(l/n2). It is an easy consequence of the comparison theorem that the
sequence {Z^=oan'"}N=i converges uniformly to ip on the closed interval
[0,1—5] for 8 > 0. (The sequence actually converges uniformly to ip on
[-1,1].) Let/be in 9Ir such that ||y||„ < 1 and set gB = <5 + (l-S)f2 for 8
in @,1]; then 0 < 1 -gs < 1 -8. For fixed 8 > 0, set hN = ZJlUa-O -gif.
Then hK is in 9Ir and
xeX n=0
< sup
Therefore, lim^^ ||/2jy—(g^I/2\m = 0, implying that (gd)Vl is in 9Ir. Now
since the square root function is uniformly continuous on [0,1], we have
lim^01||/| - (gB)Y2\\oo = 0, and thus |/| is in 9Ir.
We next show that 9Ir is a lattice, that is, for/and g in 9Ir the functions
fv g and/A g are in 9Ir, where (fv g)(x) = max{/(x), ^(x)}, and (/a #)(x) =
min{f(x),g(x)}. This follows from the identities
fv g = ${f+9 + \f-g\], and /a g = ${f+g- \f-g\]
which can be verified pointwise.
Further, if x and y are distinct points in X and a and b arbitrary real
numbers, and/is a function in 9Ir such that/(jc) #/(y), then the function
g defined by
is in 9Ir and has the property that g(x) = a and #(j) = fc. Thus there exist
functions in 9Ir taking prescribed values at two points.
We now complete the proof. Take/in Cr(X) and £ > 0. Fix x0 in X. For
each x in X, we can find a gx in 9Ir such that gx(x0) =/(x0) and gx(x) =f(x).
Since / and g are continuous, there exists an open set Ux of x such that
0x(y) ^f(y)+E for all y in Ux. The open sets {Ux}xeX cover X and hence
by compactness, there is a finite subcover Uxl,Uxz,...,UXn. Let /ixo =
9x^9x2^ — ^9^- Then /ixo is in 9Ir, hxo(x0) =f{x0), and hxo(y) ^f(y)+s
for y in X.
Thus for each x0 in X there exists hXQ in 9Ir such that hxo(x0) =/(x0)
and hxo(y) </(j)+£ for y in X. Since /!xo and /are continuous, there exists
an open set Vxo of x0 such that hxo(y) >f{y)-e for y in Fxo. Again, the

48 2 Banach Algebras
family {Vxo}XoeX covers X, and hence there exists a finite subcover
vxltvx2>—tvxm- ^ we set ^ = Ki vhX2v ■"■ v^m» tnen ^ is in ^r and/(j/) —8
< k(y) ^f(y)+s for y in X. Therefore, ||/—fc^^s and the proof is
complete. ■
2.41 If [a, b~\ is a closed interval of U, then the collection of polynomials
{Hn= o an t"} with complex coefficients is a self-adjoint subalgebra of C([a,bJ)
which separates points and contains the constant function 1. Thus its closure
must be C{[a, b~\), and this is the statement of the Weierstrass theorem.
We now consider the closed self-adjoint subalgebras of C(X) containing
the constant function 1 that do not separate points and show that they can
be identified as C(Y) for some compact Hausdorff space Y.
Let X be a compact Hausdorff space and 91 be a closed subalgebra of
C(X) which contains the constant functions. For xinlwe let <px denote
the multiplicative linear functional in Mm defined cpx(J) =/(x). The follow-
following proposition is of interest even in the nonself-adjoint case.
2.42 Proposition If t\ is the map defined from X to Mm so that tj(x) — <px,
then t] is continuous.
Proof If {xa}a£A is a net in X which converges to x, then \yTHaeAf{x^) =
f(x) for/in 91. Therefore, limasA(pXa(f) = <px(f) and hence limae/1?/(xa) =
t](x) in the topology of Mm. Thus, t\ is continuous. ■
In general, t\ is neither one-to-one nor onto. The latter property, however,
holds if 91 is self-adjoint.
2.43 Proposition If 91 is self-adjoint, then r\ maps X onto Mm.
Proof Fix <p in Mm and set Kf = {xeX :/(x) = <p(f)}. First of all, each
Kf is a closed subset of X, since/is continuous. Secondly, we want to show
that not only is each Kf nonempty but that the collection of sets {Kf :/e 91}
has the finite intersection property. Suppose
Kfl n Kfz n ■■■ n Kfn = 0
for some functions/i,/2,...,/„ in 91. Then the function
does not vanish on X. Moreover, g is in 91 since the latter is a self-adjoint
algebra. But g{x) > 0 for x in X and the fact tbat X is compact implies that

The Stone-Weierstrass Theorem 49
there exists e>0 such that 1 ^ 9(x)l\g\a;> ^ e, and hence such that
II1 -(#/|l#IDIL < 1- But then g~l is in $1 by Proposition 2.5 which implies
cp(g) ^ 0. However,
which is a contradiction. Thus the collection {Kf :fe 81} has the finite inter-
intersection property. If x is in f]femKf, then rj(x) = (p and the proof is
complete. ■
The reader should consider carefully how the self-adjointness of 81 was
used in the preceding proof. We give an example in this chapter of a sub-
algebra for which tj is not onto. Even for examples where t] is onto, the Gelfand
transform F need not be onto. It is, however, for self-adjoint subalgebras.
2.44 Proposition If 81 is a closed self-adjoint subalgebra of C(X) containing
the constant function 1, then the Gelfand transform F is an isometric iso-
isomorphism from 81 onto C(Mm).
Proof For / in 81 there exists x0 in X such that f(x0) = I/fa,, since
X is compact. Therefore,
Il/L ^ lir/IL = sup
and hence F is an isometry. Since F is known to be an algebraic homomorphism,
it remains only to prove that F is onto. The range of F is a subalgebra of
C(Mn) that contains 1, since Fl = 1, is uniformly closed since F is an
isometry, and separates points. Moreover, since by the preceding proposition
for <p in Mm there exists x in X such that r\x — q>, we have for /in 81 that
= (rf)(<p) = (r/)(j/x) =/(*) =/(*) = rc/)(j/x) =
Therefore, F/= F(/) and F81 is self-adjoint because 81 is. By the Stone-
Weierstrass theorem, we have FSl = C(Mm) and the proof is complete. ■
2.45 Lemma Let X and Y be compact Hausdorff spaces and 6 be a con-
continuous map from X onto Y. The map 6* defined by 6*f=f°6 from C(Y)
into C(X) is an isometrical isomorphism onto the subalgebra of continuous
functions on X which are constant on the closed partition {6~1(y): y e Y}
of X.
Proof That 6* is an isometrical isomorphism of C(Y) into C(X) is

SO 2 Banach Algebras
obvious. Moreover, it is clear that a function of the form /o 6 is constant
on the partition {6~1(y): y e Y}. Now suppose g is continuous on X and
constant on each of the sets B~1 (y) for y in Y. We can unambiguously define
a function/on Y such that/<>0 = g; the only question is whether this/is
continuous. Suppose {ya}aeA is a net of points in Y and y is in Y such that
lima(M;j>a = y. Choose xa in 6~1(ya) for each a in A and consider the net
{xa}aeA. In general, lim^^:*^ does not exist; however, since X is compact
there exists a subnet {xafi}PeB and an x in X such that limp^jc,^ = jc. Since
0 is continuous, we have 6(x) = y and limPeB^(x^) = #(jc) =/(j); thus/
is continuous and the proof is complete. ■
2.46 Proposition If 81 is a closed self-adjoint subalgebra of C(X) containing
the constant function 1, and tj a continuous map from X onto Mn, then r/*
is an isometrical isomorphism of C(Mm) onto 91 which is the left inverse of
the Gelfand transform, that is, rj*oT = 1.
Proof For/in 91 and x in X, we have @?*°r)/) (*) = (Tf)(*ix) =f(x).
Therefore rj* is the left inverse of the Gelfand transform. Since F maps 91
onto C(M^) by Proposition 2.44, we have that vj* maps C{M^) onto 91. ■
We state and prove the generalized Stone-Weierstrass theorem after
introducing the following terminology. For X a set and 91 a collection of
functions on X define the equivalence relation on X such that two points
jq and x2 are related if/Cx^) =f{x2) for every/in 91. This relation partitions
X into the sets on which the functions in 91 are constant. Let Hm denote this
collection of subsets of X.
2.47 Theorem Let Ibea compact Hausdorff space and 91 be a closed
self-adjoint subalgebra of C(X) which contains the constants. Then 91 is the
collection of continuous functions on X which are constant on the sets
Proof This follows by combining Lemma 2.45 and Proposition 2.46. ■
If 91 separates the points of X, then Ylm consists of one-point sets and the
usual Stone-Weierstrass theorem follows.
2.48 As we have just seen, the self-adjoint subalgebras of C{X) are all of
the form C(Y) for some compact Hausdorff space Y. This is far from true,
however, for the nonself-adjoint subalgebras. Let 91 be a closed subalgebra

The Disk Algebra SI
of C(X) which contains the constant functions. If we let © denote the smallest
closed self-adjoint subalgebra of C(X) that contains 91, then © is isometrically
isomorphic to C(Y), where Fis the maximal ideal space of©, and the Gelfand
transform F implements the isomorphism. Then F9I is a closed subalgebra
of C(Y) that contains the constant functions and, more importantly, separates
the points of Y. Therefore, rather than study 91 as a subalgebra of C(X), we
choose to study F8I as a subalgebra of C(Y). Thus we make the following
definition.
2.49 Definition Let Ibea compact Hausdorff space and 81 be a subset
of C(X). Then 91 is said to be a function algebra if 91 is a closed subalgebra
of C{X) which separates points and contains the constant functions.
The theory of function algebras is very extensive and draws on the tech-
techniques of approximation theory and complex function theory as well as those
of functional analysis. In this book we will be limited to considering only a
few important examples of function algebras.
2.50 Example Let T denote the circle group {z e C : \z\ = 1}. For n in Z
let xn t>e tne function on T defined xn(z) = z"- Then Xo = 1» X-n =
XmXn = Xm+n f°r n an<i mini.. The functions in the set
9 = \ £ anXn:ane
are called the trigonometric polynomials. Since 9* is a self-adjoint subalgebra
of C(T) which contains the constant functions and separates points, then the
uniform closure of 9* is C(T) by the Stone-Weierstrass theorem.
Let SP+ = {Y^=oanXn : «« e C}; the functions in &>+ are called analytic
trigonometric polynomials. If we let A denote the uniform closure of &>+
in C(T), then A is a function algebra, but at this point it is not obvious that
A # C(T). We prove this by showing that the maximal ideal space of A is
not T. For this we need a lemma.
2.51 Lemma If Y^=oanXn is m ^+ and w is in C, |h>| < 1, then
N I j-2n
f0 2nJ0
*-
nf0 2nJ0 \^0 I I-we
Proof Expand 1/A—we"") = J^^0(we~")m, where the series converges
uniformly for t in [0,27i]. Therefore,

52 2 Banach Algebras
T
l~we
» r-2n
since A/2ti) Jg*cfk' <ft = 1 for k = 0 and 0 otherwise. ■
For w in C, \w\ < 1, define <pw on ^+ such that
(n \ n
It is clear that (?)„, is a multiplicative linear functional on ^+. However,
since 0>+ is not a Banach algebra, that is, 0'+ is not complete, we cannot
conclude apriori that <pw is continuous. That follows, however, from the
preceding lemma since
2n
Therefore, <pw is bounded on ^+ and hence can be extended to a multiplicative
linear functional on A. Now for w in C, |w| = 1, let <pw denote the evaluation
functional on A, that is, cpw(f) =f(w) for/in A. The latter is well denned,
since A c C(T).
Now set 0 = [z e C : \z\ < 1}, let M denote the maximal ideal space of
A, and fet \j/ be the function from D to M defined by \j/(z) = cpz.
2.52 Theorem The function \p is a homeomorphism of D onto the maximal
ideal space M of A.
Proof By the remarks preceding the theorem, the function fy is well
defined. If z, and z2 are in D, then i/r(z,) = *P(z2) implies that z, = <pZl(Xi) =
<Pz2(Xi) = Z2J thus ij/ is one-to-one.
If (/) is in M, then ||#, J = 1 implies that z = <p(xt) is in D. Moreover, the
identity
n
n=0
I
= Z ocnz" =
n=0

The Disk Algebra 53
proves that <p agrees with cpz on the dense subset &+ of A. Therefore, <p = q>2
and ij> is seen to be onto M.
Since both 0 and M are compact Hausdorff spaces and \j/ is one-to-one
and onto, to complete the proof it suffices to show that ifr is continuous. To
this end suppose {zp}PbB is a net in D such that ]imPeBZp = z. Since
} = 1 and &*+ is dense in A, and since
/ JV \ / N \ N
E <*nX» I = Um( E «»V) = E
«=0
N
for every function X^=oa«&. in ^+, it follows from Proposition 1.21 that
i]/ is continuous. |
2.53 From Proposition 2.3 we know that the maximal ideal space of C(T)
is just T. We have just shown that the maximal ideal space of the closed
subalgebra A of C(T) is D. Moreover, if <pz is a multiplicative linear functional
on A, and \z\ = 1, then cpz is the restriction to A of the "evaluation at z"
map on C(J). Thus the maximal ideal space of C(T) is embedded in that of v4.
This example also shows how the maximal ideal space of a function algebra
is, at least roughly speaking, the natural domain of the functions in it. In this
case although the elements of A are functions on T, there are "hidden points"
inside the circle which "ought" to be in the domain. In particular, viewing
Xi as a function on T, there is no reason why it should not be invertible;
on D, however, it is obvious why it is not—it vanishes at the origin.
Let us consider this example from another viewpoint. The element Xi
iscontainedinbothofthealgebrasv4andC(T).InC(T)wehaveffC(T)(x,) = T,
while in A we have cA(xi) = D. Hence not only is the "v4-spectrum" of yA
larger, but it is obtained from the C(T)-speetrum by "filling in a hole." That
this is true, in general, is a corollary to the next theorem.
2.54 Theorem (Silov) If © is a Banach algebra, 91 is a closed subalgebra
of 58, and/is an element of 91, then the boundary of crm(f) is contained in
the boundary of <?%(/).
Proof If if—X) has an inverse in 91, then it has an inverse in ©. Thus
°sr(/) contains cr^(f) and hence it is sufficient to show that the boundary of
o^if) <= c<b(/). If Ao is in the boundary of <?&(/), then there exists a sequence
{/ln}"=1 contained in /%(/) such that linv^/^ = k0. If for some integer

54 2 Banach Algebras
n it were true that |[(/- A,,)1| < |1/(AO- kn)\, then it would follow that
and hence/—Ao would be invertible as in the proof of Proposition 2.7. Thus
we have lim,,^ ||(/-A?,r1|| = co.
If k0 were not in <?&(/), then it would follow from Corollary 2.8 that
||(/— X)1| is bounded for k in some neighborhood of Ao, which is a
contradiction. ■
2.55 Corollary If © is a Banach algebra, 91 is a closed subalgebra of ©,
and/is an element of 91, then am(f) is obtained by adding to <?&(/) certain
of the bounded components of C\crm{f).
Proof Elementary topology and the theorem yield this result. ■
2.56 Example We next consider an example for which the Gelfand trans-
transform is not an isometry.
In Section 1.15 we showed that /'(Z+) is a Banach space. Analogously,
if we let I1 (I) denote the collection of complex functions f on I such that
2^°=-c» l/(")l < °°» ^en with pointwise addition and scalar multiplication
and the norm ||/||j = J^™=-w |/(«)| < co, I1 (I) is a Banach space. More-
Moreover, I1 (I) can also be made into a Banach algebra in a nonobvious way.
For/and g in I1 (Z) define the convolution product
(f°0)(n)= £ f(n-k)g(k).
k=-ao
To show that this sum converges for each n in 2 and that the resulting function
is in I1 (Z), we write
E \(fog)(n)\ =
E f(n-k)g(k)
k=-oo
CO CO
E E \f(n-k)\\g(k)\
n~ — co k= — co
= I \9ik)\ I \f(n-k)\ = I/IK E
k— — co «= — co k— — co
Therefore, fog is well defined and is in I1 (I), and \\fogU < ||/|i Mi- We
leave to the reader the exercise of showing that this multiplication is asso-
associative and commutative. Assuming this, then I1 (I) is a commutative Banach
algebra.

The Algebra of Functions with Absolutely Convergent Fourier Series 55
For n in Z let en denote the function on Z defined to be 1 at« and 0 other-
otherwise. Then et is the identity element of ^(Z) and enoem = en+m for n and
m in Z.
Let M be the maximal ideal space of I1 (Z). For each z in T, let <pz be the
function defined on I1 (I) such that <?>*(/) = IT=-»/(«)*"• It is easily
verified that (pz is well defined and in M. Thus we can define a function from
T to M by setting ^(z) = <pz.
2.57 Theorem The function ^ is a homeomorphism from T onto the
maximal ideal space M of /1(Z).
Proof If Zj and z2 are in T, and <pzi = cpZz, then zt = <pzi(et) =
(pz2(ei) = z2; hence ^ is one-to-one. Suppose cp is an element of M and
z= <p(e{); then
which implies that z is in T. Moreover, since cp(en) = {.cpieiXl" = z" = <pz(en)
for n in Z, it follows that <p = cpz = ij/(z). Therefore, again \p is one-to-one
and onto and it remains only to show that tj/ is continuous, since both M
and T are compact Hausdorff spaces. Thus, suppose {zp}peB is a net of points
in T such that lim^^Zp = z. Then for /in I1 (I)-we have
K00 -<pz(f)\< E |/(«)| |z/-z"| + £ |/(«)| iv-2-l
|n|s:/V |n|>/V
< ll/lli sup |V-z"l+2 E |/(«)|-
|«|=SAf |«|> Af
Hence for e > 0, if iV is chosen such that S|«|>n l/(«)l < £/4 and j80 is then
chosen in B such that /? > /?0 implies sup^^ |zp"-z"| < e/211/ld, then
\<PZfi(f)-<PAf)\<£ for ^^)S0. Therefore, tim,eBq>zp(f) = <pz(f) and ^
is continuous and the proof is complete. ■
2.58 Using the homeomorphism ij/ we identify the maximal ideal space of
/'(Z) with T. Thus the Gelfand transform is the operator T defined from
I1 (I) to C(T) such that {Tf){z) = ^=-oof(n)z" for z in T, where the series
converges uniformly and absolutely on T to Tf. The values of/on Z can be
recaptured from Tf, since they coincide with the Fourier coefficients of Tf.
More specifically,
/(«) = — f V/)(eiV-i"' * for « in Z,
271 Jo

56 2 Banach Algebras
since
i r2* -^
dt = - > f(m)e*>»-"»dt
27tJ £
= Yn 2 f(m) relim~n)t dt =/(")'
where the interchange of integration and summation is justified since the
series converges uniformly. In particular, q> in C(T) is in the range of T if
and only if the Fourier coefficients of q> are an absolutely convergent series,
that is, if and only if
f
2n
I <p(eu)e-imdt
2n
< co.
We leave to the exercises the task of showing that this is not always the case.
Since not every function tp in C(T) has an absolutely convergent Fourier
series, it is not obvious whether l/<p does if <p does and cp(z) # 0. That this
is the case is a nontrivial theorem due to Wiener. The proof below is due to
Gelfand and indicates the power of his theory for commutative Banach
algebras.
2.59 Theorem If <p in C(T) has an absolutely convergent Fourier series
and <p(z) # 0 for z in T, then \j<p has an absolutely convergent Fourier series.
Proof By hypothesis there exists/in I1 (Z) such that F/= cp. Moreover,
it follows from Theorem 2.35 that <p(z) # 0 for z in the maximal ideal space
T of/'(Z) implies that/is invertible in ll(Z). If g =/"', then
1 = F(e0) = T{g°f) = Tg-cp or - = F#.
<P
Therefore, l/<p has an absolutely convergent Fourier series and the proof is
complete. |
2.60 Example We conclude this chapter with an example of a com-
commutative Banach algebra for which the Gelfand transform is as nice as
possible, namely an isometric isomorphism onto the space of all continuous
functions on the maximal ideal space.
I n Section 1.44 we showed that Lw is a Banach space. If/and g are elements
of L00, then the pointwise product is well defined, is in Lw, and \\fg\\K ^

The Algebra of Bounded Measurable Functions 57
II/LML- (That is, Jfw is an ideal in if00.) Thus Lw is a commutative
Banach algebra. Although it is not at all obvious, L00 is isometrically iso-
morphic to C(Y) for some compact Hausdorff space Y. We prove this after
determining the spectrum of an element of L00. For this we need the following
notion of range for a measurable function.
2.61 Definition If/ is a measurable function on X, then the essential range
M(f) of/is the set of all k in C for which {xeX: \f(x)-k\ < e} has positive
measure for every e > 0.
2.62 Lemma If/is in U>, then M (/) is a compact subset of C and ||/|| „ =
Proof If k0 is not in M(f), then there exists e > 0 such that the set
{xeX: \f(x) — ko\ < e} has measure zero. Clearly, then each k in the open
disk of radius e about k0 fails to be in the essential range of/ Therefore, the
complement of M{J) is open and hence M(f) is closed. If k0 in C is such that
K > l/(*)l +$ for almost all x in X, then the set {xeX: \f(x)-ko\ < E/2}
has measure zero, and hence sup{|A| : Xe3%(f)} ^ Halloo- Thus @%(f) is a
compact subset of C for /in Lw.
Now suppose/is in L00 and no k satisfying \k\ = \\f\\w is in M(f). Then
about every such X there is an open disk D, of radius 8A such that the set
{xeX: \f(x)-X\ < 5J has measure zero. Since the circle {k e C : \X\ = ||/|| „}
is compact, there exists a finite subcover of open disks DXi,Dh,...,DXn
such that the sets {xeX:f(x)e£>AJ have measure zero. Then the set
{jce X \f{x) e UT=iAi,} has measure zero, which implies that there exists
an e > 0 such that the set {xeX: \f(x)\ > ||/||TO— e} has measure zero.
This contradiction completes the proof. |
2.63 Lemma If/is in Lw, then cr(f) = ®{f).
Proof If k is not in o{f), then l/(f—k) is essentially bounded, which
implies that the set {xeX: \f(x) — k\ < 1/2 \\f— k||re} has measure zero. Con-
Conversely, if {xeX: \f(x) — k\ <8] has measure zero for some 5 > 0, then
lj(f—k) is essentially bounded by 1/E, and hence k is not in cr(f). Therefore
2.64 Theorem If M is the maximal ideal space of L00, then the Gelfand
transform T is an isometrical isomorphism of L00 onto C(M). Moreover,
r/= rG) for/in L00.

58 2 Banach Algebras
Proof We show first that F is an isometry. For / in If we have, combin-
combining the previous lemma and Corollary 2.36, that range F/ = M(f), and hence
I|r(/)|L = sup{|A| derange F/} = sup{|A| :26«(/)} = \\f\\w.
Therefore F is an isometry and F^L00) is a closed subalgebra of C{M).
For/in If set/=/1 + i/2. where each of/i and/2 is real valued. Since
the essential range of a real function is real and range F/j = 8%{fx) and
range F/2 = ^(/2), we have
F/= FA + iT/2 = F/, - iT/2 = F(/).
Therefore FL00 is a closed self-adjoint subalgebra of C(M). Since it obviously
separates points and contains the constant functions, we have by the Stone-
Weierstrass theorem that FL00 = C{M) and the theorem is proved. ■
Whereas in preceding examples we computed the maximal ideal space,
in this case the maximal ideal space is a highly pathological space having
22**0 points. We shall have reason to make use of certain properties of this
space later on.
2.65 It can be easily verified that /"(Z"*") (Section 1.15) is also a Banach
algebra with respect to pointwise multiplication. It will follow from one of
the problems that the Gelfand transform is an onto isometrical isomorphism
in this case also. The maximal ideal space of /CO(Z+) is denoted /?Z+ and is
called the Stone-Cech compactification of Z + .
Notes
The elementary theory of commutative Banach algebras is due to Gelfand
[41] but the model provided by Wiener's theory of generalized harmonic
analysis should be mentioned. Further results can be found in the treatises
of Gelfand, Raikov and Silov [42], Naimark [80], and Rickart [89]. The
determination of the self-adjoint subalgebras of C(X) including the generaliz-
generalization of the Weierstrass approximation theorem was made by Stone [105].
The literature on function algebras is quite extensive but two excellent sources
are the books of Browder [10] and Gamelin [40].
Exercises
2.1 Let ® = {/eC([0,l]):/'eC([0,l])} and define ||/||d = \f\a +
||/' || „. Show that 3> is a Banach algebra and that the Gelfand transform is
neither isometric nor onto.

Exercises 59
2.2 Let X be a compact Hausdorff space, K be a closed subset of X, and
3 = {/e C(X) :f(x) = 0 for x 6 X}.
Show that 3 is a closed ideal in C(X). Show further that every closed ideal
in C(X) is of this form. In particular, every closed ideal in C{X) is the inter-
intersection of the maximal ideals which contain it.*
2.3 Show that every closed ideal in 9) is not the intersection of the maximal
ideals which contain it.
2.4 Let SC be a Banach space and £($") be the collection of bounded linear
operators on SC. Show that £($") is a Banach algebra.
2.5 Show that if SC is a finite (>1) dimensional Banach space, then the
only multiplicative linear functional on £($") is the zero functional.
Definition An element T of £($") is finite rank if the range of T is finite
dimensional.
2.6 Show that if SC is a Banach space, then the finite rank operators form
a two-sided ideal in £($") which is contained in every proper two-sided ideal.
2.7 If/is a continuous function on [0,1] show that the range of/is the
essential range of/
2.8 Let/be a bounded real-valued function on [0,1] continuous except
at the point \. Let 91 be the uniformly closed algebra generated by / and
C([01]). Determine the maximal ideal space of 91.*
2.9 If A' is a compact Hausdorff space, then C(X) is the closed linear span
of the idempotent functions in C{X) if and only if X is totally disconnected.
2.10 Show that the maximal ideal space of L00 is totally disconnected.
2.11 Let X be a completely regular Hausdorff space and B(X) be the space
of bounded continuous functions on X. Show that B{X) is a commutative
Banach algebra in the supremum norm. If fiX denotes the maximal ideal
space of B(X), then the Gelfand transform is an isometrical isomorphism of
B(X) onto C(fiX) which preserves conjugation. Moreover, there exists a
natural embedding /? of X into {SX. The space /?A" is the Stone-Cech com-
pactification of X.
2.12 Let X be a completely regular Hausdorff space, Y be a compact
Hausdorff space, and <p be a continuous one-to-one mapping of X onto a

60 2 Banach Algebras
dense subset of Y. Show that there exists a continuous mapping ij/ from fix
onto Y such that p = ij/ocp. (Hint: Consider the restriction of the functions
in C(Y) as forming a subalgebra of C(fiX).)
2.13 Let © be a commutative Banach algebra and
9? = {/e © : I+ kfe <$ for I e C}.
Show that SR is a closed ideal in ©.
Definition If © is a commutative Banach algebra, then
9? = {/e © : 1 + /l/e <S for /16 C}
is the radical of © and © is said to be semisimple if *R = {0}.
2.14 If © is a commutative Banach algebra, then 91 is the intersection of
the maximal ideals in ©.
2.75 If © is a commutative Banach algebra, then © is semisimple if and
only if the Gelfand transform is one-to-one.
2.16 If © is a commutative Banach algebra, then 93/91 is semisimple.
2.17 Show that Ll([0,1])©C with the 1-norm (see Exercise 1.17) is a
commutative Banach algebra for the multiplication defined by
= Wif) + lg(t) + jj(t-x)g(x) dxi © Xp.
Show that L1 ([0,1]) © C is not semisimple.
2.18 (Riesz Functional Calculus) Let © be a commutative Banach algebra,
x be an element of ©, Q be an open set in C containing cr(x), and A be a finite
collection of rectifiable simple closed curves contained in fi such that A
forms the boundary of an open subset of C which contains <r(x). Let A(£l)
denote the algebra of complex holomorphic functions on fi. Show that the
mapping
Ja
(z)(x-z)~l dz
a
defines a homomorphism from A(Cl) to © such that <r(cp(x)) = <p(o(x))
for cp in A (fi).
2.19 If © is a commutative Banach algebra, x is an element of © with
a(x) c fi, and there exists a nonzero <p in A(Cl) such that cp(x) = 0, then
a(x) is finite. Show that there exists a polynomial p(z) such that p{x) = 0.
2.20 Show that for no constant M is it true that E^=-n|ob| <
for all trigonometric polynomialsp = ££L~Nan%n on T.

Exercises 61
2.21 Show that the assumption that every continuous function on T has
an absolutely convergent Fourier series implies that the Gelfand transform
on /' (Z) is invertible, and hence conclude in view of the preceding problem
that there exists a continuous function whose Fourier series does not converge
absolutely.*
Definition If © is a Banach algebra, then an automorphism on © is a con-
continuous isomorphism from © onto ©. The collection of all automorphisms
on © is denoted Aut(©).
2.22 If X is a compact Hausdorff space, then every isomorphism from
C(X) onto C(X) is continuous.
2.23 If X is a compact Hausdorff space and <p is a homeomorphism on X,
then (O/)(x) =f(<px) defines an automorphism O in Aut[C(X)]. Show that
the mapping (p-*<b defines an isomorphism between the group Hom(Ar)
of homeomorphisms on Zand Aut[C(X)].
2.24 If 91 is a function algebra with maximal ideal space M, then there is
a natural isomorphism of Aut(9l) into Hom(M).
2.25 If A is the disk algebra with maximal ideal space the closed unit disk,
then the range of Aut(v4) in Hom(D) is the group of fractional linear trans-
transformations on D, that is, the maps z-> jS(z — a)/(l — dz) for complex numbers
a and /? satisfying |oe| < 1 and |/?| = 1.*
Definition If 91 is a function algebra contained in C(X), then a closed subset
M of X is a boundary for 91 if \\f\\x = sup{|/(m)| : m e M} for/in 91.
2.26 If 91 is a function algebra contained in C(X); M is a boundary for 91;
/,,...,/„ are functions in 91; and t/is the open subset of Mdefined by
{xeX:\Mx)\ < 1 for i= 1,2,...,«}.
Show either that M\U is a boundary for 91 or U intersects every boundary
for 91.
2.27 (Silov) If 91 is a function algebra, then the intersection of all boundaries
for 91 is a boundary (called the Silov boundary for 91).*
2.28 Give a functional analytic proof of the maximum modulus principle
for the functions in the disk algebra A. (Hint: Show that
where the function kr(t) = TJ™=-o0r'lef'* is positive.)

62 2 Banach Algebras
2.29 Show that the Silov boundary for the disk algebra A is the unit circle.
2.30 Show that the abstract index group for a commutative Banach algebra
contains no elements of finite order.
2.31 If ©t and ©2 are Banach algebras, then ©!<§)©2 ar>d ©!<x)©2 are
Banach algebras. Moreover, if ©t and ©2 are commutative with maximal
ideal spaces Ml and M2, respectively, then Ml xM2 is the maximal ideal
space of both ©t (§)©2 and ©t®©2 (see Exercises 1.36 and 1.37 for the
definition of (§) and ®.)
2.32 If © is an algebra over C, which has a norm making it into a Banach
space such that \\fg\\ ^ ||/|| \\g\\ for /and g in ©, then 93 ©C is a Banach
algebra in the 1-norm (see Exercise 1.17) for the multiplication
with identity 0©l.
2.33 If (jo is a multiplicative linear functional on ©, then q> has a unique
extension to an element of Ma&c. Moreover, the collection of nonzero
multiplicative linear functionals on © is a locally compact Hausdorff space.
2.34 Show that Ll(U) is a commutative Banach algebra without identity
for the multiplication defined by
(f°g)(x) = f° f(x-t)g(t) dt for /and g in Ll(U).
J — CO
2.35 Show that for tin R the linear function on L1 (R) defined by
f(x)e'*'dx
= (
J—
is multiplicative. Conversely, every nonzero multiplicative linear functional
m L*(R) is of this form.* (Hint: Every bounded linear functional on L*(R)
is given by a cp in L°°(R). Show for /and g in L*(R) that
J — ao J — 0
and that implies ^>(x— ?) <?@ = <P(*) for (x, t) not in a planar set of Lebesgue
measure.)
2.36 Show that the maximal ideal space of L1 (R) is homeomorphic to R
and that the Gelfand transform coincides with the Fourier transform.

Geometry of Hilbert Space
3.1 The notion of Banach space abstracts many of the important properties
of finite-dimensional linear spaces. The geometry of a Banach space can,
however, be quite different from that of Euclidean n-space; for example, the
unit ball of a Banach space may have corners, and closed convex sets need
not possess a unique vector of smallest norm. The most important geometrical
property absent in general Banach spaces is a notion of perpendicularity or
orthogonality.
In the study of analytic geometry we recall that the orthogonality of two
vectors was determined analytically by considering their inner (or dot)
product. In this chapter we introduce the abstract notion of an inner product
and show how a linear space equipped with an inner product can be made
into a normed linear space. If the linear space is complete in the metric denned
by this norm, then it is said to be a Hilbert space. This chapter is devoted to
studying the elementary geometry of Hilbert spaces and to showing that
such spaces possess many of the more pleasant properties of Euclidean n-
space. We will show, in fact, that a finite dimensional Hilbert space is
isomorphic to Euclidean H-space for some integer n.
3.2 Definition An inner product on a complex linear space if is a function
tp from if x if to C such that:
A) <P(«i/i+«2/2,^) = <*i<p(fi,ff) + tt2<p(f2,0) for aua2 in C and
/i,/2,#inif;
63

64 3 Geometry of Hilbert Space
B) q>W,Piffi+p2ff2)=Pi<P(f,gi)+Pi<pU:9^ for fiufi2 in C and
C) <pif,g) = <p(g,f) for/and g in if; and
D) cp(f,f) Ss 0 for /in if and <?(/,/) = 0 if and only if/= 0.
A linear space equipped with an inner product is said to be an inner
product space.
The following lemma contains a useful polarization identity, the im-
importance of which lies in the fact that the value of the inner product cp is
expressed solely in terms of the values of the associated quadratic form ij/
denned by ^(/) = <p(fj) for/in if.
3.3 Lemma If if is an inner product space with the inner product <p, then
<p(f,g) = \{<pW+g,f+g) - <p(f-g,f-g) + iq>(f+ig,f+ig)
for/and gin i£.
Proof Compute. ■
An inner product is usually denoted (,), that is, (/,<?) = q>(f,g) for/
and g in ^£.
3.4 Definition If ^£ is an inner product space, then the norm || || on JS?
associated with the inner product is defined by ||/|| = (f,f)i/2 for/in i£.
The following inequality is basic in the study of inner product spaces.
We show that the norm just defined has the required properties of a norm
after the proof of this inequality.
3.5 Proposition (Cauchy-Schwarz Inequality) If / and g are in the inner
product space ^£, then
IC/",*)I< 11/11 \\g\\-
Proof For/and ^ in if and X in C, we have
|A|2||<7||2 + 2Re[I(/,<7)] + I/I2 = (f+lg,f+W
= II/+^H2 > 0.
Setting k = teie, where t is real and e'e is chosen such that e~'e(f,g)^ 0,
we obtain the inequality

The Cauchy-Schwarz Inequality 65
Hence the quadratic equation
in / has at most one real root, and therefore its discriminant must be non-
positive. Substituting we obtain
from which the desired inequality follows. ■
3.6 Observe that the property (/,/) = 0 implies that/= 0 was not needed
in the preceding proof.
3.7 Proposition If if is an inner product space, then || • | defines a norm
on if.
Proof We must verify properties (l)-C) of Definition 1.3. The fact that
Il/H = 0 if and only if /= 0 is immediate from D) (Definition 3.2) and thus
A) holds.
Since
W\\ = (¥,¥)V2 = (M(f,f)YA = \M 11/11 for k in C and /in 2,
we see that B) holds. Lastly, using the Cauchy-Schwarz inequality, we have
2 = (f+g,f+g) = (/,/) + (f,g) + (g,f) + (g,g)
= ll/ll2 + \\0\\2 + 2Re(/,<7) *S H/ll2 + ||<7||2 + 2\(fg)\
for/and g in if. Thus C) holds and || • | is a norm. ■
3.8 Proposition In an inner product space, the inner product is continuous.
Proof Let if be an inner product space and {fa}aeA
nets in £C such that limaieAfa =/and lim^g^^ = g. Then
\(f,g)-(fa,0«)\ ^ \(f-fa,g)\ + \(fa,g-gJ\
and hence lin^^C/,, ga) = (/,g).
3.9 Definition In the inner product space jSf two vectors / and g are said
to be orthogonal, denoted fL g, if (/, g) = 0. A subset 5^ of if is said to be
orthogonal if fL g for / and g in if and orthonormal if, in addition, ||/| =
1 for/in &.

