Notes on
OPERATOR THEORY
by
PETER A. FILLMORE
Indiana University


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PREFACE
These notes represent the substance of a course of lec-
tures I delivered at Indiana University in 1967. The audience
was familiar with the basic theory of Hilbert spaces and op-
erators up through the spectral theorem and the theory of
spectral multiplicity. For many interesting and helpful con-
versations I am indebted to R. G. Douglas, and to my col-
leagues Arlen Brown, J. G. Stampfli, D. M. Topping and
J..P. Williams.
P.A.F.
Bloomington, Indiana
November, 1968

CONTENTS
Section Page
I n trod u cti on . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
o. Hyponormal Oper ators .... . . . . . . . . . . . . . . . . . 7
1. Shi fts ........ . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2 . Mod e 1 s . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . 21
3. Invariant Subs paces of Simple Shifts........ 25
4. Invariant Subspaces of General Shi fts ....... 31
5. Analytic Representation of the Unilateral
Shi ft . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
6. Contractions..........  . . . . . . . . . . . . . . . . . . 49
7. Dilations ............................... 57
8. Naimark's Theorems on Dilations........... 65
9. Contractive Semi groups ................... 77
10 H . . S b 95
.. Y pe rl n v ar I an t u spa c e s .................. .
11. Invariant Subspaces for Compact Operators. .. 107
R e fer en c e s .............................. 115
Index ................................... 121
v

INTRODUCTION
Many results concerning bounded linear operators in Hil-
bert space may be regarded as contributions to a structure
theory for these operators. While it is unlikely that a theory
as comprehensi ve as that for the finite-dimensional case
exists in general, there is no doubt that substantial progress
is possible. Evidence that comes easily to mind includes (in
addition to the spectral theorem) Dunford's theory of spectral
operators and the successful analysis by numerous workers
of operators that are close (in various senses) to being normal.
One can identify several themes in this work. A natural
one is the attempt to push fini te-dimensional results to the
infinite-dimensional situation. A well-known example of this
is the spectral theorem, which results from the diagonal form
for self-adjoint matrices when finite-dimensionality is dropped.
Again, an extensive theory initiated by Livsic in 1954 is con-
cerned with results inspired by the subdiagonal form for arbi-
trary matrices. Here one comes upon a major obstacle: the
exi stence of a subdiagonal form for a matrix am01lnts to the
existence of a chain of invariant subspaces, one of each di-
mension, and in general it is not known whether invariant
subspaces must exist. Indeed, some authors are inclined to
doubt it. Even granting a good supply of invariant subspaces,
what to do wi th them is in general far from clear. Of course,
suitable discrete chains of invariant subspaces will lead (as
1

2
NOTES ON OPERATOR THEORY
in the finite-dimensional case) to a representation of the op,.
erator as an infinite subdiagonal matrix. Unfortunately such
chains need not exi st, even for self adjoint operators (be-
cause of the possible dearth of eigenvectors). Thus even
when blessed with a profusion of invariant subspaces, one
must be prepared to contend with dense chains of them.
To obtain the necessary invariant subspaces, Livsic as-
sumes that the imaginary part of the operator belongs to the
trace class.. This also implies that the non-real spectrum
consists of eigenvalues clustering only on the real axis.. Thil
discrete part of the operator may be treated by methods that
are essenti ally fini te-dimensional, leaving as the interesting l
case an operator with real spectrum. Here the reduction to
subdiagonal form consists in representing the operator, on an
L 2-space of vector-valued functions on an interval [O,E), as
the sum of a multiplication by a non-decreasing function (the
self adjoint part) and a Volterra integration operator with an
operator-valued kernel (the quasinilpotent part)..
Another important and fruitful theme, developed by Sz.-
Nagy and many others, relates the structure of a contrative
operator T to the behavior of the sequence I, TJ r 2 , 000 of its l
iterates. Perhaps the major actor in this approach is the shif1
operator, an operator that often seems the embodiment of the
typically infiniteimensional. One need not look far to ex-
plain its appearance here: if each vector x is made to corre-
spond to the sequence (a;, Tw, r 2 x, ....), then Tx corresponds
to (T, r 2 {JJ, 006)' which is the previous sequence shifted one
place to the left. Suitable modifications of this lead easily
to the following model for contractions: every contraction is

INTRODUCTION
3
I1nitarily equivalent to an operator obtained by restricting a
coisometry (the adjoint of an isometry) to an invariant sub-
space. Unhappily this elegant result is often of limited util-
ity, a fact that ari ses from the bewi Idering variety of invariant
subspaces possessed by a coi sometry. The situation is rather
frustrating and even ludicrous; with this model the invariant
. subspace problem mentioned above obviously can be settled
by determining whether the minimal non-zero invariant sub-
spaces of a coisometry must be one-dimensional. In spite of
the apparently simple nature of coisometries, even relatively
crude information of this sort is available only when the nul-
lity of the coisometry is finite. Before leaving this topic it
should be mentioned that whereas the Livisic theory attempts
to view an operator as a perturbation of a self-adjoint opera-
tor, here it is regarded as a perturbation of an isometry. Thus
a contraction T such that 1- T*T is not too big has invariant
subspaces.
This book is intended to be an introduction to structure
theory for operators, and consequently attempts to provide a
broad background. For proofs of a number of the deepest re-
sults the reader is referred to the literature. Although materi-
al is drawn from a wide variety of sources, the point of view
is usually the second of those previously outlined. There are
some new contri butions, mostly of a modest nature, such as
simplified proofs and improved points of view, as well as a
few new results. From the foregoing discussion one might ex-
pect the stars of the book to be shift operators and invariant
subs paces, and such is the case; on each page, if not explic-
itly present, they are not far behind the scenes.

4
NOTES ON OPERATOR THEORY
We conclude this introductory section with a brief outli
of the contents. After a section about hyponormal operator.
which contains material not conveniently assembled else-
where, the next two sections are devoted to the elementary
facts about shift operators and their role in model theory. 'l.
basic theorems of Rota and of de Branges and Rovayak chfI
I
acterizing the parts of the backward shift up to similarity aJI
up to unitary equivalence are obtained. With the relevance _
invariant subspaces of shift operators thus established, thil
topic is taken up in the following two sections. Beurling's
results relating the invariant subspaces of the simple shift
and inner functions are derived and extended to shifts of
countable multiplicity. This leads to the well-known formull
tion of the invariant subs pace problem as a factorization pr{J
lem for operator-valued functions.
In section 6 the N agy- Foi as model for contractions is dt
veloped. We obtain the coisometric extension in two ways:
quickly, by displaying a matrix representation, and more la-
boriously, by means of a construction that reveals in greateq
detail the structure of the extension. From this follows the
decomposi tion into uni tary and completely nonuni tary sum-
mands, as well as a related decomposition due to Foguel.
Sections 7 and 8 contain numerous results concerning unitar,
dilations of contractions and contractive operator-valued
mappings. The strong unitary dilation is obtained from the
coisometric extension, the structure of this dilation is ex-
amined, and several appli cations are given. In section 8 we
descri be the beautiful circle of results, due to Bochner, Her'....
glotz, Naimark, and Stone, dealing with positive-definite

INTRODUCTION
5
bperator-valued functions, unitary representations, and posi-
tive operator-valued measures.
The ideas developed for the study of a single contraction
(or rather, of its powers) can be applied to continuous one-
I
parameter contractive semigroups, and this is the subject of
section 9,. The coisometric extension is constructed and used
to deduce Cooper's Theorem on the structure of isometric semi-
groups. Then a quite distinct point of view is adopted, in
which the semigroup is studied in terms of its infinitesimal
generator and its cogenerator.
A number of results related through the concept of hyper-
invariance are gathered in section 10. We show that the Vol-
terra operator and certain weighted shifts are unicellular, and
that the invariant subspaces of a unicellular operator are
hyperinvariant. An operator that is quasi-similar to a unitary
operator has nontrivial hyperinvariant subspaces, and from
this several invariant subspace theorems for contractions are
deduced. The concluding section is devoted to the recent
proof of Arveson and Feldman that compact operators have
nontrivial invariant subspaces.

SECT/ON 0
HYPONORMAL OPERATORS
Throughout these notes the term "operator" will be re-
served for a bounded linear transformation on a Hilbert space,
and a "subspace" will mean a closed linear manifold. Un-
doubtedly the best-understood operators are the normal oper-
ators. Because of this a recurrent theme in operator theory is
the study of operators that in some sense are close to being
normal. An interesting class of such operators arises from
the following easily- stated but unsolved problem:
Problem: If N is a normal operator and m an invariant
subspace of dimension > 2, is there an invariant subspace
lying properly between to I and m ?
Operators such as Njm, obtained by restricting a normal
operator to an invariant subspace, are called 8ubnormal.
Thus the above problem is: does every subnormal operator
have a nontrivial invariant subspace? This is a special case
of one of the basic unsolved problems of operator theory:
Invariant Sub'8pace Problem: Does every operator have a
nontrivial invariant subspace?
Unlike the corresponding situation for self-adjoint opera-
tors, a subnormal operator need not be normal. For example,
let £2 be the Hilbert space of doubly-infinite sequences
a =: tant of complex numbers such that lIal12 = II a n l 2 < 00,
7

8
NOTES ON OPERATOR THEORY
and let U be the bilateral shift: (U a)n = an_I' Then V i
unitary, and the subspace m = ta! an := 0 for n < 01 is inv 
ant, but Vjm is not normal. On the other hand, not every 0
erator is subnormal. In fact, if T = N!m is subnormal, the
for all ::c f m,
!!T*iVll :: !j P m N *3J11 < !!N*x!1 == IINxl! = !ITxl! ,
or equivalently, TT* < T*T. Such operators are called hY1IJ
normal; they constitute a class which contains properly th..
subnormais.
In some circumstances the problem mentioned above hasl
an affirmative solution. The following theorem is due to
Wermer [58].
I. If N is a normal operator with spectrum a (N) of area zerl
and if '" is an invariant subspace of dimension > 2, then
T = N! has a nontrivial invariant subspace.
Proof: Let K be the smallest subspace containing
m and reducing N. Since NIK is a normal operator with spe...
trum contained in a(N), there is no harm in assuming that K'1
is the whole space. With this reduction it follows [24, Prob-l
lem 157] that a(N) C a(T). On the other hand, it is obviou
that the point spectrum of N includes that of T, and that th
that the same relation holds for continuous spectra. Since N
has no residual spectrum, this implies that R = a (T)" a (N
lies in the residual spectrum of T. But it is easy to see tha
any operator with residual spectrum has a nontrivial invarianl
subspace, and so the case in which a(T) = a(N) remains.
The proof is completed by showing that in this situation T 'i
actually normal; i.e., that m reduces N. Since a(N) has

. HYPONORMAL OPERATORS
9
ero area, the Hartogs- Rosenthal Theorem [271 implies that
here are rational functions r n (z) converging uniformly to z
pn a(N), so that rn(N) converges to N* in norm. This means
that it will be sufficient to show that the resolvent R'A of N
heaves m invariant. But if A is in the resolvent set p (N),
then A ( P (T) and (N - A/)Im ==: T - AI, so that (N - Al)'Jr(  m
and RAm = m.
This proof is due to J. G. Stampfli, as is the following ob-
servation.
II. If T is hyponormal, then {x! It Twit == 1\ Tll II xlI I is an in..
variant subspace.
Proof: It can be supposed that II T II :::: 1. If II Txll = \I xl{ ,
then
IIT*Tx-H2 = IIT*TxI12 - 2Re(T*Tx,x) + 11:X\j2
:s II Ta:t\ 2 - 2! Tx\\2 + II xl1 2
< 0 ,
and so T*TXJ == x. Since the converse is obvious, the above
set is a subspace. Invariance results from the following com-
putatio: if IITwH :: 11 xU == 1, then
IIT2xII > IIT*Txll > (T*T;£,x) = IITx\\2 = 1 > 1\T211,
so that II r 2 xU :;: 1"1 T;rll.
Many important properties of normal operators are valid
for hyponormals. Some of these are proved below, and others
may be found in the exercises.

10
NOTES ON OPERATOR THEORY
III. Let T be hyponormal. Then
1. T - Al and T - 1 are hyponormal,
2. Trx = AX implies T*x = A XJ and
3. Tx = AX J Ty = py, and A  f.L imply that x and yare
orthogonal.
Proof: (T-AI)(T*- >" J) = TT*-;\T*- A T + \Aj2J
<;.T*T-AT*- A T + IA!2J == (T*- A /)(T->J), so that T->J
is hyponormal. If T is invertible and rT* s T*T J then
I < T- 1 T*TT*-1, T*T- 1 r*-lT < /,
and
..
T-ir*-l <'T*-lr- 1 .
The second statement is clear from the first, and for the thi
A (, y) = (TfC, y) = (rx, T *y) = f.L (x, y).
IV. If T is hyponormal then !I Tn!! = I! T!l n for n = 1,2, ...,
and consequently r(T) = I! TI!. (Here r(T) denotes the spec....
tral radi us of T.)
Proof: The proof is by induction. First
IIT n x!1 2 = (T*Tnx J Tn-Ix) < IIT*TnfCll !!Tn- 1 a:!1
< IIT n + 1 xll !!Tn-lxll ,
so that !!TnI12:s IITn+lIIIITn-ll! for all n2 1. If !IT k !! =
II T!l k for 1 < k < n, then
!!T!!2n == !!Tn!!2 S IIT n + l l1 !ITn-lll = IIT n + 1 !! !ITlln-l ,
and consequently J!Tlln+l < IIT n + 1 !1. The other inequaliy irl
true for any operator T. The second statement results from till
fact that r(T) is the limit of !I Tnl! tin for any T [24, Probe 7.,

o. HYPONORMAL OPERATORS
11
v. If T is hyponormal then H<T-AI)-lll ::: lid, where d is
the distance from A to the spectrum a (T) of T.
Proof: It is clear that r«T - >"/)-1) :::: 1/ d for any opera-
tor T. But (T - AI)-l is hyponormal by III, and so IV applies.
For the next result, recall that W (T) denotes the closure
of the numerical range {(Tx, x) 111 wit = 1 I, and that if (T) IS
convex and contains aCT) [24, Problems 166 and 169].
VI. If T is hyponormal, then W (T) is equal to the convex
bull of a(T).
Proof: By the above remarks we need only show that the
convex hull of a(T) contaivs W eT). This will follow if it is
shown that any closed half-plane which contains a (T) also
contains W (T). By translation and rotation this reduces to
showing that
Re aCT) < 0 implies He W eT) ::s o.
Let \\:xll = 1 and Ta; = (a + ib):x + y with a, b real and x or-
thogonal to y. Then from V
l!(T - cl)xl1 > dist(c, a( T)) > c
for all c > 0, so that
c 2 s II (T - ol)x\\ 2 = (a-- 0)2 + 1J2 + !tylt 2
and therefore 2ao S a 2 + b 2 + Hyt\ 2. Since this holds for all
c > 0, Re (T;£, x) = a < 0 follows.
REMARKS. 1. Halmos (201 and Bram [91 have shown that
the fOllowing condition characterizes subnormality: for every
integer n  1 and choice of vectors xl> X 2 ' ..., n' the matrix

12
NOTES ON OPERATOR THEORY
. .
( Tt:x " T J a:: . )
J 
is posi ti ve defini te.
2. For further information on hyponormality the reader ml
consult the papers of Stampfli, from which many of the abov"
results have been taken.
3. The important classes of Laurent and Toeplitz opera1
tors are defined as follows. Let L 2 = L 2(>...), where A is L9J
besgue measure on the unit circle, and let H 2 be the subsp't
of f £ L 2 all of whose negative Fourier coefficients vanish
The Laurent operators are those of the form Mf = <PI for
f ( L 2, where r:p £ L 00, and the Toeplitz operators are the cqt
pressions T<p = PMcpIH 2 , where P is the projection on H 2 ,
Basic information on these operators is contained in [131 andl
[24, Ch. 20].
4. The class of operators A such that A and A * A corl
mute is introduced and studied by Brown in (12].
E{J]ercises. 1. If A is an operator and m an invariant sub-
space, then (Ajm)* == PA*!m, where P is the projection
(orthogonal) on m. If A is normal, then A!m is normal if an
only if m reduces.
2. If T is hyponormal and m an invariant subspace, thel
Tim is hyponormal. If T!m is normal, then m reduces.
3. The span of the eigenvectors of a hyponormal operato,
T is a reducing subspace in which T is normal.
4. If T is hyponormal and quasi nilpotent then T = O.
(T is called quasinilpotent if r (T) = 0.)
5. An isolated point of the spectrum of a hyponormal op-
erator is an eigenvalue.

o. HYPONORMAL OPERATORS
13
6. A compact hyponormal operator is normal.
7. If T is hyponormal and a(T) c Izl181 = I}, then T
is unitary. (Hint: both T and r- 1 are contractions.) If a(T)
is real then T is self-adjoint.
8. Every invariant subspace of a compact normal operator
reduces. If every invariant subspace of a compact operator C
reduces, then C is normal. (Hint: there is a minimal invari...
ant subspace m such that \I C lmll = IICII.)

