Banach Algebra Techniques in Operator Theory (Pure and Applied Mathematics 49)

Banach Algebra Techniques in Operator Theory RONALD G. DOUGLAS Department of Mathematics State University of New York at Stony Brook Stony Brook, New York

1972 ACADEMIC PRESS New York and London

COPYRIGHT 1972, BY ACADEMIC PRESS, INC. ALL RIGHTS RESERVED NO PART OF THIS BOOK MAY BE REPRODUCED IN ANY FORM, BY PHOTOSTAT, MICROFILM, RETRIEVAL SYSTEM, OR ANY OTHER MEANS, WITHOUT WRITTEN PERMISSION FROM THE PUBLISHERS.

ACADEMIC PRESS, INC. 111 Fifth Avenue, New York, New York 10003

United Kingdom Edition published by

ACADEMIC PRESS, INC. (LONDON) LTD. W. EVIJIAU, 1,4311UUti IN NV 1 arill LA %/VW.

"IA /ft 0

11-

11.71V

A

LIBRARY OF CONGRESS CATALOG CARD NUMBER:

78-187253

AMS (MOS) 1970 Subject Classifications 47B05, 47B15, 47B30, 47B35, 46L05, 46J15, 46-02, 47-02 PRINTED IN THE UNITED STATES OF AMERICA

To my family Nan, Mike, Kevin, Kristin

Contents

PREFACE ACKNOWLEDGMENTS SYMBOLS AND NOTATION

1.

Banach Spaces The Banach Space of Continuous Functions Abstract Banach Spaces The Conjugate Space of Continuous Linear Functionals Examples of Banach Spaces: c o , 1 1 , and 1' Weak Topologies on Banach Spaces The Alaoglu Theorem The Hahn-Banach Theorem The Conjugate Space of C([0,11) The Open Mapping Theorem The Lebesgue Spaces! LI and The Hardy Spaces: HI and Notes Exercises

2.

1 2 5 7 8 10 12 18 23 23 26 27 27

Banach Algebras The Banach Algebra of Continuous Functions Abstract Banach Algebras Abstract Index in a Banach Algebra The Space of Multiplicative Linear Functions The Gelfand Transform

32 34 36 39 40 vii

viii

Contents The Gelfand-Mazur Theorem The Gelfand Theorem for Commutative Banach Algebras The Spectral Radius Formula The Stone-Weierstrass Theorem The Generalized Stone-Weierstrass Theorem The Disk Algebra The Algebra of Functions with Absolutely Convergent Fourier Series The Algebra of Bounded Measurable Functions Notes Exercises

3.

42 44 45 46 50

51 54 56 58 58

Geometry of Hilbert Space Inner Product Spaces 63 The Cauchy-Schwarz Inequality 64 The Pythagorean Theorem 66 Hilbert Spaces 66 Examples of Hilbert Spaces: CTM. /2 , L 2 , and H 2 66 The Riesz Representation Theorem 72 75 The Existence of Orthonormal Bases 76 The Dimension of Hilbert Spaces 77 Notes Exercises 78

4.

Operators on Hilbert Space and C*-Algebras The Adjoint Operator Normal and Self-adjoint Operators Projections and Subspaces Multiplication Operators and Maximal Abelian Algebras The Bilateral Shift Operator C*-Algebras The Gelfand-Naimark Theorem The Spectral Theorem The Functional Calculus The Square Root of Positive Operators The Unilateral Shift Operator The Polar Decomposition Weak and Strong Operator Topologies W*-Algebras Isomorphisms of L'-Spaces Normal Operators with Cyclic Vectors Maximal Abelian W*-Algebras *-Homomorphisms of C*-Algebras The Extended Functional Calculus The Fuglede Theorem Notes Exercises

81 84 86 87 89 91 92 93 93 95 96 97 100 101 104 105 108 110 112 114 115 115

Contents

5.

Compact Operators, Fredholm Operators, and Index Theory The Ideals of Finite Rank and Compact Operators Approximation of Compact Operators Examples of Compact Operators: Integral Operators The Calkin Algebra and Fredholm Operators Atkinson's Theorem The Index of Fredholm Operators The Fredholm Alternative Volterra Integral Operators Connectedness of the Unitary Group in a W*-Algebra Characterization of Index Quotient C*-Algebras Representations of the C*-Algebra of Compact Operators Notes Exercises

6.

ix

121 124 125 127 128 130 131 132 134 138 139 142 144 145

The Hardy Spaces The Hardy Spaces: H', H 2 , and IP 149 Reducing Subspaces of Unitary Operators 151 Beurling's Theorem 153 The F. and M. Riesz Theorem 154 155 The Maximal Ideal Space of H"' The Inner—Outer Factorization of Functions in H 2 158 159 The Modulus of Outer Functions 161 The Conjugates of H' and L'/Ho The Closedness of H"+C 163 Approximation by Quotients of Inner Functions 163 164 The Gleason—Whitney Theorem 164 Subalgebras between If' and V' Abstract Harmonic Extensions 166 168 The Maximal Ideal Space of H°D+ C 170 The Invertibility of Functions in H"+ C 171 Notes Exercises 172

7. Toeplitz Operators Toeplitz Operators The Spectral Inclusion Theorem The Symbol Map The Spectrum of Self-adjoint Toeplitz Operators The Spectrum of Analytic Toeplitz Operators The C*-Algebra Generated by the Unilateral Shift The Invertibility of Toeplitz Operators with Continuous Symbol

177 179 179 183 183 184 186

x

Contents The Invertibility of Unimodular Toeplitz Operators and Prediction Theory The Spectrum of Toeplitz Operators with Symbol in H' + C The Connectedness of the Essential Spectrum Localization to the Center of a C*-Algebra Locality of Fredholmness for Toeplitz Operators Notes Exercises

187 189 194 196 199 200 203

References

208

INDEX

213

Preface

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

xii

Preface

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

Acknowledgments

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

Symbols and Notation

A

51

Auto )

61

C C" C(X) L. co (r) ET

1 66 1 26 7 93

D

52

exp RO )

a.

36 127

FA

130 173

g

34

4 h

40

182 1-1 1 26 1101 161

H2 70 H 149 "

He I-1'

161 26

Ho' 1-1'+C(T)

27 163

j

130

ker

83

NZ + ) PM /2(r)

7 54

If°

23

67 /2(7) 89 I°°(1 + ) 7 L1 23 L2 68 L" 23 21 ) 121 E0 121 RO ) Ai3 36 2(

Symbols and Notation

xvi

Ms M, Mc° M(X)

39 87 155 20

P g g+ PC Pai

177 51 51 176 86

QC

176

R r() gi( ) ran PO Pe()

4 41 57 83 41 191

a( ) E

41 4

T

26

Tio TK Z( )

177 126 179

U U+ UT)

89 96 136

w()) W( ) 9B

115 115 101

1 0 Lo 01 0 Op

©

1 0 ( , )

7 1 6,82,119 10 24 29 30,71 31,62,79,118 64 108

1

flannel Spaces

1.1 We begin by introducing the most representative example of a Banach space. Let X be a compact Hausdorff space and let C(X) denote the set of continuous complex-valued functions on X. For 11 and 12 in C(X) and 2 a complex number, we define: ( 1 ) (A +./2) (x) =A (x) +/2 (x) (2) (1f1 )(x) = Afi (x); and (3) (A .i2)(x) = .ii (x)./2. (x). With these operations C(X) is a commutative algebra with identity over the complex field C. Each functionfin C(X) is bounded, since it follows from the fact thatfis continuous and X is compact that the range of f is a compact subset of C. Thus the least upper bound of Ill is finite; we call this number the norm of f and denote it by

ilf II. = sup WWI : x e X}. The following properties of the norm are easily verified:

(1) VII., = 0 if and only if f = 0; (2) 11 2.iiim = 1 2 1 111L ;

(3)Ilf+g II. II/11m+ ligil.; and (4)ilfg11.. 11/11. ilglico •

2 1 Banach Spaces

We define a metric p on C(X) by p (f, g) = ilf— gil o, . The properties of a metric, namely, (1) p (f, g) = 0 if and only if f = g, (2) p (f, g) = p (g , f), and (3) p(f,h) < p(f, g) + p (g , h),

follow immediately from properties (1)—(3) of the norm. It is easily seen that convergence with respect to the metric p is just uniform convergence. An important property of this metric is that C(X) is complete with respect to it. 1.2 Proposition If Xis a compact Hausdorff space, then C(X) is a complete

metric space. Proof If {fn },T_ 1 is a Cauchy sequence, then

!Mx) fm(x)1 < lifn—frn11.0 = P(f.,fm) —

for each x in X. Hence, {fn (x)},T_ 1 is a Cauchy sequence of complex numbers for each x in X, so we may define f(x) = limn , .3 f„(x). We need to show that f is in C(X) and that lim..„. IV fill co = 0. To that end, given s > 0, choose N such that n, in .: N implies Ilfn—fm11. < E- For xo in X there exists a neighborhood U of x o such that I fN (x 0 )—fN (x)1 < e for x in U. Therefore, —

I f(x0 ) — f(x)I < lim I fn (x 0 ) — fN (x0 )1 + Ifpi(x0)—fN(x)1

+ lim Ifiv(x)—./.(x)1 n -+

3e which implies f is continuous. Further, for n N and x in X, we have I fn (x) — f(x) I = I fn (x) — lim fin (x) I = lim I f, (x) — fm (x)I m

177-4 CO

< rim sup Ilf, —Lilco

E_

Y77-4, GO

Thus, lim„,,, Ilfn—f11. = 0 and hence C(X) is complete. III We next define the notion of Banach space which abstracts the salient properties of the preceding example. We shall see later in this chapter that every Banach space is isomorphic to a subspace of some C(X). 1.3 Definition A Banach space is a complex linear space .1 with a norm

II ii satisfying

Abstract Banach Spaces 3

(1) 0111 = 0 if and only if f = 0, (2) 112111 = 121 11111 for 2 in C and f in X, and ( 3 ) Ilf+gll < 11111+ IWO for f and g in X, such that X is complete in the metric given by this norm. lA Proposition Let X be a Banach space. The functions a: X x .T --> X defined a(f,g)=f+g, s: C x A' A'' defined s(2,f) . Af, and n: X —> R + defined n(f) =

11111

are continuous. Proof Obvious. 1111 1.5 Directed Sets and Nets The topology of a metric space can be described in terms of the sequences in it that converge. For more general topological spaces a notion of generalized sequence is necessary. In what follows it will often be convenient to describe a topology in terms of its convergent generalized sequences. Thus we proceed to review for the reader the notion of net. A directed set A is a partially ordered set having the property that for each pair cc and 16 in A there exists y in A such that y .--- a and y ...-- /3. A net is a function a ---> fla on a directed set. If the /1,„ all lie in a topological space X, then the net is said to converge to A in X if for each neighborhood U of 2 there exists a u in A such that 2„ is in U for a > a u . Two topologies on a space X coincide if they have the same convergent nets. Lastly, a topology can be defined on X by prescribing the convergent nets. For further information concerning nets and subnets, the reader should consult [71]. We now consider the convergence of Cauchy nets in a Banach space. 1.6 Definition A net {f.}.„ in a Banach space X is said to be a Cauchy net if for every c > 0, there exists a o in A such that a l , a 2 ; a o implies

Ilf., -1.2 II < E

•

1.7 Proposition In a Banach space each Cauchy net is convergent. Proof Let {ft }„ EA be a Cauchy net in the Banach space C. Choose a l such that cc ..- a / implies 111.-f.,11 < 1. Having chosen fockl in,= 1 in A, choose an+ / -.-- an such that a .-. an+ i implies 1 Ilf. -L.+ ,11 < n+1 .

4 1 Banach Spaces

The sequence {fan }„'__ 1 is clearly Cauchy and, since .ff is complete, there exists fin A' such that limn ,, fan = f It remains to prove that limx€A fx =f Given E > 0, choose n such that l/n < B/2 and lifan —f II < 42. Then for a ? an we have

Ilia — f II < life —fail + Ilfan — f II < 11n + 6/2 < E. ■ We next consider a general notion of summability in a Banach space which will be used in Chapter 3. 1.8 Definition Let {fa} a E A be a set of vectors in the Banach space ,ff . Let .9- = {Fc A : F finite}. If we define F 1 ...‹. F2 for F1 c F2, then .97 is a directed set. For each F in .97, let gF = EG„ F fa . If the net {gF } FE , converges to some g in ,ff , then the sum E ct " fa is said to converge and we write g = LEAL. 1.9 Proposition If {L},„ EA is a set of vectors in the Banach space A' such

that EaE Allfait converges in IR, then E,,,, A fc, converges in ,T.

Proof It suffices to show, in the notation of Definition 1.8, that the net {gF} F , s,-- is Cauchy. Since LEA Vali converges, for E > 0, there exists F0 in .97 such that F .... Fo implies

I FIlfell — ceI ; I L < E• Thus for Fl , F2 _; Fo we have IIgF1 — gF2tI = IIa E fee — E Lit eFi

a EF2

= II E ice— E fell aeF1 \F2 ceEFAFi

E 11f, II + E II.feell

a EF21F2 a EF2kf 1

E llfall — E IILII < E.

a EFIU F2

aEFG

Therefore, {gF} FE , is Cauchy and Ec„ A fa converges by definition.

■

We now state an elementary criterion for a normed linear space (that is, a complex linear space with a norm satisfying (1)—(3) of Definition 1.3) to be complete and hence a Banach space. This will prove very useful in verifying that various examples are Banach spaces.

The Conjugate Space of Continuous Linear Functionals 5

1.10 Corollary A normed linear space is a Banach space if and only if for every sequence {f„},71. 1 of vectors in the condition El. i [Ifn i! < oo implies the convergence of E nc'l l Proof If X is a Banach space, then the conclusion follows from the preceding proposition. Therefore, assume that {g,}71. i is a Cauchy sequence

in a normed linear space X in which the series hypothesis is valid. Then we may choose a subsequence {g„j1,1 such that fic:', 1 < co as follows: Choose n i such that for i,j n 1 we have fig i — gJ < 1; having chosen {nk }11... I choose nN+1 > nN such that i,j n N + 1 implies ligi — < 2-N . -1 for k > 1 and f1 = g, 1 , then Xici°,- 111.411 < co, and If we set fk = a the hypothesis implies that the series I i°,1. 1 fk converges. It follows from the definition of convergence that the sequence {g„k }r.... 1 converges in X and hence so also does {g„} 1 . Thus is complete and hence a Banach space. ■ In the study of linear spaces the notion of a linear functional is extremely important. The collection of linear functionals defined on a given linear space is itself a linear space and this duality is a powerful tool for studying either space. In the study of Banach spaces the corresponding notion is that of a continuous linear functional. 1.11 Definition Let X be a Banach space. A function 9 from X to C is a bounded linear functional if: (1) 02 1 fl + 2 212) =

2 19(f1) + 2 20f2) for f1 , f2 in and 11,12 in C;

and (2) There exists Al such that 19(f)I < M 11111 for everYi in X.

1.12 Proposition Let 9 be a linear functional on the Banach space X. The following statements are equivalent: is bounded; (2) 9 is continuous; (3) 9 is continuous at 0. (1)

q

Proof (1) implies (2). If {fa L EA is a net in X converging to f, then

linkcEAllfa - f II = 0. Hence, limj^p (fa) - 9(f)1 = Urn 1 (fa 1)1 lim OtE A aeA aeA —

- f = 0,

which implies that the net {9 ff.)} ae A converges to 9 (I). Thus 9 is continuous. (2) implies (3). Obvious.

6 1 Banach Spaces

(3) implies (1). If cp is continuous at 0, then there exists b > 0 such that 11111 < 6 implies IT (f) I < 1. Hence, for any nonzero g in X we have

2

2 iigii ( b g) < 6 I 2 IIgII and thus cp is bounded. ■

We next define a norm on the space of bounded linear functionals which makes it into a Banach space.

1.13 Definition Let f"* be the set of bounded linear functionals on the Banach space'. For yo in '*,let

ik pli=sup{

kowl 11111

Then X* is said to be the conjugate or dual space of X.

1.14 Proposition The conjugate space X* is a Banach space. Proof That X* is a linear space is obvious, as are properties (1) and for the norm. To prove (3) we compute

11(pi +c9211

(2)

l(Ti +492)(i)1 sup = = sup iTi(f)+ TAD'

f 0

IIfII



sup

fro

IT' WI IIfII

f 0

Ilf II



± sup IT2(f)I

foci IIfII

11 c'1 II ± II CP2.11. Finally, we must show that X* is complete. Thus, suppose {(pn }'°_ 1 is a Cauchy sequence in X*. For each fin X we have I Pn(f) Pin(f)I < II Pn Pm II 11f11 so that the sequence of complex numbers (f )},7= 1 is Cauchy for each f in X . Hence, we can define Of) = lim n con (f). The linearity of y9 follows from the corresponding linearity of the functionals T n . Further, if N is chosen so that n, Al implies II Con — (Pm II < 1, then for f in X we have (

—

(

{

IT(f)I

iTin — CoN(DI + ISPADI lim 'Con(i') — copi(f)I+ Icctiv(f )1 Jim sup c II T. — CP iv 11 11f1I + ! ITN!! VII

(I + 11 (ppi II) VII-

(

—

(

Examples of Banach Spaces: c o , 1', and 1' 7

Thus cp is in .T* and it remains only to show that lim n 11 — 9 „II = 0. Given E > 0, choose N such that n, m N implies 119n — Com11 < E. Then for f in g' and m,n :3; N, we have 1(9 — 9.) (f) 1 < 1(9 — 9m) Cf) 1 + 1 (9m — 9n) (f )1 < 1 — COW' + II/II. Since lim,n , 1(9— 9,,)(f)I = 0, we have 119 — con II< E. Thus the sequence 19n ln°_ 1 converges to cp and .T* is complete. 1111 The reader should compare the preceding proof to that of Proposition 1.2. We now want to consider some further examples of Banach spaces and to compute their respective conjugate spaces. 1.15 Examples Let 1" (Z + ) denote the collection of all bounded complex functions on the nonnegative integers V. Define addition and multiplication pointwise and set 11f II.= sup {1f(n)1 : n ell. It is not difficult to verify that / 0 (V) is a Banach space with respect to this norm, and this will be left as an exercise. Further, the collection of all functions f in 1 (1 + ) such that f(n) = 0 is a closed subspace of r (V) and hence a Banach space; we denote this space by c o (11. In addition, let P(71 + ) denote the collection of all complex functions 9 on 1 + such that E„.°_ 0 I cp (n)I < co. Define addition and scalar multiplication pointwise and set 11911 1 = E„pc= 0 I 49(01_ Again we leave as an exercise the task of showing that / 1 (71 + ) is a Banach space for this norm. We consider now the problem of identifying conjugate spaces and we begin with c o (e). For cp in / 1- (71 + ) we define the functional on c o (Z + ) such that 0(f) = En"_ 0 9 (n)f(n) for f in co (Z + ); the latter sum converges, since

= n=o 9 (n)f(n)

n=o

190011RO

CO

<

Eo19(n)1 = U11.119111.

n=

Moreover, since 0 is obviously linear, this latter inequality shows that 0 is in c o (e)* and that 119111 11011, where the latter is the norm of 0 as an element of c o (1 + )*. Thus the map a (9) = cp from / 1 (71 + ) to c o (1 + )* is well defined and is a contraction. We want to show that a is isometric and onto co (Z To that end let L be an element of c o (1 + )* and define the function 9 L on 1+ so that 9L (n)= L(en ) for n in 1 + , where en is the element of c o (r)

8 1 Banach Spaces

defined to be 1 at n and 0 otherwise. We want to show that 0 L = L, and that II (PLIIi < liLli . For each N in 1 + consider the element

L(en)

fN

of c o (e), where 0/0 is taken to be 0. Then IliN0 1 and an easy computation yields

I I L I I

I L UN) I =

L(en) L(en) = n=0 IL(en)I

IL(en)I =

I (Pi(n) I ;

hence 9/, is in 1 1 (1 + ) and II 9L11 1 < 111,11 • Thus the map fl(L) = (p i, from c o (1 +)* to 1 1 (11 is also well defined and contractive. Moreover, let L be in c o g 1* and g be in c o (71 +); then lira Ilg

N-+

N

E g(n)e„I1 ❑ , = 0

n=0

and hence we have L(g) = lim

g(n)L(en)} = lim

N---) co n=0

=

g(n)9L(n)}

N-4 co n=0

o g (n) CPL(n) = OL(g).

Therefore, the composition a ofi is the identity on c o (71 +)*. Lastly, since = 0 implies co = 0, we have that a is one-to-one. Thus a is an isometrical isomorphism of 1 1 (71 + ) onto c o (71 + )*. Consider now the problem of identifying the conjugate space of 1 1 (r). For f in l (e) we can define an element J of 1 1 (11* as follows: j(9) En 0 f(n) (n). We leave as an exercise the verification that this identifies P(71 + )* as r(71 + ). 1.16 We return now to considering abstract Banach spaces. If a sequence of bounded linear functionals {co n },,m_ o in X* converges in norm to 9, then it must also converge pointwise, that is, lim n . = 9(f) for each f in

The following example shows that the converse is not true. For k in l + and fin l i (71 + ) define Lk (f) =1(k). Then Lk is in 1 1 (71 + )* and 114,11 = I for each k. Moreover, lim k , Lk (f) = 0 for each f in 1 1 (Z + ). Thus, the sequence {Lk } k°1 0 converges "pointwise" to the zero functional 0 but 114 — 0 11 = 1 for each k in 71+.

Weak Topologies on Banach Spaces 9

Thus, pointwise convergence in .T* is, in general, weaker than norm convergence; that is, it is easier for a sequence to converge pointwise than it is for it to converge in the norm. Since the notion of pointwise convergence is a natural one, we might expect it to be useful in the study of Banach spaces. That is indeed correct and we shall define the topology of pointwise convergence after recalling a few facts about weak topologies. 1.17 Weak Topologies Let X be a set, Y be a topological space, and .97 be a family of functions from X into Y. The weak topology on X induced by .97 is the weakest or smallest topology .9 - on X for which each function in .97 is continuous. Thus .9 is the topology generated by the sets { f (U) : f e U open in Y}. Convergence of nets in this topology is completely characterized f(x) for every f in .97 . Thus .9 by lim,, EA xo, = x if and only if is the topology of pointwise convergence. If Yis Hausdorff and .97 separates the points of X, then the weak topology is Hausdorff. 1.18 Definition For each f in A' let ., denote the function, on ff* defined = 9(f). The w*-topology on ,T* is the weak topology on X* induced by the family of functions fj:f e 1}. 1.19 Proposition The w*-topology on ,T* is Hausdorff. Proof If cp i 0 9 2 , then there exists f in 2. such that 9 1 (f) 0 9 2 (f).

Hence, j(co i ) /(9 2 ) so that the functions { ff} separate, the points of X*. The proposition now follows from the remark at the end of Section 1.17. 1111

We point out that the w*-topology is not, in general, metrizable (see Problem 1.13). Next we record the following easy proposition for reference. 1.20 Proposition A net {9„}„, EA in X* converges to 9 in X* in the w*topology if and only if lim a €, 9,c (f) = 9(f) for every fin

The following shows that the w*-topology is determined on bounded subsets of .T* by a dense subset of .T and this fact will be used in subsequent chapters. 1.21 Proposition Suppose ." is a dense subset of A' and {9a}cceil is a uniformly bounded net in .T* such that Iim cce A 49ce(f) = {p (f) for f in .14'.

10 1 Banach Spaces

Then the net {(pa } a „ converges to 9 in the w*-topology. Proof Given g in ,T and e > 0, choose f in di such that Ill — gll < el3M, where /V - --- suptII49 1, II Pa II : a e Al. If ao is chosen such that a ?.-- a o implies I (Pce (f) Co (f)I < EP, then for a _..... a o , we have (

—

Icoa (g)— 9(g)I -‘. I9 a (g)— 9,c (f)I + Icp„(f)— co (DI + Iyo (f)— co (g)I

11c9a1111f—g11 + E/ 3 + II (PII lif- 911 < E. Thusto' ,, a,3 ae A converges to (p in the w*-topology. 1111 1.22 Definition The unit ball of a Banach space X is the set {f e X : Ivo -..‹.. 1} and is denoted (ff), . 1.23 Theorem (Alaoglu) The unit ball (**), of the dual of a Banach space is compact in the w*-topology. Proof The proof is accomplished by identifying (ff*), with a closed subset of a large product space the compactness of which follows from Tychonoff's theorem (see [71]). For each/ in (ff), let C I f denote a copy of the closed unit disk in C and let P denote the product space X fE(x) , Cff. By Tychonoff's theorem P is compact. Define A from (.T*), to P by A(9) = 9 1 (ff) 1 - Since A(cp 1 ) = A(49 2) implies that the restrictions of co, and (p 2 to the unit ball of .T are identical, it follows that A is one-to-one. Further, a net {9,}a.4 in .T* converges in the w*-topology to a co in ff* if and only if lim a , coa (f) = co (f) for f in A' if and only if firn, . A (N(f) = 9(f) forf in (d) i if and only if Iim„ E A A(TO (f) = A((p)(f) forf in (T) 1 . This latter statement is equivalent to Iim„ E A A(acc) = AM in the topology of P. Thus, A is a homeomorphism between (.q(*), and the subset A [(1. *),] of P. We complete the proof by showing that A [(ff*),] is closed in P. Suppose {A((pa)}„ , A is a net in A [(T*) 1 ] that converges in the product topology to t/i in P. If f, g, and f+ g are in (ff) „ then J/ (f g) = lim A(9„)(f+ g) = lim A(9„) (f) + lim A(coj (g) aeA

= Ign +

aeA

aeA

1,14g).

Further, if f and If are in (.1) 1 , then (Af ) = Hill A(Coa)(Af) = lim c° aeA

aeA

= A litn Coct(f) = ilk(1)aeA

The Alaoglu Theorem II

Hence IA determines an element of (.T*) 1 by the relationship i(f) = forfin . Since /(f) = (f) for f in (ff)„ we see not only that kif is in (ff *) 1 but, in addition, A(1;) = fr. Thus A [(T*),] is a closed subset of P, and therefore (ff*), is compact in the w*-topology.

•

The importance of the preceding theorem lies in the fact that compact spaces possess many pleasant properties. We shall also use it to show that every Banach space is isomorphic to a subspace of some C(X). Before doing this we need to know something about how many continuous linear functionals there are on a Banach space. This and more is contained in the Hahn-Banach theorem. Although we are only interested in Banach spaces in this chapter, it is more illuminating to state and prove the Hahn-Banach theorem in slightly greater generality. To do this we need the following definition. 1.24 Definition Let 6 be a real linear space and p be a real-valued function defined on 6. Then p is said to be a sublinear functional on 6 if p(f+ < p(f) + p(g) for f and g in e and p(ill) = 2p(f) for f in and positive A,.

Let e be a real linear space and let p be a sublinear functional on e.. Let S be a subspace of e and cp be a real linear functional on such that yo (f) < p (f) for f in .97 . Then there exists a real (f) for f in g and (1)(g) p(g) linear functional el on e such that Co(f) for g in e. 1.25 Theorem (Hahn Banach)

{14. Take f Proof We may assume without loss of generality that not in and let g (g+21.f:fle IR, g E ,}. We first extend cr, to g and to do this it suffices to define OW appropriately. We want (1)(g + )f) < p(g + for all Ain ER and gin .97. Dividing by 1/1,1 this can be written 1(f-h) < p (f- h) and 44-f+ h) < p(-f+ h) for all h in g or equivalently, -p(-f+ h) + c0 (h) < (D(f) <

h) + TO)

for all h in g. Thus a value can be chosen for (1)(f) such that the resultant c1) on g has the required properties if and only if sup { -p(-f h) + o(h)} < inf fp (f - k) + yo(k)}.

he,"

However, for h and k in ..97 , we have (h)

(k) = (h k) < p(h - k) < p (f - k) p (h -f),

12 1 Banach Spaces

so that

— pl(h —f) + 9(h) p(f k) 9(k). Therefore, 9 can be extended to D on g such that 1(h) < p (h) for h in g. Our problem now is to somehow obtain a maximal extension of 9. To that end let g denote the class of extensions of 9 to larger subspaces satisfying the required inequality. Hence an element of g consists of a subspace g of 8 which contains .9°F. and a linear functional O w on g which extends co and satisfies ay (g) p(g) for g in g. There is a natural partial order defined on g, where (gi, Ow i ) < 0 such that Th + (W)

8

=

{

f E : II f— Th II < e}

c

clos {7I(X)01.

Therefore, (g), c — Th±clos{TE(T)0} c clos{TRAI N E so that (g) 1 clos{ TRAIT, where r = 2N/s. Except for the fact that this is the closure, this is what we need to prove. Thus we want to remove the closure. Let f be in (Y) 1 . There exists g 1 in (%),. with II f-- Tg i II < 1. Since f— Tg 1 is in Min, there exists g 2 in (X),. /2 with II f— Tgi Tg2I1 < Since f—Tg i —Tg 2 is in Mi/4, there exists g 3 in (g),. /4 with Ilf—Tg l — Tg 2— Tg 3 I1 l

ilr1

.-- ' ' -. - - -. " a . i -

s

I --

II, S[f] --, Tg for g in [f]. Since g 1 and g 2 in [f] imply that g i —g 2 is in mil, we have Tg 1 = Tg 2 and hence S is well defined. Obviously, S is linear and

the inequality

. Mil < II TO inf HMI = II TO HUH lInfill = inf ge[f] ge[f]

which holds for [f] in fffrit, shows that S is bounded. Moreover, if S[f] ---= 0, then Tf = 0, which implies that f is in . and [ f] = [0]. Therefore, S is one-to-one. Lastly, S is onto, since T is, and hence the preceding theorem demonstrates that S is an open map. Since the natural homomorphism n from I" to Xi., is open and T = Sir, we obtain that T is open. 1111 We conclude this chapter with some classical examples of Banach spaces due to Lebesgue and Hardy. (It is assumed in what follows that the reader is familiar with standard measure theory.) 1.44 The Lebesgue Spaces Let p be a probability measure on a a-algebra

99 of subsets of a set X. Let . 1 denote the linear space of integrable complex functions on X with pointwise addition and scalar multiplication, and let .A be the subspace of null functions. Hence, a measurable function f on X is in Y 1 if Lai dp < Co and is in ./11 if ix vi du = O. We let E denote the quotient space .29 1 /.A1 with the norm II [f] II 1 — fx If I dit. That this satisfies the properties of a norm (that is, (1)-(3) of Section 1.1) is easy; the completeness is only slightly more difficult. Let {Efi}ncl- I be a sequence in V such that Lic°-= 1 II [fn] II 1 < M < cic) Choose representatives f, from each Un ]; then the sequence fEnN-, 1 ILliicv°= 1 is an increasing sequence of nonnegative measurable functions having the property that the integrals N

fx



N

E 1II Uni II i <M E If i ) 641 -- n=1

n (n = 1

24 1 Banach Spaces

are uniformly bounded. Thus, it follows from Fatou's lemma that the function h n= 1 Ifn i is integrable. Therefore, the sequence {E:L I f„} ,T, 1 converges almost everywhere to an integrable function k in 2 1 . Finally, we have that N

[k] —

n=1

ELI

1

fX

Efn

co

—

fn

dp, 00

E

n.=-N-F1 JC

LT_ u ]

N

cc

ILI d 0, then show that there exists 9 in X* such that 9(y) = 0 for y in ./V,

9(x) . 1, and III ii --- lid1.15 Show that if we define the function AO = 9(f) for f in X and co in ff*, then/is in X** and that the mapping f --> j is an isometrical isomorphism of .q. into X**.

Exercises 29

Definition A Banach space is said to be reflexive if the image of X is all of X**. 1.16 Show that X is reflexive for .T finite dimensional but that none of the spaces c o (11, / 1 (1 + ), / "(Z + ), CO, 1]), and Li ([0, 1]) is reflexive. 1.17 Let X and g be Banach spaces. Define the 1-norm

Ilfeglli = 11111 + Ilg I

and the co-norm Ilfe gll. -- sup { VII IIgll } on the algebraic direct sum e g. Show that X eg is a Banach space with respect to both norms and that the conjugate space of .Teg with the 1-norm is ff* 0 (* with the co-norm. 1.18 Let X and g be Banach spaces and II II be a norm on X e g making it into a Banach space such that the projections 7r, : X (1) g -+ X and n 2 : X 0 g -4 g are continuous. Show that the identity map between X C) g in the given norm and X 0 g with the 1-norm is a homeomorphism. Thus the norm topology on X eg is independent of the norm chosen. 1.19 (Closed Graph Theorem) If T is a linear transformation from the Banach space X to the Banach space Y such that the graph {(f, Tf) : f E X } of T is a closed subspace of X eg, then T is bounded. (Hint: Consider the map f -9 (f Ti)-) 1.20 (Uniform Boundedness Theorem) Let X be a Banach space and {con }:= 1 be a sequence in ff* such that sup { I co n (f)I : n E il < CO for f in ff. Show that sup 111Conli : n c 1+ 1 < cc.* (Hint: Baire category.) 1.21 If X is a Banach space and {y9„}„DD_ 1 is a sequence in'* such that {49,,(.f)}„"= 1 is a Cauchy sequence for each f in ff, then lim„_ co n exists in the w*-topology. Moreover, the corresponding result for nets is false. 1.22 Show that if X is a Banach space and 9 is a (not necessarily continuous) linear functional on ff, then there exists a net {(p,},, EA in 1* such that lim„ EA co„(f) . 9(f) for f in X. 1.23 Let X and g be Banach spaces and T be a bounded linear transformation from A' onto g. Show that if A = ker T, then Xi., is topologically isomorphic to g. ,

Definition If ff is a Banach space, then the collection of functions {q E r} defines a weak topology on X called the w-topology. 1.24 Show that a subspace A of the Banach space X is norm closed if and only if it is w-closed. Show that the unit sphere in X is w-closed if and only

if X is finite dimensional.*

30 1 Banach Spaces

1.25 Show that if X is a Banach space, then X is w*-dense in X**. 1.26 Show that a Banach space X is reflexive if and only if the w- and w*topologies coincide on X*. 1.27 Let X and g be Banach spaces and T be in 2(X, g). Show that if is in g* then f p(Tf) defines an element of X*. Show that the map T* = is in Q(J* , X*). (The operator T* is called the adjoint of T.) 1.28 If X and g are Banach spaces and T is in 2.(X, g), then T is one-toone if and only if T* has dense range. 1.29 If X and g are Banach spaces and T is in QM &), then T has a closed range if and only if T* has a closed range. (Hint; Consider first the case when T is one-to-one and onto.)

Definition If .11 is a subspace of the Banach space X, then the annihilator = {9 E .T * C9(X) =0 for X E X i of X is defined as 1.30 If X is a Banach space and Al is a closed subspace of X, then X* is naturally isometrically isomorphic to 1.31 If a" is a Banach space and .A is a subspace of X*, then there exists a subspace X of X such that X i Ai if and only if .Ac is w*-closed. 1.32 If the restriction of a linear functional on the Banach space M(X) of complex regular Borel measures on X to the unit ball of M(X) is continuous in the relative w*-topology, then there exists a function f in C(X) such that 9 (p) f f dp

for p in M(X).*

(Hint: Obtain/ by evaluating cp at the point measure 6„ at x and use the fact that measures of the form El= al b x. for r. 1 i(xii 1 are w*-dense in the unit ball of M(X).) 1.33 (Grothendieck) A linear functional 9 in X** is w*-continuous if and only if the restriction of 9 to (X*), is continuous in the relative w*-topology.* (Hint: Embed X in C(X), extend 9 to M(X) via the homomorphism from M(X) to M(X)/X'- = X*, and show that the function f obtained from the preceding problem is in X.) 1.34 (Krein—Srnur yan) If X is a Banach space and Al is a subspace of then X is w*-closed if and only if X n (X*), is w*-closed.* (Hint: Show that A' is the intersection of the zero sets of a collection of w*-continuous linear functionals on X*.)