66 3 Geometry of Hilbert Space
This notion of orthogonality generalizes the usual one in Euclidean space.
It is now possible to extend various theorems from Euclidean geometry to
inner product spaces. We give two that will be useful. The first is the familiar
Pythagorean theorem, while the second is the result relating the lengths of
the sides of a parallelogram to the lengths of the diagonals.
3.10 Proposition (Pythagorean Theorem) If {fuf2,...,f,} is an orthogonal
subset of the inner product space ^£, then
I/,
Proof Computing, we have
I/,
= I
= £(/-,/*)= Ell/J2.
1 i
3.11 Proposition (Parallelogram Law) If/ and g are in the inner product
space if, then
Proof Expand the left-hand side in terms of inner products. ■
As in the case of normed linear spaces the deepest results are valid only if
the space is complete in the metric induced by the norm.
3.12 Definition A Hilbert space is a complex linear space which is com-
complete in the metric induced by the norm.
In particular, a Hilbert space is a Banach space.
3.13 Examples We now consider some examples of Hilbert spaces.
For n a positive integer let C" denote the collection of complex ordered
H-tuples {x : x = (x1,x2,---,xn), x,eC}. Then C" is a complex linear space
for the coordinate-wise operations. Define the inner product (,) on C" such
that {x,y) = YH=iX,y,. The properties of an inner product are easily verified
and the associated norm is the usual Euclidean norm ||jc||2 = (S"=i l^il2O2-
To verify completeness suppose {xk}j?L l is a Cauchy sequence in C". Then

Examples of Hilbert Spaces: C", I2, L2, and H2 67
since \x,k-xtm\ ^ H^-x™^, it follows that {xtk}™=1 is a Cauchy sequence
in C for 1 ^ i ^ n. If we set x = (xl,x2,---,xk), where xt = limt_oox,.'t, then
x is in C" and lim^^ JC* = jc in the norm of C". Thus C" is a Hilbert space.
The space C" is the complex analog of real Euclidean n-space. We show
later in this chapter, in a sense to be made precise, that the C"'s are the only
finite-dimensional Hilbert spaces.
3.14 We next consider the "union" of the C"'s. Let S£ be the collection of
complex functions on Z+ which take only finitely many nonzero values.
With respect to pointwise addition and scalar multiplication, if is a complex
linear space. Moreover, (/,#) = 2^°=0/(«)#(«) defines an inner product on
if, where the sum converges, since all but finitely many terms are zero. Is if
a Hilbert space? It is if i£ is complete with respect to the metric induced by
the norm ||/|2 = (£"=o |/(n)|2)%- Consider the sequence {/k}"=1 contained
in £^, where
jar «<*,
/*(«) =
[0 n> k.
One can easily show that {fk}%L t is Cauchy but does not converge to an
element of £C. We leave this as an exercise for the reader. Thus ^£ is not a
Hilbert space.
3.15 The space JSf is not a Hilbert space because it is not large enough. Let
us enlarge it to obtain our first example of an infinite-dimensional Hilbert
space. (This example should be compared to Example 1.15.)
Let /2(Z+) denote the collection of all complex functions <p on Z+ such
that Xr=o W(n)\2 < °°- Then I2(l+) is a complex linear space, since
For/and g in /2(Z+), define (f,g) — Z"=o/(«)#(«)• Does this make sense,
that is, does the sum converge? For each N inZ+, the n-tuples
Fs = A/@I,1/AI,-, 1/(^I) and GN = (\g@)\,W)\,-,\g(N)\)
lie in CN. Applying the Cauchy-Schwarz inequality, we have
(«j| = \(FN,GN)\ ^ \\FN\\\\GN\\
= (i i/(«)i2YY £ \0(
\«=0 / \«=0

68 3 Geometry of Hilbert Space
Thus the series Z"=o/(«)#(«) converges absolutely. That (,) is an inner
product follows easily.
To establish the completeness of /2(Z+) in the metric given by the norm
|| ||2, suppose {/*}"= i is a Cauchy sequence in /2(Z+). Then for each n in
Z+, we have
and hence {fk(n)}^=1 is a Cauchy sequence in C for each « in Z+. Define
the function/on Z+ to be/(«) = limt_„/*(«). Two things must be shown:
that/is in /2(Z+) and that lim^ ||/-/*||2 = 0. Since {/*}£=! is a Cauchy
sequence, there exists an integer K such that for fc > Kv/e have ||/*—/K||2 ^ 1.
Thus we obtain
< \ E »l2f
b=0
^ limsup ||/*-/K|2 + ||/K||2 ^ 1 + ||/*||2,
and hence / is in /2(Z+). Moreover, given e > 0, choose M such that
k, j > M implies ||/* —/J || < e. Then for k > M and any iV, we have
E l/(«)-/*(«)|2 = lim E I/J'(«)-/*(«)|2
n=0
Since iV is arbitrary, this proves that ||/—/*||2 ^ e and therefore /2(Z+) is
a Hilbert space.
3.16 The Space L2 In Section 1.44 we introduced the Banach spaces L1
and L00 based on a measure space (X, £f, [i). We now consider the correspond-
corresponding L2 space, which happens to be a Hilbert space.
We begin by letting if2 denote the set of all measurable complex functions
/onl which satisfy J* |/|2 d\i < oo. Since the inequality |/+#|2 ^ 2|/|2 +
21#|2 is valid for arbitrary functions/and g on X, we see that if2 is a linear
space for pointwise addition and scalar multiplication. Let Jf2 be the sub-
space of functions/in if2 for which J^j/|2 d/i =0, and let L2 denote the
quotient linear space if 2

Examples ofHilbert Spaces: C", I2, L2, and H2 69
If/and g are in if2, then the identity
= t{(\f\+\ff\J-\f\2-\s\2}
shows that the function/# is integrable. If we define <p(f,g) = \xfQ dp for
/and g in if2, then cp has all the properties of an inner product except one;
namely, cp(J,f) = 0 does not necessarily imply/= 0. By the remark follow-
following the proof of the Cauchy-Schwarz inequality, that inequality holds for cp.
Thus, if/,/', g, and g' are functions in if2 such that/—/' and g—g' belong
to Jf'L, then
\<p(f,ff)-<p(f',g')\ *s \<P(f-f,g)\ + \<p(f',ff-ff')\
< <P<J-f'J-f')<p{9,9)
+ <p(J',f')<P@-0',g-g') = o.
Therefore, cp is a well-defined function on L2. Moreover, if <p(C/],C/]) = 0,
then Jxl/|2 df1 = 0 and hence [/] = [0]. Thus cp is an inner product on
1} and we will denote it from now on in the usual manner. Furthermore, the
associated norm on L2 is defined by ||[/]||2 =(jx\f\2 diiI/z. The only
problem remaining before we can conclude that I2 is a Hilbert space is the
question of its completeness. This is slightly trickier than in the case of I}.
We begin with a general inequality. Take /in if2 and define g on Xsuch
that g(x) =f(x)l\f(x)\ if/(x) # 0 and g(x) = 1 otherwise. Then g is measur-
measurable, \g(x)\ = 1, and fg = j/j. Moreover, applying the Cauchy-Schwarz
inequality, we have
ll/lli =[\f\dn=\fg dti = \(f,g)\ ^ \\f\\2\\g\\2 = ||/||2.
JX JX
Therefore, 11/11^ 11/112 for/in J^2.
We now prove that L2 is complete using Corollary 1.10. Let {[/„]}"= t
be a sequence in 1} such that Y£=i IIL/nllU < Af < 00. By the preceding
inequality ££L 11| [/,] ||, ^ M, and hence, by the proof in Section 1.44,
there exists / in if1 such that £"=i/,(*) =/(*) for almost all x in X.
Moreover
r n 2 ? ( n \2 11 r n ~i
Jjr «=i Jjr\«=i / IIL"=1 J
/ n
\n=i
IIC/JII:
and since lim^...^ |Z^=i/.W|2 = I/W|2 f°r almost all x in X, it follows
from Fatou's lemma that j/j2 is integrable and hence/is in if2. Moreover,

70 3 Geometry of Hilbert Space
since the sequence {(I^=1 \fn\J}N=i is monotonically increasing, it follows
that k = linv-codJlLi \fn\J is an integrable function. Therefore,
N-+tx>
L/1-Il/J =
«=1 2
= lim
\JX
im(f
->co \JX
f-
=0
by the Lebesgue dominated convergence theorem, since \Y^=N+ifn\2 ^ ^
for all JV and lim,^ |Z^=N+i/nW|2 = 0 for almost all x in X. Thus Z?
is a Hilbert space. Lastly, we henceforth adopt the convention stated in
Section 1.44 for the elements of I?; namely, we shall treat them as functions.
3.17 The Space H2 Let T denote the unit circle, ju the normalized
Lebesgue measure on T, and L2(T) the Hilbert space denned with respect
to fi. The corresponding Hardy space H2 is denned as the closed subspace
for n = 1,2,3,...
where /„ is the function xn(e") = e"". A slight variation of this definition is
{/eL2(T):(/,yJ =
for n =-1,-2,-3,...}.
3.18 Whereas in Chapter 1 after denning a Banach space we proceeded to
determine the conjugate space, this is unnecessary for Hilbert spaces since
we show in this chapter that the conjugate space of a Hilbert space can be
identified with the space itself. This will be the main result of this chapter.
We begin by extending a result on the distance to a convex set to subsets
of Hilbert spaces. Although most proofs of this result for Euclidean spaces
make use of the compactness of closed and bounded subsets, completeness
actually suffices.
3.19 Theorem If X is a nonempty, closed, and convex subset of the Hilbert
space 2/C, then there exists a unique vector in Jf of smallest norm.
Proof If £ = inf{||/||:/eJO, then there exists a sequence {/n}"=0
in Jf such that limn^ \fA = ^- Applying the parallelogram law to the
vectors fJ2 andfjl, we obtain
Jn Jn
= 2
+ 2
L
Since W is convex, (/n+/J/2 is in Jf and hence ||(/n+/J/2||2 > 82. There-

Examples of Hilbert Spaces: C", P, L\ and H2
fore, we have ||/n-/J|2 ^ 2||/J2 + 2||/J2-4,52, which implies
limsup II f— /1112 ^ 282 + 282 — 482 = 0.
«,«?-> CO
Thus {/,}"=! is a Cauchy sequence in Jf and from the completeness of M*
and the fact that Jf is a closed subset of ^ we obtain a vector/in Jf such
that limn^xfn =/. Moreover, since the norm is continuous, we have
f-g
2
2
- 2
f
2
2
+ 2
g
2
2
f+g
2
Having proved the existence of a vector in Jf of smallest norm we now
consider its uniqueness. Suppose/and g are in Jf with ||/|| = ||#|| = 8. Again
using the parallelogram law, we have
82 82 .
since ||(/+<7)/2| ^ 8. Therefore,/= g and uniqueness is proved. |
If n is a plane and / is a line in three-space perpendicular to n and both %
and / contain the origin, then each vector in the space can be written uniquely
as the sum of a vector which lies in n and a vector in the direction of /. We
extend this idea to subspaces of a Hilbert space in the theorem following
the definition.
3.20 Definition lf.J'is a subset of the Hilbert space Jif, then the orthogonal
complement of Jl, denoted Jl1, is the set of vectors in 2f? orthogonal to
every vector in M.
Clearly JlL is a closed subspace of M', possibly consisting of just the zero
vector. However, if Jl is not the subspace {0} consisting of the zero vector
alone, then Jl1 ^ Jf.
3.21 Theorem If Jt is a closed subspace of the Hilbert space #C and / is
a vector in .?f, then there exist unique vectors g in Jt and h in Ji1 such that
f=g+h.
Proof If we set Jf = {/— k : keJt}, then Jf is a nonempty, closed
and convex subset of 2?C Let h be the unique element of Jf with smallest
norm whose existence is given by the previous theorem. If k is a unit vector
m Jl, then h — {h,k)k is in X, and hence
||/7||2 ^ \\h-(h,k)k\\2= \\h\\2~^)(h,k)-(h,k)jhj)+(h,k)(hj)\\k\\2.
Therefore, \{h,k)\2 ^ 0, which implies (h,k) = 0, and hence h is in JlL. Since
h is in Jf, there exists g in Jl such that/= g+h and the existence is proved.

72 3 Geometry ofHilbert Space
Suppose now that/= gl +ht = g2 + h2, where gt and g2 are in M while
hi and h2 are in ML. Then g2 — gx =hx~h2 is in Jtc^Jl^, and hence
g2-gx is orthogonal to itself. Therefore, \\g2-gi\\2 = (g2-gl,g2-g1) =0
which implies gt = g2. Finally, hy= h2 and the proof is complete. ■
3.22 Corollary If M is a subspace of the Hilbert space 2ft, then
iLL
Proof That clos .# cz JKLL follows immediately for any subset Ji of
2tf. If/is in JiLL, then by the theorem/= g+h, where # is in clos^ and
h is in ML. Since/is in MLL, we have
0 = (f,h) = (g+h,h) = (M) = \\h\\2.
Therefore, h = 0 and hence/is in clos .#. ■
If g is a vector in the Hilbert space Jtf, then the complex functional denned
9g(f) = W,0) for/in 3ff is clearly linear. Moreover, since \<pg(f)\ ^ ||/|| ||^||
for all /in #C, it follows that <pg is bounded and that \\cpg\\ ^ ||^||. Since
\\ff\\2 = (pg(ff)<\\(pg\\Wff\\ ^ have y| < |«,J and hence ||^!| = k||. The
following theorem states that every bounded linear functional on jjf is of
this form.
3.23 Theorem (Riesz Representation Theorem) If <p is a bounded linear
functional on #C, then there exists a unique # in #C such that <p(/) = (J,g)
for /in Jf.
Let Jf be the kernel of <p, that is, Jf = {feM> :<p(f) = 0}.
Since <p is continuous, Jf is a closed subspace of 2ft. If Jf = #C then
<p{f) = (f,0) for/in ^ and the theorem is proved. If Jf # ^, then there
exists a unit vector h orthogonal to X by the remark following Definition
3.20. Since h is notin jf, then (p(/z) # 0. For/in ^ the vector/-(cp{f)l(p{h))h
is in Jf since <p(f—(<p(f)l<p(h))h) = 0. Therefore, we have
Hf) =
(^h
\cp(h)
for /in Jf, and hence cp(J) = (f,g) for g =

The Riesz Representation Theorem 73
If (f,0i) = (f,ffi) for / in Je, then, in particular, (gt -g2,gx -g2) = 0
and hence gx = g2. Therefore, <p(f) = (f,g) for a unique g in 2ft. ■
Thus we see that the mapping from #C to ^* defined # -> q>g is not only
norm preserving but onto. Moreover, a straightforward verification shows
that this map is conjugate linear, that is, <paigl+IX2g2 = a1<pm+a2<pg2 for a,
and a2 in C and gx and #2 *n ^- Thus for most purposes it is possible to
identify 3V* with 2f? by means of this map.
3.24 In the theory of complex linear spaces, a linear space is characterized
up to an isomorphism by its algebraic dimension. While this is not true for
Banach spaces, it is true for Hilbert spaces with an appropriate and different
definition of dimension. Before giving this definition we need an extension
of the Pythagorean theorem to infinite orthogonal sets.
3.25 Theorem If {fa}aeA is an orthogonal subset of the Hilbert space 2/C,
then J^,eAfa converges in #C if and only if J^,eA \\fa\\1 < co and in this case
Proof Let !F denote the collection of finite subsets of A. If Xm^/,
converges, then by Definition 1.8, the continuity of the norm, and the
Pythagorean theorem we have
hf2-
ii £-ijaL
2
lim £ fa
FeW ueF
2
= lim
FeW
E/a
aeF
2= E
Fe!p aeF aeA
Therefore, if J^eAfa converges, then J^eA ||/J|2 < oo.
Conversely, suppose Y*eA 11/112< oo. Given e > 0, there exists Fo in $F
such that F^F0 implies J^eF||/J|2-Zaefo||/J|2 < e2. Thus, for Fx and
F2 in J5" such that FX,F2^ Fo, we have
2
y f - y f
= E II/JI5
aeF,\f2
I
7. \JF2
_ y
2 < a2,
where the first equality follows from the Pythagorean theorem. Therefore,
the net {ZaeF/jF&F is Cauchy, and hence J^eAfaconverges by definition. ■
3.26 Corollary If {ea}aeA is an orthonormal subset of the Hilbert space

74 3 Geometry of Hilbert Space
#f and Ji is the smallest closed subspace of 2f? containing the set {ea : a e A},
then
teA aeA
Let 2F denote the directed set of all finite subsets of A.
is a choice of complex numbers such that Y.aeA\K\2 < °°> then {kaea}aeA
is an orthogonal set and Y.aeA \\^aea\\2 < °°. Thus J^aeAkaea converges to
a vector/in &f by the theorem and since/= ^vaFt^'YMEFkaea, the vector
is seen to lie in
Since Jf contains {ea : a e A}, the proof will be complete once we show that
Jf is a closed subspace of #e. If {J.a}aeA and {^a}ae/i satisfy Y^ea \K\2 < °o
e^lA'a|2< oo, then
Hence Jf is a linear subspace of ^f.
Now suppose {/"}™=i is a Cauchy sequence contained in ^T and that
/" = IIE^4"'ea. Then for each a in A we have
e
aeA
and hence ka = linin^^ ^n) exists. Moreover, for F in J5" we have
5] Kl2 = Hm X |#>|2 < lim X l#}|2 = Um llril2 < oo.
aeF «-*oo ctej1 «-»oo aeA n-*oo
Hence,/= J]aEi4 kaea is well denned and an element of Jf. Now given e > 0,
if we choose N such that n,m^ N implies ||/"—/"|| < e, then for F in J5",
we have
X K-e>|2 = Urn X l^r-^l2 < lim sup!/™-/!2 < £2-
Therefore, for n > TV, we have
II/-/T = I l^-^l2 = Hm X K-A'«>|2 ^ e2,
aeA Fe.W aeF
and hence ^T is closed. ■
3.27 Definition A subset {ea}aeA of the Hilbert space jf is said to be an
orthonormal basis if it is orthonormal and the smallest closed subspace
containing it is tff.

The Existence of Orthonormal Bases 75
An orthonormal basis has especially pleasant properties with respect to
representing the elements of the space.
3.28 Corollary If {ea}abA is an orthonormal basis for the Hilbert space 2ff
and/is a vector in Jf, then there exist unique Fourier coefficients {Aa}aeA
contained in C such that /= J^eA Ken- Moreover, ka = (/,ej for a in A.
Proof That {Xa}aeA exists such that f='£cieAJ.ae0, follows from the
preceding corollary and definition. Moreover, if 3F denotes the collection of
finite subsets of A and /? is in A, then
l = lim ( £ Ke«,eA
(
= lim £ Ktie&ep) = lim h = V
fisF
(The limit is unaffected since the subsets of A containing /? are cofinal in #\)
Therefore the set {^.a}aeA is unique, where Xa — (/, ea) for a in A. ■
3.29 Theorem Every Hilbert space (# {0}) possesses an orthonormal basis.
Proof Let $ be the collection of orthonormal subsets E of &f with the
partial ordering Ex < £2 if Ex <= £2. We want to use Zorn's lemma to assert
the existence of a maximal orthonormal subset and then show that it is a
basis. To this end let {Ex}XeA be an increasing chain of orthonormal subsets
of Jf. Then clearly Uaea^a ls an orthonormal subset of 2/C and hence is
in S. Therefore, each chain has a maximal element and hence S itself has a
maximal element EM. Let Jl be the smallest closed subspace of #C containing
EM. If Ji' = 2f?, then £M is an orthonormal basis. If *# # ^f, then for e a
unit vector in ^#x, the set EM u {e} is an orthonormal subset of #C greater
than EM. This contradiction shows that Jl = #C and £M is the desired ortho-
normal basis. ■
Although there is nothing unique about an orthonormal basis, that is,
there always exist infinitely many if 2?C # {0}, the cardinality of an ortho-
normal basis is well defined.
3.30 Theorem If {ea}aeA and {ffi}peB are orthonormal bases for the Hilbert
space #C, then card A — card B.
Proof [f either of A and B is finite, then the result follows from the theory
of linear algebra. Assume, therefore, that card A ^ Ko and card B > Ko.

76 3 Geometry of Hilbert Space
For a in A set Ba = {jS 6 B : fe,/,) # 0}. Since ea = ZpeBfe,/p)/p by
Corollary 3.28 and 1 = ||ej2 = E/^bK^,//))!2 by Theorem 3.25, it follows
that card £a^K0. Moreover, since fp = '£cieA(ea:,fl!)ea:, it follows that
(ei»fp) # 0 for some a in ,4. Therefore, i? = UaEA2?a and hence cardi?<
Xae/icard£a ^Sae/i^o = card/4 since cardv4^K0. From symmetry we
obtain the reverse inequality and hence card A = card B. ■
3.31 Definition If Jf is a Hilbert space, then the dimension of #C, denoted
, is the cardinality of any orthonormal basis for #C.
The dimension of a Hilbert space is well denned by the previous two
theorems. We now show that two Hilbert spaces #C and Jf of the same
dimension are isomorphic, that is, there exists an isometrical isomorphism
from 3tf onto Jf which preserves the inner product.
3.32 Theorem Two Hilbert spaces are isomorphic if and only if their di-
dimensions are equal.
Proof If 2?e and Jf are Hilbert spaces such that dim^f = dimJf, then
there exist orthonormal bases {ea}aEA and {fa}aeA for 2/C and Jf, respectively.
Define the map $ from 2/C to Jf such that for g in #C, we set <&g =
'EceA(ff^a)fa. Since g = ^e^(^.ea)ea by Corollary 3.28, it follows from
Theorem 3.25 that Zxexl(^eJ|2= Ml2- Therefore, <&g is well denned and
That <P is linear is obvious. Hence, <P is an isometrical isomorphism of #C
to Jf. Thus, <&#e is a closed subspace of Jf which contains {fa : a e A} and
by the definition of basis, must therefore be all of Jf. Lastly, since (g, g) =
|#||2= ||<%||2 = ((j?g, <frg) for g in 3V, it follows from the polarization
identity that (g, h) = {<&g, <S>h) for g and h in ^. ■
3.33 We conclude this chapter by computing the dimension of Examples
3.13, 3.15, 3.16, and 3.17. For C" it is clear that the n-tuples
form an orthonormal basis, and therefore dimC = n. Similarly, it is easy
to see that the functions {eB}™=0 in /2(Z+) defined by en(m)= I if n = m
and 0 otherwise, form an orthonormal subset of /2(Z+). Moreover, since/
in /2(Z+) can be written/= E™=0/(«)e«> it follows that {en}™=0 is an ortho-
normal basis for /2(Z+) and hence that dim[/2(Z+)] = Ko.

Notes 77
In the Hilbert space L2([0,1]), the set {e2mnx}ne7L is orthonormal since
ji e2mnx dx= i if n = o and 0 otherwise. Moreover, from the Stone-
Weierstrass theorem it follows that C([0,1]) is contained in the uniform
closure of the subspace spanned by the set {e2mnx: neZ} and hence C([0,1])
is contained in the smallest closed subspace of L2([0,1]) containing them.
For / in L2([0,1]) it follows from the Lebesgue dominated convergence
theorem that lim^^ \\f— fk\\2 = 0, where
rts (fix), \f(x)\^k,
l0, \f(x)\ > k.
Since C([0,1]) is dense in L1 ([0,1]) in the L'-norm, there exists for each k in
Z+, a function <pk in C([0,1]) such that |%(^)| ^k for x in [0,1] and
IIA-^lk^l/fc2-Hence
limsup/ \fk-<pk\2dx) ^limsupjfc \fk-<pk\dx)
k-*a> \JO ) fc->oo \ JO )
limsup - dx = 0.
Jo
k-
k
Thus, C([0,1]) is a dense subspace of L2([0,1]) and hence the smallest closed
subspace of L2([0,1]) containing the functions {e2mnx :nel} is L2([0,1]).
Therefore, {e2mnx}ne7L is an orthonormal basis for L2([0,1]). Hence,
dim{L2([0,1])} = K0 and therefore despite their apparent difference, /2(Z+)
and L2([0,1]) are isomorphic Hilbert spaces.
Similarly, since a change of variables shows that {xn}ne% is an ortho-
normal basis for L2(T), we see that {xn}nez+ is an orthonormal basis for
H2 and hence dim/f2 = Ko also.
We indicate in the exercises how to construct an example of a Hilbert
space for all dimensions.
Notes
The definition of a Hilbert space is due to von Neumann and he along
with Hilbert, Riesz, Stone, and others set forth the foundations of the subject.
An introduction to the geometry of Hilbert space can be found in many
textbooks on functional analysis and, in particular, in Stone [104], Halmos
[55], Riesz and Sz.-Nagy [92], and Akhieser and Glazman [2].

y
{/: A - C : £ |/(«)|2 < ooj.
78 3 Geometry of Hilbert Space
Exercises
3.1 Let A be a nonempty set and let
IHA) = {/: A
Show that /2(y4) is a Hilbert space with the pointwise operations and with
the inner product (f,g) = Y.a* a f(a) &(<*)■ Show that diml2(A) = cardv4.
3.2 Let i? be a normed linear space for which the conclusion of the parallel-
parallelogram law is valid. Show that an inner product can be defined on 2£ for
which the associated norm is the given norm.
3.3 Show that the completion of an inner product space is a Hilbert space.
3.4 Show that C([0,1]) is not a Hilbert space, that is, there is no inner
product on C([0,1]) for which the associated norm is the supremum norm.
3.5 Show that C([0,1]) is not homeomorphically isomorphic to a Hilbert
space.*
3.6 Complete the proof began in Section 3.14 that the space 3? denned
there is not complete.
3.7 Give an example of a finite dimensional space containing a closed
convex set which contains more than one point of smallest norm.*
3.8 Give an example of an infinite dimensional Banach space and a closed
convex set having no point of smallest norm.*
3.9 Let <p be a bounded linear functional on the subspace Jl of the Hilbert
space Jf. Show that there exists a unique extension of cp to Jf having .the
same norm.
3.10 Let 2tf and Jf be Hilbert spaces and Jf@Jf denote the algebraic
direct sum. Show that
«/*!,£!>, </*2,£2» = (hl,h2)+{kl,k2)
defines an inner product on 3fC ©Jf", that 3fC ©Jf~ is complete with respect to
this inner product, and that the subspaces Jf © {0} and {0} © Jf are closed
and orthogonal in 3fC © Jf.
3.11 Show that each vector of norm one is an extreme point of the unit ball
of a Hilbert space.
3.12 Show that the H>*-closure of the unit sphere in an infinite-dimensional
Hilbert space is the entire unit ball.

Exercises 79
3.13 Show that every orthonormal subset of the Hilbert space 3V is con-
contained in an orthonormal basis for Jf.
3.14 Show that if Jl is a closed subspace of the Hilbert space Jf, then
3.15 {Gram-Schmidt) Let {/„}"= t be a subset of the Hilbert space Jf
whose closed linear span is Jf. Set ex =/i/|/i|| and assuming {ek}k=1 to
have been denned, set
« \ / n
fn+i~ Z (fn+i>ek)ek
k=l
where en+1 is taken to be the zero vector if
n
fn+l = Z (fn+Uek)ek-
k=l
Show that {en}™= t is an orthonormal basis for Jf.
3.76 Show that L2([0,1]) has an orthonormal basis {en}%L0 such that en is
a polynomial of degree «.
3.77 Let if be a dense linear subspace of the separable Hilbert space Jf.
Show that <£ contains an orthonormal basis for jf. Consider the same
question for nonseparable Jf .*
3.18 Give an example of two closed subspace Jl and Jf of the Hilbert
space Jf for which the linear span
Jl + Jf = {/+ g :fe Jl, g e ^V}
fails to be closed.* (Hint: Take Jl to be the graph of an appropriately chosen
bounded linear transformation from Jf to Jf and Jf to be Jf~©{0}, where
3.79 Show that no Hilbert space has linear dimension Ko. (Hint: Use the
Baire category theorem.)
3.20 If Jif is an infinite-dimensional Hilbert space, then dimJf coincides
with the smallest cardinal of a dense subset of Jf.
3.27 Let Jif and Jf be Hilbert spaces and let Jf (x) jf denote the algebraic
tensor product of jf and Jf considered as linear spaces over C. Show that
(m n \ m n
l_lni{i9Ki> l_, "j Q9«J \ — I., I., Vlh"j )(KhKj )

80 3 Geometry of Hilbert Space
defines an inner product on Jf® Jf. Denote the completion of this inner
a
product space by 2ff ®Jf. Show that if {ea}aeA and {fp}p<sB are orthonormal
bases for W and Jf, respectively, then fc®/#}feB6/)xj is an orthonormal
basis for 2/e ® Jf.
3.22 Let (X,^,fi) be a measure space with ^ finite and ipbea bounded
linear functional on L1 (X). Show that the restriction of cp to L2 (Z) is a bounded
linear functional on l3(X) and hence there exists g in L2(JT) such that
<p(f) = ixfS d[x for /in L2(X). Show further that g is in ^(X) and hence
obtain the characterization of L1 {X)* as L^CX). (Neither the result obtained
in Chapter 1 nor the Radon-Nikodym theorem is to be used in this problem.)
3.23 (von Neumann) Let fi and v be positive finite measures on (X, S?)
such that v is absolutely continuous with respect to /i. Show that/->|x/d]u
is well denned and a bounded linear functional on L2 (ji + v). If cp is the function
in l}(n + \) satisfying \xfq> d(n + v) = $xfd(i> then (l-<p)fa is in l}(ji) and
v(E)= f!^
Je 9
for £ in if.
3.24 Interpret the results of Exercises 1.30 and 1.31 under the assumption
that SC is a Hilbert space.

4
Operators on Hithert
Space and C*-Algebras
4.1 Most of linear algebra involves the study of transformations between
linear spaces which preserve the linear structure, that is, linear transformations.
Such is also the case in the study of Hilbert spaces. In the remainder of the
book we shall be mainly concerned with bounded linear transformations
acting on Hilbert spaces. Despite the importance of certain classes of un-
unbounded linear transformations, we consider them only in the problems.
We begin by adopting the word operator to mean bounded linear trans-
transformation. The following proposition asserts the existence and uniqueness
of what we shall call the "adjoint operator."
4.2 Proposition If T is an operator on the Hilbert space Jf, then there
exists a unique operator S on Jf such that
(Tf,g) = (f,Sg) for/and g in jf.
Proof For a fixed g in 3f consider the functional <p defined <p(f) =
{Tf,g) for/in 3C. It is easy to verify that cp is a bounded linear functional
on Jf, and hence there exists by the Riesz representation theorem, a unique
h in 34? such that <p(f) = (f,h) for/in Jf. Define Sg = h.
Obviously S is linear and (Tf, g) = (/, Sg) for/and g in 2%. Setting/= Sg
81

82 4 Operators on Hubert Space and C*-Algebras
we obtain the inequality
\\Sg\\2 = (Sg,Sg) = (TSg,g) ^ \\T\\ \\Sgt y\\ for g in Jf.
Therefore, ||^S'|| ^ \\T\\ and 5*is an operator on 2/e.
To show that S is unique, suppose 5*0 is another operator on #f satisfying
(f,Sog) = (Tf,g) for/and g in 3f. Then (f,Sg~Sog) = 0 for/in 2/e which
implies Sg—Sog = 0. Hence S = So and the proof is complete. ■
4.3 Definition If T is an operator on the Hilbert space Jif, then the adjoint
of T, denoted T*, is the unique operator on 2/e satisfying (Tf,g) = (/ T*g)
for/and g in Jf.
The following proposition summarizes some of the properties of the
involution T-* T*. In many situations this involution plays a role analogous
to that of the conjugation of complex numbers.
4.4 Proposition If Jf is a Hilbert space, then:
A) T** = (T*)* = Tfor Tin £pf);
B) ||7l = ||r*||fornn£pf);
C) (aS+pT)* = aS* + pT* and (ST)* = T*S* for a, p in C and S, T
in £pf);
D) (J*) = G1)* for an invertible Tin £(Jf); and
E) ||r|2 = ||r*r|| for
Proof A) If/and g are in 2/e, then
= (^, T*f) = G^,/) = (/, Tg),
and hence T** = T.
B) In the proof of Proposition 4.2 we showed ||r**|| ^ |r*|| **\\T\\.
Combining this with A), we have ||r|| = || T*\\.
C) Compute.
D) Since T*^'1)* = {T~l T)* = 1= (TT'1)* = (T'^T* by C), it
follows that T* is invertible and (T*)'1 = (J)*.
E) Since \\T*T\\ ^ ||r*|| ||r|| = ||r||2 by B), we need verify only that
\\T*T\\ ^ ||r||2. Let {/J™=1 be a sequence of unit vectors in 2/e such that
fJ = || 71. Then we have
limsup|r*7/J| > limsup(r*7/n,/n) = lim ||7/J|2 2
which completes the proof.

The Adjoint Operator 83
4.5 Definition If T is an operator on the Hilbert space Jf, then the kernel
of T, denoted ker T, is the closed subspace {/e Jf : Tf= 0}, and the range
of T, denoted ran T, is the subspace {Tf:fe 3#>}.
4.6 Proposition If T is an operator on the Hilbert space #C, then ker T =
(ran T*I and ker T* = (ran TI.
Proof It is sufficient to prove the first relation in view of A) of the last
proposition. To that end, if / is in ker T, then (T*g,f) = (#, 7/) = 0 for
g in M', and hence/is orthogonal to ran T*. Conversely, if /is orthogonal to
ran T*, then (Tf, g) = (/, T*g) = 0 for g in JP, which implies 7/= 0. There-
Therefore, /is in ker 71 and the proof is complete. ■
We next derive useful criteria for the invertibility of an operator.
4.7 Definition An operator T on the Hilbert space #f is bounded below
if there exists e > 0 such that || Tf\ > e ||/|| for /in 2/e.
4.8 Proposition If T is an operator on the Hilbert space 2tf, then T is
invertible if and only if T is bounded below and has dense range.
Proof If T is invertible, then ran T = M' and hence is dense. Moreover,
|| Tf\\ > pL_ || T~J 7/| = pi^ ll/l for /in AT
and therefore T is bounded below.
Conversely, if T is bounded below, there exists e > 0 such that
\\Tf\\ > «11/11 for/in Jf. Hence, if {Tfn}™=1 is a Cauchy sequence in ranT,
then the inequality
Wfn-fJ <~
implies {/,}™=1 is also a Cauchy sequence. Hence, if /= linv,^/,, then
7/= lim^oo 7/n is in ran T and thus ran T is a closed subspace of Jf. If we
assume, in addition, that ran T is dense, then ran T = Jf. Since 7" being
bounded below obviously implies that T is one-to-one, the inverse trans-
transformation T~l is well denned. Moreover, if g = Tf, then
\\T-l9\\ = \\f\\<-\\Tf\\=-\\g\\
£ £
and hence J is bounded. ■

84 4 Operators on Hilbert Space and C*-Algebras
4.9 Corollary If T is an operator on the Hilbert space M' such that both
T and T* are bounded below, then T is invertible.
Proof If T* is bounded below, then ker T* = {0}. Since (ran TI =
ker T* = {0} by Proposition 4.6, we have (ran TI = {0}, which implies
clos[ranr]=(ranrI1 = {0}1 = 3tf by Corollary 3.22. Therefore, ranT
is dense in #P and the result follows from the theorem. ■
4.10 If T is an operator on the finite dimensional Hilbert space C" and
{ej}"= j is an orthonormal basis for C", then the action of T is given by the
matrix {fly}"j=i, where fly = (Tet, e}). The adjoint operator T* has the
matrix {blj}"tj=l, where bi} = a}i for i,j= 1,2,...,«.
The simplest operators on C" are those for which it is possible to choose
an orthonormal basis such that the corresponding matrix is diagonal, that is,
such that fly = 0 for i #_/. An operator can be shown to belong to this class
if and only if it commutes with its adjoint. In one direction, this result is
obvious and the other is the content of the so called "spectral theorem" for
matrices.
For operators on infinite dimensional Hilbert spaces such a theorem is no
longer valid. Hilbert showed, however, that a reformulation of this result
holds for operators on arbitrary Hilbert spaces. This "spectral theorem" is
the main theorem of this chapter.
We begin by defining the relevant classes of operators.
4.11 Definition If T is an operator on the Hilbert space Jf, then:
A) T is normal if TT* = T* T;
B) T is self-adjoint or hermitian if T = T*;
C) T is positive if (Tff) ^ 0 for /in W; and
D) T is unitary if T* T = TT* = /.
The following is a characterization of self-adjoint operators.
4.12 Proposition An operator T on the Hilbert space Jf is self-adjoint if
and only if (Tff) is real for/in 3V.
Proof If T is self-adjoint and/is in Jf, then
(Tff) = (/ T*f) = </, 7JT) = (WJ)
and hence (Tff) is real. If (Tff) is real for/in 2tf, then using Lemma 3.3,
we obtain for /and g in Jf that (Tf g) = (Tg,f) and hence T=T*. ■

Normal and Self-adjoint Operators 85
4.13 Corollary If P is a positive operator on the Hilbert space #e, then
P is self-adjoint.
Proof Obvious. ■
4.14 Proposition If T is an operator on the Hilbert space #e, then T* T
is a positive operator.
Proof For / in Jf we have (T*Tf,f) = \\Tf\\2 ^0, from which the
result follows. ■
When we speak of the spectrum of an operator T denned on the Hilbert
space Jf, we mean its spectrum when T is considered as an element of the
Banach algebra £(^f) and we use o(T) to denote it. On a finite-dimensional
space I is in the spectrum of T if and only if k is an eigenvalue for T. This
is no longer the case for operators on infinite-dimensional Hilbert spaces.
In linear algebra one shows that the eigenvalues of a hermitian matrix
are real. The generalization to hermitian operators takes the following form.
4.15 Proposition If T is a self-adjoint operator on the Hilbert space Jif,
then the spectrum of T is real. Furthermore, if T is a positive operator, then
the spectrum of T is nonnegative.
Proof If k = a + ifi with a, /? real and jS#O, then we must show that
T— k is invertible. The operator K = (T—a)/p is self-adjoint and T—k is
invertible if and only if K—i is invertible, since K—i — (T~k)/p. Therefore,
in view of Proposition 4.9, the result will follow once we show that the operators
K—i and (K—i)* = K+i are bounded below. However, for/in Jf, we have
/||2 = ((K±i)f(K±i)f) = \\Kff + i(Kf,f) ± i(f,Kf) + ll/l2
and hence the spectrum of a self-adjoint operator is real.
If we assume, in addition, that T is positive and k < 0, then
\\(T-k)ff = llTfll2 - 2k(Tff) + }} \ff > k2 H/ll2.
Since (T-k)* = (T-k), then T-k is invertible by Proposition 4.9 and the
proof is complete. ■
We consider now a special class of positive operators which form the
building blocks for the self-adjoint operators in a sense which will be made
clear in the spectral theorem.

86 4 Operators on Hilbert Space and C*-Algebras
4.16 Definition An operator P on the Hilbert space Jf is a projection if
P is idempotent (P2 = P) and self-adjoint.
The following construction gives a projection and, in fact, all projections
arise in this manner.
4.17 Definition Let Jl be a closed subspace of the Hilbert space Jf. Define
PM to be the mapping PJtf= g, where/= g+h with g in Jl and h in J(L.
4.18 Theorem If Jl is a closed subspace of Jf, then PM is a projection
having range Ji. Moreover, if P is a projection on Jf, then there exists a
closed subspace ^#(=ran/)) such that P = PM.
Proof First we prove that PM is an operator on Jf. If fuf2 are vectors
in ^f and At, A2 complex numbers, then/ = #t +/?t and/2 = g2 +h2, where
gt,g2 are in ^# and hl,h2 are in ^1. Moreover
where Ajg2 + A2g2 is in «# and (Ajhl +k2h2) is in ^1. By the uniqueness of
such a decomposition, we have
and hence PM is a linear transformation on Jf. Moreover, the inequality
ll^/ill2 = ll#il|2 ^ ll#il|2 + ll^il|2 = ll/il|2
shows that PM is bounded and has norm at most one. Therefore, P is an
operator on M'. Moreover, since
(Pm/i./i) = (Pi,P2+ h-,) = (g,,g2) = (ffi +h,,qi) = (ft,P,uf-i),
we see that PM is self-adjoint. Lastly, if/is in Ji, then/=/+0 is the required
decomposition of/and hence PMf=f. Since ran 7^= Jl, it follows that
PM2 = P^ and hence PM is idempotent. Therefore P^ is a projection with
range Ji.
Now suppose P is a projection on #C and set ,# = ran P. If {P/,}"=i is
a Cauchy sequence in Jf converging to g, then
# = lim Pfn = lim P2fn = P[lim P/J = Pg.
Thus # is in ^# and hence Jl is a closed subspace of jf. If g is in ^x, then
||Pg||2 = (Pg,Pg) = (g,P2g) = 0, since P2# is in ^#, and hence Pg = 0.
If/is in Jf, then/= Pg+h, where A is in ^x and hence PMf=Pg =
P2g + Ph = Pf. Therefore, P' = PM. ■

Multiplication Operators and Maximal Abelian Algebras 87
Many geometrical properties of closed subspaces can be expressed in
terms of the projections onto them.
4.19 Proposition If Jf is a Hilbert space, {Jt$l=v are closed subspaces
of Jf, and {Pt}t=i are the projections onto them, then Pt +P2-\ \-Pn = I
if and only if the subspaces {J<ft}"= t are pairwise orthogonal and span J^,
that is, if and only if each/in J^ has a unique representation/=ft +/2 h yfn,
where / is in Mt.
Proof If Pl + P2 -i \-Pn = I, then each / in M' has the representation
f=Pif+P2f+--+Pnf and hence the Jf, span 3f. Conversely, if the
{J/i}1=l span Jf and the sum Pl+P2-\ yPn ls a projection, then it
must be the identity operator. Thus, we are reduced to proving that
Pl+P2-i \-Pn is a projection if and only if the subspaces {J/f}"=l are
pairwise orthogonal, and for this it suffices to consider the case of two sub-
subspaces.
Therefore, suppose Pl and P2 are projections such that Pl +P2 is a
projection. For/in Jit, we have
)+(Pi r,Pif)+(p2f, p1f>+
= (Pi/,/)+(/, Pif)+(p2/./>+(
since /\/=/and thus 2||P2/|2 = 2(P2f,P2f) = 2(P2f,f) = 0. Hence,
/and therefore ^#t is orthogonal to Ji2.
Conversely, if Pl and P2 are projections such that the range of Pt is
orthogonal to the range of P2, then for/in Jf, we have
since P2Pyf= PiP2f= 0. ■
The proof shows that the sum of a finite number of projections onto
pairwise orthogonal subspaces is itself a projection.
We next consider some examples of normal operators other than those
defined by the diagonalizable matrices on a finite-dimensional Hilbert space.
4.20 Example Let (X, £f, n) be a probability space. For <p in L?(p) define
the mapping Mv on l3(p) such that M^/= (jo/for/in L2(^), where (p/denotes

88 4 Operators on Hilbert Space and C*-Algebras
the pointwise product, and let 9tfi = {M^ : q> e L°(ji)}. Obviously M^ is
linear and the inequality
X (£ ^ < IVB.II/II2
shows that Mv is bounded. Moreover, if En is the set
{xeX:\cp(x)\ > MU-l/n},
then
and hence |MJ| = flipH^. For/and g in L2(^u) and <p in L™^) we have
= f (<p/H ^ = f /(^) ^ = {f, M9g),
Jx Jx
which implies M^* = M^. Lastly, the mapping defined by *P(<p) = M^ from
L™^) to 9JJ is obviously linear and multiplicative. Therefore, *¥ is a *-iso-
metrical isomorphism of Ju°(n) onto 9I. (The terminology *- is used to denote
the fact that conjugation in U°(fi) is transformed by ¥ to adjunction in
Since L™^) is commutative, it follows that M^ commutes with M^* and
hence is a normal operator. For <p in L°(pi) the operator M^ is self-adjoint
if and only if Mv = A/w* and hence if and only if tp = ^ or tp is real. Since
M^2 = ^2, the operator M^ is idempotent if and only if <p2 = qj or <p is a
characteristic function. Therefore, the self-adjoint operators in 9I are the
M^, for which q> is real, and the projections are the M^ for which q> is a
characteristic function.
Let us now consider the spectrum of the operator Mv. 1 f cp — k is invertible
in n°(fi), then M^-A=M(^_A) is invertible in £(L2(^j)) with inverse
A/(o)_A)-,, and hence a^M^) d%(cp). To assert the converse inclusion we
need to know that if M^ — X is invertible, then its inverse is in 9JJ. There are
at least two different ways of showing this which reflect two important
properties possessed by 9Ji.
4.21 Definition If 2ff is a Hilbert space, then a subalgebra STC of
is said to be maximal abelian if it is commutative and is not properly con-
contained in any commutative subalgebra of

The Bilateral Shift Operator 89
4.22 Proposition The algebra £R = {M^ : cp e Lx(n)} is maximal abelian.
Proof Let T be an operator on L2 (/i) that commutes with 9tfi. If we set
x\j = 7T, then \j/ is in L2(^) and Tcp = TM^I = M^Tl = tj/cp for <p in L™^).
Moreover, if En is the set {xel: \tj/(x)\ ^ ||T|| + 1/n}, then
J|2 5* |77£J2 = ||«/,/£J2 =
Therefore, \\IEn\\2 = 0 and hence the set {x e X: |^(x)| > || T\\} has measure
zero. Thus ^ is in L00^) and T(p = M^,(p for <p in L°(^). Since C(X) is
dense in L2(^) as proved in Section 3.33 and C(X) a E°(ix), it follows that
T'= Mj, is in SR. Therefore, 5R is maximal abelian. |
4.23 Proposition If Jf is a Hilbert space and 21 is a maximal abelian sub-
algebra of £pf), then o-(r) = or^r) for Tin S&.
Proof Clearly a{T) a am(T) for T in 9T. If A is not in c(T), then
(r— X)~1 exists. Since G"— X)~l commutes with S& and 91 is maximal abelian,
we have (T— X)~x is in 91. Therefore A is not in <rm(T) and hence a^{T) =
a(T). M
4.24 Corollary If <p is in L°°(/4 then o-(M^)
Since SR = {M^,: ^ e L™^)} is maximal abelian, it follows from
the previous result that 0-^G") = c{T). Since £R and Lw(n) are isomorphic
we have om{T) = oLaill){T) and Lemma 2.63 completes the proof. ■
We give the alternate approach after giving an example of a subalgebra
1 of £pf) and an operator Tin 91 for which crm(T) # a{T).
4.25 Example Let I2 (Z) denote the Hilbert space consisting of the complex
functions/on Z such that ££=-«, |/(«)|2 < oo. Define U on /2(Z) such that
(Uf)(ri) =f{n- 1) for/in /2(Z). The operator U is called the bilateral shift.
It is clearly linear and the identity
- I \(Uf)(n)\2= I |/(«-l)|2 = ||/||2
n= — co n= — co
for/ in /2(Z) shows that U preserves the norm and, in particular, is bounded.