SECTION 1.
SH 1FT S
The shift operators are of fundamental importance in many
parts of operator theory. In thi s section they are introduced
with some of their basic properties.
Let K be a Hilbert space, and let f (K) :::: K E9 K E9... be
the Hilbert space of all sequences x = {x n I:=: 0 of vectors
x n € K such that II xl! 2 = 2,.: 0 II a'J n 11 2 < 00. The unilateral
'shift operator U + on e (K) is defined by
U + (a'JO' a'J I' · ..) = (0, X O ' Xl' . · · ) ·
The multiplicity of U+ is the cardinal number n == dim K. It
is easy to see that U: (X o ' xl' ...) == (Xl' a'J 2 ' ...) (this opera-
tor is called the backward shift), and that unilateral shift
operators are unitarily equivalent if and only if they have the
same multiplicity.
Recall that an operator V is an isometry if IIVa'JI! = !lxll
for all vectors x. This is equivalent to (V x, Vy) = (:.r,y) for
all x and y, and to V*V = I.
I. The operator V on J{ is unitarily equivalent to a unilater-
al shift operator if and only if V is an isometry with
00
n V n J{ = { 0 } .
n=O
The multi pli ci ty is dim (VJ{ )1.
15

16
NOTES ON OPERATOR THEORY
Proof: Let Y be such an isometry, and let Ko = (V}{)1
and Kn =0 yn Ko for n 2: 1. It will be shown that these sp,
are mutually orthogonal and span J{. Since Kn C vI{ for
n > 1, we have Ko 1 Kn for n  1. Hence
Ke = VfKo 1 yf'K n = K f + n
for n > 1 and e .2 o. Thus 1Kn fl. Suppose z 1 Kn for alII
n > o. Now J{ =: VJ{ $Ko. so VJ{ = V 2 J{ $ vK o = V 2 J( $'
and therefore J{ = V 2 J{ $ K 1 E9 Ko. Repeating gives 
J{ = VnJ{ E9Kn_l $... eKo for all n > 1 .
Consequently Z f V n J{ for n > 0, and so Z == o.
j
Now let U+ be the shift on e; <Ko), and define W: £ <Ko\
 J{ by W(x o ' a;l J :t 2J ...) == :2':0 + VX 1 + V 2 x 2 + .... Then W .
evidently unitary from e <K o ) onto H, and
VW(OJxl,tX2'...) == VX o + V2cX1 + ...
JVU+(aJ o , a'J 1 ' :X 2 ' ...) = W(O, :to' xl' ...) = VX o + V2Xl + ...,
so that W*VW =:: V .
+
Such operators are called pure isometrie's, or simply uni-4
lateral shift operators. The next result is referred to by.mant
authors as the Wold decomposition..
II. Structure of Isometries. [37, sX; 22] If V is an isomet
there is a unique reducing subspace m such that vIm is unil
tary and YI,«l is a pure isometry.
Proof: Let'  = n; = 0 V n J(. Then m is an invariant suJ.
space and if xl m. VtX 1 Vm = n°O VnJ{ == m so m te-
n=l '
duces V. Next, vim js an isometry and Vm = m, so vIm is

91. SHIFTS
17
nitary. Finally, vim! is an isometry and n: = 0 Vn(m!) = tol,
o Vim 1 is a shift by I.
Let K be a Hi Ibert space, let e 2 (K) =  oc m K be the
8ilbert space of two-way sequences x = (..., fe_ 1 , O,'xl' ...)
of vectors from K with Ilx\1 2 == Ioo I\ x n ll 2 < 00, and define the
bilateral shift U on e 2 (K) by
u (. · ., x _ l' O' :t l' ...) = (. .. , x _ l' x 0' x l' .. .) ·
The multiplicity of U is dim K. Note that U is unitary.
III. If V is an isometry on J(, there is a Hilbert space K ::) J(
and a uni tary operator U on K such that VJ( c J( and U IJ( = v.
Proof: Use II and extend the unilateral shift summand to
the corresponding bilateral shift.
DEFINITION. A subspace m of J{ is wandering for the
unitary operator W on J{ if the subspaces IWnm} 00 are paIr-
wise orthogonal, and complete if they span H.
LEMMA. If m is a wandering subspace for the bilateral
shift U on f2(K), then dim m < dim K.
Proo f: (Halperin). If K is infini te-dimensional, then
dim m < dim e 2 (K) = dim K, as required. So let xl' ..., x k be
n orthonormal basis of K; then IVnx i I n = 0, .:t 1,...; 1 < i < k I
lS an orthonormal basis of e 2 (K). If 11la} is an orthonormal
basis for m, then 1UnYa} is an orthonormal set in e 2 (K), so
l\:l:i112  In,a I (:l:i' unYa)12 by Bessel's inequality. Hence
dim K =  lI:z: i ll 2 > . I(:z:., Uny )1 2
 na"  a
, , to-
=  .1(Un xi , Ya)!2 =  IIY a l1 2 = dim m .
n,a, a

18
NOTES ON OPERATOR THEORY
IV. Bilateral shifts are unitarily equivalent if and only if .
have the same multiplicity.
V. The operator W on J{ is unitarily equivalent to a bilatl
al shift if and only if it is unitary and has a complete want
ing subspace m. The multiplicity is dim m.
We conclude this section with a result like II.
LEMMA. Let W be unitary on J(, let mo be an invarian
subspace, and let m k = W-k(mo) for k = 0, + 1,.... Then
1) m k C '«n for all k < n,
2) the spaces m_oo = n m k and moo = U '«k reduce W
and
3) W!(m oo e m_oo) is a bilateral shift of multiplicity
dim OR 1 smo).
Proof: If k S n, then W n - k mo c mo and consequently
m k = W-kmo c w-nmo = mno Since W shifts the family
tm k lone step back, and W* = W- 1 shifts it one step forwal
it is clear that moo and '«_00 are reducing. Finally,
and
m 00 e m -00 =  (i) (m k + 1 e m k ) ,
k
w-k(mt e mo) = m k + 1 e m k ,
so W[<m oo e m_oo) is a bilateral shift by V.
Notice that mo reduces W if and only if moo = m_oo
VI. If W is unitary on K, there is a reducing subs pace m
such that Wrm is a bilateral shift and such that every invar,
and subspace of wlm 1 is reducing.

gl. SRI FTS
19
Proof: Consider a maximal orthogonal family tma I of re-
ucing subspaces such that each WPRa is a bilateral shift.
r m ==  €F ma  then by the lemma every invariant subspace
1
,f W\m reduces.
Exercises. 1. If U is a shift (unilateral or bilateral) of
hultiplicity n, then Uk is a shift of multiplicity kn. If Va
Is a shift of multiplicity na' then I ffi Va is a shift of multi-
,li ci ty :2:. na ·
2. Define T on L 2 [O,1] by (Tf)(t) = V 2 f(2t) on [0, ]
nd (Tf)(t) = 0 on [, 1]. Th en Tis a uni lateral shi ft of
,nfinite multiplicity.
3. The operawr (Tf)(t) = j2f(2t) on £2[0,00] is a bi-
lateral shi ft.
4. Carry out the decomposition VI for the operator W on
l2(p) defined by (Wf)(8) = i()!({J), where fl is a positive
orel measure on the unit circle.
5.. Is the space m of VI unique?
REMARKS.. [24] contains basic information on shifts.
any of the results here and in Sections 3 and 4 can be
found in [22]. For material related to VI see (19] and [57].

SECTION 2.
MODELS
If m is an invariant subspace for the unilateral shift U+,
Ihen u 1m is again a unilateral shift (by 1. I). Since m 1 is
+
Invariant for U*, it is natural to ask what can be said about
+ 1
.he operators u+*!m. The remarkable answer, due to Rota [41],
IS provided by the following construction..
Let T be an operator on a Hilbert space K with II TIJ < 1.
I>efine a map S: K -i> e (K) by Sx = (x, T, T 2 x, ...). Then S
Is linear, one-to-one, and bounded, since
HSX!\2 = 2I\Tnx\\2 < IITI12nllxl\2 = (1-IITI12)-111:t11 2 .
tt is clearly bounded below, and so   range S is a closed
ubspace of £ (K). Moreover ST = US, and therefore :R is
Jnvariant for U*, and the operators T and U* 19{ are similar.
+ +
Thus any strict contraction is similar to an operator of
he form u 1m. With a little more care the argument can be
ade to yield considerably more. The hypothesis II TII < 1
Was used only to ensure that S is bounded, but for this it is
,obviously sufficient to have the convergence of the series
I I! Tn 11 2 . The latter will be the case if lim II Tn 11 2 / n < 1.
Since lim \1 Tn111/n always exists and equals the spectral ra-
di Us r (T), the following has been proved:
I: Rota's Theorem. Any operator with spectral radius less
than one is similar to a part of the backward shift.
21

22
NOTES ON OPERATOR THEORY
COROLLARY 1. For any operator T, r(T) = infllS-
the infimum being taken over all invertible operators S.
Proof: For any E > 0 the operator (r(T) + £)-1 T has
spectral radius less than one, and an application of the t
rem leads to the conclusion that r(T) 2: inf\IS- 1 TSI\. Th
?ther inequality follows from r(T) = r(S-lrS) < IIS-1TS\t.
COROLLARY 2. (S. Hildebrandt). For any operator 
co (a(T)) ==: n I W (S-lrS) I S invertible} ,
where co denotes convex hull and W the closure of the nt
merical range.
Proof: For any operator A, W (A) is convex and contal
a(A) [24]. Then a(T) = a(S-lrS) c w (,s--lrS), and one if
elusion follows. The proof of the other inclusion is based
the simple fact that any closed convex subset of the plan1
the intersection of all open di scs containing it. Let D be)
an open disc containing co (a(T)) with center A and radiul
Then the spectrum of (l/r)(T - AT) lies in the open unit dis.
so by Corollary 1 there is an invertible operator S such t
IIO/r)S-l(T -Al)SII < 1. It follows that W «l/r)S-l(T-
is contained in the open unit disc, or equivalently that
W (S-lTS) C D. This proof is due to J. P. Williams.
From the point of view of model theory, the above con-j
struction suffers the defect that, among all possible ways
representing a given operator as a part of a backward shift.
it may not gi ve the simplest. For example, the parts of th
shift of multiplicity one which arise in this way are all ol\EI
dimensional. The following refinement of Rota's technique

92. MODELS
23
lie to de Branges and Rovnyak [11, Appendix], helps to clear
 this objection, and at the same time replaces similarity by
t i tary equi val en ceo
 Theorem. The operator T on J( is unitarily equivalent to
.. art of the backward shift if and only if II T II s 1 and Tn  0
Ifro ng1 y ·
Proof: Let T be a contraction such that Tn  0 strongly.
.et R be the non-negative square root of 1- T*T (notice that
.- T *T 2: 0 since lIT 1\ < 1), let K = (range R)-, and define
r; J{  e (K) by
Wi.V = (Rx, RTx, RT 2 x, . ..) .
then \\RTkx!!2 == (T*kR 2 r k x, x) = !!Tkx(j2 - !!Tk+lx!l2, so
11
2 !!RT nx l\2 = \\xI1 2 _1\T n + 1 x!\2 -+ Ili.Vl\2 .
k=O
frherefore W is unitary from J{ onto  = range W. Obviously,
1fT = U: W, and consequently 9{ is invariant for U, and T
tm d U  (:R are uni tarily equi valent.
COROLLARY. If I! TIJ < 1, then T is unitarily equivalent
to a part of the backward shift.
This procedure gets by with the shift of smallest possible
tUlti:liCiy, For xampl, applying it with T = U, .one gets
 IdentIty. AgaIn, T IS a part of the backward shIft of
tmultiplicity n if and only if 1- T*T is of rank at most n.
These results lead to the following reformulation of the
Invariant subspace problem: are the minimal non-zero invariant

24
NOTES ON OPERATOR THEORY
subspaces of the backward shift one-dimensional? This 
been verified for shifts of finite multiplicity.
Exercise. (Sarason [43]). If V is the Volterra operatd
defined on L 2 [0,1] by (Vf)(t) = fot f, then T == (1- V)(l +1
is unitarily equivalent to a part of the backward shift of 
plicityone.

SECTION 3
INVARIANT SUBSPACES OF SIMPLE SHIFTS
The results of the previous section make the nature of
lIe invariant subspaces of shift operators a matter of great
litterest. The reducing subspaces will be investigated first,
after establishing a useful alternative representation of the
Ihifts. Much of the material in this and the next section can
te found in Helson [28].
The functions e (0) = e inO , n ( Z, form a complete ortho-
n
ormal set in L 2 (A) = L 2 , where A is normalized Lebesgue
easure on the uni t circle.
Proof: By elementary calculus the set lent is orthonormal.
or any continuous function 1 there is a sequence IPnl of
rigonometric polynomials (linear combinations of the func-
ions en) such that Pn .... f uniformly (Stone-Weierstrass Theo-
rem). Then Pn -4> 1 in the L 2 norm since
I\P n '- 111 2 < max Ip n - 11 ·
rhus the closed linear span of the en contains the continuous
runctions. Since the latter are II 112-dense in L 2 the result
follow s.
Let H 2 be the closed linear span of the e wi th n > O.
11. -
\Totice that H 2 consists of the f (L 2 all of whose negative
l"ourier coefficients vanish: f.27Tf(8)e in8 d8 = 0 for all n > o.
o
25

26
NOTES ON OPERATOR THEORY
For cP (: L DO, consider the operator M cp defined on L 2.
Mcpt = CPt. It is easily seen that IIMcpl1 = Ilcp 1100' The nel
two results, which are concerned with the operator M e (wi
e(O) = e 1 (0) = e i8 ), follow from V and I of sl.
II. Me is unitarily equivalent to the bilateral shift of mull
plici ty one.
III. Me! H 2 is unitarily equivalent to the unilateral shift ct
multiplicity one.
IV. An operator on L 2 commutes with the bilateral shift I
if and only if it is of the form Mcp for some cp f L 00.
Proof: Suppose that A commutes with Me' and let cp
Ae o . It will be shown that cp f L oo and that A = Mcp. Sin
M is unitar y , A also commutes with M* = M-. It folIo
e e e
that A commutes with M for any trigonometric polynomi
p
and hence that
Ap == AMp eo = MpAe o = Mpcp == Mcpp .
The rest of the argument amounts to showing that the line
transformation Mcp is closed (it has not yet been shown t
cp f L oo ). If f f L 2 , by I there is a sequence {Pn l of tri
metric polynomials such that Pn 4 f in the L 2 norm, so th'
where g =
Then
CPPn = McpPn = AP n .... g ,
At. Now let 0 > 0 and let S = Ie !!cp (e)! > 0'
277
! IcpPn-gl2 > r !CPP n -gI 2
o is

S3. INVARIANT SUBSPACES OF SIMPLE SHIFTS 27
== !CPI2IPn-(g/CP)12 > 02lI Pn -(9/cp)!2
"k> that Pn .... g/cp in L 2 (8). But Pn .... f in L 2 and therefore
11 L 2(8), and so cpf == g a.e. in 8. By the arbitrary nature
IIf 0 this gives cpf == g a.e. in 101<;b (0)  01. From CPPn -., g
I>llows g = 0 a.e. in te \ cp (8) == 0 t, and hence At = 9 = CPt
Mq/' Thus A == Mcp ·
Letting X be the characteristic function of S, we now have
217
llAxl1 2 = fo Icpxl 2 :2 0 2 ;\(8) = 021lxl1 2 ,
'0 X = 0 for 0 > IIAIl and l\cp\!oo < \!AII. This completes the
traof. For another proof see 94.IV.
COROLLARY. A subspace of L 2 reduces the bilateral
phift if and only if it is of the form
{ f J f £ L 2 and f == 0 a. e. on S}
tor some measurable subset S of the circle.
Proof: The projection on a reducing subspace commutes
ivith Me and consequently is of the form M cp for some cp ( L 00.
l3ince Mcp is a projection, cp2 = cp and cp (8) = 0 or 1 a.e.
Let s=: t8!rjJ(8)=O}.
\'. The unilateral shift of multiplicity one is irreducible.
Proof: Let m be a subspace of H 2 which reduces U + =
a.J e IH2. Then U H 2 = U m ffiU m1c u m EBm 1 so that
+ + + + '
m e V +m c H 2 e U H 2 . With  == m e U m, it follows that
"h + +
J:!lt er s:. '" 101 or s:. = H 2 e U+H 2 . But u+lm is a unilateral

28
NOTES ON OPERATOR THEORY
shift by 91.1, so that
m == f$u ffiU2$...
+ +
as in the proof of that resul t, and consequently m = {o I all
m == H 2 .
The invariant subspace situation, which is much mort
complicated, was investigated by Beurling [8].
VI. The non-zero invariant non-reducing subspaces of th
lateral shift in L 2 are those of the form qH 2 , wi th q a m
surable function such that 1 ql == 1 a.e. The function q IS
mined by the subspace up to a constant factor.
Proof: If m is invariant and non-reducing, the'l em 
(This is true of any isometry.) Let q be a unit vector wi
q f m, q 1 em. Then q 1 enq for n > 1: fo21Tlq(e)1 2 e in
= 0 for n > 1. Conjugating, the same result holds for al
n /: +, so ]ql is constant a.e. Since Ilql! = 1, this const
IS 1.
The set { q e I Z is therefore orthonormal, and its s
n n f
reduces the shift. By the Corollary to IV the span is all
L 2 . Since qe n (m, n > 0, we have qH 2 em. On the ot
hand, (qH 2 )1 is spanned by the qe n with n < O. But q.
implies q 1 enm for all n > 0, or equivalently, qe 1 m
1 1 n
all n < 0, and so (qH 2 ) em. Therefore qH 2 = m .
For uniqueness, suppose pH 2 = qH 2 . Then p qH 2 = I
so p q ( H 2 . Similarly p q ( H 2 , and therefore p q ::: q,/p
constant.
COROLLARY. If f ( H 2 vanishes on a set of positivI
measure, then f == 0 a. e.

93. INVARIANT SUBSPACES OF SIMPLE SHIFTS 29
Proof: Suppose it is not true that f = 0 a.e. Then the
losed linear span m f of the functions fen (n > 0) is non-
era, invariant, and non-reducing (since e -n f' H 2 for suf...
. ciently large n). Therefore m f = qH 2 for some q of modu-
US one, by VI. This is impossible, since there is a set of
sitive measure on which all members of mt vanish.
II. The non-zero invariant subspaces of the unilateral shift
n H 2 are those of the form qH 2 , with q f H 2 and \ql = 1
.e. The function q is determined by the subspace up to a
onstant factor.
Proof: If m is invariant for the unilateral shift, then
[ ieWed as a subspace of L 2 it is invariant for the bilateral
hift, so m = qH 2 for some q f L 2 with Iql  1 a.e. But
H 2 2
C , so q E H .
The structure of functions q f H 2 with 1 ql = 1 a.e. is
ell-understood [30, C'h. 5]. Let q = 2 00 0 a e . The cor-
n= n n
esponding functions q (13) = ; = 0 an zn, analytic in the
pen unit disc, are called inner functions. Each inner func-
ion is a product c · B · S, where
1) c is a constant of modulus 1;
2) B is a Bl aschke product:
00 a -2 a
8(z) = zk n n n
n == 1 1 - a n Z · T 8 n 1
where k is a non-negative integer, and the an are complex
numbers satisfying 0 < I ani < 1 and I (1-1 ani) < "";
3) S is a singular function:

30
NOTES ON OPERATOR THEORY
 f 217 °0 l
8(z) = exp t - 0 ;:0 : : dIL (0) \
where p. is a positive finite Borel measure on the circle
which is singular with respect to Lebesgue measure.
Exercise. Let q (H 2 be of modulus one a.e., let q b
defined as above, and let
k(eifJ,z) = Re( eO + z )
e e - z '
for 1 zl < 1. Show that
217
q (z) = 2:;  q ( e i 0 ) k (e i 0 , z ) dO .
Use the properties k > 0 and
217
l 1 k(eie z)de = 1
217 '
o
of the kernel k, valid for all () and [zl < 1, to show that
I q (z) I < 1 for 1 z I < 1.