Exercises 31

1.35 (Banach) If X is a separable Banach space and ." is a subspace of X*, then dl is w*-closed if and only if It is w*-sequentially complete. 1.36 Let X and g be Banach spaces and X ® g be the algebraic tensor

product of X and ( as linear spaces over C. Show that if for w in X0 g a

we define bid IIwII = inf{E 1141 1

E

Xn E

W

E x,e y,

then II 'If is a norm on X ®g. The completion of X 0 is the projective a

tensor product of X and g and is denoted X &.

a

1.37 Let X and g be Banach spaces and X ®' be the algebraic tensor a

product of X and g as linear spaces over C: Show that if for w in X 0 g a

we define

OwIli --= sup

E co(xi)yi) : x i , ...,xn e x; yi,...,y. E w ; i-li E ( 1** )1;

0*)1 ; w E ® Yi 1= 1

then II ' II i is a norm on X g. The completion of X 0 g is the inductive a

tensor product of X and g and is denoted ao g

a

1.38 For a' and g Banach spaces, show that the identity mapping extends to a contractive transformation from X Qg to X g 1.39 For X and Y compact Hausdorff spaces show that C(X) g C(Y) C (X x Y). (Hint: Show that it is sufficient in defining If Il i to take cp and Ili to be extreme points of the unit ball of X* and g*.) 1.40 For X and Y compact Hausdorff spaces show that C(X) p C(Y) = C(Xx Y)

if and only if X or Y is finite. (Hint: show that there are functions h(x, y) fi (x) g i (y) for which II hiL 1 but 111/11„ is arbitrarily large.) Thus the tensor product of two Banach spaces is not unique.

2

Bunnell Algebras

2.1 In Chapter 1 we showed that C(X) is a Banach space and that every Banach space is, in fact, isomorphic to a subspace of some C(X). In addition to being a linear space, C(X) is also an algebra and multiplication is continuous in the norm topology. In this chapter we study C(X) as a Banach algebra and show that C(X) is a "universal" commutative Banach algebra in a sense which we will later make precise. We shall indicate the usefulness and power of this result in some examples. 2.2 Recall that in Section 1.1 we observed that C(X) is an algebra over C with pointwise multiplication and that the supremum norm satisfies Ilfg11„ < lifil„ kil o, for f and g in C(X). These properties make C(X) into what we will call a Banach algebra. In the study of Banach spaces the notion of bounded linear functional is important. For Banach algebras and, in particular, for C(X) the important idea is that of a multiplicative linear functional. (We do not assume the functional to be continuous because we show later that such a functional is necessarily continuous.) Except for the zero functional, which is obviously both multiplicative and linear, every multiplicative linear functional co satisfies cp (1) = 1 since co * 0 means there exists an f in C(X) with yo (f) 0 0, and then the equation go (1) yo (f) = cp (f) implies cp (1) = 1. Thus we restrict our attention to the set Mc(x) of complex multiplicative linear functionals co on C(X) which satisfy cp (1) = I. For each x in X we define the complex function 32

The Banach Algebra of Continuous Functions 33

9„ on C(X) such that 9 x (f) f(x) for f in C(X). It is immediate that 9„ is in Mc(x) , and we let tA denote the mapping from X to Mc(x) defined tA(x) = 9„. The following proposition shows that tA maps onto Mc(x) . 2.3 Proposition The map tA defines a homeomorphism from X onto Moir , where Mc(x) is given the relative w*-topology on C(X)*. Proof Let 9 be in Mc(x) and set

R= ker 9 = ffE COO 9(f) = 0}. We show first that there exists x o in X such that f(x 0 ) = 0 for each f in R. If that were not the case, then for each x in X, there would exist fx in R such that fx (x) 0 0. Since fx is continuous, there exists a neighborhood [i x of x on which L 0. Since X is compact and {U,,}„ Ex is an open cover of X, there exists Uxi , U„N with X = UZ= U. If we set g = EZ= fax„ then 0 (g) = EnN (f xn) ri (.f ) = 0, implying that g is in R. But g 0 on X and hence is invertible in C(X). This in turn implies 9(1) = 9(g)• 9(11g) 0, which is a contradiction. Thus there exists x o in X such that f(x o ) = 0 for f in R. If f is in C(X), then f— (p(f) -1 is in R since 9(f— co (f) • 1) = 9(f)— 9(f) = 0. Thus f (x0) - cP (f) = (f- cp(f) • 1 )(x 0) = 0, since 9— 9 (f). 1 is in R and therefore 9 = 9„o Since each co in Mc(x) is bounded (in fact, of norm one), we can give Mc(x) the relative w*-topology on C(X)* and consider the map tA X ) Mc(x). If x and y are distinct points of X, then by Urysohn's lemma there exists fin C(X) such that f(x) f(y). Thus —

Vx)(./c) = 9x (f) = f(x) f(y) = co y (f) = VMD, which implies that tif is one-to-one. To show that IA is continuous, let fx,J„ EA be a net in X converging to x. Then limccEA.nx,r) = f(x) for f in C(X) or equivalently limA. ti/(;)(/) (x) (f) for each f in C(X). Thus the net {(x„)},,, A converges in the w*-topology to IA (x) so that IA is seen to be continuous. Since is a one-to-one continuous map from a compact space onto a Hausdorff space, it follows that Ji is a homeomorphism. This completes the proof. ■ We next state the definition of Banach algebra and proceed to show that the collection of multiplicative linear functionals on a general Banach algebra can always be made into a compact Hausdorff space in a natural way.

34 2 Banach Algebras

2.4 Definition A Banach algebra Z is an algebra over C with identity I which has a norm making it into a Banach space and satisfying 11 1 11 = 1 and the inequality Ilfg II < V II 11g11 for f and g in Z. We let A, denote the element of Z obtained upon multiplying the identity by the complex number A,. The following fundamental proposition will be used to show that the collection of invertible elements in 95 is an open set and that inversion is continuous in the norm topology. 2.5 Proposition If f is in the Banach algebra 8 and II I invertible and

—fll< 1, then f is

1

Ilf II < 1 — 11 1 —1

—

fII .

Proof If we set ii =-- II 1 —fll< 1, then for N .... M we have N

M

n=0

(1—nn

—

n=0

(1—nn

N

N

=--

E (1-f)"

n = M+ 1

n= M + 1

Ill —flIn

ti M +1

. I N

n= M+ 1

in < 1 _ ,1 .

and the sequence of partial sums fE nN= 0 (1 —fr} irNi )= 0 is seen to be a Cauchy sequence. If g = En.= 0 0 — fr, then '`

co

fg = [1 0

—

r = lim ([1— (1—f)] i (1 —f ) (1 —1)](E r (1—f) N , co -

= lim (1 — (1 - - f) N + 1 ) = N-+ .0

n=0

1,

since lim N , 11(1 —f) N ' II = 0. Similarly, gf= 1 so that f is invertible with f -1 — g. Further, 11911 = lim

Ni co

(1—f)" < lirn N 11n i l o (1—f 1 —f li= N-> co n=0

1 1 — Ill

— f II . III

2.6 Definition For R3 a Banach algebra, let ¶ denote the collection of invertible elements in 0 and let gii , respectively, gr denote the collection of left, respectively, right invertible elements in 8 that are not invertible. The following result will be of interest in this chapter only as it concerns g but we will need the results about g i and gr in Chapter 5 when we study index theory.

Abstract Banach Algebras 35

2.7 Proposition If 3 is a Banach algebra, then each of the sets g, and g,. is open in a3. Proof If f is in 9 and 111— g 11 < 1 11 f -1 11, then 1 > Ilf Ilf-gll 'g 111-f-1 g II. Thus the preceding proposition implies that f is in g and hence g = f(f -1 g) is in g. Therefore g contains the open ball of radius

1 /11f -1 11 about each element off in 5. Thus 5 is an open set in O. If f is in WI , then there exists h in a3 such that hf = 1. If Ilf-gll < 1 /11h11, then 1 > Ilhil Ilf — gil II/if-17 911 = II 1— hgli. Again the proposition implies that k = hg is invertible and the identity (k - 'h)g = 1 implies that g is left invertible. Moreover, if g is invertible, then h = kg -1 is invertible which in turn implies that f is invertible. This contradiction shows that g is in gi so that gi is seen to contain the open ball of radius 1/11h11 about f Thus 5' 1 is open.

The proof that gr is open proceeds in the same manner. III 2.8 Corollary If a3 is a Banach algebra, then the map on f f -1 is continuous. Thus, g is a topological group. Proof If f is in g, then the inequality

1 1—f

-1

gii

1AI

f=1 Al

1 — (1

_ f_A)

so that I —flA is invertible by Proposition 2.5. Thus f— A is invertible. Therefore, A is in p(f), a(f) is bounded and hence compact, and r(f) 111 2.29 Theorem If 0 is a Banach algebra and f is in R3, then a(f) is non-

empty. Proof Consider the function F p(f) 0 defined F(A) = (f— A)"' We show that F is an analytic 0-valued function on p (f) that is bounded at infinity and use the Liouville theorem to obtain a contradiction.

42 2 Banach Algebras

First, since inversion is continuous we have for .1. 0 in p (f) that

{FP) - F(10)} A -4A.0 A - )10 lim

(f MU A) 1

{(f- ilo) -1 E(.f- ilo) A-4 Ao A - Ao

= Hill

-

-

-

1

= lim (f— AO - 1 (f— Al l = V— Aol 2 A-a A o

In particular, for (19 in the conjugate space 0*, the function („9 (F) is a complex analytic function on p(f). Further, for IAI > II/II we have, using Proposition 2.5, that 1 -f1.1, is invertible and (1 - -f) A

, N

.,

)

an wn = a—") n x n (e 1

n=0 2n

0

n=0

1- we

-it

dt.

Proof Expand 1/(1 - we -1`) =En,',. 0 (we')m, where the series converges

uniformly for t in [0, 217]. Therefore,



c,

52 2 Banach Algebras

1

27r i'

27r Jo

1

1 N aD

2

anXn)fri) 1 — we n= 0

df = .>:

27r n-o

wm 27r e, "0 — not dt

an

m-0 wn,

since (1/27-0 jreikt dt = 1 for k = 0 and 0 otherwise.

■

For w in C, jwi < 1, define cp„ on 9+ such that

(

N

N

Cow E an xn) = E an wn. n=0

n=0

It is clear that p„, is a multiplicative linear functional on 9 + . However, since g+ is not a Banach algebra, that is, g+ is not complete, we cannot conclude apriori that cp,,„, is continuous. That follows, however, from the preceding lemma since

1

1I

1

N

Pw( il a n Zn1 -,n=0

n=0

r 27r

= a, wnl N

nY,n)(e)

I— we1- " 1

2n (n=o

dt



I dt. 1 —we - "I o I j 00 27r

Therefore, (p„ is bounded on 9+ and hence can be extended to a multiplicative linear functional on A. Now for w in C, 1 wi =-- 1, let (p„ denote the evaluation functional on A, that is, (p w (f)= f(w) for fin A. The latter is well defined, since A c C(T). Now set Iii --= (z e C : Izi ‘, 1}, let M denote the maximal ideal space of A, and li-t kli be the function from ® to M defined by /r (z) = ()o z . 2.52 Theorem The function tt/ is a homeomorphism of

a onto the maximal

ideal space M of A. Proof By the remarks preceding the theorem, the function tp is well defined. If z 1 and z 2 are in ED, then tp(z 1 ) = 1,(1(z 2 ) implies that z, = (p zi (x,) = (p z2 (x l ) = z 2 ; thus iii is one-to-one. if Ca is in M, then ))x f )) = I implies that z = cp(x f ) is in a. Moreover, the

identity N

(y

co an n=0

xn) = i °CR() n=-7-0

0( I A n N

N

= V Ot n n=0

Zn =

Coz ( an Xn n =0

The Disk Algebra 53

proves that (p agrees with (p z on the dense subset g + of A. Therefore, (p coz and 0 is seen to be onto M. Since both f and M are compact Hausdorff spaces and 0 is one-to-one and onto, to complete the proof it suffices to show that 0 is continuous. To this end suppose {zp } p E B is a net in ® such that lim e EB Zp = Z. Since suPp€BIliPz i, = 1 and g_F is dense in A, and since Jim Pzp(

fiEB

I o ,z„) E an Zn) lirn( /3E13 n=

Zn

;



n=0

N = Pz(I an Xn n=0 (

for every function En8=o an y,, in g+ , it follows from Proposition 1.21 that IP is continuous. ■ 2.53 From Proposition 2.3 we know that the maximal ideal space of C(T) is just T. We have just shown that the maximal ideal space of the closed subalgebra A of C(T) is n. Moreover, if (p z is a multiplicative linear functional on A, and Iz( = 1, then (p z is the restriction to A of the "evaluation at z" map on C(T). Thus the maximal ideal space of C(T) is embedded in that of A. This example also shows how the maximal ideal space of a function algebra is, at least roughly speaking, the natural domain of the functions in it. In this case although the elements of A are functions on 11, there are "hidden points" inside the circle which "ought" to be in the domain. In particular, viewing x i as a function on T, there is no reason why it should not be invertible; on (17, however, it is obvious why it is not—it vanishes at the origin. Let us consider this example from another viewpoint. The element x i is contained in both of the algebras A and C(T). In C(T) we have ac(T)(Xi) = T, while in A we have crA (x l ) = U. Hence not only is the "A-spectrum" of x i larger, but it is obtained from the COD-spectrum by "filling in a hole." That this is true, in general, is a corollary to the next theorem. 2.54 Theorem (Silov) If 93 is a Banach algebra, 9f is a closed subalgebra of 0, and f is an element of W, then the boundary of crw (f) is contained in the boundary of uz (f). Proof If (f— A) has an inverse in 91, then it has an inverse in 13. Thus a w (f) contains o 3 (f) and hence it is sufficient to show that the boundary of o-su (f) c oW). If A 0 is in the boundary of (r 9i (f), then there exists a sequence {A.,},T= 1 contained in p 9i (f) such that limn , co An . If for some integer -

54 2 Banach Algebras

11 it were true that 11(f— An)

-1

11 < f 1/(4— /101, then it would follow that

11(f— Ao) — (f— An)ii < lliiti— Ani l 11 and hence f— A o would be invertible as in the proof of Proposition 2.7. Thus we have limn-.Go 110e— Ani 9 = .. If Ao were not in uz (f), then it would follow from Corollary 2.8 that 11U— AY 1 V is bounded for A in some neighborhood of A0 , which is a contradiction. II 2.55 Corollary If $ is a Banach algebra, 9,1 is a closed subalgebra of B,

and f is an element of 91, then cr9i (f) is obtained by adding to crq3 (f) certain of the bounded components of qu z (f). Proof Elementary topology and the theorem yield this result. II 2.56 Example We next consider an example for which the Gelfand trans-

form is not an isometry. In Section 1.15 we showed that 1 1 (V) is a Banach space. Analogously, if we let / 1 (71) denote the collection of complex functions f on Z such that LT— — co I f(n) 1 < co, then with pointwise addition and scalar multiplication and the norm 1111 l = LT— — co IRO < co, 1 1 (71) is a Banach space. Moreover, 1 1 (71) can also be made into a Banach algebra in a nonobvious way. For f and g in / 1 (Z) define the convolution product ( f 0 g)(n) = i f(n — k) g (k). k = — co

To show that this sum converges for each n in Z and that the resulting function is in 1 1 (7/), we write n

foll)(n 1 = i1 . c

(

)

=

k

i l c f(n — k) g(k) < n i co k = — co ci

i c

co

l. fin — 101 I 0)1

i 10)1 in —1./0-101 = °fillk --=E— co10)1 — co

k= — co

= IlfIli °gill. Therefore, fog is well defined and is in 1 1 (71), and Ilfogll 1 -4 11f I I IIgh. We leave to the reader the exercise of showing that this multiplication is associative and commutative. Assuming this, then / 1 (1) is a commutative Banach algebra.

The Algebra of Functions with Absolutely Convergent Fourier Series 55

For n in 1 let e,, denote the function on 1 defined to be 1 at li and 0 otherwise. Then e 1 is the identity element of / I (1) and en o em = en± ,n for n and m in Z. Let M be the maximal ideal space of 1 1 (4 For each z in T, let co z be the function defined on 1 1 (71) such that (p z (f) = En"=„ f(n) z". It is easily verified that ca z is well defined and in M. Thus we can define a function from T to M by setting tfr(z) = (p z . 2.57 Theorem The function if, is a homeomorphism from T onto the

maximal ideal space M of / 1 (Z),

Proof If z 1 and z 2 are in T, and cp zi = (p z ,, then z 1 = (p z ,(e 1 ) --= (p z ,(e i ) = z 2 ; hence lk is one-to-one. Suppose (p is an element of M and

z = v(e 1 ); then 1

1

= 'kill'  140 (e1)1 = IA`

1

> = 1,

lz 11 1 Iv(e 1)1 Ile 111 which implies that z is in T. Moreover, since (p (en ) = [(p (e 1 )]" = z" = (p z (en ) for n in 1, it follows that co = co z =111(z). Therefore, again i is one-to-one and onto and it remains only to show that t/i is continuous, since both M and T are compact Hausdorff spaces. Thus, suppose {ze } 0EB is a net of points in T such that lim e EB Z ii --= Z. Then for f in / 1 (Z) we have - -

-

-

-

Icroz,(f) — (z WI In' 0. 2.62 Lemma If f is in if, then a (f) is a compact subset of C and A e M(f)}-

sump,' :

Proof If A o is not in M(f), then there exists e > 0 such that the set {x e X : 1 f(x) — AOI < 6} has measure zero. Clearly, then each A in the open

disk of radius E about A o fails to be in the essential range off. Therefore, the complement of a (f) is open and hence a (f) is closed. If Ao in C is such that A o I f(x) I + 6 for almost all x in X, then the set {x e X : I f(x) — A 0 1 < 612} has measure zero, and hence sup { I AI : A e 01) Thus a (f) is a compact subset of C for f in L". Now suppose f is in L" and no A satisfying 121 = IIf II is in M(f). Then about every such A there is an open disk D, of radius 6 2 such that the set {x e X : 1f(x)— Al < 8 2 ) has measure zero. Since the circle {A e C : 1AI = 11f110 is compact, there exists a finite subcover of open disks D2 1 , D 22 , D2. such that the sets {x e X : f(x) e D 2 } have measure zero. Then the set {x e X : f(x) E U ni= 1 D 2 } has measure zero, which implies that there exists an E > 0 such that the set {x e X : If(x) I > el has measure zero. This contradiction completes the proof. II 2.63 Lemma If f is in V°, then cr(f) = R(f).

Proof If A is not in cr(f), then 11(f— A) is essentially bounded, which implies that the set {x e X : 1 f(x) — Al < 1 /2 11 f— A11.0) has measure zero. Conversely, if {x e X : I f(x)— Al < 6) has measure zero for some b > 0, then 1/(f— A) is essentially bounded by 1/c5, and hence A is not in cr (f). Therefore (f) = Min. ■ 2.64 Theorem If M is the maximal ideal space of L", then the Gelfand transform IT is an isometrical isomorphism of if onto C(M). Moreover, If = [(f) for f in L" .

58 2 Banach Algebras

Proof We show first that F is an isometry. For f in Lc° we have, combining the previous lemma and Corollary 2.36, that range Ff = M(f), and hence II RD II m = sup WI : 2 e range

rf} =

sup flAl e

a WI VII co.

Therefore F is an isometry and F(L") is a closed subalgebra of C(M). For f in set f = fi -Fif2 , where each off,. and f2 is real valued. Since the essential range of a real function is real and range Ft; = (ft ) and range Ff2 = (f2 ), we have

rf = rfi + rf2 = TY; rf2 Therefore FL' is a closed self-adjoint subalgebra of C(M). Since it obviously separates points and contains the constant functions, we have by the StoneWeierstrass theorem that FL' = C(M) and the theorem is proved. II Whereas in preceding examples we computed the maximal ideal space, in this case the maximal ideal space is a highly pathological space having 220 points. We shall have reason to make use of certain properties of this space later on. 2.65 It can be easily verified that Iw(Z 1 ) (Section 1.15) is also a Banach algebra with respect to pointwise multiplication. It will follow from one of the problems that the Gelfand transform is an onto isometrical isomorphism in this case also. The maximal ideal space of Pc(r) is denoted )62 + and is called the Stone—tech compactification of Z ± . Notes The elementary theory of commutative Banach algebras is due to Gelfand [41] but the model provided by Wiener's theory of generalized harmonic analysis should be mentioned. Further results can be found in the treatises of Gelfand, Raikov and gilov [42], Naimark [80], and Rickart [89]. The determination of the self-adjoint subalgebras of C(X) including the generalization of the Weierstrass approximation theorem was made by Stone [105]. The literature on function algebras is quite extensive but two excellent sources are the books of Browder [10] and Gamelin [40].

Exercises Let g = {f e C([0, I]) : f' e C([0, 1])} and define Ilf Ild Ilf II + Show that g is a Banach algebra and that the Gelfand transform is neither isometric nor onto. 2.1

Exercises 59

2.2 Let X be a compact Hausdorff space, K be a closed subset of X, and

3 --= ffe C(X) :f(x) = 0 for x e K}. Show that 3 is a closed ideal in C(X). Show further that every closed ideal in C(X) is of this form. In particular, every closed ideal in C(X) is the intersection of the maximal ideals which contain it.* 2.3 Show that every closed ideal in g is not the intersection of the maximal

ideals which contain it. 2.4 Let a' be a Banach space and .2(T) be the collection of bounded linear

operators on X. Show that .2(g) is a Banach algebra.

2.5 Show that if A' is a finite (> 1) dimensional Banach space, then the

only multiplicative linear functional on £ (SC) is the zero functional. Definition An element T of .V(T) is finite rank if the range of T is finite dimensional. 2.6 Show that if X is a Banach space, then the finite rank operators form

a two-sided ideal in 2(ff) which is contained in every proper two-sided ideal. 2.7 Iff is a continuous function on [0, 1] show that the range of f is the

essential range off. 2.8 Let f be a bounded real-valued function on [0, 1] continuous except at the point 1. Let 91 be the uniformly closed algebra generated by f and

C([0 11). Determine the maximal ideal space of 91.* 2.9 If X is a compact Hausdorff space, then C(X) is the closed linear span of the idempotent functions in C(X) if and only if X is totally disconnected. 2.10 Show that the maximal ideal space of 1.,' is totally disconnected. 2.11 Let X be a completely regular Hausdorff space and B(X) be the space of bounded continuous functions on X. Show that B(X) is a commutative Banach algebra in the supremum norm. If PX denotes the maximal ideal space of B(X), then the Gelfand transform is an isometrical isomorphism of B(X) onto C(f3X) which preserves conjugation. Moreover, there exists a natural embedding f3 of X into igX. The space fiX is the Stone—tech compactification of X. 2.12 Let X be a completely regular Hausdorff space, Y be a compact Hausdorff space, and (p be a continuous one-to-one mapping of X onto a

60 2 Banach Algebras

dense subset of Y. Show that there exists a continuous mapping from flAr onto Y such that p = tp o (p. (Hint: Consider the restriction of the functions in C(Y) as forming a subalgebra of C(f3X))

2.13 Let 0 be a commutative Banach algebra and = {fe : I+ Afec9 for A e C}. Show that 91 is a closed ideal in 0.

Definition If B is a commutative Banach algebra, then tie : 1+A:ft for 2 e C} is the radical of 0 and 0 is said to be semisimple if = {O}.

2.14 If 0 is a commutative Banach algebra, then 91 is the intersection of the maximal ideals in 0.

2.15 If 93 is a commutative Banach algebra, then 0 is semisimple if and only if the Gelfand transform is one-to-one.

2.16 If 0 is a commutative Banach algebra, then 93/91 is semisimple. 2.17 Show that 12 ([0, 1]) C with the 1-norm (see Exercise 1.17) is a commutative Banach algebra for the multiplication defined by

[(f01)(gep)](t) = 1pf(t)+ itg(t)+ f(t—x)g(x)dx}10 Show that 1,1 ([0, 1]) e C is not semisimple.

2.18 (Riesz Functional Calculus) Let 0 be a commutative Banach algebra, x be an element of 0, n be an open set in C containing or(x), and A be a finite collection of rectifiable simple closed curves contained in n such that A forms the boundary of an open subset of C which contains o- (x). Let A (S2,) denote the algebra of complex holomorphic functions on D. Show that the mapping A

(p(z)(x—z)' dz

defines a homomorphism from A (n) to 0 such that for (p in A 0.

(7 (Cp (3)) =

(p(o- (x))

2.19 If 0 is a commutative Banach algebra, x is an element of 0 with (r(x) c 0, and there exists a nonzero co in A(SM) such that co (x) 0, then o- (x) is finite. Show that there exists a polynomial p(z) such that p(x)= 0. 2.20 Show that for no constant M is it true that EnN. -N Ian! < for all trigonometric polynomials p =E,,N= N anYn on T.

Exercises 61

2.21 Show that the assumption that every continuous function on T has

an absolutely convergent Fourier series implies that the Gelfand transform on / 1 (Z) is invertible, and hence conclude in view of the preceding problem that there exists a continuous function whose Fourier series does not converge absolutely. * Definition If B is a Banach algebra, then an automorphism on 93 is a continuous isomorphism from 2 onto 93. The collection of all automorphisms on 93 is denoted Aut(3). 2.22 If X is a compact Hausdorff space, then every isomorphism from C(X) onto C(X) is continuous. 2.23 If X is a compact Haus(lorff space and go is a homeomorphism on X, then (cly)(x) =f(yox) defines an automorphism (13 in Aut [C(X)]. Show that

the mapping 9 -*it, defines an isomorphism between the group Hom(X) of homeomorphisms on X and Aut[C(X)]. 2.24 If 91 is a function algebra with maximal ideal space M, then there is

a natural isomorphism of Aut(2I) into Hom(M). 2.25 If A is the disk algebra with maximal ideal space the closed unit disk,

then the range of Aut(A) in Hom(E) is the group of fractional linear transformations on ED, that is, the maps z ---> fl(z — a)/(1— az) for complex numbers a and fl satisfying lal < 1 and Ifil = 1.* Definition If 91 is a function algebra contained in C(X), then a closed subset M of X is a boundary for 21 if VII.. sup {Ifin# : m e MI for f in 91. 2.26 If 21 is a function algebra contained in C(X); M is a boundary for 91; ft , ...,f,, are functions in 21; and U is the open subset of M defined by

fx e X : If(x)I < 1 for i = 1, 2, ..., nl. Show either that M\U is a boundary for 91 or U intersects every boundary for 21. 2.27 (gilov) If 91 is a function algebra, then the intersection of all boundaries

for I is a boundary (called the gilov boundary for 21).* 2.28 Give a functional analytic proof of the maximum modulus principle for the functions in the disk algebra A. (Hint: Show that

1 f 2ir

Area) = — 27r

where the function kr (t). y,,,co_ _

09

0

kr (0 — 0 f(eill) dt,

rneint

is positive.)

62 2 Banach Algebras

2.29 Show that the Silov boundary for the disk algebra A is the unit circle. 2.30 Show that the abstract index group for a commutative Banach algebra contains no elements of finite order. 2.31 If 13 and 0 2 are Banach algebras, then 0 1 602 and $, g 0 2 are Banach algebras. Moreover, if 0 1 and 0 2 are commutative with maximal ideal spaces M 1 and M2, respectively, then M 1 x M2 is the maximal ideal space of both 0 1 60 2 and 0 1 0 0 2 (see Exercises 1.36 and 1.37 for the definition of O and j.) 2.32 If 0 is an algebra over C, which has a norm making it into a Banach space such that Ilfgli -..C. II f II ligli for f and g in $, then 0 (i) C is a Banach algebra in the 1-norm (see Exercise 1.17) for the multiplication

(fZDA)(g@ii) = (fg+ Ag+ pp@ ilit with identity °Cll. 2.33 If 9 is a multiplicative linear functional on 3, then (p has a unique extension to an element of Mq3 433,c . Moreover, the collection of nonzero multiplicative linear functionals on 0 is a locally compact Hausdorif space. 2.34 Show that L1 (1I) is a commutative Banach algebra without identity for the multiplication defined by (f 0 g)(x) = f lf(x— t) g(t) dt

for f and g in Ll. (IMF).

2.35 Show that for t in III the linear function on L1 (IR) defined by apt (f) = 5 f(x) e" dx

is multiplicative. Conversely, every nonzero multiplicative linear functional in ii. (IMF) is of this form. * (Hint: Every bounded linear functional on Ll. (R) is given by a 9 in Lm (R). Show for f and g in Ll. (R) that co foo f(x — t) g(t) f(p(x— t) (p (t) — 9(x)} dt dx = 0

and that implies 9(x— t) 9 (t) = (p (x) for (x, t) not in a planar set of Lebesgue measure.) 2.36 Show that the maximal ideal space of Li (R) is homeomorphic to FR and that the Gelfand transform coincides with the Fourier transform.

3

Geometry of Hilbert Space

3.1 The notion of Banach space abstracts many of the important properties of finite-dimensional linear spaces. The geometry of a Banach space can, however, be quite different from that of Euclidean n-space; for example, the unit ball of a Banach space may have corners, and closed convex sets need not possess a unique vector of smallest norm. The most important geometrical property absent in general Banach spaces is a notion of perpendicularity or orthogonality. In the study of analytic geometry we recall that the orthogonality of two vectors was determined analytically by considering their inner (or dot) product. In this chapter we introduce the abstract notion of an inner product and show how a linear space equipped with an inner product can be made into a normed linear space. If the linear space is complete in the metric defined by this norm, then it is said to be a Hilbert space. This chapter is devoted to studying the elementary geometry of Hilbert spaces and to showing that such spaces possess many of the more pleasant properties of Euclidean nspace. We will show, in fact, that a finite dimensional Hilbert space is isomorphic to Euclidean n-space for some integer n.

3.2 Definition An inner product on a complex linear space £f is a function (p from .2' x 2' to C such that : (1) cp(a l f, ± a 2 /2 ,g) = a l (p(fl ,g)+ a 2 cp(f2 ,g) for c( 1 ,c( 2 in C and ft,f2, g in 2'; 63

64 3 Geometry of Hilbert Space

(2) 9(f, fl i g i + fl2 112 ) = p, cp(f, gi ) + 13 2 9(f, g2 ) for Ill, /3 2 in C and .1, gi, 92 in - 2 2 ; (3) 9(f, g) = 9(g,f) for f and g in'; and (4) 9(f,f) ..-- 0 for f in ..,22 and 9(f,f) = 0 if and only if f = 0. A linear space equipped with an inner product is said to be an inner product space. The following lemma contains a useful polarization identity, the importance of which lies in the fact that the value of the inner product 9 is expressed solely in terms of the values of the associated quadratic form tfr defined by iii(f) = 9 (f,f) for f in .2 9 . 13 Lemma If .2 2 is an inner product space with the inner product go, then

q (f, g) = 'I f go (f+ g, f+ g) - q (f- g, f- g) + kp(f+ ig, f+ ig) - i9(f-ig, f- ig)} for f and g in ..25' .

Proof Compute. ■ An inner product is usually denoted (,), that is, (f,g) = 9(f,g) for f and g in P. 3.4 Definition If .2 2 is an inner product space, then the norm II II on Y associated with the inner product is defined by 11f11 =-_.(f,f)" for f in -CP . The following inequality is basic in the study of inner product spaces. We show that the norm just defined has the required properties of a norm after the proof of this inequality. 3.5 Proposition (Cauchy-Schwarz Inequality) If f and g are in the inner product space .2 2 , then 1(fg)

1

11111 11g11.

Proof For f and g in Z' and A in C, we have 1 2 1 2 11911 2 + 2 Re [[(f, g)] +

11f11 2 = (f Filthf Filg) — 11f-1-411 2 O. -

-

Setting A = te e, where t is real and e i° is chosen such that e -ie (f, g) ..... 0, we obtain the inequality Ogre + 2 1(A9)1t + 11f11 2 o.

The Cauchy—Schwarz Inequality 65

Hence the quadratic equation 11911 2 t 2 + 2 001 t +

11f11 2 = 0

in I has at most one real root, and therefore its discriminant must be nonpositive. Substituting we obtain [2 Ki, 9)0 2 — 4 11g11 2 11f11 2

< 0,

from which the desired inequality follows. ■

3.6 Observe that the property (f,f) = 0 implies that f = 0 was not needed in the preceding proof.

3.7 Proposition If .22 is an inner product space, then II* II defines a norm on Y.

Proof We must verify properties (1)—(3) of Definition 1.3. The fact that VII = 0 if and only if f --= 0 is immediate from (4) (Definition 3.2) and thus

(1) holds. Since

Ililf II = (Af,iliP = (11)1 (1,D) % = IAI VII

for I in C and f in .29 ,

we see that (2) holds. Lastly, using the Cauchy—Schwarz inequality, we have II f+ g11 2 = (f+ g, f+ g) = (1,f) + (f, g) + (g,f) + (g,g)

= 11f 11 2 + 111111 2 + 2 Re (f, 0 < 11f11 2 + 11911 2 + 2 l(f, g) 1 < 11f11 2 + 11911 2 + 2 11f11 114 < (VII + 11911) 2 for f and g in .22 . Thus (3) holds and II' II is a norm. ■

3.8 Proposition In an inner product space, the inner product is continuous. Proof Let .29 be an inner product space and ffa l,„ A and {gG}aEA be nets in .29 such that lim„EAL =f and limaEA the = g. Then

I(i,g) — (L, ga)I < 1(f—fa, 9) 1 + l(fo,,g—g.)1

< Ilf—fall 11911 + Ilfall 119 — gall , and hence lim„ EA (f,,, go) --= (I, g). ■

3.9 Definition In the inner product space ..T two vectors f and g are said to be orthogonal, denoted f 1 g, if (f, g) = 0. A subset „V of 2' is said to be orthogonal if f 1 g for f and g in 99 and orthonormal if, in addition, VII ---I for f in 99.

66 3 Geometry of Hilbert Space

This notion of orthogonality generalizes the usual one in Euclidean space. It is now possible to extend various theorems from Euclidean geometry to inner product spaces. We give two that will be useful. The first is the familiar Pythagorean theorem, while the second is the result relating the lengths of the sides of a parallelogram to the lengths of the diagonals. is an orthogonal

3.10 Proposition (Pythagorean Theorem) If {f i. J2 , subset of the inner product space .22 , then

2 ,=,

Proof Computing, we have

2

(E f-

i f

E = I i,1

= (fad =

+ I Vbfi) I



j

=

3.11 Proposition (Parallelogram Law) If f and g are in the inner product space 21 , then

Ilf+g11 2 + Ilf-g11 2

2

1If 1 2 2 1Ig11 2

Proof Expand the left-hand side in terms of inner products.

■

As in the case of normed linear spaces the deepest results are valid only if the space is complete in the metric induced by the norm.