90 4 Operators on Hilbert Space and C*-Algebras
If we define A on I2 (I) such that (Af)(n) =/(n+l) for/in /2(Z), we have
f,g)= f (Uf)(n)g~W= £ f(n-\)g!ri)
Therefore, A = U* and a computation yields UU* = U*U = /or C/ = U*.
Thus C/ is a unitary operator.
From Proposition 2.28 we have o{U) a D and o{U~l) = a{U*) <=. D.
If X is in D, 0< |A| < 1, then (U-^U'1 = X((l/X)-U~l). Since I/A is
not in 0, the operator X((l/X)—U~1) is invertible and hence so is U—X.
Therefore o(U) is contained in T. For fixed 0 in [0,2k] and n in Z+, let/,
be the vector in I2(l) denned by /„(£) = Bn + l)/2 e~M for |fc| < n and 0
otherwise. A straightforward calculation shows that /„ is a unit vector and
that hmn^oo(C/-e'e)/n = 0. Therefore U-ew is not bounded below, and
hence e1'" is in a(U). Thus a(U) = T.
Let 91 + denote the smallest closed subalgebra of fi(/2(Z)) containing
/ and U. Then 9l+ is the closure in the uniform norm of the collection of
polynomials in U, that is,
U =clos{£ anU": anecj.
If Jl denotes the closed subspace
{/e I2(I) :f(k) = 0 for A; < 0}
of/2(Z), then UJi a M, which implies p{U)Ji <= ^# for each polynomial p.
If for A in 9l+ we choose a sequence of polynomials {pn}^°= t such that
linin-^oo^Ct/) = A, then 4f= limn_xpn{U)f implies v4/in Ji, if/is, since
each pn(U)f is in ^. Therefore AM <= M for y4 in S&+. We claim that C/
is not in 9l+. If e0 is the vector in /2(Z) denned to be 1 at 0 and 0 otherwise,
then
(l7-1eo)(-l) = (£7*eo)(-l)=l/0.
Hence V~xJi <$. Ji which implies that C/-1 is not in 9l+. Therefore, 0 is
in crm+(U), implying crm+(U) # <r(U).
The algebra 91 + can be shown to be isometrically isomorphic to the disk
algebra denned in Section 2.50 with U corresponding to %, and hence
ffauC^) = O. We return to this later.
The algebra 9f+ is not maximal abelian, since it is contained in the
obviously larger commutative algebra {M^ : cp eLw(J)}. Equally important

C*-Algebras 91
is that 91+ is not a self-adjoint subalgebra of fi(/2(Z)), since, as we will soon
establish, such algebras also have the property of preserving the spectrum.
We consider the abstract analog of a self-adjoint subalgebra.
4.26 Definition If 9t is a Banach algebra, then an involution on 91 is a
mapping T-*T* which satisfies:
(i) r** = rfor Tin 9t;
(ii) (aS + fiT)* = dS* +pT* for S, Tin 91 and a, fi in C; and
(iii) (ST)* = T*S* for S and T in 91.
If, in addition, || T* T\\ = || T\\2 for Tin 9T, then 91 is a C*-algebra.
A closed self-adjoint subalgebra of £pf) for Jif a Hilbert space is a
C*-algebra in view of Proposition 4.4. Every C*-algebra can, in fact, be
shown to be isometrically isomorphic to such an algebra (see Exercise 5.26).
All of the various classes of operators whose definitions are based on the
adjoint can be extended to a C*-algebra; for example, an element T in a
C*-algebra is said to be self-adjoint if T= T*, normal if 7T* = T* T, and
unitary if T* T = TT* = I.
We now give a proof of Proposition 4.15 which is valid for C*-algebras.
Our previous proof made essential use of the fact that we were dealing with
operators denned on a Hilbert space.
4.27 Theorem In a C*-algebra a self-adjoint element has real spectrum.
Proof Observe first that if 7" is an element of the C*-algebra 91, then the
inequality
implies that ||r|| ^ ||r*|| and hence ||r| = ||r*||, since T** = T. Thus
the involution on a C*-algebra is an isometry.
Now let H be a self-adjoint element of 91 and set U = exp iH. Then from
the fact that H is self-adjoint and the definition of the exponential function,
it follows that U* = exp(i//)* = exp(— iH). Moreover, from Lemma 2.12
we have
UU* = exp(i7/)exp(-/#) = exv(iH-iH) = 1= exp(-ii/)exp(/i/) = U*U
and hence U is unitary. Moreover, since 1 = ||/|| = ||C/*C/|| = ||t/||2 we see
that \\U\\ = \\U*\\ = HE/1| = 1, and therefore a(U) is contained in T. Since
cr(U) = exp(io(H)) by Corollary 2.37, we see that the spectrum of H must
be real. ■

92 4 Operators on Hilbert Space and C*-Algebras
4.28 Theorem If © is a C*-algebra, 91 is a closed self-adjoint subalgebra
of 33, and T is an element of 9t, then am(T) = om(T).
Proof Since ow(T) contains om(T), it is sufficient to show that if T— X
is invertible in 93, then the inverse (T— X)~x is in 31. We can assume X = 0
without loss of generality. Thus, T is invertible in 33, and therefore T* T
is a self-adjoint element of 91 which is invertible in 33. Since a^{T* T) is
real by the previous theorem, we see that crm(T* T) = o^{T* T) by Corollary
2.55. Thus, T* T is invertible in 91 and therefore
Y~1 — (f~1 (T*)~l) T* = (T* T)~1 T*
is in 91 and the proof is complete. ■
We are now in a position to obtain a form of the spectral theorem for
normal operators. We use it to obtain a "functional calculus" for continuous
functions as well as to prove many elementary results about normal operators.
Our approach is based on the following characterization of commutative
C*-algebras.
4.29 Theorem (Gelfand-Naimark) If 91 is a commutative C*-algebra and
M is the maximal ideal space of 91, then the Gelfand map is a *-isometrical
isomorphism of 91 onto C(M).
Proof If r denotes the Gelfand map, then we must show that F(T) =
T{T*) and that HHTlIU = ||r||. The fact that T is onto will then follow from
the Stone-Weierstrass theorem.
If T is in 91, then H=i(T+T*) and K=(T-T*)/2i are self-adjoint
operators in 91 such that T= H+iK and T* = H-iK. Since the sets c(H)
and o{K) are contained in R, by Theorem 4.27, the functions T(H) and
are real valued by Corollary 2.36. Therefore,
T(T) = r(H) + ir(K) = r(H) - iT(K) = r(H-iK) = T(T*),
and hence F is a *-map.
To show that F is an isometry, let T be an operator in 9t. Using Definition
4.26, Corollary 2.36, Theorem 2.38, and the fact that T* T is self-adjoint,
we have
||7l2 = || FT || = \\(T*TJk\\l/2k = li
k~-*

The Functional Calculus 93
Therefore T is an isometry and hence a *-isometrical isomorphism onto
C(M). M
If 91 is a commutative C* algebra and Jis in 91, then Tis normal, since
T* is also in 9f and the operators in 9t commute. On the other hand, a normal
operator generates a commutative C*-algebra.
4.30 Theorem (Spectral Theorem) If Jf is a Hilbert space and Tis a normal
operator on Jif, then the C*-algebra GT generated by T is commutative.
Moreover, the maximal ideal space of GT is homeomorphic to o(T), and
hence the Gelfand map is a *-isometrical isomorphism of GT onto C(cr(T)).
Proof Since T and T* commute, the collection of all polynomials in T
and T* form a commutative self-adjoint subalgebra of fi(^f) which must be
contained in the C*-algebra generated by T. It is easily verified that the
closure of this collection is a commutative C*-algebra and hence must be
GT. Therefore GT is commutative.
To show that the maximal ideal space M of GT is homeomorphic to cr(T),
define \j/ from M to a{T) by \j/{q>) = r(T)(cp). Since the range of r(T) is
cr(T) by Corollary 2.36, tj/ is well defined and onto. If <p1 and cp2 are elements
of M such that ^(.Vi) = <H<P2)> then
T{T) (<Pi) = nT)(tp2), 9i(T) = (p2(T),
and
<px(T*) = T{T*){<px) = r(D(,?1) = T{T){tp2) = T{T*){<p2) =
and hence qjj and <p2 agree on all polynomials in T and T*. Since this col-
collection of operators is dense in GT, it follows that (?\—<p2 and therefore
\j/ is one-to-one. Lastly, if {(pJaeA is a net in Af such that limag/, <pa = (p,
then
and hence ij/ is continuous. Since M and o-G") are compact Hausdorif spaces,
then ^ is a homeomorphism and the proof is complete. ■
4.31 Functional Calculus If 71s an operator, then a rudimentary functional
calculus for Jean be defined as follows: for the polynomialp(z) — Y%=o<*n^'->
define p(T) = Z^=oa« T"- The mappingp -*p(T) is a homomorphism from
the algebra of polynomials to the algebra of operators. If Jf is finite
dimensional, then one can base the analysis of T on this functional calculus.
In particular, the kernel of this mapping, that is, {p(z) :p(T) =0}, is a

94 4 Operators on Hilbert Space and C*-Algebras
nonzero principal ideal in the algebra of all polynomials and the generator
of this ideal is the minimum polynomial for T. If Jti? is not finite dimensional,
then this functional calculus may yield little information.
The extension of this map to larger algebras of functions (see Exercise
2.18) is a problem of considerable importance in operator theory.
If T is a normal operator on the Hilbert space Jf, then the Gelfand map
establishes a *-isometrical isomorphism between C(cr(T)) and GT. For cp
in C(cr(T)), we define (p(T) = F~1(p. It is clear that if q> is a polynomial in z,
then this definition agrees with the preceding one. Moreover, if A is an
operator on Jf that commutes with T and T*, then A must commute with
every operator in GT and hence, in particular, with (p(T) for each <p in
C(a(T)).
In the remainder of this chapter we shall obtain certain results about
operators using this calculus and then extend the functional calculus to a
larger class of functions.
4.32 Corollary If T is a normal operator on the Hilbert space 3fC, then
T is positive if and only if a{T) is nonnegative and self-adjoint if and only
if c{T) is real.
Proof By Proposition 4.15, the spectrum of T is nonnegative if T is
positive. Conversely, if 71s normal, o(T) is nonnegative, and T is the Gelfand
transform from GT to C(a(T)), then T(r) $s 0. Thus there exists a con-
continuous function q> on o(T) such that Y(T) = |<p|2. Then
T= Lv(.TJl<p(T)} = <p(T)*<p(T)
and hence Tis positive by Proposition 4.14.
If T is self-adjoint, then the spectrum of T is real by Proposition 4.15.
If 71s normal and has real spectrum, then \j/ = T(T) is a real-valued function
by Corollary 2.36, and hence T= ij/(T) = ij/(T) = ty{T)* = T*. Therefore,
T is self-adjoint. ■
The preceding proposition is false without the assumption that T is
normal, that is, there exist operators with spectrum consisting of just zero
which are not self-adjoint.
The second half of the preceding proposition is valid in a C*-algebra,
while the first half allows us to define a positive element of a C*-algebra to
be a normal element with nonnegative spectrum.
We now show the existence and uniqueness of the square root of a positive
operator.

The Square Root of Positive Operators 95
4.33 Proposition If P is a positive operator on the Hilbert space Jf, then
there exists a unique positive operator Q such that Q2 = P. Moreover,
Q commutes with every operator which commutes with P.
Proof Since the spectrum cr(P) is positive, the square root function .
is continuous on o(P). Therefore yfP is a well-defined operator on Jf that
is positive by Corollary 4.32, since o(yfP) = yfo(P)- Moreover, (y/PJ = P
by the definition of the functional calculus, and yfp commutes with every
operator which commutes with P by Section 4.31. It remains only to show
that y/P is unique.
Suppose Q is a positive operator on Jf satisfying Q2 = P. Since QP =
QQ2 = Q2Q = PQ, it follows from the remarks in Section 4.31 that the
C*-algebra 91 generated by P, yjP, and Q is commutative. If T denotes the
Gelfand transform of 21 onto C(Mn), then T(yfp) and T((Q) are non-
negative functions by Proposition 2.36, while tQTJ = T(P) = JT@2,
since T is a homomorphism. Thus r(,/P) = T{Q) implying Q = ^JP and
hence the uniqueness of the positive square root. ■
4.34 Corollary If T is an operator on 3V, then T is positive if and only if
there exists an operator S on Jff such that T= S*S.
Proof If T is positive, take S = yjY. If T= S*S, then Proposition 4.14
yields that T is positive. ■
Every complex number can be written as the product of a nonnegative
number and a number of modulus one. A polar form for linear transformations
on C" persists in which a positive operator is one factor and a unitary operator
the other. For operators on an infinite-dimensional Hilbert space, a similar
result is valid and the representation obtained is, under suitable hypotheses,
unique. Before proving this result we need to introduce the notion of a partial
isometry.
4.35 Definition An operator V on a Hilbert space 3tf is a partial isometry
if \Vf\ = Il/H for/orthogonal to the kernel of V; if, in addition, the kernel
of Fis {0}, then V is an isometry. The initial space of a partial isometry is the
closed subspace orthogonal to the kernel.
On a finite-dimensional space every isometry is, in fact a unitary operator.
However, on an infinite-dimensional Hilbert space this is no longer the case.

96 4 Operators on Hilbert Space and C*-Algebras
Let us consider an important example of this which is related to the bilateral
shift.
4.36 Example Define the operator U+ on /2(Z+) by (C/+/)(n) =
/(«— 1) for n > 0 and 0 otherwise. The operator is called the unilateral shift
and an easy calculation shows that U+ is an isometry. Moreover, since the
function e0 denned to be 1 at 0 and 0 otherwise is orthogonal to the range
of U+, we see that U+ is not unitary. A straightforward verification shows
that the adjoint U+* is denned by (£/+ */)(«) =/(«+1).
Let us next consider the spectrum of U+. Since U+ is a contraction,
that is, \\U+\\ = 1, we have o(U+) a D. Moreover, for z in D the function
fz defined by fz(n) = z" is in /2(Z+) and U+*fz = zfz. Thus z is in o(U+)
and hence a(V+) = D.
The question of whether a partial isometry exists with given subspaces
for initial space and range depends only on the dimension, as the following
result shows.
4.37 Proposition If Jl and Jf are closed subspaces of the Hilbert space Jf
such that dim.J' = dim Jf, then there exists a partial isometry V with initial
space Jl and range Jf.
Proof If Ji and Jf have the same dimension, then there exist ortho-
normal bases {eJaeA and {fa}aeA for Jl and Jf with the same index set.
Define an operator F on Jf as follows: for g in Jf write g — h+YiaeAXaea
with /*±«# and set Vg = Ym^aKL- Then the kernel of V is ^x and
ll^ll = ll#ll f°r 9 m -^- Thus, F is a partial isometry with initial space Ji,
and range Jf. ■
We next consider a useful characterization of partial isometries which
allows us to define partial isometries in a C*-algebra.
4.38 Proposition Let V be an operator on the Hilbert space Jf. The follow-
following are equivalent:
A) V is a partial isometry;
B) V* is a partial isometry;
C) VV* is a projection; and
D) V* V is a projection.
Moreover, if F is a partial isometry, then VV* is the projection onto the
range of V, while F*Fis the projection onto the initial space.

The Polar Decomposition 97
Proof Since a partial isometry V is a contraction, we have for / in Jf
that
((/-F*F)/,/) = (/,/) - (V*Vf,f) = Il/H2 - ||F/||2 5* 0.
Thus /—F*Fis a positive operator. Now if/is orthogonal to kerF, then
\\Vf\\ = H/ll which implies that ((/-F*F)//) = 0. Since \\{I-V*Vf-f\2 =
((/-F*F)//) = 0, we have G-F*F)/=0 or V*Vf=f. Therefore, F*F
is the projection onto the initial space of V.
Conversely, if V*V is a projection and /is orthogonal to ker (F*F),
then V*Vf=f Therefore,
IIF/II2 = (V*Vff) = (//) = Il/H2,
and hence F preserves the norm on ker (F*F)X. Moreover, if V*Vf=0,
then 0 = (V*Vff) = \Vf\\2 and consequently ker(F*F) = kerF. Therefore,
F is a partial isometry, and thus A) and D) are equivalent. Reversing the
roles of V and F*, we see that B) and C) are equivalent. Moreover, if F* F
is a projection, then (VV*J = V(V*V)V* = VV*, since V{V*V) = F.
Therefore, FF* is a projection, which completes the proof. ■
We now obtain the polar decomposition for an operator.
4.39 Theorem If T is an operator on the Hilbert space Jf, then there
exists a positive operator P and a partial isometry F such that T = VP. More-
Moreover, F and P are unique if ker P = ker F.
Proof If we set P = (T* T)\ then
\\pf\\2 = (/y,/>/) = (P2//) = (r* r//) = || iff for /in w.
Thus, if we define F on ran/1 such that VPf— Tf, then F is well defined and
is, in fact, isometric. Hence, F can be extended uniquely to an isometry
from closfranP] to Jf. If we further extend F to ^f by defining it to be the
zero operator on [ran/1]1, then the extended operator Fis a partial isometry
satisfying T—VP and ker F = [ran P~\L = ker P by Proposition 4.6.
We next consider uniqueness. Suppose T = WQ, where W is a partial
isometry, Q is a positive operator, and kerW= kerQ, Then P2 = T*T =
QW*WQ = Q2, since W*Wis the projection onto
[kerfT]1 = [keriQ]1 = clos[rang]-
by Propositions 4.38 and 4.6. Thus, by the uniqueness of the square root,
Proposition 4.33, we have P= Q and hence WP=VP. Therefore, W= V

98 4 Operators on Hilbert Space and C*-Algebras
on ran P. But
[ran/*]1 = kerP = kexW = kerF,
and hence W—V on [ran/1]1. Therefore, V—W and the proof is
complete. ■
Although the positive operator will be in every closed self-adjoint sub-
algebra of £pf) which contains T, the same is not true of the partial isometry.
Consider, for example, the operator T = M^My in fi(L2(T)), where <p is a
continuous nonnegative function on T while \jj has modulus one, is not con-
continuous but the product <p\jj is.
In many instances a polar form in which the order of the factors is reversed
is useful.
4.40 Corollary If T is an operator on the Hilbert space Jf, then there
exists a positive operator Q and a partial isometry W such that T — QW.
Moreover, Wand Q are unique if ranW = [kerg]1.
Proof From the theorem we obtain a partial isometry V and a positive
operator P such that T* — VP. Taking adjoints we have T — PV*, which is
the form we desire with W—V* and Q — P. Moreover, the uniqueness also
follows from the theorem since ranW = [kerg]1 if and only if
kerF = kerPF* = [ranW]1 = [kerig]11 = ker P. ■
It T is a normal operator on a finite-dimensional Hilbert space, then the
subspace spanned by the eigenvectors belonging to a certain eigenvalue
reduces the operator, and these subspaces can be used to put the operator in
diagonal form. If T is a not necessarily normal operator still on a finite-
dimensional space, then the appropriate subspaces to consider are those
spanned by the generalized eigenvectors belonging to an eigenvalue. These
subspaces do not, in general, reduce the operator but are only invariant for it.
Although no analogous structure theory exists for operators on an infinite-
dimensional Hilbert space, the notions of invariant and reducing subspace
remain important.
4.41 Definition If T is an operator on the Hilbert space 3V and Ji is a
closed subspace of Jf, then Jl is an invariant subspace for T if TM <= M
and a reducing subspace if, in addition, T(J?1) a JiL.
We begin with the following elementary facts.

The Polar Decomposition 99
4.42 Proposition If T is an operator on Jf, Ji is a closed subspace of
Jf, and P^ is the projection onto Ji, then ^# is an invariant subspace for T
if and only if PM TPM = 77^ if and only if JiL is an invariant subspace for
T*; further, Ji is a reducing subspace for 71 if and only if PM T= TPM if
and only if Ji is an invariant subspace for both T and T*.
Proof If Ji is invariant for T, then for /in jf, we have TPMfm Ji
and hence PM TPMf= TPMf\ thus PM TPM = TPM, Conversely, if PM TPM =
77^, then for/in Ji, we have Tf= TPMj = P^ 77^/= P* 7f, and hence
7/ is in Ji. Therefore, TJi cz Ji and ^# is invariant for T. Further, since
/— PM is the projection onto JiL and the identity
T*(I~PM) = {1-PM)T*{I-PM)
is equivalent to P^ T* = P^ T*PM, we see that ^.#x is invariant for T* if
and only if Ji is invariant for T. Finally, if Ji reduces T, then PMT —
PM TPM — TPM by the preceding result, which completes the proof. ■
4.43 In the remainder of this chapter we want to extend the functional
calculus obtained in Section 4.31 for continuous functions on the spectrum
to a larger algebra of functions. This larger algebra of functions is related
to the algebra of bounded Borel functions on the spectrum.
Before beginning let us give some consideration to the uses of a functional
calculus and why we might be interested in extending it to a larger algebra of
functions. Some of the details in this discussion will be omitted.
Suppose T is a normal operator on the Hilbert space Jf with finite
spectrum cr(T) = {XX,X2, •■■,XN} and let <p -* cp(T) be the functional calculus
defined for <p in C(cr(T)). If A, is a point in the spectrum, then the character-
characteristic function 7Ui) is continuous on cr(T) and hence in C{a{T)). If we let
Et denote the operator I{K){T), then it follows from the fact that mapping
cp -* (p(T) is a *-isomorphism from C(a(T)) onto GT, that each Et is a
projection and that Ex+E2-\ \-EN = I. If Ji, denotes the range of E,,
then the {Ji,}"=l are pairwise orthogonal, their linear span is Jif and Ji,
reduces T by the preceding proposition. Moreover, since
1=1
we see that T acts on each Ji, as multiplication by A,. Thus the space Jf
decomposes into a finite orthogonal direct sum such that T is multiplication
by a scalar on each direct summand. Thus the functional calculus has enabled
us to diagonalize Tin the case where the spectrum of Tis finite.

100 4 Operators on Hilbert Space and C*-Algebra
If the spectrum of T is not discrete, but is totally disconnected, then a
slight modification of the preceding argument shows that Jf can be de-
decomposed for such a T into a finite orthogonal direct sum of reducing
subspaces for 7" such that the action of Ton each direct summand is approxi-
approximately (in the sense of the norm) multiplication by a scalar. Thus in this case
T can be approximated by diagonal operators.
If the spectrum of 7" is connected, then this approach fails, since C(a(T))
contains no nontrivial characteristic functions. Hence we seek to enlarge
the functional calculus to an algebra of functions generated by its character-
characteristic functions. We do this by considering the Gelfand transform on a larger
commutative self-adjoint subalgebra of £pf). This algebra will be obtained
as the closure of GT in a weaker topology. Hence we begin by considering
certain weaker topologies on £pf).
4.44 Definition Let jf be a Hilbert space and £pf) be the algebra of
operators on Jif. The weak operator topology is the weak topology denned
by the collection of functions T-* (Tf,g) from £pf) to C for/and g in Jf.
The strong operator topology on £(^f) is the weak topology denned by the
collection of functions T-* Tf from £(^f) to Jf for /in Jf.
Thus a net of operators {TJaeA converges to T in the weak [strong]
operator topology if
]im(Tj,g) = (Tf,g) [lim7;/= 7JT] for every fandginJf.
txEA heA
Clearly, the weak operator topology is weaker than the strong operator
topology which is weaker than the uniform topology. We shall indicate
examples in the problems to show that these topologies are all distinct.
The continuity of addition, multiplication, and adjunction in the weak
operator topology is considered in the following lemma. We leave the corre-
corresponding questions for the strong operator topology to the exercises.
4.45 Lemma If Jf is a Hilbert space and A and B are in £(Jf), then the
functions:
A) a(S, T) = S+ T from £(Jf) x £(^f) to
B) P(T) = AT from £pf) to £(Jf),
C) y(T)=TB from £(^f) to £pf), and
D) S(T)=T* from fl(jf) to £pf),
are continuous in the weak operator topology.

W*-Algebras 101
Proof Compute. ■
The enlarged functional calculus will be based on the closure of the
C*-algebra GT in the weak operator topology.
4.46 Definition If Jf is a Hilbert space, then a subset 91 of fi(^f) is said
to be a W*-algebra on Jf if 91 is a C*-algebra which is closed in the weak
operator topology.
The reader should note that a W*-algebra is a C*-subalgebra of
which is weakly closed. In particular, a W*-algebra is an algebra of operators.
Moreover, if $ is a *-isometrical isomorphism from the W*-algebra 91
contained in fi(Jf) to the C*-algebra 33 contained in fi(Jf), then it doqs
not follow that 23 is weakly closed in fi(Jf). We shall not consider such
questions further and refer the reader to [27] or [28].
The following proposition shows one method of obtaining W*-algebras.
4.47 Proposition If M' is a Hilbert space and SEW is a self-adjoint sub-
algebra of £pf), then the closure 91 of SEW in the weak operator topology is
a W*-algebra. Moreover, 91 is commutative if SEW is.
Proof That the closure of SEW is a W*-algebra follows immediately from
Lemma 4.45. Moreover, assume that SEW is commutative and let {SJaeA
and {Tp}p&B be nets of operators in SEW which converge in the weak operator
topology to <S and T, respectively. Then for/and g in M' and /? in B, we have
(ST,f,g) = Km(SaTtf,g) = lim(TpSafg) = (TpSf,g).
(EA eA
(KEA
Therefore, STp — TpS for each jS in B and a similar argument establishes
ST = TS. Hence, 91 is commutative if SEW is. ■
4.48 Corollary If Jf is a Hilbert space and T is a normal operator on Jf,
then the W*-algebra 2CT generated by T is commutative. Moreover, if AT
is the maximal idea space of 2CT, then the Gelfand map is a *-isometrical
isomorphism of 2CT onto C(AT).
Proof This follows immediately from the preceding proposition along
with Theorem 4.29. ■
4.49 If T is a normal operator on the separable Hilbert space Jif with
spectrum A contained in C, then we want to show that there is a unique Lw

102^ 4 Operators on Hilbert Space and C*-Algebras
space on A and a unique *-isometrical isomorphism T* : 2CT -> L00 which
extends the functional calculus r of Section 4.31, that is, such that the
accompanying diagram commutes, where the vertical arrows denote inclusion
■C(A)
mT—> u°
maps. Thus, the functional calculus for T can be extended to 2CT and
We begin with some measure theoretic preliminaries concerning the
following illustrative example. Let A be a compact subset of the complex
plane and v be a finite positive regular Borel measure on A with support A.
(The latter condition is equivalent to the assumption - that the inclusion
mapping of C(A) into L°°(v) is an isometrical isomorphism.) Recall that for
each <p in L°°(v) we define Mv to be the multiplication operator denned on
L2(v) by Mipf= <pf and that the mapping q>-± M^ is a *-isometrical iso-
isomorphism from L°°(v) into fi(L2(v)). Thus we can identify the elements of
L°°(v) with operators in £(L2(v)).
The following propositions give several important relationships between
v, C(A), L°°(v), and fi(L2(v)). The first completes the presentation in Section
4.20.
4.50 Proposition If (X, £f, p) is a probability space, then Lw(ji) is a maximal
abelian W*-algebra in Z(L2(ji)).
Proof In view of Section 4.19, only the fact that L°°(ju) is weakly closed
remains to be proved and that follows from Proposition 4.47, since the weak
closure is commutative and hence must coincide with L°°(ju). ■
The next result identifies the weak operator topology on Lw(ji) as a
familiar one.
4.51 Proposition If (X, £f, n) is a probability space, then the weak oper-
operator topology and the H>*-topology coincide on L™(ju).
Proof We first recall that a function / on X is in L1 (ji) if and only if it
can be written in the form/= gh, where g and h are in I?(p).
Therefore a net {(pa}aEA in LK(ji) converges in the w*-topology to <p if

W*-Algebras 103
and only if
lim (pxfdp = (pfdp
aeA JX JX
if and only if
lim(M^#,/z) = lim tpagh dp = cpgh dp = (M^g,^
aeA oleA JX JX
and therefore if and only if the net {MvJaeA converges to M^ in the weak
operator topology. ■
The next proposition shows that the W*-algebra generated by multipli-
multiplication by z on L2(v) is Lw(v).
4.52 Proposition If X is a compact Hausdorff space and /lisa finite positive
regular Borel measure on A', then the unit ball of C(X) is u>*-dense in the
unit ball of Lw(p).
Proof A simple step function ^ in the unit ball of LK(p) has the form
\jj = Xr=ta,/£,, where |a,| ^ I for i—\,2,...,n, the {Et}"=1 are pairwise
disjoint, and \J"=l El = X. For i = \,2,...,n, let K, be a compact subset of
£,. By the Tietze extension theorem there exists q> in C(X) such that
|| <p || „ ^ 1 and <p(x) = a, for jcin Kt. Then for /in L}(p), we have
r fto-w dp ^ t \fw<p~^\dp
JX JX
= t f \f\\<P~^\dp<2t f \f\dp.
i=t JE,\K, i=l JB,\fC,
Because p is regular, for fu...,fm in l}(p) and e > 0, there exist compact
sets K, c E, such that
1.
\fj\dp <— for j= 1,2,...,m.
This completes the proof. ■
4.53 Corollary If X is a compact Hausdorff space and p is a finite positive
regular Borel measure on X, then C(X) is n*-dense in l^(p)-
Proof Immediate. ■

104 4 Operators on Hilbert Space andC*-Algebras
We now consider the measure theoretic aspects of the uniqueness problem.
We begin by recalling a definition.
4.54 Definition Two positive measures v1 and v2 defined on a sigma
algebra (X, £f) are mutually absolutely continuous, denoted vt ~ v2, if vt
is absolutely continuous with respect to v2 and v2 is absolutely continuous
with respect to v,.
4.55 Theorem If vl and v2 are finite positive regular Borel measures on
the compact metric space X and there exists a *-isometrical isomorphism
$ : r0^!)-*!^) which is the identity on C(X), then vt ~ v2, LK(vt) =
L00^), and 0 is the identity.
Proof If £ is a Borel set in X, then $(/£) is an idempotent and there-
therefore a characteristic function. If we set $(/£) = IF, then it suffices to show
that E = Fv2—a.e. (almost everywhere), since this would imply v^is) = 0
if and only if IE = 0 in L^) if and only if lF = 0 in LK(v2) if and only if
v2 (F) = 0, and therefore if and only if v2 (E) = 0. Thus we would have
vt ~ v2 and ^(v^ = Lc0(v2), since the L00 space is determined by the sets
of measure zero. Finally, $ would be the identity since it is the identity on
characteristic functions and the linear span of the characteristic functions
is dense in If.
To show that E = Fv2—a.e. it suffices to prove that Fa E v2—a.e., since
we would also have (X\F) c (X\E) v2—a.e. Further, we may assume that E
is compact. For suppose it is known for compact sets. Since vt and v2 are
regular, there exists a sequence of compact sets {Kn}™= l contained in E such
that E= U^°=1 Knvt—a.e. and E= \J™=1 Kn v2—a.e. Thus, since $ and
d) are *-linear and multiplicative and hence order preserving, $ preserves
suprema and we have
IF = sup$(/Kn) ^ sup/Kn = /,,» k. = h v2—a.e.
n n
and therefore F is contained in E v2—a.e. Therefore, suppose E is closed
and for n in Z+ let cpn be the function in C(X) defined by
\-n-d{x,E) if d(x,E) ^-,
0 if d(x,E) > -,

Normal Operators with Cyclic Vectors 105
where d{x,E) = 'mf{p{x,y): yeE} and p is the metric on X. Then
/£< cpn for each n and the sequence {cpn}™=1 converges pointwise to 1E.
Since $ is order preserving and the identity on continuous functions, we have
IF ^ <pn v2—a.e. and thus F is contained in E v2—a.e. ■
After giving the following definition and proving an elementary lemma
we obtain the functional calculus we want under the assumption the operator
has a cyclic vector.
4.56 Definition If J? is a Hilbert space and 91 is a subalgebra of fi(Jf),
then a vector/in Jf is cyclic for SH if clos[9l/'] = Jf and separating for 91
if Af= 0 for A in 91 implies 4 = 0.
4.57 Lemma If ^f is a Hilbert space, 91 is a commutative subalgebra of
fi(Jf), and/is a cyclic vector for 91, then/is a separating vector for 91.
Proof If B is an operator in 91 and Bf = 0, then BAf= ABf= 0 for
every A in 91. Therefore, we have 91/ <= ker £, which implies 5 = 0. ■
The following theorem gives a spatial isomorphism between 2CT and Lw,
if 7" is a normal operator on Jf having a cyclic vector. (Note that such an
^f is necessarily separable.)
4.58 Theorem If T is a normal operator on the Hilbert space Jf such that
GT has a cyclic vector, then there is a positive regular Borel measure v on C
having support A = cr(T) and an isometrical isomorphism y from Jf onto
L2(v) such that the map T* defined from 2BT to £(L2(v)) by r*/l = y^1
is a *-isometrical isomorphism from 2CT onto L^v). Moreover, T* is an
extension of the Gelfand transform T from GT onto C(A). Lastly, if vt is
a positive regular Borel measure on C and F\* is a *-isometrical isomorphism
from 2CT onto L^v^ which extends the Gelfand transform, then vt ~ v,
L00^) = LK(v), and r\* = T*.
Proof Let / be a cyclic vector for GT of norm one and consider the
functional defined on C(A) by ^(Vp) = (<p(T)ff). Since ^ is obviously
linear and positive and
= \(<p(T)ff)\
there exists by the Riesz representation theorem (see Section 1.38) a positive

106 4 Operators on Hilbert Space and C*-Algebras
regular Borel measure v on A such that
<pdv = (<p(T)fJ) for cp in C(A).
I
Now suppose that the support of v were not all of A, that is, suppose there
exists an open subset V of A such that v(V) = 0. By Urysohn's lemma there
is a nonzero function q> in C(A) which vanishes outside of V. Since, however,
we have
\\<p(T)f\\2 = (<p(T)f,<p(T)f) = (\<p\2(T)f,f) = f M2 dv = 0,
and / is a separating vector for GT by the previous lemma we arrive at a
contradiction. Thus the support of v is A.
If we define y0 from GT/to L2(v) such that yo(<p{T)f) = <p, then the
computation
IMI22 = f \<P\2 dv = (\cp\2(T)f,f) = \\<p(T)f\\2
Ja
shows that y0 is a well-defined isometry. Since GT/ is dense in Jf by
assumption and C(A) is dense in L2(v) by Section 3.33, the mapping y0 can
be extended to a unique isometrical isomorphism y from 2f? onto L2(v).
Moreover, if we define T* from 2CT into £(L2(v)) by F*(A) = yAy~l, then
r* is a *-isometrical isomorphism of 2CT into £(L2(v)). We want to first
show that r* extends the Gelfand transform r on Cr. If tj/ is in C(A), then
for all (jo in C(A), we have
and since C(A) is dense in L2(v), it follows that T*(ij/(T)) = M^ = r(ij/(T)).
Thus F* extends the Gelfand transform.
To show that r*BCT) = L?(v), we note that since T* is defined spatially
by y, it follows from Proposition 4.51 that T* is a continuous map from
2CT with the weak operator topology to Lw(y) with the H>*-topology. There-
Therefore, by Corollary 4.53, we have
r*BBT) = r*(weakoperclos[GT])
= w*-clos[C(A)] = L™(v),
and thus T* is a *-isometrical isomorphism mapping 2CT onto Lw(y).
Finally, if vl is a positive regular Borel measure on A and I,* is a
*-isometrical isomorphism from 2CT onto L^O^) which extends the Gelfand
transform, then r*r*-1 is a *-isometrical i^'-morphism from L™(vi) onto

Normal Operators with Cyclic Vectors 107
V°{v) which is the identity on C(A). Hence, by Theorem 4.55, we have
v, ~v, L°(yl) = L°(v), and r*rf-1 is the identity. Therefore the proof
is complete. ■
4.59 The preceding result gives very precise information concerning normal
operators possessing a cyclic vector. Unfortunately, most normal operators
do not have a cyclic vector. Consider the example:
* = L2([0,l])©L2([0,l])
and T= Mg@Mg, where g{t) = t. It is easy to verify that GT has no cyclic
vector and this is left as an exercise. Thus the preceding result does not apply
to T. Notice, however, that
«r = iK © M^ : cp e C([0,1])} and 2BT = {M^ © M^ : cp e I»([0,1])},
and thus there still exists a *-isometrical isomorphism F* from 2CT onto
£"([0,1]) which extends the Gelfand transform on GT. The difference is
that F* is no longer spatially implemented.
To see how to obtain F* in the general case, observe that/= l©0 is
a separating vector for 2BT and that the space Ji = clos [SKT/] is just
£2([0,1]) © {0}; moreover, Jl is a reducing subspace for Tand the mapping
A-+ A\Ji defines a *-isometricalisomorphism from WT onto the W/*-algebra
generated by T\Ji. Lastly, the operator T\Ji is normal and &T\M has a
cyclic vector; hence the preceding theorem applies to it.
Our program is as follows: For a normal Twe show 9BT has a separating
vector, that 2CT and 2Ct-|^ are naturally isomorphic, and that the theorem
applies to T\Ji. Thus we obtain the desired result for arbitrary normal
operators. To show that 2CT has a separating vector requires some preliminary
results on W*-algebras.
4.60 Proposition If 91 is an abelian C*-algebra contained in fi(^f), then
there exists a maximal abelian W*-algebra in fi(Jf) containing 81.
Proof The commutative C*-subalgebras of fi(Jf) which contain 81 are
partially ordered by inclusion. Since the norm closure of the union of any
chain is a commutative C*-algebra of fi(Jf) containing 9t, then there is a
maximal element by Zorn's lemma. Since the closure of such an element in
the weak operator topology is commutative by Proposition 4.47, it follows
that such an element is a W*-algebra. ■

108 4 Operators on Hilbert Space and C*-Algebras
The following notion is of considerable importance in any serious study
of W*-algebras.
4.61 Definition If 91 is a subset of fi(Jf), then the commutant of 91,
denoted 91', is the set of operators in fi(Jf) which commute with every
operator in 91.
It is easy to show that if 91 is a self-adjoint subset of fi(Jf), then 91' is a
W*-algebra.
The following proposition gives an algebraic characterization of maximal
abelian W*-algebras.
4.62 Proposition A C*-subalgebra 91 of fi(Jf) is a maximal abelian
W*-algebra if and only if 91 = 91'.
Proof If 91 = 91', then 91 is a W*-algebra by our previous remarks.
Moreover, by definition each operator in 91' commutes with every operator
in 91 and therefore 91 is abelian. Moreover, if A is an operator commuting
with 91, then A is in 91' and hence already in 91. Thus 91 is a maximal abelian
W*-algebra.
Conversely, if 91 is an abelian W*-algebra, then 9t <= 91'. Moreover, if
Tis in 91', then T=H+iK, where H and K are self-adjoint and in 91'. Since
the W*-algebra generated by 91 and either H or K is an abelian W*-algebra,
then for 91 to be maximal abelian, it is necessary for H and K to be in 91 and
hence 91 = 91'. ■
4.63 Lemma If 91 is a C*-algebra contained in fi(Jf) and /is a vector
in Jf, then the projection onto the closure of 91/ is in 91'.
Proof In view of Proposition 4.42 it suffices to show that clos [91/"] is
invariant for both A and A* for each A in 91. That is obvious from the
definition of the subspace and the fact that 91 is self-adjoint. ■
The following proposition shows one of the reasons for the importance
of the strong operator topology.
4.64 Proposition If Jf is a Hilbert space and {Pa}aeA is a net of positive
operators on Jf such that 0 ^ Pa =$ Pp =$ / for a and /? in A with a =$ /?, then
there exists P in fi(Jf) such that 0 ^ Pa ^ P ^ / for a in A and the net
{Pa}aeA converges to P in the strong operator topology.