SECTiON 4.
I INVARIANT SUBSPACES OF GENERAL SHIFTS
Shifts of arbitrary countable multiplicity will be consider-
il3d in this section. The procedure is much the same as in the
t receding section, except that it will be necessary to use
irect sums of the L 2 - spaces of the circle. Speci fi cally, let
I{ be a separable Hilbert space, and consider functions de-
"Rined on the unit circle with values in K. Such a function 1
ill be called measurable if the complex- valued function
If( . ), x) on the circle is measurable in the usual sense for
ach x f K.
The separability of K is used in the following lemma.
I. Lemma. If f and 9 are measurable, so is (/( · ), 9 ( · )).
L Proof: If Ib a I is an orthonormal basis of K, then f(O) =
Fa (f (8), b a) b a' and so (I «()), 9 «(})) = la (I «()), b a )(b a' 9 «()))
is measurable.
DEFINITION. L 2(K) consists of all measurable functions
f on the unit circle to K such that
IlfI122(K) = 2  277 IIf(O)lli dO < 00 ,
functions which differ only on a set of measure zero being
!identifie d Th " " ' Ib t . h h .
· IS IS a I er space WIt respect to t e Inner
produ ct
31

32
NOTES ON OPERATOR THEORY
f 277
(f, g) L 2(K) = i7T 0 (f( ()), g «()))K d ()
(completeness will be clear presently).
Functions in L 2(K) admit two kinds of orthogonal exIt
sion, as follows.
1) Let !bal be an orthonormal basis of K. If f (L2(J
the functions 1 a (f)) = (I «()), b a) are called the coordinate
functions of f. Of course, f«()) = I fa«())b a (convergence I
the norm of K. The second of these equations shows thatl
coordinate functions are in L 2. In fact,
1 277 ! 217
Ilflli 2(K) = i7T Ilf«())IIR d() =  i7T Ifa«())12<J
o a 0
= $ II fa Iii 2
a
Thus the map f -+ I ffi f is an isometr y from L 2(K) iI
a a
a $L 2 . To see that it is actually onto, let  $f a ($1
Then
$llfall2 = i7T j27TIlfa«())12dO
o
i finite, so that Ilfa«())12 < "" a.e., and therefore the fu l
tlon f(f)) =  fa(f))b a is defined a.e. It is measurable, 8
SInce
f 277
l7T 0 IIf(O)IIR dO = Illfall2
< 00 ,
it is in L 2(K). This proves

9 4 . fNVARfANT SUBSPACES OF GENERAL SHlFTS 33
'I. Lemma. The map f.... I ffi fa is an isometry from L 2(K) on-
&0 !.a ffi L 2. Th erefore L 2 (K) is com pI ete.
2) The second kind of expansion is as a Fourier series
avith vector coefficients.
,n. Lemma. Each f E L 2 (K) admits a unique expansion
+00
f(()) = 
x e ikO
k '
- 00
",ith XkEK and IlfI122(K) = IlIxkllK' This is to be under-
stood in the weak sense: for each x f K, the Fourier expan-
ion of (f( · ), x) is I (x k ' x)e ikO .
Proof: x  1/217 i 217 (f (0), x)e -ik8 dO is a bounded conju-
o
gate linear functional on K, so there exi sts a unique [l;k f K
such that
(X k , x) = f 277 (f( (), x )e ik () d() .
o
Hence  (x k ' x)e ik () is the Fourier expansion of (f(O), x),
1\(f( . ),x)112 := k !(;x k ' x)[2 9 and
IIfl1 2 =I Ilfal12 = I 1(xk,ba)12 =llxkI12.
a a k
COROLLARY. The map f  {x k } is an isometry from
rL 2 (K) onto £2(K).
Multiplication by a bounded measurable complex-valued
Unction defines a bounded operator in L 2 (K). In particular,
It fOllows . 1 f h e' F . . h
eaSI y rom t e loregolng ourler expanSIon t at
multiplication by e is the bilateral shift U of multiplicity

34
NOTES ON OPERATOR THEORY
dim K. The subspace
H 2 (K) = If! X k = 0 for all k < 01
is invariant under U, and U+ = U I H 2 (K) is the unilateral
shift of multiplicity dim K. Notice that f (H 2 (K) if and 0
iff (H 2 for all a. Another helpful observation is that
a
can be isometrically embedded in H 2 (K), by identifying [»'
with the function identically equal to x. This identificati
will often be made in order to simplify the notation.
Invariant subspaces will now be investigated in terms
this realization of the shift operators. As before, the redu
subspaces will be treated first, by means of a characteriz
tion of all operators which commute with the shift. Of cou'
multiplication by any L 00 function commutes with the shi
but if dim K > 1 there are others. For example, if dim K
then any operator defined on L 2(K) = L 2 e L 2 by means
2 x 2 matrix of L 00 functions commutes with the shift. Sue
matrix may be viewed as a function on the circle with val
in the bounded operators on K. The burden of the followi
theorem is that every commuting operator is of thi s type.
DEFINITION. An operator-valued function A: circle
33(K) is measurable if A(. )x is measurable for all x ( K.
in addition the function IIA(. )11 is essentially bounded, 0
can define a bounded operator A on L 2 (K) as follows:
(A f)(8) = A( 0) f (0) .
It is easily seen that A f is measurable, and
f 277
II Af 11 2 '" i1T II (Af)(O)11 2 aO
o

9 4 . INVARIANT SUBSPACES OF GENERAL SHIFTS 35
= i7T 27TIIA(e)f(e)112de
< (ess sup IIA(O)11 2 ) . 11/11 2 ,
so \\A \1 < ess sup IIA(O)II. Observe that A commutes with
multiplications, and in particular with U.
IV. Theorem. If A is an operator in L 2 (K) that commutes
with the bilateral shift U, there exists an essentially unique
essentially bounded measurable operator-valued function A
such that A = A.
The proof will be reduced to the case of a unitary opera-
tor A, by means of:
LEMMA. If the operator A commutes with the normal op-
erator N, then A is a linear combination of four unitary opera-
tors each of which commutes with N.
Proof: Let A = R + is with Rand S self- adjoint. Then
Rand S commute with N, since N is normal. It can be as-
sumed that Rand S are contractions, and then R + i (1- R2)Y2
and is i (1- S2) are unitary, commute with N, and have sum
2A.
Proof of Theorem: By the lemma it will suffice to consider
a unitary operator A that commutes with the shift. Let
Ibal be an orthonormal basis of K, and 9a a function in the
e .
qUlvalence class Ab for each a. Define A(O)b = g (0).
a a a
If cP f L DQ
(A(cpb a ), Ab(3) = (cf>b a , b(3) = l7T  27Tcf> (e)(b a , b(3)de ,

36
NOTES ON OPERATOR THEORY
and on the other hand, since A commutes with McpJ
(A(ifJ b a ), Ab(3) = (ifJAb a , A b(3) = i" 1 2 " ifJ (e)(ga(e), g{3(e1
o
Therefore (ga«(}),gf3«(})) = (b a , bf3) for () outside a set N t
of measure zero. Since K is separable N = U N af3 has m
sure zero, and
(A«(})b a , A«(})b{3) = (b a , b(3) for all a, f3 and all () IN.
For an arbitrary element x = I cab a of K, define A(O) =I
I caA(O)b a . This series converges for each 0 , N, and del
fines an isometry by the previous equation.
Now Aba = Aba for each a; since both operators are l
linear and bounded, they agree on K. The Fourier expansi
of f f L 2 (K) ma y be written as I+ oo Unx with x f K;
-00 n n
since both operators commute with U, this gives At = At.
Suppose A = B. Then A(O)b a = B(O)"b a except on a s.
N a of measure zero. Let N = UNa' and let K be a set of
measure zero outside of which both A and B are bounded.
If () I N U K J then A«(}) and B«(J) agree on a dense subm"
fold of K, and therefore on K.
COROLLARY. A subspace m of L 2 (K) reduces U if t
only if there is a measurable operator-valued function P sl
that P«()) is a projection a.e. and such that
J11 = {f \ f ( 0) f m ( 0) a. e.} ,
where m(O) = P(O)K.

94. rNvARrANT SUBSPACES OF GENERAL SHIFTS 37
Proof: If m reduces U and p is the projection on m,
then p == P by the theorem. In the previous proof it was
shown that A(O) is an isometry a.e., and a similar argument
shows that P(O) is a projection a.e. Finally, the following
are equivalent: f f m, PI = f, (Pf)(O) = f(O) a.e., P(O) f(O)
== f (()) a. e., and f «()) ( m (0) a. e.
REMARKS. 1) The theorem and its corollary reduce to
93.IV and its corollary when K is taken to be one-dimensional,
i.e. the complex numbers.
2) One cannot simply define A(O) by means of A(O) 9 (0) =
(Ag)(O). For this one would need, with the exception of a set
of measure zero independent of g, that the value of Ag at ()
depends only on the value of g at O.
3) The situation which results if K is allowed to be in-
separable is evidently not well understood. [55].
4) A similar theorem is valid for any normal operator N.
The circle is replaced by a (N) and Lebesgue measure by a
measure determined by the spectral family of N and a family
of vectors cyclic for the commutant of N. ([36, Ch. V and
VIII]; [17])
DEFINITION. An essentially bounded measurable opera-
tor-valued function A is called analytic if AH 2 (K) C H 2 (K).
In this Case the restriction A I H 2 (K) is denoted by A+.
V. Theorem. An operator A on H 2 (K) commutes with the uni-
lateral shift V if and only if A = A for some analytic O p -
+ +
erator-valued function A.

38
NOTES ON OPERATOR THEORY
Proof: Extend A to L 2 (K) by setting A1Un = unA.
n < 0 and :x f K. Then Al commutes with U, so Al = A
by IV. Therefore A is analytic and A = A+.
Exercise. A is analytic if and only if AX c H 2 (K).
VI. Lemma. Let A be an analytic operator-valued function
Then the following statements are equivalent:
1. 1 and U * commute;
+ +
2. A K c K; and
+
3. A is constant a. e.
Proof: If A+ and U: commute, then U: A+K = A+U:KI
to}, so AK is contained in K, the null space of U+*. CODI
versely, suppose A+K c K. Then A+U; = 0 = U: A+ on ,
while for x f K and n > 0,
A+U+*U:;c = A+U+ n - 1 ;c = U+*U+A+u+n-l = U:A+l{n;c J
so A+ and U+* commute on U+ n K .
Suppose A K c K, and define an operator C on K by
+
C = 1+! K. If {baJ is an orthonormal basis of K, then for
each a there is a set N a of measure zero such that A(O)hal
Cb a , (), N a . Since K is separable N = uNa is of measul
zero, and A(tJ) = C for () , N. Conversely, if A«(}) = C a.tI
then for all :x f K, (A:x)«(}) = A(O)x = Cx a.e. so that Ax ( I
COROLLARY. The subspace m of H 2 (K) reduces U+ I
and only if m = H2(f!.) for some subspace  of K.
Proof: Let p be the projection on a reducing subspace
so that pK c K by V and VI. This implies that f!. == pK ill

S4. INVARIANT SUBSPACES OF GENERAL SHIFTS 39
closed. Since  c m and m is invariant it follows that
82(2) c m. On the other hand, if I Unxn f m, then
!. U nx n = P (I Unx n ) =  Un PX n f H2() .
The invariant subspace of the bilateral shift will be an-
alyzed in the next two lemmas.
VII. Lemma. Each invariant subspace m of the bilateral shift
admits a uni q ue decomposition m = m EB n such that m is
00 00
reducing, and n is invariant with n;=o Unj( = {ol.
Proof: Let m = n o oo Unm and n = m em. Then Um
00 00 00
= m so m reduces. This im p lies the invariance of n f while
00 00
nUnn = {OJ is obvious, as is the uniqueness.
Since moo is taken care of by the corollary to IV, the next
lemma deals wi th the summand n. First recall that an opera-
tor V which is isometric on a subspace g of a Hilbert space
J(, and zero on the orthogonal complement, is called a partial
isometry. The subspace  is called the initial space of V,
and :f = VJ{ the final space. The projections on the initial
and final spaces are gi ven by V *V and VV * respecti vely.
The following statements are easily seen to be equi valent:
V is a partial isometry, V*V is a projection, VV* is a pro-
jection, and VV*V = V.
VIII. Lemma. If n is an invariant subspace of the bilateral
shift U such that n: = 0 Unrc. = {ai, then there are a sub-
space f of K and a partial isometry V on L 2(K), such that
the initial space of V is L2(), V commutes with U, and
n == V H 2 (!£ ) .

40
NOTES ON OPERATOR THEORY
Proof: The hypotheses and 91.1 imply that U I j( is a
unilateral shift, so that j{ =  m Un o' where o = j{ .
Since o is wandering for U, a lemma of 91 gives dim 
dim K. Let  be a subspace of K with the dimension of
and W an isometry from  onto O. Put V = 0 on L2(f)
and
00 00
V (  Un:J:n) = 
-00 -00
unWx
n
if x n f f for all n. Then Vis a partial isometry that caDI
mutes wi th U, and
00 00
VH2()=  mUnWf =  (B Uno = j{.
o 0
REMARK. If dim O = dim K, V can be taken to be an
isometry.
The situation regarding uniqueness is as follows. If f.1
V' is another such pair, then V 'll2(f "') = VH 2 (f) implies
v'f'" = V, and consequently there is a partial isometry "
on K with initial space  '" and final space  such that vi
= VW o on K. If W is defined on L 2 (K) by W (I Unx n ) 
1 UnWO.Tn ' it follows readily that V' = VW and that W carl
. '" 1
mutes WIth U. In fact W = W, where W(O) = W o for all ().
Summing up:
IX. Theorem. Each invariant subspace of the bilateral Shi l
U on L 2 (K) is of the form m €I) VH2(), where m reduc
00 00
U,  is a subspace of K, and V is a partial isometry that
commutes with U and has initial space L 2(). The space

94. INVARIANT SUBSPACES OF GENERAL SHIFTS 41
m is unique. If f' is a subspace of K and V' is a partial
00
isometry such that V' commutes with V, the initial space of
V' is L 2( '), and V 'H 2 (f:.') = VH 2 (J!.), then V" = VW for
some constant partial isometry We.
The statement that W is constant means that it commutes
with V, and that the corresponding operator-valued function
is constant a.e. If K is one-dimensional, this result reduces
easily to VI of the previous section. The two summands do
not occur simultaneously, and V is either zero or multiplica-
tion by a function of modulus one a.e.
Exercise. Let V be as in the theorem, so that V = V
for some operator-valued function V. Show that, almost every-
where, V(8) is a partial isometry with initial space .
x. Theorem. (Lax (31]; Halmos [22]). The subspace m of
H 2 (K) is invariant for the unilateral shift U+ if and only if
it is the range of a partial isometry V that commutes with
U+. The partial isometry V is determined by m up to right-
multiplication by a constant partial isometry.
Proof: If m is invariant for U+ ' then viewed as a sub-
space of L 2 (K), it is invariant for U, and so IX applies. A
non-zero reducing subspace of U cannot lie in H 2 (K), so that
3R"" == to} and  = V I H2() for some partial isometry V I
that commutes with U and has L2(f) as initial space. It
ollows that H 2 (K) is invariant for V I and that V = VI IH 2 (K)
IS as required.
The uniqueness statement follows in much the same way
as that of IX, except that the form of the initial space of V

42
NOTES ON OPERATOR THEORY
is not specified in X. Thi s is unnecessary since a partial
isometry V in H 2 (K) that commutes wi th U + must have
initial space of the form H2(f:.) for some  C K. This is
shown by the following exercise, together with the corollart
to VI.
Exerci'se. (H. G. Douglas) If U is an isometry and V a
partial isometry that commutes with U, Ithe the initial
space of V reduces U.
Uniqueness can also be deduced from the following gent
eral resul t.
XI. Theorem. If V 1 and V 2 are partial isometries in H 2 (Kt
each commuting wi th U+' then range vIe range V 2 if antf
only if V 1 = V 2 W for some partial isometry W that com-
mutes with U+. Equality holds if and only if W is constan
Proof: The converse is trivial, so suppose that range
V t C range V 2 . Then it is easy to see that W = V 2 V t is,
partial isometry with V t = V 2 W. To show that Wand U+
commute, observe first that V 2 (U+ W - WU+) = 0 since U+
commutes with V 2 and V 2 W. On the other hand, range W is
a subspace of the initial space of V 2 ' and the latter is in-
vari ant for U+, so that the range of U+ W - WU+ is containedl
in the initial space of V 2 . These two statements imply thai
U +w - WU+ = o.
If equality holds, the reverse inclusion implies in the
same way that W* commutes with U , and then W is const.
+
by VI.
This theorem permi ts an important reformulation of the il
variant subspace problem. The first reformulation was to slJl

9 4 . fNVARfANT SUBSPACES OF GENERAL SHfFTS 43
that each invariant subspace of U; of dimension at least two
contained a further invariant subspace. Taking orthogonal
complements and keeping XI and V in mind, the problem be-
comes one of finding non-constant factorizations for analytic
partial-isometry valued functions. In this form the problem has
been thoroughly analyzed by Potapov [39], for the case in
which K is finite-dimensional. For an interesting discussion,
see [28, Lecture VIII].
For the invariant subspace problem, it is actually suffi-
cient to consider analytic unitary-valued functions. To see
thi s, recall that if Tis a proper contraction on K, then
00
m ={  unrn:c!:c (K}
is the subspace of H 2 (K) introduced by Rota (2). It is in-
variant for U;, so the orthogonal complement n = H 2 (K) e m
is invariant for U+ (and for U, if j{ is regarded as a subspace
of L 2 (K)). A look at the proof of VIII shows that if
o = jt e uTl
s a complete wandering subspace for U, then the partial
Isometry V constructed there can be taken to be unitary. An
equivalent condition is that the smallest subspace containing
j( and reducing U is all of L 2(KL Such an invariant sub.
S p .
ace IS said to have full range.
Notice first that T*x - Ux (11 for all x (K. Since
T*x - Ux ( H 2 (K) ,
it III
Ust be shown that T*x - Ux 1 m. If y ( K ,

44
NOTES ON OPERATOR THEORY
(UnTny, T*x)
= J_ j 27T(e in ()T n y T*x)d8
217 '
o
_ J_ f 27T(einOTn+l y x)dO
217 '
o
= (Unr n + ly, x),
and therefore
(i VnTny, T*a;- Va:) = i (UnTny, T*a;) - i (UnTny, Va:)
000
00 00
= !, (UnT n + ly, x) - !, (U n - 1 T n y, x) + (Ty, x) _ (y, I
1 1
= 0 .
Now suppose that! is orthogonal to I:' 00 m Vnfo' T&I
U n { 1 T*x - Ux for all x € K and all integers n, so that
o = (T*a;-Va:,Vnf) = i7T j27T e - in ()(T*a:_ i()a:,!«())dt
o
(T*{)J- eiO;x, {(e)) = 0 a.e.,
(T - e -i8 I) {( 8) = 0 a. e.,
and
{(8) = 0 a.e.,
SInce T IS a proper contraction.