3.12 Definition A Hilbert space is a complex linear space which is complete in the metric induced by the norm. In particular, a Hilbert space is a Banach space.

3.13 Examples We now consider some examples of Hilbert spaces. For n a positive integer let C" denote the collection of complex ordered n-tuples fx : x = (x 1 , x 2 , ..., x,), x, e C}. Then Cn is a complex linear space for the coordinate-wise operations. Define the inner product ( ,) on Cn such that (x,y) = En,_ i x, y,. The properties of an inner product are easily verified and the associated norm is the usual Euclidean norm II x 112 (En,= ix,12)Y2. To verify completeness suppose IxTcl i is a Cauchy sequence in Cn. Then



Examples of Hilbert Spaces: V, P, L 2 , and H 2 67

since 14' xim I < II xk — xi' 11 22 it follows that {e}iT=1 is a Cauchy sequence in C for 1 ...-c, i < n. If we set x = (x 1 , x2, • • • , xk), where xi = lim k ,,x, xik , then x is in C" and lim k , 00 xk = x in the norm of C". Thus C" is a Hilbert space. The space C" is the complex analog of real Euclidean n-space. We show later in this chapter, in a sense to be made precise, that the C"'s are the only finite-dimensional Hilbert spaces. —

3.14 We next consider the "union" of the C"s. Let .0 be the collection of complex functions on r which take only finitely many nonzero values. With respect to pointwise addition and scalar multiplication, .0 is a complex linear space. Moreover, (f, g) = EnaD_ o f(n) g(n) defines an inner product on 2', where the sum converges, since all but finitely many terms are zero. Is 22 a Hilbert space? It is if 2' is complete with respect to the metric induced by the norm 11111 2 = (fiT= o I f(n) 12)1/• Consider the sequence ffij ic,°. 1 contained in 2', where

10

fk (n) = (1)n

n < k, n > k.

One can easily show that Ifk U,I I is Cauchy but does not converge to an element of Y. We leave this as an exercise for the reader. Thus 2' is not a Hilbert space. 3.15 The space .0 is not a Hilbert space because it is not large enough. Let us enlarge it to obtain our first example of an infinite-dimensional Hilbert space. (This example should be compared to Example 1.15.) Let 1 2 (V) denote the collection of all complex functions cp on l + such that LT— o I co 0012 < CO. Then 12(7 +) is a complex linear space, since

1(1+ g) (n) 1 2 < 21 1(n) 1 2 + 21 g(n)1 2 . For f and g in 1 2 (71 + ), define (f, g) = LT_ 0 f(n) g(n). Does this make sense, that is, does the sum converge? For each N in 71 + , the n-tuples and FN = (if( 0 )1,1f( 1 )1, • • • 2 1 f(N)I) G N --= (19(0 )1,19( 1 )12-..,19(N)1) lie in C N . Applying the Cauchy—Schwarz inequality, we have

N Ifto gom = I(FN, GN)I < IIFNII IIGNII

n=0

= (i If(n)I 2 n=0

V2 (N

)

...) 1/2

E 190'01 2 < 11111 2 11g11 2.

n=0

68 3 Geometry of Hilbert Space

Thus the series I:_ o f(n)g(n) converges absolutely. That ( ,) is an inner product follows easily. To establish the completeness of 2 (1 + ) in the metric given by the norm 11 11 2, suppose {/* };:_ 1 is a Cauchy sequence in / 2 (71 + ). Then for each n in Z + , we have 1.ik(n) — fj(n)1

and hence {fk (n)}k 1 is a Cauchy sequence in C for each n in Z +. Define the function f on l + to be f(n) lirnk _.fk (n). Two things must be shown: that f is in / 2 (1 + ) and that lim k _,, 11 f—f 11 2 = 0. Since ffklT_ is a Cauchy sequence, there exists an integer K such that for k K we have II.fk —/K112 1 . Thus we obtain

n=o

If(n)1 2

N

o ifin) —Pc (T) I 21 % + N I PC (n)1 2} 1A

n=-

n=0

N

co



Ilfk — P11 2

lim ti 1 fk ( 1) — ft( (11)1 2 n=0

} + n=0 E ific(n)1 2

} Y2

limsup Ilfk —fig 2 + IlfK 11 2 < 1 + IIfK II 2, k-' co and hence f is in 1 2 (j + ). Moreover, given e > 0, choose M such that k,j M implies 11.fk —P II < E. Then for k M and any N, we have N

1E(n) —fk (n)1 2 f(n) —fk (n)1 2 urn .1–) co n=0 n=0

lim sup Ilff —fk II2 2 E 2 • j–) co

Since N is arbitrary, this proves that Ilf —fk 11 2 < E and therefore / 2 (1 + ) is a Hilbert space. 3.16 The Space L 2 In Section 1.44 we introduced the Banach spaces /41 and if based on a measure space (X, .99 , it). We now consider the corresponding L2 space, which happens to be a Hilbert space. We begin by letting 2 2 denote the set of all measurable complex functions f on X which satisfy fx J f J 2 < co . Since the inequality If g1 2 < 21f 1 2 + 2IgJ 2 is valid for arbitrary functions f and g on X, we see that ..99 2 is a linear space for pointwise addition and scalar multiplication. Let .A( 2 be the subspace of functions f in ..T 2 for which Sx if 12 du = 0, and let L2 denote the quotient linear space .222/./1(2.

Examples of Hilbert Spaces: C",1 2, L 2, and H 2 69

If f and g are in 27 2 , then the identity

ifgi + toll +10 2 1f12 — igi2}

shows that the function f# is integrable. If we define co (f, g) = fx f# dp for f and g in 27 2 , then (p has all the properties of an inner product except one; namely, co (f, f) = 0 does not necessarily imply f = 0. By the remark following the proof of the Cauchy-Schwarz inequality, that inequality holds for (p. Thus, if f, f', g, and g' are functions in 272 such that f—f' and g— g' belong to .A 2 , then

Is o (f, g) — c 0 U ' , 01 < lc 0 (f—F , 9)1 + lc 0 (I , g g')i .

—

< (09 or fc f n co (th 0

+ (I) (r, Jr) co (g gl, .. q ... g 1 ) ... 0.

Therefore, 9 is a well-defined function on L2 . Moreover, if (p([1],[f]) = 0, then jx 1/1 2 dp = 0 and hence [f] = [0]. Thus co is an inner product on L2 and we will denote it from now on in the usual manner. Furthermore, the associated norm on L2 is defined by Ili.f] 112 =0 x Ifi2 die. The only problem remaining before we can conclude that L2 is a Hilbert space is the question of its completeness. This is slightly trickier than in the case of Li . We begin with a general inequality. Take f in 27 2 and define g on X such that g(x) .--..-- f(x)/if(x)1 if f(x) # 0 and g(x) = 1 otherwise. Then g is measurable, 1 g(x)I = 1, and f# = I fl. Moreover, applying the Cauchy-Schwarz inequality, we have .

11f111 =x f Ifl clii = fx fP dii = 1(1; 01 < 11f11 2 11g11 2 = 11f11 2. Therefore, VII 1 11f11 2 for f in Y 2 We now prove that L2 is complete using Corollary 1.10. Let af,j},7. 1 be a sequence in L2 such that E c°-1 11[4]112 < M < cc. By the preceding inequality E;,°=111F.Q11 1 < M, and hence, by the proof in Section 1.44, there exists f in 27 1 such that E,T_ , fn (x) = f(x) for almost all x in X. Moreover .

N2

N

2

E f. di" < fX(1n=1 if.1)2 dii = Lt i ILI] 2

fX1n=1

k.

Since C([0, 1]) is dense in Li ([0, 1]) in the L'-norm, there exists for each k in 1 + , a function (pk in C([0, I]) such that lcp k (x)1 c k for x in [0, 1] and Ilfk— %II < 1/k 2 . Hence lim sup (f .fk V k 12 6) lim sup (k f k—) co

0

k--) co

0

9k1

lim sup f dx O. k.-)oo

k

Thus, C([0, 1]) is a dense subspace of L2 ([0, 1]) and hence the smallest closed subspace of L2 ([0, 1]) containing the functions {e 2 ' : n e 1} is L2 ([0, 1]). Therefore, fe 2 '1 7, Ez is an orthonormal basis for L2 ([0, 1]). Hence, dim (L2 ([0, 1])} _ o and therefore despite their apparent difference, 1 2 (V) and L2 ([0, 1]) are isomorphic Hilbert spaces. Similarly, since a change of variables shows that {x,,},, Ez is an orthonormal basis for L2 (T), we see that t y,,,I nez + is an orthonormal basis for H 2 and hence dim H 2 = N o also. We indicate in the exercises how to construct an example of a Hilbert space for all dimensions.

Notes The definition of a Hilbert space is due to von Neumann and he along with Hilbert, Riesz, Stone, and others set forth the foundations of the subject. An introduction to the geometry of Hilbert space can be found in many textbooks on functional analysis and, in particular, in Stone [104], Halmos [55], Riesz and Sz.-Nagy [92], and Akhieser and Glazman [2].

78 3 Geometry of Hilbert Space

Exercises 3.1 Let A be a nonempty set and let

/ 2 (A) = If: A ---> C :

If(a)1 2 < cot.

Show that 1 2 (A) is a Hilbert space with the pointwise operations and with the inner product (f, g) --- E„ GA f(a)g(a). Show that dim / 2 (A) = card A. 3.2 Let 2' be a normed linear space for which the conclusion of the parallel-

ogram law is valid. Show that an inner product can be defined on 2' for which the associated norm is the given norm. 3.3 Show that the completion of an inner product space is a Hilbert space. 3.4 Show that C([0, 1]) is not a Hilbert space, that is, there is no inner

product on C([0, 1]) for which the associated norm is the supremum norm. 3.5 Show that C([°, 1]) is not homeomorphically isomorphic to a Hilbert

space.* 3.6 Complete the proof began in Section 3.14 that the space 2' defined

there is not complete. 3.7 Give an example of a finite dimensional space containing a closed

convex set which contains more than one point of smallest norm.* 3.8 Give an example of an infinite dimensional Banach space and a closed

convex set having no point of smallest norm.* 3.9 Let co be a bounded linear functional on the subspace Jd of the Hilbert space Ye. Show that there exists a unique extension of co to Ye having the

same norm. 3.10 Let Ye and JC be Hilbert spaces and Yee X - denote the algebraic

direct sum. Show that

(0 1 , k i >, 01 2 , k2 >) = (12 1 , h 2 ) + (k i , k 2 ) defines an inner product on Ye IOX- , that YeCor is complete with respect to this inner product, and that the subspaces Ye ,CD {0} and WI e s are closed and orthogonal in Ye 0 if -

3.11 Show that each vector of norm one is an extreme point of the unit ball

of a Hilbert space. 3.12 Show that the w*-closure of the unit sphere in an infinite-dimensional

Hilbert space is the entire unit ball.

Exercises 79

3.13 Show that every orthonormal subset of the Hilbert space ?r is contained in an orthonormal basis for le. 3.14 Show that if II is a closed subspace of the Hilbert space Ye, then -. dim Ye. dim ./d ^. 3.15 (Gram-Schmidt) Let {f„} nw=1 be a subset of the Hilbert space .Ye whose closed linear span is Ye. Set e 1 = AIDA] and assuming {ek }Z =1 to have been defined, set n

en+ 1 -- (frt+ 1

—

E (i.+1, ek)ekV

k= 1

n

L-Fi—

E Un+1,ek

k=1

ek

9

where en+i is taken to be the zero vector if .

fi+ 1 =

(L+19ek)ekk= 1

Show that {en }:_ i is an orthonormal basis for Ye.

3.16 Show that L2 ([0, ID has an orthonormal basis fe,r_ o such that en is a polynomial of degree n. 3.17 Let 2' be a dense linear subspace of the separable Hilbert space ,_ff . Show that 2' contains an orthonormal basis for Ye. Consider the same question for nonseparable Ye.*

3.18 Give an example of two closed subspace ./d and .AI of the Hilbert space Ye for which the linear span

J12+.K= = {f+g:feJN,ge.)31 } fails to be closed.* (Hint: Take dl to be the graph of an appropriately chosen bounded linear transformation from X° to j( and .)1/ to be S- Ci{t)}, where Ye =---- yfoyir.)

3.19 Show that no Hilbert space has linear dimension N o . (Hint: Use the Bake category theorem.)

3.20 If Ye is an infinite-dimensional Hilbert space, then dimes coincides with the smallest cardinal of a dense subset of Ye. 3.21 Let Ye and if be Hilbert spaces and let Ye0X- denote the algebraic a tensor product of Ye° and' considered as linear spaces over C. Show that rn

n

i-i

.1=1

rn

E hi () ki , E hi ' 0 kl) = E E

(

irt

i=i j=1

(hi, hi ')(ki , ki')

80 3 Geometry of Hilbert Space

defines an inner product on .YeOfi. Denote the completion of this inner a

product space by APOIc. Show that if ea } aE A and {f0 } 0EB are orthonormal bases for .Ye and S", respectively, then tec,043 1 („, p)eAxB is an orthonormal basis for .ff(:),Y1. {

3.22 Let (X, 9', p) be a measure space with p finite and (p be a bounded linear functional on Li (X). Show that the restriction of co to L2 (X) is a bounded linear functional on L2 (X) and hence there exists g in L2 (X) such that co (f) = Ix fd dp for f in L2 (X). Show further that g is in If (X) and hence obtain the characterization of Li. (X)* as if(X). (Neither the result obtained in Chapter 1 nor the Radon—Nikodym theorem is to be used in this problem.) 3.23 (von Neumann) Let p and v be positive finite measures on (X, .9') such that v is absolutely continuous with respect to p. Show that f--+ f x f dp is well defined and a bounded linear functional on L2 (p + v). If co is the function in L2 (// + v) satisfying fx /0 d(p+v) = fx f dp, then (1 — (p)/9 is in Li (,ti) and v(E) = f

1—(P

E P

du

(

for E in .99 .

3.24 Interpret the results of Exercises 1.30 and 1.31 under the assumption that X is a Hilbert space.

Operators on Hilbert 4 Spam and C*-Algebras

t

4.1 Most of linear algebra involves the study of transformations between linear spaces which preserve the linear structure, that is, linear transformations. Such is also the case in the study of Hilbert spaces. In the remainder of the book we shall be mainly concerned with bounded linear transformations acting on Hilbert spaces. Despite the importance of certain classes of unbounded linear transformations, we consider them only in the problems. We begin by adopting the word operator to mean bounded linear transformation. The following proposition asserts the existence and uniqueness of what we shall call the "adjoint operator."

4.2 Proposition If T is an operator on the Hilbert space Yr, then there exists a unique operator

S on Ye such that

(Tf, g) — (f, Sg) for f and g in .e. Proof For a fixed g in "f consider the functional 'p defined co (f) = (Tf, g) for f in Ye. It is easy to verify that 'p is a bounded linear functional on Ye, and hence there exists by the Riesz representation theorem, a unique h in Ye such that (p (f) .--: (f, h) for fin If. Define Sg = h. Obviously S is linear and (Tf, g) = (f, Sg) for f and g in Ye. Setting f = Sg 81

82 4 Operators on Hilbert Space and C*-Algebras

we obtain the inequality

IIS911 2 = (Sg, Sg)

(TSg, g)

II TIlilsgli ligII

for g in Ye.

and S is an operator on P. Therefore, 11 S11 < To show that S is unique, suppose S o is another operator on Ye satisfying (f, So g) (Tf, g) for f and g in le. Then (f, Sg So g) 0 for f in Ye which implies Sg — So g = 0. Hence S So and the proof is complete. • 4.3 Definition If T is an operator on the Hilbert space Ye, then the adjoint of T, denoted T*, is the unique operator on Ye satisfying (Tf, g) = (f, T* g) for f and g in Ye. The following proposition summarizes some of the properties of the involution T T*. In many situations this involution plays a role analogous to that of the conjugation of complex numbers. 4.4 Proposition If Ye is a Hilbert space, then:

(1) T** = (T*)* T for T in .2(Ye); (2) 11 T11 = 11 T* I for T in 2(Ye); (3) (aS + fl T)* = as* - fl T* and (ST)* = T*S* for a, )6' in C and S, T in .2 (Ye) (4) (T *) 1 (T 1 )* for an invertible T in .2(Ye); and (5 ) 11 T11 2 = II T* TII for T in .2 (Ye). ;

Proof (1) If f and g are in Ye, then (f,T**g) = (T* f, g) (g, T* f) = (Tg, f) = (f, Tg), and hence T** = T. (2) In the proof of Proposition 4.2 we showed 11 T**11 11 T * II < II T II Combining this with (1), we have 11 T11 I T. (3) Compute. (4) Since T*(T 1 )* (T 1 T)* = I (TT 1 )* = (T 1 )* T* by (3), it follows that T* is invertible and (T*) (T 1 )*. (5) Since 11 T* Til < II T * II II T II =II T 11 2 by (2), we need verify only that 11 T * TI1 1 7 11 2 • Let fien l,c°_ 1 be a sequence of unit vectors in Ye such that lim n 11 TL 11 ---- 11 T11- Then we have -

-

IIT * TII

lim sup 11 T' T1 II n—) oo

which completes the proof. •

-

lim sup (T* Tf„,f,i) = lin' II TfnII 2 T 11 2 , n oo

co

The Adjoint Operator 83

4.5 Definition If T is an operator on the Hilbert space Ye, then the kernel of T, denoted ker T, is the closed subspace {f e Yr : Tf = 0}, and the range --X '77 c...4 w '77 IJI 1, uGUULcu 1 dal 1

I

.

Lisc buunictvc t 1 J . W cfe, I.

4.6 Proposition If T is an operator on the Hilbert space cYr , then ker T = (ran T*) 1 and ker T* (ran T) 1 . Proof It is sufficient to prove the first relation in view of (1) of the last proposition. To that end, if f is in ker T, then (T*g,f). (g, Tf) = 0 for g in cKP , and hence f is orthogonal to ran T*. Conversely, iff is orthogonal to ran T*, then (Tf, g) (f,T*g) — 0 for g in A', which implies Tf 0. Therefore, f is in ker T and the proof is complete. III

We next derive useful criteria for the invertibility of an operator. 4.7 Definition An operator T on the Hilbert space .1( is bounded below if there exists E > 0 such that II/ 11 for fin Ye4.8 Proposition If T is\En operator on the Hilbert space Ye, then T is invertible if and only if T is bounded below and has dense range. Proof If T is invertible, then ran T = Ye' and hence is dense. Moreover, IlTfll

1

IIT -1 TfIl =

I I -1 11

1

for fin Ye

II T 1 II

and therefore T is bounded below. Conversely, if T is bounded below, there exists E > 0 such that 11 Tf eilfil for f in P. Hence, if { Tf,},T= 1 is a Cauchy sequence in ran T, then the inequality

114 f.11 -

II T.f. Tf.11

, then implies fa°3_ 1 is also a Cauchy sequence. Hence, if f = Tf is in ran T and thus ran T is a closed subspace of Ye. If we P. Since T being assume, in addition, that ran T is dense, then ran T bounded below obviously implies that T is one-to-one, the inverse transformation T 1. is well defined. Moreover, if g Tf, then -

T

and hence

-

1

911 =

is bounded. IN

< -,11

=

F

-

I1911

84 4 Operators on Hilbert Space and C*-Algebras 4.9 Corollary If T is an operator on the Hilbert space AP such that both T and T* are bounded below, then T is invertible. Proof If T* is bounded below, then ker T* = {0}. Since (ran T) 1 = ker T* — {0} by Proposition 4.6, we have (ran T) 1 = (Oh which implies clos [ran 71 = (ran T) 11 . {0} 1 = A" by Corollary 3.22. Therefore, ran T is dense in Ybe and the result follows from the theorem. III 4.10 If T is an operator on the finite dimensional Hilbert space C" and {ei }7. 1 is an orthonormal basis for C", then the action of T is given by the matrix {chi }i j=1 , where aii = (Te l , e3 ). The adjoint operator T* has the matrix f b u nj ..= 1 , where b 1 . a ji for i, j ,------ 1, 2, ..., n. The simplest operators on C" are those for which it is possible to choose an orthonormal basis such that the corresponding matrix is diagonal, that is, such that a ii = 0 for i j. An operator can be shown to belong to this class if and only if it commutes with its adjoint. In one direction, this result is obvious and the other is the content of the so called "spectral theorem" for matrices. For operators on infinite dimensional Hilbert spaces such a theorem is no longer valid. Hilbert showed, however, that a reformulation of this result holds for operators on arbitrary Hilbert spaces. This "spectral theorem" is the main theorem of this chapter. We begin by defining the relevant classes of operators. ,

4.11 Definition If T is an operator on the Hilbert space (1) (2) (3) (4)

.re, then:

T is normal if TT* = T* T; T is self-adjoint or hermitian if T = T*; T is positive if (Tf,f).-- 0 for f in Ye; and T is unitary if T* T = TT* = I.

The following is a characterization of self-adjoint operators.

4.12 Proposition An operator T on the Hilbert space AP is self-adjoint if and only if (Tf,f) is real for f in A. Proof If T is self-adjoint and f is in Ye, then (T.f.n = (.f, T *D = (I; T.f) = (Tf.f) and hence (Tf,f) is real. If (Tf,f) is real for f in , then using Lemma 3.3, we obtain for f and g in Ye that (Tf, g) = (Tg,f) and hence T = T*. •

Normal and Self-adjoint Operators 85

4.13 Corollary If P is a positive operator on the Hilbert space A, then P is self-adjoint. Proof Obvious. 4.14 Proposition If T is an operator on the Hilbert space A'', then T* T is a positive operator.

Proof For f in Ye we have (T* Tf,f) = II Tf 11 2 0, from which the

result follows. ■

When we speak of the spectrum of an operator T defined on the Hilbert space A, we mean its spectrum when T is considered as an element of the Banach algebra .2(.X) and we use cr (T) to denote it. On a finite-dimensional space A is in the spectrum of T if and only if A is an eigenvalue for T. This is no longer the case for operators on infinite-dimensional Hilbert spaces. In linear algebra one shows that the eigenvalues of a hermitian matrix are real. The generalization to hermitian operators takes the following form.

4.15 Proposition If T is a self-adjoint operator on the Hilbert space .* ' , then the spectrum of T is real. Furthermore, if T is a positive operator, then the spectrum of T is nonnegative. Proof If A= a+ifl with a, p real and f3 # 0, then we must show that T — A is invertible. The operator K = (T — 01[3 is self-adjoint and T — A is invertible if and only if K — i is invertible, since K — i = (T — A)/$. Therefore, in view of Proposition 4.9, the result will follow once we show that the operators K — i and (K — i)* — K -Fi are bounded below. However, for f in Ye', we have

1 2 --- F- i(Kf,f) ± V, Ki) + 11f11 2 11Kf11 2 + 11f11 2 11f11 2

ii(K±O.P1 2 = ((K ± Of (K± Of) ----= 11Kf

and hence the spectrum of a self-adjoint operator is real. If we assume, in addition, that T is positive and A < 0, then ii ( 1' — A)f11 2 = IITfII 2 — 2A(Tf,f) +

2 2 11f11 2 A 2 11f11 2 •

Since (T — A)* = (T — A), then T — A is invertible by Proposition 4.9 and the proof is complete. • We consider now a special class of positive operators which form the building blocks for the self-adjoint operators in a sense which will be made clear in the spectral theorem.

86 4 Operators on Hilbert Space and C*-Algebras

4.16 Definition An operator P on the Hilbert space Ye is a projection if P is idempotent (P 2 P) and self-adjoint. The following construction gives a projection and, in fact, all projections arise in this manner. 4.17 Definition Let Yd be a closed subspace of the Hilbert space le. Define P to be the mapping P4 f g, where f= g+h with g in J/ and h in t'. 4.18 Theorem If ./N is a closed subspace of Ye, then Pai is a projection having range .1d. Moreover, if P is a projection on Ye, then there exists a closed subspace Yd(—ranP) such that P = Proof First we prove that P4,1 is an operator on Ye. If f1 ,f2 are vectors in Ye and A l , 12 complex numbers, then f1 = g1 +h1 and f2 = g2 +h 2 , where g 1 , g 2 are in A' and h i. , h 2 are in Ye'. Moreover Al f1 + 2 2 h (1 1 g 1 + A2 92) + (11 h 1 + 12 h2),

where Al g 2 +1 2 g 2 is in di' and (A i h 1 + 12 h2) is in Ye. . By the uniqueness of such a decomposition, we have PARA + it2f2) = 11g1 + 2 2 92 = A 1 PALA + 11 2 Pdft f2 and hence Pig is a linear transformation on Ye. Moreover, the inequality 11P.S1 11 2 = II II 2 11 9111 2 + II h l 11 2 = 11./111 2 shows that Pig is bounded and has norm at most one. Therefore, P is an operator on Ye. Moreover, since

— (91, g2 h2) (919 92) (91 + h1, 92) = P.4th), we see that PA is self-adjoint. Lastly, iffis in Yd, thenf= f+0 is the required decomposition of f and hence Pat f = f Since ran Pat = J/, it follows that PA 2 = P and hence PA is idempotent. Therefore P is a projection with range W. Now suppose P is a projection on ye and set ./d = ran P. If {Pf„},7., 1 is a Cauchy sequence in Ye converging to g, then g lim Pf, urn P 2f„ = PE lim Pfj = Pg. n

CO

n-;

CO

n-3 co

Thus g is in Yll and hence .A' is a closed subspace of Ye. If g is in Ye- , then pg y pg) "9 p 2g) u since P e g is in Yd, and hence Pg = 0. II

2 (Pg,

(g,

If f is in Ye, then f= Pg+h, where h is in .dil l and hence P 2 g + Ph = Pf. Therefore, P = 1111

Pit

Pg =

Multiplication Operators and Maximal Abelian Algebras 87

Many geometrical properties of closed subspaces can be expressed in terms of the projections onto them.

4.19 Proposition If AP is a Hilbert space, (XX = are closed subspaces of .Ye', and (Pi n_ 1 are the projections onto them, then P 1 + P2 + • • I if and only if the subspaces (X i n=i are pairwise orthogonal and span .1tP, that is, if and only if eachfin AP has a unique representationf f, +f2 + • • • +f,„ where fi is in .1

Proof If P 1 + P2 4- • • • +P„ = I, then each f in Ye has the representation P 1 f+ P2 f+ • • • +PH /. and hence the X i span Ye. Conversely, if the {Jf 1 }i=1 span le and the sumPi +P2 ± • • P„ is a projection, then it must be the identity operator. Thus, we are reduced to proving that Pi + P 2 + -•-+Pn is a projection if and only if the subspaces (X i n= 1 are pairwise orthogonal, and for this it suffices to consider the case of two subspaces. Therefore, suppose P 1 and P2 are projections such that P 1 +P 2 is a projection. For fin X j , we have ((Pi +P2).f.f) = ((Pi +P 2 ) 2f,f) (P1 P1 + (P 1 f, P2 + (19 2f, P1 + (P2f, P2 f) = (P1

.f9D

(A P2

((P1+ P 2)f.f)

f)

(P2

f9D + (P2 f9D

+ 2 (P2f.f),

since P 1 f f and thus 2 II P2 f11 2 2(P2 f, P 2f) = 2(P2 f,f) 0. Hence, f and therefore X 1 is orthogonal to M2 . Conversely, if P 1 and P2 are projections such that the range of P 1 is orthogonal to the range of P2, then for fin AP, we have (Pi + P2) 2f = (Pi + P2) Pi f + (Pi+P2)P2i= Pi 2f+ P1 2f

= (P + POI;

since P2 Pi f= PIP2f= O. ■ The proof shows that the sum of a finite number of projections onto pairwise orthogonal subspaces is itself a projection. We next consider some examples of normal operators other than those defined by the diagonalizable matrices on a finite-dimensional Hilbert space.

4.20 Example Let (X, , p) be a probability space. For co in If (p) define the mapping Mc, on L2 (p) such that Mc, f (pf for f in L2 (p), where (pfdenotes

88 4 Operators on Hilbert Space and C*-Algebras

the pointwise product, and let 9J1 (Mc : cp E L°°(p)}. Obviously /14 -,,, is linear and the inequality •

I m pf112

IW12 dp)

i

•

11.9 11f112

(11(pILIf1)2

shows that Mc, is bounded. Moreover, if En is the set {x E X:

ICO(X)I

then

=



IC04,,12 dp

r

110103

[fx (IbP — -1717 ) 2 (OWL

and hence

nI2

dpr

— —1 )11k112 n

C011.0 • For f and gin L2 (p) and go in L°°(p) we have

f,g) = (cof)g dp f f(C,dg)dp = (f MM g), which implies M,* = Mo . Lastly, the mapping defined by 'I' (co) Mc, from L°°(p) to 9)1 is obviously linear and multiplicative. Therefore, 4' is a *-isometrical isomorphism of L°°(p) onto 9)1. (The terminology *- is used to denote the fact that conjugation in Op) is transformed by 41 to adjunction in (I-2 (P))• ) Since If (p) is commutative, it follows that Mc, commutes with M9 * and hence is a normal operator. For co in L°°(p) the operator Mg, is self-adjoint if and only if Mc M,* and hence if and only if co = 0 or (p is real. Since N4 2 A4,p 2, the operator M 4, is idempotent if and only if co 2 (p or co is a characteristic function. Therefore, the self-adjoint operators in 931 are the M9 for which (p is real, and the projections are the Mc, for which co is a characteristic function. Let us now consider the spectrum of the operator Mc,. If co-- A is invertible in Lf(p), then Mc,— A = M ac, _ 1) is invertible in .2(L2 (p)) with inverse Mk,_ A) -1, and hence °WO c .1(4 To assert the converse inclusion we need to know that if Mc A is invertible, then its inverse is in 931. There are at least two different ways of showing this which reflect two important properties possessed by 9)1. , —

4.21 Definition If .Ye is a Hilbert space, then a subalgebra of VII is said to be maximal abelian if it is commutative and is not properly contained in any commutative subalgebra of 1( ).

The Bilateral Shift Operator 89

4.22 Proposition The algebra 971 = {M c, : cp c (p)} is maximal abelian. Proof Let T be an operator on L2 (pi) that commutes with WiL If we set T1, then 0 is in L2 (p) and Tip = TM,p 1 1119 T1 kkp for co in If(p). Moreover, if En is the set {x E X : 11P(x)i 11711 + lin}, then

11 Th1114.112

IITIE,,112 — 110 En112 — (f 11P/E,21 2 dp

)1/2

x

(

11 TI1 +n1)(f x 14.124) 1/2 071 + 1n)147,02.

Therefore, II/En112 = 0 and hence the set {x E X: I 0 (x) I > 11 711} has measure zero. Thus 0 is in L" ' (p) and lip = M ,fr cp for co in If (,u). Since C(X) is dense in L2 (p) as proved in Section 3.33 and C(X) c MO, it follows that T = M, is in 9)1. Therefore, Wil is maximal abelian. 1111 4.23 Proposition If AP is a Hilbert space and 91 is a maximal abelian subalgebra of 2(r), then o- (T) =-- o91 (T) for T in U. Proof Clearly or (T) cr9i (T) for T in 91. If A is not in o- (T), then (T — A) -1 exists. Since (T — A) -1 commutes with 91 and 91 is maximal abelian, we have (T — A) - 1 is in 91. Therefore A is not in or 9i (T) and hence cr 9i (T) =-a En- III

4.24 Corollary If co is in If (p), then o- (iii,p ) = M(co). Proof Since 9i1 = {14 -4, : c If(,u)} is maximal abelian, it follows from the previous result that o-9:,1 (T) c r (T). Since WI and L (p) are isomorphic we have crga (T) = o L .Q 0) (T) and Lemma 2.63 completes the proof. ■

We give the alternate approach after giving an example of a subalgebra 91 of 2(10 and an operator T in 21 for which o-91 (T) c (T). 4.25 Example Let l 2 (Z) denote the Hilbert space consisting of the complex functions f on 7L such that E;;0,_ _ I f(n)1 2 < oo . Define U on / 2 (1) such that (Uf)(n) f(n — 1) for f in / 2 (1). The operator U is called the bilateral shift. It is clearly linear and the identity

oo I( Un(n= 1 (1112 2 = J n )1 2 = 03

0o

Ifin— 0 1 2 11f11 2

for f in 1 2 (1) shows that U preserves the norm and, in particular, is bounded.

90 4 Operators on Hilbert Space and C*-Algebras

If we define A on / 2 (Z) such that (Af) (n) = f(n + 1) for f in / 2 (1), we have (Uf, g) =n i .0(Uf) (n) g(n) = ic f(n 1) g(n) n= — co := i f(n) n= — co

g(n +1) = (I; Ag).

Therefore, A = U* and a computation yields UU* = U* U = /or U -1 = U*. Thus U is a unitary operator. From Proposition 2.28 we have Q(U) c ® and Q(U -1 ) = of (U*) ®. If A is in n, 0 < IAI < 1, then (U—A)U -1 = 2((112)—U -1 ). Since 1/A is not in ®, the operator A ((l/A)— U -1 ) is invertible and hence so is U— A. Therefore a (U) is contained in T. For fixed 0 in [0,2n] and n in l + , let jer, be the vector in / 2 (Z) defined by fn (k) = (2n +1) - 1/2 e - " for WI < n and 0 otherwise. A straightforward calculation shows that f., is a unit vector and that lim„,,,(U — e` °) f,, = 0. Therefore U— el° is not bounded below, and hence e ° is in o- (U). Thus Q(U) = T. Let 21 + denote the smallest closed subalgebra of .2 (l 2 (1)) containing I and U. Then VI + is the closure in the uniform norm of the collection of polynomials in U, that is, N 9. f+ = ClOSII a n U n : an E n=0

(4.

If ./# denotes the closed subspace ff E 1 2 (1) : f(k) = 0 for k < 0}

of / 2 (1), then U./# c JN, which implies p(U)./d c .A for each polynomial p. If for A in 21+ we choose a sequence of polynomials fp,,g1 1 such that lirn„, o, pn (U) = A, then Af = limn , 09 pn (U)f implies Af in ./d, if f is, since each pn (U)f is in .A. Therefore A.A c dl for A in 21 + . We claim that U -1 is not in 91 + . If e 0 is the vector in / 2 (Z) defined to be 1 at 0 and 0 otherwise, then (U-1e0)(— 1) (U* eo )(— 1)

1 0 0.

Hence U -1 Jd c: J# which implies that U -1 is not in 21 + . Therefore, 0 is in o! 4 (U), implying o-94 (U) 0 o! (U). The algebra 2I 4 can be shown to be isometrically isomorphic to the disk algebra defined in Section 2.50 with U corresponding to x, and hence o- 94 (U) -,--- ®. We return to this later. The algebra 91 + is not maximal abelian, since it is contained in the obviously larger commutative algebra (M c, : cp E L"(1)}. Equally important

C*-Algebras 91

is that 91 + is not a self-adjoint subalgebra of 2(1 2 (1) , since, as we will soon establish, such algebras also have the property of preserving the spectrum. We consider the abstract analog of a self-adjoint subalgebra.