Maximal Abelian W*-Algebras 109
Proof If Q is in £(JT) and 0 < Q < /, then 0 < Q2 < Q < / since Q
commutes with (J-QfA by Proposition 4.33 and since
((c- e2)/,/) = (ea- eI^. (/- QYAf) > o for /in jr.
Further, for each / in JT the net {(f,//)}KX is increasing and hence a
Cauchy net. Since for /? ^ a, we have
lie,-wii2 = ((pP-pyfj) < ((pP-p«)f,f) = (pPf,f)~(paf,f),
it therefore follows that {/•«/}« eyl is a Cauchy net in the norm of JT. If we
define Pf= \imasAPaf, then Pis linear,
||/>/|| < lim||/>a/|| < Il/H, and 0 < \im(PafJ) = (Pf,f).
aeA aeA
Therefore, P is a bounded positive operator, 0 < Pa < P < /, for a in yl,
and {/>a}aeyl converges strongly to P, and the proof is complete. ■
The converse of the following theorem is also valid but is left as an exercise.
4.65 Theorem If 91 is a maximal abelian H^*-algebra on the separable
Hilbert space Jf, then 91 has a cyclic vector.
Proof Let $ denote the set of all collections of projections {Ea}aeA
in 91 such that:
A) For each a in A there exists a nonzero vector/, in Jf such that Ea is
the projection onto clos [9I/J; and
B) the subspaces {clos[9I/J}aeyl are pairwise orthogonal.
We want to show there is an element {Ea}aeA in S such that the span of the
ranges of the Ea is all of Jf.
Order S by inclusion, that is, {Ea}aeA is greater than or equal to {Fp}PeB
if for each /?0 in B there exists a0 in A such that FPo = Eao. To show that &
is nonempty observe that the one element set {/>cios[M/]} is in $ for each non-
nonzero vector / in Jf. Moreover, since the union of a chain of elements in $
is in $, then $ has a maximal element {Ea}aeA by Zorn's lemma.
Let J5" denote the collection of all finite subsets of the index set A partially
ordered by inclusion and let {Pf)fbS; denote the net of operators defined
by Pf = HaeFEa. By the remark after Proposition 4.19, each PF is a pro-
projection and hence the net is increasing. Therefore, by the previous proposition
there exists a positive operator P such that 0 < PF < P < / for every F in J5"
and {PF}Fe& converges to P in the strong operator topology. Therefore,
{PF2}Fe& converges strongly to P2 and P is seen to be a projection.

110 4 Operators on Hilbert Space and C*-Algebras
The projection P is in 91 since 91 is weakly closed and the range J'off
reduces 91 since 91 is abelian. Thus, if/is a nonzero vector in M^, then
clos[9I/] is orthogonal to the range of each Ea. If we let Ep denote the
projection onto clos[9I/], then {Ea}aeAum is an element in g larger than
{Ea}aeA. This contradiction shows that J(L = {0} and hence that P = I.
Since Jf is separable, dim ran £„ > 0, and dimJf = £aeyldimran.Ea,
we see that A is countable. Enumerate A such that A = {a,ua2,...} and set
f=l4>i2~ifj\fa.l Since EaJ=2-%/\\fJ, we see that the range of
Ea% is contained in clos[9I/]. Therefore, since the ranges of the Ea% span Jf,
we see that/is a cyclic vector for 91 and the proof is complete. ■
The assumption that Jf is separable is needed only to conclude that A
is countable.
The following result is what we need in our study of normal operators.
4.66 Corollary If 91 is an abelian C*-algebra denned on the separable
Hilbert space Jf, then 91 has a separating vector.
Proof By Proposition 4.60 91 is contained in a maximal abelian W*-
algebra 93 which has a cyclic vector / by the previous theorem. Finally, by
Lemma 4.57 the vector/is a separating vector for 93 and hence also for the
subalgebra 91. ■
The appropriate setting for the last two results is in W*-algebras having
the property that every collection of pairwise disjoint projections is countable.
While for algebras denned on a separable Hilbert space this is always the
case, it is not necessarily true for H^*-algebras denned on a nonseparable
Hilbert space Jf; for example, consider £(Jf).
Before we can proceed we need one more technical result concerning
*-homomorphisms between C*-algebras.
4.67 Proposition If 91 and 93 are C*-algebras and $ is a *-homomorphism
from 91 to 93, then ||$|| < 1 and <b is an isometry if and only if <b is one-to-one.
Proof If H is a self-adjoint element of 91, then (£H is an abelian C*-
algebra contained in 91 and <£(GH) is an abelian *-algebra contained in 93.
If \}i is a multiplicative linear functional on the closure of <£(GH) then \}i°<b
defines a multiplicative linear functional on (£H. If ip is chosen such that
\4*(<b{Hj)\ = \\®{H)\, which we can do by Theorem 4.29, then we have

*-Hotnotnorphisms of C*-Algebras 111
> \ij/(<5(H))\ = ||$(/7)|| and hence $ is a contraction on the self-
adjoint elements of 91. Thus we have for T in 91 that
||712 = ||r*r||
and hence ||$|| < 1.
Now for the second statement. Clearly, if <b is an isometry, then it is
one-to-one. Therefore, assume that <£ is not an isometry and that T is an
element of 91 for which ||J|| = 1 and \<S>(T)\ < 1. If we set A = T*T, then
||yl|| = 1 and \\^(A)\\ = 1-e with e > 0. Let / be a function in C([0,1])
chosen such that /(I) = 1 and f(x) = 0 for 0 < x < 1 — e. Then using the
functional calculus on <£A we can define f(A) and since by Corollary 2.37
we have a(f(A)) = range T(f(A)) =f((r(A)), we conclude that 1 is in
<r(/(/4)) and thus f(A) # 0. Since d> is a contractive *-homomorphism, it is
clear that <&(p(A)) = p{<b{A)) for each polynomial and hence <&(J(A)) =
f(Q>(A)). Since, however, we have ||fl>(/4)|| = 1 —e, it follows that (t(<D(A)) c
[0,1 — e] and therefore
ff(®(/-(^))) =/(ff(®(^))) «= A[0,l-a]) = {0}.
Since <&(A) is self-adjoint, we have <&(A) = 0, which shows that $ is not
one-to-one. ■
Now we are prepared to extend the functional calculus for an arbitrary
normal operator on a separable Hilbert space.
4.68 Let JP be a separable Hilbert space and 7* be a normal operator on Jf.
By Corollary 4.66, the abelian W*-algebra 2Br has a separating vector /
If we set Jl = clos [2Br/]» then Jl is a reducing subspace for each ^4 in
2Br, and hence we can define the mapping <b from 2Br to £(.#) by
$(^4) = A \M for ^4 in 2Br. It is clear that $ is a *-homomorphism. We use
the previous result to show that $ is a *-isometrical isomorphism.
4.69 Lemma If 91 is a C*-algebra contained in 2(jf), f is a separating
vector for 91 and M is the closure of 91/, then the mapping $ defined
<$>{A) = A\J( for A in 91 is a *-isometrical isomorphism from 91 into £(^).
Moreover, a (A) = a{A\JC) for ^ in 91.
Proo/ Since <b is obviously a *-homomorphism, it is sufficient to show
that $ is one-to-one. If A is in 91 and <&(A) = 0, then Af= 0, which implies
A = 0 since / is a separating vector for 91. The last remark follows from

112 4 Operators on Hilbert Space and C*-Algebras
Theorem 4.28, since we have
Finally, we shall need the following result whose proof is similar to that of
Theorem 1.23 and hence is left as an exercise.
4.70 Proposition If 91 is a W*-algebra contained in £(^f), then the unit
ball of 91 is compact in the weak operator topology.
Our principal result on normal operators can now be given.
4.71 Theorem (Extended Functional Calculus) If T'\% a normal operator on
the separable Hilbert space Jf with spectrum A, and F is the Gelfand trans-
transform from dT onto C(A), then there exists a positive regular Borel measure v
having support A and a *-isometrical isomorphism F* from 2Br onto L°°(v)
which extends F. Moreover, the measure v is unique up to mutual absolute
continuity, while the space L^v) and F* are unique.
Proof Let/be a separating vector for 2Br, Jl be the closure of 2Br/
and <b be the *-isometrical isomorphism denned from 2Br into £(Jf) by
<b{A) = A\M as in Section 4.68. Further, let 2B^ be the W*-algebra gener-
generated by T\J(. Since <b is denned by restricting the domain of the operators,
it follows that 3> is continuous from the weak operator topology on 2Br
to the weak operator topology on 2.{M), and hence $BBj) c 2B^. More-
Moreover, it is obvious that if Fo is the Gelfand transform from &T\jt onto
Since T\M is normal and has the cyclic vector/ there exist by Theorem
4.58 a positive regular measure v with support equal to a(T\Jf) = A by
the previous lemma and a *-isometrical isomorphism Fo* from 2B^ onto
V°(v) which extends the Gelfand transform Fo on (£r|^. Moreover, Fo* is
continuous from the weak operator topology on 2B^ to the w*-topology on
L^v). Therefore, the composition F* = F0*o$ is a *-isometrical isomor-
isomorphism from 2Br into L^v) which is continuous from the weak operator
topology on 2Br to the w*-topology on L^v) and extends the Gelfand
transform F on (£r.
The only thing remaining is to show that F* takes 2Br onto l?(v). To
do this we argue as follows: Since the unit ball of 2Br is compact in the weak
operator topology, it follows that its image is w*-compact in l^(v) and
hence w*-closed. Since this image contains the unit ball of C(A), it follows

The Extended Functional Calculus 113
from Proposition 4.52 that r* takes the unit ball of 2Br onto the unit ball
of L^Cv). Thus, T* is onto.
The uniqueness assertion follows as in Theorem 4.58 from Theorem
4.55. ■
4.72 Definition If T is a normal operator on the separable Hilbert space
Jiif, then there exists a unique equivalence class of measures v on A such that
there is a *-isometrical isomorphism r* from 2Br onto n°(v) such that
r*(q>(T)) = (p for (p in C(A). Any such measure is called a scalar spectral
measure for T. The extended functional calculus for T is denned for q> in
L°°(v) such that r*(#>G)) = (p. If /A is a characteristic function in L^v),
then I&(T) is a projection in 2BT called a spectral projection for T, and its
range is called a spectral subspace for T.
We conclude this chapter with some remarks concerning normal operators
on nonseparable Hilbert spaces and a proposition which will enable us to
make use of certain aspects of the extended functional calculus for such
operators. We begin with the proposition which we have essentially proved.
4.73 Proposition If 91 is a norm separable C*-subalgebra of 2Cf), then
Jf = Y*oLeA®^a sucn that each Jfa is separable and is a reducing sub-
space for 91.
Proof It follows from the first three paragraphs of the proof of Theorem
4.65 that Jf = Y*eA ® ^a> where each 34?a is the closure of 9l/a for some/a
in Jf. Since 91 is norm separable, each Jfa is separable and the result
follows. ■
4.74 From this result it follows that if T is normal on 2/e, then each
Ta = T\jfa is normal on a separable Hilbert space and hence has a scalar
spectral measure va. If there exists a measure v such that each \a is absolutely
continuous with respect to v, then Theorem 4.71 can be shown to hold for T.
If no such v exists, then the functional calculus for T is usually based on the
algebra of bounded Borel functions on A. In particular, one defines (p(T) =
HaeA®<p(Ta) for each Borel function (p on A. The primary deficiencies in
this approach is that the range of this functional calculus is no longer 2Br
and the norm of (p(T) is less accessible.
Sometimes the spectral measure £(■) for T is made the principal object

114 4 Operators on Hilbert Space and C*-Algebras
of study. If A is a Borel subset of A, then the spectral measure is denned by
as A
and is a projection-valued measure such that (E(A)f,f) is countably additive
for each/in Jf. Moreover, the Stieltjes integral can be denned and T = |A z dE.
We shall not develop these ideas further except to show that the range of each
spectral measure for T lies in 2Br.
4.75 Proposition If T is a normal operator on the Hilbert space #C and
E(-) is a spectral measure for T, then E(A) is in 2Br for each Borel set A in A.
Proof Let T= Y^eA® Ta be the decomposition of T relative to which
the spectral measure £(■) is denned, where each Ta acts on the separable
space Jfa with scalar spectral measure jua. If BF denotes the collection of
finite subsets of A, then (JFeSr Y^er®^a is a dense linear manifold in Jf.
Thus if A is a Borel subset of A, it is sufficient to show that for fuf2, ••-,/„
lying in £aeFo ® 3^a for some Fo in !F and e > 0, there exists a continuous
function (p such that 0 < (p < 1, and \\((p(T)-E(A))ft\\ < e for i = 1,2,...,«.
Since
\\(<p(T) -E(A))M2= I \\(<p(Ta) -
F
(\9-h\2\P,Ji\2dh,
this is possible using Proposition 4.52.
We conclude this chapter with an important complimentary result. The
following ingenious proof is due to Rosenblum [93].
4.76 Theorem (Fuglede) If T is a normal operator on the Hilbert space Jf
and X is an operator in £(Jf) for which TX = XT, then X lies in 3Br'.
Proof Since 2BT is generated by T and T*, the result will follow once it
is established that T*X=XT*.
Since TkX = XT* for each k^O,it follows that exp(i%T)X = Xexp(ilT)
for each k in C. Therefore, we have X = exp(ilT)Xexp(-ilT), and hence
F(A) = exp(iir*)Xexp(-i/l7'*)
= exp [i(lr+ XT*)] X exp [- i(lr+ AT*)]
by Lemma 2.12, since TT* = T*T. Since 1T+XT* is self-adjoint, it follows

Exercises 115
that exp[i(lr+ AT*)] and exp[-i(lr+Ar*)] are unitary operators for A
in C. Thus the operator-valued entire function F(X) is bounded and hence by
liouville's theorem must be constant (see the proof of Theorem 2.29).
Lastly, differentiating with respect to A yields
= 0.
Canceling A and then setting A = 0 yields T*X = XT*, which completes
the proof. ■
Notes
The spectral theorem for self-adjoint operators is due to Hilbert, but
elementary operator theory is the joint work of a number of authors including
Hilbert, Riesz, Weyl, von Neumann, Stone, and others. Among the early
works which are still of interest are von Neumann's early papers [81], [82],
and the book of Stone [104]. More recent books include Akhieser and
Glazman [2], Halmos [55], [58], Kato [70], Maurin [79], Naimark [80],
Riesz and Sz.-Nagy [92], and Yoshida [117].
We have only introduced the most elementary results from the theory of
C*- and W*-algebras. The interested reader should consult the two books
of Dixmier [27], [28] for further information on the subject as well as a guide
to the vast literature on this subject.
Exercises
4.1 If T is a linear transformation denned on the Hilbert space 3tf, then
T is bounded if and only if
Definition If T is an operator denned on the Hilbert space #e, then the
numerical range W(T) of T is the set {(Tf,f) :fe 34?, \\f\\ = 1} and the
numerical radius w(T) is sup{|A| : A e W(T)}.
4.2 (Hausdorff-Toeplitz) If T is an operator on 3V, then W(T) is a
convex set. Moreover, if Jf is finite dimensional, then W(T) is compact.
(Hint: Consider W(T) for T compressed to the two-dimensional subspaces
ofjf.)

116 4 Operators on Hilbert Space and C*-Algebras
4.3 If T is a normal operator on 3V, then the closure of W(T) is the closed
convex hull of o(T). Further, an extreme point of the closure of W{T) belongs
to W(T) if and only if it is an eigenvalue for T.
4.4 If T is an operator on 3f, then o(T) is contained in the closure of
W(T). (Hint: Show that if the closure of W(T) lies in the open right-half
plane, then T is invertible.)
4.5 If 7* is an operator on the Hilbert space 2/e, then r(T) < w(T) < ||r||
and both inequalities can be strict.
4.6 (Hellinger-Toeplitz) If S and T are linear transformations denned on
the Hilbert space 3V such that (Sf,g) = (/, Tg) for/and ginMf, then S and
T are bounded and T = S*.
4.7 If T is an operator on Mf, then the graph {</, 2/> :/e Mf} of T is a
closed subspace of Jf © Jf with orthogonal complement {< — T*g, gy-.ge 34?}.
4.8 If Tis an operator on 3/e, then T\% normal if and only if || Tf\\ = || T*f\
for / in Jf. Moreover, a complex number k is an eigenvalue for a normal
operator T if and only if 1 is an eigenvalue for T*. The latter statement is not
valid for general operators on infinite-dimensional Hilbert spaces.
4.9 If S and T are self-adjoint operators on jtf, then ST is self-adjoint if
and only if S and T commute. If P and Q are projections on 3V, then .Pg
is a projection if and only if P and Q commute. Determine the range of PQ
in this case.
4.10 If 3V and jf are Hilbert spaces and A is an operator on Jf © jf, then
there exist unique operators Allt A12, A21, and A22 in
, jf), and £(jf), respectively, such that
In other words A is given by the matrix
^22
Moreover show that such a matrix defines an operator
^.7/ If Jf and Jf are Hilbert spaces, /4 is an operator on £(Jf, Jf), and
is the operator on Jf © jf denned by the matrix
/ A
0 0

Exercises 117
then J is an idempotent. Moreover, J is a projection if and only if A = 0.
Further, every idempotent on a Hilbert space £? can be written in this form
for some decomposition «Sf =
Definition If Tt and T2 are operators on the Hilbert space J^1 and J^2,
respectively, then Tl is similar (unitarily equivalent) to T2 if and only if there
exists an invertible operator (isometrical isomorphism) S from Jft onto
3V2 such that T2S = STt.
4.12 If Jf is a Hilbert space and J is an idempotent on jf with range Jt,
then J and /^ are similar operators.
4.13 If (X,Sf,n) is a probability space and (p a function in E°(n), then A
is an eigenvalue for M^ if and only if the set {x e X: <p(x) = X) has positive
measure.
4.14 Show that the unitary operator U denned in Section 4.25 has no eigen-
eigenvalues, while the eigenvalues of the coisometry U+* denned in Section 4.36
have multiplicity one.
4.15 If 91 is a C*-algebra, then the set & of positive elements in 91 forms
a closed convex cone such that §P c\ — 0* = {0}. (Hint: Show that a self-
adjoint contraction H in 91 is positive if and only if ||/— H\ < 1.)
4.16 If 91 is a C*-algebra, then an element H in 91 is positive if and only
if there exists T in 91 such that H=T*T.* (Hint: Express T*T as the
difference of two positive operators and show that the second is zero.)
4.17 If P and Q are projections on Jf such that H^P—Gil < 1> then
dim ran P = dim rang. (Hint: Show that (I-P) + Q is invertible, that
ker P n ran Q = {0}, and that P[ran Q] = ran P.)
4.18 An operator V on Jf is an isometry if and only if V* V = I. If V is an
isometry on Jf, then V is a unitary operator if and only if V* is an isometry
if and only if ker V* = {0}.
4.19 If Jf and Jf are Hilbert spaces and A is an operator on Jf © Jf given
by the matrix
then Jf © {0} is an invariant subspace for A if and only if A21 = 0, and
#f © {0} reduces A if and only if A21 = A12 = 0.

118 4 Operators on Hilbert Space and C*-Algebras
4.20 If U+ is the unilateral shift, then the sequence {U+"}™=1 converges
to 0 in the weak operator topology but not in the strong. Moreover, the
sequence {U*"}%L, converges to 0 in the strong operator topology but not
in the norm.
4.21 Show that multiplication is not continuous in both variables in either
the weak or strong operator topologies. Show that it is in the relative strong
operator topology on the unit ball of
4.22 If Jf is a Hilbert space, then the unit ball of £(jf) is compact in the
weak operator topology but not in the strong. (Hint: Compare the proof of
Theorem 1.23.)
4.23 If 21 is a H^-algebra contained in £(Jf), then the unit ball of 91 is
compact in the weak operator topology.
4.24 If 91 is a *-subalgebra of 2.C4?), then 9lB) = {gJ]:Je 91} is a
*-subalgebra of £(jf ©34?). Similarly, 9I(JV) is a *-subalgebra of £(Jf © ■ ■ • © 34?)
for any integer N. Moreover, 9I(JV) is closed in the norm, strong, or weak
operator topologies if and only if 91 is. Further, we have the identity
4.25 If 91 is a *-subalgebra of Jf, A is in 91", xux2,...,xN are vectors in Jf,
and e > 0, then there exists B in 91 such that H^Xj—-&C||| <efori= 1,2,...,N.
(Hint: Show first that for M a subspace of 34?®---®34?, we have
dos[9qN)JT] = clos[9I(JV).#].)
4.26 {von Neumann Double Commutant Theorem) If 91 is a *-subalgebra
of £pf), then 91 is a H^-algebra if and only if 91 = 91".*
4.27 If 91 is a C*-subalgebra of £pf), then 91 is a H^*-algebra if and only
if it is closed in the strong operator topology.
4.28 If S and T are operators in the Hilbert spaces 3V and X, respectively,
then an operator S® 7* can be defined on Jf ®jf in a natural way such that
= ||,S'||||r||andE'®r)* = 5'*®r*.
In Exercises D.29-4.34) we are considering linear transformations defined on
only a linear subspace of the Hilbert space.
Definition A linear transformation L defined on the linear subspace 3iL of
the Hilbert space 3V is closable if the closure in 3V®3V of the graph
{</,!/> :fe 3ij} of L is the graph of a linear transformation L called the

Exercises 119
closure of L. If L has a dense domain, is closable, and L = L, then L is said
to be closed.
4.29 Give an example of a densely denned linear transformation which is
not closable. If T is a closable linear transformation with QsT = Jf, then
T is bounded.
4.30 If L is a closable, densely denned linear transformation on Jf, then
these exists a closed, densely denned linear transformation Mon / such
that (Lf,g) = (f, Mg) for/in <2L and g in 3>M. Moreover, if N is any linear
transformation on Jf for which (Lf,g) = (f,Ng) for/in ^t and # in @N,
then S>jv c 3>m an<* -% = ^ f°r ^ in ^jv- (Hint: Show that the graph of M
can be obtained as in Exercise 4.7 as the orthogonal complement of the graph
ofL.)
Definition If L is a closable, densely denned linear transformation on Jf,
then the operator given in the preceding problem is called the adjoint of L
and is denoted L*. A densely denned linear transformation H is symmetric
if (#/,/) is real for/in 3)H and self-adjoint if H = H*.
4.31 If T is a closed, densely denned linear transformation on Jf, then
T = T**. * (This includes the fact that 3>T = ,S>r»»). If H is a densely denned
symmetric transformation on Jf, then H is closable and H* extends H.
4.32 If H is a densely denned symmetric linear transformation on ffl with
range equal to Jf, then H is self-adjoint.
4.33 If 7" is a closed, densely denned linear transformation on Jf, then
T* T is a densely denned, symmetric operator. (Note: T* Tf is denned for
those/for which 2/is in S^.)
4.34 If r is a closed, densely denned linear transformation on ffl, then
T*T is self-adjoint. (Hint: Show that the range of I+T*T is dense in &
and closed.)
4.35 If 91 is a commutative H^-algebra on Jf with 2/e separable, then 91
is a *-isometrically isomorphic to l?(n) for some probability space (X, y, n).
4.36 Show that there exist H^*-algebras 91 and © such that 91 and 93 are
*-isomorphic but 91' and 93' are not.
4.37 If 91 is an abelian H^*-algebra on Jf for Jf separable, then 91 is maximal
abelian if and only if 91 has a cyclic vector.

120 4 Operators on Hubert Space and C*-Algebras
4.38 Give an alternate proof of Fuglede's theorem for normal operators
on a separable Hilbert space as follows: Show that it is enough to prove that
EiA^ XE(A2) = 0 for At and A2 disjoint Borel sets; show this first for Borel
sets at positive distance from each other and then approximate from within
by compact sets in the general case.
4.39 {Putnam) If 7\ and T2 are normal operators on the Hilbert spaces
J^ and Jf2, respectively, and X in 2,{#e^#e2) satisfies T2X=XT1, then
T2*X= XT)*. (Hint: Consider the normal operator [J1 r°] on ^©^
together with the operator [£ %~\).
4.40 If 7\ and T2 are normal operators on the Hilbert spaces Jft and Jf 2,
respectively, then Tl is similar to T2 if and only if Tl is unitarily equivalent
to T2.
4.41 Let X be a compact Hausdorff space, #? be a Hilbert space, and <E> be
a ^isomorphism from C(X) into fi(Jf). Show that if there exists a vector/
in &f for which <&(C(Xj) is dense in J'f, then there exists a probability measure
ju on X and an isometrical isomorphism ¥ from L2 (/i) onto Jf such that
^M^1!'* = $(<£)) for (p in CCA'). (Hint: repeat the argument for Theorem
4.58.)
4.42 Let X be a compact Hausdorff space, Jf be a Hilbert space, and <D
be a *-isomorphism from C(A') into ii(Jf), then there exists a *-homo-
morphism <D* from the algebra 38(X) of bounded Borel functions on X
which extends G>. Moreover, the range of G>* is contained in the von Neumann
algebra generated by the range of <D. (Hint: use the arguments of 4.74 and
4.75 together with the preceding exercise.)

5
Compact Operators* Fredhotm
Operators^ and Index Theory
5.1 In the preceding chapter we studied operators on Hilbert space and
obtained, in particular, the spectral theorem for normal operators. As we
indicated this result can be viewed as the appropriate generalization to
infinite-dimensional spaces of the diagonalizability of matrices on finite-
dimensional spaces. There is another class of operators which are a general-
generalization in a topological sense of operators on a finite-dimensional space.
In this chapter we study these operators and a certain related class. The
organization of our study is somewhat unorthodox and is arranged so that
the main results are obtained as quickly as possible. We first introduce the
class of compact operators and show that this class coincides with the norm
closure of the finite rank operators. After that we give some concrete examples
of compact operators and then proceed to introduce the notion of a Fredholm
operator. We begin with a definition.
5.2 Definition If Jf is a Hilbert space, then an operator Tin £(jf) is a
finite rank operator if the dimension of the range of Tis finite and a compact
operator if the image of the unit ball of Jf under 7* is a compact subset of Jf.
Let £g(jf), respectively, £C(^f) denote the set of finite rank, respectively,
compact operators.
In the definition of compact operator it is often assumed only that
121

122 5 Compact and Fredholtn Operators, Index Theory
T[CfI] has a compact closure. The equivalence of these two notions
follows from the corollary to the next lemma.
5.3 Lemma If 3V is a Hilbert space and Tis in 2C4?), then T is a continu-
continuous function from 34? with the weak topology to 34? with the weak topology.
Proof If {fa}aeA is a net in 34* which converges weakly to/and g is a
vector in 34?, then
and hence the net {Tfa}aeA converges weakly to Tf. Thus T is weakly con-
continuous. ■
5.4 Corollary If 3f is a Hilbert space and J is in 2C4?), then T\_Cf)(\
is a closed subset of Jf.
/Voo/ Since CtfI is weakly compact and T is weakly continuous, it
follows that r[(jf)J is weakly compact. Hence, T[CV)^\ is a weakly
closed subset of 34? and therefore is also norm closed. ■
The following proposition summarizes most of the elementary facts about
finite rank operators.
5.5 Proposition If 34" is a Hilbert space, then £g(Jf) is the minimal
two-sided *-ideal in 2C4?).
Proof If S and T are finite rank operators, then the inclusion
ran(^+ T) c ranS + ranT
implies that S+T is finite rank. Thus, £g(Jf) is a linear subspace. If S is
a finite rank operator and T is an operator in 2CV), then the inclusion
ranSTc: ranS shows that 2%C^) is a left ideal in 2C€). Further, if T is a
finite rank operator, then the identity
ranT* = T* [(ker r*)x] = r*[closranr]
which follows from Corollary 3.22 and Proposition 4.6, shows that T* is
also a finite rank operator. Lastly, if S is in 2(#F) and T is in 2%CV), then
T* is in 2%C4?) which implies that T*S* is in Cg(Jf) and hence that ST=
(T*S*)* is in 2%Ctf). Therefore, 2%C4?) is a two-sided *-ideal in 2C4?).
To show that £5(Jf) is minimal, assume that 3 is an ideal in

The Ideals of Finite Rank and Compact Operators 123
not @). Thus there exists an operator T # 0 in 3, and hence there is a non-
nonzero vector/and a unit vector g in Jf such that Tf=g. Now let h and k be
arbitrary nonzero vectors in #e and ^ and B be the operators denned on Jf
by ^/ = {l,g)k and £/ = (l,h)f for / in Jf. Then, 5 = ^r£ is the rank one
operator in 3 which takes h to k. It is now clear that 3 contains all finite rank
operators and hence £gpf) is the minimal two-sided ideal in £(jf). ■
The following proposition provides an alternative characterization of
compact operators.
5.6 Proposition If Jf is a Hilbert space and T is in 2C4?), then T is com-
compact if and only if for every bounded net {fa}aeA in Jiif which converges
weakly to/it is true that {Tfa}aeA converges in norm to Tf.
Proof If T is compact and {fa}aeA is a bounded net in Jiif which con-
converges weakly to/ then {Tfa}aeA converges weakly to Tf by Lemma 5.3 and
lies in a norm compact subset by the definition of compactness. Since any
norm Cauchy subnet of {Tfa}aeA must converge to Tf, it follows that
liniag^ Tfa = Tf in the norm topology.
Conversely, suppose 7* is an operator in 2(Jf) for which the conclusion
of the statement is valid. If {Tfa}aeA is a net of vectors in T\_Ct){\, then
there exists a subnet {fap}peB which converges weakly to an / Moreover,
since each/ap is in the unit ball of Jf, it follows that {Tfaf}peB converges in
norm to Tf Therefore, TK^OJ is a compact subset of Jf and hence T is
compact. ■
5.7 Lemma The unit ball of a Hilbert space Jf is compact in the norm
topology if and only if Jf is finite dimensional.
Proof If Jf is finite dimensional, then Jf is isometrically isomorphic
to C" and the compactness of (^)i follows. On the other hand if Jf is
infinite dimensional, then there exists an orthonormal subset {eM}"=1 con-
contained in (jf )t and the fact that ||eM-em|| = y/2 for « # w shows that (Jf)t
is not compact in the norm topology. ■
The following property actually characterizes compact operators on a
Hilbert space, but the proof of the converse is postponed until after the next
theorem. Whether this property characterizes compact operators on a Banach
space is unknown.

124 5 Compact and Fredholm Operators, Index Theory
5.8 Lemma If Jf is an infinite-dimensional Hilbert space and T is a
compact operator, then the range of T contains no closed infinite-dimensional
subspace.
Proof Let Jt be a closed subspace contained in the range of T and let
Pjt be the projection onto M. It follows easily from Proposition 5.6 that the
operator P t( T is also compact. Let A be the operator defined from ffl to
Jl by Af= PJt Tf for / in jf. Then A is bounded and onto and hence by
the open mapping theorem is also open. Therefore, y4[(^f)i] contains the
open ball in J( of radius b centered at 0 for some b > 0. Since the closed
ball of radius b is contained in the compact set PM r[(^f),], it follows from
the preceding lemma that M is finite dimensional. |
We are now in a position to show that £G(Jf) is the norm closure of
. The corresponding result for Banach spaces is unknown.
5.9 Theorem If Jf is an infinite-dimensional Hilbert space, then
is the norm closure of
Proof We first show that the closure of £g(#) is contained in
Firstly, it is obvious that £g(Jf) is contained in £(£(jf). Secondly, to prove
that £(£(Jf) is closed, assume that {i<^}™= t is a sequence of compact operators
which converges in norm to an operator K. If {fa}aeA is a bounded net in
Jf that converges weakly to /, and
M = sup{l,||/J:«ei4},
then choose N such that \\K— KN\\ <e/3M. Since KN is a compact operator,
we have by Proposition 5.6 that there exists ce0 in A such that \\KNfa—KNf\\ <
e/3 for a > ce0. Thus we have
Wa-Kf\\ < \\(K-KN)fa\\ + \\KNfa-KNf\\ + \\(KN-K)f\\
and hence K is compact by Proposition 5.6. Therefore, the closure of
is contained in £(£(Jf).
To show that £gpf) is dense in £C(jf), let K be a compact operator
on Jf and let K= PVbe the polar decomposition for K. Let Jf = Y^^a®^*
be a decomposition of Jf into separable reducing subspaces for P given by
Proposition 4.73, and set Pa = P\3^a. Further, let 9Ca be the abelian W*-

Examples of Compact Operators; Integral Operators 125
algebra generated by Pa on Jf a and consider the extended functional calculus
obtained from Theorem 4.71 and defined for functions in 15° (ya) for some
positive regular Borel measure va with support contained in [0, ||.Pl|]. If
Xe denotes the characteristic function of the set (e, ||.P||], then & is in L°°(vJ
and Eae — xc(PJ is a projection on 34?a. If we define \j/t on [0, ||.P||] such that
ijjt(x) = \jx for e < x< \\P|| and 0 otherwise, then the operator Qae = il/e(Pa)
satisfies Qa£Pa = PaQa = Ea\ Thus we have
ran(X © Eac) = ™np{Y. ® <?«*) c ran/> = ranK
aeA aeA
and therefore the range of the projection ^eA®Eae is finite dimensional
by Lemma 5.7. Hence, Pe = P(Y.*eA® Ea) is in £S(^) and thus so is PtV.
Finally, we have
\\K-PtV\\ = \\PV-PeV\\ < \\P-Pt\\ =
= sup sup Ijx-x^WL < e,
aeA \\P\\
and therefore the theorem is proved. H
Notice that in the last paragraph of the preceding proof we used only
the fact that the range of K contained no closed infinite-dimensional sub-
space. Thus we have proved the remaining half of the following result.
5.10 Corollary If Jf is an infinite-dimensional Hilbert space and T is an
operator on Jf, then T is compact if and only if the range of T contains no
closed infinite-dimensional subspaces.
5.11 Corollary If Jf is an infinite-dimensional Hilbert space, then
is a minimal closed two-sided *-ideal in £(Jf). Moreover, if #C is separable,
then £G(Jf) is the only proper closed two-sided ideal in
Proof For the first statement combine the theorem with Proposition 5.5.
For the second, note by the previous corollary that if T is not compact, then
the range of T contains a closed infinite-dimensional subspace M. A simple
application of the open mapping theorem yields the existence of an operator
S on Jf such that TS = PJt, and hence any two-sided ideal containing T
must also contain I. This completes the proof. ■
5.12 Example Let K be a complex function on the unit square [0,1] x
[0,1] which is measurable and square-integrable with respect to planar

126 5 Compact and Fredholm Operators, Index Theory
Lebesgue measure. We define a transformation TK on L2 ([0,1]) such that
(TKf)(x) = AK(x,y)f0>) dy for /in L2([0,1]).
Jo
The computation
fVx/)(*)|2«k = fi [lK(x,y)f(y)dy
Jo Jo I Jo
dx
11 I I \K(x,y)\2dydx
'o Jo
which uses the Cauchy-Schwarz inequality, shows that TK is a bounded
operator on L2([0,1]) with
\\TK\\
The operator !TK is called an integral operator with kernel K. We want to
show next that TK is a compact operator.
If we let $ denote the mapping from L2([0, l])x [0,1]) to £(L2([0,1]))
denned by $>(K) = TK, then $ is a contractive linear transformation. Let
Si be the subspace of L2([0,1] x [0,1]) consisting of the functions of the
form
where each /• and gt is continuous on [0,1]. Since &> is obviously a self-
adjoint subalgebra of the algebra of continuous functions on [0,1] x [0,1]
which contains the identity and separates points, it follows from the Stone-
Weierstrass theorem that C([0,1] x [0,1]) is the uniform closure of Si.
Moreover, an obvious modification of the argument given in Section 3.33
shows that S is dense in L2([0,1] x [0,1]) in the L2-norm. Thus the range
of O is contained in the norm closure of O(S) in £(L2([0,1])).
Let/and g be continuous functions on [0,1] and let T be the integral
operator with kernel f(x)g(y). For h in L2([0,1]), we have
(Th)(x) = j*f(x)g(y)h(y) dy =
\
and hence the range of T consists of multiples of/ Therefore, T is a rank one
operator and all the operators in <bCi) are seen to have finite rank. Thus

The Calkin Algebra and Fredholm Operators 127
we have by Theorem 5.9 that
O(L2([0,1] x [0,1])) c: clos$(S) c: clos£g(L2([0,1])) = £C(L2([0,1J)),
and hence each integral operator TK is compact.
We will obtain results on the nature of the spectrum of a compact operator
after proving some elementary facts about Fredholm operators.
If #? is finite dimensional, then £(£(Jf) = £(JT). Hence, in the remainder
of this chapter we assume that ffi is infinite dimensional.
5.13 Definition If #C is a Hilbert space, then the quotient algebra
£(Jf)/£C(^) is a Banach algebra called the Calkin algebra. The natural
homomorphism from £(Jf) onto £(Jf)/£C(Jf) is denoted by n.
That the Calkin algebra is actually a C*-algebra will be established later
in this chapter.
The Calkin algebra is of considerable interest in several phases of analysis.
Our interest is in its connection with the collection of Fredholm operators.
The following definition of Fredholm operator is convenient for our purposes
and will be shown to be equivalent to the classical definition directly.
5.14 Definition If Jf is a Hilbert space, then T in £(Jf) is a Fredholm
operator if n(T) is an invertible element of £(^f)/£C(Jf). The collection
of Fredholm operators on Jf is denoted
The following properties are immediate from the definition.
5.15 Proposition If Jf is a Hilbert space, then &(#?) is an open subset
of 2(J^), which is self-adjoint, closed under multiplication, and invariant
under compact perturbations.
Proof If A denotes a group of invertible elements in
then A is open by Proposition 2.7 and hence so is &(#?) = ft (A), since n
is continuous. That J^Jf) is closed under multiplication follows from the
fact that 7i is multiplicative and A is a group. Further, if T is in ^(Jf) and
K is compact, then T+K is in J^C^f) since n{T) = n(T+K). Lastly, if Tis
in ^(JiT), then there exist S in £(Jf) and compact operators /^ and K2
such that ST=I+Kl and TS = I+K2. Taking adjoints we see that n(T*)
is invertible in the Calkin algebra and hence J^C^f) is self-adjoint. ■
The usual characterization of Fredholm operators is obtained after we
prove the following lemma about the linear span of subspaces. If M and Jf

128 5 Compact and Fredholm Operators, Index Theory
are closed subspaces of a Hilbert space, then the linear span Jl+Jf is, in
general, not a closed subspace (see Exercise 3.18) unless one of the spaces
is finite dimensional.
5.16 Lemma If Jf is a Hilbert space, J'isa closed subspace of Jf, and
Jf is a finite-dimensional subspace of Jf, then the linear span Jl+Jf is a
closed subspace of Jf.
Proof Replacing Jf, if necessary, by the orthogonal complement of
Jf njV in Jf, we may assume that Jl n Jf = {0}. To show that
J? + jr = {f+g:feJ/, geJf}
is closed, assume that {/,}"= 1 and {gn}%L t are sequences of vectors in Jt
and Jf, respectively, such that {/M+fif«}™=1 is a Cauchy sequence in #?.
We want to prove first that the sequence {gn}™= t is bounded. If it were not,
there would exist a subsequence {gnk}£L i and a unit vector hinJf such that
lim ||gf,J = oo and lim -J—^ = h.
k-*«> k-*«> \\9nk\\
(This depends, of course, on the compactness of the unit ball of jV.) However,
since the sequence {(l/||fif^||)(/nk + fifMk)}r=i would converge to 0, we would
have
This would imply that h is in both Jl and Jf and hence a contradiction.
Since the sequence {fif«}™=1 is bounded, we may extract a subsequence
{0«k}r=i sucn that lirtit-co gnk = g for some g in Jf. Therefore, since
is a Cauchy sequence, we see that {fnJ%L i is a Cauchy sequence and hence
converges to a vector/in Jl. Therefore, limn-,co(fn+g^)=f+g and thus
Jf+A~ is a closed subspace of Jf. ■
The following theorem contains the usual definition of Fredholm operators.
5.17 Theorem (Atkinson) If Jf is a Hilbert space, then T in £(Jf) is a
Fredholm operator if and only if the range of T is closed, dim ker T is finite,
and dim ker 7"* is finite.