SECTION 5.
ANALYTIC REPRESENTATION OF THE
UNILATERAL SHIFT
In thi s section a brief alternative treatment of some of
the results of the previous sections will be given. This point
of view has been developed by de Branges and Rovnyak [10,
11].
Let K be a Hilbert space. The space e (K) is naturally
identified with the space K(z) of formal power series
:= 0 zn xn with vector coefficients :1J o , :1J 1 , ... € K such that
 l!xn l 1 2 < 00. In K(z) the unilateral shift U+ is "multiplica-
tion by z." Let D be the open unit disc in the complex plane.
I. If (z) = I zn x is in K(z), then for each W f D, I wnx
n n
converges to an element f(/w) of K.
Proof: For all integers k and m wi th k < m,
m
11 I w n xnll2 < (2lwln II:1J n ll)2 < (IlwI2n)(211:1JnI12)
n=:k
m
< (1_lwI 2 )-1 I 11 x n 11 2 -+ 0
n=k
as k
, m -+ 00.
COROLLARY. IIf(w)11 < (1 - Iw]2)- r2 llf(z)ll.
45

46
NOTES ON OPERATOR THEORY
Exercise. If P is the projection on K, considered asl
subspace of K(z) in the obvious way, then
few) = P(I-wU+*)-l f (z) .
If Bo' B 1 , .... are operators on K, theformalpowerse
B(z) = 2 znBn with operator coefficients acts formally 0
power series fez) = I zn xn with vector coefficients by th
usual rule for multiplication of series: B(z) fez) = 2 zn Yn
n
where Y n = 2k=0 Bkx n _ k for n > o.
II. 8(z) is a bounded operator on K(z) of norm at most M
and only if for every w f D the series B(te) = I wnBn C
verges in the operator norm to a bounded operator on K 0
norm at most M.
Proof: Suppose IIB(z)f(z)11 < M Ilf(z)11 for all fez) f
Then M211xI12  IIB(z)xI1 2 = IIIBnxI12 for all x€K, so
IIBn]1 < M for n > 0, and consequently I wnBn converg:c
in the operator norm to a bounded operator B(w) on K for'
w € D. Now
(f ( w) , x )K == (P(I- 'tC U+* ) - 1 f ( z), :r:)K = (f ( z), (1- W z ) - 1:r:)4
Substituting fez) = B(z)(/- in z)-ly with Y € K gives
(B(w)y,x) = (1-lwI 2 )(B(z)(I-wz)-l y , (l-wz)-l x ) ,
and therefore
!(B(W)y,X)!2 < (1-]w!2)2M211(I-wz)-lYI1211(I-wz)-1:t(l1
Since H(l-wz)-lyH 2 is easily computed to be (1_l w I 2 )-tl
it follows that !!B(w)!! SM. '

95. ANALYTIC REPRESENTATION
47
Conversely, suppose that IlB(w)11 s M for all W f D. Let
I(z) =' I zn ICn be an element of K(z) and B(z)f(z) = I zn yn .
'BY hypothesis IIIw n Y n ll 2 < M 2 11Iw n IC n ll 2 for all wiD. Let-
Jting w =' ri O and integrating this relation over 0 gives
I,2nllYnl12 < M 2 I r2nllICnll2 for 0 < r < 1. Therefore
I lIy n ll 2  M 2 I II ICn112, or equivalently,
II B ( z) f ( z) 11 2 < M 211 {( z) 11 2 .
Exercise. B( w) = P(l- wU+* )-1 B(z)1 K .
[II. An operator B on K(z) is of the form B(z) if and only if
it commutes with the unilateral shift.
Proof: Suppose B commutes with the shift. For :J) f K let
Bx = l zn x , and define B :J) = X n ' so that B:J) =  znB :J).
n n n
These operators are linear and bounded by IIBII, and if ((z)
= l zn xn is an element of K(z), then
00 00 00
Bf(z) =  zkBx k =  zk (  ZnBn:J)k ) :: B(z) {(z)
k=O k=O n::O
since B commutes wi th the shi ft.

SECT/ON 6. CONTRACT/ON S
In section 2 it was shown that any contraction whose
powers tend strongly to zero is part of a backward shi ft.. The
backward shift is a coisometry (the adjoint of an isometry),
so it is natural to ask what operators can be obtained as parts
of general coisometries.. It will be shown that all contractions
arise in this way, that the coisometry corresponding to a given
contraction T may be taken to be minimal in a certain sense,
and that then it is unique up to unitary equivalence. This co-
I
isometry C T will be referred to as the minimal coisometnc
eaJtension of T. Recall that an invariant subspace for an op-
erator is said to be full or to have full range if the smallest
reducing subspace that contains it is the whole space. The
result then goes as follows:
I. Theorem. If T is a contraction on a Hilbert space H,
there are a Hilbert space K containing H and a coisometry
C on K such that J{ is a full invariant subspace for C and
GIJ( = T. If (, D) is another such pair there is an isometry
J from K onto  such that DJ = JC and J is the identity
on H.
Proof: Let S = (/_ TT*)Y2 and let S be the closure of
the range SJ( of S. The operator matrix (top, p.. 50) defines
an operator C on the space K = H e S e S e ... (elements of
J( being treated as column vectors). This operator satisfies
49

50
NOTES ON OPERATOR THEORY
T S 0 "0 0
o 0 I 0 0
o 0 0 I 0
o 0 0 0 I
CC* = I and consequently is a coisometry. The space J{.
identified with the subspace J{ e {ol e {Ol E9 ..., is obvious.
invariant with the property G I J{ = T. The computations
C*(xeOEBOE9...) = T*xG;)SxeOe....
C*(OmyEBOm...) = OeOeyEBOm...
C*(OeOeymOEB ...) = OEBO$O$ymOEl1...
and so forth, for x (J( and y (S, show that }{ is full. It
follows that the elements I; = 0 G*k ilJk , for all n > 0 an't
XO" xl" ...., ilJ n £ H, are dense in K. Define an operator J II
this dense submanifold of K to the corresponding dense SI
manifold of  by
J ( i C*k Xk ) =
k=O
n
I D*kk ·
k=O
Then J is unambiguously defined and isometric, for
II D*kkI12 = I (D*k xk , D*m{Xm)
k,m
'= I Wm-kJ: k , J: m ) +  (X k ' Dk-mot:,.I
k < m k>m

=
96. CONTRACTIONS
 (Tm-kx a: ) + , (x rk-m{C )
.. k' m  k' m
k<m k>m
-
51
= III C*ka: k I1 2 .
Therefore J has a unique isometric extension from K onto .
It is obvious that J is the identity on J{ and that JC* = D*J.
Multiplication of the last equation by J* on the right and left
gives C*J* = J*D*, and so DJ  JC as required.
It should be noticed that the uniqueness result, together
with 92011, implies that the isometry C} is pure (i..e., a uni-
lateral shi ft) if and only if Tn.... 0 strongly..
A proof of this theorem along the lines of the special case
92.11 has been given by R. G. Douglas [15], and goes as fol-
lows. If, as in the proof of 92.11, R = (1- T*T)Y2 , :R = (RJ{)-,
and W: J{ ... H2(g{) is defined by
w x = (Rw, R Tx, R T 2 iJJ, ...) ,
then IIWiVl!2 = IlxU 2 - lim !ITnxIl2. Now IT*nr n } is a decreas-
ing sequence of positive contractions, which therefore con...
verge-s strongly to a positive contraction. If A is the positive
square root of this contraction, then
II A x l1 2 = (A 2 x, a;) = lim!! r n xl! 2 ,
and therefore 1\ W x \12 + 11 A:x:11 2 == II x 11 2 . With  = (AJ{)-, thi s
means that V: :.t -+ Wx ED Ax is an isometry from J( into
H2(5{) $. The mapping A:c -+ AT:c is readily seen to define
an isometry G on  such that GA = AT. Since WT = U* W,
, +
It follows that VT = W+* (j) GW, and therefore that T is uni-
t 'I
8.rI y equivalent to (U; e1G)!VJ{. The operator U; (BG need

52
NOTES ON OPERATOR THEORY
not be a coisometry, but this can be remedied by means
extension theorem 91.111 for isometries.. If the isometry G
extended to a unitary operator F, then U: El1 F is the des
coi sometry.
The ideas in thi s proof are often more useful than tho
in the first one. For example, an analogous theorem for 0
parameter semigroups of contractions may be proved in tb,
way, with the help of some basic information about infini
mal generators (cf. $9.111)..
The positive contraction A constructed above embodi
useful information about the contraction T. For example,
is not difficult to see that T is similar to an isometry if
only if A is invertible. It is also useful, as pointed out
R.G. Douglas, in proving the result, due to Nagy and Foi
[52], that any contraction has a largest unitary summand.
contraction is called completely nonunitary if this summa
vanishes.
II. Theorem. Let T be a contraction, and let A and A*
the positive contractions defined by A 2 = str.lim T*nr n
A = str. lim TnT *n. Then 1..1 == {xl Ax == A *:1: = :1J J is the
est reducing subspace of T on which T is unitary..
Proof: It is obvious that'll is a closed subspace. No
that T*A 2 r == A 2 and TA;T* == A;. The following elem
tary facts will also be useful: for any contraction T and
tor x, the statements II Txl! = Ilxl! and T*Tx = x are equi
lent; if T > 0 they are equivalent to Tx = x. ,
Now A 2 < T*T implies II Ax l1 2 < II Txl1 2 < IIxl12 for 
vector x, so that if x f 'U, then II Txll = II xii and T *Tx = 

96. CONTRACTIONS
53
In particular T is isometric on 'U, and in the same way, so
is T*. This will imply that T is unitary on 'U, once it is
shown that 1.1 reduces. For the latter, if x € 'U then
11 A T x ,,2 == (T * A 2 T :x J x) = (A 2 x J :x)
_ \\ x \\ 2 ?: \\ T x II 2 > II A T x \\ 2 ,
and so IIATxl! = "Txll. Therefore ATx = TrIJ by the remark
above, since 0 < A S I. The same inequalities give II Txl! =
Ilxll, so that T*Tx =  and
A;Tx = TAT*Tx = TA == Tx,
and therefore A*Tx = Tx, again using the above remark.
Consequently 11 is invariant for T. In the same way 1.1 is
invariant for T*, which proves that 'U reduces T.
If the vector xis a member of a subspace which reduces
T and in which T is unitary, then 11 rnxll = I\xl\ for all n > 0,
and so \I Axll 2 ::: lim 11 r n :x11 2 = }1 x}12 . Then A 2 x == x by the
remark above, which implies A = x since A is non-negative.
Dually, A *x == x, and consequently x (: 'U.
This decomposition is often useful in reducing questions
about contractions to the case of completely nonunitary con-
tractions. Another useful decomposition, due to Foguel [18],
runs as follows.
III. Theorem. Let T be a contraction, and let
Z( T) = {xl Tn:.v -+ 0 weakly J .
Then 2eT) = zeT*), zen is a reducing subspace, and T IS
unitary on Z(T)l.

54
NOTES ON OPERATOR THEORY
Proof: The first conclusion follows from the equivalJ
of the conditions (Tnx,:.v)... 0 and Tnx  0 weaklYG In 
direction this is trivial; for the other, let C be the mini.
coisometric extension of T* on K :) J{. Then since
(c*n x , x) = (x, Cnx) = (x, r*n x ) = (Tn x, x)
(Tn,) -+ 0 implies (C*nJ) -+ o. Now CkC*n = c*(nl
for n > k, and therefore lim (G*n x , C*k{fJ) = 0 for all k I
The relation (C*n x , y) -+ 0 now follows, first for y in th
closed linear span of the vectors C*k x (since they are u.
fonnly bounded), second for y orthogonal to this span, a.
consequently for all Y l K. If '!I ( J(, then
(Tnx, y) = (x, T *f1y) = (x, Cny) = (C*n x , y) ... 0 .
Since Z(T) is a closed subspace invariant for
T, the condition Z(T) ::: Z(T*) implies that Z(T) reduce.
To see that T is unitary in Z(T)l, it will first be shown.
(1- T*T)}{ c Z(T). For this,
!(T*n(!- T*T)a:,y)/2 = /(x,(l- T*T)T n Y)1 2
< II iXI1 2 II Tny - T *TT n YI12
< 211112(11 T n Yl12 _ Re(Tny, T*TTnyl
= 2]lxI1 2 (IIT n YI12 _IIT n + 1 YI12),
and the latter converges to zero, since
00
I <II r n Y11 2 - II T n + lY112) = I1Yl12 - lim 11 r n Yl12 < 00 
n=O
Now suppose that x is orthogonal to Z<T). Then so is

96" CONTRACTIONS
55
(/_ T*T);c, since Z(T) reduces T, and therefore (1- T*T);c =
o. This means that T is isometric on Z(T)l. In the same way
SO is T*, and the proof is finished..
COROLLARY" If T is a completely nonunitary contrac-
tion, then Tn -+ 0 weakly..
As to the connection between these two decompositions,
it is of course true that Z(T)l c 11. That equality need not
hold can be seen by taking T to be a bilateral shift, or even
a direct summand (restriction to a reducing subspace) of a
bilateral shi ft, in whi ch case both Z( T) and 'l1 are the entire
space" A unitary operator is said to be absolutely continuous
if its spectral measure is absolutely continuous with respect
to Lebesgue measure on the unit circle, and singular if its
spectral measure is singular with respect to Lebesgue mea-
sure.. It is not difficult to see that any unitary operator is
uniquely the direct sum of an absolutely continuous unitary
and a singular unitary, and that a unitary operator is absolute-
ly continuous if and only if it is a direct summand of a bilater-
al shifto With these facts, the above remarks mean that
TIZ(T)l is singular.
J.P.. Williams has pointed out that Foguel's decomposition
permi ts an easy proof of the following result:
If PI' P2 J ""OJ P k are projections, then (PtP2 ... Pk)n con-
erges weakly to Pt 1\ P 2 1\ .... 1\ P k (the projection on the
Intersection of the ranges of the Pi). For if T = P t P 2 ... P k ,
then Tn -t 0 weakly on Z(T), and T is the identity on Z(T)l
(since it is isometric there). Thus Tn converges weakly to
the P . 1
rOJ ection on Z( T) = (PI /\ P 2 1\ ..  /\ Pk)J{.

56
NOTES ON OPERATOR THEORY
Halperin [26] shows that the convergence is actually
strong.
Exercises. 1. The contraction T is an isometry if a4
only if A == T.
2. The contraction Tis simi lar to an isometry if and
only if A is invertible.
3. The null space of A is invariant for T.
4. [f T is normal, then A is a projection.
5. (f Tis a contraction, then 1/ Tnxll :: II xII if and onll
if T*Tx = x.
6. A vector :J; is in It if and only if II Tnxl\ = II xii ani
II T*nxl! n \1 :1;11 for all n  O.
7. If U is a unitary operator, ,there is a unique reducil
suspace JR such that U \ '}IT is absolutely continuous and
U I m 1 i s sin gu 1 ar.
8. A unitary operator is absolutely continuous if and <J
if it is a direct summand of a bilateral shi ft..
9. If C is a coisometry and m a fun invariant subspaJI
the isometry C* I m 1 is pure.

SECTION 7.
DlLAT10NS
If T is an operator on a Hilbert space R, and P is the
projection on a subspace m, the operator PTlm is called the
compression of T to m, and T is called a dilation of PTlm.
Recall that (PTI'«)* = PT*pR. If ( ) is the matrix of T
with respect to the decomposition J{ ==  $l, then PTI ==
A.
It is obvious that any compression of a unitary operator
is a contraction. Conversely:
I. Any contraction has a unitary dilation.
Proo f: If Tis a contraction on J{, then
w = [-*J
. 1i 1L
IS unitary on J( $J{, where R = (1- T*T)72 and S =: (/- TT*)72.
This is verified by computation, with the help of the relation
TR = ST. To see the latter, observe that TR 2 = S2 r , so that
Tp(R2) =: p(S2)T for all polynomials p. Since the polynomj-
als in ;;e2 are uniformly dense in C [0, 1], it follows that T R
=:. ST.
In similar fashion, compressions of projections may be
identified as the class of all posi ti ve contractions.
II. Any positive contraction can be dilated to a projection.
57

58 NOTES ON OPERATOR THEORY
Proof: If A is a positive contraction on J{, the opera
[(A_2)Y2
(A-A2) ]
I-A
on J{ Ea J{ is a proj ection.
These results, due respectively to Halmos and Micha
are discussed in [24, Problem 177]. If W is a unitary dil'
tion of T = PW[J{, it need not be true that r 2 = PW 2 1J{. '
unitary dilation W such that Tn = PWn\J( for all n > 0 WI
be called strong. Of course then T*n = PW*nlJ{ for all n
A result of the preceding section makes it easy to see th '
any contraction T has a strong unitary dilation. For this
call the two constructions in that section of the minimal
isometric extension e, taking place on the spaces J{ ffi H 2 '-
and H 2 (:R) Ea (where R = (/- T*T)Y2, S:= (/- TT*)Y2,
(RH)-, S = (sJ{)-). From the uniqueness it follows that
is a decomposition
J{ Ea H 2 (S) = H 2 (:R') Ea j=' ,
'.
where :R' is isomorphic to :R and j=' to j=. and e == u *
+ '
where U + is the unilateral shift on H2(;R') and F' is uni
on j='. Then W = U* EaF' on K = L2(9{') Eaj=' is a stron:
unitary dilation of T. This is clear since the compressio .
W n to H 2 (:R') Eaj=' is en for all n > o.
A uniqueness statement, similar to that for the coisorn
rie extension and proved in the same way, is valid for str
unitary dilations. Call a dilation W on a space K .:) J{ mi'
mal if the smallest subspace containing J{. and reducing
W is all of K. The minimality of the dilation just constru .