4.26 Definition If 91 is a Banach algebra, then an involution on 91 is a mapping T -÷ T* which satisfies: (i) T** = T for T in 91; = S* +PT* for S, T in 9I and a, )6' in C; and (ii) (aS+ (iii) (ST)* T* S* for S and T in 9f. If, in addition, II T* TII = II TI1 2 for T in 91, then 91 is a C*-algebra. A closed self-adjoint subalgebra of 2 (*) for ,_Ybe a Hilbert space is a C*-algebra in view of Proposition 4.4. Every C*-algebra can, in fact, be shown to be isometrically isomorphic to such an algebra (see Exercise 5.26). All of the various classes of operators whose definitions are based on the adjoint can be extended to a C*-algebra; for example, an element T in a C*-algebra is said to be self-adjoint if T T*, normal if TT* = T* T, and unitary if T* T = TT* = L We now give a proof of Proposition 4.15 which is valid for C*-algebras. Our previous proof made essential use of the fact that we were dealing with operators defined on a Hilbert space.

4.27 Theorem In a C*-algebra a self-adjoint element has real spectrum. Proof Observe first that if T is an element of the C*-algebra 91, then the inequality

I 1 2 = I * I

IIT * I 11 Th implies that II TII < " and hence II T = II T* since

T** = T. Thus

the involution on a C*-algebra is an isometry. Now let H be a self-adjoint element of 91 and set U exp ill. Then from the fact that H is self-adjoint and the definition of the exponential function, it follows that U* = exp(iH)* exp(— iH). Moreover, from Lemma 2.12 we have

UU* exp(iH)exp( iH) = exp (in — iH) = I = exp( iH) exp (in) = U* U —

and hence U is unitary. Moreover, since 1 II/ 11 = II U * Ull = I1U112 we see that II Ull = U * II 9 = 1, and therefore o- (U) is contained in T. Since cr(U) exp(io(H)) by Corollary 2.37, we see that the spectrum of H must be real. •

92 4 Operators on Hilbert Space and CLAIgebras

4.28 Theorem If 0 is a C*-algebra, 91 is a closed self-adjoint subalgebra of 0, and T is an element of 91, then cr9i (T) = cr Q3 (T). Proof Since cr9,(T) contains a (T), it is sufficient to show that if T — A is invertible in 0, then the inverse (T — A) 1 is in 91. We can assume A = 0 without loss of generality. Thus, T is invertible in 0, and therefore T* T is a self-adjoint element of 91 which is invertible in 0. Since cr9i (T* T) is real by the previous theorem, we see that o-91 (T* T) = o (T* T) by Corollary 2.55. Thus, T* T is invertible in 91 and therefore -

T

-

1

= (T

-

1

(T*)

-

1)

T* = (T* Ty 1 T*

is in 91 and the proof is complete. ■ We are now in a position to obtain a form of the spectral theorem for normal operators. We use it to obtain a "functional calculus" for continuous functions as well as to prove many elementary results about normal operators. Our approach is based on the following characterization of commutative C *-algebras. 4.29 Theorem (Gelfand Naimark) If 91 is a commutative C*-algebra and M is the maximal ideal space of 91, then the Gelfand map is a *-isometrical isomorphism of 91 onto C(M). -

Proof If F denotes the Gelfand map, then we must show that F(T) = T. The fact that F is onto will then follow from F(T*) and that II FM11.0 = II TII the Stone--Weierstrass theorem. If T is in 91, then f I = i(T + T*) and K (T— T*)12i are self-adjoint operators in 21 such that T = H+ iK and T* = H — 1K. Since the sets or (H) and or(K) are contained in R, by Theorem 4.27, the functions F(H) and IT(K) are real valued by Corollary 2.36. Therefore,

F(T) = F(H) + iF(K) = F(H) — iF(K) = F(H — 1K) = F(T*), and hence F is a *-map. To show that F is an isometry, let T be an operator in 91. Using Definition 4.26, Corollary 2.36, Theorem 2.38, and the fact that T* T is self-adjoint, we have °Tr = II T * T II = ii (T * n 2h r k = lim II(T * n 24 1 1Ak k--* oo

= IIRT*T)11.0 = 11F(T*)r(T)11c0 = 111FM1 2 11.0 = 111"(7)112-

The Functional Calculus 93

Therefore r is an isometry and hence a *-isometrical isomorphism onto C(M). 1111 If I is a commutative C* algebra and T is in 91, then T is normal, since T* is also in 91 and the operators in % commute. On the other hand, a normal operator generates a commutative C*-algebra. 4.30 Theorem (Spectral Theorem) If is a Hilbert space and T is a normal operator on AP, then the C*-algebra CT generated by T is commutative. Moreover, the maximal ideal space of CT is homeomorphic to a (T), and hence the Gelfand map is a *-isometrical isomorphism of E T onto C(cr(T)). Proof Since T and T* commute, the collection of all polynomials in T and T* form a commutative self-adjoint subalgebra of 2(YP) which must be contained in the C*-algebra generated by T. It is easily verified that the

closure of this collection is a commutative C*-algebra and hence must be CT- Therefore C T is commutative. To show that the maximal ideal space M of CT is homeomorphic to a(T), define from M to o (T) by “co) = F(T) (co). Since the range of F(T) is o (T) by Corollary 2.36, tp is well defined and onto. If (p i and (p2 are elements of M such that ;/ (co l ) = kli(c o 2 ), then -

-

F(T) (Co 1) = F(T)(CO2),

Col ( 1) = cp2(T),

and

co 1 (r) = r(T *)(Coi) = r(T)(Coi) = r(T)(CO2) = F(T *)(CO2)

(P2(T* ), and hence co i and co 2 agree on all polynomials in T and T*. Since this collection of operators is dense in C T , it follows that (p i = (p 2 and therefore /i is one-to-one. Lastly, if tcoaLEA is a net in M such that lim E A Coo, — Co, then lin 1 IP OP a) = li ln i"( 1) OP a) = r (T)( (P) = kW) aE A aE A and hence qi is continuous. Since M and a(T) are compact Hausdorff spaces, then 1p is a homeomorphism and the proof is complete. 111

4.31 Functional Calculus If Tis an operator, then a rudimentary functional calculus for T can be defined as follows: for the polynomial p(z) = EnN = 0 a n e, define p(T) =-= 0 anT. Tn. The mapping p --+ p(T) is a homomorphism from the algebra of polynomials to the algebra of operators. If .Ye is finite dimensional, then one can base the analysis of T on this functional calculus. In particular, the kernel of this mapping, that is, (p(z) : p(T) = 01, is a

94 4 Operators on Hilbert Space and C*-Algebras

nonzero principal ideal in the algebra of all polynomials and the generator of this ideal is the minimum polynomial for T. If .rt" is not finite dimensional, then this functional calculus may yield little information. The extension of this map to larger algebras of functions (see Exercise 2.18) is a problem of considerable importance in operator theory. If T is a normal operator on the Hilbert space A, then the Gelfand map establishes a *-isometrical isomorphism between C (cr (T)) and C T . For co in C(cr(T)), we define cp (T) F -1 cp. It is clear that if cp is a polynomial in z, then this definition agrees with the preceding one. Moreover, if A is an operator onP that commutes with T and T*, then A must commute with every operator in CT and hence, in particular, with cp (T) for each cp in C(0. (T)). In the remainder of this chapter we shall obtain certain results about operators using this calculus and then extend the functional calculus to a larger class of functions.

4.32 Corollary If T is a normal operator on the Hilbert space , then T is positive if and only if cr (T) is nonnegative and self-adjoint if and only if o(T) is real. Proof By Proposition 4.15, the spectrum of T is nonnegative if T is positive. Conversely, if Tis normal, cr (T) is nonnegative, and F is the Gelfand transform from to C (cr (T)), then F(T) 0. Thus there exists a continuous function co on r(T) such that f(T) = I (p 1 2 . Then T = (T)][(10 ( 1)] = 49 (T) * Co(T) and hence T is positive by Proposition 4.14. If T is self-adjoint, then the spectrum of T is real by Proposition 4.15. If T is normal and has real spectrum, then = I"(T) is a real-valued function by Corollary 2.36, and hence T (T) (T) (T)* = T*. Therefore, T is self-adjoint.

•

The preceding proposition is false without the assumption that T is normal, that is, there exist operators with spectrum consisting of just zero which are not self-adjoint. The second half of the preceding proposition is valid in a C*-algebra, while the first half allows us to define a positive element of a C*-algebra to be a normal element with nonnegative spectrum. We now show the existence and uniqueness of the square root of a positive operator.

The Square Root of Positive Operators 95

4.33 Proposition If P is a positive operator on the Hilbert space r, then there exists a unique positive operator Q such that Q 2 = P. Moreover, Q commutes with every operator which commutes with P.

Proof Since the spectrum o (P) is positive, the square root function .J/ is continuous on cr(P). Therefore .FP is a well-defined operator on le that is positive by Corollary 4.32, since crWil = Vo (P). Moreover, (f.i)) 2 = P -

-

by the definition of the functional calculus, and commutes with every operator which commutes with P by Section 4.31. It remains only to show that VP is unique. Suppose Q is a positive operator on Ye' satisfying Q 2 = P. Since QP = QQ 2 = Q 2 Q P Q , it follows from the remarks in Section 4.31 that the C*-algebra 2,1 generated by P, JP, and Q is commutative. If F denotes the Gelfand transform of 9t onto C(M9/ ), then r(/75) and F(Q) are nonnegative functions by Proposition 2.36, while r(. JP) 2 =r(P)= 1"(Q) 2 , since F is a homomorphism. Thus F(.JP) = F(Q) implying Q JP and hence the uniqueness of the positive square root. ■

4.34 Corollary If T is an operator on AP, then T is positive if and only if there exists an operator S on le such that T S*S. Proof If T is positive, take S VT. If T S*S, then Proposition 4.14 yields that T is positive. ■ Every complex number can be written as the product of a nonnegative number and a number of modulus one. A polar form for linear transformations on C" persists in which a positive operator is one factor and a unitary operator the other. For operators on an infinite-dimensional Hilbert space, a similar result is valid and the representation obtained is, under suitable hypotheses, unique. Before proving this result we need to introduce the notion of a partial isometry.

4.35 Definition An operator V on a Hilbert space Yf is a partial isometry if II vfII = IIf II for f orthogonal to the kernel of V; if, in addition, the kernel of V is (0}, then V is an isometry. The initial space of a partial isometry is the closed subspace orthogonal to the kernel. On a finite-dimensional space every isometry is, in fact a unitary operator. However, on an infinite-dimensional Hilbert space this is no longer the case.

96 4 Operators on Hilbert Space and C*-Algebras

Let us consider an important example of this which is related to the bilateral shift.

4.36 Example Define the operator U + on / 2 (1 + ) by (U + f)(n) f(n— 1) for n> 0 and 0 otherwise. The operator is called the unilateral shift and an easy calculation shows that U + is an isometry. Moreover, since the function e 0 defined to be 1 at 0 and 0 otherwise is orthogonal to the range of U + , we see that U+ is not unitary. A straightforward verification shows that the adjoint U+ * is defined by (U + *f)(n) f(n+1). Let us next consider the spectrum of U. Since U+ is a contraction, that is, II U+ 1, we have cr(U + ) c II Moreover, for z in ED the function L defined by fz (n)= z" is in l 2 (7l + ) and u.„*.fz . ffz . Thus z is in o (U + ) and hence o (U + ) = D. The question of whether a partial isometry exists with given subspaces for initial space and range depends only on the dimension, as the following result shows. .

-

4.37 Proposition If ./d and Al are closed subspaces of the Hilbert space Ye such that dim £ dim .K, then there exists a partial isometry V with initial space ./d and range .Ac.

Proof If ./N and .Ar have the same dimension, then there exist orthonormal bases fra L EA and {L} aeA for .A' and X with the same index set. Define an operator V on Ye as follows: for g in Ye write g h +L E A il a ea with h 1 di and set Vgf - ce• Then the kernel of V is JN -L and . Thus, V is a partial isometry with initial space Yll, IIV.9 II ---- ligii for g in and range • We next consider a useful characterization of partial isometries which allows us to define partial isometries in a C*-algebra.

4.38 Proposition Let V be an operator on the Hilbert space Ye. The following are equivalent:

(1) (2) (3) (4)

V is a partial isometry; V* is a partial isometry; VV* is a projection; and V*V is a projection.

Moreover, if V is a partial isometry, then VV* is the projection onto the range of V, while V*V is the projection onto the initial space.

The Polar Decomposition 97

Proof Since a partial isometry V is a contraction, we have for f in if that ((I — V * V)f f) (f,f) — (V* Vf, f)

11f11 2 - IIVfII 2

0.

Thus I— V*V is a positive operator. Now if f is orthogonal to ker V, then IIVfII = 11f11 which implies that ((/— V* V) f, = 0. Since II(/— V* VY/2./ 11 2 '='--((/— V* V)f, f) = 0, we have (I— V* V)f = 0 or V* Vf = f. Therefore, V* V is the projection onto the initial space of V. Conversely, if V* V is a projection and f is orthogonal to ker (V* V), then V* Vf -,---- f Therefore,

IIVfII 2 = (v*vf,f)

(f,f)

- 11f11 2 ,

and hence V preserves the norm on ker(V*V) ± . Moreover, if V* Vf = 0, then 0 = (V* Vf, f) = 11vf11 2 and consequently ker(V* V) = ker V. Therefore, V is a partial isometry, and thus (1) and (4) are equivalent. Reversing the roles of V and V*, we see that (2) and (3) are equivalent. Moreover, if V* V is a projection, then (VV*) 2 = V(V* V)V* = VV*, since V(V* V) = V. Therefore, VV* is a projection, which completes the proof. • We now obtain the polar decomposition for an operator.

4.39 Theorem If T is an operator on the Hilbert space Ye, then there exists a positive operator P and a partial isometry V such that T VP. Moreover, V and P are unique if ker P — ker V. Proof If we set P (T* T) 1/2 , then 11Pf11 2 = (Pf Pf) (P 2 ff) = (T * Tff) 11Tf11 2 for f in le. Thus, if we define V on ran P such that V Pf = Tf, then 17 is well defined and is, in fact, isometric. Hence, V can be extended uniquely to an isometry from clos [ran P] to Ye. If we further extend V to Ye by defining it to be the zero operator on [ran /3 ] 1 , then the extended operator V is a partial isometry satisfying T = VP and ker V = [ran P] 1 ----. ker P by Proposition 4.6. We next consider uniqueness. Suppose T = WQ, where W is a partial isometry, Q is a positive operator, and ker W = ker Q, Then P 2 = T* T QW*WQ = Q 2 , since W*W is the projection onto [ker W]' = [ker 0 1 = clos [ran 0 by Propositions 4.38 and 4.6. Thus, by the uniqueness of the square root, Proposition 4.33, we have P = Q and hence WP = VP. Therefore, W V

98 4 Operators on Hilbert Space and C*-Algebras

on ran P. But [ran P]' .---- ker P .---- ker W ker V, and hence W V on [ran P] '. Therefore, V W and the proof is complete. III Although the positive operator will be in every closed self-adjoint subalgebra of .2(Ye) which contains T, the same is not true of the partial isometry. Consider, for example, the operator T -, Mc, MM in .2 (L2 where co is a continuous nonnegative function on T while V/ has modulus one, is not continuous but the product 0/ is. In many instances a polar form in which the order of the factors is reversed is useful.

4.40 Corollary If T is an operator on the Hilbert space Ye, then there exists a positive operator Q and a partial isometry W such that T = QW. Moreover, W and Q are unique if ran W [ker Q]'.

Proof From the theorem we obtain a partial isometry V and a positive operator P such that T* .---- VP. Taking adjoints we have T = PV*, which is the form we desire with W = V* and Q = P. Moreover, the uniqueness also follows from the theorem since ran W = [ker 0 1 if and only if ker V = ker W* = [ran W]' = [ker 0' = ker P. ■ It T is a normal operator on a finite-dimensional Hilbert space, then the subspace spanned by the eigenvectors belonging to a certain eigenvalue reduces the operator, and these subspaces can be used to put the operator in diagonal form. If T is a not necessarily normal operator still on a finitedimensional space, then the appropriate subspaces to consider are those spanned by the generalized eigenvectors belonging to an eigenvalue. These subspaces do not, in general, reduce the operator but are only invariant for it. Although no analogous structure theory exists for operators on an infinitedimensional Hilbert space, the notions of invariant and reducing subspace remain important.

4.41 Definition If T is an operator on the Hilbert space Ye and .1% is a closed subspace of Ye, then ./d is an invariant subspace for T if TA c ./d and a reducing subspace if, in addition, T(./d 1 ) ./d 1 . We begin with the following elementary facts.



The Polar Decomposition 99

4.42 Proposition If T is an operator on *, ./W is a closed subspace of Ye , and Pat is the projection onto A, then ./d is an invariant subspace for T if and only if Pat TPA = TPA if and only if ./d ± is an invariant subspace for T*; further, Jd is a reducing subspace for T if and only if P T = TPA if and only if AI' is an invariant subspace for both T and T*. Proof If A' is invariant for T, then for f in AP, we have TPA f in ./N and hence Pal TPA f = TP.At f; thus Pig TPA = TPA . Conversely, if PA TP4,i = TP,g , then for f in A', we have 7f = TP,it j = PA TP,g f --,-- P Tf, and hence Tf is in 4'. Therefore, TA c A' and A' is invariant for T. Further, since

I— PA is the projection onto A" and the identity T* (I— px ) = (1— P,g )T* (I— PA) is equivalent to Pai T* = PA T* P t , we see that .t' is invariant for T* if and only if Jd is invariant for T. Finally, if Jd reduces T, then P T PA TPA TPA by the preceding result, which completes the proof. • 4.43 In the remainder of this chapter we want to extend the functional calculus obtained in Section 4.31 for continuous functions on the spectrum to a larger algebra of functions. This larger algebra of functions is related to the algebra of bounded Borel functions on the spectrum. Before beginning let us give some consideration to the uses of a functional calculus and why we might be interested in extending it to a larger algebra of functions. Some of the details in this discussion will be omitted. Suppose T is a normal operator on the Hilbert space Ye with finite spectrum cr(T) = fA 1 , A2, -, AN} and let co --* co (T) be the functional calculus defined for co in C(a(T)). If A, is a point in the spectrum, then the characteristic function /1;4 is continuous on c r(T) and hence in C(o(T)). If we let E1 denote the operator /tx , } (T), then it follows from the fact that mapping go --> co (T) is a *-isomorphism from C(o(T)) onto CT, that each Et is a projection and that E l + E2 + ••• + EN = I. If A' denotes the range of E1 , then the fAri=1 are pairwise orthogonal, their linear span is ,_e and A, reduces T by the preceding proposition. Moreover, since T = z(T) = {

N N A l 1(1 , 1 1(T) = I Al E1, ,- i 1=1

I

we see that T acts on each A as multiplication by A 1 . Thus the space Ye decomposes into a finite orthogonal direct sum such that T is multiplication by a scalar on each direct summand. Thus the functional calculus has enabled us to diagonalize T in the case where the spectrum of T is finite.

100 4 Operators on Hilbert Space and C*-Algebra

If the spectrum of T is not discrete, but is totally disconnected, then a slight modification of the preceding argument shows that A' can be decomposed for such a T into a finite orthogonal direct sum of reducing subspaces for T such that the action of T on each direct summand is approximately (in the sense of the norm) multiplication by a scalar. Thus in this case T can be approximated by diagonal operators. If the spectrum of T is connected, then this approach fails, since C(cr(T)) contains no nontrivial characteristic functions. Hence we seek to enlarge the functional calculus to an algebra of functions generated by its characteristic functions. We do this by considering the Gelfand transform on a larger commutative self-adjoint subalgebra of (Ye). This algebra will be obtained as the closure of t T in a weaker topology. Hence we begin by considering certain weaker topologies on 2(Ye). ,

4.44 Definition Let A be a Hilbert space and ,e (°) be the algebra of operators on ,e. The weak operator topology is the weak topology defined by the collection of functions T g) from WY() to C for f and g in Ye.

The strong operator topology on 2(r) is the weak topology defined by the collection of functions T Tf from 2(Ye) to Ye for f in Ye. Thus a net of operators {Ta } ae A converges to T in the weak [strong] operator topology if lim(Ta f, g) aEA

g)

Taf Tf] aEA

for every f and g in Ye.

Clearly, the weak operator topology is weaker than the strong operator topology which is weaker than the uniform topology. We shall indicate examples in the problems to show that these topologies are all distinct. The continuity of addition, multiplication, and adjunction in the weak operator topology is considered in the following lemma. We leave the corresponding questions for the strong operator topology to the exercises. 4.45 Lemma If Ye is a Hilbert space and A and B are in .2(e), then the

functions: (1) a(S, T) = S + T from 2(Ye) x WY() to .204 (2) f3(T) = AT from 2(Ye) to 2(Ye), (3) y(T) TB from 2(Ye) to 2(Ye), and (4) (5(T) T* from .2.(Ye) to 2(Yel,

are continuous in the weak operator topology.

W*-Algebras 101

Proof Compute. •

The enlarged functional calculus will be based on the closure of the C*-algebra 3 T in the weak operator topology. 4.46 Definition If Ye is a Hilbert space, then a subset 9.1 of 2(Ye) is said to be a W*-algebra on Ye if 91 is a C*-algebra which is closed in the weak operator topology.

The reader should note that a W*-algebra is a C*-subalgebra of 2(Ye) which is weakly closed. in particular, a W*-algebra is an algebra of operators. Moreover, if (1) is a *-isometrical isomorphism from the W*-algebra 91 contained in (Ye) to the C*-algebra 3 contained in 2(X), then it does not follow that 3 is weakly closed in .2(S). We shall not consider such questions further and refer the reader to [27] or [28]. The following proposition shows one method of obtaining W*-algebras. 4.47 Proposition If .Ye is a Hilbert space and gji is a self adjoint subalgebra of 2(Ye), then the closure VT of 931 in the weak operator topology is a W*-algebra. Moreover, 91 is commutative if 931 is. -

Proof That the closure of 931 is a W*-algebra follows immediately from Lemma 4.45. Moreover, assume that J1 is commutative and let {S a l aGA

and {T3 } pB be nets of operators in gji which converge in the weak operator topology to S and T, respectively. Then for f and g in Ye and # in B, we have (ST p f g) lim(Sa Tfl f,g) lim(T13 Sa f, g) (Tfl Sf, g). aeA

aeA

Therefore, STS T S for each # in B and a similar argument establishes ST = TS. Hence, 91 is commutative if 9:31 is. ■ 4.48 Corollary If if is a Hilbert space and T is a normal operator on Ye, then the W*-algebra 01 generated by T is commutative. Moreover, if AT is the maximal idea space of U T , then the Gelfand map is a *-isometrical isomorphism of V T onto C(A T). Proof This follows immediately from the preceding proposition along

with Theorem 4.29. • 4.49 If T is a normal operator on the separable Hilbert space if with spectrum A contained in C, then we want to show that there is a unique L"

102, 4 Operators on Hilbert Space and C*-Algebras

space on A and a unique *-isometrical isomorphism F : IBT --4 La) which extends the functional calculus F of Section 4.31, that is, such that the accompanying diagram commutes, where the vertical arrows denote inclusion

1 I

CT

---). - C(A)

r* I' -' Lc°

maps. Thus, the functional calculus for T can be extended to U T and VT = { (P( T) : (i9 e ElWe begin with some measure theoretic preliminaries concerning the following illustrative example. Let A be a compact subset of the complex plane and v be a finite positive regular Borel measure on A with support A. (The latter condition is equivalent to the assumption. that the inclusion mapping of C(A) into L" (v) is an isometrical isomorphism.) Recall that for each cp in L°(v) we define Mg, to be the multiplication operator defined on L2 (v) by M4 f = (pf and that the mapping co --+ M, is a *-isometrical isomorphism from L' (v) into 1 (L2 (v)). Thus we can identify the elements of L' (v) with operators in £' (L2 The following propositions give several important relationships between v, C(A), L' (v), and 2(L2 (v)). The first completes the presentation in Section 4.20. 4.50 Proposition If (X, 6e, p) is a probability space, then If (p) is a maximal abelian W*-algebra in .2(L2 (y)).

Proof In view of Section 4.19, only the fact that L'(p) is weakly closed remains to be proved and that follows from Proposition 4.47, since the weak closure is commutative and hence must coincide with L"(p). II The next result identifies the weak operator topology on L"(p) as a familiar one. 4.51 Proposition If (X, 6P, p) is a probability space, then the weak operator topology and the w*-topology coincide on L'0,/).

Proof We first recall that a function f on X is in L1 (µ) if and only if it can be written in the form f = gh, where g and h are in L2 (1i). Therefore a net to 1 ac A in Lf(p) converges in the w*-topology to go if t, co

W*-Akebras 103

and only if lirn .1 co ce f diu = js cp. dy .EA .3( x if and only if

dp = (Mv g,h) x lim(M,pcs,h) = firn f (p a gh dy f cogh aEA .3(

ae A

and therefore if and only if the net (M,p. L GA converges to My, in the weak operator topology. III The next proposition shows that the W*-algebra generated by multiplication by z on L2 (v) is 1°°(v).

4.52 Proposition If X is a compact Hausdorff space and p is a finite positive regular Borel measure on X, then the unit ball of C(X) is w*-dense in the unit ball of If(p).

Proof A simple step function 0 in the unit ball of L°°(lc) has the form 0 = 1,7. , , a, 4, , where la,' -^... I for i = 1, 2, ..., n, the (E,}", = 1 are pairwise disjoint, and U7, 1 E, = X. For i = 1, 2, ..., n, let K, be a compact subset of E. By the Tietze extension theorem there exists cp in C(X) such that IIVL < 1 and v(x) = a, for x in K. Then for fin Li (p), we have x f(49- 0

du

fxIf I iv—Iiii dit II ci,u. — Vil diu 0, there exist compact sets K, c E, such that E

fe,\K,

141 dti < 2/7

for j = 1, 2, ..., m.

This completes the proof. •

4.53 Corollary If X is a compact Hausdorff space and p is a finite positive regular Borel measure on A, then C(X) is u*-dense in If (p).

Proof Immediate. •

104 4 Operators on Hilbert Space and C*-Algebras

We now consider the measure theoretic aspects of the uniqueness problem. We begin by recalling a definition.

4.54 Definition Two positive measures v, and v 2 defined on a sigma algebra (X, .9) are mutually absolutely continuous, denoted v 1 — v 2 , if v i is absolutely continuous with respect to v 2 and v 2 is absolutely continuous with respect to v 1 . 4.55 Theorem If v 1 and v 2 are finite positive regular Borel measures on the compact metric space X and there exists a *-isometrical isomorphism : L"(v i )--L"(v 2 ) which is the identity on C(X), then v i v 2 , L°°(v i ) ----/°°(v 2 ), and (1) is the identity.

Proof If E is a Borel set in X, then 0(4) is an idempotent and therefore a characteristic function. If we set I (IF ) = IF, then it suffices to show that E F v 2 —a.e. (almost everywhere), since this would imply v 1 (E) = 0 if and only if IF = 0 in L"(1, 1 ) if and only if IF = 0 in If(v 2 ) if and only if v2 (F) = 0, and therefore if and only if v 2 (E) = 0. Thus we would have v i ti v 2 and L°°(v i ) Lf(v 2 ), since the If space is determined by the sets of measure zero. Finally, (1) would be the identity since it is the identity on characteristic functions and the linear span of the characteristic functions is dense in L°°. To show that E = F v 2 —a.e. it suffices to prove that F E v 2— a.e., since we would also have (XV) c (X\E) v 2 —a.e. Further, we may assume that E is compact. For suppose it is known for compact sets. Since v 1 and v 2 are regular, there exists a sequence of compact sets fICI_ 1 contained in E such that E = U nw- 1 Kt, v 1 —a.e. and E U,,"_ 1 KJ, v 2 —a.e. Thus, since 4:1) and 0 -1 are *-linear and multiplicative and hence order preserving, ID preserves suprema and we have IF

= sup (1)(4.) sup /K . =

ff

1

K„ = IE

v2—a.e.

and therefore F is contained in E v 2—a.e. Therefore, suppose E is closed and for n in Z ± let co n be the function in C(X) defined by

—n • d(x,E)

if d(x,E)

1

,

co n (x) = if d(x,E)

1

-,

Normal Operators with Cyclic Vectors 105

where d(x,E) = inf{p(x,y) : y e E} and p is the metric on X. Then IF < con for each n and the sequence {co n },T_ 1 converges pointwise to / E . Since 4 is order preserving and the identity on continuous functions, we have IF -."C Pn V2— a.e. and thus F is contained in E v 2 a.e. II —

(

After giving the following definition and proving an elementary lemma we obtain the functional calculus we want under the assumption the operator has a cyclic vector. 4,56 Definition If Ye is a Hilbert space and 91 is a subalgebra of .2 (..3f), then a vector f in Ye is cyclic for uhf if clos [91f] = ,re and separating for ti if Af = 0 for A in VI implies A = 0. 4.57 Lemma If

Ye is a Hilbert space, 91 is a commutative subalgebra of

.2 on, and f is a cyclic vector for 9{, then f is a separating vector for 91

Proof If B is an operator in 9 and Bf = 0, then BAf = ABf = 0 for every A in 21. Therefore, we have 91f c ker B, which implies B = 0. 1111

The following theorem gives a spatial isomorphism between U T and L', if T is a normal operator on ye having a cyclic vector. (Note that such an .rt is necessarily separable.) 4.58 Theorem If T is a normal operator on the Hilbert space .rt such that

CT has a cyclic vector, then there is a positive regular Borel measure v on C having support A = o! (T) and an isometrical isomorphism y from ye onto L2 (v) such that the map f* defined from VT to 2(L2 (v)) by F*A = yily -1 is a *-isometrical isomorphism from 1.1.; 7, onto L'(v). Moreover, F* is an extension of the Gelfand transform F from CT onto C(A). Lastly, if v 1 is a positive regular Borel measure on C and F i * is a *-isometrical isomorphism from U T onto if(v 1 ) which extends the Gelfand transform, then v 1 — v, L°°(v 1 ) = 1°°(v), and 1"1 * = T*. Proof Let f be a cyclic vector for CT of norm one and consider the functional defined on C(A) by Ip(co) --, (v(T)f,f). Since ip is obviously

linear and positive and Iiii(V)i = i(V(T)fpi < liV(T)11 VP = 110100 , there exists by the Riesz representation theorem (see Section 1.38) a positive

106 4 Operators on Hilbert Space and C*-Algebras

regular Borel measure v on A such that i (p dv - (9 (T) J1; jf)

J A

for 9 in %._,‘"iA) .

Now suppose that the support of v were not all of A, that is, suppose there exists an open subset V of A such that v (V) = 0. By Urysohn's lemma there is a nonzero function (p in C(A) which vanishes outside of V. Since, however, we have 11P(I)fII 2 = ((1)( 1).f(P(T)f) = (1C0 1 2 (T)fAP = f 191 2 dv = 0, A

and f is a separating vector for C T by the previous lemma we arrive at a contradiction. Thus the support of v is A. If we define y o from C T f to L2 (v) such that y o (co(T)f) = 9, then the computation 119112 2 = I IVI 2 dv = (191 2 (nff) = 11(1)(nf 1 2 A

shows that y o is a well-defined isometry. Since C T f is dense in Ye by assumption and C(A) is dense in L2 (v) by Section 3.33, the mapping y o can be extended to a unique isometrical isomorphism y from .Ye' onto L2 (v). Moreover, if we define T* from V T into .2(L2 (v)) by F* (A) = y Ay -1 , then F* is a *-isometrical isomorphism of V T into & (L2 (v)). We want to first show that F* extends the Gelfand transform T on CT. If 1p is in C(A), then for all q' in C(A), we have [F * OP (nA (P = YIP (T) Y -1 (P = YIP ( 7) (P (nf = Y[OP(P)(T) fi = ikv = gfr (P and since C(A) is dense in L2 (v), it follows that F* Op (T)) = 11.4 = nip (n). Thus T* extends the Gelfand transform. To show that F* (./1.; T) = Lc ° (v) , we note that since F* is defined spatially by y, it follows from Proposition 4.51 that F* is a continuous map from T8T with the weak operator topology to L°°(v) with the w*-topology. Therefore, by Corollary 4.53, we have ,

F*(T8 T) = F*(weak oper clos [C T]) . w*-clos [C(A)] = if (v), and thus F* is a *-isometrical isomorphism mapping V T onto I.' (v). Finally, if v 1 is a positive regular Borel measure on A and F1 * is a *-isometrical isomorphism from 928 T onto if(v 1 ) which extends the Gelfand transform, then F* Ti 1 is a *-isometrical i ,;-- -norphism from L°°(v i ) onto

Normal Operators with Cyclic Vectors 107

L°°(v) which is the identity on C(A). Hence, by Theorem 4.55, we have v 1 N v, L°°(v i ) = L"(v), and F* r r 1 is the identity. Therefore the proof

is complete. a 4.59 The preceding result gives very precise information concerning normal operators possessing a cyclic vector. Unfortunately, most normal operators do not have a cyclic vector. Consider the example: = L2 ([0, 1]) ,C) L2 ([0, 11) and T. M9 Mg , where g(t) = t. It is easy to verify that has no cyclic vector and this is left as an exercise. Thus the preceding result does not apply to T. Notice, however, that CT = Mtp Ej Mc, : rp e Ca°, 1])} and 9 T (M,i,10 Mc, : co L'([0, 1])},

and thus there still exists a *-isometrical isomorphism F* from V T onto if°([0, 1]) which extends the Gelfand transform on CT. The difference is that f* is no longer spatially implemented. To see how to obtain F* in the general case, observe that f = 1 e 0 is a separating vector for UT and that the space Jl = clos[0 71] is just L2 ([0, 1]) CI (01; moreover, Jd is a reducing subspace for T and the mapping A -÷ A 1./# defines a *-isometrical isomorphism from VD T onto the W*-algebra generated by T Lastly, the operator T I Ji is normal and 1 T1at has a cyclic vector; hence the preceding theorem applies to it. Our program is as follows: For a normal T we show 91 7, has a separating vector, that V T and U TIai are naturally isomorphic, and that the theorem applies to TI.A. Thus we obtain the desired result for arbitrary normal operators. To show that IBT has a separating vector requires some preliminary results on W*-algebras. 4.60 Proposition If 91 is an abelian C*-algebra contained in .2.(10, then there exists a maximal abelian W*-algebra in 2(4e) containing 91. Proof The commutative C*-subalgebras of 2(1f) which contain 91 are

partially ordered by inclusion. Since the norm closure of the union of any chain is a commutative C*-algebra of .2.(e) containing %, then there is a maximal element by Zorn's lemma. Since the closure of such an element in the weak operator topology is commutative by Proposition 4.47, it follows that such an element is a W*-algebra.

•

108 4 Operators on Hilbert Space and C*-Algebras

The following notion is of considerable importance in any serious study of W*-algebras. 4.61 Definition If 91 is a subset of 2(.09 ), then the commutant of 91,

denoted 91', is the set of operators in 2(.4r) which commute with every operator in 91. It is easy to show that if 9f is a self-adjoint subset of .2 (P), then 91' is a W*-algebra. The following proposition gives an algebraic characterization of maximal abelian W*-algebras. 4.62 Proposition A C*-subalgebra 91 of 2(10 is a maximal abelian

W*-algebra if and only if 91 Proof If 91 = 91', then 91 is a W*-algebra by our previous remarks.