Atkinson's Theorem 129
Proof If T is a Fredholm operator, then there exists an operator A in
and a compact operator K such that AT =I+K. If/is a vector in
the kernel of I+K, then (I+K)f= 0 implies that Kf=-f, and hence /is
in the range of .K. Thus,
kerT c ker,4r= ker(/+#) c ranK
and therefore by Lemma 5.8, the dimension of ker T is finite. By symmetry,
the dimension of ker 7"* is also finite. Moreover, by Theorem 5.9 there
exists a finite rank operator F such that \\K— F\\ <\. Hence for/in kerF,
we have
M|| \\Tf\\ £ \\ATf\\ = \\f+Kf\\ = \\f+Ff+Kf-Ff\\
> I/I - \\Kf-Ff\\ > ll/H/2.
Therefore, T is bounded below on kerF, which implies that T(kerF) is a
closed subspace of Jf (see the proof of Proposition 4.8). Since (ker.F)x is
finite dimensional, it follows from the preceding lemma that ran T=
T(kerF)+ 7T(kerF)x] is a closed subspace of Jf.
Conversely, assume that the range of T is closed, dim ker T is finite, and
dim ker T* is finite. The operator To denned Tof=Tf from (ker!T)x to
ran!T is one-to-one and onto and hence by Theorem 1.42, is invertible. If
we define the operator S on Jf such that Sf= !T0~ V for /in ran !T and Sf= 0
for /orthogonal to ran!T, then Sis bounded, ST~I-PU and TS = I-P2,
where /^ is the projection onto ker T and P2 is the projection onto (ran !T)X =
ker!T*. Therefore, n(S) is the inverse of n(T) in £(Jf)/£C(Jf), and hence
T is a Fredholm operator which completes the proof. ■
5.18 As we mentioned previously, the conclusion of the preceding theorem
is the classical definition of a Fredholm operator. Early in this century
several important classes of operators were shown to consist of Fredholm
operators. Moreover, if T is a Fredholm operator, then the solvability of the
equation Tf= g for a given g is equivalent to determining whether g is
orthogonal to the finite-dimensional subspace ker 7"*. Lastly, the space of
solutions of the equation Tf= g for a given g is finite dimensional.
At first thought the numbers dim ker T and dim ker T* would seem to
describe important properties of T, and indeed they do. It turns out, how-
however, that the difference of these two integers is of even greater importance,
since it is invariant under small perturbations of T. We shall refer to this
difference as the classical index, since we shall also introduce an abstract
index. We will eventually show that the two indices coincide.

130 5 Compact and Fredholm Operators, Index Theory
5.19 Definition If Jf is a Hilbert space, then the classical index j is the
function defined from ^(Jf) to 1 such that j(T) = dim kerT- dim kerT*.
For n in Z, set J^ = {7^6 &(#?): y(T) = «}■
We first show that J^q is invariant under compact perturbations, after
which we obtain the classical Fredholm alternative for compact operators.
5.20 Lemma If Jf is a Hilbert space, T is in &0, and K is in
then T+K is in J^.
Since T is in J^o. there exists a partial isometry V by Proposition
4.37 with initial space equal to ker T and final space equal to ker T*. For
/in ker Tand g orthogonal to ker T, we have (T+V)(f+g) = Tg + Vf, and
since Tg is in ran T and F/is orthogonal to ran T then GH- V) (f+ g) = 0
implies f+g = 0. Thus 7"+F is one-to-one. Moreover, since 7"+Fis onto,
it follows that 7"+ F is invertible by the open mapping theorem.
Let F be a finite rank operator chosen such that ||/^-F|| <l/\\(T+V)~i\\.
Then T+V+K-F is invertible by Proposition 2.7, and hence T+K is the
perturbation of the invertible operator S= T+V+K-F by the finite rank
operator G = F— V. Now !T+ K is a Fredholm operator by Proposition
5.15, andj(T+K)=j(S+G)=j(I+S-1G), since
and
kerftS+G)*) = ker((I+S~1G)*S*) = S
Thus it is sufficient to show that j(I+ S~i G) = 0.
Since S~1G is a finite rank operator the subspaces ran^^'G) and
ran^-'G)* are finite dimensional, and hence the subspace Jl spanned by
them is finite dimensional. Clearly, (I+S~lG)J?c:J[, (I+S~XG)*J?<=:J!,
and (/+S~ 1G)/=/for/orthogonal to J(. If A is the operator on Jt defined
by Ag = (I+S~1G)g for g in ^M, then ker A = keril+S'^) and ker-4* =
ker^Z+if-'G)*). Since A is an operator on a finite-dimensional space, we
have dim ker A = dimker/1*, and therefore
dimCZ+S-'G) = dim((/+,S'-1G)*).
Thus, j(I+S~1G) = 0 and the proof is complete. ■
After recalling the definition of generalized eigenspace we will prove a
theorem describing properties of the spectrum of a compact operator.

The Fredholm Alternative 131
5.21 Definition If Jf is a Hilbert space and T is an operator in
then the generalized eigenspace Sk for the complex number k is the collec-
collection of vectors/such that (T—k)"f=0 for some integer n.
5.22 Theorem (Fredholm Alternative) If K is a compact operator on the
Hilbert space Jf, then o(K) is countable with 0 the only possible limit point,
and if k is a nonzero element of a(K), then k is an eigenvalue of K with finite
multiplicity and A is an eigenvalue of K* with the same multiplicity. More-
Moreover, the generalized eigenspace S\ for k is finite dimensional and has the
same dimension as the generalized eigenspace for K* for 1,
Proof If k is a nonzero complex number, then — kl is invertible, and
hence K—k is in & by Proposition 5.15. Therefore, if k is in o(K), then
ker(A— k) # {0}, and hence k is an eigenvalue of K of finite multiplicity.
Moreover, since j(K— k) = 0, we see that A is an eigenvalue of K* of the
same multiplicity.
If $x # ker(K—k)N for any integer N, then there exists an infinite ortho-
normal sequence {kBj}^=1 such that kBj is in Sn+l QSn. Since ||/T/:MJ|2 =
|A|2+||(/(:-A)/cMj||2 and the sequence {kBj}™=1 converges weakly to 0, it
follows from Proposition 5.6 that 0 = lim,=c0 flA'A:,, || > \k\ which is a contra-
contradiction. Thus Sk = ker(K—k)N for some integer N. Moreover, since (K—kf
is a compact perturbation of (—k)NI, (K—kf is a Fredholm operator with
index 0. Therefore,
dimker[(Jf-A)w] = dim ker[(Je*-*)"],
and since Sx = ker(K—k)N for some N by the finite dimensionality of Sx,
it follows that the dimension of the generalized eigenspace for K for k is
the same as that of the generalized eigenspace for K* for 1,
Finally, to show that o(K) is countable with 0 the only possible limit
point, it is sufficient to show that any sequence of distinct eigenvalues con-
converges to 0. Thus let {/lM}™=1 be a sequence of distinct eigenvalues and let
fB be an eigenvector of kn. If we let Jt^ denote the subspace spanned by
{/,,/2, ■•■,/n}, then Jlx g jB g jC c..., since the eigenvectors for distinct
eigenvalues are linearly independent. Let {fifM}™=1 be a sequence of unit
vectors chosen such that gn is in Jln and gn is orthogonal to J(B-V If h is a
vector in Jf, then h = Z™=i(^. 0«H«+fi'o. where go is orthogonal to all the
{«[.}-= i- Since \\h\\2 = Z"=il(^0j|2 + ll0ol|2 by Theorem 3.25, it follows
that limM-co(g(M,/!) = 0. Therefore, the sequence {fifB}™=1 converges weakly

132 5 Compact and Fredholtn Operators, Index Theory
to 0, and hence by Proposition 5.6 the sequence {KgB}™=1 converges to 0
in norm. Since gn is in J(n, there exist scalars {aj"= t such that gn = £JL t ajt,
and hence
Kgn = £ «t*ft = t «tkfi = 4 £ ok ft + "
i ii 1
where /;„ is in Jtn_^. Therefore,
Um|A,,|2 < limCKH^P + ll/y2) = U
n-+ co n-+ co «-♦ co
and the theorem follows. ■
5.23 Example We now return to a special integral operator, the Volterra
integral operator, and compute its spectrum. Let
f 1 if x > y
K(x,y) = for (x,y) in [0,1] x [0,1]
{0 if x < y
and let V be the corresponding integral operator denned on L2([0,1]) in
Section 5.12. We want to show that o(V) = {0}. If A were a nonzero number
in a(V), then since Fis compact by Section 5.12 it follows that X is an eigen-
eigenvalue for V. If/is an eigenvector of V for the eigenvalue X, then jo/(j) dy =
Xf(x). Thus, we have
(*)l < \'\f(y)\dy < C\f(y)\dy
Jo Jo
using the Cauchy-Schwarz inequality. Hence, for x1 in [0,1] and n > 0,
we have
^ J/fe)! dx2 < ^p
_
|A|n+1 n!
Since

Volterra Integral Operators 133
it follows that /= 0. Therefore, a nonzero X cannot be an eigenvalue, and
hence a(V) = {0}.
We digress to make a couple of comments.
5.24 Definition If Jf is a Hilbert space, then an operator T in £(Jf) is
quasinilpotent if o(T) = {0}.
Thus the Volterra operator is compact and quasinilpotent.
5.25 Example Let T be a quasinilpotent operator and 91 be the com-
commutative Banach subalgebra of £(Jf) generated by /, T, and (T— X) for
all nonzero X. Since T is not invertible, there exists a maximal ideal in 91
which contains T, and thus the corresponding multiplicative linear func-
functional (p satisfies (p(I) = 1 and (p(T) = 0. Moreover, since the values of a
multiplicative linear functional on 91 are determined by its values on the
generators and these are determined by its value at T, it follows from Corollary
2.36 that the maximal ideal space of 91 consists of just (p. In particular, this
example shows that the Gelfand representation for a commutative Banach
algebra may provide little aid in studying this algebra.
We now return to the study of Fredholm operators and define the abstract
index.
5.26 Definition If Jf is a Hilbert space, then the abstract index i is defined
from J^(Jf) to Aewyee^) such that i = yon, where y is the abstract index
for the Banach algebra
The following properties of the abstract index are immediate.
5.27 Proposition If #C is a Hilbert space and i is the abstract index, then
i is continuous and multiplicative, and i(T+K) = i(T) for T in ^(Jf) and
K in
Proof Straightforward. ■
5.28 We have denned two notions of index on the collection of Fredholm
operators: the classical index j from ^(Jf) to Z and the abstract index /
from J^pf) to A. Our objective is to show that these two notions are essen-
essentially the same. That is, we want to produce an isomorphism a from the

134 5 Compact and Fredholtn Operators, Index Theory
additive group Z onto A such that the following diagram commutes.
To produce a we will show that for each n, the set &„=] 1 (n) is connected.
Since i is continuous and A is discrete, / must be constant on 8Fn. Thus the
mapping denned by a(n) = i(T) for T in J% is well denned. The mapping
a is onto, since i is onto. Further, consideration of a special class of operators
shows that a is a homomorphism. Finally, the fact that S^o is invariant under
compact perturbations will be used to show that ker/ = ^0 and hence that
a is one-to-one.
Once this isomorphism is established, then the following results are
immediate corollaries: j is continuous and multiplicative, and is invariant
under compact perturbation.
We begin this program with the following proposition.
5.29 Proposition If 91 is a fF*-algebra of operators in 2C4?), then the
set of unitary operators in 91 is arcwise connected.
Proof Let U be a unitary operator in 91 and let £0^ be the H^*-algebra
generated by U. As in the proof of Theorem 4.65 there exists a decomposi-
decomposition of 2/e = Y,aSA®^a such that each 3Va reduces U and Ua= U\34?a
has a cyclic vector. Moreover, by Theorem 4.58 there exists a positive regular
Borel measure va with support contained in T and a functional calculus
denned from /."(vj onto WVa by an isometrical isomorphism from L2(va)
onto Jfa.
If we define the function ^ on T such that ^(e") = / for — n< l < n,
then \j/ is in each Lw(va), and there exists a sequence of polynomials {/?„}"= t
such that \\pn\\ < 7i and
(Use the Stone-Weierstrass theorem to approximate a function which agrees
with ^ on {e": —n + (\jn) </< 7i} and is continuous.) Consider the sequence
of operators
aeA

Connectedness of the Unitary Group in a W*-Algebra 135
in W&u and the operator H = Y*ea ® ^(UJ denned on Jf. Using the identi-
identification of 34?a as L2(va), it is easy to check that the sequence {pn{Uj}^=l
converges to ^(i/J in the strong operator topology. Since the operators
are uniformly bounded, it follows that the sequence converges strongly
to H. Therefore, H is in %RV and moreover e'H = U, since ^(Va) = Ua for
each a in A. If we define the function UA = eaii for k in [0,1], then for kx
and k2 in [0,1], we have
11^,-^J = Ilexp/A,//-expiry/1| = |exp/A1//G-exp/(A2-A1)y/)||
aeA
< ||1 -exp/Uz-A,)^ = |exp/Ai -exp/A2|,
and therefore Uk is a continuous function of L Thus the unitary operator
U is arcwise connected to Uo = I by unitary operators in 3C. ■
The following corollary is now easy to prove.
5.30 Corollary If Jf is a Hilbert space, then the collection of invertible
operators in i*pf) is arcwise connected.
Proof Let T be an invertible operator in £(Jf) with the polar decom-
decomposition T= UP. Since T is invertible, U is unitary and P is an invertible
positive operator. For k in [0,1] let Ux be an arc of unitary operators con-
connecting the identity operator Uo to U = U, and Px = (l — k) 1+ kP. Since
each Pk is bounded below, it is invertible and hence UA PA is an arc con-
connecting the identity operator to T. ■
Much more is true; a theorem of Kuiper [73] states that the collection
of invertible operators in £pf) for a countably infinite-dimensional Jtf is
a contractible topological space.
5.31 Corollary If Jf is a Hilbert space and T is an invertible operator
in 2(JV), then i(T) is the identity in A.
Proof Since n(T) is in the connected component Ao of the identity
in A, it is clear that i(T) is the identity in A. ■

136 5 Compact and Fredholm Operators, Index Theory
We now consider the connectedness of &n.
5.32 Theorem If Jf is a Hilbert space, then for each n in Z the set
is arcwise connected.
Proof Since ($rnBtf'))* = Jsr_M(Jf), it is sufficient to consider n > 0.
If n ^ 0 and 7 is in 3Fn, then dim ker 7 > dim ker 7*; thus there exists a
partial isometry V with initial space contained in ker 7 and range equal to
ker 7* = (ran 7)x. For each e > 0 it is clear that T+eV is onto and hence is
right invertible and
ker(T+eV) = ker(T) 0 init(F).
Hence it is sufficient to prove that if S and 7 are right invertible with
dim ker S = dim ker 7, then S and 7 can be connected by an arc of right
invertible operators in !Fn.
Let S and 7 be right invertible operators with dim ker S = dim ker 7.
Let U be a unitary operator chosen such that ker SU = ker 7 and t/^ be an
arc of unitary operators such that Uo = I and Ut = U. Then SU is connected
to S in &„ and hence we can assume that kernels of our two right invertible
operators are equal.
Hence, assume that S and 7 are right invertible operators with ker S =
ker 7. By Proposition 4.37, there exists an isometry W with range W =
(ker^)x = (ker7)x. Then the operators SW and TW are invertible and
hence by Corollary 5.30 there exists an arc of operators Jk for 0 < k < 1
such that /0 = SW and Jt = TW. Since WW* is the projection onto the
range of W, we see that SWW* = S and TWW* = T. Hence JkW* is an
arc of operators connecting S and 7, and the proof will be completed once
we show that each JkW* is in JsrM. Since (/jfT*)^/;'1) = /, it follows that
JkW* is right invertible and hence ker((JkW*)*)= {0}. Further since
ker(JkW*) = kerW* we see that j(JkW*) = n for all k. This completes the
proof. ■
5.33 Recall now the unilateral shift operator U+ on /2(Z+) introduced in
Section 4.36. It is easily established that kerU+ = {0}, while kert/+* =
{e0}. Since U+ is an isometry, its range is closed, and thus U+ is a Fredholm
operator and j(U+) = — 1.
Now define
£/+" n > 0,
£/*-" n < 0.

Connectedness of the Unitary Group in a W*-Algebm 137
Since for n > 0 we have ker Uty = {0} and
ker £/<?>* = kert/*<"> = V{eo.«i,.,Vi}>
it follows thaty(f/+)) = -«■ Similarly since U^* = t/;(B), we have j(U^) =
— n for all integers «. The following extension of this formula will be used
in showing that the map to be constructed is a homomorphism.
If m and n are integers, then j(U<?>t/(+M)) =j(Ut+))+j(V^)). We prove
this one case at a time. If m > 0 and n > 0 or m < 0 and n < 0, then t/
t/(f+B), and hence
> E/<">) = -m - n =
If m < 0 and n > 0, then
and again the formula holds. Lastly, if m^O and « <0, then
ker£/<+m>£/<+"> = kerf/*"" = Vfe^-.e-,-.}
and
kerCt/^t/^]* = ker(t/+mt/rT = kerC/;"t/*m = \/{eo.-.«V«-i},
and hence
The next lemma will be used in the proof of the main theorem to show
that each of the &n is open.
5.34 Lemma If Jf is a Hilbert space, then each of the sets J^o and
\}«**&« is open in £(*»)■
Proof Let T be in 3Fq and let F be a finite rank operator chosen such
that T+F is invertible. Then if X is an operator in C(Jf) which satisfies
|| T- X || < l/Kr+ZO"!, then A"+F is invertible by the proof of Proposi-
Proposition 2.7, and hence A' is in &0 by Lemma 5.20. Therefore, 3F0 is an open set.
If T is a Fredholm operator not in J^q, then there exists as in the proof
of Lemma 5.20 a finite rank operator F such that 2"+ F is either left or right
invertible. By Proposition 2.7 there exists e > 0 such that if X is an operator
in fi(Jf) such that ||7'+F-A'|| <e, then X is either left or right invertible
but not invertible. Thus A' is a Fredholm operator of index not equal to 0
and therefore so is X—F by Lemma 5.20. Hence \Jn^0^n is also an open
subset of

138 5 Compact and Fredhohn Operators, Index Theory
We now state and prove the main theorem of the chapter.
5.35 Theorem If Jf is a Hilbert space, j is the classical index from
onto Z, and / is the abstract index from &{#?) onto A, then there exists an
isomorphism a from Z onto A such that a°j = /.
Proof Since the dimension of Jf is infinite, we have
isomorphic to Jf, and hence there is an operator on Jf unitarily equivalent
to /© U%°. Therefore, each &„ is nonempty and we can define a(n) — i(T),
where !Tis any operator in &'„. Moreover, a is well defined by Theorem 5.32.
Since / is onto, it follows that a is onto. Further, by the formula in Section
5.33, we have
a(m+n) = /(
= i(I® t/(-m))i(/© C/<-">) = a(m) ■ a(n),
and hence a is a homomorphism.
It remains only to show that a is one-to-one. Observe first that n{
is disjoint from n(\Jn+0&„), since if Tis in &0 and S is in &„ with n(S) =
n(T), then there exists Kin £C(Jf) such that S+K=T. However, Lemma
5.20 implies that j(T) = 0, and hence 71(^0) is disjoint from 7t(UM#0 J^J.
Since J^q and \Jn^o^n are open and n is an open map, it follows that
71(^0) and ^(Um^o^J are disjoint open sets. Therefore, ni&o) is an open
and closed subset of A and hence is equal to the connected component Ao
of the identity in A. Therefore, n takes J^q onto Ao and hence / takes J^o
onto the identity of A. Thus a is an isomorphism. ■
We now summarize what we have proved in the following theorem.
5.36 Theorem If Jf is a Hilbert space, then the components of
are precisely the sets {^n: neZ}. Moreover, the classical index defined by
j(T) = dim ker T - dim ker T*
is a continuous homomorphism from &(#e) onto Z which is invariant under
compact perturbation.
We continue now with the study of 2f£(jP) and £(Jf)/£e(Jf) as C*-
algebras. (Strictly speaking, £&(jf) is not a C*-algebra by our definition,
since it has no identity.) This requires that we first show that the quotient
of a C*-algebra by a two-sided ideal is again a C*-algebra. While this is
indeed true, it is much less trivial to prove than our previous results on

Quotient C*-Algebras 139
quotient objects. We begin by considering the abelian case which will be
used as a lemma in the proof of the main result.
5.37 Lemma If 31 is an abelian C*-algebra and 3 is a closed ideal in 31,
then 3 is self-adjoint and the natural map n induces an involution on the
quotient algebra 91/3 with respect to which it is a C*-algebra.
Proof Let X be the maximal ideal space of 31 and M be the maximal
ideal space of the commutative Banach algebra 31/3. Each m in M defines a
multiplicative linear function m°n on 31. The map \j/(m) = m°n is a homeo-
morphism of M onto a closed subset of X. Further, this homeomorphism \j/
defines a homomorphism *F from C(X) to C(M) by
*¥{k)=k°il/ and ker^F = {k e C(X): k{xjj{m)) = 0 for me M}.
Moreover, ^oT^ = ra/3o7i and by the Gelfand-Naimark theorem D.29),
we know that ra is a *-isometrical isomorphism. Therefore, kerrc =
r^'(ker^) and thus the kernel is self-adjoint and n induces an involution
on 31/3. Moreover, it is clear that 31/3 is *-isometrically isomorphic to
C(M) and hence a C*-algebra. ■
We now proceed to the main result about quotient algebras.
5.38 Theorem If 31 is a C*-algebra and 3 is a closed two-sided ideal in 31,
then 3 is self-adjoint and the quotient algebra 31/3 is a C*-algebra with
respect to the involution induced by the natural map.
Proof We begin by showing that 3 is self-adjoint. For T an element of
3, set// = T*T. For k > 0 the element kH2 is positive, since it is the square
of a self-adjoint element, and therefore kH2 + / is invertible in 31. More-
Moreover, rearranging the identity (kH2 + I)(kH2 + I)~i = / shows that the
element defined by
is in 3. Moreover, if we set S^ = TU^+T, then SfS^ = (kH2 + I)~2H and
from the functional calculus for (£H, we have
US/Sill = \\(kH2+I)-2H\\ < supl * ■ xea(H)
x

140 5 Compact and Fredholm Operators, Index Theory
where the last inequality is obtained by maximizing the function q>(x) =
x(Ax2+l)~2 on U. Taking adjoints of the equation Sx = TUX + T and re-
rearranging yields
lim \\T*+UXT*\\ = lim ||^|| ^ lim —^ = 0.
Since each — Ux T* is in 3, and 3 is closed, we have T* in 3 and hence 3
is self-adjoint.
Now 91/3 is a Banach algebra and the mapping (^4 + 3)* = /4* + 3 is
the involution induced by the natural map. Since we have ||C4 + 3)*|| =
m + 3||, it follows that
UG4+3)*U+3)|| < ||C4+3)*|| M+3|| = M + 3||2,
and hence only the reverse inequality remains to be proved before we can
conclude that 91/3 is a C*-algebra.
Returning to the previous notation, if we set ft = (£H n 3, then ft is a
closed two-sided self-adjoint ideal in the abelian C*-algebra (£H, and hence
(£H/ft is a C*-algebra by the previous lemma. If we consider the subalgebra
7r((£H) = (£H/3 of 91/3, then there is a natural map n' from (£H/M onto (£H/3.
Moreover, it follows from the definition of M and the quotient norm that
n' is a contractive isomorphism. Therefore, for A in (£H, we have
and hence
M+ ail^ ^ M + 3||fa/3 ^ PeH/s(A + Z) = PzH/*(A+ ft) = \\A+ ft|CH/
Thus, n' is an isometry and GH/3 is an abelian C*-algebra. Lastly, it follows
from the functional calculus for (£H/3 and the special form of the function
^{x)=Xx2l(\+Xx2), that
\\n(U,)\\= k\\n(H)\\2(\+X\\n(H)fy\
To complete the proof we use the identity T= SX—TUX to obtain
Setting k = 1/C ||7i(#)||2), we further obtain the inequality

Quotient C*-Algebras 141
and finally
||7r(r>||2 < \\n(H)\\ = \\n{r>*n{T)\\-
Therefore, 91/3 is a C*-algebra. ■
We now consider the algebra CG(Jf) which plays a fundamental role in
the study of C*-algebras. The following result has several important con-
consequences. A subset © of £(Jf) is said to be irreducible if no proper closed
subspace is reducing for all S in S>.
5.39 Theorem If 91 is an irreducible C*-algebra contained in C(Jf) such
that 91 n £e(Jf) ^ 0, then £e(Jf) is contained in 91.
Proof If K is a nonzero compact operator in 91, then (K+K*) and
(l/i)(K—K*) are compact self-adjoint operators in 91. Moreover, since at
least one is not zero, there exists a nonzero compact self-adjoint operator
H in 91. If A is a nonzero eigenvalue for H which it must have, then the pro-
projection onto the eigenspace for k is in (£H and hence in 91 using the functional
calculus. Moreover, since this eigenspace is finite dimensional by Theorem
5.22, we see that 91 must contain a nonzero finite rank projection.
Let £ be a nonzero finite rank projection in 91 of minimum rank. Con-
Consider the closed subalgebra 9I£ = {EAE: A e 91} of 91 as a subalgebra of
EJ4?. If any self-adjoint operator in 9I£ were not a constant multiple of a
scalar, then 9l£ and hence 91 would contain a spectral projection for this
operator and hence a projection of smaller rank than E. Therefore, the
algebra 9l£ must consist of scalar multiples of E. Suppose the rank of E is
greater than one and x and y are linearly independent vectors in its range.
Since the closure of {Ax: A e 3f} is a reducing subspace for 91, it follows
that it must be dense in Jf. Therefore, there must exist a sequence {/4B}™= j
in 91 such that lim,,^ ||/4Bx—j|| =0, and hence lim,,^ \\EAnEx—y\\ = 0.
Since x and y are linearly independent, the sequence {EAnE}^L1 cannot
consist of scalar multiples of E. Therefore, E must have rank one.
We next show that every rank one operator is in 91 which will imply
by Theorem 5.9 that £C(Jf) is in 91. For x and y in Jf, let Ty%x be the rank
one operator defined by Tyx(z) = (z,x)y. For a unit vector x0 in E^f, if
{/4B}™=1 is chosen as above such that lim,,.^ |/4Bx0—j|| = 0, then the
sequence {AnTXOyXo}^L1 is contained in 91 and Hmn^ooAnTXOiXo = T^xo.
Similarly, using adjoints, we obtain that TXo^x is in 91 and hence finally that
TywX = T^Xo TXOtX is in 91. Thus £G(Jf) is contained in 91 and the proof is
complete. ■

142 5 Compact and Fredholm Operators, Index Theory
One of the consequences of this result is that we are able to determine
all representations of the algebra
5.40 Theorem If $ is a *-homomorphism of QfS.(MT) into £(Jf), then
there exists a unique direct sum decomposition Jf = ^0®YM£a ® ^*->
such that each JfB reduces $(CG(Jf)), the restriction <&(T)\X'o = 0 for
T in £G(Jf), and there exists an isometrical isomorphism Ua from Jf onto
Jfa for a in ^4 such that <&(T)\jfa = Ua TUa* for T in
Proo/ If $ is not an isomorphism, then ker<£ is a closed two-sided
ideal in £C(Jf) and hence must equal £C(jf), in which case <S>(T) = 0 for
T in £C(Jf). Thus, if we set Jf0 = Jf, the theorem is proved. Hence, we
may assume that <£ is an isometrical isomorphism.
Now let {e,},e/ be an orthonormal basis for Jf and let Pt be the pro-
projection onto the subspace spanned by ef. Then Et = <£(/\) is a projection
on Jf. Choose a distinguished element 0 in / and define Vt on Jf for i in /
such that f-fCEjejAye,) = Aoe,-. It is obvious that F; is a partial isometry
with V[Vi* = Pi and F;*Fj = Po. Hence Wt is a partial isometry on Jf and
Wi*Wl = Eo and WtW-* = E{. Let {x0a}oe^ be an orthonormal basis for the
range of Eo and set jc" = WiX0". It is easy to see that each x? is in the range
of Et and that {-*f},-ei, aeA is an orthonormal subset of 3f. Let Jfa denote
the closed subspace of jf spanned by the {-*f}ie/. The {^a}aeA are pair-
wise orthogonal and hence we can consider the closed subspace £ae,, © 3fa
of Jf. Lastly, let jf0 denote the orthogonal complement of this subspace.
We want to show that the subspaces {^"a}ae^ u @} have the properties ascribed
to them in the statement of the theorem.
Since V{Vj* is the rank one operator on Jf taking es onto et, it is clear
that the norm-closed *-algebra generated by the {V^Ul is £G(Jf). There-
Therefore, $(£G(Jf)) is the norm-closed *-algebra generated by the {ff;},-e/ and
hence each Jfa reduces $(£G(Jf)). If we define a mapping Ua from Jf to
Jfa by Ua ex = x", then Ua extends to an isometrical isomorphism and
<S>(T)\CfTa = UaTUa*. Therefore, $ is spatially implemented on each Jfa.
Lastly, since each Jfa reduces <£(£(£(Jf)), it follows that Jf0 also is a
reducing subspace and since
while /—Ete/^i is the projection onto jf 0, we see that ^>G)|^To = ° f°r

Representations of the C*-Algebra of Compact Operators 143
Such a result has a partial extension to a broader class of C*-algebras.
5.41 Corollary If 91 is a C*-algebra on & which contains fie(Jf) and
$ is a *-homomorphism of 91 into C(jf) such that $|£G(Jf) is not zero
and <£(9l) is irreducible, then there exists an isometrical isomorphism U
from Jf onto Jf such that <&(A) = £//4£/* for A in 31.
If $(CC;(Jf)) is not irreducible, then by the preceding theorem
there exists a proper closed subspace jp of Jf such that the projection P
onto JP commutes with the operators <b(K) for K in CG(Jf), and there
exists an isometrical isomorphism U from Jf onto jPsuch that <&(K)\jp=
UKU* for K in 2d(J^). (The alternative leads to the conclusion that
<S>\2&(JF) = 0.) Then for A in 91 and # in 2£CP), we have
[/><I>(,4)-<I>(,4)P]<b(K) = P<t>(A)®(K) - ®(A)<t>(K)P
= P<b(AK) - <fr(AK)P = 0,
since AK is in £(£(Jf). If {Ea}aeA is a net of finite rank projections in Jf
increasing to the identity, then{$(£a)|jf'}ae^ converges strongly to P, and
thus we obtain P$(A)P= <b{A)P for every A in 91. Since $(91) is self-
adjoint and is assumed to be irreducible, this implies Jf' = Jf. Lastly, for
A in 91 and # in &&(#?), we have
<£(/O[<£(,4)-£/,4£/*] = <&{KA) - (UKU*)(UAU*)
= UKAU* - UKAU* = 0,
and again using a net of finite projections we obtain the fact that <£(/4) =
UAU* for A in 91. ■
These results enable us to determine the *-automorphisms of i*(Jf) and
5.42 Corollary If Jf is a Hilbert space, then $ is a *-automorphism of
2(Jif) if and only if there exists a unitary operator U in £(Jf) such that
<!>(A) = UAU* for A in
/Voo/ Immediate from the previous corollary. ■
Such an automorphism is said to be inner and hence all *-automorphisms
of £(Jf) are inner. A similar but significantly different result holds for

144 5 Compact and Fredholm Operators, Index Theory
5.43 Corollary If Jf is a Hilbert space, then O is a *-automorphism of
CG(Jf) if and only if there exists a unitary operator U in C(Jf) such that
®(K) = UKU* for K in
Proof Again immediate. ■
The difference in this case is that the unitary operator need not belong to
the algebra and hence the automorphism need not be inner. The algebra
£(£(Jf) has the property, however, that in each *-isomorphic image of the
algebra every *-automorphism is spatially implemented by a unitary operator
on the space.
We conclude with an observation concerning the Calkin algebra.
5.44 Theorem If $ is a *-isomorphism of the Calkin algebra £(jf )/£(£(,?f)
into £(Jf), then $(£(Jf)/£e(Jf)) is not a H^-algebra.
Proof If $(£(Jf)/£e(Jf)) were a H^-algebra, then the group of in-
invertible elements would be connected by Proposition 5.29, thus contradicting
Theorem 5.36. ■
Notes
The earliest results on compact operators are implicit in the studies of
Volterra and Fredholm on integral equations. The notion of compact oper-
operator is due to Hilbert, while it was F. Riesz who adopted an abstract point
of view and formulated the so called "Fredholm alternative." Further study
into certain classes of singular integral operators led Noether to introduce
the notion of index and implicitly the class of Fredholm operators. The
connection between this class and the Calkin algebra was made by Atkinson
[4]. Finally, Gohberg and Krein [48] systematized and extended the theory
of Fredholm operators to approximately its present form. The connection
between the components of the invertible elements in the Calkin algebra
and the index was first established by Cordes and Labrouse [23] and Coburn
and Lebow [22].
Further results including more detailed historical comments can be found
in Riesz and Sz.-Nagy [92], Maurin [79], Goldberg [51], and the expository
article of Gohberg and Krein [48]. The reader can also consult Lang [74]
or Palais [85] for a modern treatment of a slightly different flavor. Again
the results on C*-algebras can be found in Dixmier [28]. The proof of
Theorem 5.38 is taken from Naimark [80], whereas the short and clever
proof of Lemma 5.7 is due to Halmos [58].

Exercises 145
Exercises
5.7 If Jf is a Hilbert space and T is in £(>f), then T is compact if and
only if (T*T)l/2 is compact.
5.2 If T is a compact normal operator on Jf, then there exists a sequence
of complex numbers {k^^L t and a sequence {£„}"= t of pairwise orthogonal
finite rank projections such that lim,,^^^ = 0 and
IN
T- £ XnEn = 0.
5.3 If Jf is a Hilbert space, then £(£(Jf) is strongly dense in £(Jf).
5.4 If Jf is a Hilbert space, then the commutator ideal of £(Jf) is £(jf).
5.5 If .Kj and K2 are complex functions in L2([0,1] x [0,1]) and Tl and
7 the integral operators on L2 ([0,1]) with kernels Kt and K2, respectively,
then show that 7\* and 7\ 7 are integral operators and determine their
kernels.
5.6 Show that for every finite rank operator F on L2([0,1]), there exists
a kernel K in L2([0,1] x [0,1]) such that F=TK.
5.7 If TK is an integral operator on I? ([0,1]) with kernel K in
and {/,}"=! is an orthonormal basis for L2([0,1]), then the series
I \{TKfn,fn)\2
converges absolutely to §l$l\K(x,y)\2dxdy. (Hint: Consider the expansion
of K as an element of L2([0,1] x [0,1]) in terms of the orthonormal basis
5.8 Show that not every compact operator on L2([0,1]) is an integral
operator with kernel belonging to L2([0,1] x [0,1]).
5.9 (JVeyl) If T is a normal operator on the Hilbert space Jf and K is
a compact operator on Jf, then any k in a(T) but not in a(T+K) is an
isolated eigenvalue for T of finite multiplicity.
5.10 If T is a quasinilpotent operator on Jf for which T+ T* is in
then T is in

146 5 Compact and Fredholtn Operators, Index Theory
5.11 If T is an operator on Jf for which the algebraic dimension of the
linear space Jf/ranT is finite, then T has closed range.*
5.12 If T is an operator on Jf, then the set of k for which T— k is not Fred-
holm is compact and nonempty.
5.13 (Gohberg) If T is a Fredholm operator on the Hilbert space Jf, then
there exists e > 0 such that
a = dimker(r-A)
is constant for 0<|A|<e and a^dimkerT.* (Hint: For sufficiently
small k, {J-k)\S£ is right invertible, where ££ = C]^0T"^C is closed, and
ker(r-A) <= JSP.)
5.14 If T is an operator on Jf, then the function dimker(!T— k) is locally
constant on the open set on which T—k is Fredholm except for isolated
points at which it is larger.
5.75 If H is a self-adjoint operator on Jf and K is a compact operator on
Jf, then a{H+K)\a{H) consists of isolated eigenvalues of finite multiplicity.
5.76 If Jf is a Hilbert space and n is the natural map from 2(Jif) to the
Calkin algebra £(Jf)/£C(Jf) and Tis an operator on Mf, then n(T) is self-
adjoint if and only if T= H+ iK, where H is self-adjoint and K is compact.
Further, n{T) is unitary if and only if T= V+K, where K is compact and
either V or V* is an isometry for which /— VV* or /— V* V is finite rank.
What, if anything, can be said if n(T) is normal?**
5.77 (Weyl-von Neumann). If H is a self-adjoint operator on the separable
Hilbert space Jf, then there exists a compact self-adjoint operator K on Jf
such that H+ K has an orthonormal basis consisting of eigenvectors.* (Hint:
Show for every vector x in #C there exists a finite rank operator F of small
norm such that H+F has a finite-dimensional reducing subspace which
almost contains x and proceed to exhaust the space.)
5.18 If U is a unitary operator on the separable Hilbert space Jf, then
there exists a compact operator K such that V+K is unitary and Jf has an
orthonormal basis consisting of eigenvalues for V+K.
5.19 If t/+ is the unilateral shift on /2(Z+), then for any unitary operator V
on a separable Hilbert space Jf, there exists a compact operator 7£ on
/2(Z+) such that t/+ +7<: is unitarily equivalent to C/+ © K on /2(Z+)© Jf.*
(Hint: Consider the case of finite-dimensional Jf with the additional re-
requirement that K have small norm and use the preceding result to handle the
general case.)

Exercises 147
5.20 If Ft and F2 are isometries on the separable Hilbert space Jf and at
least one is not unitary, then there exists a compact perturbation of Ft which
is unitarily equivalent to F2 if and only if dimkerF,* = dimkerF2*.
Definition If 91 is a C*-algebra, then a state cp on 91 is a complex linear
function which satisfies (p(A*A) > 0 for A in 91 and <p(I) = 1.
5.21 If <p is a state on the C*-algebra 91, then (A,B) = <p(B*A) has the
properties of an inner product except (A, A) = 0 need not imply A = 0.
Moreover, <p is continuous and has norm 1. (Hint: Use a generalization of
the Cauchy-Schwarz inequality.)
5.22 If <p is a state on the C*-algebra 91, then 9i = {A e 91: p(/4*/4) = 0}
is a closed left ideal in 91. Further, q> induces an inner product on the quotient
space 91/91, such that n(B)(A + 9l) = BA + 91 defines a bounded operator for
B in 91. If we let nv(E) denote the extension of this operator to the completion
#?v of 91/91, then nv defines a *-homomorphism from 91 into S.(J^V).
5.23 If 91 is a C*-algebra contained in £(Jf) having the unit vector/as a
cyclic vector, then cp(A) = (Af,f) is a state on 91. Moreover, if nv is the
representation of 91 given by cp on 3^v, then there exists an isometrical iso-
isomorphism \j/ from Jf'v onto Jf such that A = ij/n^A)^*.
5.24 (Kreiri) If £C is a self-adjoint subspace of the C*-algebra 91 con-
containing the identity and (p0 is a positive linear functional on £C (that is,
(Po(A) 5= 0 for A > 0), satisfying #>0G) = 1> then tnere exists a state 9) on
91 extending q>0. (Hint: Use the Hahn-Banach theorem to extend (p0 to (p
and prove that <p is positive.)
5.25 If 91 is a C*-algebra and A is in 9t then there exists a state q> on 91 such
that <p(/4*/4) = ||/4||2. (Hint: Consider first the abelian subalgebra generated
by A*A.)
5.26 If 91 is a C*-algebra, then there exists a Hilbert space Jf and a *-iso-
metrical isomorphism n from 91 into £(Jf). Moreover, if 91 is separable in
the norm topology, then Jf can be chosen to be separable. (Hint: Find a
representation nA of 91 for which |7r^(/4)|| = \\A\\ for each A in 91 and con-
consider the direct sum.)
5.27 The collection of states on a C*-algebra 91 is a weak *-compact convex
subset of the dual of 91. Moreover, a state q> is an extreme point of the set
of all states if and only if 7^(91) is an irreducible subset of £(Jfv). Such states
are called pure states.