97. DILATIONS
59
folloWS from that of C. Moreover for any such dilation, the
elements I = -n WkaJ k , rJ: k (J{. form a dense submanifold,
and this fact implies that the minimal strong unitary dilation
is unique up to a unitary equi valence leaving the elements
ments of J{ fixed.
III. Theorem. Any contraction T on a Hilbert space J{ has
an essentially unique minimal strong unitary dilation W. The
dilation space may be decomposed as
K = ( i m Wn,) m}( m ( i m w*ns) ,
n=l n=l
where  ' and S are wandering subspaces for W, 9\' is iso-
morphic to «1- T*T)J{)-, and S = «1- TT*)J{)-.
Proof: Only the last statement will be considered. This
follows from the relations
K== £2(9\') e H 2 (9\')) eeJ{ fBH 2 (S),
00
L 2 (9{ ') e H 2 (9\') = .  fB wn:i{ , ,
n=l
and
00
H 2 (S) = !. fB W*ns
n=1
IV. L 2 (,)1 n L 2 (S)1 is the largst subspace of J{ which
reduces T and in which T is unitary.
Proof: Suppose j( reduces T and T/j( is unitary. Then
1- T*T = 0 on n. and so j{ C j=' == L2(:i{ ,)1 from the con-
struction of the coisometric extension. Dually n c L 2(S)1.
1'he space L 2( ,)1 n L 2(Sl obviously reduces W, and by

60
NOTES ON OPERATOR THEORY
III is a subspace of Je It follows that T = W on this spi
and consequently T is unitary there.
v. The minimal strong unitary dilation of a completely n<tl
unitary contraction is absolutely continuous.
Proof: Let W = J 0 217 eitdE t be the minimal dilation 0
The spaces L 2 (:R') and L 2 (S) reduce W, and in each W
a bilateral shift. Therefore the measure 'IE(. )z112 is abs
lutely continuous if z is a vector in either of these spse '
If z = :x + y is an element of L 2(;R ') + L 2(S), then since
II E( · )z 11 2 < (II E( · )xll + II E( · )yll)2, IIE(.)z 11 2 is absolu1l
continuous. But when T is completely nonunitary this rot
fold is dense by IV, and the result follows.
In many cases W is actually a bilateral shift.
VI. In each of the following situations, the minimal stronl
uni tary di I ation is a hi 1 ateral shi ft:
(i) Tn -+ 0 strongly, or dually;
(ii) T is completely nonunitary and the rank of 1- T* ,
is infini te, or dually.
Proof: In case (i) it has been observed previously th
the minimal coisometric extension is pure, which implies
the minimal strong unitary dilation is a shift.
In case (ii) W has a direct summand which is a bilat
shift of infinite multiplicity by III. But W is absolutely
tinuous by V. By a well-known technique of multiplicity
ry, the shift summand of W will "absorb" the rest of W
to unitary equivalence), and so W is a bilateral shift.

97. DILATIONS
61
The minimal dilation W of a completely nonunitary con-
traction T need not be a shifto The question of the spectral
type of W when both 1- T*T and [- TT* are of finite rank
haS been resolved by Nagy and Foias [53]. Those W that
can occur are precisely as follows: the bilateral shift re-
stricted to reducing subspaces of the form
L 2 (1l1) e... $ L2(ll n ) ,
where Ili is Lebesgue measure on a subset Si of the circle,
8 1  8 2 J ... J S1" and at least half the Si are the whole
circle.
The results of this section are due mainly to Nagy and
Foias, and can be found, aJong with many other things, in
their series of papers on contractions. The present treatment,
with its geometrical flavor, is essentially that of Douglas [15].
Other relevant papers are (20, 25, and 44].
It should be mentioned that the existence of the minimal
strong unitary dilation can be established in the same way as
that of the minimal coisometric extension, namely, by display-
Ing a matrix for it. The matrix in question is
0
I
I
R -T*
T S
l
l
0
acting on (In <0 (g) ;R) @ J( E9 (In> 0 ED S), where R == (I-T*T)1h,

62 NOTES ON OPERATOR THEORY
S = (1- TT*)1f2,  = (RJ{), and S '" (sHr. Here the (0,'
location of the matrix contains T.
Now several applications to von Neumann's theory of
spectral sets will be given [38]. A closed subset X of t
complex plane is ca1led a 8pectral set for an operator T
it contains the spectrum of T, and if, for any rational run
tion r with poles lying outside of X,
Ilr(T)11 < supI!r(a)11 Z f Xl .
VII. Theorem. The closed unit disc D is a spectral set I
any contraction T.
Proof: Let f be holomorphic in a region containing D.
and let W be a strong unitary dilation of T. If f(a) = J
I: = 0 an an, then I: = 0 an Tn converges in norm to a bo .
ed operator f (T), and
f(T) = ! an(PW n I J{) == Pf(W) I J{ .
Now from the spectral mapping theorem,
a(f(W)) = f(a(W)) C If(a) I a f D I ,
and since the normality of f(W) implies that its norm and
spectral radius are equal, it follows that
Ilf(T)11 < I\f(W)11 < supIlf(z)11 a f D I .
For the next application, recall that W (T) denotes th
closure of the numeri cal range I (T [l, a?) III [lll = II of the o
stor T.

97. DILATIONS
63
VIII. Theorem. (J. P. Williams [59].) If C is a closed convex
set containing a (T) in its interior, there is an invertible op-
- 1
era tor S such that W (S- T S) C C.
Proof: The following fact will be needed: If H is a con-
vex spectral set of T, then W (T) C H. It suffices to prove
this for H a closed half-plane, and by translation and rotation
it can be assumed that H = tz IRe z > OJ. It must be shown
that Re (Tx, aJ) > 0 for all vectors [1]. Since 1(1- 8)(1 + 8 )-1!
is bounded by 1 on H, it follows from the defini tion of spec-
tral set that 11(/- T)(/+ T)-1!1  1. This is equivalent to
11(/- T)x(1 < II (I + T):xl! for all [.C; squaring and expanding
gives He (T:x,:x) :2: 0 as required.
Now let V be the interior of C, let I be a conformal map
of the open unit disc D onto V, and let 9 be the inverse map.
Then D contains the spectrum of 9 (T), and so by 92.1 there
is an inverti ble operator S such that II S- 1 9 (T)SI! =:: r < 1.
Let D 1 be the closed disc of radius r. By VII D 1 is a spec-
tral set of g(S-1 TS ) = S-1 g (T)S. But I is a uniform limit
of polynomials on D l' from which it follows readily that
!(D t ) is a spectral set for l(g(S-1 TS )) = s-lrs. Since C
contains I(D}) it is also a spectral set, and because it is
conVex the assertion of the first paragraph gives W (S-1 T S)
c c.

SECTION 8
NAIMARK'S THEOREMS ON DILATIONS
A complex-valued function 1> defined on a group G IS
called po'sitive-definite if the inequality
n
1 -
 1> (g-: g.)A . A. > 0
J t t J -
i,j = 1
holds for every choice of group elements gl' ..., gn and com-
plex numbers A 1 ' ..., An. I t follow s that
1> (e) > 0 an d cp (g - 1)  1> (g) ,
the former by taking n  1, and the latter by taking n = 2,
gt = e (the identity), g2 = g, A 1 == 1, and A 2 = A.
A function U: g -+ U(g) assigning to each group element
a unitary operator on a Hilbert space K is called a unitary
repre'sentation of G if U(e) = I and U(gh) = U(g) U(h) for
all g, h f G.
These concepts are intimately related. If U is a unitary
representation and x is a vector, a simple computation shows
that the function 1> (g) = (U(g)3J, x) is positive-definite. On
the other hand:
I. Theorem. If 1> is a positive-definite function on a group G,
there is a unitary representation U of G and a vector x such
that  (g) = (U(g)x, x) for all g f G.
65

66
NOTES ON OPERATOR THEORY
Proof: Let K 1 be the complex vector space of all co.
plex-valued functions  on G which vanish except on a rt
subset. For , TJ { K l' define
(,TJ)1 = 2 cp(h-lfl)(g) TJ(h) ·
g,h
This is a bilinear functional which satisfies (, 0 1  0 51
cp is posi tive-definiteo The relation cp (g- 1) = cp (g ) evidCW
implies that (,1J)1 == (7J'I. Consequently the C.B.S. in
equality
I (, 7J ) 11 2 < (  , e) 1 (7J , 7J ) 1
is valid. It follows that Ko = {I(,;, 0 1 = 01 is a subspactl
K 1 ,andthat
(+Ko,7J+Ko) = (,7J)1
is a well-defined bilinear functional which makes the quotJ
space K l/KO a pre-Hilbert space. Let K be the completiJ.
For k { G define a linear transformation Uk on K 1 by
(U ke )(fl) = e <k- 1 fl). Then (U ke, Uk"") 1 = <e, 1]) l' so Uk 1.
leaves Ko invariant and induces a linear isometry on Kt(1
which has a unique linear isometric extension U(k) to K.
is easy to see that U(e) = I and U(g)U(h) = U(gh) for all
fl, h E G, and so U is a uni tary representation. If e i s de
by c;(e) = 1 and g(g) = 0 for all g I: e, then
(U(k)(e + Ko), e + Ko) = (Uk e ,e)l = 2CP (h- 1g )e(k- 1 g).
= cp(k),
and the proof is complete.

98. NAIMARK'S THEOREMS ON DILATIONS 67
Let J{ be a complex Hilbert space. An operator-valued
function A: G -+ 93(J{) is called positive-definite if the in-
equality
n
 (A( g -:-l g .)x., x.) > 0
 J 1, 1, 1-
i,j= 1
holds for every choice of group elements gl' ..., gn and vec-
tors xl' ..., x n . If J{ is one-dimensional this concept reduces
to the previous one, and as in that case
A(e) > 0 and A(g-l) = A(g)*
for all g (G. Again, much as before, such functions arise by
compressing a unitary representation on a space K to a sub-
space H, and conversely, such a function has a unitary dila-
tion. Here K 1 consists of all functions g: G -+ J( which van-
ish except on a finite subset, and
(g, 17) 1 =  (A(h-1g)g(g),g (h)) .
g,h
For each X (:}{ define x (K 1 by g{J/e) = x and g(g) = 0
for g i: e. Then {lJ .... 'x + Ko is a linear transformation from
}{ into K, and because
(a: + KOdy + Ko) = «(a:'(y)1 = I (A(h-lg)(/g), (y(h))
= (A(e)a:, y) ,
}( may be considered to be a subspace of K provided that
A(e) == I. If this identification is made, then
(U(k){X,y) = (U k g x ,gy)l = (A(k)x,y)

68 NOTES ON OPERATOR THEORY
for all [,c, y (Je This is equivalent to A(k) = PU(k)IJ{, wi
P is the projection of K onto H. This proves:
II. Theorem. If G is a group, J{ is a complex Hilbert sp
and A: G  C](H) is positive-definite with A(e) = /, ther
a Hilbert space K J J{ and a unitary representation U of:
on K such that A = PU I H.
It is not difficult to prove a uniqueness statement siml
to those of the preceding section, namely: the smallest 81
space containing J{ and reducing U is all of K; if V
is any unitary dilation of A on a space f which is mini '
in this sense, there is an isomorphism W of K onto .f s
that V(g)W = WU(g) for all g € G and W leaves the elem
of H fixed.
This theorem has numerous interesting consequences,,,
which a few will now be presented. Of course the trick is,.
find interesti ng posi tive-defini te functions. In the case of;
group of integers, there is a well-known connection betwe
positive-definite functions and functions analytic in the d:.
with non-negative real part. With II this leads to the folIo
theorem.
III. Theorem. Let D be the open unit disc in the complex;
plane, let J( be a complex Hilbert space, and let T: D.... t
be analytic and satisfy He T(2) > 0 in D and T(O)  I.
there is a Hilbert space K ':::) J{ and a unitary operator U
such that
T(z) = P(l + zU)(1 - zU)-l

98" NAIMARK'S THEOREMS ON DILATIONS 69
for all Z f D. Conversely, the function defined by this equa-
tion is analytic with Re T(z) > 0 and T(O) = I.
proof.; Since T is analytic in D it admits a strongly con-
vergent expansion T(z) = I: = 0 An zn in D. Let A be the
operator-valued function defined on the group of integers by
A( n) = %A for n > 0, A(O) = I, and A(n) = %A* for n < O.
n -n
Then A is posi ti ve-defini te, for if x k f J( for - n < k :$ n,
y (e) = 'i. = _ n e -eke x k ' and 0 < r < 1, computation shows
that
217
L f (ReT(reiO)y(O),y(O))dO=!. rlk-el(A(k-O:ck,:ce).
217 0 k, e
Consequently there is a Hilbert space K :> J{ and a unitary
operator U on K such that A(n) = PUn\J{ for all n. There-
fore
00
T(a) = I + 2  A(n)zn
n=1
00
= P(I + 2 !. Un zn) I J{
n=1
 PCI + zU)(I- zU)-1 I J{
For the converse, if U = J02" e iO dEO ' then
f 217
(He T(z)x, x) = Re(l+ zeiO)(l_ zeiO)-ldll Eoxl! 2
o
> 0
SInce the integrand is non -negati ve for all z f D and all fJ.
When }{ is one-dimensional, this result reduces to Her-
glotz's Theorem, whi ch runs as follows: iff is an alyti c in D

70
NOTES ON OPERATOR THEORY
and satisfies He f > 0 there and f(O) > 0, then
2"
[(z) =! (e i8 + z)(e i8 _ z)-l d p. (8)
o
for some positive finite Borel measure Jl on [0,211].
The existence of a strong unitary dilation for a contra
tien can be deduced from this theorem, in essentially the
same way that Nagy [48] proved it. The key observation i
that the conditions II Tlj < 1 and Re (I + zT)(I- zT)-l > ,
for Iz\ < 1 are equivalent. For if x f J{ and y = (1- zT)
then
(Re (I + zT)(/- zT)-l x , aJ) = He «I + zT)y, (1- zT)y),
= IIYl 1 2 - Iz1211 Ty 11 2 ,
which implies the assertion. Therefore if T is a contract.
there is K :) J{ and a unitary operator U on K such that
(I + zT)(I- zT)-l = P (I + zU)(/- zU)-lj J{. Expanding
sides and comparing coefficients gives Tn = p Unl J( for
n > o.
A similar application was noticed recently by C. A. B
[5]. For an operator T, let w (T) denote the numerical rad
sup II (Tx, )' ! II xII = 1 I. Here the basic observation is that.!
conditions w (T) :s 1 and Re (/- zT)-l > 0 for [z[ < 1
equivalent. To see this let aJ f J{ and y = (/- zT)-lx. T.: I
(Re(l-zT)-lx,) = Re(y,(/-zT)y) = IlyIl2_ Rez (Ty,y:
I
which implies the assertion. '
IV. Theorem (Berger). Let T be an operator on H. Then
weT) .$ 1 if and only if there is a unitary operator U on I

98. NAIMARK'S THEOREMS ON DILATIONS 71
space K 'J J{ such that Tn = 2PU n I J{ for all n > o.
Proof: By III and the above remarks the conditions
w(T) < 1
and
(1- zT)-l = P([ + zU)(I- zU)-l\ J{
for \z\ < 1 are equivalent. On expanding and equating coef-
ficients the latter condition is seen to be equivalent to Tn ==
2PU n I J( for n > O.
One interesting consequence of IV is the following power
inequality for the numerical radius.
COROLLARY. w(r ffl ) < w(T)m for any operator T and
positive integer m.
Proof: By a change of scale it can be assumed that w(T)
.5 1, so that Tn = 2PU n ! J{ for all n > o. The latter equa..
tion then holds with T replaced by T rn and U by Urn, so
that w(T m ) < 1 by the converse part of IV.
This corollary has been generalized by Berger and Starn..
pfli [6], as follows: if f is analytic in I zl < 1, continuous
in ! zl < 1, and f(O) : 0, and if w(T) < 1, then w(f(T)) .$
fhax ! f (e iO )!.
COROLLARY. If w(T) < l then IITnll < 2 for all n 2 o.
Theorem II can be refined considerably in the case of a
lOcally compact abelian group, owing to the following descrip-
tion of the posi ti ve..defini te function s on su ch a group.

72
NO TES ON OPERATOR THEORY
V. Bochner's Theorem. If cp is a continuous positive..def
function on a locally compact abelian group G, there is a -
unique positive regular Borel measure fl on the dual group
such that for all g f G,
cp(g) =ly(g)dp.(y) ·
r
This was proved by Herglotz for the group of integers
(see above), by Bochner for the real line, and ,by Weil in t:
general case. An easily accessible proof may be found in .:
Rudin [42], In view of the connection between positive..
definite functions and unitary representations, it is not sur
prising that this theorem can be made 00 say something a
unitary representations.
VI. Stone's Theorem. Let V be a weakly continuous unit
representation on a Hilbert space J{ of a locally compact
abelian group G. Then there is a unique regular spectral m
sure E on the Borel sets of the dual group r such that
(V(g):£,y) = 1 y (g)d(E(y):£,y)
r
for all :c, y { J{ and {J € G.
The proof will only be sketched. (For background mate",
rial on positive operater-valued measures, the reader may"
suIt Berberian's excellent account [4], especially sections,'
1-6. A P.O. measure is a positive operator-valued fune
F on a a..algebra of sets such that (F(. )x, x) is a measu '
,
for all :x; a spectral measure is a projection-valued p.O.:i
measure.) For each x (J{ the function cp (g) = (U(g):c,) j:'

98. NAIMARK'S THEOREMS ON DILATIONS 73
continuous and positive-definite, so there is a regular Borel
measure Ilx on Y such that (U(g), x) = f r Y (g)dllx(Y). The
polarization identity then provides a regular complex measure
for each pair w, Y f J{ such that
Jlx,y
(U(g)x,y) = /r y(g)dJ1. x ,y(Y) ·
This measure is unique, since any regular complex measure
which annihilates the functions y -. y (g) for all 9 ( G is zero
[42, po 17]. The uniqueness now implies a standard way [21,
p. 70] the existence of a unique regular spectral measure E
such that J.L = (E(. )x, y) for all {lJ, Y f J{. This concludes
x,y
the sketch of the proof.
Before proceeding, it should be noticed that Stone's Theo-
rem in the case of the group of integers amounts to the spec-
tral theorem for unitary operators, and indicates how the latter
theorem may be deduced from Herglotz's Theorem.
Now recall the statement of II, that a positive-definite
operator-valued function A on a group G can be dilated to
a unitary representation V: A = PU I J<, or equivalently,
(A(g) x, y) = (U(g)w, y) for all x, Y f J{. If G is locally com-
pact and abelian, then by VI
(A{g)x, y) = 1 Y (g)d{E{y)x, y)
r
for a spectral measure E on r. Now the function defined by
F = FE! J{ is a P. O. measure, and so the following has been
proved..