Moreover, by definition each operator in 91' commutes with every operator in 91 and therefore 91 is abelian. Moreover, if A is an operator commuting with 9f, then A is in 91' and hence already in 9f. Thus 91 is a maximal abelian W*-algebra. Conversely, if 91 is an abelian W*-algebra, then 91 91'. Moreover, if T is in 9', then T H+ iK, where H and K are self-adjoint and in 91'. Since the W*-algebra generated by 91 and either H or K is an abelian W*-algebra, then for 91 to be maximal abelian, it is necessary for H and K to be in 91 and hence 9f = 9f'. • 4.63 Lemma If 91 is a C*-algebra contained in 2(. f) and f is a vector

in if, then the projection onto the closure of 9ff is in 9' . Proof In view of Proposition 4.42 it suffices to show that clos Ulf] is invariant for both A and A* for each A in 91. That is obvious from the

definition of the subspace and the fact that 21 is self-adjoint. • The following proposition shows one of the reasons for the importance of the strong operator topology. 4.64 Proposition If X' is a Hilbert space and (Pa L EA is a net of positive

operators on P such that 0 < P a < / for a and 16 in A with a (3, then there exists P in 201t) such that 0 < Pa < 1 ^,I for a in A and the net (Pa l a EA converges to P in the strong operator topology. )

-

Maximal Abelian W*-Algebras 109

Proof If Q is in 2(Ye) and 0 < Q < I, then 0 < Q 2 < Q < I since Q commutes with (I Q)'12 by Proposition 4.33 and since —

Q 2 )./4) == (Q(/ -- 01, (I — Q) 0 for f in W. Further, for each f in Ye the net f(P,./J)}„A is increasing and hence a

Cauchy net. Since for /3 ..; a, we have 11(Pp

—

Pa)f 11 2 =

—

P0 2.1,1) <

PaKi) = p .1;.1)

—

(Pc, ff),

it therefore follows that {Pa A EA is a Cauchy net in the norm of Ye. If we define pl.-- lima EA Pa f, then P is linear, 11P.f11 < lin111Pafll aeA

and

0

lim(Pa f,f) = (Pf,f)• aEA

Therefore, P is a bounded positive operator, 0 < P a < P < I, for a in A, and {Pa } aeA converges strongly to P, and the proof is complete. III The converse of the following theorem is also valid but is left as an exercise.

4.65 Theorem If 91 is a maximal abelian W*-algebra on the separable Hilbert space .W, then 91 has a cyclic vector. Proof Let in 21 such that:

e denote the set of all collections of projections {E,},,A

(1) For each a in A there exists a nonzero vector fa in .' such that Ea is the projection onto dos NIL]; and (2) the subspaces {dos [91fa] } a , A are pairwise orthogonal. We want to show there is an element {Ea } a€A in g such that the span of the ranges of the Ea is all of Ye. Order by inclusion, that is, {Ea } a EA is greater than or equal to {Fp } fiEB if for each fl o in B there exists ao in A such that Fp° =-- Ea o . To show that g is nonempty observe that the one element set {P ChM } is in 6 for each nonzero vector f in Ye. Moreover, since the union of a chain of elements in is in then has a maximal element {Ea} a „4 by Zorn's lemma. Let 9; denote the collection of all finite subsets of the index set A partially ordered by inclusion and let {PF } FE9, denote the net of operators defined by Pr = EF Ea By the remark after Proposition 4.19, each PF is a projection and hence the net is increasing. Therefore, by the previous proposition there exists a positive operator P such that 0 < PF < P < I for every F in F and {PF } Fea, converges to P in the strong operator topology. Therefore, {PF 2 } F , 0 converges strongly to P 2 and P is seen to be a projection.

e

e,

e

e

•

110 4 Operators on Hilbert Space and C*-Algebras

The projection P is in 91 since 91 is weakly closed and the range di of P reduces 91 since 91 is abelian. Thus, if f is a nonzero vector in X i , then _ rcW,c^ 1 _ _1 4_ 41_ —1_ WC I irilIgC UI cauit L. II WC ICI r, p uenute CIOS LW A IS VI projection onto clos PIP, then {E a }„ EA u(p) is an element in e larger than {Ea }„ A. This contradiction shows that ./N 1 = {0} and hence that P = I. Since Ye is separable, dim ran Ea > 0, and dim .W = EcteA dim ran Ea, we see that A is countable. Enumerate A such that A = cx2, ...} and set I= E i ,.. 1 2 -1LAIL,11. Since Ecc , f = 2 - ULLA, we see that the range of E, is contained in clos [91f]. Therefore, since the ranges of the Ea , span Ye, we see that f is a cyclic vector for 91 and the proof is complete. III -

—C

77

TL' __- 1-a 7r,

A

.1_ _

E

The assumption that Ye is separable is needed only to conclude that A is countable. The following result is what we need in our study of normal operators. 4.66 Corollary If 91 is an abelian C*-algebra defined on the separable

Hilbert space Ye, then 91 has a separating vector.

Proof By Proposition 4.60 91 is contained in a maximal abelian W*algebra 93 which has a cyclic vector f by the previous theorem. Finally, by Lemma 4.57 the vector f is a separating vector for 93 and hence also for the

subalgebra 91. III The appropriate setting for the last two results is in W*-algebras having the property that every collection of pairwise disjoint projections is countable. While for algebras defined on a separable Hilbert space this is always the case, it is not necessarily true for W*-algebras defined on a nonseparable Hilbert space Ye; for example, consider £('). Before we can proceed we need one more technical result concerning *-homomorphisms between C*-algebras. 4.67 Proposition If 91 and 23 are C*-algebras and 1 is a *-homomorphism from 91 to 23, then 11 0 1i < 1 and ID is an isometry if and only if +13 is one-to-one. Proof If H is a self-adjoint element of 91, then EH is an abelian

C*-

algebra contained in 91 and J(C H) is an abelian *-algebra contained in 93. If /i is a multiplicative linear functional on the closure of 43(C H) then ri defines a multiplicative linear functional on CH If V/ is chosen such that = litK101 , which we can do by Theorem 4.29, then we have .

*-11omomorphisms of C*-Algebras 111

and hence CI is a contraction on the selfilHii - itP(D(H))1 adjoint elements of 91. Thus we have for T in 21 that

II T 112 == II T* TII

i tD(T * = 11 43 (T) * O(nii = 11 4) (nii 2

and hence 11 0 0 < 1 . Now for the second statement. Clearly, if ID is an isometry, then it is one-to-one. Therefore, assume that (13 is not an isometry and that T is an element of 21 for which II Til = 1 and 110(7)11 < 1. If we set A = T* T, then All 1 and II 1 (A) II = 1 — e with e> 0. Let f be a function in C([0, I]) chosen such that f(1) = 1 and f(x) = 0 for 0 x E. Then using the functional calculus on E A we can define f(A) and since by Corollary 2.37 we have a (f(A)) = range T(f(A)) .f(o-(A)), we conclude that 1 is in (TWA)) and thus f(A)  0. Since cl) is a contractive *-homomorphism, it is clear that Co(p(A)) p(cl)(A)) for each polynomial and hence CM)) = f(43(A)). Since, however, we have 110(A)I1 = 1 — e, it follows that o (D(A)) [0, 1— e] and therefore -

-

(f(A))) =

(4) (A))) f([0 , 1 — c]) = {0} .

Since (1)(A) is self-adjoint, we have el(A) = 0, which shows that (13 is not one-to-one. III Now we are prepared to extend the functional calculus for an arbitrary normal operator on a separable Hilbert space. 4.68 Let Ye be a separable Hilbert space and T be a normal operator on Ye. By Corollary 4.66, the abelian W*-algebra T3 T has a separating vector f If we set di clos [0 7-f], then di is a reducing subspace for each A in aBT, and hence we can define the mapping cico from 0 1 to 2(.4") by 43(A) = Aldi for A in UT . It is clear that (13 is a *-homomorphism. We use the previous result to show that ID is a *-isometrical isomorphism. If 21 is a C*-algebra contained in ,e(Ye), f is a separating vector for 91 and X is the closure of 9if, then the mapping CI defined cI^(A) = Aidi for A in 91 is a *-isometrical isomorphism from 21 into 2(./1). Moreover, o (A) o (A[di) for A in 91. 4.69 Lemma

-

-

Proof Since 113 is obviously a *-homomoiphism, it is sufficient to show that tto is one-to-one. If A is in 21 and +11)(A) = 0, then Af = 0, which implies A = 0 since f is a separating vector for 21. The last remark follows from

112 4 Operators on Hilbert Space and C*-Algebras

Theorem 4.28, since we have Q2 ( ) (A) = cr (A) = Q c, (90 (A I .) = Q (Jo (Al di). III Finally, we shall need the following result whose proof is similar to that of Theorem 1.23 and hence is left as an exercise. 4.70 Proposition If 1I is a W*-algebra contained in 2(. f), then the unit

ball of 91 is compact in the weak operator topology.

Our principal result on normal operators can now be given. 4.71 Theorem (Extended Functional Calculus) If T is a normal operator on

the separable Hilbert space Ye with spectrum A, and F is the Gelfand transform from C T onto C(A), then there exists a positive regular Borel measure v having support A and a *-isometrical isomorphism F* from UT onto If' (v) which extends F. Moreover, the measure v is unique up to mutual absolute continuity, while the space 1.f(v) and F* are unique. Proof Let f be a separating vector for OT,

be the closure of U T f and cici be the *-isometrical isomorphism defined from BT into 2(11) by (13 (A) = A I X as in Section 4.68. Further, let 9Bff be the W*-algebra generated by TI.11. Since t is defined by restricting the domain of the operators, it follows that CI is continuous from the weak operator topology on UT to the weak operator topology on 2(4'), and hence cicI( 3 7 ) c993 it . Moreover, it is obvious that if To is the Gelfand transform from C TI .,ft onto C(A), then F = To 00. Since T I is normal and has the cyclic vector f, there exist by Theorem 4.58 a positive regular measure v with support equal to o (T1.11) = A by the previous lemma and a *-isometrical isomorphism F from 93 it onto L°°(v) which extends the Gelfand transform To on C TIsit . Moreover, G* is continuous from the weak operator topology on Usti to the w*-topology on (v). Therefore, the composition 1* = ro *oe• is a *-isometrical isomorphism from UT into If (v) which is continuous from the weak operator topology on U T to the w*-topology on If (v) and extends the Gelfand transform F on C T . The only thing remaining is to show that F* takes UT onto If (v). To do this we argue as follows: Since the unit ball of UT is compact in the weak operator topology, it follows that its image is w*-compact in 12 ° (v) and hence w*-closed. Since this image contains the unit ball of C(A), it follows -

The Extended Functional Calculus 113

from Proposition 4.52 that F* takes the unit ball of U T onto the unit ball of If (v). Thus, F* is onto. The uniqueness assertion follows as in Th eorem 4.5 from 8 Theor em 4.55. ■ 4.72 Definition If T is a normal operator on the separable Hilbert space W, then there exists a unique equivalence class of measures v on A such that there is a *-isometrical isomorphism T* from 07- onto If (v) such that 1"*(q)(T))=-- q) for q) in C(A). Any such measure is called a scalar spectral measure for T. The extended functional calculus for T is defined for co in 12'(v) such that F*((p(T))= co. If ./6, is a characteristic function in ifp(v), then 16,(T) is a projection in UT called a spectral projection for T, and its range is called a spectral subspace for T. We conclude this chapter with some remarks concerning normal operators on nonseparable Hilbert spaces and a proposition which will enable us to make use of certain aspects of the extended functional calculus for such operators. We begin with the proposition which we have essentially proved. 4.73 Proposition If 2 is a norm separable C*-subalgebra of .e(Ye), then Ye = L EA Ye c, such that each Yect is separable and is a reducing subspace for 9f.

Proof It follows from the first three paragraphs of the proof of Theorem 4.65 that Ye = ylo,„ where each Yec, is the closure of 9Xf, for some ft in Ye. Since 2 is norm separable, each Yect is separable and the result follows. ■ 4.74 From this result it follows that if T is normal on Ye, then each TlYe a, is normal on a separable Hilbert space and hence has a scalar spectral measure v cc . If there exists a measure v such that each v c, is absolutely continuous with respect to v, then Theorem 4.71 can be shown to hold for T If no such v exists, then the functional calculus for T is usually based on the algebra of bounded Borel functions on A. In particular, one defines (p (T) ---L EA C)(p(TO for each Borel function co on A. The primary deficiencies in this approach is that the range of this functional calculus is no longer O T and the norm of co (T) is less accessible. Sometimes the spectral measure E(.) for T is made the principal object

114 4 Operators on Hilbert Space and C*-Algebras

of study. If A is a Borel subset of A, then the spectral measure is defined by E(A) = /A(T) = 0 WO

aeA

and is a projection-valued measure such that (E(A)f,f) is countably additive for eachfin Jr. Moreover, the Stieltjes integral can be defined and T = f A z dE. We shall not develop these ideas further except to show that the range of each spectral measure for T lies in UT.

4.75 Proposition If T is a normal operator on the Hilbert space Ye° and E(.) is a spectral measure for T, then E(A) is in az, for each Borel set A in A. Proof Let T =L EA OTa be the decomposition of T relative to which the spectral measure E(.) is defined, where each Tc, acts on the separable denotes the collection of space 0 , with scalar spectral measure pa . If finite subsets of A, then UF E. L er aye% is a dense linear manifold in Ye° . G

Thus if A is a Borel subset of A, it is sufficient to show that for f i ,f2 , lying in E G, EF. Yea, for some F0 in g7 and e > 0, there exists a continuous function co such that 0 c rp < 1, and 11(co(T)—E(AW;11 < e for i 1, 2, ..., n. Since

11(CP (T) — E(o)M11 2 I 11(co(To—h(Tcc))Poa cceF0

—

cceFo A

ip-4,121P,rafil2

I

2

diAcr ,

this is possible using Proposition 4.52. III We conclude this chapter with an important complimentary result. The following ingenious proof is due to Rosenblum [93].

4.76 Theorem (Fuglede) If T is a normal operator on the Hilbert space Ye° and X is an operator in 2(.re) for which TX = XT, then X lies in WT.'. Proof Since au, is generated by T and T*, the result will follow once it is established that T* X XT*. Since Tk X = XI* for each k 0, it follows that exp(iA T) X = Xexp(i)T) for each A in C. Therefore, we have X = exp(aT)X exp( — gT), and hence F(A) = exp (i),T*)X exp(— i) T*) = exp [i T+ AT*)] X exp [ — i

(

T± AT*)]

by Lemma 2.12, since TT* = T* T. Since IT+ .1..T* is self-adjoint, it follows

Exercises 115

that exp [i()T+ AT*)] and exp — i(1T+ AT*)] are unitary operators for 2 in C. Thus the operator-valued entire function F(A) is bounded and hence by jouville's theorem must be constant (see the proof w. I new ein 2.29). Lastly, differentiating with respect to 2 yields F'01) exp (i),T*) X exp(— aT*) + exp (i2P') X exp(— illT*)(—aT*) 0. Canceling 2 and then setting 2 = 0 yields T* X = XT*, which completes the proof. 111 Notes The spectral theorem for self-adjoint operators is due to Hilbert, but elementary operator theory is the joint work of a number of authors including Hilbert, Riesz, Weyl, von Neumann, Stone, and others. Among the early works which are still of interest are von Neumann's early papers [81], [82], and the book of Stone [104]. More recent books include Akhieser and Glazman [2], Halmos [55], [58], Kato [70], Maurin [79], Naimark [80], Riesz and Sz.-Nagy [92], and Yoshida [117]. We have only introduced the most elementary results from the theory of C*- and W*-algebras. The interested reader should consult the two books of Dixmier [27], [28] for further information on the subject as well as a guide to the vast literature on this subject. Exercises 4.1 If T is a linear transformation defined on the Hilbert space Ye, then T is bounded if and only if sup {1(Tf,f)I :f

ye, Ilfil = 1 } < Do. '

Definition If T is an operator defined on the Hilbert space .W, then the numerical range W(T) of T is the set {(Tf,f) :fe II f II = 1} and the numerical radius w (T) is sup { 1 2 1 : 2 E W(T)} .

4.2 (Hausdorff—Toeplitz) If T is an operator on Ye, then W(T) is a convex set. Moreover, if Ye is finite dimensional, then W(T) is compact. (Hint: Consider W(T) for T compressed to the two-dimensional subspaces of Ye.)

116 4 Operators on Hilbert Space and C*-Algebras

4.3 If T is a normal operator on Ye, then the closure of W(T) is the closed convex hull of olT). Further, an extreme point of the closure of W(T) belongs to W(T) if and only if it is an eigenvalue for T. 4.4 If T is an operator on .W, then u(T) is contained in the closure of W(T). (Hint: Show that if the closure of W(T) lies in the open right-half plane, then T is invertible.) 4.5 If T is an operator on the Hilbert space Ye, then r(T) w(T) 1 71 and both inequalities can be strict. 4.6 (Hellinger—Toeplitz) If S and T are linear transformations defined on the Hilbert space Ye such that (Sf, g) = (f,Tg) for f and g in Ye", then S and T are bounded and T S. 4.7 If T is an operator on Ye, then the graph {< f, Tf) : f e Ye} of T is a closed subspace of Ye()Ye with orthogonal complement {< T*g, g> : g E Ye}. 4.8 If Tis an operator on Ye, then T is normal if and only if II Tf. II =III ' *.1 . 1 for f in .W. Moreover, a complex number A is an eigenvalue for a normal operator T if and only if is an eigenvalue for T*. The latter statement is not valid for general operators on infinite-dimensional Hilbert spaces. 4.9 If S and T are self-adjoins operators on .W, then ST is self-adjoint if and only if S and T commute. If P and Q are projections on Ye, then PQ is a projection if and only if P and Q commute. Determine the range of PQ in this case. 4.10 If Ye and S. are Hilbert spaces and A is an operator on Yealf, then there exist unique operators A 11 , Al2, A21, and A22 in 2 (. (e), 2(r, Ye), 2(ee, X), and 2(S. ), respectively, such that A = . In other words A is given by the matrix A11 Al2 [

A21 A22

Moreover show that such a matrix defines an operator 'per''. 4.11 If Ye and 1 are Hilbert spaces, A is an operator on 2(.1C, Ye), and J is the operator on Yea t- defined by the matrix -

Exercises 117

then J is an idempotent. Moreover, J is a projection if and only if A = 0. Further, every idempotent on a Hilbert space 2 can be written in this form cis..

sm .ri

_

ar• rlorri

-

•••• 1 a-4n; t; ^In co L.101 Llk .1 II

—

Cr"%iir 1/4:1-

Definition If Ti and T2 are operators on the Hilbert space Ye t and Ye2, respectively, then T 1 is similar (unitarily equivalent) to T2 if and only if there exists an invertible operator (sometrical isomorphism) S from Ye, onto Ye2 such that T2 S = ST1 . 4.12 If Ye is a Hilbert space and J is an idempotent on Ye with range di, then J and P are similar operators. 4.13 If (X, 9, p) is a probability space and (p a function in If(p) , then A is an eigenvalue for M4, if and only if the set {x E X: co (x) = A} has positive

measure.

4.14 Show that the unitary operator U defined in Section 4.25 has no eigenvalues, while the eigenvalues of the coisometry U,,_* defined in Section 4.36 have multiplicity one. 4.15 If 91 is a C*-algebra, then the set g of positive elements in 91 forms a closed convex cone such that 6 n = {0}. (Hint: Show that a selfadjoint contraction H in 91 is positive if and only if 1.) —

4.16 If 91 is a C*-algebra, then an element H in 91 is positive if and only if there exists T in 91 such that H = T* T.* (Hint: Express T* T as the

difference of two positive operators and show that the second is zero.) 4.17 If P and Q are projections on Ye such that II P Q II < 1, then dim ran P = dim ran Q. (Hint : Show that (I P)+ Q is invertible, that ker P n ran Q = {0}, and that P [ran Q] = ran P.) —

—

4.18 An operator V on .W is an isometry if and only if V* V = I. If V is an isometry on Ye, then V is a unitary operator if and only if V* is an isometry if and only if ker V* = {0}. 4.19 If Ye and 1 are Hilbert spaces and A is an operator on Yea S' given

by the matrix

[

A11

Al2

A 21 A22

then .W 0 {0} is an invariant subspace for A if and only if A21 = 0, and Ye 0 {0} reduces A if and only if A21 = A l2 = 0.

118 4 Operators on Hilbert Space and C*-Algebras

4.20 If U + is the unilateral shift, then the sequence {U + 1,71 1 converges

to 0 in the weak operator topology but not in the strong. Moreover, the sequence {U:n},7.... 1 converges to 0 in the strong operator topology but not in the norm. 4.21 Show that multiplication is not continuous in both variables in either

the weak or strong operator topologies. Show that it is in the relative strong operator topology on the unit ball of 2(Ye). 4.22 If Ye is a Hilbert space, then the unit ball of 2(Ye) is compact in the

weak operator topology but not in the strong. (Hint : Compare the proof of Theorem 1.23.)

4.23 If 91 is a W*-algebra contained in 2(Ye), then the unit ball of 91 is

compact in the weak operator topology. 4.24 If 91 is a *-subalgebra of £(. P), then 91 (2) = {[oI] : A E 91} is a *-subalgebra of 2(Ye ®°). Similarly, 91(N) is a *-subalgebra of 2(Ye C) • • • (I) Ye)

for any integer N. Moreover, 91 (N) is closed in the norm, strong, or weak operator topologies if and only if 91 is. Further, we have the identity 9I(N)" = 9 (N).

4.25 If 21 is a *-subalgebra of Ye, A is in 91", x l , x2 , ..., xN are vectors in Ye, and e > 0, then there exists Bin 21 such that IlAx i — 13xi ll < e for i = 1, 2, ..., N. (Hint: Show first that for .11 a subspace of Ye° • • • 0 Y e , we have dos [91'('N) di] = dos [91 (N) X ]. ) 4.26 (von Neumann Double Commutant Theorem) If 91 is a * subalgebra of 2(Ye), then 91 is a W*-algebra if and only if 91 = 91".* -

4.27 If 91 is a C*-subalgebra of

£('), then 91 is a W*-algebra if and only

if it is closed in the strong operator topology. 4.28 If S and T are operators in the Hilbert spaces Ye and YC, respectively, then an operator S 0 T can be defined on Yealf in a natural way such that Is® Til = 11S1111Thi and (S ® T)* = S* 0 T*.

In Exercises (4.29-4.34) we are considering linear transformations defined on only a linear subspace of the Hilbert space. Definition A linear transformation L defined on the linear subspace ?Y the Hilbert space le is closable if the closure inYe() Ye of the graph {< f, Lf> : f c g,+ } of L is the graph of a linear transformation L called the

Exercises 119

closure of L. If L has a dense domain, is closable, and L = L, then L is said to be closed. 4.29 Give an example of a densely defined linear transformation which is not closable. If T is a closable linear transformation with gT = Ye, then T is bounded. 4.30 If L is a closable, densely defined linear transformation on Ye, then these exists a closed, densely defined linear transformation M on Ye such that (Lf, g) = (f, Mg) for f in g,i, and g in gm. Moreover, if N is any linear transformation on Ye for which (Lf,g) = (f, Ng) for f in g i, and g in g N , then 1N c and Ng = Mg for g in gx. (Hint: Show that the graph of M can be obtained as in Exercise 4.7 as the orthogonal complement of the graph of L.)

Definition If L is a closable, densely defined linear transformation on .W, then the operator given in the preceding problem is called the adjoint of L and is denoted L*. A densely defined linear transformation H is symmetric if (Hf,f) is real for f in gli and self-adjoint if H = H*. 4.31 If T is a closed, densely defined linear transformation on .W, then T = T**. * (This includes the fact that gT = gT*4). If H is a densely defined symmetric transformation on Ye, then H is closable and H* extends H. 4.32 If H is a densely defined symmetric linear transformation on range equal to Ye, then H is self-adjoint.

Ye with

4.33 If T is a closed, densely defined linear transformation on Ye, then T* T is a densely defined, symmetric operator. (Note: T* Tf is defined for those f for which Tf is in g 7-4.) 4.34 If T is a closed, densely defined linear transformation on A', then T* T is self-adjoint. (Hint: Show that the range of I+ T* T is dense in .' and closed.) 4.35 If 91 is a commutative W*-algebra on .W with Ye separable, then 91 is a *-isometrically isomorphic to L"(,u) for some probability space (X, 99, µ). 4.36 Show that there exist W*-algebras 91 and 0 such that 91 and 0 are *-isomorphic but 91' and V are not. 4.37 If 91 is an abelian W*-algebra on Ye for Ye separable, then 91 is maximal abelian if and only if 91 has a cyclic vector.

120 4 Operators on Hilbert Space and C*-Algebras

4.38 Give an alternate proof of Fuglede's theorem for normal operators on a separable Hilbert space as follows: Show that it is enough to prove that E(A 1 ) XE(A 2 ) = 0 for Al and A2 disjoint Borel sets; show this first for Borel sets at positive distance from each other and then approximate from within by compact sets in the general case. 4.39 (Putnam) If T1 and T2 are normal operators on the Hilbert spaces Yei and .W2 , respectively, and X in 2(Yel , .W2 ) satisfies T2 X = XTi , then T2 *X = XT1 *. (Hint: Consider the normal operator [1: 1 T°2 ] on Yei C)Ye2 together with the operator [X g]). 4.40 If T1 and T2 are normal operators on the Hilbert spaces Ye i and .W 2,, respectively, then T 1 is similar to T2 if and only if T 1 is unitarily equivalent to T2 . 4.41 Let X be a compact Hausdorff space, Ye be a Hilbert space, and 4 be a *-isomorphism from C(X) into 2(°). Show that if there exists a vector f in If for which J(C(X)) is dense in Yt°, then there exists a probability measure u on X and an isometrical isomorphism 'I' from L 2 (u) onto Ye such that kflitio (II* =0:1:1(4)) for co in C(X). (Hint: repeat the argument for Theorem 4.58.) 4.42 Let X be a compact Hausdorff space, X2 be a Hilbert space, and b be a *-isomorphism from C(X) into 2 (Ye), then there exists a *-homomorphism (I)* from the algebra 2(X) of bounded Borel functions on X which extends (D. Moreover, the range of 0* is contained in the von Neumann algebra generated by the range of (D. (Hint: use the arguments of 4.74 and 4.75 together with the preceding exercise.)

Compact Operators, Fredhohn Operators, and Index Theory

5.1 In the preceding chapter we studied operators on Hilbert space and obtained, in particular, the spectral theorem for normal operators. As we indicated this result can be viewed as the appropriate generalization to infinite-dimensional spaces of the diagonalizability of matrices on finitedimensional spaces. There is another class of operators which are a generalization in a topological sense of operators on a finite-dimensional space. In this chapter we study these operators and a certain related class. The organization of our study is somewhat unorthodox and is arranged so that the main results are obtained as quickly as possible. We first introduce the class of compact operators and show that this class coincides with the norm closure of the finite rank operators. After that we give some concrete examples of compact operators and then proceed to introduce the notion of a Fredholm operator. We begin with a definition. 5.2 Definition If Ye is a Hilbert space, then an operator T in (Ye) is a finite rank operator if the dimension of the range of T is finite and a compact operator if the image of the unit ball of Ye under T is a compact subset of Ye. Let .e(Yel, respectively, 2C(Ye) denote the set of finite rank, respectively, compact operators. In the definition of compact operator it is often assumed only that 121

122 5 Compact and Fredholm Operators, Index Theory

T[(.W) 1 ]

has a compact closure. The equivalence of these two notions follows from the corollary to the next lemma.

Ye is a Hilbert space and T is in .e (.W), then T is a continu-

5.3 Lemma If

ous function from Ye with the weak topology to Proof If ffaccA is a net in vector in Ye, then lim(Tft , g) = aEA

aEA

Ye with the weak topology.

Ye which converges weakly to

f and g is a

T*g) = (f,T*g) = (T f, g),

and hence the net tinuous. III

{TX,L cA converges weakly to Tf. Thus T is weakly con-

5.4 Corollary If

.W is a Hilbert space and T is in £(.;(e), then T[(Ye) i ]

is a closed subset of Ye.

Proof Since (.W) 1 is weakly compact and T is weakly continuous, it follows that T[(.°) 1 ] is weakly compact. Hence, T[(.') 1 ] is a weakly closed subset of Ye and therefore is also norm closed. III The following proposition summarizes most of the elementary facts about finite rank operators.

5.5 Proposition If

Ye is a Hilbert space, then 2(yta)

is the minimal

two-sided *-ideal in 2(re). Proof If S and T are finite rank operators, then the inclusion ran (S+

T) c

ran S + ran T

implies that S+ T is finite rank. Thus, .ea(.ee) is a linear subspace. If S is a finite rank operator and T is an operator in 2((e), then the inclusion ran ST c ranS shows that Qa(.W) is a left ideal in t(Ye). Further, if T is a finite rank operator, then the identity ran T* = T* [(ker T*)1 T* [clo s ran 1] which follows from Corollary 3.22 and Proposition 4.6, shows that T* is also a finite rank operator. Lastly, if S is in 2(Ye) and T is in £a (.W), then T* is in *CYO which implies that T*S* is in *(ee) and hence that ST = (T*S*)* is in 2nreo). Therefore, 2we) is a two-sided *-ideal in 2(Ye). To show that 2a((e) is minimal, assume that 3 is an ideal in £(°)

The Ideals of Finite Rank and Compact Operators 123

not (0). Thus there exists an operator T 0 in 3, and hence there is a nonzero vector f and a unit vector g in Ye such that Tf = g. Now let h and k be arbitrary nonzero vectors in Jr and A and B be the operators defined on Jr by Al ----= (1, g)k and B1 = (1, h)f for 1 in Ye. Then, S ATB is the rank one operator in 3 which takes h to k. It is now clear that 3 contains all finite rank operators and hence 2R (Ye) is the minimal two-sided ideal in 2(Ye). ■ The following proposition provides an alternative characterization of compact operators. 5.6 Proposition If Ye is a Hilbert space and T is in £(. 9), then T is compact if and only if for every bounded net ff,J,„ A in Ye which converges weakly to f it is true that {Tfcr } G, eA converges in norm to Tf.

Proof If T is compact and ILL, EA is a bounded net in Ye which converges weakly to f, then {Tfc,} c,„ converges weakly to Tf by Lemma 5.3 and lies in a norm compact subset by the definition of compactness. Since any norm Cauchy subnet of {Tf,},, EA must converge to Tf, it follows that lim,„ cA Tf in the norm topology. Conversely, suppose T is an operator in Q(e) for which the conclusion of the statement is valid. If {Tf„} a€A is a net of vectors in T[(Ye) 1 ], then there exists a subnet {f ts } flEB which converges weakly to an f. Moreover, since each ft, is in the unit ball of Ye, it follows that {Tfati }p e B converges in norm to Tf. Therefore, T[(W) 1 ] is a compact subset of Ye and hence T is compact. III 5.7 Lemma The unit ball of a Hilbert space Ye is compact in the norm topology if and only if Ye is finite dimensional.

.W is finite dimensional, then ye is isometrically isomorphic to (En and the compactness of (°) 1 follows. On the other hand if Ye is Proof If

infinite dimensional, then there exists an orthonormal subset {e,,}7_ 1 contained in (Y e) 1 and the fact that lien — end' Ng for n 0 In shows that (Ye) 1 is not compact in the norm topology. 111 The following property actually characterizes compact operators on a Hilbert space, but the proof of the converse is postponed until after the next theorem. Whether this property characterizes compact operators on a Banach space is unknown.

124 5 Compact and Fredholm Operators, Index Theory

5.8 Lemma If W is an infinite-dimensional Hilbert space and T is a compact operator, then the range of T contains no closed infinite-dimensional subspace.

Proof Let 4' be a closed subspace contained in the range of T and let PA( be the projection onto 4% It follows easily from Proposition 5.6 that the operator P x T is also compact. Let A be the operator defined from W to art by Af = P.At Tf for f in Ye. Then A is bounded and onto and hence by the open mapping theorem is also open. Therefore, A [O, contains the open ball in di of radius 6 centered at 0 for some 6 > 0. Since the closed ball of radius 6 is contained in the compact set P iit T[(*e),], it follows from the preceding lemma that X is finite dimensional. ■ We are now in a position to show that .earta) is the norm closure of £'R (WP). The corresponding result for Banach spaces is unknown.

5.9 Theorem If .W is an infinite-dimensional Hilbert space, then 2C(re) is the norm closure of 2a (Ye). Proof We first show that the closure of 2a(Ye) is contained in 2C (Ye).

ta(ye)

is contained in £€(.'). Secondly, to prove Firstly, it is obvious that that 2C(Ye) is closed, assume that {K}°1 1 is a sequence of compact operators which converges in norm to an operator K. If {f,} 0, EA is a bounded net in Ye that converges weakly to f, and

M = sup {1, Ilia : a E 4 } , -

then choose N such that I1K— KN II<e/3M. Since KN is a compact operator, we have by Proposition 5.6 that there exists a o in A such that II KN f, KN f II < e/3 for a ,-- ao . Thus we have —

iiK.fa

—

Kf II < IKK — KALI' + °Ka; — K N f II + ii(K Iv — K) f II
0 the element 2H 2 is positive, since it is the square of a self-adjoint element, and therefore 2R 2 + I is invertible in 91. Moreover, rearranging the identity (2H 2 + Ir) (2H2 + 1 ) 1 = / shows that the element defined by .

= (A.H 2 +

/ = —(A.H 2 +

is in 3. Moreover, if we set S2 = TU2 + T, then SA * SA = (2H 2 + I) from the functional calculus for H , we have liSA * SAli = 11(21/2+1121/11

R

2H

sup { (2x2 + 1)2 . X E Q(H) 9 sup { (Ax2 + 1)2 : x ot ‘- 16 (32)1/2

and



140 5 Compact and Fredholm Operators, index Theory

where the last inequality is obtained by maximizing the function co (x) = x(.1.x 2 + 1) -2 on R. Taking adjoints of the equation S A = TUA + T and rearranging yields = 0. lim 3 4(3.1)A

lim 11 T * U AT* = lim 11 SAII .1.-+ co A-4. co

Since each — UA T* is in 3, ands is closed, we have T* in 3 and hence 3 is self-adjoint. Now 91/3 is a Banach algebra and the mapping (A +3)* = A* + 3 is the involution induced by the natural map. Since we have II(A+- t3 ) * 11 = 11A + 311, it follows that 11(A+ 3) * (A+ 3)11

11(A+ 3) * I111A+ 3 11 = 11A+ 3 1 2,

and hence only the reverse inequality remains to be proved before we can conclude that 91/3 is a C*-algebra. Returning to the previous notation, if we set St = CH r) 3, then St is a closed two-sided self-adjoint ideal in the abelian C*-algebra CH, and hence CH /St is a C*-algebra by the previous lemma. If we consider the subalgebra n(C H ) = CH /3 of 91/3, then there is a natural map 7C' from CH/S1 onto CH /3. Moreover, it follows from the definition of Si and the quotient norm that it is a contractive isomorphism. Therefore, for A in CH, we have '

ucHig (A+ =

s(A+ Z)

and hence 11A+ StileHim

11A+ 3.11e11/3

p e ,,, / 3 (A + 3) = p eHig (A $) = II A+ St II efigt

Thus, it' is an isometry and C H /3 is an abelian C*-algebra. Lastly, it follows from the functional calculus for C H /-3 and the special form of the function = Ax2 /(1+.01..x 2 ), that IIn(UA)II = 2 1 74 11 4 2 ( 1 + lin(H)0 2 ) -

To complete the proof we use the identity T= SA — TUA to obtain iin(S A)11 + Setting A

x( 3 11 it

(

H

)

lin(U

3

4(3A) 1/4 +

All M°On( 11 )11 2 I +A 117(11)112

11 2 ) , we further obtain the inequality

3(1174 4H)11) 1/2 Iln( n11

+

41 Il

n(T)11

Quotient C*-Algebras 141

and finally Iln(n112 11n(H)11 = lin(T) * n(n11• Therefore, 91/Zs' is a C*-algebra. IIII We now consider the algebra 2E(Ye) which plays a fundamental role in the study of C*-algebras. The following result has several important consequences. A subset Z of Q(i9) is said to be irreducible if no proper closed subspace is reducing for all S in S. 5.39 Theorem If 91 is an irreducible C*-algebra contained in Q(Ye) such that 91 n 2C (Ye) 0 0, then 2C( Ye) is contained in 21.