148 5 Compact and Fredholm Operators, Index Theory
5.28 Show that there are no proper closed two-sided ideals in
(Hint: Assume 3 were such an ideal and show that a representation of
£G(Jf)/3 given by Exercise 5.26 contradicts Theorem 5.40.)
5.29 If 51 is a Banach algebra with an involution, no identity, but satisfying
|| T*T\\ = || T||2 for T in 51, then 51 © C can be given a norm making it into
a C*-algebra. (Hint: Consider the operator norm of 91 © C acting on 51.)

6
The Hardy Spaces
6.1 In this chapter we study various properties of the spaces H1, H2, and
//"in preparation for our study of Toeplitz operators in the following
chapter. Due to the availability of several excellent accounts of this subject
(see Notes), we do not attempt a comprehensive treatment and proceed in
the main using the techniques which we have already introduced.
We begin by recalling some pertinent definitions from earlier chapters.
For n in Z let ■£„ be the function on T denned by xB(e/e) = e'"9. For p = 1,2, oo,
we define the Hardy space:
f C2n 1
Hp = \feLp(T):\ f(e'e)xn(eie) dO = 0 for n > 0}.
I Jo J
It is easy to see that each Hp is a closed subspace of the corresponding
LP(T), and hence is a Banach space. Moreover, since {z«}neZ is an ortho-
normal basis for L2(T), it follows that H2 is the closure in the L2-norm of
the analytic trigonometric polynomials ^+. The closure of ^+ in C(T) is
the disk algebra A with maximal ideal space equal to the closed unit disk D.
Lastly, recall the representation of L™ (T) into £ (L2 (T)) given by the mapping
(p-*Mv, where Mv is the multiplication operator denned by Mlj>f=(pf
for / in I? (T).
We begin with the following result which we use to show that H™ is an
algebra.
149

150 6 The Hardy Spaces
6.2 Proposition If (p is in L°°(T), then H2 is an invariant subspace for
Mv if and only if (p is in H°°.
Proof If Mv H2 is contained in H2, then (p ■ 1 is in H2, since 1 is in H2,
and hence 9) is in i?00. Conversely, if 9) is in T/00, then q>&'+ is contained in
#2, since for p = Y!j=oajXj in ^+, we have
PW)*. «=!«; PW, <# = 0 for n > 0.
JO j=0 JO
Lastly, since H2 is the closure of SP+, we have q>H2 contained in H2 which
completes the proof. ■
6.3 Corollary The space H™ is an algebra.
Proof If p and \ji are in 7/00, then M^H2 = MV(M^H2) cM,i?2c
H2 by the proposition, which then implies that qn]/ is in H™. Thus 7/00 is
an algebra. ■
The following result is essentially the uniqueness of the Fourier-Stieltjes
transform for measures on T.
6.4 Theorem If ju is in the space M(T) of Borel measures on T and
Jt Xr,dfi = 0 for n in Z, then ju = 0.
Since the linear span of the functions {/„}„<= z is uniformly dense
in C(T) and M(T) is the dual of C(T), the measure ju represents the zero
functional and hence must be the zero measure. ■
6.5 Corollary If/ is a function in L1 (T) such that
0 = 0 for «inZ,
Jo
jo
then/= 0 a.e.
Proof If we define the measure ju on T such that fi(E) = J£/(el9) dG,
then our hypotheses become JT xndfi = O for n in Z, and hence ju = 0 by the
preceding result. Therefore,/= 0 a.e. ■
6.6 Corollary If / is a real-valued function in Hl, then /= a a.e. for
some a in U.

Reducing Subspaces of Unitary Operators 151
Proof If we set a = A/2ti) JoVC^) dO, then a is real and
f
Since /— a is real valued, taking the complex conjugate of the preceding
equation yields
\2\f-«) In dO = [2\f- a)x-nde = 0 for n>0.
Jo Jo
Combining this with the previous identity yields
I (/— a)xnap = 0 for alln,
Jo
and hence /= a a.e. |
6.7 Corollary If both /and /are in H \ then /= a a.e. for some a in C.
Proof Apply the previous corollary to the real-valued functions i(/+/)
and i (/-/)// which are in Hl by hypothesis. ■
We now consider the characterization of the invariant subspaces of
certain unitary operators. It was the results of Beurling on a special case of
this problem which led to much of the modern work on function algebras
and, in particular, to the recent interest in the Hardy spaces.
6.8 Theorem If ju is a positive Borel measure on T, then a closed sub-
space J( of I? (ju) satisfies X\ ^ = ^ if and only if there exists a Borel subset
E of T such that
M = LE2(pi) = {/e I}(ii):f(e") = 0 for e" $ E}.
Proof If M = L£2 (jj), then clearly x% JK — •&■ Conversely, if x% JM = ^,
then it follows that J( = X-\"tiJ^ — X-i^ anc^ hence Jt is a reducing
subspace for the operator MX1 on I? (ju). Therefore, if F denotes the pro-
projection onto M, then F commutes with MX1 by Proposition 4.42 and hence
with Mv for q> in C(T). Combining Corollary 4.53 with Propositions 4.22
and 4.51 allows us to conclude that F is of the form Mv for some (p in L00 (p),
and hence the result follows. ■
The role of H2 in the general theory is established in the following
description of the simply invariant subspaces for MXi.

152 6 The Hardy Spaces
6.9 Theorem If ju is a positive Borel measure on T, then a nontrivial
closed subspace Jt of L2(ju) satisfies Xi^ <^ <M and On^oXn^ — {0} if
and only if there exists a Borel function (p such that \(p\2dfi = d6/2n and
M = (pH2.
Proof If (f> is a Borel function satisfying \g>\2 dfi = d6/2n, then the
function yVf= (pf is ju-measurable for/in H2 and
>=ll/l|22.
Thus the image Jl of H2 under the isometry *P is a closed subspace of I? (ju)
and is invariant for MXi, since ZiC^/) = ^(Xif)- Lastly, we have
r ~i
{0}
0X^ = ^1 f]xnH2]=
and hence M is a simply invariant subspace for MXl.
Conversely, suppose J is a nontrivial closed invariant subspace for
MXl which satisfies P)«==oXn^ = {0}- Then££ = MQi\<M ^ nontrivial and
Xn^C = Xn^OXn+i^> since multiplication by Xi is an isometry on L2(ju).
Therefore, the subspace £™=o ®Xn^ 's contained in ^, and an easy argu-
argument reveals ^e(Z™=o©Z»-^) to be C)r»oXn^ and hence {0}.
If (p is a unit vector in JS?, then ^o is orthogonal to xnj{ and hence to xn 9
for « > 0, and thus we have
0 = (<P,XnV) = I 1^1 X«dfi for k > 0.
Jt
Combining Theorem 6.4 and Corollary 6.6, we see that \(p\2dfi = dGj2n.
Now suppose ££ has dimension greater than one and <p' is a unit vector in ££
orthogonal to (p. In this case, we have
,Xm(p') = I <P<P~'Xn-
0 = (xn<P,Xm(p') = I <P<P~'Xn-mdn f°r n,m>0,
and thus Jt^^v = 0 for A: in Z, where </v = (pqj'di*. Therefore, q>(p' = 0 ju a.e.
Combining this with the fact that \(p\2dfi = \<p'\2dn leads to a contra-
contradiction, and hence S£ is one dimensional. Thus we obtain that (p^+ is dense
in M and hence M = <pH2, which completes the proof. ■
The case of the preceding theorem considered by Beurling will be given
after the following definition.
6.10 Definition A function <p in 7/00 is an inner function if \(p\ = 1 a.e.

Betiding1 s Theorem 153
6.11 Corollary (Beurling) If TXI = MXl\H2, then a nontrivial closed sub-
space Jl of H2 is invariant for TXl if and only if Jl = q>H2 for some inner
function (p.
Proof If (p is an inner function, then (p0>+ is contained in H™, since
the latter is an algebra, and is therefore contained in H2. Since q>H2 is the
closure of <pSP+, we see that <pH2 is a closed invariant subspace for TX1.
Conversely, if Jl is a nontrivial closed invariant subspace for rzi, then
Jl satisfies the hypotheses of the preceding theorem for dfi = d6/2n, and
hence there exists a measurable function <p such that Jl = <pH2 and
\(p\2de/2n = d6/2n.
Therefore, \(p\ = 1 a.e., and since 1 is in H2 we see that (p = (p-l is in H2;
thus (p is an inner function. ■
A general invariant subspace for MXi on L2(ju) need not be of the form
covered by either of the preceding two theorems. The following result enables
us to reduce the general case to these, however.
6.12 Theorem If ju is a positive Borel measure on T, then a closed in-
invariant subspace Jl for MXl has a unique direct sum decomposition Jl =
Jtx © Jl2 such that each of Jl± and Jl2 is invariant for MXI, X\ Jl\ = Jl\,
Proof If we set Jl^ = f)n^oxnJl, then Jl\ is a closed invariant sub-
subspace for MXI satisfying X\^\ = -M\- To prove the latter statement observe
that a function/is in Jlx if and only if it can be written in the form ing for
some g in Jl for each n > 0. Now if we set Jl2 = Jl Q Jlu then a function/
in Jl is in Jl2 if and only if (f,g) = 0 for all g in J/v Since 0 = (f,g) =
(XxfliQ) and Zi^i = */#j, it follows that x^fis in */#2 and hence Jl2 is
invariant for M^. If/is in f]n^oXn-^2-> then it is in ./#, and hence/= 0.
Thus the proof is complete. ■
Although we could combine the three preceding theorems to obtain a
complete description of the invariant subspaces for MXI, the statement would
be very unwieldy and hence we omit it.
The preceding theorems correspond to the multiplicity one case of certain
structure theorems for isometries (see [66], [58]).
To illustrate the power of the preceding results we obtain as corollaries
the following theorems which will be important in what follows.

154 6 The Hardy Spaces
6.13 Theorem (F. and M. Riesz) If/is a nonzero function in H2, then the
set {e" £ T:/(e") = 0} has measure zero.
Proof Set E = {e" e T:/(e") = 0} and define
JK = {g e H2: flf(c") = 0 for e" £ E}.
It is clear that J( is a closed invariant subspace for 7^, which is nontrivial
since / is in it. Hence, by Beurling's theorem there exists an inner function
(/> such that M = (pH2. Since 1 is in H2, it follows that cp is in Jl and hence
that E is contained in {e" e T: <p(e") = 0}. Since |#>| = la.e., the result
follows. ■
6.14 Theorem (F. and M. Riesz) If v is a Borel measure on T such that
JtZ«^v = 0 for k > 0, then v is absolutely continuous and there exists / in
H1 such that dv=fd9.
Proof If fi denotes the total variation of v, then there exists a Borel
function \j/ such that dv = \j/dn and \\j/\ = la.e. with respect to p.. If J(
denotes the closed subspace of L?(n) spanned by {/„:«> 0}, then
Jt
and hence ^ is orthogonal to M in L2(/i).
Suppose J( = Jtx ©*/#2 is tne decomposition given by Theorem 6.12.
If E is the Borel subset of T given by Theorem 6.8 such that Jf^ = L£2(ju),
then we have
Ai(£)
= f
since ^/E is in J, and \j/ is orthogonal to M. Therefore, M^ = {0} and
hence there exists a ju-measurable function tp such that J( = ^o//2 and
|(/)|2 dfi = J0/27I by Theorem 6.9. Since ^ is in M, it follows that there exists
g in H2 such that Xi = <P0 a.e. with respect to ju, and since (p ^ 0 ju a.e., we
have that ju is mutually absolutely continuous with Lebesgue measure. If
/is a function in L^T) such that dv =fd6, then the hypotheses imply that
/ is in H1, and hence the proof is complete. ■
Actually, the statements of the preceding two theorems can be com-
combined into one: an analytic measure is mutually absolutely continuous with
respect to Lebesgue measure.

The Maximal Ideal Space ofHm 155
6.15 We now turn to the investigation of the maximal ideal space Mx
of the commutative Banach algebra //".We begin by imbedding the open
unit disk D in Mm. For z in D define the bounded linear functional q>z on
H1 such that
Since the function 1/A -ze~l") is in L°°(T) and H1 is contained in L\T),
it follows that tpz is a bounded linear functional on H1. Moreover, since
1/A —ze~ie) = Z"=oe"'"9z" and tne latter series converges absolutely, we see
that
Thus, if p is an analytic trigonometric polynomial, then (pz(p)=p(z) and
hence cpz is a multiplicative linear functional on 3P+. To show that q>z is
multiplicative on H™ we proceed as follows.
6.16 Lemma If/ and g are in H2 and z is in D, then/g is in H1 and
Proof Let {/?„}"= t and {#„}"= t be sequences of analytic trigonometric
polynomials such that
lim \\f-pn\\2 = lim \\9-qJi\2 = 0.
k-*co n-*oo
Since the product of two L2-functions is in L1, we have
Wf9-p»q«h < Wf9-p«9\\i+ \\p«9-p«q«\\i
and hence l^^^ \\fg—puqu\\i = 0. Therefore, since each pnqtt is in H1 we
have fg in H1. Moreover, since cpz is continuous, we have
<Pz(fg) = I'm (pz(p«q«) = I'm %(/?„) lim ^z(gJ = (pz(J)(pz{Q)- ■
«-*CO «-*CO K-+CO
With these preliminary considerations taken care of, we can now imbed
D in M^.
6.17 Theorem For z in D the restriction of cpz to H™ is a multiplicative
linear functional on i?00. Moreover, the mapping FfromD into M^ denned
by F(z) = cpz is a homeomorphism.

156 6 The Hardy Spaces
Proof That cpz restricted to //°° is a multiplicative linear functional
follows from the preceding lemma.
Since for a fixed/in Hl, the function (pz(f) is analytic in z, it follows
that F is continuous. Moreover, since (pz(%,) = z, it follows that F is one-
to-one. Lastly, if {tpzJaEA ls a net in Mm converging to cpz, then
limza = lim^O^) = (pz{Xi) = z,
and hence F is a homeomorphism. ■
From now on we shall simply identify D as a subset of Mx. Further, we
shall denote the Gelfand transform of a function / in H™ by /. Note that
/|D is analytic. Moreover, for /in H1 we shall let / denote the function
defined on D by/(z) = cpz(f). This dual use of the A-notation should cause
no confusion.
The maximal ideal space Mx is quite large and is extremely complex.
The deepest result concerning M^ is the corona theorem of Carleson, stating
that D is dense in M^. Although the proof of this result has been somewhat
clarified (see [15], [39]) it is still quite difficult and we do not consider it
in this book.
Due to the complexity of Af„ it is not feasible to determine the spectrum
of a function / in i?00 using /, but it follows from the corona theorem that
the spectrum of/is equal to the closure of/(D). Fortunately, a direct proof
of this result is not difficult.
6.18 Theorem If cp is a function in H™, then q> is invertible in Hw if and
only if 0|O is bounded away from zero.
Proof If q> is invertible in i?00, then (p is nonvanishing on the compact
space Mw and hence |0(z)| > e > 0 for z in D. Conversely, if |0(z)| > e > 0
for z in D and we set \j/{z) = l/<Hz), then \j/ is analytic and bounded by 1/e
on D. Thus ^ has a Taylor series expansion \j/(z) = Z™=ofl«z"' which con-
converges in D. Since for 0 < r < 1, we have
it follows that Zr=oK|2 < 1/e2. Therefore, there exists a function/in H2
If <P = Z™=o^«Z« is the orthonormal expansion of q> as an element of
H2, then 0(z) = :C°=(Az" for z in D. Since <p-(z)$(z) = 1, it follows that

The Maximal Ideal Space ofH™ 157
^ D. Therefore, ZS.o&<A*«-*)*" =
for z in D, and hence the uniqueness of power series implies that
{ 1 if n = 0,
Z **c»-* =
0 if n > 0.
Since
lim
N-tac
<P~ Z *»
= lim
2 M-»co
M
- z «m
= o,
we have that
lim
TV-* go
/A- \ / N
- Z fe»z» Z c»
\« = 0 / \m = 0
= o,
which implies that
II
lim
Therefore,
0 for cn =
i.
1 if k = 0,
0 if k 5* 0,
and hence <p/= 1 by Corollary 6.6. It remains only to show that/is in L°°(T)
and this follows from the fact that the functions {/.}re@,i) are uniformly
bounded by 1/e, where/r(eir) =/(reit), and the fact that lim,^ ||/-/r||2 = 0
Thus/is an inverse for cp which lies in H™. ■
The preceding proof was complicated by the fact that we have not in-
investigated the precise relation between the function / on D and the function
/on T. It can be shown that for/in H1 we have limr_>1/(re1'') =/(e") for
almost all e" in T. We do not prove this but leave it as an exercise (see Exer-
Exercise 6.23).
Observe that we proved in the last paragraph of the preceding proof
that if/is in H2 and/ is bounded on D, then/is in H°°.
We give another characterization of invertibility for functions in W"
which will be used in the following chapter, but first we need a definition.
6.19 Definition A function/in H2 is an outer function if clos|73s+] = H2.
An alternate definition is that outer functions are those functions in H2
which are cyclic vectors for the operator Txt which is multiplication by Xi
onH2.

158 6 The Hardy Spaces
6.20 Proposition A function cp in H™ is invertible in H™ if and only if
cp is invertible in L00 and is an outer function.
Proof If l/(/> is in //", then obviously cp is invertible in L00. Moreover,
since
clos [>£?+] = cpH2 =>
it follows that cp is an outer function. Conversely, if l/cp is in L00 A), and cp
is an outer function, then cpH2 = clos[cp&'+'] = H2. Therefore, there exists
a function \j/ in H2 such that cp\j/ = 1, and hence l/cp = ^ is in H2. Thus,
l/cp is in i?00 and the proof is complete. ■
Note, in particular, that by combining the last two results we see that
an outer function can not vanish on D. The property of being an outer func-
function, however, is more subtle than this.
The following result shows one of the fundamental uses of inner and
outer functions.
6.21 Proposition If/is a nonzero function in H2, then there exist inner
and outer functions cp and g such that/= cpg. Moreover,/is in H°° if and
only if g is in H™.
Proof If we set M = closQ/3s+], then J is a nontrivial closed in-
invariant subspace for TXl and hence by Beurling's Theorem 6.11 is of the
form cpH2 for some inner function cp. Since/is in jU, there must exist g in
H2 such that/= cpg. If we set Jf = closQ/^V], then again there exists an
inner function \j/ such that Jf = \j/H2. Then the inclusion/^. = cpg&>+ a
cp\j/H2 implies cpH2 = closC/"^] c: cp\j/H2, and hence there must exist h in
H2 such that cp = cpij/h. Since cp and \j/ are inner functions, it follows that
$ = h and therefore \j/ is constant by Corollary 6.7. Hence, clos[fif^+] =
H2 and g is an outer function. Lastly, since |/| = \g\, we see that/is in W"
if and only if g is. ■
We next show that the modulus of an outer function determines it up to
a constant as a corollary to the following proposition.
6.22 Proposition If g and h are functions in H2 such that g is outer, then
\h\ < \g\ if and only if there exists a function kin H2 such that h = gk and

The Modulus of Outer Functions 159
Proof If h = gk and \k\ < 1, then clearly \h\ < \g\. Conversely, if g is
an outer function, then there exists a sequence of analytic trigonometric
polynomials {/?„}"= i such that lim^^ ||l-p«fif||2 = 0. If \h\ < \g\, then we
have
Wn-hPmU = ^
and hence {pnh}™=1 is a Cauchy sequence. Thus the sequence {pnh}™=l
converges to a function k in H2, and
^IK < lim ||ff|2||*-jPll/i||2 + lim |ffA-l||2||/i||2 =0.
K-+CO It-* 00
Therefore, gk = h and the proof is complete. ■
6.23 Corollary If g^ and g2 are outer functions in H2 such that \g^\ =
\g2\, then g1 = kg2 for some complex number of modulus one.
Proof By the preceding result there exist functions h and kin H2 such
that \h\, \k\ < 1, g1 = hg2, and g2 = kgt. This implies g± = khgu and by
Theorem 6.13 we have that kh=\. Thus h = R, and hence both h and E
are in H2. Therefore, h is constant by Corollary 6.7 and the result follows. ■
The question of which nonnegative functions in L2 can be the modulus
of a function in H2 is interesting from several points of view. Although an
elegant necessary and sufficient condition can be given, we leave this for the
exercises and obtain only those results which we shall need. Our first result
shows the equivalence of this question to another.
6.24 Theorem If/is a function in L2 (T), then there exists an outer function
g such that j/j = |#|a.e. if and only if c!osQ/3s+] is a simply invariant sub-
space for MX1.
Proof If |/| = \g\ for some outer function g, then /= q>g for some
unimodular function cp in L00 A). Then
clos [/£?+] = clos[^fif^+] = <pclos [#£?+] = cpH2,
and hence clos [/^+] is simply invariant.
Conversely, if clos [/^+] is simply invariant for MXI, then there exists a
unimodular function cp in L00 (T) by Theorem 6.9 such that clos 0/^+] =
cpH2. Since/is in clos[/^+], there must exist a function g in H2 such that

160 6 The Hardy Spaces
f= cpg. The proof is concluded either by applying Proposition 6.21 to g or
by showing that this g is outer. ■
6.25 Corollary If/is a function in L2(T) such that |/| > e > 0, then there
exists an outer function g such that \g\ = \f\.
Proof If we set Ji — clos [/&+], then MXI Ji is the closure of
{fp:pe<?+,p@) = 0}.
If we compute the distance from/to such an fp, we find that
f r\i-
2.TC Jo
and hence / is not in MXi Ji. Therefore, Ji is simply invariant and hence
the outer function exists by the preceding theorem. ■
We can also use the theorem to establish the following relation between
functions in H2 and H1.
6.26 Corollary If/is a function in H1, then there exists g in H2 such
that \g\2 = |/| a.e.
Proof Iff— 0, then take g = 0. If/is a nonzero function in H1, then
there exists h in L2 such that \h\2 = \f\. It is sufficient in view of the theorem
to show that clos[/^+] is a simply invariant subspace for MXl. Suppose it
is not. Then x~ N h is in clos [_/&'+] for N > 0, and hence there exists a sequence
of analytic trigonometric polynomials {/?„}"= t such that
lim \\pnh-X-Nh\\2 =0.
K-+CO
Since
In Jo
\de
we see that h2x~N is in the closure, clos^h2&'+'], of h23?+ in L^T). Since
there exists a unimodular function q> such that /= <ph2, we see that the
function X-Nf=(p(X-Nh2) is in q>clost\h2^+] = clos 1[/^s+] e H1 for
N>0. This implies/= 0, which is a contradiction. Thus clos[^+] is
simply invariant and the proof is complete. ■

The Conjugates ofH1 andLmIHom 161
6.27 Corollary If/is a function in. H1, then there exist functions gt and
g2 in H2 such that l^l = \g2\ = (|/|)I/2 and/=
Proof If g is an outer function such that \g\ = (|/j)I/2, then there exists
a sequence of analytic trigonometric polynomials {/?„}"= t such that
lim||ap.-l||2 =0.
K-+CO
Thus we have
WfP»-fPm\\l < \\f(P»-Pm)P»\\l + Wf(Ptt-pJPm\\l
< \\9pJ2 WgiPn-pJh + \\gpmh II0(a,-/Oil2,
and hence the sequence {^«2}™=1 is Cauchy in the I}(J) norm and there-
therefore converges to some function cp in H1. Extracting a subsequence, if
necessary, such that limfI_oo(/pII2)(e''') = (p(eh) a.e. and limB_00EpII)(efc) = 1
a.e., we see that cpg2 =/. Since \g2\ = \f\ a.e., we see that |<p| = 1 a.e.,
and thus the functions gx = cpg and g2 = g are in H2 and satisfy/= g^g2
and 1^x1 = 1^1 =(|/|)%- ■
6.28 Corollary The closure of ^+ in I}(J) is H1.
Proof [f/is in H1, then/= gtg2 with gt and g2 in H2. If {/?„}"= t and
{<7m}«°=i are sequences of analytic trigonometric polynomials chosen such
that Hm^^ Hfift-pJz = hm^^ ||sr2-?»ll2 = °> then {pnqj™=1 is a sequence
of analytic trigonometric polynomials such that lim,,-^ ||/— puqB\\ 1 = 0. ■
With this corollary we can determine the dual of the Banach space Hl.
Before stating this result we recall that Hop denotes the closed subspace
ifeH": y fa*/d9 = o| of H" for p=l,2,<x>.
6.29 Theorem There is a natural isometrical isomorphism between (H1)*
and
Proof Since // *' is contained in L1 (T), we obtain a contractive mapping
from L°°(T) into (H1)* such that
1 f2
= —
27T Jo
q> in L°°(T) and /in 771.

162 6 The Hardy Spaces
Moreover, from the Hahn-Banach theorem and the characterization of
I}(J)*, it follows that given $ in (H1)* there exists a function cp in L°°(T)
such that ||cp\\„ = ||$|| and *F(<p) = $. Thus the mapping *F is onto and
induces an isometrical isomorphism of L^CTQ/kerT onto (H1)*.
We must determine the kernel of *P. If cp is a function in ker*F, then
(71
= 0 for n > 0,
since each /„ is in H1 and hence <p is in flo™- Conversely, if cp lies in Hq00,
then [*F(<p)] (p) = 0 for each p in ^+, and hence <p is in ker *F by the preced-
preceding corollary. ■
Although ^(T) can be shown not to be a dual space, the subspace H1 is.
6.30 Theorem There is a natural isometrical isomorphism between
(C(J)/A)* and H0K
27T Jo
Proof If cp is a function in Hol, then the linear functional denned
for /inC(T)
is bounded and vanishes on A. Therefore, the mapping
<t>0{f+A) = <b(f) = l- Bnfcpd9
£tc Jo
is well denned on C(T)/A and hence defines an element of (C(T)//4)*. More-
Moreover, the mapping *F(<p) = <E>0 is clearly a contractive homomorphism of
Hol into (C(T)//4)*.
On the other hand, if $0 is a bounded linear functional on C(T)/A, then
the composition $0°n, where 7r is the natural homomorphism of C(T) onto
C(J)/A, defines an element v of C(T)* = M(T) such that
dv for /inC(T)
and ||v|| = ||<B>oII- Since this implies, in particular, that \Tgdv = 0 for g in A,
it follows from the F. and M. Riesz theorem that there exists a function cp
in Hq1 such that
= r r
2n Jo
dQ for/in C(T) and \\cp\\, = ||v|| = ||<&0||.
Therefore, the mapping *P is an isometrical isomorphism of Hol onto

Approximation by Quotients of Inner Functions 163
6.31 Observe that the natural mapping i of C(T)/A into its second dual
L^OO/tf00 (see Exercise 1.15) is i(f+A) =/+#°°. Since the natural map is
an isometry, it follows that /[C(T)//4] is a closed subspace of Eo(J)/Ha>.
Hence, the inverse image of this latter subspace under the natural homo-
morphism of L°°(T) onto E°(J)/Ha) is closed, and therefore the linear span
#°° + C(T) is a closed subspace of L°°(T). This proof that H°° + C(T) is
closed is due to Sarason [97].
The subspace Hco + C(T) is actually an algebra and is just one of a large
family of closed algebras which lie between H™ and L°°(T). Much of the
remainder of this chapter will be concerned with their study. We begin with
the following approximation theorem.
6.32 Theorem The collection 2L of functions in L°°(T) of the form \}/<p
for \j/ in i?00 and q> an inner function forms a dense subalgebra of L°°(T).
Proof That 3. is an algebra follows from the identities
and
Since M is a linear space and the simple step functions are dense in L00, to
conclude that J is dense in L°°(T) it suffices to show every characteristic
function is in clos^i!]. Thus let E be a measurable subset of T and let/
be a function in H2 such that
1 if e"e£,
2 if ««££.
The existence of such a function follows from Corollary 6.25. Moreover,
since / is bounded, it is in 7/00 and consequently so is 1 +/" for n > 0. If
l+f" = cpngn is a factorization given by Proposition 6.21, when q>n is an
inner and gn is an outer function, then \gn\ = jl +f"\ 5= i and hence l/gn is
in 77°° by Proposition 6.20. Therefore, the function 1/A +/") = (l/g„) ipn
is in 5, and since Mm^^ ||/£ —1/A +f")\\00 = 0, we see that IE is in
Thus, 3. is dense in L°°(T) by our previous remarks. ■
We next prove a certain uniqueness result.

164 6 The Hardy Spaces
6.33 Theorem (Gleason-Whitney) If $ is a multiplicative linear functional
on i?00 and Lt and L2 are positive linear functionals on L°°(T) such that
Ljtf00 = L^tf00 = 0>, then L, = L2.
If u is a real-valued function in L°°(T), then there exists an in-
vertible function <p in H™ by Proposition 6.20 and Corollary 6.25 such that
\cp\ = e". Since Lt and L2 are positive, we have
and
Multiplying, we obtain
1 = \<b(<p)\
and hence the function W(i) = L1(e)L2(e~"') defined for all real t has an
absolute minimum at t = 0. Since *F is a differentiable function of t by the
linearity and continuity of L, and L2, we obtain
Substituting r = 0 yields 0 = <F'(°) = LiM^W-^iO)^^). and hence
L^m) = L2(m) which completes the proof. ■
6.34 Theorem If 9T is a closed algebra satisfying #°° c 91 cz L00 (I), then
the maximal ideal space M^ of 91 is naturally homeomorphic to a subset
ofMK.
Proof If $ is a multiplicative linear functional on 91, then $1//°° is a
multiplicative linear functional on fl™ and hence we have a continuous
natural map rj from M^ into M^. Moreover, let $' denote any Hahn-Banach
extension of fl> to L00 (T). Since L00 (T) is isometrically isomorphic to C(ML«.)
by Theorem 2.64, $' is integration with respect to a Borel measure v on
Mt<» by the Riesz-Markov representation theorem (see Section 1.38). Since
v(ML«.) = $'(!)= 1 = II*'II = |v|(Aft«,), the functional $' is positive and
hence uniquely determined by <t>\Ha> by the previous result. Therefore, the
mapping r\ is one-to-one and hence a homeomorphism. ■
Observe that the maximal ideal space for 91 contains the maximal ideal
space for L00 (T) and, in fact, as we indicate in the problems, the latter is the
Silov boundary of 91.

Subalgebras between Hm andL 165
We now introduce some concrete examples of algebras lying between
#°° and L°°(T).
635 Definition If X is a semigroup of inner functions containing the con-
constant function 1, then the collection {ij/ip: il/eH™, <peX} is a subalgebra
of L°°(T) and the closure is denoted 9lE.
The argument that 91^ is an algebra is the same as was given in the proof
of Theorem 6.32.
We next observe that H°° + C(T) is one of these algebras.
6.36 Proposition If £(/) denotes the semigroup of inner functions
{*„: n > 0}, then ffl^, = H<° + C(T).
Proof Since the linear span H°° + C(T) is closed by Section 6.31, we
have #°° + C(T) =closco[#co + £?]. Lastly, since
the result follows. ■
The maximal idea! space of 91^ can be identified as a closed subset of
Ma by Theorem 6.34. The following more abstract result will enable us to
identify the subset.
6.37 Proposition Let A" be a compact Hausdorff space, 91 be a function
algebra contained in C(X) with maximal ideal space M, and £ be a semi-
semigroup of unimodular functions in 91. If 91^ is the algebra
clos{^>: ^ £ 91, cp e £}
and Mj is the maximal ideal space of 91^, then Mz can be identified with
{m £ M: \Q(m)\ = 1 for <p e £},
where ^ denotes the Gelfand transform.
Proof If *F is a multiplicative linear functional on 9Ij;, then <F|9I is a
multiphcative linear functional on 91, and hence »7(*F) = *F|9I defines a
continuous mapping from Mz into M. If vfl and *F2 are elements of ML
such that j/OFO = t](W2), then ^^91 = «F2|9I. Further, for q> in E, we have
A

166 6 The Hardy Spaces
and thus *F, = *F2- Therefore, r\ is a homeomorphism of M^ into M. More-
Moreover, since |*F(<p)| < ||<p|| = 1 and
1
we have |*F(<p)| = 1 for *F in Mz and cp in E; therefore, the range of r\ is con-
contained in
{m e M: \(p(m)\ = 1 for cp e E}
and only the reverse inclusion remains.
Let m be a point in M such that |0(w)| = I for every cp in E. If we define
<F on {ij/cp: ^ e 91, <p e E} such that ^(ij/cp) = $(m)@(m), then <F can easily
be shown to be multiplicative, and the inequality
|«P(^| = |#(#«)| \Q(m)\ = \Um)\ < ll^ll = il^ll
shows that *P can be extended to a multiplicative linear functional on 91^.
Since t\^¥) = m, the proof is complete. ■
6.38 Corollary If E is a semigroup of inner functions, then the maximal
ideal space Mz of SIj can be identified with
{meM^: \(p(m)\ = Iforcpel,}.
Proof Since L°°(T) = C(X) for some compact Hausdorff space X, the
result follows. ■
Using the Gleason-Whitney theorem we can determine the Gelfand
transform in the following sense.
6.39 Theorem There is a homeomorphism r\ from Af„ into the unit ball
of the dual of L°°(T) such that \j/(m) = ti(m){\j/) for \j/ in any algebra 91 lying
between H°° and L°°(T) and minMs.
Proof For m in M^ let j/(w) denote the unique positive extension of
m to L°(J) by Theorem 6.33. Since a multiplicative linear functional on 91
extends to a positive extension of m on L°°(T), we have $(w) = rj(m)(il/)
for ^ in 91. The only thing to prove is that t\ is a homeomorphism. Recall that
the unit ball of L00^)* is w*-compact. Thus if {ma}aeA is a net in M^ which
converges to m, then any subnet of [t] (ma)}a eA has a convergent subnet whose

Abstract Harmonic Extensions 167
limit is a positive extension of m and hence equal to rj(m). Therefore, r\ is
continuous and hence an into homeomorphism. |
We now adopt the notation <p(m) = r](m)((p) for cp in L°°(T). The re-
restriction $|O will be shown to agree with the classical harmonic extension
of a function in L00 (T) into the disk. We illustrate the usefulness of the pre-
preceding by proving the following result showing the unique position occupied
by H^ + CiT) in the hierarchy of subalgebras of L°°(T).
6.40 Corollary If 91 is an algebra lying between #°° and L°°(T), then
either « = #°° or <tt contains //°° + C(T).
Proof From Theorem 6.34 it follows that the maximal ideal space of
91 can be identified as a subset M^ of Mx. If the origin in D is not in M^,
then X\ is invertible in 91 (£, ^ 0), and hence C(T) is contained in 81, whence
the result follows. Thus suppose the origin in D is in Mm. Since
1 f2
7T \
2n Jo
defines a positive extension of evaluation at 0, it follows that 0@) =
(l/2n)jln<pdt. If <p is contained in « but not in #°°, then (l/2n)jl*<pxndt # 0
for some « > 0, and hence 0 ^ ^z«@) = 0@) #„(()) = 0. This contradiction
completes the proof. ■
One can also show that either M^ is contained in A/00\D or 91 = Hw.
Before we can apply this to //CO + C(T) we need the following lemma on
factoring out zeros.
6.41 Lemma If q> is in Hx and z is in D such that 0(z) = 0, then there
exists \ji in H °° such that q> = (/, —z)\}/.
Proof If 0 is in H00, then <p0(z) = 0(zH(z) = 0. If dtp = Z"=o«»Z» is
the orthonormal expansion of Ocp viewed as an element of H2, then
f fl.2" = fij(f) = 0
by Section 6.15, and hence
/ 1 \ / CO CO \cO
\ l~zXl/ \«=0 n = 0 / « = o

168 6 The Hardy Spaces
Therefore, we have
Xi<P
for A: =1,2,3,...,
and hence the function/^/(l — z/,) is in//".Thus setting^ = Xi<pIO- — zXi)>
we obtain (#i — z) \j/ = q>. ■
6.42 Corollary The maximal ideal space of H CO + C(T) can be identified
with M00\D.
From the preceding corollary, we have
{meM^. faim)] = 1}.
It remains to show that this latter set is A/00\D. Let m be in Ma such that
|#i(m)| < 1 and set %i(m) = z. If <p is in Hw, then ^-^(z) 1 vanishes at z,
and hence by the preceding lemma we have cp — <p(z) 1 = (#, — z) ^ for some
^ in i?00. Evaluating at m in M^, we have
= 0,
and hence ^(w) = ^(z). Therefore, w = z and the proof is complete. ■
In the next chapter we shall be interested in determining when functions
in //CO + C(T) are invertible. From this point of view, the preceding result
seems somewhat unfortunate since the only portion of the maximal ideal
space of H°° over which we have some control, namely D, has disappeared.
We shall show, however, that the question of invertibility of functions in
i?°° + C(T) can be answered by considering the harmonic extension of the
function on D. Our motivation for introducing the harmonic extension is
quite different from that considered classically. We begin by determining a
more explicit representation for 0 on D.
6.43 Lemma If z = reie is in D and <p in L°°(T), then
«= - oo &t Jo

The Maximal Ideal Space of H^ + C 169
where
Kit) =
l-r2
and
l r*
a = — i (
2n Jo
<PX-n dt.
l—2rcost+r2
Moreover, \\<p\\x < \\<p\\x.
Proof The function A:r is positive and continuous; moreover, since
(^l re j n=—oo
it follows that
1 f2*
==- M
27T Jo
Therefore, we have
\Kh =
Lastly, since <p{z) = J£L _xanrMeinB where z = rew, for <p in //°° it
follows that this defines a positive extension of evaluation at z. The unique-
uniqueness of the latter by the Gleason-Whitney theorem completes the proof. |
6.44 Lemma The mapping from Z/^ + CfT) to C(D>) defined by ^>->
0| B is asymptotically multiplicative, that is, for q> and \]/ in //°° + C(T) and
e > 0, there exists K compact in B such that
\0(z)\j/(z)-(p4>(z)\ < e for z in B\K.
Proof Since /7°° + C(T) = clos[UB>0X-n//°°], it is sufficient to estab-
establish the result for functions of the form xn <P f°r <P in H™. If q> = S"=o an Xn
is the Fourier expansion of <p, then for z = re", we have
-*-.(*
Z
A:=0
l £
lk-n+1
Thus, if |l-r| < 8, then |(Cj)(z)-tBG)^(z)| < e.
Since for ^>! and cp2 in //°° and z in B, we have

170 6 The Hardy Spaces
the result follows. ■
An abstract proof could have been given for the preceding lemma, where
the compact set K is replaced by a set {zeD: |x\(z)| < 1—8}. Moreover,
a similar result holds for the algebras 31 x and can be used to state an in-
vertibility criteria for functions in 31 x in terms of their harmonic extension
on ID (see [30], [31]).
6.45 Theorem If <p is in //°° + C(T), then <p is invertible if and only if
there exist 8, e > 0 such that
s £ for 1 — 8 < r < 1.
Proof Using the preceding lemma for £ > 0 there exists 8 > 0 such that
for 1 — 8 < r < 1, we have
<P
whence the implication follows one way if £ is chosen sufficiently small.
Conversely, let <p be a function in //°° + C(T) such that
|0(relV)| > £ > 0 for 1 - 8 < r < 1.
Choose \]/ in //°° and an integer N such that \\(p — x_jv^IU < £/3. Then there
exists ^! > 0 such that for 1 —52 < r < 1, we have
Therefore, for 1 — 8t < r < 1 we have using Lemma 6.43 that
and hence |$(re")| > e/3 if we also assume r > 1—8. Let zu...,zN be the
zeros of the analytic function \j/(z) on ID counting multiplicities. (Since $(z)
is not zero near the boundary, the number is finite.) Using Lemma 6.41
repeatedly we can find a function 6 in //°° such that i]/ = p6, where
P = (Xi-Zi)(Xi-z2)-- v'/i-zj-

Notes 171
Since \j) = pS, we conclude that 8 does not vanish on ID and is bounded away
from zero in a neighborhood of the boundary. Therefore, 6 is invertible in
Hx by Theorem 6.18. Since p is invertible in C(T), it follows that \j/ = p6,
and hence also X-jv^ is invertible in //°° + C(T).
Lastly, since lim,-,!-||<o —<0r|2 = 0, where <pr(e") = Q(reu), we have
>£ a.e., and hence |(x-w^)(ea)| >2e/3a.e. Therefore,
and hence <p is invertible in //°° + C(T) by Proposition 2.7. ■
We conclude this chapter by showing that the harmonic extension of a
continuous function on T solves the classical Dirichlet problem.
6.46 Theorem If <p is a continuous function on T, then the function q>
defined on the closed disk to be ^ on ID and cp on 3D = T is continuous.
Proof If p is a trigonometric polynomial, then the result is obvious.
If <p is a continuous function on T and {pn}%= t is a sequence of trigonometric
polynomials such that limB_0C ||^>—/?J =0, then lim,,^ \\<p — ]5j| =0, since
|$(z) — pn(z)\ ^ ||^>— pn\\x by Lemma 6.43. Therefore, <p is continuous
on D. ■
Notes
The classical literature on analytic functions in the Hardy spaces is quite
extensive and no attempt will be made to summarize it here. Some of the
earliest and most important results are due to F. Riesz [90] and F. and
M. Riesz [91]. The proofs we have presented are, however, quite different
from the classical proofs and largely stem from the work of Helson and
Lowdenslager [62]. The best references on this subject are the books of
Hoffman [66] and Duren [39]; in addition, the books of Helson [61] and
Gamelin [40] should be mentioned.
As we suggested in the text, the interest of the functional analyst in the
Hardy spaces is due largely to Beurling [6], who pointed out their role in
the study of the unilateral shift. The F. and M. Riesz theorem occurs in [91],
but our proof stems from ideas of Lowdenslager (unpublished) and Sarason
[99]. Besides the work of Helson and Lowdenslager already mentioned, the
study of the unilateral shift was extended by Lax [75], [76], Halmos [56],
and more recently by Helson [61] and Sz.-Nagy and Foia§ [107].