74
NOTES ON OPERATOR THEORY
VII. Theorem. If A is a positive-definite operator-valued
function on a locally compact abelian group G, there is a
unique regular P. O. measure F on the dual group r such
that
(A(g)x, y) = 1 y (g)d(F(y)x, y)
r
for all x, y (J{ and g ( G.
Of course this result includes Bochner's Theorem. Al-
though F is not a spectral measure, it has the advantage
that an extension space is not used. The proof raises the
following question: can a P. O. measure alwaY'8 be dilated
to a spectral measure? In the context of locally compact
abelian groups an easy affirmative answer is possible. Fo,
if F is a regular P.O.measure on r, the operator-valued
function A on G defined by
(A(g)x,y) =  y(g)d(F(y)x,y)
is easily seen to be positive-definite, so that as above,
(A(g)x,y) = 1 y(g)d(E(y)x,y)
r
for a spectral measure E on an extension space. ComparinJ
these equations gives F = PE IJ( as required. An "abstral'
version of this result runs as follows:
VIII. Theorem. Let J( be a Hilbert space, X a set, and S
a ring of subsets of X with £), X (S, and let F be a funcut
on S such that
(i) the values of F are positive operators on J(,

98. NAIMARK'S THEOREMS ON DILATIONS 75
(ii) F() = 0 and F(X) = I, and
(iii) F(S U T) = F(S) + F(T) if Sand T are disjoint.
Then there is a Hilbert space K ')}{ and a function E on S
such that the valuns of E are projections in K, E satisfies
(ii) and (iii), and F = PEl J{.
The argument is much like that used to prove II, and will
only be outlined. Let K 1 consist of all S-measurable simple
functions on X with values in J{. If g,." f K 1 , and if X =
S 1 U ... U Sn is a measurable parti tion such that g and TJ
are constant in each Sk' say with values [JJk and 'Yk' then
define
(g, "')1 =  (F(Sk)[JJk' Yk) ·
As before Ko = {c;j(t",g)l = OJ is a subspace, K 1 /K o is a
pre-Hilbert space, and K is its completion. The constant
functions provide an embedding of }( into K, and E(S) g is
defined to be equal to t" on Sand 0 on X - S.
Except as otherwise noted, the theorems of this section
are due to Naimark [34, 35]. In [51] may be found a general
dilation theorem of Nagy which includes both II and VIII, as
well as an excellent discussion of many related topics.

SECTION 9
CONTRACTIVE SEM1GROUPS
Basic references for this subject are Hille-Phillips [29]
and Dunford-Schwartz [16]. The appendix of Lax-Phillips
[32] contains a brief and elegant account.
DEFINITION. A family T = t T(t)! t > O} of operators on
a Hilbert space J( is called a semigroup if
T(O) = I
and
T(J8+ t) =: T(s)T(t) for all 8, t > 0 .
It is uniformly continuous if t .... T(t) is continuous in the
operator norm, and strongly continuous if t 4 T(t)x is con-
tinuous in the norm of J( for all x (H. (Because of the
semigroup property it is sufficient to assume continuity at
t = 0.) If T(t) is a contraction for all t > 0, the semigroup
IS a contractive semigroup.
As examples we mention the following:
1. Let A be an operator and T(t) == etA. It is shown be-
low that these are precisely the uniformly continuous semi-
groups.
2. Unitary semigroups. A semigroup such that all T(t)
are unitary amounts to a unitary representation of the group
77

78
NOTES ON OPERATOR THEORY
of real numbers, and these are described by Stone's Theoret
98. VI.
3. Translation semigroups. Let K be a separable HiIb,.
space, and denote by L 2 (R+,K) the Hilbert space of weak.
measurable functions f from [0,00) into K such that
IIfl1 2 = 1 00 Ilf(s)\1 2 ds < 00 .
o
The (backward) transl ationsemigroup is defined on L 2(R +, I
by (B(t)f)(s) = f(s+t). It is easily seen that
(B(t)*f)(s) = 
o for 0 < 8 < t
f (8 - t) for t < 8 .
This semi group, consisting of isometries, is the forward traJ.
lation semi group. The bilateral translation semi group, definl
analogously on L2(R, K), is unitary. These semigroups are
strongly (but not uniformly) continuous.
4. The restriction to an invariant subspace of a semi-
group is again a semigroup, as is a direct sum of semigroup4,
The adjoint of a semigroup is a semi group; it is easy to se.
that the adjoint of a strongly continuous contractive semi-
group is strongly continuous.
I. Theorem. {T(t)l is a uniformly continuous semigroup on
J{ if and only if there is A f (J{) with T(t) = etA for all
t > O.
Proo f: For B £ 93(J{), e B is defined by means of the us.
al uniformly convergent power series. If Band C commut
the estimate

9 9 . CONTRACTIVE SEMIGROUPS
79
It e B - e ell < eM It B - C II
follows readily, where M = max{\(B\\,ltcHL Hence e tB is
uniformly continuous. That it is a semigroup is proved just
as in the scalar case.
If { T(t)} is a uniformly continuous semi group, then
II ..!. r t T(s )ds - /11 -+ 0 as t -+ 0 +
t J o
because the integrand is continuous, and so there is a > 0
such that Jot T(s)ds is invertible for 0 < t  a. If 0 < h <
t :s a,
k (T(h)-J) itnS)dS =  f tT(h)T(S)dS - tT(S)dS}
f f t+ h t }
- i ) h T(s)ds -  T(s)ds
_ k {t+hT(S)dS _ h T(S)dS}
which converges in the operator norm as h .... 0 + (to T(t) - I).
Hence A = lim 1/ h (T(h) -I) exists in :B(J{), and therefore
It T(t) "" lirn k (T(t+h)- T(t)) "" AT(t) ·
The semi group T 1 (t) = etA satisfies the same differential
equation, which implies that T(s - t)T 1 (t) is constant on
[O,s] for any 8 > 0, and consequently that T(8) = T 1 (8).
It should be noticed that the theorem makes sense and
the proof is valid with (J{) replaced by any Banach algebra
wi th iden ti ty.

80
NOTES ON OPERATOR THEORY
The remainder of this section will be devoted to eluci-
dating the structure of strongly continuous contractive semi.
groups. This may be thought of as the continuous analogue
of the study carried out in sections 1 and 6 for (the powers
of) a single contration. Two methods will be employed:
adapt the methods of the discrete case, and extend the pro-
cedures used in the proof of I above. In each the notion of
the infinitesimal generator plays an important role.
DEFINITION. The infinite simal generator A of the
semigroup {T(t)} is defined by
Ax = lim h I (T(h)-l)x
hO+
on the set :D = i)(A) of vectors x for which this limit exists
It is clear that i) is a linear manifold and that A is lin-
ear on . In general :D will be a proper submanifold and A
will fail to be bounded. For the backward translation semi-
group of example 3, :D consists of the differentiable function
f f L 2 for whi ch f' f L 2, and A is differentiation. The fol-
lowing basic properties of A will be needed.
II. Lemma. Let {T(t)} be a strongly continuous semigroup
with infinitesimal generator. A. Then:
1. A commutes with T(t) for all t > 0;
2. d T(t);c = T(t)A;c and It II T(t);c1l2 =
2Re (A T(t)x, T(t)x) for all x € :D(A) ;
3. foa T(s)xds (i)(A) for all x f J{ and a  0, and
a
A fo T(s)xds = (T(a)-l)x; and
4. A is closed and densely defined.

99. CONTRACTIVE SEMIGROVPS
81
Proof: The first statement is that if :JJ c :D(A) then
T(t):.c (:D(A) and AT(t)x = T(t)Ax, and thi s is clear from the
defini tion. For the second, if x ( :D(A) then
i (T(t+h)- T(t)) = T(t) i(T(h)-l) -+ T(t)A
_ (T(t- h)- T(t)) = T(t- h) i (T(h)-l) -+ T(t)A
as h  0 +. The second part of 2 follows from the first and
the fact that
(f(t),g(t))' = (f'(t),g(t)) + (f(t),g '(t)) .
Just as in the proof of I,
i (T(h)-l) f a T(s)d8 -+ (T(a)-l) ,
o
which is 3. Since 1/aJ:aT(8)xd8x as a..., 0+, 3 implies
o
that :D(A) is dense. To see that A is closed, notice first
that integrating the relation in 2 gi ves
t
1 T(s)Ad8 = (T(t) - l)a:
o
for x f :D(A). If :JJ n f :D(A), :JJ n  x, and A:JJ n  y, letting n ..., 00
In J o t T(s)Aa:nds = (T(t) - l)a: n gives
! tT(s)yds = (T(t) - J)a: .
o
Dividing by t and letting t ..., 0 + gives A:JJ = y as required.
III. Theorem. Any strongly continuous contractive semi group
can be extended to a strongly continuous coisometric semi-
group.

82
NOTES ON OPERATOR THEORY
Proof: Let {T(t) J be a strongly continuous contractive
semigroup and A its infinitesimal generator. The hermitiat
symmetric bilinear form defined on  = (A) by
(x, Y)l = -(Ax, y) - (z, Ay)
is positive-definite by 11(2) since HT(t)xll is non-increasing"
It follows that n = {w f  I (x, x)1 = oJ is a submanifold of
9), and that the bilinear fonn induced on :D/n makes it a pr..
Hilbert space. Let K be the completion. The positive con-
tractions T(t)*T(t) are decreasing and therefore converge
strongly to a positive contraction. Let C be the positive
square root of this contraction, so that lim IIT(t)x\l2 ::: IIC:.v!12 j
and let £ = (CJ()-. Now define
I:  -+ L 2 (8 + , K) G) 
by I:l = Ww  ex, where (Wx)(t) = T(t)[1]. Then
\I W:l11 2 = lim f nil T(t):cll  dt
n-+oo 0
n
= -lirn l 2Re(AT(t):c, T(t):c)dt
= -lirn  n d II T(t):c1l 2 dt
= \I [1]11 2 - lim \I T(n):.v11 2
_ 1\ xl1 2 - II 0[1]\\2
by 11(2), so that  is an isometry on 5). Since j) is dense
in J{, I has a unique isometric extension to all of J{. Now

99. CONTRACTIVE SEMIGROUPS
83
if x ( then T(s) f j) and I T(s)m = WT(s):lJ G) CT(s);x. But
(WT(s)w)(t) = T(t)T(s)x = T(t + s)x = (B(s)Wx)(t) ,
where B is the translation semigroup of example 3. In addi-
tion, V(s): Cx.... CT(s)x is a well-defined isometry on C,
since
II CT(s )x11 2 = lim II T(t)T(s ):1:11 2 = lim 1\ T(t)xI1 2 = II C xii 2 ,
and so V(s) has a unique isometric extension to all of .
Therefore
I T(s)rr = B(s)Wx $ V(s)Cx = (8(s) ffi yes))  x
for all x (, so that by continuity
2 T(s)  (8(s) $V(s))I
on }(. Thi s implies that  =  J{ is an invariant subspace
for the semigroup B EB V, and that T and (B E9 V) I:R are uni-
rarily equivalent. (Since V(s)C = CT(s) it follows easily
that V is a strongly continuous semigroup.)
The final step in the proof is, as in the discrete case,
the extension of the isometric semigroup V to a unitary semi-
group. This is accomplished by means of the following con-
tinuous analogue of the Wold decomposition @l.I1, due to
J. L. B. Cooper [14]. The ingenious proof given below was
discovered by James Deddens. Other interesting proofs may
be found in Masani [33] and Sz.-Nagy [50].
IV. Theorem. Let V = {V(t)} be a strongly continuous iso-
metric semigroup on J{. Then there are Hilbert spaces K and

84
NOTES ON OPERATOR THEORY
f and a strongly continuous unitary semigroup V = IU(t)} cJ
f, such that V is unitarily equivalent to B* eD, where B-
is the forward translation semigroup on L 2 (R +, K) (c.f. ex...
ample 3).
Proof: Apply the above reasoning to the strongly conti -: .
ous coisometric semigroup T(t) = V(t)*. Thus there are Hi
bert spaces K and f, a strongly continuous isometric semi:.
group V t on f, and an isometry I from J{ into L 2 (R+,K)
such that
l T(s) = (B(8) e V t (8) I .
In this case it will be shown tht V l is unitary and tat IJI
= L 2(R +, K) e f>; the theorem wIll then follow on takIng adJ
joints.
Since Tis coi sometri c the operators T(t)* T(t) are pro-
jections, and therefore so is C. The equation T(t)*C 2 T(t) I
C 2 implies that C 2 commutes with T(t), again because T(i
is a coisometry, and hence that C commutes with T(t). Thul
f == C J{ reduces T and V t = T! f; since V t is isometri<*i
and T I f is coisometric, both must be unitary.
Let P be the projection of L 2 (R+,K) ef on f.. Then
P  = C, so that for 3J f f, I 3J = f fSj C3J = f E9;]} for some
f l L 2 (R+,K). But f = 0 since  is an isometry, and this
gives IJ{ == m $f where m is a subspace of L 2 (R+,K)
invariant for B. Now B! m and T 1(1- C)J{ are unitarily
equivalent. Since the restriction of a coisometry to an invalt
ant subspace is a coisometry if and only if the subspace r
duces, it follows that m reduces B. To get m = L 2 (R+,K>
it will suffice to show that:m contains the step functions

99. CONTRACTIVE SEMIGROUPS
85
with range in i1. Since m reduces we need only consider func-
tions of the fonn f = 4>[ 0 t a] (V , where 4>[ 0, a] is the character-
istic function of [O,a], a > 0, and (V €:D. Now
9 = WT(b)x-B(b)*WT(2b)x E m for b > 0
(where Wis as on p. 82), and
=: j T (t + 1> ):l, t < b
I 0, b<t.
g (t)
Let E > O. Since
2
H(I- T(s))xI1 1 == -2Re(A(I- T(s»x, (1- T(s))x)
== -2Re «1- T(s))Ax, (1- T(s))x)
converges to 0 as s -+ 0+, there is an integer N > 0 such
2
that 1\ x- T(s)x!ll < E/ a for 8 < 2a/N. If b = a/N and
h = IN B«n- l)b)*g, then h € m and
n=1
Ilh-fl\2 = Ja\lh(t)-XIIdt
o
N f nb 2
= 2 Ilg(t-(n-1)b)-xll t dt
n = 1 ( n- 1) b
(b 2
= N J,.. II T(t+ l> )x- w!I I dt
o
< Nb(Ela) = E .
lIence f f m and the proof is complete.
COROLLARY. Any strongly continuous isometric semi-
group can be extended to a strongly continuous unitary semi-
group.

86
NOTES ON OPERATOR THEORY
COROLLARY.Any strongly continuous contractive semi
group can be dilated to a strongly continuous unitary semi
group.
REMARKS. 1. The effect of these results is to reduce
the study of contractive semigroups to the study of invariazt
subspaces of coisometric semigroups. In the case of the
translation semigroup B, a further reduction is possible, jul
as in the discrete case: the Fourier transforms of the inva,rf
ant subspaces of B may be expressed in terms of certain
operator-valued functions which are analytic in a half-planet
[30, 31].
2. The uniqueness situation is the same as in the dis-
crete case 96.1. It is also worth noting that the invariant 816
space  J{ in the proof of III is full. For the preceding stell
function argument actually shows that (P IJ{)- is always a A
full invariant subspace of L 2(R +, K). In addition, if V on t
is the unitary extension of V obtained from Cooper's Th
rem, and if Q is the projection on j=, then it is almost obvi,
ous that (QJ()- is a full invariant subspace of :f. These
facts imply that I J{ is full.
Eerci8e8. 1. The translation semigroups are strongly
continuous.
2. The adjoint of a strongly continuous contractive sem..-
group is strongly continuous.
3. If tis a continuous mapping from [0, oc) into a Ban"
space, then
1 j a+t
t f ( 8 )d 8 -+ f ( a) as t -II O.
a

99. CONTRACTIVE SEMIGROUPS
87
4. If C is a coisometry and m an invariant subspace,
then C I m is a coisometry if and only if m reduces C.
5. A strongly continuous contractive semigroup is uni-
tarIly equivalent to a part of a backward translation semi-
group if and only if T(t) -. 0 strongly as t -J> 00.
6. A semigroup T is unitarily equivalent to a backward
translation semigroup if and only if it is strongly continuous,
coi sometric, and T(t)  0 strongly as t .... 00.
In the remainder of thi s section the second method men-
tioned above will be considered. Here the procedure is to
characteri ze the i nfini tesimal generators, and to develop
ways of recovering the semigroup from its generator. Again
only strongly continuous contractive semigroups will be con-
sidered.
DEFINITION. . A linear transformation A in a Hilbert
space J( is called accretive 1 if it is densely defined and if
Re(A:x,x) < 0 for all Xl :D(A),
and maximal accretive if it is accretive and admits no proper
accretive extension (in J().
If A is accretive, then for any x ( j)(A)
!!(A + [)xl!2 = !!AJJI1 2 + !!!12 + Re(Ax,:1:)
and consequently
II (A + [)xI1 2  II A x l1 2 + It xi! 2 $ II (A -1)a:I! 2 ,
So thatA -l is one-to-one and S = (A + 1)(A _1)-1
;----
See the remark on tenninology on p. 93.