Proof If K is a nonzero compact operator in 91, then (K+K*) and (110(K — K*) are compact self-adjoint operators in 21. Moreover, since at

least one is not zero, there exists a nonzero compact self-adjoint operator H in 91. If A. is a nonzero eigenvalue for H which it must have, then the projection onto the eigenspace for A is in CH and hence in 91 using the functional calculus. Moreover, since this eigenspace is finite dimensional by Theorem 5.22, we see that 21 must contain a nonzero finite rank projection. Let E be a nonzero finite rank projection in 21 of minimum rank. Consider the closed subalgebra 91 E = { EAE: A € 91} of 91 as a subalgebra of Ere. If any self-adjoint operator in 91 E were not a constant multiple of a scalar, then 91 E and hence 91 would contain a spectral projection for this operator and hence a projection of smaller rank than E. Therefore, the algebra 91 E must consist of scalar multiples of E. Suppose the rank of E is greater than one and x and y are linearly independent vectors in its range. Since the closure of {Ax: A E .W} is a reducing subspace for 91, it follows that it must be dense in W. Therefore, there must exist a sequence {A„}„_ 1 in 91 such that lim„,11A„x—yll = 0, and hence lim„, co llEA„ Ex —yll = 0. Since x and y are linearly independent, the sequence {EA„E},71 1 cannot consist of scalar multiples of E. Therefore, E must have rank one. We next show that every rank one operator is in 21 which will imply by Theorem 5.9 that 2C(Ye) is in 91. For x and y in Ye, let 7;,,„ be the rank one operator defined by 7;,,,(z) = (z, x) y. For a unit vector x 0 in Ere, if {A n }:_ 1 is chosen as above such that lim„„ 11 A n x 0 — Yll = 0, then the sequence {A.Tx„,„,,}„'_.. 1 is contained in 91 and limn A„7:„,,, x0 = Ty Xo. Similarly, using adjoints, we obtain that Tx0 ,,„ is in 21 and hence finally that Ty. ,e = 7 3 0 7;0 ,„ is in 91. Thus .VE(Ye) is contained in 21 and the proof is complete. 111 ,

1

,,„

142 5 Compact and Fredholm Operators, Index Theory

One of the consequences of this result is that we are able to determine all representations of the algebra .NCO. 5.40 Theorem If 1 is a *-homomorphism of K(Ye) into WS), then

there exists a unique direct sum decomposition S = 0 L E A Yir cc , such that each j" reduces (1)(QE(Ye)), the restriction faco(T)11 0 = 0 for T in Mpee), and there exists an isometrical isomorphism U G, from Ye onto .11 „ for a in A such that (1)(7)1Ac c, = TU ,* for T in 2C(Ye). .

-

G

-

Proof If ID is not an isomorphism, then ker 43 is a closed two-sided

ideal in 2C(Ye) and hence must equal QC(re), in which case (1)(7) = 0 for T in tC( ). Thus, if we set YC 0 = 1, the theorem is proved. Hence, we may assume that 1 is an isometrical isomorphism. Now let {ei } iE1 be an orthonormal basis for Ye and let Pt be the projection onto the subspace spanned by e i . Then Ei =c1)(P3 is a projection on . Choose a distinguished element 0 in I and define Vi on Ye for i in I such that Vi (EjEl Aj ej )-- It is obvious that Vi is a partial isometry with Vi Vi* = Pi and Vi* Vi = Po . Hence Wi is a partial isometry on dir and Wi* W, = E0 and Wi = Ei . Let {xo"}„ EA be an orthonormal basis for the range of E0 and set .4 wpcocc. It is easy to see that each xi" is in the range of Ei and that {.4} 1EI , , Eit is an orthonormal subset of dir . Let itc a denote the closed subspace of yr spanned by the {4} ic1 . The {S'„} ce EA are pairwise orthogonal and hence we can consider the closed subspace A—doce A0dra of if. Lastly, let '' o denote the orthogonal complement of this subspace. We want to show that the subspaces f dr c1oct€A L.) to}have the properties ascribed to them in the statement of the theorem. Since Vi Vi * is the rank one operator on Ye taking ej onto ei , it is clear that the norm-closed *-algebra generated by the {V i} iE , is 2C(Ye). Therefore, +11)(Q.E(Ye)) is the norm-closed *-algebra generated by the {Wi} t " and hence each Ac a reduces (Q (Ye)). If we define a mapping Uct from Ye to dr ce by U" e i 4, then UG, extends to an isometrical isomorphism and D(T)I" UG, TUce *. Therefore, 1 is spatially implemented on each Ara . Lastly, since each .11 a reduces cl)(2C(Ye)), it follows that s 0 also is a reducing subspace and since ,

-

Ei ID (T) (E

-

for T in 2C(Ye), ilD(T) i./ while /—E, E , Ei is the projection onto Yr o , we see that sli(T)Idr o = 0 for T in 2C(Ye). 1111 I./

Representations of the C' Algebra of Compact Operators 143

Such a result has a partial extension to a broader class of C*-algebras. 5.41 Corollary If 91 is a C*-algebra on Ye which contains QE(Ye) and 1' is a *-homomorphism of 91 into 2(YC) such that (1312 (Ye) is not zero and 0(91) is irreducible, then there exists an isometrical isomorphism U from Ye onto ir such that c1(A) = U AU* for A in 91. Proof If ilD(M(Ye)) is not irreducible, then by the preceding theorem there exists a proper closed subspace 1' of ir such that the projection P onto A" commutes with the operators (I)(K) for K in MVO, and there exists an isometrical isomorphism U from Ye onto ir'such that fico(K)1Y('= U KU * for K in 2C(Ye). (The alternative leads to the conclusion that cIcol.M(Ye) = 0.) Then for A in 91 and K in M(Ye), we have [PO (A) — 119 (A) Pi eto(K) = P (I9 (A) +ID (10 — 41) (A) +ID (K) P = Pc I)(AK) — 4: 0 (AK) P = 0, since AK is in 2E(Ye). If {E„},, EA is a net of finite rank projections in Ye increasing to the identity, then {c1)(4)11 }„ EA converges strongly to P, and thus we obtain AKA) P = t(A) P for every A in 91. Since 0(91) is selfadjoint and is assumed to be irreducible, this implies yr = Jr. Lastly, for A in 91 and K in M(Ye), we have -1

CI (K) [0 (A) — U AU *] = cro(K A) — (U KU *)(U AU *) = UKAU* — UKAU* = 0, and again using a net of finite projections we obtain the fact that 41)(A) = U AU* for A in 9,t. ■ These results enable us to determine the *-automorphisms of i (°) and 2C(Ye). 5.42 Corollary If .re is a Hilbert space, then CI is a *-automorphism of Q(.*) if and only if there exists a unitary operator U in 2(Ye) such that (13 (A) = U AU* for A in .V(Ye). Proof Immediate from the previous corollary. ■ Such an automorphism is said to be inner and hence all *-automorphisms of 2(Ye) are inner. A similar but significantly different result holds for 2C(Ye).

144 5 Compact and Fredholm Operators, Index Theory

5.43 Corollary If if is a Hilbert space, then (1) is a *-automorphism of QC(Ye) if and only if there exists a unitary operator U in .e (.W) such that (1)(K) = UKU* for K in 2E(Ye). Proof Again immediate. ■

The difference in this case is that the unitary operator need not belong to the algebra and hence the automorphism need not be inner. The algebra 2C(Ye) has the property, however, that in each *-isomorphic image of the algebra every *-automorphism is spatially implemented by a unitary operator on the space. We conclude with an observation concerning the Calkin algebra. 5.44 Theorem If (1) is a *-isomorphism of the Calkin algebra 2 into 2(1), then c.e(Ye)/2C(Ye)) is not a W*-algebra.

vevee(yo

Proof If (1)(2(Ye9/2E(*)) were a W*-algebra, then the group of in-

vertible elements would be connected by Proposition 5.29, thus contradicting Theorem 5.36. ■ Notes

The earliest results on compact operators are implicit in the studies of Volterra and Fredholm on integral equations. The notion of compact operator is due to Hilbert, while it was F. Riesz who adopted an abstract point of view and formulated the so called "Fredholm alternative." Further study into certain classes of singular integral operators led Noether to introduce the notion of index and implicitly the class of Fredholm operators. The connection between this class and the Calkin algebra was made by Atkinson [4]. Finally, Gohberg and Krein [48] systematized and extended the theory of Fredholm operators to approximately its present form. The connection between the components of the invertible elements in the Calkin algebra and the index was first established by Cordes and Labrouse [23] and Coburn and Lebow [22]. Further results including more detailed historical comments can be found in Riesz and Sz.-Nagy [92], Maurin [79], Goldberg [51], and the expository article of Gohberg and Krein [48]. The reader can also consult Lang [74] or Palais [85] for a modern treatment of a slightly different flavor. Again the results on C*-algebras can be found in Dixmier [28]. The proof of Theorem 5.38 is taken from Naimark [80], whereas the short and clever proof of Lemma 5.7 is due to Halmos [58].

Exercises 145

Exercises 5.1 If Ye is a Hilbert space and T is in 2(Yr), then T is compact if and only if (T*T) / is compact. 1

2

5.2 If T is a compact normal operator on If, then there exists a sequence of complex numbers RIG° 1 and a sequence {En }:1 1 of pairwise orthogonal finite rank projections such that lim, co A„ = 0 and N

lira AL+ CO

T— E An E„ = 0. n=1

5.3 If Ye is a Hilbert space, then 2E(Ye) Is strongly dense in 2(Ye). 5.4 If Ye is a Hilbert space, then the commutator ideal of 2(Ye) is £('). 5.5 If K 1 and K2 are complex functions in L2 ([0, I] x [0, 1]) and T 1 and T2 the integral operators on L2 ([0, 1]) with kernels K 1 and K2, respectively, then show that T1 * and T1 T2 are integral operators and determine their kernels. 5.6 Show that for every finite rank operator F on L2 ([0, 1]), there exists a kernel K in L2 ([0, 1] x [0, 1]) such that F= TK. 5.7 If TK is an integral operator on L2 ([0, 1]) with kernel Kin L2 ([0, I] x [0, 1]) and {f,j,,"_ 1 is an orthonormal basis for L2 ([0, 1]), then the series CO

I I(Tici„,f,)1 2

n=1

converges absolutely to II, f 1K(x,y)1 2 dx dy. (Hint: Consider the expansion of K as an element of L2 ([0, I] x [0, 1]) in terms of the orthonormal basis

fft,(x)f.(y)},T,.. 1.) 5.8 Show that not every compact operator on L2 ([0, 1]) is an integral operator with kernel belonging to L2 ([0, I] x [0, 1]). 5.9 (Weyl) If T is a normal operator on the Hilbert space W and K is a compact operator on ye, then any A, in u(T) but not in ff(T+ 10 is an isolated eigenvalue for T of finite multiplicity. 5.10 If T is a quasinilpotent operator on Ye° for which T+ T* is in QC(), then T is in .M(e).

146 5 Compact and Fredholm Operators, index Theory

5.11 If T is an operator on Ye for which the algebraic dimension of the linear space Ye/ran T is finite, then T has closed range.* 5.12 If T is an operator on Ye, then the set of A for which T — A is not Fredholm is compact and nonempty. 5.13 (Gohberg) If T is a Fredholm operator on the Hilbert space Ye, then there exists s > 0 such that = dim ker (T— A) is constant for 0 < 1.11 < s and a dim ker T.* (Hint: For sufficiently small A, (T— 42 is right invertible, where 2 = nnc°›ciTn Ye is closed, and ker(T A) c 2.) —

5.14 If T is an operator on Ye, then the function dim ker (T— A) is locally constant on the open set on which T — 2 is Fredholm except for isolated points at which it is larger. 5.15 If H is a self-adjoint operator on Ye and K is a compact operator on Ye, then u(H+ K)/u(H) consists of isolated eigenvalues of finite multiplicity. 5.16 If Ye is a Hilbert space and n is the natural map from £(.Ye) to the Calkin algebra 2(Ye)/2E(Ye) and T is an operator on Ye, then n (T) is selfadjoint if and only if T = H +1K, where H is self-adjoint and K is compact. Further, m(T) is unitary if and only if T = V+ K, where K is compact and either V or V* is an isometry for which I — VV* or 1—V*V is finite rank. What, if anything, can be said if n(T) is normal?** 5.17 (Weyl—von Neumann). If H is a self-adjoint operator on the separable Hilbert space Ye, then there exists a compact self-adjoint operator K on Ye such that H+ K has an orthonormal basis consisting of eigenvectors.*(Hint: Show for every vector x in Ye there exists a finite rank operator F of small norm such that H+ F has a finite-dimensional reducing subspace which almost contains x and proceed to exhaust the space.) 5.18 If U is a unitary operator on the separable Hilbert space Ye, then there exists a compact operator K such that U+ K is unitary and ye has an orthonormal basis consisting of eigenvalues for U+ K. 5.19 If U. is the unilateral shift on / z (i + ), then for any unitary operator V on a separable Hilbert space Ye, there exists a compact operator K on (Z + ) such that U. +K is unitarily equivalent to U. 0 V on / (1 + )C) Jr.* (Hint: Consider the case of finite-dimensional Ye with the additional requirement that K have small norm and use the preceding result to handle the general case.)

Exercises 147

are isometries on the separable Hilbert space Ye and at least one is not unitary, then there exists a compact perturbation of V I which is unitarily equivalent to V2 if and only if dim ker V I * = dim ker V2 * . 5.20 If V1 and

V2

Definition If 91 is a C*-algebra, then a state 9 on 91 is a complex linear function which satisfies rp (A* A) 0 for A in 91 and 9(1) = 1. 5.21 If co is a state on the C*-algebra 91, then (A, B) = 9(B* A) has the properties of an inner product except (A, A) = 0 need not imply A = 0.

Moreover, 9 is continuous and has norm 1. (Hint: Use a generalization of the Cauchy—Schwarz inequality.) 5.22 If 9 is a state on the C*-algebra 21, then 91 = {A e 91: co(A*A) = 0} is a closed left ideal in 91. Further, 9 induces an inner product on the quotient space 91/91, such that m(B)(A + 91) = BA + 91 defines a bounded operator for B in 91. If we let ir k, (B) denote the extension of this operator to the completion Ye v of 91/91, then 7Cp defines a *-homomorphism from 21 into 2(Y e gp ). 5.23 If 91 is a C*-algebra contained in 2(1) having the unit vector f as a cyclic vector, then 9(A) = f) is a state on 91. Moreover, if it , is the representation of 91 given by 9 on Ye y , then there exists an isometrical isomorphism from Ye v onto c such that A = Orv (A) . 5.24 (Krein) If 2 is a self-adjoint subspace of the C*-algebra 91 con-

taining the identity and 9 0 is a positive linear functional on So (that is, 9 0 (A) ?; 0 for A 0), satisfying q (I) = 1, then there exists a state 9 on 91 extending 9 0 . (Hint : Use the Hahn—Banach theorem to extend co o to 9 and prove that 9 is positive.) 5.25 If 91 is a C*-algebra and A is in 9I then there exists a state 9 on 91 such that c o (A* A) = IIA11 2 . (Hint: Consider first the abelian subalgebra generated by A* A.) 5.26 If 91 is a C*-algebra, then there exists a Hilbert space

Ye and a *-iso-

metrical isomorphism it from 91 into 2 (Ye). Moreover, if 91 is separable in the norm topology, then Ye can be chosen to be separable. (Hint: Find a representation ir A of 91 for which II n A (A)II = II A II for each A in 91 and consider the direct sum.) 5.27 The collection of states on a C*-algebra 91 is a weak *-compact convex subset of the dual of 91. Moreover, a state 9 is an extreme point of the set

of all states if and only if nip (%) is an irreducible subset of 2(Ye p ). Such states are called pure states.

148 5 Compact and Fredholm Operators, index Theory

5.28 Show that there are no proper closed two-sided ideals in £E (WP).

(Hint: Assume 3 were such an ideal and show that a representation of 2Cpee)/3 given by Exercise 5.26 contradicts Theorem 5.40.) 5.29 If 91 is a Bariach algebra with an involution, no identity, but satisfying

1 7' 411 11 = 11 711 2 for T in 91, then 91 0 C can be given a norm making it into a C*-algebra. (Hint: Consider the operator norm of 91 0 C acting on 91.)

6 The Hardy Spaces

6.1 In this chapter we study various properties of the spaces H', H 2 , and H" in preparation for our study of Toeplitz operators in the following chapter. Due to the availability of several excellent accounts of this subject (see Notes), we do not attempt a comprehensive treatment and proceed in the main using the techniques which we have already introduced. We begin by recalling some pertinent definitions from earlier chapters. For n in 1 let xn be the function on T defined by A(e i°) = For p =1,2, co, we define the Hardy space:

en°.

2n H P = If e If (T): i f(e)z„(e w) dO = 0 for n > 01. It is easy to see that each HP is a closed subspace of the corresponding H(T), and hence is a Banach space. Moreover, since {x„},,, z is an orthonormal basis for L2 (T), it follows that H 2 is the closure in the L2 -norm of the analytic trigonometric polynomials g + . The closure of g+ in C(T) is the disk algebra A with maximal ideal space equal to the closed unit disk D. Lastly, recall the representation of If (T) into £ (L2 (T)) given by the mapping co —* M, , where My is the multiplication operator defined by Mgp f = cof for f in E(T). We begin with the following result which we use to show that H" is an algebra. 149

150 6 The Hardy Spaces

6.2 Proposition If co is in L" (T), then H 2 is an invariant subspace for Mg, if and only if co is in H" .

Proof If M H 2 is contained in H 2 , then co • I is in H 2 , since I is in H 2 , then cog, is contained in and hence co is in II' . Conversely, if co is in H 2 , since for p = 0 of x in 4_ we have ,

,

2rc

0

N 1227

(cop) x„ de = E a j (pxj ,„ dO = 0 j=o o

for n > 0.

Lastly, since H 2 is the closure of .9 4_, we have c o H 2 contained in H 2 which completes the proof. III 6.3 Corollary The space H" is an algebra.

Proof If cp and cif are in H , then Mai, H2 = mv (mil, H2) M4, H2 H 2 by the proposition, which then implies that colif is in H°°. Thus H is an algebra. 1111 The following result is essentially the uniqueness of the Fourier—Stieltjes transform for measures on T. 6.4 Theorem If p is in the space M(T) of Borel measures on T and f r dp, = 0 for n in Z, then p O.

Proof Since the linear span of the functions {x„}„, / is uniformly dense in C(T) and M(T) is the dual of C(T), the measure u represents the zero functional and hence must be the zero measure.

■

6.5 Corollary If f is a function in V(T) such that fo2rt f(ei e)x,i (e) dO = 0 for /I in Z, then f = 0 a.e.

Proof If we define the measure p on T such that p(E) = f E f(e) dO, then our hypotheses become IT dp = 0 for n in 1, and hence p = 0 by the preceding result. Therefore, f = 0 a.e.

■

6.6 Corollary If f is a real-valued function in H 1 , then f = a a.e. for some a in R.

Reducing Subspaces of Unitary Operators 151

Proof If we set a (1/27c) S f(eie ) dO, then a is real and Jo

( f—a) c10 = 0

X.1

for n O.

Since f— a is real valued, taking the complex conjugate of the preceding equation yields rir

(f— cc) in dO =

27r

(f— a) x_,, dO = 0

for n 0 .

Combining this with the previous identity yields

1.

.21c

(f—a)x„ dO = 0

for all n,

and hence f = a a.e. ■

6.7 Corollary If both f and f are in W, then f a a.e. for some a in C. Proof Apply the previous corollary to the real-valued functions l(f+ j) and l(f—j)/i which are in H 1 by hypothesis. III

We now consider the characterization of the invariant subspaces of certain unitary operators. It was the results of Beurling on a special case of this problem which led to much of the modern work on function algebras and, in particular, to the recent interest in the Hardy spaces. 6.8 Theorem If it is a positive Borel measure on T, then a closed subspace di of L2 (j) satisfies x 1 di' = ' if and only if there exists a Borel subset E of T such that LE (Y) = ff E 1-2 0*.fie d) = 0 for e" El .

Proof If = L E (p), then clearly x i A = . Conversely, if Xi A = A" and hence ' is a reducing then it follows that ' = X - 1 Xi = subspace for the operator Mx , on L2 (p). Therefore, if F denotes the projection onto X, then F commutes with Mx , by Proposition 4.42 and hence with My for co in C(T). Combining Corollary 4.53 with Propositions 4.22 and 4.51 allows us to conclude that F is of the form My for some co in Lc° (p),

and hence the result follows. 111 The role of H 2 in the general theory is established in the following description of the simply invariant subspaces for Mx,.

152 6 The Hardy Spaces

6.9 Theorem If it is a positive Bore! measure on T, then a nontrivial closed subspace di' of L2 (p) satisfies x i di'c de and nn>o Xn = { 0 } if and only if there exists a Borel 'function co such that I cm-1 2 d0/2n eind dif = (pH'.

Proof If co is a Borel function satisfying Icor dp = d012n, then the function klff = cof is p-measurable for f in H 2 and / 2/r = 27r 0 11 2 clO = Ilf 112 2 . INT112 2 I (OF -0—

Thus the image dZ of H 2 under the isometry PI' is a closed subspace of L 2 (p) and is invariant for Mx „ since x 1 (xPf) = P(x i f). Lastly, we have

n

xndif

=

n xnH2] =

n -0

{

0

}

and hence de is a simply invariant subspace for Mx ,. Conversely, suppose dl is a nontrivial closed invariant subspace for Mx , which satisfies n n >0 xn {0}. Then 2 = 0 xi di' is nontrivial and )6, 2 = Xn dl O xn+ t dl, since multiplication by x i is an isometry on L2 (14. Therefore, the subspace E't. 0 L2 is contained in d, and an easy argument reveals e (LT_ 0 0 x„ 2) to be n„> 0 x„ dl and hence {0}. If (p is a unit vector in 2, then co is orthogonal to xn di and hence to xn for n > 0, and thus we have = (Co, xn (P) =

0

Icor Xn

for n > 0.

Combining Theorem 6.4 and Corollary 6.6, we see that 191 2 dp = d012n. Now suppose 2 has dimension greater than one and (p' is a unit vector in 97 orthogonal to co. In this case, we have 0

=

(x.p,x.co') = coCo'xn--m dp

for n,m 0,

and thus ST xk dv = 0 for k in Z, where dv (p0' dp. Therefore, TO' = Op a.e. Combining this with the fact that Icol 2 dp = 14)1 2 dµ leads to a contradiction, and hence is one dimensional. Thus we obtain that yog + is dense in di and hence dl = (pH', which completes the proof. ■ The case of the preceding theorem considered by Beurling will be given after the following definition.

6.10 Definition A function co in

is an inner function if kpl

1 a.e.

Beurling's Theorem 153

6.11 Corollary (Beurling) If Tx , = MxI1H2, then a nontrivial closed subspace X of H 2 is invariant for Tx , if and only if X = 0/ 2 for some inner function co. Proof If 9 is an inner function, then 9g + is contained in H", since the latter is an algebra, and is therefore contained in H 2 . Since rp H 2 is the closure of cog + , we see that coH 2 is a closed invariant subspace for Tx ,. Conversely, if di' is a nontrivial closed invariant subspace for Ti „ then di satisfies the hypotheses of the preceding theorem for dp = dB/2n, and hence there exists a measurable function co such that di . coH 2 and

191 2 c10/2n = do 9127r . Therefore, 191 = I a.e., and since 1 is in H 2 we see that co = q l is in H 2 ; thus co is an inner function. II1 A general invariant subspace for Mx , on L2 (p) need not be of the form covered by either of the preceding two theorems. The following result enables us to reduce the general case to these, however. 6.12 Theorem If p is a positive Bore! measure on T, then a closed invariant subspace di for Mx , has a unique direct sum decomposition di' = ,ff, $0 .1i2 such that each of M 1 and eit 2 is invariant for Mx1 , Xi Xi = Xi,

and

n,

n o

,„x 2 = { 0 } .

Proof If we set di i = nn>oXnX, then X i is a closed invariant sub-

space for Mx , satisfying Xi t 1 = X i . To prove the latter statement observe that a function f is in X i if and only if it can be written in the form x„ g for some g in X for each n > 0. Now if we set X2 = di 9 x,, then a function f in di is in t2 if and only if (f, g) = 0 for all g in di p Since 0 = (f, g) = (x 1 f, x i g) and x i ./i i ../1 1 , it follows that x l f is in def 2 and hence ./g2 is invariant for Mx ,. If f is in n 1 in> 0 Xn Af21 then it is in X i and hence f = O. Thus the proof is complete. ■ .

Although we could combine the three preceding theorems to obtain a complete description of the invariant subspaces for Mx „ the statement would be very unwieldy and hence we omit it. The preceding theorems correspond to the multiplicity one case of certain structure theorems for isometries (see [66], [58]). To illustrate the power of the preceding results we obtain as corollaries the following theorems which will be important in what follows.

154 6 The Hardy Spaces

6.13 Theorem (F. and M. Riesz) Iff is a nonzero function in H 2 , then the set fe"e T: fie") = 01 has measure zero. Proof Set E {e" c T: f(e") = 0} and define = {g H 2 : g (e") = 0 for e E} . It is clear that A' is a closed invariant subspace for Tx , which is nontrivial since f is in it. Hence, by Beurling's theorem there exists an inner function yo such that X = coH 2 . Since 1 is in H 2 , it follows that cp is in X and hence that E is contained in {e" E T: rp (e") = 0}. Since kol = 1 a.e., the result follows. III 6.14 Theorem (F. and M. Riesz) If v is a Borel measure on T such that S T xn dv = 0 for n > 0, then v is absolutely continuous and there exists f in H 1 such that dv = f do. Proof If p denotes the total variation of v, then there exists a Bore! function 1i such that dv dp and IC = 1 a.e. with respect to p. If X denotes the closed subspace of L2 (.1) spanned by {xn : n > 0}, then dp

tri) =

x„ dv = 0,

and hence IT is orthogonal to de' in L2 (/1). Suppose X = X i 0 X2 is the decomposition given by Theorem 6.12. If E is the Borel subset of T given by Theorem 6.8 such that X i = LE2 then we have

p(E) = f

1 citt

II11 2

( 11,0E) = 0 ,

since file is in X i and 17/ is orthogonal to A'. Therefore, X i = {0} and hence there exists a p-measurable function c9 such that X = coH 2 and 1C0 1 2 dp det12n by Theorem 6.9. Since x i is in A', it follows that there exists g in H 2 such that x i = cog a.e. with respect to p, and since 4)  0 p a.e., we have that p is mutually absolutely continuous with Lebesgue measure. If f is a function in Ll (T) such that dv = f dO, then the hypotheses imply that f is in 11 1 , and hence the proof is complete. IIII Actually, the statements of the preceding two theorems can be combined into one: an analytic measure is mutually absolutely continuous with respect to Lebesgue measure.

The Maximal Ideal Space of

155

6.15 We now turn to the investigation of the maximal ideal space M c, of the commutative Banach algebra H. We begin by imbedding the open unit disk in M. For z in ID define the bounded linear functional coz on

such that I 1 27r f(eie) a (f) Cozki = 271 Jo 1— ze - to

for f in H 1 .

Since the function 1/(1— ze -1°) is in (T) and H 1 is contained in /2(1), it follows that co z is a bounded linear functional on H 1 . Moreover, since 1/(1 — ze - t e) = 0 e - ul e zn and the latter series converges absolutely, we see that

( 9z (f) = I zk —

00



a

27r jop k

k=0

do)

Thus, if p is an analytic trigonometric polynomial, then p z (p) = p (z) and hence co z is a multiplicative linear functional on ,9+ . To show that p z is multiplicative on we proceed as follows. 6.16 Lemma If f and g are in H 2 and z is in ED, then fg is in 11 1 and

(Mfg) 9,( e) (MOProof Let fp nLI., and {gn}:° polynomials such that

lim I f pn II 2

11-4. CO



1

be sequences of analytic trigonometric



—

ti-s CO

I g — q.11 2 = 0 .

Since the product of two L2 -functions is in L', we have

+ 11p. g Ilfg —pn qnlli < Ilfg < lif—pn11211g112 + 11/3.112 Ilg and hence lim n _. co II fg — P n qn II 1 = 0. Therefore, since each p,,q,, is in H 1 we have fg in 11 1 . Moreover, since (p z is continuous, we have Coz (fg) = lim (P2(p.q.) = llm coz(13,-,) 11TH c9z(qO co

n-4 co

n -+ co

92(f)C9z(g). 111

With these preliminary considerations taken care of, we can now imbed in Mco . 6.17 Theorem For z in BF the restriction of 9 2 to Hc° is a multiplicative linear functional on H". Moreover, the mapping F from into M oc, defined by F(z) = p z is a homeomorphism.

156 6 The Hardy Spaces

Proof That (p z restricted to Hc° is a multiplicative linear functional follows from the preceding lemma. Since for a fixed f in H', the function 'o z (f) is analytic in z, it follows that F is continuous. Moreover, since yo z ( z,) = z, it follows that F is oneto-one. Lastly, if 19 9.},, EA is a net in M op converging to 9 z , then limz„ = lim co z .(x i ) = 9.(Xi) =

aEA

Z,

acA

and hence F is a homeomorphism. III From now on we shall simply identify D as a subset of M oo . Further, we shall denote the Gelfand transform of a function f in H" by .11 Note that :fl D is analytic. Moreover, for f in H' we shall let I denote the function defined on D by j(z) = 9 z (f). This dual use of the "-notation should cause no confusion. The maximal ideal space M oo is quite large and is extremely complex. The deepest result concerning MOD is the corona theorem of Carleson, stating that D is dense in A 1 co . Although the proof of this result has been somewhat clarified (see [15], [39) it is still quite difficult and we do not consider it in this book. Due to the complexity of M c° it is not feasible to determine the spectrum of a function f in H. ' using f, but it follows from the corona theorem that the spectrum off is equal to the closure of RD). Fortunately, a direct proof of this result is not difficult. 6.18 Theorem If 9 is a function in H', then 9 is invertible in Hc° if and only if 01 D is bounded away from zero. Proof If 9 is invertible in H", then (P is nonvanishing on the compact space M c° and hence 10(z)I..›.- e > 0 for z in D. Conversely, if KO (z)I .,>, e > 0 for z in D and we set tp(z) = 11(p(z), then ti/ is analytic and bounded by I/e on D. Thus ip has a Taylor series expansion tp(z) = E nc°_ o a n ?, which converges in D. Since for 0 < r < 1, we have

l an i2 r 2. _ _ E 2n 0 n=0 co

I

j' 271

lip (reit) 1 2 dt c 1 , e

it follows that LT_ o Ia n ! 2 -...5. 1/e 2 . Therefore, there exists a function f in H 2 such that f, E" an Li If co --= LT. 0 b n z„ is the orthonormal expansion of 9 as an element of H 2 , then 0(z) = Ec°. 0 bn e for z in D. Since 0(z) tP(z) = 1, it follows that

The Maximal Ideal Space of H° 157

(E n". 0 b n z") (LT. 0 an e)= I for z in D. Therefore, E n"... 0 (EZ =0 b k an _ of = 1 for z in ED, and hence the uniqueness of power series implies that n

1

b k an _ k =

k=0

( 1

if n = 0,

0

if n > 0.

Since

9— vi bhx.

lim

n=0

Pol—+ OC

M

2

=

lim

M ■ co —

1 mE= amxm 0 —

. 2

0,

we have that lirn l(Pi — (iv i bnX)( o am Xm) iv-..0

0,

1

which implies that 2/V

lim i (9f— l) + N-3 co I

N

----n= N+1

0

for

en =

1

I ak b,i _ k .

k=n— N

Therefore, —I .1 2r(PfXk dO

2ir 0

(I

0

if k

=

0,

if k  0,

and hence cof = 1 by Corollary 6.6. It remains only to show that f is in L' (T) and this follows from the fact that the functions {fr }„ E(0 , 1) are uniformly bounded by 1/e, where fr (eit ) = Are), and the fact that iimi._. i II f—fil 2 = 0 Thus f is an inverse for co which lies in H". III The preceding proof was complicated by the fact that we have not investigated the precise relation between the function f on D and the function f on T. It can be shown that for f in H 1 we have limr , i )(re') --=f(e ir) for almost all e i t in T. We do not prove this but leave it as an exercise (see Exercise 6.23). Observe that we proved in the last paragraph of the preceding proof that if f is in H 2 and j is bounded on ED, then f is in H". We give another characterization of invertibility for functions in II' which will be used in the following chapter, but first we need a definition. 6.19 Definition A function f in H 2 is an outer function if clos[f.9 4 ] = H 2 . An alternate definition is that outer functions are those functions in H 2 which are cyclic vectors for the operator Tx , which is multiplication by x l on H2.

158 6 The Hardy Spaces

6.20 Proposition A function 9 in H' is invertible in II' if and only if 9 is invertible in If and is an outer function.

Proof If 1/(p is in H", then obviously 9 is invertible in If. Moreover,

since

1 clos[9g+] = 9 H2 9(— H 2 ) = H 2 , Co

it follows that 9 is an outer function. Conversely, if 1/9 is in UM, and 9 is an outer function, then 9H 2 = dos [9g + ] = H 2 . Therefore, there exists a function IA in H 2 such that (ptp = 1, and hence 1/(p = t ff is in H 2 . Thus, 1/(p is in Hc° and the proof is complete. III i

Note, in particular, that by combining the last two results we see that an outer function can not vanish on D. The property of being an outer function, however, is more subtle than this. The following result shows one of the fundamental uses of inner and outer functions. 6.21 Proposition If f is a nonzero function in H 2 , then there exist inner

and outer functions 9 and g such that f = (pg. Moreover, f is in H°3 if and only if g is in H". Proof If we set X = clos [PA, then X is a nontrivial closed in-

variant subspace for Tx , and hence by Beurling's Theorem 6.11 is of the form 9H 2 for some inner function 9. Since f is in X, there must exist g in H 2 such that f = (pg. If we set Ai = clos[gg i, then again there exists an inner function IA such that K = OP. Then the inclusion fg+ = (pgg+ c 901 2 implies 9H 2 = dos[fgAc 9'141 2 , and hence there must exist h in H 2 such that 9 = 9tPh. Since 9 and IP are inner functions, it follows that = h and therefore IP is constant by Corollary 6.7. Hence, clos[g.9,1 = H 2 and g is an outer function. Lastly, since IfI =i g I, we see that f is in Hc° if and only if g is. 111 i

We next show that the modulus of an outer function determines it up to a constant as a corollary to the following proposition. 6.22 Proposition If g and h are functions in H 2 such that g is outer, then

Ihl Igi if and only if there exists a function k in H 2 such that h = gk and lki - 0, then there exists an outer function g such that Ig I= Ill-

Proof If we set di = clos Efg41, then Mxi di is the closure of {fp: p E g+ ,p(0) = 0}. If we compute the distance from f to such an fp, we find that 2 r2ir 1 r27r L ? Ilf—fP112 2 = i If 1 2 11— pl 2 do — 27, I 1 1 —p1 2 dO ?- E 2 , n o and hence f is not in Mx , X. Therefore, X is simply invariant and hence the outer function exists by the preceding theorem. 1111 We can also use the theorem to establish the following relation between functions in H 2 and I-P. 6.26 Corollary If f is a function in H 1 , then there exists g in H 2 such that 1W = if1a.e.