172 6 The Hardy Spaces
The study of H™ as a Banach algebra largely began in [100] and the
deepest result that has been obtained is the corona theorem of Carleson
[14], [15].
The material covered in Theorem 6.24 is closely connected with what is
called prediction theory (see [54]). The algebra //°° + C(T) first occurred in
a problem in prediction theory [63] and most of the results presented here
are due to Sarason [97]. The study of the algebras between Hx and L°°(T)
was suggested to the author [31] in studying the invertibility question for
Toeplitz operators. Theorem 6.32 is taken from [34] in which it is shown
that quotients of inner functions are uniformly dense in the measurable
unimodular functions. The Gleason-Whitney theorem is in [43]. The role of
the harmonic extension in discussing the invertibility of functions in H™ + C(T)
is established in [30].
Exercises
6.1 If <p and \j/ are unimodular functions in L°°(T), then q>H2 = \]/H2 if
and only if ip = h}i for some X in C
Definition A function/in H1 is an outer function if clos1[/^'+] = H1.
6.2 A function/in H1 is outer if and only if it is the product of two outer
functions in H2.
6.3 A closed subspace Jt of H1 satisfies Xi •& ^ •& if a"d only if M =
q>Hl for some inner function q>. State and prove the corresponding criteria
for subspaces of £/(¥).*
6.4 If/is a function in H1, then/= <pg for some inner function q> and
outer function g in H1.
6.5 If/ is a nonzero function in H1, then the set {e" e T: /(e") = 0} has
measure zero.
6.6 An analytic polynomial is an outer function in either H1 or H2 if
and only if it does not vanish on the interior of ID.
6.7 Show that rotation on D induces a natural representation of the circle
group in Aut(H°°) = Hom(MJ. Show that the orbit of a point in MK under
this group of homeomorphisms is closed if and only if it lies in D.*
Definition If ^ is the Gelfand transform of xi on Mm, then the fiber Fx
of Mx over k in T is defined by Fh= {me Mx: j^ (m) = X}.

Exercises 173
6.8 The fibers Fx of Mx are compact and homeomorphic.
6.9 If <p is in //°° and A is in T, then <p is bounded away from zero on a
neighborhood of k in {A} u D if and only if (p does not vanish on FA.
6.iO If <p is in //" and a is in the range of 0\F^ for some k in T, then there
exists a sequence {zn}™=1 in ID such that lim,,.,^ zn = X and lim,,^ @(zn) = a.
6.// Show that the density of ID in Mx is equivalent to the following state-
statement: For <pl5<p, ...,<P;v in H™ satisfying Yt=i \Qt(z)\ > £ for z in ID, there
exists \j/u\}/2,...,\}/N in //°° such that Y.i'=i(Pi^i= 1- Prove this statement
under the additional assumption that the ^>, are in //°° + C(T).
6.72 If 91 is a closed algebra satisfying ^c^lcr (T), then the maximal
ideal space of L™ is naturally embedded in Mm as the Silov boundary of 91.
6.13 (Newman) Show that the closure of ID in Mx contains the Silov
boundary.
Definition An isometry U on the Hilbert space #? is pure if f]n^0 Uf =
{0}. The multiplicity of U is dim kerf/*.
6.14 A pure isometry of multiplicity one is unitarily equivalent to TXl on H2.
6.15 A pure isometry of multiplicity N is unitarily equivalent to £i <f<N @ TX1
6.16 (von Neumann-Wold) If U is an isometry on the Hilbert space Jf,
then Jf = 3ff j © 3ff2 such that ^f x and Jf2 reduce t/, U\JV2 is a pure
isometry, and C/|^f 2 is unitary.
6.17 If C/ is an isometry on the Hilbert space jf, then there exists a unitary
operator W on a Hilbert space jf containing ^f such that WM1 a #F and
= U.
6.18 (Sz.-Nagy) If T is a contraction on the Hilbert space Jf, then there
exists a unitary operator W on a Hilbert space JT containing Jf such that
TN = P^WN\j^ for N in Z+. (Hint: Choose operators 5 and C such that
(o ?) is a coisometry and then apply the previous result to the adjoint.)
6.19 (von Neumann) If T is a contraction on the Hilbert space Jf, then
the mapping denned *¥(p)= p(T) for each analytic polynomial p extends
to a contractive homomorphism from the disk algebra to £pf). (Hint:
Use the preceding exercise.)

174 6 The Hardy Spaces
6.20 Show that the w*-topology on the unit ball of L00 (T) coincides with
the topology of uniform convergence of the harmonic extensions on com-
compact subsets of ID. (Hint: Evaluation at a point of ID is a w*-continuous
functional.)
6.21 If/is a function in H1 and/r(elV) =f(re") for 0 < r < 1 and e" in T,
then lim,.,! ||/—fr\\t = 0. (Hint: Imitate the proof of Theorem 6.46.)
6.22 If/is a function in L1 (T) for which
then the harmonic extension / satisfies
= 0 and F(t) = J' f(eie) dQ,
where
\-r2
Kit) =
l+r2-2r cos?"
(Hint: Observe that
ei») = ±- f" kr(9-t)dF(t)
2n J-n
and use integration by parts.)
6.23 (Fatou) If /is a function in L^T) and /is its harmonic extension of
/to ID, then limr_2/(re6) =/(e{1) a.e.* (Hint: Show that
and that limf-n/fre'8) exists and is equal to F'F), whenever the latter de-
derivative exists.)
6.24 If q> is a function in J/°° and $ is its Gelfand transform on Mm, then
|0| is subharmonic, that is, if $ is the harmonic extension of a real-valued
function to Mx such that \]/ > \cp\ on T, then $ > |$j on MK.
6.25 A function /in Z/1 is an outer function if and only if the inequality
|/| > \g\ on T implies |/| > |^| on M^ for every fif in /71.
6.26 (Jensen's Inequality) If/is a function in H1, then
^- F"log\f(eu)\dt.*
2n Jo

Exercises 175
(Hint: Assume that/is in A and approximate log(|/j + e) by the real part
u of a function gin A; show that log|/@)| — w@) < £ and let £ tend to zero.)
6.27 (Kolmogorov-Kreiri) If /i is a positive measure on T and /i0 is the
absolutely continuous part of \i, then
inf f |l-/|2rf/i = inf f |l-
(Hint: Show that if F'vs, the projection of 1 onto the closure of Ao in I?(fi),
then \—F= 0 a.e./is, where /is is the singular part of
6.25 A function/in /f2 is outer if and only if
infi rii-
heA0^1t Jo
(Hint: Show that /is outer if and only if 1 — /@)//is the projection of 1 on
the closure of Ao in L2(\f\2d6).)
6.29 {Szego) If fi is a positive measure on T, then
inf f |l-/j2^
e^o Jt
where A is the Radon-Nikodym derivative of \i with respect to Lebesgue
measure.* (Hint: Use Exercise 6.27 to reduce it to /i of the form wd6; use
the geometric-arithmetic mean inequality for one direction and reduce to
the case \h\2 dQ for h an outer function in the other.)
6.30 If 91 is a closed algebra satisfying H °° c 91 c L°°(T), then 91 is gen-
generated by //°° together with the unimodular functions w for which both u
and u are in 91. (Hint: If/is in 91, then/+21|/|| = ug, where w is unimodular
and g is outer.)
6.31 If F is a group of unimodular functions in L°° (T), then the maximal
ideal space Mr of the subalgebra 9lr of If (T) generated by Hx and T can
be identified by
MT - {meM^: \u(m)\ = 1 for ueT}.
6.32 Is every 9tr of the form 91 x for some semigroup £ of inner functions ?* *
6.33 Show that the closure of Hco+Hco is not equal to L°°(T).* (Hint: If
argz were in the closure of //°°+5°0, then there would exist q> in //°° such
that ze*1 would be invertible in Hx.)

176 6 The Hardy Spaces
6.34 If m is in Fx and \i is the unique positive measure on ML«> such that
Q(m)=\ Qdp. for <pinL°°(T),
JML«.
then fi is supported on M^nF^.* (Hint: Show that the maximum of |$|
on Fx is achieved on MLm n Fx for ^o in //°°.)
6.35 If ^> is a continuous function and \]/ is a function in L°° (T), then ^
and ^(^ are asymptotically equal on ID and equal on M00\D.
6.56 If <p is a function in L°° (T), then the linear functional on H1 defined by
Lf= (l/2n)$lKf(pd6 is continuous in the w*-topology on H1 if and only if
6.37 Show 'that the collection PC of right-continuous functions on T posses-
possessing a limit from the left at every point of T is a uniformly closed self-adjoint
subalgebra of L°° (T). Show that the piecewise continuous functions form a
dense subalgebra of PC. Show that the maximal ideal space of PC can be
identified with two copies of T given an exotic topology.
6.38 If <p is a function in PC, then the range of the harmonic extension of
q> on Fh is the closed line segment joining the limits of q> from the left and
right at X.
6.39 Show that QC = [//°° + C(T)] n [//°° + C(T)] is a uniformly closed
self-adjoint subalgebra of L°° (T) which properly contains C(T). Show that
every inner function in QC is continuous but that QCnH™ # A.* (Hint:
There exists a real function q> in C(T) not in Re>4; if ^ is a real function
in L2(T) such that <p + U}i is in //2 then e«"+* is in //" and en is in gC.)
6.40 If m is a unimodular function in QC, then \u\ = 1 on M00\D. Is the
converse true?**
6.41 Show that M00\D is the maximal ideal space of the algebra generated
by //°° and the functions w in L°° (T) for which |u| has a continuous extension
to D. Is this algebra //°° + C(T)?**
6.42 Show that PCnQC= C(T). (Hint: Consider the unimodular func-
functions in the intersection and use Exercises 6.38 and 6.40.)
6.43 Show that there is a natural isometrical isomorphism between //°°
and (L'OT)///,I)*. Show that the analytic trigonometric polynomials 0>+
are w*-dense in H°°.

7
Toeplitz Operators
7.1 Despite considerable effort there are few classes of operators on Hilbert
space which one can declare are fully understood. Except for the self-adjoint
operators and a few other examples, very little is known about the detailed
structure of any class of operators. In fact, in most cases even the appropriate
questions are not clear. In this chapter we study a class of operators about
which much is known and even more remains to be known. Although the
results we obtain would seem to fully justify their study, the occurrence of
this class of operators in other areas of mathematics suggests they play a
larger role in operator theory than would at first be obvious.
We begin with the definition of this class of operators.
7.2 Definition Let P be the projection of L2 (T) onto H2. For <p in L00 (T)
the Toeplitz operator Tv on H2 is defined by T<pf= P(cpf) for/in H2.
7.3 The original context in which Toeplitz operators were studied was not
that of the Hardy spaces but rather as operators on /2(Z+). Consider the
orthonormal basis {xn: neZ+} for H2, and the matrix for a Toeplitz oper-
operator with respect to it. If <p is a function in L°° (T) with Fourier coefficients
0(») = (\l2n)ll«q>x-ndt, then the matrix {^,«}m,B6Z+ for T9 with respect to
{*„: neZ+} is
am,n = (T Xn,Xm) = z- (PXn-mdt = Q(m-n).
2tc Jo
177

178 7 Toeplitz Operators
Thus the matrix for Tv is constant on diagonals; such a matrix is called a
Toeplitz matrix, and it can be shown that if the matrix defines a bounded
operator, then its diagonal entries are the Fourier coefficients of a function
inL°°(T) (see [11]).
We begin our study of Toeplitz operators by considering some elementary
properties of the mapping f from L°°(T) to £(//2) defined by £(<p) = Tv.
7.4 Proposition The mapping £, is a contractive *-linear mapping from
L°°(T)into 2(H2).
Proof That £ is contractive and linear is obvious. To show that £(<p)* =
£(<p), let/and g be in H2. Then we have
(T9f,g) = (P(<tr),g) = (J,cpg) = (f,P(<pg)) = (/, T9g) = (T^fg),
and hence £(<?)* = T* = T^ =
The mapping £ is not multiplicative, and hence £ is not a homomorphism.
We see later that £ is actually an isometric cross section for a *-homomorph-
ism from the C*-algebra generated by {T^. <pe L°°(T)} onto L°°(T), that is,
if a is the *-homomorphism, then ao£ is the identity on L°°(T).
In special cases, £ is multiplicative, and this will be important in what
follows.
7.5 Proposition If q> is in L°°(T) and i]/ and 0 are functions in Hx, then
Ty T$ = T^ and TB T<p = Tev.
Proof If/is in H2, then ^/is in //2 by Proposition 6.2 and hence T#f=
Ptyf) = ^/ Thus
T9 Tt f = T9W) = P(wfcO = 7W/ and j; 7; = 7^.
Taking adjoints reduces the second part to the first. ■
The converse of this proposition is also true [11] but will not be needed
in what follows.
Next we consider a basic result which will enable us to show that £ is an
isometry.
7.6 Proposition If cp is a function in L00 (T) such that Tv is invertible, then
<p is invertible in L° (T).
Proof Using Corollary 4.24 it is sufficient to show that Mv is an in-
invertible operator if Tv is. If Tv is invertible, then there exists £ > 0 such that

The Symbol Map 179
llr«-/ll > £ 11/11 for/in H2. Thus for each n in 1 and/in H2, we have
WM^xJn = \WxJW = \\<pf\\ > \\P(9f)\\ = \Kf\\ > e\\f\\ = e\\Xnf\\-
Since the collection of functions {#„/:/£//2, neZ} is dense in L2(T), it
follows that WM^gW > e\\g\\ for g in L2(T). Similarly, \\MJ\\ > b||/||,
since 7^ = 7^* is also invertible, and thus Mv is invertible by Corollary 4.9,
which completes the proof. ■
As a corollary we obtain the spectral inclusion theorem.
7.7 Corollary (Hartman-Wintner) If <p is in L00 (T), then ^(<p) = a{Mv) <=
Prao/ Since Tv — X = Tv _A for X in C, we see by the preceding proposi-
proposition that <t(Mv) <= o-fr^). Since the identity 0t{(p) = <r(Mv) was established in
Corollary 4.24, the proof is complete. ■
This result enables us to complete the elementary properties of £.
7.8 Corollary The mapping £, is an isometry from L°°(T) into £(//2).
Proof Using Proposition 2.28 and Corollaries 4.24 and 7.7, we have for
cp in L00 (I) that
TJ = sup{|A|: Aeff(^)}
(p)} = \\(p\\x,
and hence £ is isometric. ■
7.9 Certain additional properties of the correspondence are now obvious.
If Tv is quasinilpotent, then M{(p) a a(Tv) = {0}, and hence Tv = 0. If Tv
is self-adjoint, then M{q>) a o-G^) a U and hence cp is a real-valued function.
We now exhibit the homomorphism for which £ is a cross section.
7.10 Definition If S is a subset of L00 (T), then 2E") is the smallest closed
subalgebra of £(//2) containing {7^: ^ g S}.
7.11 Theorem If (£ is the commutator ideal in £(L°° (T)), then the mapping
4 induced from L°°(T) to £(L°° (¥))/(£ by ^ is a *-isometrical isomorphism.
Thus there is a short exact sequence
@) ->■ £ ->■ 2(L°° (T)) -A- L00 (T) ->■ @)
for which £ is an isometrical cross section.

180 7 Toeplitz Operators
Proof The mapping £c is obviously linear and contractive. To show
that £c is multiplicative, observe for inner functions q>l and q>2 and functions
\j/x and \j/2 in ff°°, that we have
Since 7^ T*2 — T*2T^t is a commutator and G is an ideal, it follows that the
latter operator lies in (L Thus £c is multiplicative on the subalgebra
& = {ij/ip: \]/ e Hx, cp an inner function}
of L00 (T) and the density of & in U° (T) by Theorem 6.32 implies that f c is a
*-homomorphism.
To complete the proof we show that ||7^ + ^11 > \\TJ for <p in L°°(T)
and K in (£ and hence that £c is an isometry. A dense subset of operators in
(£ can be written in the form
n m
where ^,- is in £(L°°(T)), the functions <pu (pt',and Pl} are inner functions,the
functions ^i,^/, and a,-; are in //°°, and square brackets denote commutator.
If we set
e = n kiwi*
i=i
7=1
then 6 is an inner function and KFf) = 0 for/in H2.
Fix £ > 0 and let / be a function in H2 chosen such that ||/|| = 1 and
IIT<pf\\ > II T<p\\ ~e- If <Pf= 9i +02> where gt is in H2, and g2 is orthogonal
to H2, then since 0 is inner we have that 6gt is in H2 and orthogonal to Qg2-
Thus
> 1100.11 = Il0.ll = IVII > Iirj
and therefore ||7^ + ^|| > ||7Vll' wnicn completes the proof.
A direct proof of this result which avoids Theorem 6.32 can be given
based on a theorem due to Bunce [12]. In this case the spectral inclusion
theorem is then a corollary.

The Symbol Map 181
The C*-algebra Z(JU°(J)) is a very interesting one; the preceding result
shows that its study largely reduces to that of the commutator ideal (£ about
which very little is known. We can show that (£ contains the compact oper-
operators, from which several important corollaries follow.
7.12 Proposition The commutator ideal in the C*-algebra Z(C(J)) is
££(//2). Moreover, the commutator ideal of Z(LX(J)) contains ££(ff2).
Proof Since the operator TX1 is the unilateral shift, we see that the
commutator ideal of Z(C(T)) contains the nonzero rank one operator
T*iTxi-TxiT*f Moreover, the algebra 2(C(T)) is irreducible since TXl
has no proper reducing subspaces by Beurling's theorem. Therefore, Z(C{Tj)
contains ££(//2) by Theorem 5.39.
Lastly, since the image of TX1 in Z(C(T))/2(£(H2) is normal and generates
this algebra, it follows that Z(C(J))/Q(£(H2) is commutative, and hence
£G(//2) contains the commutator ideal of Z(C(J)). To complete the identi-
identification of £(£(//2) as the commutator ideal in Z(C(J)), it is sufficient to
show that £(£(//2) contains no proper closed ideal. If 3 were such an ideal,
then it would contain a self-adjoint compact operator H. Multiplying H by
the projection onto the subspace spanned by a nonzero eigenvector, we
obtain a rank one projection in 3. Now the argument used in the last para-
paragraph of the proof of Theorem 5.39 can be applied, and hence £&(H2) is
the commutator ideal in Z(C(J)). Obviously, £(£(//2) is contained in the
commutator ideal of 2(L°° (¥))• ■
7.13 Corollary There exists a *-homomorphism f from the quotient al-
algebra Z(LX 00)/££(H2) onto L°°(T) such that the diagram
0 or)) * > sCl00 00)/£e(#2)
commutes.
Proof Immediate from Theorem 7.11 and the preceding proposition. ■
7.14 Corollary If q> is a function in L°° (T) such that Tv is a Fredholm
operator, then cp is invertible in L00 (T).
Proof If Tv is a Fredholm operator, then Tt(J^) is invertible in
Z(n°(J))IQ(£(H2)
by Definition 5.14, and hence <p = (fom)^) is invertible in L°°(T).

182 7 Toeplitz Operators
7.15 Certain other results follow from this circle of ideas. In particular,
it follows from Corollary 7.13 that ||r,,+*:|| > ||rj for <p in L°°(T) and
K in £(£(//2), and hence the only compact Toeplitz operator is 0.
Let us consider again the Toeplitz operator TXt. Since the spectrum of
TX1 is the closed unit disk we see, in general, that the spectrum of a Toeplitz
operator Tv is larger than the essential range of its symbol q>. It is this phenom-
phenomenon which shall largely concern us. In particular, we are interested in de-
determining criteria for a Toeplitz operator to be invertible and, in addition,
for obtaining the spectrum. The deepest and perhaps the most striking result
along these lines is due to Widom and states that the spectrum of a Toeplitz
operator is a connected subset of C. This will be proved at the end of this
chapter.
We now show that the spectrum of a Toeplitz operator cannot be too
much larger than the essential range of its symbol. We begin by recalling
an elementary definition and lemma concerning convex sets.
7.16 Definition If E is a subset of C, then the closed convex hull of E,
denoted h(E), is the intersection of all closed convex subsets of C which
contain E.
7.17 Lemma If E is a subset of C, then h(E) is the intersection of the
open half planes which contain E.
Proof Elementary plane geometry. ■
The lemma and the following result combine to show that o{T,^ is con-
contained in h{0t(q>)).
7.18 Proposition If cp is an invertible function in L°° (T) whose essential
range is contained in the open right half-plane, then Tv is invertible.
Proof If A denotes the subset {z e C: \z— 11 < 1}, then there exists an
£ >0 such that tM{ap) = {ez: ze&t{(p)} a A. Hence we have fle^o —1|| < 1
which implies ||/— Tetp\\ < 1 by Corollary 7.8, and thus Te<p = eTv is invertible
by Proposition 2.5. ■
7.19 Corollary (Brown-Halmos) If <p is a function in L00 (T), then a(TJ c
Proof By virtue of Lemma 7.17 it is sufficient to show that every open
half-plane containing 8ft Up) also contains o(T^. This follows from the

The Spectrum of Analytic Toeplitz Operators 183
proposition after a translation and rotation of the open half-plane to co-
coincide with the open right half-plane. ■
We now obtain various results on the invertibility and spectrum of cer-
certain classes of Toeplitz operators. We begin with the self-adjoint operators.
7.20 Theorem (Hartman-Wintner) If cp is a real-valued function in L°°(T),
then
GG^) = [essinf^j, ess sup ^>].
Proof Since the spectrum of Tv is real it is sufficient to show that T^—X
invertible implies that either q> — X > 0 for almost all e" in T or q>—X < 0
for almost all elt in T. If Tv—X is invertible for X real, then there exists g in
H2 such that {Tv — X)g = 1. Thus there exists h in H02 such that {q> — X)g =
l+h. Since (<p — X)g = l + h is in H2, we have (<p — X)\g\2 = (l+h)g is in
Hl, and therefore (<p — X)\g\2 = a for some a in U by Corollary 6.6. Since
g # 0 a.e. by the F. and M. Riesz theorem, it follows that <p — X has the sign
of a and the result follows. ■
Actually much more is known about the self-adjoint Toeplitz operators.
In particular, a spectral resolution is known for such operators up to unitary
equivalence (see Ismagilov [68], Rosenblum [94], and Pincus [86]).
The Toeplitz operators with analytic symbol are particularly amenable to
study. If <p is in Hx, then the operator Tv is the restriction of the normal
operator Mv on L2(T) to the invariant subspace H2 and hence is what is
called a subnormal operator.
7.21 Theorem (Wintner) If cp is a function in Hx, then Tv is invertible if
and only if q> is invertible in H™. Moreover, if 0 is the Gelfand transform
of <p, then oiTy) = clos[0(D>)].
Proof If q> is invertible in H™, then there exists \j/ in /7°° such that
cpxj/ = 1. Hence /= T$Tip= 7^7^ by Proposition 7.5. Conversely, if Tv is
invertible, then cp is invertible in L°°(T) by Proposition 7.6. If \]/ is 1/cp, then
7^ Tv =T^ = I by Proposition 7.5, and hence T# is a left inverse for Tv.
Thus T$=T~l and therefore 1 = Tv T# 1 = <pP(i]/). Multiplying both sides
by l/cp = \j/, we obtain P(\J/) = \j/ implying that \j/ is in /7°° and completing
the proof that cp is invertible in Hx. The fact that a{T^) = clos[^(ID)]
follows from Theorem 6.18. ■

184 7 Toeplitz Operators
This result yields especially nice answers to the questions of when Tv is
invertible and what its spectrum is for analytic (p. It is answers along these
lines that we seek to determine. We next investigate the Toeplitz operators
with continuous symbol and find in this case that an additional ingredient
of a different nature enters into the answer. Our results on these operators
depend on an analysis of the C*-algebra 2(C(T)).
We begin by showing the £, is almost multiplicative if the symbol of one
of the factors is continuous.
7.22 Proposition If <p is in C(T) and \]/ is in L°°(T), then Tv T#- T^ and
T$ Tv- T#v are compact.
Proof If \ji is in L°°(T) and/is in //*, then
and hence T^Tx_l—T^x_t is a rank one operator.
Suppose TfTx_n—Tyx_n has been shown to be compact for every \j/ in
L°°(T) and -N < n < 0. Then we have
and hence is compact. Since T$ TXn = T^Xn for « > 0 by Proposition 7.5, it
follows that 7^ Tp— T$p is compact for every trigonometric polynomial p.
The density of the trigonometric polynomials in C(T) and the fact that £
is isometric complete the proof that 7^ Tv— T^v is compact for \]/ in L°°(T)
and q> in C(T). Lastly, since
V^ tp i \ji ■• qnjr) == * y * <p * yip
we see that this operator is also compact. ■
The basic facts about £(C(T>) are contained in the following theorem
due to Coburn.
7.23 Theorem The C*-algebra 2(C(T)) contains £<S.(H2) as its com-
commutator and the sequence
@) -»se(//2) -»a:(C(T)) -» aj) -»@)
is short exact; that is, the quotient algebra £(C(T))/£(£(//2) is *-isometric-
ally isomorphic to C(T).

The C*-Algebra Generated by the Unilateral Shift 185
Proof It follows from the preceding proposition that the mapping £
of Corollary 7.13 restricted to 2(C(T))/££(/72) is a *-isometrical isomorph-
isomorphism onto C(T), and hence the result follows. ■
Combining this with the following proposition yields the spectrum of a
Toeplitz operator with continuous symbol.
7.24 Proposition (Coburn) If q> is a function in L°°(T) not almost every-
everywhere zero, then either ker Tv = {0} or ker Tv* = {0}.
Proof If /is in ker Tv and g is in ker Tv*, then <pf and <pg are in H02.
Thus <pfg and <pfg are in Hol by Lemma 6.16, and therefore <pfg is 0 by
Corollary 6.7. If neither / nor g is the zero vector, then it follows from the
F. and M. Riesz theorem that <p must vanish for almost all e" in T, which
is a contradiction. ■
7.25 Corollary If <p is a function in L°°(T) such that Tv is a Fredholm
operator, then Tv is invertible if and only if j(T^) = 0.
Proof Immediate from the proposition. ■
Thus the problem of determining when a Toeplitz operator is invertible
has been replaced by that of determining when it is a Fredholm operator and
what is its index. If cp is continuous, then this is readily done. The result is
due to a number of authors including Krein, Widom, and Devinatz.
7.26 Theorem If ^ is a continuous function on T, then the operator Tv
is a Fredholm operator if and only if q> does not vanish and in this casej(^)
is equal to minus the winding number of the curve traced out by <p with
respect to the origin.
Proof First, Tv is a Fredholm operator if and only if cp is invertible in
C(T) by Theorem 7.23. To determine the index of Tv we first observe that
JiJq) =J(T$) if (p and \j/ determine homotopic curves in C\{0}. To see this,
let $ be a continuous map from [0, l]xT to C\{0} such that 3>@,e') =
<p(e") and <J>(l.elV) = \l/(e") for e" in T. Then the mapping ?-> TOt is norm
continuous and each TOt is a Fredholm operator. Since j(TCc) is continuous
and integer valued, we see that j(Tv) =j{Tf).
If n is the winding number of the curve determined by cp, then <p is homo-
topic in C\{0} to xn- Since j(TXn) = -n, we have j(Tv) = -n and the result
is completely proved. ■

186 7 Toeplitz Operators
7.27 Corollary If cp is a continuous function on T, then Tv is invertible
if and only if cp does not vanish and the winding number of the curve de-
determined by cp with respect to the origin is zero.
Proof Combine the previous theorem and Corollary 7.25. ■
7.28 Corollary If cp is a continuous function on T, then
o(TJ = St(tp) u{leC: ^fa>,A) # 0},
where it{q>,X) is the winding number of the curve determined by (p with
respect to X.
In particular, the spectrum of Tv is seen to be connected since it is formed
from the union of 3k ((p) and certain components of the complement.
In this case the invertibility of the Toeplitz operator Tv depended on q>
being invertible in the appropriate Banach algebra C(T) along with a topo-
logical criteria. In particular, the condition on the winding number amounts
to requiring <p to lie in the connected component of the identity in C(T) or
that q> represent the identity in the abstract index group for C(T). Although
we shall extend the above to the larger algebra HW + C(T), these techniques
are not adequate to treat the general case of a bounded measurable function.
We begin again by identifying the commutator ideal in 2(/7°° + C(T))
and the corresponding quotient algebra.
7.29 Theorem The commutator ideal in %{H°° + C(T)) is ££(/72) and the
mapping £K = tio£ from /7°°+C(T) to 2(//°° + C(T))/£G(//2) is an iso-
metrical isomorphism.
Proof The algebra 2(/7°° + C(T)) contains £G(//2), since it contains
£(C(T)) and thus the mapping £K is well defined and isometric by Corollary
7.13 and the comments in Section 7.15. If cp and i]/ are functions in //°° and
/ and g are continuous, then
Ty — r(<?+/H, + Tv+f Tg — Ti<p+f)g
Tg — Ti<p+r)g
by Proposition 7.5, and the latter operator is compact by Proposition 7.22.
Thus the commutator ideal is contained in £G(//2) and hence is equal to it.
Thus £K is multiplicative, since

Invertibility of Unimodular Toeplitz Operators and Prediction Theory 187
Therefore, £K is an isometrical isomorphism, which completes the
proof. ■
Note that 2(/fo° + C(T)) is not a C*-algebra, and hence the fact that
Tv is a Fredholm operator and n{T^) has an inverse in £(/72)/£(£(//2) does
not automatically imply that n(T^) has an inverse in Z(HX + C(T))/£G(/72).
To show this we must first prove an invertibility criteria due to Widom and
based on a result of Helson and Szego from prediction theory.
7.30 Theorem If <p is a unimodular in If (T), then the operator Tv is left
invertible if and only if dist(^,/7°°) < 1.
Proof If dist((p,Hx) < 1, then there exists a function \j/ in /7°° such
that H^-^L < 1. This implies that || 1 — ^IU < 1 and hence ||/-7^|| < 1.
Thus TfTv = T^ is invertible in £(//2) by Proposition 2.5 and therefore
Tv is left invertible.
Conversely, if Tv is left invertible, then there exists £ > 0 such that
IIVI >e\f\ for/in H2. Thus \\P(<pf)\\ > 4f\ = e|l<l»/ll, and hence
Ik/I2 = W-P)(<pf)\\2+\\P(<Pf)\\2 > \\(I-P)(cpf)\\2 + E\\cpf\\2.
Therefore, we have \\(I-P)(<pf)\\ <A-5)||/|| for / in /72, where 5 =
1 -A -e)'a > 0. If/is in H2 and g is in //02, then
If
271 Jo
(pfgdt
'o
h\V
If for h in Hol we choose/in H2 and g in H02 by Corollary 6.27, such that
h —fg and \h\± = H/IMIfllUs then we obtain
<>2n
Thus, the linear functional defined by 3>(ft) = (l/2rc)Jo*A<orff for ft in .Ho1
has norm less than one. Therefore, it follows from Theorem 6.29 (note that
we are using (//</)* = I^O")///00), that there exists \j/ in Hx such that
\<p — ^||„ < 1, which completes the proof. ■
7.31 Corollary If <p is a unimodular function in L°°(T), then Tv is in-
invertible if and only if there exists an outer function \j/ such that || q> — \j/ \\ „ < 1.
Proof If ll^-^lco < l,then |^| >£>0fore= l-||^-^|U and hence
T^ is invertible by Theorem 7.21 and Proposition 6.20. Since 7^* Tv is in-
invertible, we see that Tv is invertible. Conversely, if Tv is invertible and \j/

188 7 Toeptitz Operators
is a function in //" such that ||<0 —^||K < 1, then 7^ is invertible since
?V* Tv is, and hence ^ is an outer function. ■
As a result of this we obtain a very general spectral inclusion theorem.
7.32 Theorem If 2 is a closed subalgebra of £(//2) containing 3;(//°°)
and Tv is in 2 for some <p in L°°(T), then a{T^ = ^(T^).
Proof Obviously a(Tv) is contained in ax(T9). To prove the reverse
inclusion suppose Tv is invertible. Using Corollary 6.25, we can write q> =
mj/, where u is a unimodular function and i]/ is an outer function. Conse-
Consequently, \j/ is invertible in If(J), and by Proposition 6.20 we obtain that
T^1 is in 3;. Thus Tu = T^T^1 is in X, moreover, Tu and hence TB is in-
invertible in £(//2). Employing the preceding corollary there exists an outer
function 0 such that ||0—ull^ < 1. Since TuTe is in X and ||/- Tu Te\\ =
Hi-well^ < 1, we see that (r,,^) is in % by Proposition 2.5. Since
is in %, the proof is complete. ■
This leads to a necessary condition for an operator to be Fredholm.
7.33 Corollary If 31 is a closed subalgebra of L00 (T) containing H °° + C(T)
and <p is a function in 21 for which Tv is a Fredholm operator, then cp is in-
invertible in 91.
Proof If Tv is a Fredholm operator having index n, then TXn<j> is in-
invertible by Propositions 7.5 and 7.24, and the function %n<p is also in 91.
Therefore, by the preceding result, T~^, is in £(91), and hence using Theorem
7.11 we see that xn<p is invertible in p[£Cl)] = 91. Thus <p is invertible
in 91, which completes the proof. ■
The condition is also sufficient for HX + C(J).
7.34 Corollary If <p is a function in //°° + C(T), then Tv is a Fredholm
operator if and only if <p is invertible in //°° + C(T).
The result follows by combining the previous result and Theorem
7.29. ■

The Spectrum of Toeplitz Operators with Symbol in Hx + C 189
We need one more lemma to determine the spectrum of Tv for <p in
//°° + C(T). Although this lemma is usually obtained as a corollary to the
structure theory for inner functions, it is interesting to note that it also follows
from our previous methods.
7.35 Lemma If <p is an inner function, then H2 Q <pH2 is finite dimen-
dimensional if and only if cp is continuous.
Proof If <p is continuous, then Tv is a Fredholm operator by Theorem
7.26, and hence H2 Q <pH2 = ker^*) is finite dimensional. Conversely, if
H2 © <pH2 is finite dimensional, then Tv is a Fredholm operator and hence
<p is invertible in //°° + C(T). We need to show that this implies that q> is
continuous.
By Theorem 6.45 there exist e,8 > 0 such that |0(re")| > e for 1-8 <
r < 1, and hence 0 has at most finitely many zeros Zi,z2» ■••>zw ln ^b counted
according to multiplicity. Using Lemma 6.41 we obtain a function \j/ in //°°
such that i]/ nJL i (Xi —Zj) = (p. Thus \j/ does not vanish on ID and is bounded
away from zero on a neighborhood of the boundary. Therefore, \j/ is in-
invertible in Hx by Theorem 6.18. Moreover, since
|e"-Zj| = |l-z,e*| for eu in T,
we have
= 1.
Thus the function nJ=i(Xi— zj)l(.l ~ zj Xi) is a continuous inner function,
and 6 = i]/YlI-=1(\. — ZjX1) has modulus one on T. Since 6 is invertible in
Hx, it follows that B = 6~1 is in H™ and hence that 9 is constant. Thus
and therefore is continuous. ■
7.36 Theorem If <p is a function in /7°° + C(T), then Tv is a Fredholm
operator if and only if there exist 8,8 > 0 such that |$(re")| > e for 1—8 <
r < 1, where <p is the harmonic extension of q> to ID. Moreover, in this case
the index of Tv is the negative of the winding number with respect to the
origin of the curve ^(f-e'1) for 1 — 8 < r < 1.
Proof The first statement is obtained by combining Corollary 7.34 and
Theorem 6.45.

190 7 Toeplitz Operators
If q> is invertible in Hn + C(T) and e, 8 > 0 are chosen such that
for 1— 8 < r < 1, then choose ^ in //°° and a negative integer A' such that
Il^-Xiv^lloo < £/3- If we set <P* = X<p + A -A)xjv^ for 0 < A < 1, then each
(^ is in //°° + C(T) and ||<p-<P;iL <s/3. Therefore, |0jt(re?")l >2e/3 for
1 — 5 < r < 1, and hence each (px is invertible in H™ + C(T) by Theorem 6.45.
Hence each TV3i is a Fredholm operator and the winding number of the curve
(Px(relt) is independent of k and r for 0 < X < 1 and 1 — 8 < r < 1. Thus it is
sufficient to show that the index of 7^ = 7^ is equal to the negative of the
winding number of the curve (Xn^)^1)-
Since TXKl!l = TXN T$ we see that T$ is a Fredholm operator. If we write
\j/ = \}io\}il, where ^o is an outer function and \j/t is an inner function, then
T$o is invertible and hence 7^ is a Fredholm operator. Thus \j/t is con-
continuous by the previous lemma which implies Xn ^i is in C(T) and we conclude
by Theorem 7.26 that
Since there exists by Lemma 6.44 a^,>0 such that 8 > 8t > 0 and
\x^(.re") - jfoJ.OtfoKo-e")! < | for 1 - 8, < r < 1,
it follows that
fo ^ £ for 1 - «i < i- <
Using the fact that ^0 does not vanish on D and Theorem 6.46, we obtain
the desired result. |
We conclude our results for Tv with <p in H^ + OJ) by showing that
the essential spectrum is connected. Although we shall show this for arbitrary
<p, a direct proof would seem to be of interest.
7.37 Corollary If cp is a function in H™ + C(T), then the essential spectrum
of Tv is connected.
Proof From the theorem it follows that X is in the essential spectrum
of Tv for <p in /700 + C(T) if and only if <p- k is not invertible in Hx + C(T).
Hence, by Theorem 6.46 we have that X is in the essential spectrum of Tv if

The Spectrum of Toeplitz Operators with Symbol in H™ + C 191
and only if X is in
"): 1 > r > 1 - 3} for each 1 > 8 > 0.
Since each of these sets is connected, the result follows. ■
We now take up the proof of the connectedness of the essential spectrum
for an arbitrary function in L°° (T). The proof is considerably harder in this
case and we begin with the following lemma.
7.38 Lemma If <p is an invertible function in L°° (T), then Tv is a Fredholm
operator if and only if T1/(p is and moreover, in this case, 7G^) = —j(TUv).
Proof If we set cp = u\j/ by Corollary 6.25, where ^ is an outer function
and u is unimodular, then Tl/<p = Tu Tw = T*lt T^* TUy by Proposition 7.5.
Since 7\M is invertible by Theorem 7.21, the result follows. ■
The proof of connectedness is based on the analysis of the solutions fx
and gx of the equations T<?_A/A = 1 and ri/(<?_A)^ = 1. Since we want to
consider X for which these operators are not invertible, the precise definition
of fx and gx is slightly more complicated.
7.39 Definition If cp is a function in L°°(T), then the essential resolvent
PeiTy) for Tv is the open set of those X in C for which Tv_x is a Fredholm
operator. If X is in pe{T^) and jC"v-a) = "» then
The basic result concerning the/A and 0^ is contained in the following.
7.40 Proposition If <p is a function in L°°(T) and X is in p^T,,), then
fkQx = 1 and
-t/a(z) = A(z) • P< i(z) for z in D.
oA [q> — X)
Proof There exist functions ma and z;A in H02 such that Xn((p—X)f}i =
1 + Wjt and X-n9)J(<p-X) = ! + »/• Multiplying these two identities, we
obtain
where/A0A is in /71 and u^+vx+uxvx is in //</ and thus/j^ = 1. We also
)^A(z) = 1 by Lemma 6.16.