88
NOTES ON OPERATOR THEORY
is a contraction defined on (A-I) :D(A). This contraction.
be referred to as the Cayley transform of A. Since 8 -[ =e
2(A _/)-1 and 8 + 1 = 2A(A -/)-1 , it follows that S-I if
one-to-one and
(8 + 1)(8- 1)-1 = A .
This implies that if A 1 and A 2 are accretive with Cayley
transforms 8 1 and 8 2 , then A 1 is a (proper) extension of
A 2 if and only if 8 1 is a (proper) extension of 8 2 .
V. Let A be a densely defined linear transformation in J{.
The following conditions are equivalent:
1. A is maximal accretive;
2. A is accretive and (A -I) :D(A) = H ;
3. A  (8 + [)(8 -1)-1 for some everywhere defined cot
traction S of which 1 is not an eigenvalue; and
4. A and A * are closed and accretive.
Proof: That 1 and 2 are equivalent and imply 3 is clear:
from the preceding discussion. If A = (8 + /)(8 _/)-1 is as;,
in 3, and x = (8 -/)y (:D(A), then
Re(A,x) = Re«S+/)y,(8-/)y) = IISy!\2 _\\y\\2 $ 0
so that A is accretive; it is maximal because S is every-
where defined. Thus 3 implies 1.
Again consider A = (S + 1)(8 -1)- 1 as in 3. If 8*x = ""
then
IISx-X\\2 = 118x\\2 - 2Re(Sx,x) + l\x!\2
= \!Sx11 2 - 2Re(x,S*x) + \\xl\2
= Il 8xl12 - 1\ I! 2 < 0 ,

99. CONTRACTIVE SEMIGROUPS
89
SO that Sx = x and :r = O. Hence 1 is not an eigenvalue of S*,
and therefore
(S* + /)(8* - 1)-1 = -[ + 2 (S* -1)- 1
= [_1+2(8-/)-1)*
== A*
is maximal accretive. To prove 4 it will now suffice to show
that a maximal accretive transformation is closed. Since
(A _1)-1 is bounded with domain (A -I) :D(A), an accretive
transformation A is closed if and only if (A -/):J)(A) is
closed. Hence maximal accretive transformations are closed
by 2.
Finally assume 4. If Y is orthogonal to (A-l):D(A), then
A *y = y, so that y = 0 since A * is accretive. Therefore
(A-/):D(A) is dense in H. Since A is closed this gives
(A -I) j)(A)  J{ by the above rem ark.
VI. An accretive linear transform has a maximal accretive
extension.
Proof: Let A be accretive with Cayley transform S. By
the foregoing it is sufficient to extend S to an everywhere
;/
defined contraction So of which 1 is not an eigenvalue.
Define So on the closure of (A -I) :D(A) by continuity and
On the orthogonal complement by So == o. If Sox == x, then
as in the preceding proof *:;c = x, so that for any Y l j)(A),
(:1;, (A -/)y) = (So aJ, (A -/)y)
= (x, S(A -l)y)
= (x, (A + l)y) ,

90
NOTES ON OPERATOR THEORY
(x, y) = 0, and :.c = 0 because (A) is dense.
VII. The infinitesimal generator of any strongly continuous
contractive semigroup is maximal accretive.
Proof: Since \I T(t)xll is nonincreasing, A is accretive
by 11(2). The generator of the semigroup !e-tT(t)} is A-l_"
so that by 11(3)
e-tT(t)- '" (A-I) it e-ST(s)xds
o
for any :E (J{. Since y = fo 00 e-ST(s)xds converges and A
is closed, this gives (A -l)y = - {£. Thus (A -I) 9) (A ) = J{,
so that A is maximal by V.
REMARK. (1- A)-l is given by the Laplace transfonn
fo 00 e-ST(s)ds. More generally,
(,v-A)-l '" loo e-AST(s)ds for ReA> 0 .
The next step is the converse: any maximal accretive
transformation A is the generator of a unique strongly con-i
tinuous contractive semigroup. The semigroup in question if;
simply etA; the difficulty is to make sense of the exponen-.
.
tial. There are several ways to do this, in each of which tbfJ
idea is to construct approximating semigroups e tB ; where f;
is a bounded approximation to A. The following is a sketl
of the method of Hille and Yosida.
For 'A > 0 it follows just as above that 'A/- A has an
everywhere defined inverse R'A which is bounded by 1/'\.
The operators

99. CONTRACTIVE SEMIGROUPS
91
B A = A 2 R A - Al
converge strongly to A on :D(A) as A -+ 00, and the semigroups
TA(t) = exp (tB A ) are strongly continuous and consist of con-
tractions. Since
T A (t) - T 11 (t)
= 1 1 A.. T>..(ts)T (t(l-s))ds
o ds 11
1
= 1 tT>..(ts)T (t(l-'s))(B A -B )ds ,
o 11 Il
it follows that
II T A (t)x- TIl (t)a:11 < t II B A x- B /LxiI
and hence that a limiting strongly continuous contractive
semlgroup T(t) exists. If A t is its generator, then for
[lJ f 9)(A)
h h
(T>..(h}-l)x = B>..1 T>..(s)xds = 1 T>..(s)B>..xds;
o 0
letting A -+ 00 this gives
(T(h)-[):r: = 1 h T(s)A:r:ds
o
which implies that A C At. Equality follows since A is
maximal accretive and At is accretive (V). Uniqueness may
be proved just as in I. This proves
VIII. Theorem. A maximal accretive operator is the infini-
tesimal generator of a unique strongly continuous contrac-
tive semi group.

92
NOTES ON OPERATOR THEORY
IX. If A is the infinitesimal generator of the strongly con:4
tinuous contractive semigroup IT(t)}, then A* is the infini,;
tesimal generator of IT (t)* L
Proof: Let B be the generator of IT(t)*1. If x l :D(A) ana
Y f :D(B), then
(Ax, y) = lim k «T(h)-I)x, y)
- Ii m  ( x, ( T (Ii,) * - l)y) = (x, By) ,
so that B C A *. Equali ty follows since B is maximal accreirc,:
tive and A* is accretive (by V and VII).
To properties of a semigroup T there are corresponding:;
properties of the infinitesimal generator A and of the Caylt
transform S = (A + 1)(A _/)-1 (S is called the cogenerator
of T). For example:
X. Let T be a strongly continuous contractive semigroup wf
infinitesimal generator A and cogenerator S. The followinl
statements are equivalent:
1. T is isometri c;
2.. A c - A * (equivalently, iA is symmetric);
3. Re (Aaa, x) = 0 for all x € (A); and
4. S is an isometry.
Proof: If T is isometric, then for x f :D(A)
(T(h)*- 1)[C == - T(h)*( T(h)- l)x
so that A C -A* (using IX). Then
Re(Ax,x) = -Re(A*a:,x) = -Re(Ax,x)

9. CONTRACTIVE SEMIGROUPS
93
and consequently Re (A:l,:l) ==: O. Since
II (A :t /)wIl 2 = IIA3JI! 2 + II xl/ 2 + 2Re(A, x) ,
S is an isometry if and only if He (AX', x) = 0 for all :;v ( g)(A).
Finally, if the latter holds, then d/ dt II T(t):vl! 2 == 0 for all
x (. J{ by II (2), and so T is isometric.
COROLLARY. The following are equivalent: T is unitary,
A is skew-adjoint, and S is unitary.
These facts may be used to obtain an alternative proof of
the extension theorem III. For if S1 is the minimal coisomet-
rie extension of the contraction S (96.1), it is easy to see
that 1 is not an eigenvalue of S1 J so that 8 1 is the cogen-
erator of a coisometric semigroup T 1 which extends T. A
proof of Cooper's Theorem along these lines can also be
given [54].
REMARK. Let T be a strongly continuous contractive
semigroup with generator A and cogenerator S. Let R =
(/- 8*8)72 and  = (RJ{)-. Then the coefficient space K
employed in the proof of the extension theorem III is isomor-
phic to :R. For if x ((A), one computes easily that
II R(A -l)xI1 2 = 211 xii: '
So that :l -to (1/y2)R(A -l)x induces the desired isometry.
REMARK ON TERMONOLOGY. Many authors prefer to
define the infinitesimal generator by the equation Brx ==
t h i m (i/h)(l- T(h»x. Accreti ve then means rm Bx, x) > 0,
o+ -
and the Cayley transform is defined as S == (il- B)(il + 8)-1
On (if + B) g)(B).

94
NOTES ON OPERATOR THEORY
E::cercises. 1. Let T be a strongly continuous contrac.
tive semigroup with cogenerator S. A subspace m is invari-
and for T if and only if it is invariant for S, and in this case
S I m is the cogenerator of TIm.
2. An everywhere defined accretive linear transformatiQit
is bounded.
3. Complete the proof of VIII.

SECTION 10
HYPER1NVAR1ANT SUBSPACES
A subspace m of a Hilbert space J{ is said to be hyper-
invariant for an operator T on J{ if sm c m for all operators
S that commute wi th T. This concept is related to invari ant
subspaces for algebras of operators: the hyperinvariant sub-
spaces of T are the invariant subspaces of the (strongly
closed) operator algebra
IT}'= {S!ST =TS} .
Of course a scalar operator T = cl has only the trivial hyper-
invari ant subspaces to J and J{. When J{ is fini te-dimensional
the converse assertion is true: a nonscalar operator has non-
trivi al hyperinvari ant su bspaces. More generally, a well-
known result of Burnside asserts that the only algebra of
n x n matrices without nontrivial invariant subspaces is the
algebra of all n x n matrices. In general the question is
open, even for compact operators.
Problem. Does every non scalar (compact) operator have
a nontrivial hyperinvariant subspace?
The corresponding problem for operator algebras is:
Problem. Is 93(J{) the only strongly closed operator alge-
bra wi thout nontrivial invariant subspaces?
95

96
NOTES ON OPERATOR THEORY
An affinnative answer for the latter problem (which seem's
quite unlikely) would imply an affirmative answer for the for-
mer, and hence for the invariant subspace problem. This sec-
tion is concerned wi th several existence theorems for hyper-
invariant subspaces. Beyond this, very little is known
concerning these interesting and important problems.
Before proceeding, consider the case of a normal operator
N. Because the spectral projections for N commute with any
operator in {N}', the spectral subspaces are hyperinvariant.
On the other hand, since A ( {N}' implies A * ( IN}' (Fug-
lede's Theorem), the projection P on a hyperinvariant sub-
space commutes with any operator in {N}'. It is well-known
that when }( is separable, such projections must be spectral.
Thus in general it may be useful to think of the hyperinvariant
subspaces as assuming the role of the spectral subspaces.
The following result, due to Rosenthal and Stampfli,
shows that certain invariant subspaces must be hyperinvariant
solely by virtue of their position in the lattice of all invariant
subspaces.
I. Theorem. Let S be a countable family of invariant sub-
spaces for an operator T, wi th the property that for any in-
variant subspaces m ( Sand Jr , S, either men or j( C '".
Then S consi sts of hyperinvariant subspaces.
Proof: Observe first that for any operator Sand !;\1 > IISII,
the operators Sand (8- AI)-l have the same invariant sub"
spaces. For the series _,\ -n-1 Sn converges to (S _ A/)- f
in the operator norm, and so SJJl c m implies
(S - Al)- t m em.

910. HYPERINVARlANT SUBSPACES
97
On the other hand, if (S - A/)-1 m c m then m c (8 - AI) m. If
the inclusion were proper there would be a unit vector x ( m
orthogonal to (S - A/)aJ, so that
o == «S- A/)x, x) > ["1-! (Sx, x)1 > I;\I-IISI[ > 0 ,
a contradiction. Thus m = (8- Al)'Jr{ and Sm em.
Now suppose that 8 commutes with T and let m € s.
Then the subspaces (8 - AI)m, IAI > Ilslt, are invariant for
T. If (S-A/)m I S for some \AI > ]\SII, then by hypothesis
m c (S - AI)  or (8 - AI)  c , so that (8 - A/)-1 m c m or
(S-Al)m c 'Jrl, and in either case Sm c:m. If (S-A/)'Jrl ( S
for all IAI > 11SII, then (S-At/)m = (S-A 2 1)m for some
Al I: A 2 since S is countable, and therefore
m = (S-A 2 I)-1(S-A t l)m == (!+(A 2 -A t )(S-A 2 /)-1)m ,
so that again s'Jrl c m. Hence m is hyperinvariant.
COROLLARY. If the in,rariant subspaces of T are count..
able in number, then every invariant subspace is hyperinvari-
ant.
COROLLARY. If m is an invariant subspace of T that is
comparable wi th every other invariant subspace of T, then 'Jrl
is hyperinvariant.
COROLLARY. If the invariant subspaces of T form a
chain, then every invariant subspace is hyperinvariant.
An operator such that the invariant subspaces form a chain
(i.e., for any invariant subspaces m and m, either men or
n c )IT) is called unice llular. Wh en J{ is fini te-dimensional,

98
NOTES ON OPERATOR THEORY
the unicellular operators are those of the form Al + N, where
N is cyclic and nilpotent (in other words, cyclic with one
point spectrum). For arbitrary K, it is not known whether the
spectrum of a unicellular operator must reduce to a point.
However (40]:
II. A unicellular operator on a separable Hilbert space has a
cycli c vector.
Proof: Let T be unicellular and let tma \ a E A I be the
family of all proper invariant subspaces of T. The problem
is to show that U ma :I. J{. Let Ixil be a countable dense
subset of u ma ' and let i E ma(i)' If mf3 c u ma(i) is false
for some (3, then u ma(i) C m since tma} is a chain, and
then U ma C mf3  J(. In the other case U ma = u ma(i) I: J{
by the Baire Category Theorem.
We now discuss several examples of unicellular operators.
The chain of invariant subspaces of the first will be shown
to have the order type of a closed interval, and that of the
second the order type of the positive integers with + 00 ad-
joined. It is not known whether the order type of the integers
with + 00 adjoined occurs for some unicellular operator.
The Volterra operator is defined on L 2(0, 1] wi th Lebes-
gue measure by (Vf)(t) = J. t f. It is clear that the subspaces
o
£2[8.. 1], 0 < a < 1, are invariant, and it will be shown that
these are the only invariant subspaces. The proof we give is
due to G. K. Kalisch, as simplified by M. Schreiber [45]. Tbe
formula
t
(Vnf)(t) = 1 ! (t- s)n-l!(s)ds
(n - I)! 0
will be needed. We note in passing that using this to estim. te

910. HYPERINVARIANT SUBSPACES 99
I (Vnf)(t) I 2 by means of the Cauchy-Schwarz inequality leads
to II Vnl\ < l/n!, so that V is quasinilpotent.
Recall that the convolution of functions f, g f L 1[0, b] is
defined by
t
(f '" g)(t) = f f(t-'s)g(s)ds ·
o
If e is the constantly 1 function on [0,1], then Vf == e * f,
so that Vnf = e * f, where e is the n..fold convolution of
n n
e wi th itself. The argument will be based on the following
theorem of Titchmarsh (56] ([31] contains an interesting
proof): if t * 9 = 0 a.e. in [0, b) and 0 l S(f), then g = 0 a.e.
in [0, 0]. Here the 8upport S(f) of f is the complement of the
largest open set in which f = 0 a.e.
III. f f L 2[a, 1] is a cyclic vector for V ! L 2(a, 1] if and only
if a l S(f).
Proof: Assume g f L 2 [0 , 1] is such that (Vnt, g) = 0 for
all n > o. If h(t) = g(l- t), then
1
(VnfJg) =! (Vnf)(t)h(l-t)dt
o
= «Vnf) * h)(l)
- (en * f * h)(l)
= (vn(f * h))(l)
1
1 1 (1_8)n-1(1 * h)(s)ds =
(n-l)! 0
1
1 r tn-t(f '" h)(l-t)dt ,
(n-l)! J O
=

100
NOTES ON OPERATOR THEORY
,
sQ>t)1at f * h = 0 a.e. in [0,1]. Now assume in addition that
.t/ 0 in [0, a] and that a ( S(f). If {1 is defined in [0, 1- a]
-by f 1 (t) = f (t + a), then
(f * h)(t + a) = ({1 * h)(t)
in [ 0, 1 - a], so th at f 1 * h = 0 a. e. in [0, 1 - a]. But
o ( S(f 1)' and therefore h = 0 a.e. in [0, 1- a] by Titchmarsh's
Theorem. Thus g == 0 a.e. in [a, 1], which proves that f is
a cyclic vector for V I £2[a, 1]. The converse is clear.
IV. Theorem. m is an invariant subspace of V if and only if
m = L 2 [a,1] forsorne a([O,l].
Proof: For f ( L 2[0, 1], let a(f) = inf S(f). Let m be in-
variant. Then from III it follows that L 2[a(f), 1] c m for all
f (m. This implies that L2(a, 1] em, where
a = i nf I a (f) I f l m} .
Since the other inclusion is clear, m = L 2[8, 1] as required.
Consider now the weighted shift operator S, defined on
e by
S(x o ' xl' ...) == (0'''"0 O' '\l X l' · ..) ,
where 1,\ } is a bounded sequence. It is clear that for n  0
n
the subspaces
m n := {I x l e and ;ck = 0 for all k < n}
are invariant for S. Under suitable restrictions on the weigbts
{A 1 it will be shown that these are the only nonzero invari"
n
ant subspaces.

9 1 0. HYPERINVARIANT SUBSPACES 101
v. If An -+ 0 then S is quasinilpotent.
Proof: If x = (X a , xl' .. .), then snx = y, where Y k = 0
for k < nand Yn+k == Ak A k+l ... A k + n - 1 x k for k > O.
It follows that
Ilsnll = sup\Ak A k+l ... Ak+n-tl ,
and hence that II S nil :s 110 111 '" I1n-l' where
/lk == SUp{IA k \,!A k + 1 !,...1 ·
If An ... 0 then J1 n .... 0 and ( J1 0 III ... fl n -l) 11n -+ 0, so that
S is quasinilpotent.
VI. Let m be a subspace of e. If there exists y (e such
that ! :In.1 < Y n!!x!! for all x f m and n > 0, then m is finite-
dimensional.
Proof: Let e 1 , ..., e k be an orthonormal set in m, and let
fn = ("e, ..., e) for all n > O. If a is any complex k-
vector, then :c = I a i e i f m, and so
!(a,f n )! = 1:£ aie! = !x n ! < Ynl!x!! -= Yn!!al! ·
This implies that
k
! le12 = IIf n ll 2 S y
i= 1
Summing over n gives k < IIYIl2) so that dim m 5 \\ yI1 2 < 00 .
\/11. Theorem. (Nikolskii). Let S be the weighted shift oper-
ator with weights IAn I, and assume that IIAn11 is non-increas..
ing, An I: 0 for all n > 0, and I/An!P < 00 for some p ( (0,00).