Proof If f = 0, then take g = 0. If f is a nonzero function in 1-1', then there exists h in L2 such that 1h1 2 = If I. It is sufficient in view of the theorem to show that clos[hg + ] is a simply invariant subspace for Mx1 . Suppose it is not. Then x_. N h is in dos [hg.,] for /V > 0, and hence there exists a sequence of analytic trigonometric polynomials {p,r },T_ 1 such that inn ii/inh — X-Nhii2 = 0 . Since

1 1 2 i1 1 IIX—Nh — Pnh112 2 = ih2 X- 2N —2h 2 P,X-N+ h 2 P,,2 1d0 2ir 0

I r 2n

= — i 1h 2 X—N — h 2 (213, — Ph2 XN) I dO 2n 0

= 11 112 X- N — h 2 (2Pn — Pn2 XN) III, we see that h 2 x_N is in the closure, clos i [h 2 g+ ], of h 2 .9+ in V(T). Since there exists a unimodular function 9 such that f = g o 1, 2 , we see that the function x_ N f = (p(x_ N h 2 ) is in 9 dos i [h 2g ] = C1OS i [ fg +] c 111 for N > 0. This implies f 0, which is a contradiction. Thus clos[hg + ] is simply invariant and the proof is complete. III

The Conjugates of H' and

161

6.27 Corollary If f is a function in H', then there exist functions g, and g 2 in H 2 such that Igil = 1g21 Ur and f = g1g2•

Proof If g is an outer function such that Ig1 = (If D 1/2 , then there exists a sequence of analytic trigonometric polynomials {p,i }71. 1 such that

lim II gP. — 1 1 12 = o.

11-4 Co

Thus we have

lifp.2 —fp.2 111

Ilf(pn—pm)phIli + Ilf(pn—p,OP.111

119P.11211g(pn—p.)112 + Ilgp.11211g(p.-1412, and hence the sequence ffp n2 V_ I is Cauchy in the L 1 (T) norm and therefore converges to some function co in 11'. Extracting a subsequence, if necessary, such that lim„, 09 (fp,r 2)(e 1 ) = Q(e i 9 a.e. and lim n , (gp,i )(eit) = 1 a.e., we see that 9g 2 = f Since Ig 2 1 = If I a.e., we see that 191 = 1 a.e., and thus the functions g, = Qg and g 2 g are in H 2 and satisfy f = g, g 2 and 1911 = 1921 = Ur. ■ 6.28 Corollary The closure of Y+ in L1 (T) is H 1 .

Proof 1ff is in 11', then f = g,g 2 with g, and g 2 in H 2 . If {pn };',1. 1 and are sequences of analytic trigonometric polynomials chosen such If j that limn , 091191 — Pn112 = 1192 — 17,112 = 0, then {pn g},c.,° is a sequence of analytic trigonometric polynomials such that lim n , c9 II f—p n qn ll i = 0. III nN n00_

With this corollary we can determine the dual of the Banach space 11'. Before stating this result we recall that Ho p denotes the closed subspace lie HP:

1 —

2rc

.1' 2nf dO = 0 of HP

for p = 1,2, co.

6.29 Theorem There is a natural isometrical isomorphism between (11')* and Ifc)(11)/H o '.

Proof Since H is contained in L1 (T), we obtain a contractive mapping IP from /NT) into (H 1 )* such that

1 l 2n

{111 (9)] (f) = 27z 0 (Pi dO

for ( in /.4"(T) and f in H1.

162 6 The Hardy Spaces

Moreover, from the Hahn–Banach theorem and the characterization of L' (T)r, it follows that given 4:1) in (H 1)* there exists a function 9 in If (T) such that 1191100 = IP II and il(p) = O. Thus the mapping tP is onto and induces an isometrical isomorphism of if Mike' . Y. onto (H l )*. We must determine the kernel of T. if 9 is a function in ker 'F, then 1 r 2n for n,- 0, 2ir jo 9Xn dO = ['F ( p)](Xn) = 0 (

since each x,, is in H ' and hence 9 is in Ho ". Conversely, if 9 lies in H o ', then [TM] (p) = 0 for each p in g 4_, and hence p is in ker P by the preceding corollary. 11111 Although L' (T) can be shown not to be a dual space, the subspace H' is.

6.30 Theorem There is a natural isometrical isomorphism between (C(T)/A)* and H0 1 .

Proof If co is a function in H 0 1 , then the linear functional defined 1 i 21r

(DU) = 27c 0 f9 dO for f in C(T) —

is bounded and vanishes on A. Therefore, the mapping 1 f 2n 0 0 (f+ A) = 41:1(1) = — fp dO 2n 0

is well defined on C(T)/A and hence defines an element of (C(T)/A)*. Moreover, the mapping 'F (go) = IN is clearly a contractive homomorphism of Ho ' into (C(T)/A)*. On the other hand, if 0 0 is a bounded linear functional on C(T)/A, then the composition 1 on, where n is the natural homomorphism of C(T) onto C(T)/A, defines an element v of C(T)* = M(T) such that

(Dal+ A) =0(1)=ff dv for f in C(T) and II114 = 110 0 11. Since this implies, in particular, that f ir g dv = 0 for g in A, it follows from the F. and M. Riesz theorem that there exists a function co in H o ' such that

I f 2g

(D O (f+ A) = — 27r

0

f9 dO for f in C(T) and 1191 1 = II v II = II ( DoII -

Therefore, the mapping 'I' is an isometrical isomorphism of Ho ' onto (C(T)/A)*• 1111

Approximation by Quotients of Inner Functions 163

6.31 Observe that the natural mapping i of C(T)/A into its second dual L"(T)/H" (see Exercise 1.15) is i(f+ A) = f+ H". Since the natural map is an isometry, it follows that i [C(T)fA] is a closed subspace of if (T)/H" ) . Hence, the inverse image of this latter subspace under the natural homomorphism of /4" (T) onto if (OM' is closed, and therefore the linear span H"+C(T) is a closed subspace of INT). This proof that 11"+C(T) is closed is due to Sarason [97]. The subspace H " + C(T) is actually an algebra and is just one of a large family of closed algebras which lie between H" and If (T). Much of the remainder of this chapter will be concerned with their study. We begin with the following approximation theorem. .

6.32 Theorem The collection 9 of functions in L"(T) of the form O for tiff in H" and co an inner function forms a dense subalgebra of L"(T).

Proof That 9 is an algebra follows from the identities Nil 01)(112i/52) = (11111/ 2)(9192)

and IP 1 01 .

4- 111 2

92 = 011 192 ± 111 2 90 (91 92) •

Since 9 is a linear space and the simple step functions are dense in If, to conclude that .9 is dense in L' (T) it suffices to show every characteristic function is in clos„,[9]. Thus let E be a measurable subset of T and let f be a function in H 2 such that

if(e)i =

r 2

if e" E E, if e" 0 E.

The existence of such a function follows from Corollary 6.25. Moreover, since f is bounded, it is in H" and consequently so is 1 +f for n > 0. If 1+p ,_- co n g ,, is a factorization given by Proposition 6.21, when co n is an inner and g n is an outer function, then 'g n i = 11 +fni .?_. 1 and hence 1/g,, is in H" by Proposition 6.20. Therefore, the function 1/(1-1-fl = (1/g,,) Co n is in 9, and since lim,, , c0114 -1 /( 1 +P)11.0 = 0, we see that 1E is in clos,„[2]. Thus, .9 is dense in If (T) by our previous remarks. III We next prove a certain uniqueness result.

164 6 The Hardy Spaces

6.33 Theorem (Gleason-Whitney) If 43 is a multiplicative linear functional

on H" and L 1 and L2 are positive linear functionals on L°° (T) such that L i lH' = L 2 IH' = 01), then L, = L2. Proof If u is a real-valued function in L'IT), then there exists an invertible function 9 in II' by Proposition 6.20 and Corollary 6.25 such that

I 91 = e". Since L 1 and L2 are positive, we have

Li(1(P1) = L1(?)

1 0 (9)1 = iLi (9)1

and 0

1)

r - 1 1-e2 e) 9

l ) = L2 P1 ----C. L2 (TE

(e - u).

Multiplying, we obtain 1= 1 0 (9)1 (t)

1 (-)

Li(eu) L2 (e -u )

9

and hence the function 'IT) = 14 1 (etu) L2 (C I') defined for alt real t has an absolute minimum at t = 0. Since 'P is a differentiable function of t by the linearity and continuity of L, and L2, we obtain 'P'(t) = L 1 (uetu) L2 (e ru) — L 1 (en) L2(ue -ru). Substituting t ,--- 0 yields 0 = ‘111 (0) = L, (4)L2 (1) - L, (1) L2 (u), and hence L i (u) = L2 (U) which completes the proof. 111 -

634 Theorem If 2 is a closed algebra satisfying II' c 91 c If (T), then

the maximal ideal space A4.2, of 91 is naturally homeomorphic to a subset of M. Proof If 4) is a multiplicative linear functional on 91, then 4:101H" is a multiplicative linear functional on II' and hence we have a continuous

natural map n from Mg, into M. Moreover, let 4)' denote any Hahn-Banach extension of 0 to L"(T). Since if (T) is isometrically isomorphic to C(M L .) by Theorem 2.64, 11' is integration with respect to a Borel measure v on ML . by the Riesz-Markov representation theorem (see Section 1.38). Since v (ME-) = V( 1 ) = I = ii di' ii = iv' (ML.), the functional di' is positive and hence uniquely determined by 0111" by the previous result. Therefore, the mapping q is one-to-one and hence a homeomorphism. III Observe that the maximal ideal space for 91 contains the maximal ideal space for If (T) and, in fact, as we indicate in the problems, the latter is the ilov boundary of 91.

Subalgebras between H' and L' 165

We now introduce some concrete examples of algebras lying between H' and L°° (V). 6.35 Definition If E is a semigroup of inner functions containing the constant function 1, then the collection E H", 9 e 1} is a subalgebra of if (T) and the closure is denoted 9I E . The argument that 94 is an algebra is the same as was given in the proof of Theorem 6.32. We next observe that Hc° + C(T) is one of these algebras. 6.36 Proposition If E(x) denotes the semigroup of inner functions {xn : n 0}, then 94 (x) = H"+C(T).

Proof Since the linear span H" + C (T) is closed by Section 6.31, we have Hc° +C(U) = clos,,[11'+g]. Lastly, since H" + g = {1,14n : e the result follows.

n 0},

■

The maximal ideal space of 94 can be identified as a closed subset of M co by Theorem 6.34. The following more abstract result will enable us to identify the subset. 6.37 Proposition Let A' be a compact Hausdorff space, 91 be a function algebra contained in C(X) with maximal ideal space M, and E be a semigroup of unimodular functions in 2L If 91 1 is the algebra dos {On: VI G91, 9 e 1} and Mx is the maximal ideal space of 91 1 , then ME can be identified with {m e M: 10(01 = 1 for go c E}, where 0 denotes the Gelfand transform.

Proof If

is a multiplicative linear functional on 91 1 , then ‘11. 191 is a multiplicative linear functional on 91, and hence //OP) = ‘191 defines a continuous mapping from ME into M. If 'P i and 'P2 are elements of M I such that n (' 1 ) = r/('' 2 ), then 'P i 191 = ‘1/ 2 191. Further, for 9 in 1, we have 1 qf =

= 'P1 ((P) 1 = W2(9) = q12(i-P)

166 6 The Hardy Spaces

and thus 'P t = '11. 2 . Therefore, i is a homeomorphism of M x into M. Moreover, since I 111 (9)1 < II Coll =-- 1 and

IT(Q)1

--- 1 '11 (01

—

we have IIP (go) I = l for '1/ . in Mx and (p in 1; therefore, the range of is contained in 111/1 a Air

tvral

—

1 fnr

a 3 1. .

and only the reverse inclusion remains. Let m be a point in M such that 10(01 = 1 for every 9 in E. If we define on {ii/O: E QC, 9 E E} such that 'F (k Co) = t (m) (m), then 'F can easily be shown to be multiplicative, and the inequality 'F(101 = it(m)110(m)1 = shows that ‘11 can be extended to a multiplicative linear functional on 91 1 . Since n(T) = m, the proof is complete. ■ 6.38 Corollary If E is a semigroup of inner functions, then the maximal

ideal space Mx of 9I x can be identified with {m c M : 10 (m)I = 1 for 9 E El . Proof Since INT) = C(X) for some compact Hausdorff space X, the

result follows. ■ Using the Gleason—Whitney theorem we can determine the Gelfand transform in the following sense. 6.39 Theorem There is a homeomorphism n from M 09 into the unit ball of the dual of L"(T) such that ik (m) = q (in) (0 for tp in any algebra 91 lying between H and E° (T) and m in M. Proof For in in M co let q(m) denote the unique positive extension of

m to L" (T) by Theorem 6.33. Since a multiplicative linear functional on VI extends to a positive extension of in on if CO, we have 1( 171) = r1 (n) ( 1P) for tP in 91. The only thing to prove is that // is a homeomorphism. Recall that the unit ball of If (T)* is w*-compact. Thus if {m.,,L EA is a net in M c° which converges to in, then any subnet of {q(mc,)}„ A has a convergent subnet whose E

Abstract Harmonic Extensions 167

limit is a positive extension of m and hence equal to n (m). Therefore, ri is continuous and hence an into homeomorphism. ■ We now adopt the notation Co (m) = rJ (m) (9) for Q in INT). The restriction 01 D will be shown to agree with the classical harmonic extension of a function in (T) into the disk. We illustrate the usefulness of the preceding by proving the following result showing the unique position occupied by H"+C(T) in the hierarchy of subalgebras of if (T).

6.40 Corollary If 91 is an algebra lying between H" and 1,"(T), then either 91 = H" or 91 contains H"+C(T). Proof From Theorem 6.34 it follows that the maximal ideal space of 91 can be identified as a subset M 2, of M. If the origin in D is not in M v, , then x, is invertible in 2f (2,  0), and hence C(T) is contained in 91, whence the result follows. Thus suppose the origin in D is in M. Since 1 12n

co ds

_ 2n

defines a positive extension of evaluation at 0, it follows that 0(0) = (1/2n) J ori co dt. If Q is contained in 9.1 but not in H", then (1/2n) Z ir (att dt  0 for some n 0, and hence 0  (an(0 ) = 0(0)2(0) = 0. This contradiction completes the proof. III One can also show that either /14 2, is contained in M., D or 91 = H". Before we can apply this to H" + C(T) we need the following lemma on factoring out zeros. .

6.41 Lemma If (p is in 11" and z is in D such that 0(z) = 0, then there exists ti/ in H" such that Q = (x i — z)1,11. Proof If 0 is in H then pi (z) = (z) (z) = 0. If 09 = Ll.. 0 an xn is the orthonormal expansion of 09 viewed as an element of H 2 , then co I an ? = 09(z) = 0 n=0

by Section 6.15, and hence CO 1 (Op, = (E an x,„ e - x). a n e = O. 1— zx i n=o n= 0 n-0 CO

CO

168 6 The Hardy Spaces

Therefore, we have 2n 1 f 2ni l(P ( X19 d dB = 4 , ) = 2.7c 1 1 94 - t jo Xk 1— zil f 1— l — zX i 0 zii

2ir

_

(Xk—

19, 1 _ 1. ,-)ci ) =

o

for k = 1,2, 3, ..., and hence the function k i 9/(1— zjc 1 ) is in H" . Thus setting tp = 2 1 91(1— z2 1 ), we obtain (x — z) VI = 9. III -

6.42 Corollary The maximal ideal space of H ' + C(T) can be identified with M o9 \ED.

Proof From the preceding corollary, we have MH-4-con = {in € Moo: 121(01 = 1 1' It remains to show that this latter set is M oo \ED. Let m be in M oo such that 121(01 < I and set 2 1 (m) = z. If 9 is in II' , then 9 — 0 (z)1 vanishes at z, and hence by the preceding lemma we have 9— 0 (z)1 = (x, — z) i/i for some IP in II'O. Evaluating at m in M„, we have

ep (m) — t (z) = (2 1 (m) — z) (z) = 0, and hence (p(m) = Co(z). Therefore, m = z and the proof is complete. III In the next chapter we shall be interested in determining when functions in H" + C(T) are invertible. From this point of view, the preceding result seems somewhat unfortunate since the only portion of the maximal ideal space of H" over which we have some control, namely ED, has disappeared. We shall show, however, that the question of invertibility of functions in H" + C (T) can be answered by considering the harmonic extension of the function on D. Our motivation for introducing the harmonic extension is quite different from that considered classically. We begin by determining a more explicit representation for 0 on D. 6.43 Lemma If z = re is in ED and 9 in L"(T), then

1

co

0 ( z ) = E an rIni eit' = — in=

-

00

f2n

27r 0

9 (el') kr (0 — t) dt,

The Maximal Ideal Space of IP° + C 169 -

where 1-2

kr (t) ,

1

—

and an = — I f 27 9x- n dt. 2r cost + r 2 2ir 0

Moreover, 00110 E for z in ID, there exists th, tfr 2 , ...,N in H" such that Er_ i Bpi Vi z = 1. Prove this statement under the additional assumption that the Cp i are in H'+C(T). 6.12 If 91 is a closed algebra satisfying Hc° c 91 c L°° (T), then the maximal ideal space of L' is naturally embedded in M94 as the gilov boundary of 91 s

6.13 (Newman) Show that the closure of ID in Moc, contains the Silov

boundary. Definition An isometry U on the Hilbert space Ye is pure if {0}. The multiplicity of U is dim ker U.

n

n

,.

0

CO =

6.14 A pure isometry of multiplicity one is unitarily equivalent to T xt on H . 2

6.15 A pure isometry of multiplicity Nis unitarily equivalent to

E i „ci.- 191 on T, then (k > 101 on M oo .

6.25 A function f in W is an outer function if and only if the inequality If I 191 on T implies 1f1 on Moo for every g in W.

141

6.26 (Jensen's Inequality) If f is a function in II 1 , then

log 1 j(0) 1 < s-r

1 f2n 0

log If(eil)Idt.*

Exercises 175

(Hint: Assume that f is in A and approximate log(1f1+ E) by the real part u of a function g in A; show that logi' 1 J 0:91 — 0 (0) < E and let E tend to zero.) 6.27 (Kolmogorov-Krein) If p is a positive measure on T and p a is the absolutely continuous part of p, then inf f 1 1 —f1 2 dtta.* inf 1 11 —f1 2 dit =feA0 T .

feA 0 T

(Hint: Show that if F is the projection of 1 onto the closure of A o in L2 (a), then 1-F = 0 a.e. its , where its is the singular part of it.) 6.28 A function f in H 2 is outer if and only if

inf — 1 5 27 11-W11 2 dB = 1/(0)1 2 .*

heyl o 2n o

(Hint: Show that f is outer if and only if 1 -1(0)If is the projection of 1 on the closure of A 0 in L2 (1/1 2 C1(9 )•)

6.29 (SzegO) If p is a positive measure on T, then

inf f 11 -f1 2 dp = exp (1 f Tr log h do), 0 eA0 IT 2n where h is the Radon-Nikodym derivative of it with respect to Lebesgue measure.* (Hint: Use Exercise 6.27 to reduce it to ,u of the form w dB; use the geometric-arithmetic mean inequality for one direction and reduce to the case 1h1 2 d0 for h an outer function in the other.) 6.30 If 91 is a closed algebra satisfying H" c 21 c Et) (1), then 21 is generated by H" together with the unimodular functions u for which both u and ft are in 21. (Hint: Iff is in 21, then f+ 211f II = ug, where u is unimodular and g is outer.) 6.31 If F is a group of unimodular functions in La' (T), then the maximal ideal space Mr of the subalgebra 21r of Lc° (T) generated by H" and F can

be identified by

Mr ---- OW E Moo : Iil(m)I = 1 for u E F} . 6.32 Is every 91r of the form 21 1 for some semigroup I of inner functions?** 6.33 Show that the closure of H +Fp is not equal to I,' (T).* (Hint: If arg z were in the closure of H" +R", then there would exist 9 in H" such that ze 4' would be invertible in II' .)

176 6 The Hardy Spaces

6.34 If m is in FA, and p. is the unique positive measure on M L.. such that

0 (m) =

f

0 du for v in If (7),

iilice

then p. is supported on Mi. n FA..* (Hint: Show that the maximum of I01 on FA is achieved on ML .0 n FA, for v in H".) ..----, 6.35 If (/) is a continuous function and ifr is a function in UM, then (p ' and $ are asymptotically equal on ED and equal on 114 /113. -

00

6.36 If (1) is a function in L`c (T), then the linear functional on H 1 defined by Lf = (1/27r)irfipd0 is continuous in the w*-topology on 11 1 if and only if cp is in H" + C(T). 6.37 Show 'that the collection PC of right-continuous functions on T possessing a limit from the left at every point of T is a uniformly closed self-adjoint subalgebra of If' (7). Show that the piecewise continuous functions form a dense subalgebra of PC. Show that the maximal ideal space of PC can be identified with two copies of T given an exotic topology. 6.38 If (1) is a function in PC, then the range of the harmonic extension of cp on FA is the closed line segment joining the limits of cp from the left and right at 2,. 6.39 Show that QC = [H' +C(11)]n[H" + C(T)] is a uniformly closed self-adjoint subalgebra of if (T) which properly contains C(T). Show that every inner function in QC is continuous but that QC n H" 0 A.* (Hint: There exists a real function v in C(T) not in Re A; if tfr is a real function in L2 (T) such that (p +NI is in H 2 then ev +11fr is in H" and e* is in QC.) 6.40 If u is a unimodular function in QC, then VII = I on Moo /D. Is the converse true?** 6.41 Show that ilf,,\D is the maximal ideal space of the algebra generated by H and the functions u in if (T) for which PI has a continuous extension Is this algebra 11" + C(T)?** to

a

6.42 Show that PC n QC .-- C(T). (Hint: Consider the unimodular functions in the intersection and use Exercises 6.38 and 6.40.) 6.43 Show that there is a natural isometrical isomorphism between H" and (L1 (T)/1/0 1 )*. Show that the analytic trigonometric polynomials g+ are w*-dense in H.

7

Toeplitz Operators

7.1 Despite considerable effort there are few classes of operators on Hilbert space which one can declare are fully understood. Except for the self-adjoint operators and a few other examples, very little is known about the detailed structure of any class of operators. In fact, in most cases even the appropriate questions are not clear. In this chapter we study a class of operators about which much is known and even more remains to be known. Although the results we obtain would seem to fully justify their study, the occurrence of this class of operators in other areas of mathematics suggests they play a larger role in operator theory than would at first be obvious. We begin with the definition of this class of operators.

7.2 Definition Let P be the projection of L2 (T) onto H 2 . For 9 in ED (T) the Toeplitz operator Tq, on H 2 is defined by f = P(9f) for f in H 2 . 7.3 The original context in which Toeplitz operators were studied was not that of the Hardy spaces but rather as operators on 1 2 (4). Consider the orthonormal basis {x,„: n E Z + } for H 2 , and the matrix for a Toeplitz operator with respect to it. If 9 is a function in L 0 (T) with Fourier coefficients 0(n) = (1/27r)5grr9x.... n dt, then the matrix fa„,,,J,,,,„ Ez , for T4, with respect to

{x,„: n E LE} is am , n --=

=

1 —

a

r

= (m—n). 177

178 7 Toeplitz Operators

Thus the matrix for Tv is constant on diagonals; such a matrix is called a Toeplitz matrix, and it can be shown that if the matrix defines a bounded operator, then its diagonal entries are the Fourier coefficients of a function in UM (see [11]). We begin our study of Toeplitz operators by considering some elementary properties of the mapping from ED (T) to (H 2 ) defined by (9) = Tv,. 7.4 Proposition The mapping is a contractive *-linear mapping from

(T) into (H 2 ). Proof That is contractive and linear is obvious. To show that (9)* = (ip), let f and g be in H 2 . Then we have -

(Tv f, g) = (P(ipf), g) (f, 9g) = (f, P (9g)) (f, Ty g) = (Tv * f, g), -

and hence (gyp)* = Tv * = Tv =

111

The mapping is not multiplicative, and hence is not a homomorphism. We see later that is actually an isometric cross section for a *-homomorphism from the C*-algebra generated by {4: 9 c Lc° (T)} onto If° (T), that is, if a is the *-homomorphism, then ao is the identity on Lcc(T). In special cases, is multiplicative, and this will be important in what follows. 7.5 Proposition If 9 is in L'`) (T) and 1// and B are functions in H , then Tv = TTY and To = Tov .

Proof Iff is in H 2 , then cif is in H 2 by Proposition 6.2 and hence T,fr f = POW) = tiff. Thus

7; f = Tv (W) P (Off) =

Top f

and Tv T = Tom .

Taking adjoints reduces the second part to the first. • The converse of this proposition is also true [11] but will not be needed in what follows. Next we consider a basic result which will enable us to show that is an isometry. 7.6 Proposition If 9 is a function in Lc° (T) such that Tq is invertible, then 9 is invertible in L' (T). ,

Proof Using Corollary 4.24 it is sufficient to show that M4, is an invertible operator if Tv, is. If Tv, is invertible, then there exists E > 0 such that

The Symbol Map 179

I T ,f II ->-- II 11I for f in H g

E

2

. Thus for each PI in Z and fin H 2 , we have

= ii9X.iii = ii(Pf IIP(c'f)II = II Tg,f EIIf II = EIlxnfII. Since the collection of functions {x n f: f E H 2 , 11 E Z} is dense in L2 (T), it follows that II My g II ENO for g in L2 (T). Similarly, IlMof II 11111, since To = Tq,* is also invertible, and thus Mg, is invertible by Corollary 4.9, which completes the proof.

•

As a corollary we obtain the spectral inclusion theorem. 7.7 Corollary (Hartman—Wintner) If 9 is in Q (Tv).

(1), then a(9)= c(M ) 4

Proof Since T4 — A = Tcp _A for A in C, we see by the preceding proposition that o- (M,) c 6(4). Since the identity 1(9) = a(M,) was established in ,

Corollary 4.24, the proof is complete. •

This result enables us to complete the elementary properties of 7.8 Corollary The mapping c is an isometry from Lc° (T) into 2(H 2). Proof /Using Proposition 2.28 and Corollaries 4.24 and 7 .7, we have for 9 in (T) that

ii(PL

040 r(T9) =

A E cr(4)}

sup {1 ,11: e ROM =

and hence is isometric. ■ 7.9 Certain additional properties of the correspondence are now obvious. If Tg, is quasinilpotent, then M(9) c a (TO = {0}, and hence Tq, = 0. If Tq, is self-adjoint, then (9) c cy(T9 ) c G and hence 9 is a real-valued function. We now exhibit the homomorphism for which is a cross section. 7.10 Definition If S is a subset of 12° (T), then t(S) is the smallest closed subalgebra of 2 (H 2 ) containing {Tg,: 9 E S}. 7.11 Theorem If E is the commutator ideal in Z(L`c (T)), then the mapping induced from L'`) (T) to Z(L°c(T))/E by is a *-isometrical isomorphism. Thus there is a short exact sequence

(0) —> —> Z(L' (T)) for which is an isometrical cross section.

(T) —> (0)

180 7 Toeplitz Operators

Proof The mapping

is obviously linear and contractive. To show that is multiplicative, observe for inner functions 9 1 and 9 2 and functions , that we have th and IP 2 in 1 (-T1n(Yfr202)

--

(Pi ( P14202)

= er aT.ell 24 2 - TIII1V1.1 2 1 2 T 1.1 7

1 ()

= 1 9*,(Ti, i T(P2 — 4 ,7, 2 ) 4 2 1

Tvt(Tiyi T(P*2 Tqt To1)TI2

Ttfr , is a commutator and E is an ideal, it follows that the Since To , T:2 - latter operator lies in E. Thus is multiplicative on the subalgebra tif H , 9 an inner function}

.2 =

of E° (T) and the density of .2 in L' (T) by Theorem 6.32 implies that is a *-homomorphism. To complete the proof we show that II T+KII HTvII for 9 in If (T) and K in E and hence that is an isometry. A dense subset of operators in ( can be written in the form K =

E

i= 1

T„,,,.„

where A l is in Z(Lf (T)), the functions 9 s , 9: , and A i are inner functions, the functions tfr i , 'Pi, and a u are in H", and square brackets denote commutator. If we set 0

n,rn

= i=

then 0 is an inner function and K(Of) 0 for f in H 2 . Fix E> 0 and let f be a function in H 2 chosen such that Ilf II = 1 and II Tcp f 1140 —E* If 9f= g 1 +g 2 , where g 1 is in H 2 , and g 2 is orthogonal to H 2 , then since 0 is inner we have that 0g 1 is in H 2 and orthogonal to 0g 2 . Thus ii(Tcp 4

-

I 0 (OD

Tcp

= P(9 19f)11

11 og 1 11 = 11g 1 11 = 11 7;1'11

7;11 — E,

and therefore II T9 + KII -;II TvII, which completes the proof.

•

A direct proof of this result which avoids Theorem 6.32 can be given based on a theorem due to Bunce [12]. In this case the spectral inclusion theorem is then a corollary.

The Symbol Map 181

The C*-algebra Z(Lf (10) is a very interesting one; the preceding result shows that its study largely reduces to that of the commutator ideal E about which very little is known. We can show that E contains the compact operators, from which several important corollaries follow. 7.12 Proposition The commutator ideal in the C*-algebra Z(C(T)) is ..VE(H 2). Moreover, the commutator ideal of Z(L" (1)) contains 2E(H 2). Proof Since the operator Tri is the unilateral shift, we see that the

commutator ideal of Z(C(T)) contains the nonzero rank one operator 711 7'xi — Txi T. Moreover, the algebra t(C(T)) is irreducible since Txi has no proper reducing subspaces by Beurling's theorem. Therefore, Z(C(T)) contains 2E(H 2) by Theorem 5.39. Lastly, since the image of Txi in Z(C(T))/.2E(H 2 ) is normal and generates this algebra, it follows that Z(C(T))/2E (H 2 ) is commutative, and hence .£E(H 2 ) contains the commutator ideal of Z(C(11)). To complete the identification of .2E(H 2 ) as the commutator ideal in Z(C(T)), it is sufficient to show that ..VE(1/ 2 ) contains no proper closed ideal. If 3 were such an ideal, then it would contain a self-adjoint compact operator H. Multiplying H by the projection onto the subspace spanned by a nonzero eigenvector, we obtain a rank one projection in Z. Now the argument used in the last paragraph of the proof of Theorem 5.39 can be applied, and hence 2CH 2 ) is the commutator ideal in Z(C(T)). Obviously, 2E(H 2) is contained in the commutator ideal of Z(Lf (T)). ■ 7.13 Corollary There exists a *-homomorphism C from the quotient algebra Z(E'D (T))1 2E (H 2) onto Lf (T) such that the diagram

Z (ED (T))_>.' Z (if (T ))/ (H 2 ) L' (T)

commutes. Proof Immediate from Theorem 7.11 and the preceding proposition. MI

7.14 Corollary If 9 is a function in I,' (T) such that Tv is a Fredholm operator, then 9 is invertible in E°(T). Proof If T c, is a Fredholm operator, then n(T9) is invertible in

Z(E° (T ))12 E (H 2 ) by Definition 5.14, and hence 9 = go 7r) (TT) is invertible in If° (T). •

182 7 Toeplitz Operators

7.15 Certain other results follow from this circle of ideas. In particular, it follows from Corollary 7.13 that II Tv 1 1(11> 040 for 9 in U(T) and - -

K in 2E (H 2 ), and hence the only compact Toeplitz operator is 0. Let us consider again the Toeplitz operator Txi . Since the spectrum of Txi is the closed unit disk we see, in general, that the spectrum of a Toeplitz operator Tv is larger than the essential range of its symbol 9. It is this phenomenon which shall largely concern us. In particular, we are interested in determining criteria for a Toeplitz operator to be invertible and, in addition, for obtaining the spectrum. The deepest and perhaps the most striking result along these lines is due to Widom and states that the spectrum of a Toeplitz operator is a connected subset of C. This will be proved at the end of this chapter. We now show that the spectrum of a Toeplitz operator cannot be too much larger than the essential range of its symbol. We begin by recalling an elementary definition and lemma concerning convex sets.

7.16 Definition If E is a subset of C, then the closed convex hull of E, denoted h (E), is the intersection of all closed convex subsets of C which contain E. 7.17 Lemma If E is a subset of C, then h (E) is the intersection of the open half planes which contain E. Proof Elementary plane geometry. • The lemma and the following result combine to show that cr(T,p) is contained in hP(9)).

7.18 Proposition If 9 is an invertible function in Et) (T) whose essential range is contained in the open right half-plane, then Tv, is invertible. Proof If A denotes the subset {z E C: IZ- II < 1}, then there exists an E > 0 such that ER (9) = {ez: z E A. Hence we have il Ecp —111 < 1 which implies III 71911 < 1 by Corollary 7.8, and thus 71,, = ET,,, is invertible by Proposition 2.5. MI

am} c

—

7.19 Corollary (Brown—Halmos) If 9 is a function in if (T), then a (TO c h(m (9))Proof By virtue of Lemma 7.17 it is sufficient to show that every open half-plane containing a (9) also contains cr(T9). This follows from the

The Spectrum of Analytic Toeplitz Operators 183

proposition after a translation and rotation of the open half-plane to coincide with the open right half-plane. • We now obtain various results on the invertibility and spectrum of certain classes of Toeplitz operators. We begin with the self-adjoint operators. 7.20 Theorem (Hartman—Wintner) If 9 is a real-valued function in UM,

then (7(4) =

[ess inf 9, ess sup 9].