192 7 Toeplitz Operators
Since the functions X->%J((p — X) and X->x-n/((p — X) are analytic re-
revalued functions, it follows from Corollary 7.8 that fx and gx are analytic
/72-valued functions. Differentiating the identity xn(<P~fyfx= l+w* with
respect to X yields -%„/,. +Xn(9~Z)A' = w/ and hence fx = (<p - X)fx' -
1-nux', where w/ lies in //02. Multiplying both sides of this equation by
gxl(<p — X), we obtain
J A'(
Since fx'gk is in Hl and u/(l + SA) is in Hol, we have P{l/((p — X)} =fx9x-
Finally, again using the identity proved in the first paragraph, we reach the
equation// =/jP{l/(9-A)} and observing that evaluation at z in ID com-
"mutes with differentiation with respect to A we obtain the desired results. ■
It is possible to solve this latter equation to relate the fx which lie in the
same component of pe(
7.41 Corollary If q> is a function in L°° (T) and X and Xo are endpoints of a
rectifiable curve F lying in pe(T^), then
and
for z in ID.
For each fixed z in ID we are solving the ordinary first order linear
differential equation dx(X)/dX = F(X)x(X), where F(X) is an analytic function
in X. Hence the result follows for fx and the corresponding result for gx is
obtained by using the identity/A(z)^(z) =1. ■
We now show that no curve lying in peG^) can disconnect
1.42 Proposition If <p is a function in L°°(T) and C is a rectifiable simple
closed curve lying in pe{T^, then M{q>) lies either entirely inside or entirely
outside of C.

The Spectrum of Toeptttz Operators with Symbol in H^ + C 193
Proof Consider the analytic function defined by
F(z) = — P{ \{z)dn for each z in D.
2ni Jc [<p — n)
If Xo is a fixed point on C, then it follows from the preceding corollary that
hSz) = hSzo) exp{2raF(z)} for z in D.
Therefore, exp{2niF(z)} = 1 whenever fXo(z) # 0, and hence for all z in D,
since fXg is analytic. Thus the function F(z) is integervalued and hence equal
to some constant A'.
Now for each elt in T, the integral
1
equals the winding number of the curve C with respect to the point <p(elt).
Thus the function defined by
is real valued. Since
is constant, we see that i]/ is constant a.e., and hence the winding number
of C with respect to 3k (<p) is constant. Hence 3k {ip) lies either entirely inside
or entirely outside of C. ■
The remainder of the proof consists in showing that we can analytically
continue solutions to any component of C\pe(Tv) which does not contain
3%{(p). Before doing this we need to relate the inverse of TXn(<p_X) to the func-
functions fx and gx.
7.43 Lemma If <p is a function in L°°(T), k is in pe(Tv), k is in /700, and
> — X)}, then hx is in H2 and TXni9^^hx = k.
Proof If we set hx = T^^^k, then there exists /in H02 such that
Xn(<p-X)hx = k+1. Multiplying by x-ngxl(<p-X) = l + h> where vi is in
//02, we obtain
0A = jX-n0^
<9 — A

194 7 Toeplitz Operators
and hence gxhx = P{x-ngxk/((p — X)}, since l(l+vx) is in Hal. We now
obtain the desired result upon multiplying both sides by fx and again using
the identityfxgx=l. ■
We need one more lemma before proving the connectedness result.
7.44 Lemma If Q is a simply connected open subset of C, C is a rectifiable
simple closed curve lying in Q, and Fx(z) is a complex function onfixD
such that Fx{zo) is analytic on Q for zo in ID, FXo(z) is analytic on ID for Xo
in Q, and FXo is in H2 for Xo in C, then Fx is in H2 for X in the interior of C
and H^lla
Proof An analytic function \]/ on ID is in H2 if and only if
is finite, since /=S™=o^(")(°)(fI0"Zii is in #2 and /=^5 moreover
If ao,al3 ...,aA are arbitrary complex numbers, then
is an analytic function of X for 0 < r < 1. Moreover, since
fc = O ""
it follows that
N
fe=0
and hence the result follows. ■
7.45 Theorem If <p is a function in L°°(T), then the essential spectrum of
Tv is a connected subset of C.
Proof It is sufficient to prove that if C is a rectifiable simple closed curve
lying in p£{T^) with M{q>) lying outside, then T(f_x is a Fredholm operator
for A in the interior of C. Let Q be a simply connected open set which contains
C and the interior of C and no point of the essential spectrum ae(Tv) exterior
to C lies in Q. We want to show that ae(Tv) and Q are disjoint.

The Connectedness of the Essential Spectrum 195
Fix Xo in C. For each X in Q let T be a rectifiable simple arc lying in Q
with endpoints Xo and X and define
l<p-H
and
tf
1
for z in ID. Since P{\l((p — /i)} (z) is analytic on the simply connected region Q,
it follows that Fx(z) and Gk{z) are well defined. Moreover, since Fx{z) =
fx(z) for X in some neighborhood of C by Corollary 7.41, it follows from the
preceding lemma that Fx is in H2 for all X in Q. Similarly, Gk is in H2 for
each A in Q.
If we consider the function \]/(X) = T^^^^F^—l, then \]/(X) lies in H2
for A in Q, is analytic, and vanishes on the exterior of C in Q. Thus
for X in Q, and in a similar manner we see that Tx_ni<p_^-1 Gk = 1 for X in
Q. If & is a function in //°% then
Hk{z) = Fk{z
defines a function satisfying the hypotheses of the preceding lemma. Thus,
Hx is in H2 and rZn(<?_A)//A = k for A in Q. Moreover, we have
|//j2 < sup ||//,j|2 = sup ||r-^_w|| ||*||2.
i.EC A,eC
Therefore, if A: is in H2 and {&,}"= i is a sequence of functions in //" such
that lim^^^ ||A:—Aj||2 =0, then the corresponding functions {H/}f=l for a
fixed A in Q form a Cauchy sequence and hence converge to a function //A
such that TXni<p_i}Hx = k. Thus we see that TXn(ip_^ is onto for X in Q. Since
the same argument applied to I on Q yields that Tx_n(^_^ is onto for X in Q,
we see that rZri(<?_A) is invertible, and hence that Tv—X is a Fredholm oper-
operator, which completes the proof. ■
7.46 Corollary (Widom) If cp is a function in £*■(¥), then a{T^ is a con-
connected subset of C.
Proof By virtue of Proposition 7.24, the spectrum of Tv is formed from
the union of the essential spectrum plus the X for which 7^ — X is a Fredholm

196 7 Toeplitz Operators
operator having index different from zero. Since Tv—k is a Fredholm oper-
operator for X in each component of the complement of the essential spectrum
and the index is constant, it follows that the spectrum is obtained by taking
the union of a compact connected set and some of the components in the
complement—and hence is connected. ■
Despite the elegance of the preceding proof of connectedness, we view
it as not completely satisfactory for two reasons: First, the proof gives us no
hint as to why the result is true. Second, the proof seems to depend on showing
that the set of some kind of singularities for a function of two complex
variables is connected, and it would be desirable to state it in these terms.
We conclude this chapter with a result of a completely different nature
but of a kind which we believe will be important in the further study of
Toeplitz operators. It involves a notion of "localization" and suggests that
in order to understand certain phenomena concerning Toeplitz operators it
is necessary to consider other representations of the C*-algebra 2(L°°(T)).
We begin with a result concerning C*-algebras having a nontrivial center.
The center of an algebra is the commutative subalgebra consisting of those
elements which commute with all the other elements in the algebra. In the
proof we make use of the fact that an abstract C*-algebras has a *-isometric
isomorphic representation as an algebra of operators on some Hilbert space.
Also we need to know that every ^isomorphism on C{X) can be extended
to the Borel functions on X. Although we have not proved these results
in the text, outlines of the proofs were given in the Exercises in Chapter
4 and 5.
7.47 Theorem If 2 is a C*-algebra, 31 is a C*-algebra contained in the
center of 2 having maximal ideal space Mm, and for x in Mm, 3X is the closed
ideal in 2 generated by the maximal ideal {A e 31: A(x) = 0} in-31, then
In particular, if <bx is the *-homomorphism from 2 onto 2/3x, then
UteMs, © ®x is a *-isomorphism of 2 into YxeMn © ^P*. Moreover, T
is invertible in 2 if and only if 0>x(T) is invertible in 2/3x for x in Mm.
Proof By Proposition 4.67 it is sufficient to show that supxeMffl ||
||r|| for Tin 2. Fix Tin 2 and suppose that ||r|| -supxeMstt ||3>X(I)| = £ > 0.
For xo in Mm let Oxo denote the collection of products &4, where S is in
2 and A is in 31 such that A vanishes on a neighborhood of xo. Then Oxo

Localization to the Center of a C*-Algebra 197
is an ideal in Z contained in 3Xo, and since the closure of OXt> contains
we see that 3Xo = clos[Oxj. Thus there exists S in % and A in 91 such that
A vanishes on an open set UXo containing xo and ||<E>Xo(r)H +e/3 > || T+ SA |.
Choose a finite subcover^ {Ux.}?=1 of M^ and the corresponding operators
Si}^! in X and {Ai}^1 in St'such that
and y?j vanishes on Ux..
Let $ be a ^isomorphism of X into £(jf) for some Hilbert space Jf
and let ^o be the corresponding ^isomorphism of C{Mm) into £(jf) defined
by ^o = 3)°r~1, where F is the Gelfand isomorphism of 21 onto
Since 3>(9l) is contained in the commutant of $(£), it follows that i
is contained in the commutant of $(£), and hence there exists a *-homo-
morphism \]/ from the algebra of bounded Borel functions on Mm into the
commutant of $ (£) which extends ^o.
Let {AJjL! be a partition of Afg, by Borel sets such that A,- is contained
in Ux. for i= 1,2,...,N. Then
and hence
\\®(T)t(IAt)\\ < mr+StA^ U{IA)\\ < ||r|| -1 for each 1 < i < TV.
Since the {^(/Al)}f=i are a family of commuting projections which reduce
0>(r) and such that £f=1 ^(/Ai) = /^, it follows that
= sup
and hence ||CE>C")|| < || T|| — e/2. Since $ is an isometry, this is a contradiction.
If T is invertible in X, then clearly 3>X(T) is invertible in 2/3x for x in Afg,.
Hence suppose <&X(T) is invertible in Z/3X for each x in Mm. For xo in Mm there
exists S in 2 by Theorem 4.28 such that 3>XoET-/) = 0. Repeating the
argument of the first paragraph, we obtain a neighborhood O^ on which
\\<bx(ST-I)\\ < i for x in OXa. From Proposition 2.5 we obtain that <&X(ST)
is invertible and flcD^ST)!! < 2 for x in Oxo. Since <&X(T) is invertible,
<&JCET)~1$JCE') is its inverse, and hence ||<E>XC")-11| is bounded for x in

198 7 Toeplitz Operators
OXt>. A standard compactness argument shows that supxeMstt II^CO11| < oo.
Therefore YXEMm ® ®X(T) is invertible in £xeMsi © 3:/3x, and hence T is
invertible in X by Theorem 4.28. ■
7.48 The context in which we want to apply this result is the following.
Although the center of the C*-algebra 3:(L°°(T)) is equal to the scalars, the
quotient algebra £(L°°(T))/£(E(//2) has a nontrivial center which contains
3:(C(T))/££(//2) = C(J) by Proposition 7.22. Thus we can "localize" the
algebra 3:(L°°(T))/£G:(//2) to the points of T.
For k in T let 3^ be the closed ideal in X(L°°(T)) generated by
and let 3^ be the quotient algebra Z (L00 (T))/3^, and ^ be the natural *-homo-
morphism from 3:(L°°(T)) to 3^.
7.49 Theorem The C*-algebras 3^ are all *-isomorphic. If <t> is the *-homo-
morphism defined by $ = X^t©*^ &om ^(L°°(T)) to ^eT©^, then
the sequence
@)—1
is exact at 2(L°°(T)).
Proof Since ^(^(T)) is an irreducible algebra in £(//2), every non-
nonzero closed ideal contains ££(//2) by Theorem 5.39. Thus ££(//2) is con-
contained in the kernel of $ and hence <1> induces a *-homomorphism <bc from
the quotient algebra Z(L00(¥))/££(//2) into ^eT©^. Since
2(C(T))/£G:(//2)
is contained in the center of 3;(L00(r))/£e(//2)) the preceding theorem
applies, and we conclude that Oc is a *-isomorphism. Rotation by X on T
obviously induces an automorphism on !X(L°°(T)) taking 30 onto 3^, and
hence there exists a ^isomorphism from 20 onto ^. ■
The usefulness of this result lies in the fact that it reduces all questions
concerning operators in ^(^(T)) modulo the compacts to questions con-
concerning 3^. Unfortunately, we do not know very much about the algebras
3^. The following proposition shows that the operators in %x depend only
on local properties of the defining functions.
Recall that MOO\1D is fibered by the circle such that

Locality of Fredholmness for Toeptttz Operators 199
and that the Silov boundary of H°° can be identified with the maximal ideal
space AfLco of L°°(T). Let us denote the intersection Fx n ML«> by dFx.
7.50 Proposition If {q>lJ}lJ = l are functions in L°°(T) with Gelfand trans-
transform 0tJ on MLCO and X is in T, then O^XfL x U% 1 TVij) depends only on the
functions WX
Proof It is sufficient to show that ^(T^) = 0 for <p in L°°(T) such that
(f>\dFx = 0. By continuity and compactness, it follows that for e > 0 there
exists an open arc U of T containing X such that || <(>/(,!!«, < e. If \j/ is a con-
continuous function on T which equals 1 on the complement of U and vanishes
at X, then
where
II^(?WO)|| < ||r^j| < 6, *A(r^oA_#)) = o,
since
Wd-ti = o and ^(ro/TV^) = ^(r*lTXj^(r#) = o,
since T^ is in 3^. Therefore, we have 11^C^I1 < e for e > 0 and hence
^W = o. ■
As corollaries, we obtain the following results proved originally by
"localizing" in //°° + C(T).
7.51 Corollary If q> is a function in L°°(T), then J^ is a Fredholm operator
if and only if for each X in T there exists ij/ in L°°(T) such that Tt is a Fredholm
operator and <p = ij/ on 5FA.
From Theorem 7.49 and the definition, it follows that Tv is a
Fredholm operator if and only if ^G^,) is invertible for each X in T. If for
X in T there exists \j/ in L°°(T) such that T^ is a Fredholm operator and
$ = $ on d/^, then <!>.*( 7^) is invertible in 2^ and equal to ^G^) by the
previous proposition. Thus the result follows from Theorem 7.47. ■
7.52 Corollary If q> and ij/ are functions in L°°(T) such that for each X
in T either 0 — G^dF^^O for some 0, in H°° or {j/ — 62\dFx = 0{or some
92 in //°°, then T^T^-T^ is compact.
Since T^T^-T^ is compact if and only if *^(r,,)
<t>^(r^) for each A in T by Theorem 7.49, the result is seen to follow from
Proposition 7.5. ■

200 7 Toeplitz Operators
Notes
In an early paper [109] Toeplitz investigated finite matrices which are
constant on diagonals and their relation to the corresponding one- and
two-sided infinite matrices. The fundamental theorem in this line of study
was proved by Szego (see [54]), and most of the early work concerned this
type of question. In [116] Wintner determined the spectrum of analytic
Toeplitz matrices, and he and Hartman set the tone for much of the work
in this chapter in two papers [59] and [60] some twenty years later. The
first systematic study of Toeplitz operators emphasizing the mapping (p-tT^
was made by Brown and Halmos in [11]. What might be called the algebra
approach to these problems was first made explicit in [29] and [30] and was
based on the earlier papers [17] and [18] of Coburn.
A "vast literature exists for Wiener-Hopf operators beginning with the
fundamental paper of Wiener and Hopf [115]. Most of the early work con-
concerns the study of explicit operators, and a good exposition of that along
with a bibliography can be found in Krein [72]. The studies of Toeplitz
operators and Wiener-Hopf operators had parallel developments until
Rosenblum observed [94] using Laguerre polynomials that the two classes
of operators were unitarily equivalent. Subsequently, Devinatz showed in
[25] that the canonical conformal mapping of the unit disk onto the upper
half-plane establishes the unitary equivalence between a Toeplitz operator and
the Fourier transform of a Wiener-Hopf operator. Thus a given result can
be stated in the context of either Toeplitz operators or of Wiener-Hopf
operators.
The spectral inclusion theorem is due to Hartman and Wintner [60],
although the proof given here of Proposition 7.6 first occurs in [110]. Corol-
Corollaries 7.8 and 7.19 as well as the remarks following 7.9 are due to Brown and
Halmos [11]. The existence of the homomorphism in Theorem 7.11 as well
as its role in these questions is established in [31]. In [103] Stampfli observed
that a proof of Coburn in [17] actually yields Proposition 7.22. The analysis
of the C*-algebra 2(C(T)) was made by Coburn in [17] and [18] while its
applicability to the invertibility problem for Toeplitz operators with con-
continuous symbol was observed in [29]. Proposition 7.24 was proved by Coburn
in [16]. The content of Corollary 7.27 is the culmination of several authors
including Krein [72], Calderon, Spitzer, and Widom [13], Widom [110],
and Devinatz [24]. The proof given here first appears in [29] and indepen-
independently in [3], where Atiyah used the matrix analog in a proof of the periodicity
theorem. A related proof was given by Gohberg and Fel'dman [45]. The study

Notes 201
of Toeplitz operators with symbol in //°°+C(T) was made in [30] with
complements in [77] and [103].
The invertibility criteria stated in Theorem 7.30 and its corollary were
given independently by Widom [111] and Devinatz [24] and are based on
a study in prediction theory by Helson and Szego [64]. The spectral inclusion
theorem given in Theorem 7.32 is based on an extension by Lee and Sarason
[77] of a result of the author [31]. The question of the connectedness of the
spectrum of a Toeplitz operator was posed by Halmos in [57] and answered
by Widom in [113] and [114]. The proof of Theorem 7.45 is a slight adapta-
adaptation of that in [114] to cover the essential spectrum and avoiding certain
measure theoretic considerations as well as the use of the harmonic con-
conjugate. The possibility of the essential spectrum being connected was sug-
suggested to the author by Abrahamse.
Theorem 7.47 is closely related to various central decompositions in
C*-algebra (see [96]) but its application to Toeplitz operators is new, and
these results were suggested by the earlier Corollaries 7.51 and 7.52. The
first corollary is due to Simonenko [102] and independently to Douglas and
Sarason [35] and extends a result of Douglas and Widom [38]. The second
corollary is due to Sarason [98].
Several further developments and additional topics should be mentioned.
The invertibility problem for symbols in the algebra of piecewise continuous
functions has been considered by Widom [110], Devinatz [24], and from
the algebra viewpoint by Gohberg and Krupnik [50]. The invertibility prob-
problem has also been considered for certain algebras of functions which appear
more natural in the context of the line. The algebra of almost periodic func-
functions was considered independently by Coburn and Douglas [19] and by
Gohberg and Fel'dman [46], [47]. Actually, the latter authors considered the
Fourier transforms of measures having no continuous singular part. The
problem for the Fourier transform of an arbitrary measure has been con-
considered by Douglas and Taylor [37] using a deep result [108] of the latter
on the cohomology of the maximal ideal space of the convolution algebra of
measures. Lastly, Lee and Sarason [77] and Douglas and Sarason [36] have
considered the invertibility problem for certain functions of the form <pip for
inner functions <p and ij/.
Generalizations of the notion of Toeplitz operator have been considered
by many authors: Douglas and Pearcy studied the role of the F. and M. Riesz
theorem in [33]; Devinatz studied Toeplitz operators on the //2-space of a
Dirichlet algebra [24]; Devinatz and Shinbrot [26] studied the invertibility
of compressions of operators to subspaces; and Abrahamse studied Toeplitz

202 7 Toeplitz Operators
operators defined on the //2-space of a finitely connected region in the
plane [1]. A different kind of generalization is obtained by considering
Wiener-Hopf operators. A subsemigroup of an abelian group is prescribed,
and convolution operators compressed to the corresponding L2 space of
functions supported on it are studied. Certain basic results for rather arbitrary
semigroups are obtained by Coburn and Douglas in [20]. Earlier work
involving half-spaces is due to Goldenstein and Gohberg [52] and [53]. The
case of the quarter plane is of particular importance and coincides with
Toeplitz operators defined on H2 of the bidisk. Results on this have been
obtained by Simonenko [101], Osher [84], Malysev [78], Strang [106], and
Douglas and Howe [32]. In particular, the latter authors obtain necessary
and sufficient conditions for such a Toeplitz operator to be a Fredholm
operator. They show, moreover, that while such an operator must have
index zero, it need not be invertible.
In many of the preceding contexts, various topological invariants of the
symbol enter into the determination of when the corresponding operator is
invertible, and usually these invariants must be zero. In the case of a con-
continuous symbol on the circle, a nonzero invariant corresponds to the oper-
operator being a Fredholm operator having a nonzero index. A start at establish-
establishing similar results for the other classes of operators has been made by Coburn,
Douglas, Schaeffer, and Singer in [21] where a generalized notion of Fredholm
operator due to Breuer [8], [9] is utilized. It is expected that questions of
this kind will prove important in future developments.
Although attention in this book has been confined to the scalar case, the
matrix case is perhaps of even greater importance. The function cp is allowed
to have nxn matrices as values and to operate on a C-valued //2-space.
Many of the techniques of this chapter can be carried over to this case using
the device of the tensor product. More specifically, if 31 is an algebra of scalar
functions, $lB is equal to S&® Mn, where Mn is the C*-algebra of operators
on the H-dimensional Hilbert space C, and more importantly, £($IJ is
isometrically isomorphic to £($1)® Mn. In particular, applying this to one
of the exact sequences considered in the chapter one obtains the exact sequence
@) -* £e(//2) ® Mn -* £(C(T)) <g> Mn -* C(T) <g> Mn -+ @).
(The faci that the "scalar" sequence has a continuous cross section is also
used.) Since ££(//2) ® Mn is equal to ££(//2 ® C"), one obtains that a
matrix Toepliiz operator with continuous symbol is a Fredholm operator if
and only if the determinant function does not vanish on the circle. More-
Moreover, the index can be shown to equal minus the winding number of the

Exercises 203
determinant. (This latter argument uses the fact that every continuous mapping
from T into the invertible n x n matrice is homotopic to a mapping from T to
the invertible diagonal matrices.)
This latter result is due to Gohberg and Kreln [49]. A generalization to
certain operator-valued analogs can be found in [31]. Additional results on
the matrix and operator-valued case are in [87] and [88].
Lastly, although this book does not comment on it, the study of Toeplitz
and Wiener-Hopf operators is important in various areas of physics and
probability (see [54], [69], [83]) and in examining the convergence of certain
differences schemes for solving partial differential equations (see [83]).
Exercises
7.1 If 31 and 23 are C*-algebras and p is an isometric *-linear map from
31 into 93, then a(T) a o(p(T)) for T in 31. (Hint: If T is not left invertible
in 31, then there exists {Sn}%L 2 in 31 such that || SB \\ = 1 and lirn^ «, || Sn T || = 0.)
If, in addition, 31 is commutative, then o{p{Tj) a hla{T)~] for T in 31.
7.2 If q> is a nonconstant real function in L00 (T), then 7^ has no eigenvalues.
7.3 An operator T in £(//2) is a Toeplitz operator if and only if
"T1^ onon rj-i
7.4 If q> is a nonzero function in L°°(T), then Mv and Tv have no eigen-
eigenvalues in common.
7.5 If q> is a real function, then Tv is invertible if and only if the function 1
is in its range.
7.6 If <p is in L°°(T), then W{T9) =h\m((p)\.
7.7 If <pu <p2, and <p3 are functions in L°°(T) such that TVl T^-T^ is
compact, then cpt q>2 = <p3.
Definition If {aB}^L0 is a bounded sequence of complex numbers, then the
associated Hankel matrix {a;j}?>J=o is defined by a;j = al+J for i,j in Z+.
7.8 (Nehari) If {aj^o is a bounded sequence, then the Hankel operator
H defined by the associated Hankel matrix is bounded if and only if there
exists a function <p in L°°(T) such that
2n Jo

204 7 Toeplitz Operators
Moreover, the norm of H is equal to the infimum of the norm ||^H<» f°r all
such functions <p.* (Hint: Consider the linear functional L defined on H2
by L(f) = (Hg,h), where /= gh, and show that L is continuous if H is
bounded.)
7.9 (Hartman) If {aB}^=0 is a bounded sequence, then ihe associated
Hankel operator is compact if and only if there exists a continuous function
q> on T such that
«- = T" f "
2n Jo
") * for n in R+.*
(Hint: Use Exercise 1.33 to show that a certain linear functional L is w*~
continuous if and only if 7/is compact and apply the analog of Exercise 6.36.)
7.10 An operator H in £G/2) is a Hankel operator if and only if T*t H =
HTXi.
7.11 If> is in L°°(T), then n(T9) is unitary in 2(L°° (¥))/££(H2) if and only
if q> is in QC* (Hint: Show that the Hankel operator Hv is compact if and
only if n{Tv*) is an isometry in 2(L°°(T))/fiC;(//2).)
7.72 If <p is in L°°(T), then n(Tv) is in the center of X(L00(T))/£G(T2) if
and only if q> is in QC. Are there any other operators in the center?**
7.75 Show that Tx Tv-Txv is compact for every x in L°°(T) if and only if
<pis in 7/°° + C(T).*
7.14 If <p is in ff°° + C(T), then ^ is invertible in L°°(T) if and only if 7^
has closed range.
7.75 If <p is in //°° + C(T) and A is in <r G^^D0), then either A is an eigen-
eigenvalue for Tv or I is an eigenvalue for Tv.
7.16 (Widom-Devinatz) If <p is an invertible functions in L°°(T), then T^
is invertible if and only if there exists an outer function g such that
\arggcp\ <n/2-S
for 8 > 0.
7.77 Show that there exists a natural homomorphism y of Aut[£(C(T))]
onto the group Hom+(T) of orientation preserving homomorphisms of T.
(Hint: If <p is in Hom+(T), then there exists K in ££G/2) such that T^ + K
is a unilateral shift.)
7.18 Show that the connected component Auto[2:(C(T))] of the identity
in Aut[3:(C(T))] is contained in the kernel of y. Is it equal?**

Exercises 205
7.19 If <p is a unimodular function in QC and K is appropriately chosen in
fi£(//2), then a(T) = (T^K)*T(T<!> +K) defines an automorphism on
!I(C(T)) in the kernel of y. Show that the automorphisms of this form do not
exhaust kery.* Is a in Auto[jr(C(T))]?**
7.20 If F is a pure isometry on Jf and En is the projection onto F"Jf, then
Ut = Y.™=oe""(En-~En+i) 's a unitary operator on jf for e" in T. If we set
Pt(T) = U* TUt for Tin the C*-algebra GK generated by V, then the mapping
F(e") = JS, is a homomorphism from the circle group into Aut[(£K] such that
F(e") = fit(T) is continuous for Tin (£K. Is it norm continuous? Moreover,
for V — TXl, the unitary operator Ut coincides with that induced by rotation
by e* on H2.
7.21 Identify the fixed points %v for the JS, as a maximal abelian subalgebra
of (£K. (Hint: Consider the case of F= TXI, acting on /2(Z+).)
7.22 Show that the mapping p defined by
p(T) = — P,(T)dt for rin£K
2?r Jo
is a contractive positive map from (£K onto gK which satisfies p(TF) = p{T)F
for T in (£K and /" in %v.
7.23 If Fj and F2 are pure isometries on 2tfx and Jf 2, respectively, such
that there exists a *-homomorphism $ from GKl to GK2 with ^(V^ = F2,
then $ is an isomorphism. (Hint: Show that po<& = <bop and that $|3fv,
is an isomorphism.)
7.2^ If ^4 and B are operators on ^ and Jf", respectively, and 3) is a *-homo-
morphism from &A to (£B with 3) (^4) = B, then there exists *-isomorphism
*¥ from <LA@B to e^ such that »PU®B) = ^.
7.25 If F is a pure isometry on Jf and W is a unitary operator on Jf", then
there exists a *-isomorphism ^ from (£Kl onto GK2 such that ^(F2) = F2.
7.26 (Coburri) If F2 and F2 are nonunitary isometries on H1 and H2,
respectively, then there exists a ^isomorphism *P from GKx onto (£V2 such
that ^(F!) = V2.
Definition A C*-algebra 31 is said to be an extension of £(£ by C(T), if
there exists a *-isomorphism 4> from fiGpg") into 31 and a *-homomorphism
*P from 31 onto C(T) such that the sequence
@) —> fie (Jf) -A- SI -^> C (T) —> @)

206 7 Toeplitz Operators
is exact. Two extensions Sl2 and S&2 will be said to be equivalent if there
exists a *-isomorphism 6 from Sl2 onto S&2 and an automorphism a in
Aut[£G(jf)] such that the diagram
@) |« |« jJcOD—>(®
7 *, * /""*>
£e(jf)
commutes.
7.27 If TV is a positive integer and £w is the C*-algebra generated by TXN
and £e(//2), then 2N is an extension of ££ by C(T) with ¥G^) = xi ■
Moreover, 3^ and 3:M are isomorphic C*-algebras if and only if N = M.
(Hint: Consider index in XN.)
7.28 Let K2 = l}(J)Q H2, Q be the orthogonal projection onto K2, and
define S^, = QMV\K2. If TV is a negative integer and <SN is the C*-algebra
generated by SXI) and £&(K2), then <ZN is an extension of £G by C(T) with
lFE'J(w) = xi- Moreover, the extensions £M and <5N are all inequivalent
despite the fact that the algebras 3^ and <ZN are isomorphic.
7.29 Let E be a closed perfect subset of T and /i be a probability measure
on T such that the closed support of /i is T and the closed support of the
restriction /i£ of /i to E is also E. If S&E is the C*-algebra generated by MXl
on I}(jx) and ££(L2(pE)), then Sl£ is an extension of ££ by C(T) with
*P(AfZl) = Xi- Moreover, two of these extensions are equivalent if and only
if the set E is the same.
7.30 If E is a closed subset of T, then E — Eo u {e11": k > 1}, where 2?0 is a
closed perfect subset of T. Let /i be a probability measure on T such that the
closed support of /i is T and the closed support of the restriction /i£o of ,u to
Eo is also Eo. If SEE is the C*-algebra generated by W = MXi © Y*» i © Melt-
on £2 00© I^i ©/2(Z) and ^(I^OO©!^! ©J2(z))> then 9l£ is an
extension of £(£ by C(T) with *P(W) = Xi- Moreover, two of these extensions
are equivalent if and only if the set E is the same.
7.31 Every extension S& of £(£ by C(T) is equivalent to exactly one exten-
extension of the form 3^, <»w, or 5t£.* (Hint: Assume 31 is contained in £(jf)
and decompose the representation of £(£ on Jf. Use Exercises 5.18-5.19.)
7.32 If T is an operator on Jf such that ae{T) = T and T*T-TT* is
compact, then the C*-algebra generated by T is an extension of £(£ by C(T)
with *F(r) = xi- Which one is it?

Exercises 207
7.33 If B2 is the closure of the polynomials in I2 (ID) with planar Lebesgue
measure and PB is the projection of L2(D>) onto B2, then the C*-algebra
generated by the operators {Rv: ^>eC(D>)}, where Rv = P<%MV\B2, is an
extension of C(T) by £G with ^(Rx,) = Xi- Which one is it?
7.34 If SI is an extension of £(£ by C(T)> determine the range of the mapping
from Aut(St) to Aut(C(T)).*
7.35 If X is in T and <p is in L°°(T), then ^(FJ c ^(^(r^)) c h@(Fj),
where $ is the harmonic extension of q> to the Silov boundary of H°°.
7.36 (Widom) If q> is in PC, ^># is the curve obtained from the range of q>
by filling in the line segments joining (p(e"') to (p{&1*) for each discontinuity,
then Tv is a Fredholm operator if and only if ip* does not contain the origin.
Morec»rer, in this case the index of Tv is minus the winding number of q>*.*
(Hint: Use Section 7.51 to show that Tv is a Fredholm operator in this case
and that the index is minus the winding number of q>*. If q>* passes through
the origin, then small perturbations of q> produce Fredholm operators of
different indexes.)
7.37 {Gohberg-Kmpnik) The quotient algebra Z{PC)/H(£(H2) is a com-
commutative C*-algebra. Show that its maximal ideal space can be identified as
a cylinder with an exotic topology.*
7.38 If X is an operator on H2 such that Tv*XTV-X is compact for each
inner function cp, then is X = T^ + K for some \]/ in //°° and compact oper-
operator X?**
7.39 If for each z in C, cpz is a function in L°°(T) such that <pz(e") is an
entire function in z having at most N zeros for each e" in T, then the set of z
for which TVx fails to be invertible is a closed subset of C having at most N
components.* (Repeat the whole proof of Theorem 7.45.)

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Index
Absolute continuity, of measures, 104
Absolutely convergent Fourier series, 56,61
Abstract index, 36, 62, 133, 138
Abstract index group, 36, 38
Adjoint
of bounded linear transformation, 30
of closed, densely defined linear
transformation, 119
of Hilbert space operator, 81-82
Alaoglu theorem, 10
Algebra of bounded holomorphic
functions, 26, 150
Annihilator, of subspace, 30, 80
Asymptotically multiplicative functions,
169
Atkinson theorem, 128
Automorphism
of Banach algebra, 61
of C*-algebra
inner, 143
weak, 144
Banach algebra, 34, 32-62
Banach space, 2, 1-31
Banach theorem, 13, 31
Beurling theorem, 45, 153
Bilateral shift, 89
Boundary, for function algebra, 61
Bounded below, operator, 83
Bounded variation, function, 14
Brown-Halmos theorem, 182
C*-algebra, 91
Calkin algebra, 127
Cauchy-Schwarz inequality, 64
Cech cohomology group, 38
Center, of algebra, 196
Characteristic function, 18
Circle group, 51
Classical index, 130, 138
Closed convex hull, 182
Closed graph theorem, 29
Coburn theorem, 184, 185, 205
Cohomotopy group, 39
Commutant, of algebra, 108
Commutator, 180
ideal, 145, 179
Compact operator, 121, 145
Complete metric space, 2
Conjugate space, 6
Convolution product, 54
Corona theorem, 156, 173
Cross-section map, 178
Cyclic vector, 105, 157
213

214
Index
D
Dimension, Hilbert space, 76, 79
Direct sum
of Banach spaces, 29
of Hilbert spaces, 78
Directed set, 3
Dirichlet problem, 171
Disk algebra, 51-53
Division algebra, 42
Eigenspace, generalized, 131
Essential range, 57
Essential resolvent, 191
Essential spectrum, 190, 191
Essentially bounded, 24
Exponential map, 36
Extensions, of C*-algebras, 205
Extreme point, 28, 78
Fatou theorem, 174
Fiber, of Ma, 172
Finite rank operator, 59, 121
Fourier coefficient, 55
Fourier series, 55
Fredholm alternative, 131
Fredholm integral operator, 125
Fredholm operator, 127
Fuglede theorem, 114, 120
Function algebra, 51
Functional calculus, 93, 99
extended, 112-114
Gelfand-Mazur theorem, 42
Gelfand-Naimark theorem, 93
Gelfand theorem, 44, 45
Gelfand transform, 40
Gleason-Whitney theorem, 164
Gohberg theorem, 146
Gohberg-Krupnik theorem, 207
Gram-Schmidt orthogonalization process,
79
Grothendieck theorem, 30
H
Hahn-Banach theorem, 11, 12, 78
Hankel matrix, 203
Hankel operator, 203
Hardy space, 26, 70, 149-176
Harmonic extension, 168
Hartman theorem, 183, 204
Hartman-Wintner theorem, 179, 183
Hausdorff-Toeplitz theorem, 115
Hellinger-Toeplitz theorem, 116
Helson-Szego theorem, 187
Hermitian operator, 84
Hilbert space, 63-80, 66
I
Idempotent operator, 116-117
Indicator function, see Characteristic
function
Initial space, of partial isometry, 95
Inner function, 152
Inner-outer factorization, 158
Inner product, 63
Inner product space, 64
Integral operator, 125, 132, 145
Invariant subspace, 98
simply, 151-152
Involution, of Banach algebras, 82, 91
Irreducible subset, 141
Isometry, 95, 117, 146
Isomorphism, of Hilbert spaces, 76
Jensen's inequality, 174
K
Kernel
of integral operator, 126
of operator, 83
Kolmogrov-Kreln theorem, 175
Krem-Mil'man theorem, 28
Krem-Smul'yan theorem, 30
Krein theorem, 147
Lebesgue space, 7, 23-26, 68-70
Linear functional, 5, 72

Index
215
Linear transformation
bounded, 21
closable, 118
closure of, 118
self-adjoint, 119
symmetric, 119
Localization, of C*-algebras, 196
M
Matrix operator entries, 116-117
Maximal abelian subalgebra, 88
Maximal ideal space, 44
Metric, 2 (
Metric space, 2,
Multiplication operator, 87
Multiplicative linear functional, 32, 39
i
N
Nehari theorem, 203
Net, 3
Cauchy, 3
Newman theorem, 173
Norm, 1
Hilbert space, 64
Normal element, in C*-algebra, 91
Normal operator, 84
Normed linear space, 4, 27
Numerical radius, 115
Numerical range, 115
O
Open mapping theorem, 22, 23
Operator, 81
Orthogonal, 65
Orthogonal complement, 71
Orthonormal, 65, 79
Orthonormal basis, 74-75
Outer function
in H\ 172, 174
in H2, 157
Parallelogram law, 66
Partial isometry, 95
Piecewise continuous function, 176
Polar decomposition, 97, 98
Polar form, 95
Polarization identity, 64
Polynomials
analytic trigonometric, 51
trigonometric, 51
Positive element, of C*-algebra, 94, 117
Positive operator, 84
Projection, 86
Pure isometry, 173
multiplicity, 173
Putnam theorem, 120
Pythagorean theorem, 66, 73
Quasicontinuous function, 176
Quasinilpotent operator, 133, 145
Quotient algebra, 43
Quotient C*-algebra, 139
Quotient space, 20
R
Radical ideal, of Banach algebra, 60
Radon-Nikodym theorem, 25, 80
Range, of operator, 83
Reducing subspace, 98
Reflexive Banach spaces, 28-30
Resolvent set, 41
Riemann-Stieltjes integral, 15
Riesz theorem, 18
Riesz (F. and M.) theorem, 154
Riesz functional calculus, 60
Riesz-Markov representation theorem, 20
Riesz representation theorem, 72
Self-adjoint element, of C*-algebra, 91
Self-adjoint operator, 84
Self-adjoint subset, 46
Semisimple algebra, 60
Separating vector, 105
Silov boundary, 61, 164, 173
Silov theorem, 53
Similarity, of operators, 117
Spectral inclusion theorem, 179, 188
Spectral mapping theorem, 45
Spectral measure, scalar, 113
Spectral projection, 113
Spectral radius, 41
Spectral subspace, 113

216
Index
Spectral theorem, 93
Spectrum, 41
operator, 85
Square root, of operator, 95
*-Homomorphism, 88, 110
State, 147
pure, 147
Stone-Cech compactification, 58, 59
Stone-Weierstrass theorem, 46
generalized, 50
Strong operator topology, 100
Subharmonic function, 174
Sublinear functional, 11
Subnormal operator, 183
Summability, in Banach space, 4
Symbol, of Toeplitz operator, 184
Symmetric linear transformation, see Linear
transformation, symmetric
Szego theorem, 175
Sz.-Nagy theorem, 173
Tensor product
of Banach spaces, 31
of Banach algebras, 62
of Hilbert spaces, 79
of operators on Hilbert spaces, 118
Toeplitz matrix, 177
Toeplitz operator, 177
U
Uniform boundedness theorem, 29
Unilateral shift, 96, 117, 136
Unit ball, of Banach space, 10
Unitary element, of C*-algebra, 91
Unitary equivalence, of operators, 117
Unitary operator, 84, 117
von Neumann theorem, 173
von Neumann double commutant theorem,
118
von Neumann-Wold decomposition, 173
Volterra integral operator, 132
YV
*F*-algebra, 101
w-topology, 29
n>*-topology, 9, 30, 78
Weak topology, 9
Weak operator topology, 100
Weierstrass theorem, 48
Weyl theorem, 145
Weyl-von Neumann theorem, 146
Widom-Devinatz theorem, 204
Widom theorem, 195, 207
Wiener-Hopf operator, 200
Wiener theorem, 56
Wintner theorem, 183