102
NOTES ON OPERATOR THEORY
Then m is a nonzero invariant subspace of S if and only if
m = m n for some n  o.
Proof: Let n be the least integer for which there is {J} £ 
with n  O. Then m c :»In' and it will be shown that equal.
ity holds. There is no loss of generality in assuming that
n = 0, in which case it must be shown that m = e.
Fix an integer N > 1 such that p S 2N; then
y = {IAn IN I f e .
If x (m wi th aJ a /; 0, th en because x, S;JJ, .. 0, S N JJ are linear.
ly independent elements of m, it is clear that a suitable linear
combination is a vector y f  with Yo = 1 and Yt = ... ==
YN = o. Now let 8. be orthogonal to m, so that (8, sny) = 0
for all n > O. Then
<XI
.I Ak .... A k + n - 1 Yk 2 n +k = 0
k=O
00
2n = - (A O ... An_1 )-1 !. Ak ... A k + n - t Yk 8 n + k
k=N+t
for all n > 1. But if n, k  N + 1,
I A k · II A k+n - 1
Ao .". An _ 1
<
'A N + 1 .. 0 AN + n
AO ... An_ 1
=
A '0' A N
n +n <
AO · n AN
\AnI N
jAo ... ANI
SInce 11 An! J is non-increasing, and therefore

910. HYPERINVARIANT SUBSPACES
108
lenl  BIAnlNllzl1 for all n 2: N + 1 ,
where B  I Ao ... AN' -1 lIyll- Hence dim m 1 < 0() by VI. If
m 1 /: lot there must be an eigenvector e (m 1 for S*. But
S* is quasinilpotent (because S is, by V), so S*e = 0 and
therefore en :::: 0 for n  1. Since (e, y) = 0 and Yo =: 1 it
follows that e = 0, a contradiction. Thus m 1 = fOJ and
m = e 2 
+
The rest of this section will be devoted to showing that
operators thatare close (in a suitable sense) to being uni-
tary have nontrivial hyperinvariant subspaces. Call operators
A and B quasi-similar if there are one-to-one operators P
and Q each having dense range such that AP = PB and
QA = BQ. Similar operators have isomorphic lattices of in-
variant subspaces. Although this does not seem to be true
for quasi -simi lari ty, we have:
VIII. If A and B are quasi-similar and A has a nontrivial
hyperinvariant subspace, then so does B.
Proof: There are one-to-one operators P and Q with
dense range such that AP == PB and QA == BQ. Suppose m
is hyperinvariant for A, and consider the subspace
Jl = { SQx I x (m and SB = BS l- .
It is clear that j( is hyperinvariant for B and that j{  {OJ
whenever m :j 10J.. If SB = BS then
(PSQ)A = PSBQ = PBSQ  A(PSQ) ,

104
NOTES ON OPERATOR THEORY
and therefore pSQm c m because m is hyperinvariant for A.
Hence P j( em, from which it follows that n  J{ whenever
m ;i J{.
IX. Let T be a contraction such that liT nx\\ -A 0 and 1\ T *naJ\I
,4 0 for all x  O. Then T is quasi-similar to a unitary op- ,
er ator .
Proof: As in the second proof of 96. I, let A be the non-
negative square root of the strong limit of T*nrnJ and let V
be the isometry defined on (AJ()- by V A = AT. Since
IIAx!! = lim n rnx!1 the hypothesis implies that A is one-to-
one, and hence that (AJ{)- = J{. To complete the proof it
will be show!! that Vis unitary and that there is a one-to-
one operator B with dense range such that BV = TB. By
hypothesis 0 is not an eigenvalue of T*, and therefore T
has dense range, V has dense range (from V A = AT), and
Vis uni tary.
Let A* and V* be the operators constructed from T* as
above, so that V*A* = A*T*, A* = V A*T*, and
A*AVA = A*A 2 T= Y:A*T*A 2 r = ftA*A 2 .
Since A has dense range this implies A*AV = V1 A*A . Now
B = A A is one-to-one with dense range, and
TB == TA;A =: (A*T*)*A*A
== A*V1 A*A == AAV = BV
as required.

9 1 0. HYPERINVARIANT SUBSPACES
105
.x. Theorem. [54, Ch. II]. Let T be a contraction, and sup"
pose that there are vectors Xo and Yo such that 11 rnxo II .;. 0
and II T*nYoll I- o. Then either T has a nontrivial hyperinvari-
ant subspace or T = dJ.
Proof: The subspaces {x! Tnx ..., oJ and {y \ T*ny -+ OJl
are hyperinvariant for To If both are proper then T is quasi-
similar to a unitary operator V by IX. If V is scalar so is
T; otherwise T has a nontrivial hyperinvariant subspace by
VIII.
XI. Theorem. Let A be an operator such that Re A is of fi-
nite rank and Re A < o. Then A has a nontrivial invariant
subspace..
Proof: According to 99 the Cayley transform
T == (A + 1)(A _/)-1
is an everywhere defined contraction. In addition, because
1- T*T == -4(A*-I)-1(ReA)(A-I)-1 ,
[- T*T is of finite rank. In the same way so is [- TT*. If
Tn -+ 0 strongly, then by S;2.II T is unitarily equivalent to a
part of a backward shift of finite multiplicity. By a result
mentioned at the end of 2 (and also at the end of 94),
T has a nontrivial invariant subspace. If T *n ....,. 0 strongly,
then in the same way T* has a nontrivial invariant subspace,
and hence so does T. If neither of these is the case, a non-
trivial invariant subspace exists by X. Thus in all cases T
has a nontrivial invariant subspace. The proof is completed
by showing Tm em implies Am c. Since

106
NOTES ON OPERATOR THEORY
T=I+2(A-l)-1,
Tm em implies (A-I)-1m em. If the inclusion were
proper, there would exist x I: 0 such that «A _/)-1 x, x) == 0,
so that for y = (A _/)-1:.v,
o =: (y, (A -/)y) = (y, Ay) _ \\y\1 2 < _ \Iyl\ 2 ,
and consequently y = 0 and :.v = 0, a contradiction. Hence
(A-I)-1m = m, (A-l)m =: m, and Am c m.
REMARK. This result is valid if Re A is compact and
its sequence of eigenvalues lies in r.: for some p f [1,00)
[46]. The question is open if He A is merely compact.
Exercises. 1. If P is a polynomial, the null space and
the closure of the range of p(T) are hyperinvariant subspaces
for T. When J{ is finite-dimensional every hyperinvariant
subspace is of thi s form.
2. If m is hyperinvariant for T then m 1 is hyperinvariant
for T*.
3. Quasi-similarity is an equivalence relation. If A and
B are quasi-similar, so are A* and B*.
*
4. Under the hypotheses of IX, the operators V and V*
are uni tarily equi valent.
5. Theorem VII is valid for weighted shift operators on
E!, 1 < q < 00. (Prove the following and use it to establish
the required generalization of VI: if {Tn} is a sequence of
operators on a Banach space B such that Tn ..." 1 strongly,
then a subset C of B is relatively compact if and only if
Tn  I uniformly on C.)

SECTION 11.
INVARIANT SUBSPACES FOR
COMPACT OPERATORS
In this section use will be made of the weak operator
topology on the algebra 93(J{) of all operators on J(. A net
{Tal converges to an oprator T in this topology if and only
if (Tax, y)  (Tx, y) for all [c, y (J{. A typical neighborhood
U of T is determined by vectors xl' ..., x n ' Y l' ..., Y n ( J(
as follows:
U ::::: {S I ! « T - S)X i , Y i) 1 :s 1 for 1 .s i  n I .
I. Theorem. The unit ball of 93(J{) is compact in the weak
operator topology.
Proof.. Since J{ is reflexive, the ball
B r = Lx I x (J{ and II xii  1'1, r > 0
is a weakly compact subset of J{ by Alaoglu's Theorem.
Therefore  = n 18 II xlii x ( J{ } is compact in the cartesian
product topology. This product consists of the functions
l: J{  J{ such that Ilf (:x) II :s I! xii for all x (J(, and conse-
quently contains the unit ball of ;B(J{). It is almost obvious
that the uni t ball of :B(J{) is a closed su bset of 6'1 on whi ch
the product topology and the weak operator topology coincide,
and the theorem follows.
107

108
NOTES ON OPERATOR THEORY
In outline, the method for obtaining invariant subspaces
is to produce projections En whose ranges are almost invari...
ant, in a suitable sense, and then to consider a cluster point
of JE n I in the weak operator topology (which must exist by I).
Unfortunately such a cluster point need not be a projection.
In fact:
II. Theorem. If J{ is infini te-dimensional, the closure in the
weak operator topology of the set  of all projections is the
set e+ of all posi ti ve contractions.
LEMMA. If A is a positive contraction and Q is a pro-
jection such that dim (1- Q)J{ = dim H, there is a projection
E such that QAQ = QEQ.
Proof: By 9 7 there is a space K ) J{ and a projection F
In K with PF! J{ = A and dim K =: dim J( (where P is the
projection on J{). From this and the hypothesis it follows that
that dim (K e :D) == dim (J{ e :1)), where :D = Q J(. Therefore
there is an isometry W from J{ onto K such that W I  is the
identity. If E = W*FW, then for all x, Y f}(,
(QEQx, y) :: (W*FWQx, Qy) = (FWQx, WQy) = (FQa;, Qy)
=: (AQ:JJ, Qy) = (QAQx, y) .
Proof of Theorem. It is clear that e+ contains the clo-
sure of . Consider a positive contraction A. Write
J{ = !, ffi J{k
k>l
wi th dim J{k == dim J{ for all k 2 1, and let Qn be the pro-
j ection on J{ 1 ffi... EB J(n. By the lemm a there is a projection

911. INVARIANT SUBSPACES
109
En such that Qn En Qn = Qn A Qn for all n  1. Now Qn  I
strongly, and it follows that En  A weakly, for
«En - A)} y) = «En - A) Qn x , QnY) + «En - A)QnJ (/- Qn)Y)
+ «En - A)(l- Qn)x, y)
and «En - A) Qn x , Qn Y ) = 0, so that
I {(En - A)x, Y)I :$ 21/ xii 1\ ([- Qn)y/I + 2/1 (1- Qn)11 II YII ·
This result is due to Halmos [20] and Nagy [49]. Byar-
guing in the same way with unitary dilations, it may be shown
that the closure in the weak operator topology of the unitary
operators is the entire uni t ball of 53(J{).
The construction of invariant subspaces begins with the
following lemma.
III. Let e f J(, T f 93(J{) , Rn the projection on [e, Te, ...., rne],
and d n = II Tfie - Rn_l rnel/ the distance from Tne to
[e} ..., T n - 1 e]" Then II TR n -RnTRnl! =: d n + 1 /d n .
prool: Fix x f J{ and let Rn{;C = aoe + .... + anTn e . Then
TRnx = aoTe + '" + a n T n + 1 e
RTR I'VI aT e + ".. +a r n e+aRT n+l e
n n W = 0 n-1 n n
TR x-R TR x =a (Tn+le_ R T n+l e)
n n n n n
IITRnx-RnTRnxl! - ! a n ld n + 1 = (dn+l/dn)(dnlan!)'

110
NOTES ON OPERATOR THEORY
But
lanldn = \\anTne-Rn_l(anTne)\\
= II Rn;J:- Rn_ 1 R n x l\
= \\Rn x - R n _ 1 x!! ,
and the lemma follows.
IV. If e is a cyclic vector for T such that lim inf II Tne Ill/n
= 0, then there are finite-dimensional projections Pt .:5 P2 < ...
such that Pn --+ I strongly and IITPn - PnTPnl1 --+ O. If Qn is a
P ro J "ection such that Q < P and Q J{ is invariant for T =
n - n n n
PnTPn' then \\QnTQn - TQnl\ 4 o.
Proof: lim inf(dn+l/dn) < lim infd;/n.s lim infllTnelll/n
= o. Therefore if IP lis a suitable subse q uence of IR I,
n n
III implies lIT Pn - Pn T Pn 1\ -+ O. Since e is cyclic Rn .... I, so
 --+ I too. Finally
TQn - QnTQn = TQn - QnPnTPnQn
= TQn - QnTnQn
= TQn-TnQn
= (TP n - PnTPn)Qn ,
so \lTQn-QnTQn\\ < \\TPn-PnTPn\1 40.
The next step is to construct projections Qn as in IV, but
with the additional property that if Q is a weak cluster point
of {Qn I, then Q 1= 0, I. Thi s will be done wi th the help of the
linear functional
p (A) =  [(A e, e) + (A f, f)],

911. INVARIANT SUBSPACES
111
where e and f are any orthogonal unit vectors. It is easy to
see that p (E) <  for anyone-dimensional projection E.
Since p (Pn) .... 1, it can be assumed that p (Pn) > 3/4 for all
n. If P is m-dimensional, there are P rojections P 1 < p2 <
11 n - n -
... < pm = P such that dimpk = k and pkJ{ is invariant
- 11 n n n
for PnTPn' k = 1,2,. ..,m. Since
P (pk+l) _ P (pk) = P (pk+ 1_ pk) < 1
n 11 n n - 2
there is J . < m with X < P (P j) < %. Let Q = P j . If Q is
- - n - 11 n
a weak cluster point of 1Qn l , then X < p (Q) < %, and there-
fore Q  0, I.
V. The space m == 1:1J I Q:1J = xl is closed, invariant, and dis-
tinct from J{.
Proof: Obviously m is a closed subspace, and m i: J{
since Q # I. Let lQa l be asubnet of lQn J which converges
weakly to Q. If x (m, then
IIQax-xlt2 = (x,x)-(Q a :1J,x)....O ,
and hence 11 TQax- T:1J1I .... O. But
I (Qa TQ a :1J- QTx, y)[ < I(QaTQa x - QaTx,Y)1 + !(Qa T :1J- QTx,Y)1
< II TQa x - Tx!1 Ilyll + I «Qa - Q)T:x, y)1 ,
so that QaTQa:X -+ QT:x weakly. On the other hand, QaTQax
-. TQ:1J weakly by IV, so that QTx = TQ:x = T:1J and T:1J (m.
The only point remaining is whether m # {OJ, and this is
where a compactness hypothesis is used.

112
NOTES ON OPERATOR THEORY
VI. Theorem. Let T be an operator such that (i) there is a
non-zero vector e with lim inf II Tnell1/n = 0, and (ii) the
norm-closed algebra generated by T and I contains a non-
zero compact operator C. Then T has a nontrivial invariant
subspace.
Proof: Since the subspace spanned bye, Te, ... is invari-
ant, it can be assumed that e is a cyclic vector. Consider
the set (1 of all operators A such that 1\ QaAQa - AQa II -to 0,
where {Qat is as above. For any operators A and B,
IIQaBQa-BQall < IIQaBQa-QaAQall + \IQaAQa-AQall
+ IIAQa-BQa\\
< IIQaAQa-AQall + 211A-BII ,
and consequently (1 is norm-closed. Clearly C1 is linear,
and since
Qa(AB)Qa - (AB)Qa = (QaAQa - AQa)BQa
+ (A-QaA)(QaBQa-BQa) ,
it is an algebra. But T € C! by IV, and so C € ct :
IIQaCQa- CQa l1 -to 0 .
From this and the compactness of C it follows readily
that QCQ = CQ, and hence that CQ J{ em. For CQa -+ CQ
strongly since C is compact, and therefore
QaCQa-QCQ = Qa(CQa- CQ ) + (Qa-Q)CQ,

9 11 . INVARIANT SUBSPACES
113
1«QaCQa-QCQ)x,Y)1 < II<CQa-CQ)xll IIYII + !«Qa-Q)CQX,y)1
and QaCQa -+ QCQ weakly.
If m :/: to} there is nothing to prove. If m = {OJ, then
CQ J{ = to}, so that the null space of C is nontrivial (be-
cause Q I:. 0). But C and T commute, and therefore the
null space of C is invariant for T. This completes the
proof.
COROLLARY. If T is quasinilpotent and if the norm-
closed algebra generated by T and I contains a non-zero
compact operator, then T has a nontrivial invariant sub-
space.
These results are due to Arveson and Feldman [2]; the
proof is based on earlier work of Bernstein and Robinson [7]
and Halmas [23].
VII. Theorem. If p (T) is compact for some polynomial p I: 0,
then T has a nontrivial invariant subspace.
Proof: Let C = p (1). If 0 * A ( a (C) and y is a contour
enclosing A but no other point of a (C), then
E = 2:i J (131- C)-ldz
y
is an idempotent distinct from 0 and I which commutes with
every operator commuting with C. In particular E and T
commute, so E J( is a nontrivial invariant subspace.
On the other hand, suppose that a (C) = to J. In this case
a (T) is a fini te set by the Spectral Mapping Theorem. If there
is more than one point in this set, then the above argument

114
NOTES ON OPERATOR THEORY
may be repeated. If a(T) = {A}, then T-Al is quasinilpo-
tent and VI applies, provided C # o. Since it is obvious that
T has invariant subspaces when p (T) == C == 0, the theo-
rem is proved.
COROLLARY. Any compact operator has a nontrivial in-
variant subspace.
The theorem is due to Bernstein and Robinson [7], and
the corollary to von Neumann, Aronszajn, and Smith [1] (on
any Banach space).

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116
NOTES ON OPERATOR THEORY
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118
NOTES ON OPERATOR THEORY
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INDEX
accretive transformation, 87ff
Berger, 70
Beur Ii ng, 28
bi lateral shi ft, 8, 17, 26, 33,
commutant, 26, 35
reducing subspaces, 27, 36
invariant subspaces, 28,40
Bochner, 72, 74
de Branges and Rovnyak, 23,
45
Cayley transform, 88,105
cogenerator, 92,93
coi sometry, 49, 56, 87
coisometric extension, 49,81
compact operator, 13,106,
107 ff.
completely nonuni tary con-
traction, 52, 55, 60
Cooper, 83
di lation, 65ff., 70
uni tary, 57, 58
proj ection, 57
Douglas, 51,61
Foguel, 53
Herglotz, 69
hyperinvariant subspace, 95ff.
hYPODormal operator, 8£f.
invariant subspace, 7, 23, 28,
39ff., 95£f., 112£f.
full, 43, 49
infinitesimal generator, 80ff.,
93
inner function, 29
isometry, 15, 92
structure theorem, 16
N aimark, 65ff., 75
Nikolskii, 101
numeri cal radi us, 70
power i nequ ali ty, 71
numeri cal range, 11, 22, 63,
70
operator-valued function, 34
analyti c, 37
parti al isometry, 39, 41, 42
posi ti ve -defini te fun cti on, 65,
67
P. O. measure, 72
dilation of, 74
quasinilpotent operator, 12,
99,101,113
quasi -si mi lar, 103
Rota 21
semi group, 77£f.
121

122
NOTES ON OPERATOR THEORY
shi ft-see uni lateral shi ft,
bi lateral shi ft, and weighted
shi ft
Spectral set, 62
Stone, 72
subnormal operator, 7
Sz.- Nagy, 70,75
and Foias, 52,61
uni cellular: operator, 97
unilateral shift, 15,26,34,45
commutant of, 27,37,47
invariant subspaces, 28, 41
reducing subspaces, 27, 38
uni tary
absolutely continuous, 55
dilation, 57,58,70
representation, 65
singular, 55
Volterra operator, 24
invariant subspaces, 98
von Neumann, 62
wandering subspace, 17
weighted shift operator, 100
invariant subspaces, 101
Wermer, 8
Wold decomposition, 16

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P. R. HALMOS received his Ph.D. from the University of Illinois,
and spent three years at the Institute for Advanced Study, two of
them as Assistant to John von Neulnann. He taught at the Uni-
versities of Chicago,
lichigan, and Hawaii and is presently Pro-
fessor of Mathematics at Indiana University.


F. \\T. GEHRING received his Ph.D. frolll Cambridge University,
England. He has held Visiting Professorships at Harvard and Stan-
ford Universities as we]\ as Guggenheim, Fulbright, and NSF
Fello\vships at the University of Helsinki and the Eidgcnossischc
Technische Hochschule in Zurich. He is presently Professor of
Mathen1atics at the University of :tvlichigan.