Proof Since the spectrum of Ty is real it is sufficient to show that 7; — A invertible implies that either 9 — A >: 0 for almost all e" in T or 9 — A < 0 for almost all e l' in T. If Tv — A is invertible for A real, then there exists g in H 2 such that (T9 — A) g = 1. Thus there exists h in H0 2 such that ((p — A) g = 1 + h. Since (9 — Am = 1 + h is in H 2 , we have (9 A)1912 = (1+ h)g is in H', and therefore (9 — A)191 2 = a for some a in ER by Corollary 6.6. Since g 0 0 a.e. by the F. and M. Riesz theorem, it follows that 9 — A has the sign —

of a and the result follows. MI Actually much more is known about the self-adjoint Toeplitz operators. In particular, a spectral resolution is known for such operators up to unitary equivalence (see Ismagiiov [68], Rosenblum [94], and Pincus [86]). The Toeplitz operators with analytic symbol are particularly amenable to study. If 9 is in H', then the operator Ty, is the restriction of the normal operator My on L2 (T) to the invariant subspace H 2 and hence is what is called a subnormal operator. 7.21 Theorem (Wintner) If 9 is a function in H", then Tv is invertible if and only if 9 is invertible in H". Moreover, if 0 is the Gelfand transform of 9, then c(Tv) = dos [0 (D)]. Proof If 9 is invertible in H", then there exists V/ in H such that 9iii = 1. Hence I = To. Tv = Tv T, by Proposition 7.5. Conversely, if Ty is invertible, then 9 is invertible in L' (T) by Proposition 7.6. If V/ is 1/9, then To Ty, = Tov = I by Proposition 7.5, and hence To. is a left inverse for Ty,. Thus 4. T,,,1 and therefore 1 = 7; To l = 9P(t,P). Multiplying both sides by 1/9 = ti, we obtain P(ti) = i/i implying that t/i is in H" and completing the proof that 9 is invertible in H. The fact that o- (Ty) = clos[0 (D)] -

follows from Theorem 6.18. •

184 7 Toeplitz Operators

This result yields especially nice answers to the questions of when Tv is invertible and what its spectrum is for analytic 9. It is answers along these lines that we seek to determine. We next investigate the Toeplitz operators with continuous symbol and find in this case that an additional ingredient of a different nature enters into the answer. Our results on these operators depend on an analysis of the C*-algebra Z(C(T)). We begin by showing the is almost multiplicative if the symbol of one of the factors is continuous. 7.22 Proposition If 9 is in C(T) and tp is in iNT), then Tv To — Tvo and 4Tv — Tov are compact. Proof If tfr is in L°° (T) and f is in H 2 , then To Tx _ i f = To P(x_ i f) PM,k (x_ i f

= P(k-tf)

—

(f,l)P(k-i)

T!)-1 f — (f,l) P(tfrx-1),

and hence To.Tx _ i — c _ is a rank one operator. Suppose To Tx _. — Tox _. has been shown to be compact for every tfr in L' (if) and — N

(H 2 ) —> Z(C(T)) C(T) —> (0)

is short exact; that is, the quotient algebra i(C(T))/t (H 2 ) is *-isometrically isomorphic to C(T).

The C* AlgebraGenerated by the Unilateral Shift 185

Proof It follows from the preceding proposition that the mapping

of Corollary 7.13 restricted to Z(C(T))/2E(H 2) is a *-isometrical isomorphism onto C(T), and hence the result follows. III Combining this with the following proposition yields the spectrum of a Toeplitz operator with continuous symbol. 7.24 Proposition (Coburn) If 9 is a function in V° (T) not almost everywhere zero, then either ker T4, = {0} or ker T9 * = (0}. Proof If f is in ker Tq, and g is in ker Tq,*, then of and 9# are in H 0 2 . Thus (pfg and (751g are in He by Lemma 6.16, and therefore 9f# is 0 by Corollary 6.7. If neither f nor g is the zero vector, then it follows from the F. and M. Riesz theorem that 9 must vanish for almost all e ft in T, which -

is a contradiction. MI 7.25 Corollary If 9 is a function in If° (T) such that Tq, is a Fredholm operator, then Tip is invertible if and only if j(Tp) = 0. Proof Immediate from the proposition. •

Thus the problem of determining when a Toeplitz operator is invertible has been replaced by that of determining when it is a Fredholm operator and what is its index. If 9 is continuous, then this is readily done. The result is due to a number of authors including Krein, Widom, and Devinatz. 7.26 Theorem If 9 is a continuous function on T, then the operator Tv is a Fredholm operator if and only if 9 does not vanish and in this case j(T9) is equal to minus the winding number of the curve traced out by 9 with respect to the origin. Proof First, Tq, is a Fredholm operator if and only if 9 is invertible in C(T) by Theorem 7.23. To determine the index of Tv we first observe that AT() = j(70 if 9 and i/i determine homotopic curves in C\{0}. To see this, let (I) be a continuous map from [0,1] x T to C\{0} such that (I)(0, ea) = T. Then the mapping t —> To, is norm 9(e") and 0 (1. e fr) = i (e") for eat

continuous and each To is a Fredholm operator. Since j(Tot) is continuous and integer valued, we see that j(T4) = j(4). If n is the winding number of the curve determined by 9, then 9 is homotopic in C\{0} to xn . Since j(Txn ) = —n, we have j(Tip) = — n and the result is completely proved. • t

186 7 Toeplitz Operators

7.27 Corollary If 9 is a continuous function on T, then Tv, is invertible if and only if 9 does not vanish and the winding number of the curve determined by 9 with respect to the origin is zero. Proof Combine the previous theorem and Corollary 7.25.

■

7.28 Corollary If 9 is a continuous function on T, then

a (Tcp) = M(9) u (A E C: 4(9,A) 0 0} where 4(9, )) is the winding number of the curve determined by 9 with respect to A. In particular, the spectrum of 7 1, is seen to be connected since it is formed from the union of a (9) and certain components of the complement. In this case the invertibility of the Toeplitz operator T v depended on 9 being invertible in the appropriate Banach algebra C(T) along with a topological criteria. In particular, the condition on the winding number amounts to requiring 9 to lie in the connected component of the identity in C(T) or that 9 represent the identity in the abstract index group for C(T). Although we shall extend the above to the larger algebra II' + C(T), these techniques are not adequate to treat the general case of a bounded measurable function. We begin again by identifying the commutator ideal in Z(I-1' + C(T)) and the corresponding quotient algebra. 1

7.29 Theorem The commutator ideal in Z(I-1' + C(T)) is 2E(H 2 ) and the mapping K = no from II + C(T) to Z(11 ' + C(T))/2E(H 2 is an isometrical isomorphism. .

)

Proof The algebra Z(H" + C(T)) contains 2E(11 2), since it contains

Z(C(T)) and thus the mapping 4 is well defined and isometric by Corollary 7.13 and the comments in Section 7.15. If 9 and i/i are functions in H" and f and g are continuous, then Tv+ f T‘fr+g — Top+

MIA +0 = Tv + f

To. — Top+ no. 4- Tv+ f Tg — T(p+f)g

= Tv+ f Tg — 74 + f) g

by Proposition 7.5, and the latter operator is compact by Proposition 7.22. Thus the commutator ideal is contained in .2E (H 2) and hence is equal to it. Thus K is multiplicative, since ic

((p+f) ic (t,P i g) - -

—

409 + f) WI +

g)) = n[T" To. + 9 — T(p+ f)01, 4_ 0] = 0.

Invertibility of Unimodular Toeplitz Operators and Prediction Theory 187

Therefore, K is an isometrical isomorphism, which completes the proof.

■

+ C(T)) is not a C*-algebra, and hence the fact that Note that Tv is a Fredholm operator and n(Tv) has an inverse in t(H 2)/2E(H 2 ) does not automatically imply that n(Tv) has an inverse in Z(Hm+ C(T))/ 2'E (H 2 ). To show this we must first prove an invertibility criteria due to Widom and based on a result of Helson and Szego from prediction theory.

7.30 Theorem If 9 is a unimodular in E° (T), then the operator Tv is left < 1. invertible if and only if dist (9, Proof If dist (9, Hm) < 1, then there exists a function tfr in H" such

that 119 -111 11m < 1. This implies that II 1— 01,1/11c. < 1 and hence III— T < 1. Thus To.' TQ = T,Tv is invertible in .2(H 2) by Proposition 2.5 and therefore Tv is left invertible. Conversely, if T , is left invertible, then there exists E 0 such that II T^f II E 11111 for f in H 2 . Thus P(9i) II E = E 110 and hence 119/11 2 = ll(i — P)(9.nli 2 + liP(9n11 2 11(i— P)(9.f)11 2 Ell9f 1 2 Therefore, we have II(/— P)(9f)ii < ( 1 6 ) Ilf II for f in H 2 , where 6 = 1 — (1— > 0. If f is in H 2 and g is in H 0 2 , then 2

fo

'

(pfg dt = 1(9f, 01

P)(9f),

< (1 -5) Ilf 11 119

If for h in He we choose f in H 2 and g in H0 2 by Corollary 6.27, such that = fg and Oh° Ilf II 2 119 11 2, then we obtain 1 .12n

(ph dt 4 (1 — a) IIhe ,. 227r Thus, the linear functional defined by (I)(h) = (1/27r) hcp dt for h in H0 1 has norm less than one. Therefore, it follows from Theorem 6.29 (note that we are using (H0 1 )* = L`c)(T)/Hm), that there exists 1i in 1-1" such that II — II co < 1, which completes the proof. 111 —

7.31 Corollary If 9 is a unimodular function in Lm(T), then

is in-

vertible if and only if there exists an outer function such that II cP — II < 1 . IL, and hence lb II1100 < 1, then I till E > 0 for E -1 - 119 To. is invertible by Theorem 7.21 and Proposition 6.20. Since T,fr * Tc, is invertible, we see that Tv is invertible. Conversely, if T, is invertible and Proof If II

188 7 Toeplitz Operators

is a function in H such that 119 —01 0, < 1, then To. is invertible since To.* Tv, is, and hence Iii is an outer function.

■

As a result of this we obtain a very general spectral inclusion theorem. 732 Theorem If Z is a closed subalgebra of .V(H 2 ) containing Z(1-1') and Tv is in Z for some 9 in L"(T), then c(Tq,) = c x (7 1,). Proof Obviously cr(4) is contained in cr z (Tv). To prove the reverse inclusion suppose Tv is invertible. Using Corollary 6.25, we can write 9 , utfr, where u is a unimodular function and tii is an outer function. Conse-

quently, tt, is invertible in If (T), and by Proposition 6.20 we obtain that Tv is in Z. Thus T.= T4,4 1 is in Z, moreover, T. and hence ; is invertible in 2 (11 2 ). Employing the preceding corollary there exists an outer function 0 such that 110—aL < 1. Since T.7'13 is in Z and II/— T. To ll --111 — u0L, < 1, we see that (T. To ) - ' is in Z by Proposition 2.5. Since T;;' = 11 . 1 T.

1

= 4

1

To (T.To) -

is in Z, the proof is complete. MI This leads to a necessary condition for an operator to be Fredholm. 7.33 Corollary If 21 is a closed subalgebra of L°° containing H + C(T) and 9 is a function in 91 for which Tv, is a Fredholm operator, then 9 is invertible in 91. Proof If Tv is a Fredholm operator having index n, then T zap is invertible by Propositions 7.5 and 7.24, and the function x.9 is also in 91.

Therefore, by the preceding result, T x3, 91 is in Z(91), and hence using Theorem 7.11 we see that x.9 is invertible in p [Z(21)] = 21. Thus 9 is invertible in 21, which completes the proof. MI The condition is also sufficient for II' + C(T). 7.34 Corollary If 9 is a function in H'+C(T), then T9 is a Fredholm operator if and only if 9 is invertible in H' + C(T). Proof The result follows by combining the previous result and Theorem

7.29. MI

The Spectrum of Toeplitz Operators with Symbol in + C 189

We need one more lemma to determine the spectrum of Tq, for 9 in H' +C(T). Although this lemma is usually obtained as a corollary to the nctions, it is interesting to note that it also structure theory for inner fu ws follo from our previous methods. 7.35 Lemma If 9 is an inner function, then H 2

sional if and only if 9 is continuous.

e (pH 2 is finite dimen-

Proof If 9 is continuous, then 7; is a Fredholm operator by Theorem 7.26, and hence H 2 e (pH2 ker(Tv *) is finite dimensional. Conversely, if H 2 e 9H 2 is finite dimensional, then T4, is a Fredholm operator and hence 9 is invertible in H'+C(T). We need to show that this implies that 9 is continuous. By Theorem 6.45 there exist E,S> 0 such that 10(reh)I E for 1 — S < r < 1, and hence 0 has at most finitely many zeros z l , z 2 , ...,zA in ED counted according to multiplicity. Using Lemma 6.41 we obtain a function tJi in H" such that 1 (J1 — zi) = 9. Thus IA does not vanish on ED and is bounded away from zero on a neighborhood of the boundary. Therefore, is invertible in H" by Theorem 6.18. Moreover, since

e"

6'41

=

for eh in T,

we have

rri=

e it — zj 1—

= 1.

Thus the function (x, -zi)/(1 - zJ xi) is a continuous inner function, and B = L11[17_ 1 (1-5 j x i ) has modulus one on I. Since 0 is invertible in H", it follows that = 0 -1 is in H" and hence that 0 is constant. Thus 0

fir ( X — zi — fi Xi

and therefore is continuous. IN 7.36 Theorem If 9 is a function in H'+C(T), then T4, is a Fredholm operator if and only if there exist 6,e> 0 such that lep(rek )I E for 1 —6 < r < 1, where 0 is the harmonic extension of 9 to ID. Moreover, in this case

the index of Tq, is the negative of the winding number with respect to the origin of the curve 0(re h) for 1 —6 < r < 1. Proof The first statement is obtained by combining Corollary 7.34 and Theorem 6.45.

190 7 Toeplitz Operators

If 9 is invertible in H" + C(T) and E, 6 > 0 are chosen such that 10(re% ›. E for 1-6 < r < 1, then choose tfr in H" and a negative integer N such that 119 — XN tP II c, < c/3. If we set q = .1.9 + (1 — A) XN tfr for 0 ‘.. A -...C. 1, then each co a is in H" 4- C(T) and 119 — ( P AL < E/3. Therefore, 1 0 A (re t)1 > 243 for 1—.5 < r < 1, and hence each 9 A is invertible in H" + C(T) by Theorem 6.45. Hence each Tq,,,, is a Fredholm operator and the winding number of the curve (p ),(re) is independent of A and r for 0 -.^. A -..... 1 and 1—.5 < r < 1. Thus it is sufficient to show that the index of Tcp. = Txop is equal to the negative of the winding number of the curve (X N tfr)(rez t ). Since Tx,To = Tx,,, To we see that To is a Fredholm operator. If we write tfr = kJ' itp„ where tlio is an outer function and tp, is an inner function, then Too is invertible and hence 4 is a Fredholm operator. Thus ///, is continuous by the previous lemma which implies XN C is in C(T) and we conclude by Theorem 7.26 that 1

.i(Tx,41) = .i(Tr ioki) = it (XN tP1)

-

Since there exists by Lemma 6.44 a 6 1 > 0 such that 6 > 6 1 > 0 and IXN IP (ret)

— XN yr '(re t) IP. (re") I < j



for 1 — (5, < r < 1,

it follows that it ((XN 1,1i) (re"))

= it ((XN 3 (ret )) + 4 (1,io (re"))

for I — 6 1 < r < 1.

./..

Using the fact that th does not vanish on ED and Theorem 6.46, we obtain the desired result. MI We conclude our results for T4, with 9 in 11"+ C(T) by showing that the essential spectrum is connected. Although we shall show this for arbitrary 9, a direct proof would seem to be of interest. 7.37 Corollary If 9 is a function in H" + C(T), then the essential spectrum

of 7 4, is connected.

Proof From the theorem it follows that A is in the essential spectrum of Tv, for 9 in H" 4- C(T) if and only if 9 — A is not invertible in H" + C(T). Hence, by Theorem 6.46 we have that A is in the essential spectrum of T„, if

The Spectrum of Toeplitz Operators with Symbol in IP' + C 191

and only if A is in dos {(p (rest): 1 > r > 1 — 6}

for each 1 > (5 > 0.

Since each of these sets is connected, the result follows. • We now take up the proof of the connectedness of the essential spectrum for an arbitrary function in 1.,°°(T). The proof is considerably harder in this case and we begin with the following lemma. 7.38 Lemma If 9 is an invertible function in L''' (T), then Tv is a Fredholm operator if and only if Tik, is and moreover, in this case, j(T,p) = —j(Top).

Proof If we set 9 = utfr by Corollary 6.25, where ifr is an outer function and u is unimodular, then T119 = 1"„T11 ,fr = MT: Tofr by Proposition 7.5.

Since T 110 is invertible by Theorem 7.21, the result follows. ■ The proof of connectedness is based on the analysis of the solutions h and g a of the equations Ty,_ A ft = 1 and Ti/op-),)g), -, I. Since we want to consider A for which these operators are not invertible, the precise definition of f,, and g a is slightly more complicated. 7.39 Definition If 9 is a function in If (T), then the essential resolvent pe (TO for Ty, is the open set of those A in C for which Tp _A is a Fredholm operator. If A is in pe (Tip) and j(Tip _ ),) = n, then

ft = Tx:ttp- A) I and g

a

= Txlini op _ 01) 1.

The basic result concerning the fa and g a is contained in the following. 7.40 Proposition If 9 is a function in Lc° (T) and A is in p e (T9), then fa 9a = 1 and d

fA I (z) = h(z) - P { 1 (z) for z in ED. dA 9— A.

Proof There exist functions u a and v a in H0 2 such that Li(9 — A) ii. = 1 + 172, and x_„ g A1(9— A) = 1+6 A . Multiplying these two identities, we

obtain fAg A = 1 + (14),±v A +u A v A),

where /kg ), is in H i and u ),+v 2,+uA vA is in H0 1 and thus fa, g a = 1. We also have ja (z)# a (z) = 1 by Lemma 6.16.

192 7 Toeplitz Operators

Since the functions A —p x„I (9 — A) and A —) x_ nl (9 — are analytic L° valued functions, it follows from Corollary 7.8 that f t and g 2 are analytic H 2 -valued functions. Differentiating the identity Xn(9 — = 1 +16 with respect to A yields — xnik+ X„(9 = 171,1 1 and hence fx = (9 AVA 1 where 142 ' lies in H0 2 . Multiplying both sides of this equation by g 2 /(9— A), we obtain 1

—

1

9—A 9—A

g2 = g — (X-07 2 1 )(Xn( 1

A))

/01+ ij0Since fit' g A is in H' and /01+ ij A) is in H 0 ', we have P11/(9— A)) = L' g A . Finally, again using the identity proved in the first paragraph, we reach the equation L' f P {11(9 A)) and observing that evaluation at z in D commutes with differentiation with respect to A we obtain the desired results. • It is possible to solve this latter equation to relate the fA which lie in the same component of p.(4). 7.41 Corollary If 9 is a function in 1.")(T) and A and A. are endpoints of a rectifiable curve F' lying in p e (T 9), then

Jot (z) fAo (z) exp

P 1 (z) dp} r 9—p

and (z) =

(z) exp — ji P

1

r 9—

(z) dp}

for z in D. Proof For each fixed z in D we are solving the ordinary first order linear differential equation dx(A)IdA = F(A) x(A), where F(A) is an analytic function in A. Hence the result follows for f and the corresponding result for g A is obtained by using the identity It (z) A (z) = 1. 111

We now show that no curve lying in MT() can disconnect a (9). 7.42 Proposition If 9 is a function in 1.5 (T) and C is a rectifiable simple closed curve lying in pe (Tp), then .W (9) lies either entirely inside or entirely outside of C.

The Spectrum of Toeplitz Operators with Symbol in IP' -1--C 193

Proof Consider the analytic function defined by F(z) =

.......-----■....... 1 i If 9 1 27r

1 (z) dp Jul

for each z in D.

If A0 is a fixed point on C, then it follows from the preceding corollary that h . (z) = .12,0 (;) exp{2niF(z)}

for z in ED.

Therefore, exp{2,7riF(z)} = I wheneverlAjz) 0 0, and hence for all z in ED, since is analytic. Thus the function F(z) is integervalued and hence equal to some constant N. Now for each e u in T, the integral

L.

I I dp 27ri fc co (eit) — It equals the winding number of the curve C with respect to the point 9(e u). Thus the function defined by

IP (eu) =

1 i 1 dp a i jc co (ei) — tt

is real valued. Since

PIP = P{ 1 H. dp -= —1 . f P{ 1 } clp = F 2ni c 9— tt 27ri c 9— it is constant, we see that tfr is constant a.e., and hence the winding number of C with respect to 2 (9) is constant. Hence 2 (9) lies either entirely inside or entirely outside of C.

•

The remainder of the proof consists in showing that we can analytically continue solutions to any component of C\p e (T9) which does not contain a (9). Before doing this we need to relate the inverse of T xn op _ A) to the functions fA and g A .

7.43 Lemma If 9 is a function in UM, A is in pe (T,), k is in H', and h A = /A Pfx_ n g A k1(9— A)), then h A is in H 2 and Txn ",_ A) h A = k. Proof If we set h A = T x. (9 _ A) k, then there exists 1 in H0 2 such that x„(9— 2)h A = k +1. Multiplying by x__„g,1,1(9 A) = 1 + V A , where v A is in —

H0 2 , we obtain

gAliA — -jx,g A k + 1(1+EA) 9-2

194 7 Toeplitz Operators

and hence g A lt x = P{x_ n g A kl(v—A)}, since 1(1 4- v x) is in H0 1 . We now obtain the desired result upon multiplying both sides by fi and again using the identity fi g A = 1. IN We need one more lemma before proving the connectedness result. 7.44 Lemma If f is a simply connected open subset of C, C is a rectifiable simple closed curve lying in 1/, and F, (z) is a complex function on 1 x ID such that Fx (z.) is analytic on S2 for z0 in D, 1,10 (z) is analytic on ID for k, in [2, and FA, o is in H 2 for A. in C, then Fx is in H 2 for A in the interior of C

and 111,1 1 2 < sup x. EC II1F 4112' 1

Proof An analytic function iii on ED is in H 2 if and only if

i 1c, tp(n) (0) I 20 02 is finite, since f= LT_ 0 ono) (n!) -1 xn is in H 2 and f= Iii; moreover 2/010-1/2. lif 112 = (E:=- 0 10 (0 1 If a o , al, • ••, ari are arbitrary complex numbers, then -

'1)

N Fik)

(0) —

1 inn

ak = -Fx (reft ) rk elk' a k dt 27r 0 k= 0

k = 0 k!

is an analytic function of A for 0 < r < I. Moreover, since N Flo (0) — k= 0

(4

Id

it follows that

---.. sup 2.° Gc =c)

R

F 12 (0) k!•

N 1/2 ak < sup IIF A,112(E 1%1 2 ) 9 X.EC k=0

' Fxk) (0) 2 \ i/2 -...C. sup II FAo112 , k! ) A.GC k =0 V

(

kz_di

and hence the result follows. MI 7.45 Theorem If cp is a function in L'( -1), then the essential spectrum of T,,, is a connected subset of C. Proof It is sufficient to prove that if C is a rectifiable simple closed curve lying in p e (T9 ) with R(9) lying outside, then T9 _,I. is a Fredholm operator for A in the interior of C. Let n be a simply connected open set which contains C and the interior of C and no point of the essential spectrum cr e (T9 ) exterior to C lies in 1. We want to show that cc e (T9 ) and S2 are disjoint.

The Connectedness of the Essential Spectrum 195

Fix A. in C. For each A in n let F be a rectifiable simple arc lying in with endpoints Ao and A and define

UP -t

Q

.....-0."'".'-'"•---....

F2 (z)= jA , (Z) exp

1

(z) dy}

(µ

and G A(Z) = # A o (Z) exp { —fr it

1

_ ii } (z) dp}

---------------

for z in D. Since P {11(9 — 1.1)}(z) is analytic on the simply connected region n, it follows that F2 (z) and G 2 (z) are well defined. Moreover, since FA (z) = jA (z) for Ain some neighborhood of C by Corollary 7A1, it follows from the preceding lemma that FA is in H 2 for all A in n. Similarly, G A is in H 2 for each A in [2. If we consider the function Ili (A) = Tx. (y _ ,t) F A 1, then ti (A) lies in H 2 for A in 1, is analytic, and vanishes on the exterior of C in n. Thus —

Tx.(9 _ A)FA = 1

for A in [2, and in a similar manner we see that Tx _. (9 _ 24 -1 GA = 1 for A in Q. If k is a function in II', then

. . . . . . . . ._.- - - - .-.___.. ._. . . . 1

1-12 (z) = FA (z) P{ 9_ A x,G A k (z)

defines a function satisfying the hypotheses of the preceding lemma. Thus, HA is in H 2 and Tx.(tp _ A) H A= k for A in 1. Moreover, we have li HAll 2

--.

sup II HA.11 2 — sup II Tx-J9 _Aoll II kli 2 AocC 2. 0 cC

Therefore, if k is in H 2 and {ici } j°1 1 is a sequence of functions in II' such that limi3O0 Ilk —1c1 1 2 = 0, then the corresponding functions {H/} .7,_ / for a fixed A in n form a Cauchy sequence and hence converge to a function HA such that Tic,. ( — 2) H 2, = — k. Thus we see that Tx.(tp - )) is onto for A in n. Since the same argument applied to I on n yields that Tx _„ (ip- _ A- ) is onto for A in I, we see that Tx „ (w_ A) is invertible, and hence that Tv — A is a Fredholm operator, which completes the proof. 111 9

7.46 Corollary (Widom) If 9 is a function in if (T), then 6(Tc) is a con-

nected subset of C. Proof By virtue of Proposition 7.24, the spectrum of T9 is formed from the union of the essential spectrum plus the A for which Tv — A is a Fredholm

196 7 Toeplitz Operators

operator having index different from zero. Since Tom — A, is a Fredholin operator for A. in each component of the complement of the essential spectrum and the index is constant, it follows that the spectrum is obtained by taking the union of a compact connected set and some of the components in the complement—and hence is connected. 111 Despite the elegance of the preceding proof of connectedness, we view it as not completely satisfactory for two reasons: First, the proof gives us no hint as to why the result is true. Second, the proof seems to depend on showing that the set of some kind of singularities for a function of two complex variables is connected, and it would be desirable to state it in these terms. We conclude this chapter with a result of a completely different nature but of a kind which we believe will be important in the further study of Toeplitz operators. It involves a notion of "localization" and suggests that in order to understand certain phenomena concerning Toeplitz operators it is necessary to consider other representations of the C*-algebra Z (L°°(T)). We begin with a result concerning C*-algebras having a nontrivial center. The center of an algebra is the commutative subalgebra consisting of those elements which commute with all the other elements in the algebra. In the proof we make use of the fact that an abstract C*-algebras has a *-isometric isomorphic representation as an algebra of operators on some Hilbert space. Also we need to know that every *-isomorphism on C(X) can be extended to the Borel functions on X. Although we have not proved these results in the text, outlines of the proofs were given in the Exercises in Chapter 4 and 5. 7.47 Theorem If Z is a C*-algebra, 91 is a C*-algebra contained in the

center of Z having maximal ideal space M 93 , and for x in M21, 3„ is the closed ideal in Z generated by the maximal ideal {A e 91: A(x) = 0} in 21, then

n ix = { 43} -

xe.m.,

In particular, if (I)„ is the *-homomorphism from Z onto Zgl x , then yxEm., e o x is a *-isomorphism of Z into G x€M,,* e Z/f3 x . Moreover, T is invertible in Z if and only if ' x (T) is invertible in Z/ 3 x for x in M91 . .

Proof By Proposition 4.67 it is sufficient to show that sup x€mcg iPx(T)II = 11T hi for Tin Z. Fix Tin Z and suppose that II Til —supxEm . ii (Dx(nii = E > 0 . For xe in M91 let Ox . denote the collection of products SA, where S is in and A is in 21 such that A vanishes on a neighborhood of .4. Then Ox,,

Localization to the Center of a C*-Algebra 197

is an ideal in Z contained in 3„, and since the closure of 0, contains {A e 21: A(x.) = 0} , we see that 3„ = clos[0„]. Thus there exists S in Z and A in 91 such that A vanishes on an open set U x. containing xo and ii ll)x.( 7)11 +EP > II T -1-. SA II. Choose a finite subcoverx {U„,}N 1 of M93 and the corresponding operators {Si }r_ 1 in i and Mg._ 1 in 21 such that

IITII -;

110x,(7)11+;

IIT + S A i il

and A i vanishes on Usi . Let (I) be a *-isomorphism of Z into .2(1f) for some Hilbert space .if and let th be the corresponding *-isomorphism of C(M93 ) into .e (Ye) defined by O. = (1)01- ', where F is the Gelfand isomorphism of 21 onto C(M93 ). Since 0(91) is contained in the commutant of Om, it follows that ti/ o (C(M9)) is contained in the commutant of do(a), and hence there exists a *-homomorphism 0 from the algebra of bounded Borel functions on M a into the commutant of OM which extends ;if.. Let {Ai } N_ 1 be a partition of M2, by Borel sets such that N is contained in Ux , for i = 1, 2, ..., N. Then (1)(T+ Si A i)0(tj = (I)(T) 0 (IA) and hence

I ( I (T) 0 ',al < II 0 (T + S i AM' 110 Or AMI < 11TI1

-•

2

for each 1 < i -. N.

Since the {00;60_ 1 are a family of commuting projections which reduce 1(T) and such that EN i 0 (IA,) = I, , it follows that II 131) (T)I1 =1 4/4N sup 1 0 ( 7)0 0:011 and hence II tD ( T) II < II T II —e/2. Since' is an isometry, this is a contradiction. If T is invertible in Z, then clearly O x (T) is invertible in Z/3x for x in M. Hence suppose (1:„ (T) is invertible in Z/3„ for each x in M. For x 0 in M91 there exists S in Z by Theorem 4.28 such that (1),(ST— I) = 0. Repeating the argument of the first paragraph, we obtain a neighborhood O, on which I (I) x (ST— I) il < 1 for x in 0 x.. From Proposition 2.5 we obtain that Ox (ST) is invertible and II (I)x (ST) - 1 II < 2 for x in O. Since O x (T) is invertible, (I)„(ST) -1 do x (5) is its inverse, and hence II (Dx (T) ill is bounded for x in

198 7 Toeplitz Operators

O xo . A standard compactness argument shows that supxcm.11 0 x(Tril < cc. Therefore Mei 0 (Dx(T) is invertible in E xG ,,,,„ e Z/3x , and hence T is z-...CCE

invertible in by Theorem 4.28. •

7.48 The context in which we want to apply this result is the following. Although the center of the C*-algebra Z(L'(T)) is equal to the scalars, the quotient algebra Z(12D(T))/2C(H 2 ) has a nontrivial center which contains 3"(C(T))/2C(H 2 ) = C(T) by Proposition 7.22. Thus we can "localize" the algebra Z(L'(T))/2C(H 2 ) to the points of T. For A, in T let is be the closed ideal in I(E°(T)) generated by t

E

C(T), (X/1) = 0}

and let Za be the quotient algebra Z(r) (T))/f3a , and 0A be the natural *-homomorphism from Z(U(T)) to Za . The C*-algebras Za are all *-isomorphic. If 0 is the *-homomorphism defined by 0 Ea e -F e 0,1 from ZW(T)) to E AG -6- 0 ZA, then the sequence 7.49 Theorem

(0)----->QC(H 2 )--->Z(U(T))---4--) -> I 0 Z.1. is exact at Z(I2D(T)). Proof Since Z(L'(T)) is an irreducible algebra in (H 2 ), every non-

zero closed ideal contains QC(H 2 ) by Theorem 5.39. Thus QC(H 2 ) is contained in the kernel of 0 and hence 0 induces a *-homomorphism 0, from the quotient algebra Z(If(T))/2C(H 2 ) into EA G T e ZA . Since (C(T))/2C (H 2 ) is contained in the center of Z(if (T))/2C(H 2 ), the preceding theorem applies, and we conclude that 0 c is a *-isomorphism. Rotation by A, on T obviously induces an automorphism on ZW(T)) taking 3 0 onto 3 A , and hence there exists a *-isomorphism from Z o onto ZA . • The usefulness of this result lies in the fact that it reduces all questions concerning operators in Z(V)(T)) modulo the compacts to questions concerning Z A . Unfortunately, we do not know very much about the algebras Z a . The following proposition shows that the operators in Z A depend only on local properties of the defining functions. Recall that Moc \ED is fibered by the circle such that = fm E

g i (m) =

Locality of Fredholmness for Toeplitz Operators 199

and that the gilov boundary of H" can be identified with the maximal ideal space AI L . of L" (T). Let us denote the intersection FA n ML . by OF) . 7.50 Proposition If 19, j },N, J=. 1 are functions in iNT) with Gelfand trans-

form 0, j on Mi.„ and A is in T, then (1),I (EN,_ functions {0, j I OFA } Ni, j .

n ,Tom=

depends only on the

Proof It is sufficient to show that (DA (T9) = 0 for d in L°° (T) such that 0106. 0. By continuity and compactness, it follows that for > 0 there exists an open arc U of T containing A such that 940 < E" If Vi is a continuous function on T which equals I on the complement of U and vanishes at A, then Tom = T4,1 + + where 4:11)(TridTori (1 _110) = 0, Il AV 91 u)II < II41u II < since low

= 0

and

0 and hence 0,1 (4) =0. IN As corollaries, we obtain the following results proved originally by "localizing" in H'+C(T). 7.51 Corollary If 9 is a function in UM, then Tv is a Fredholm operator if and only if for each A in T there exists VI in If (T) such that T fr is a Fredholm operator and 9 = VI on OF) .

Proof From Theorem 7.49 and the definition, it follows that Tc, is a Fredholm operator if and only if 0,1 (4) is invertible for each A in T. If for A in T there exists rp in LNT) such that T, is a Fredholm operator and = ?Ti on OF) , then 0,1 (4) is invertible in Z A and equal to 0 1 (4) by the previous proposition. Thus the result follows from Theorem 7.47. MI 7.52 Corollary If 9 and 1,/r are functions in 1."(T) such that for each A

in T either 0-0 1 leFa 0 for some 0 1 in 17 or if/ — 0 2 I aFA 0 for some 02 in If', then 4 4— 4,, is compact. Proof Since 7p — T compact if and only if O a (Tip):Da (T) (I)a (Top.) for each A in T by Theorem 7.49, the result is seen to follow from Proposition 7.5. MI

200 7 Toeplitz Operators

Notes In an early paper [109] Toeplitz investigated finite matrices which are constant on diagonals and their relation to the corresponding one- and two-sided infinite matrices. The fundamental theorem in this line of study was proved by Szego (see [54]), and most of the early work concerned this type of question. In [116] Wintner determined the spectrum of analytic Toeplitz matrices, and he and Hartman set the tone for much of the work in this chapter in two papers [59] and [60] some twenty years later. The first systematic study of Toeplitz operators emphasizing the mapping 9 —> Tv was made by Brown and Halmos in [11]. What might be called' the algebra approach to these problems was first made explicit in [29] and [30] and was based on the earlier papers [17] and [18] of Coburn. A vast literature exists for Wiener—Hopf operators beginning with the fundamental paper of Wiener and Hopf [115]. Most of the early work concerns the study of explicit operators, and a good exposition of that along with a bibliography can be found in Krein [72]. The studies of Toeplitz operators and Wiener—Hopf operators had parallel developments until Rosenblum observed [94] using Laguerre polynomials that the two classes of operators were unitarily equivalent. Subsequently, Devinatz showed in [25] that the canonical conformal mapping of the unit disk onto the upper half-plane establishes the unitary equivalence between a Toeplitz operator and the Fourier transform of a Wiener—Hopf operator. Thus a given result can be stated in the context of either Toeplitz operators or of Wiener—Hopf operators. The spectral inclusion theorem is due to Hartman and Wintner [60], although the proof given here of Proposition 7.6 first occurs in [110]. Corollaries 7.8 and 7.19 as well as the remarks following 7.9 are due to Brown and Halmos [11]. The existence of the homomorphism in Theorem 7.11 as well as its role in these questions is established in [31]. In [103] Stampfli observed that a proof of Coburn in [17] actually yields Proposition 7.22. The analysis of the C*-algebra Z(C(T)) was made by Coburn in [17] and [18] while its applicability to the invertibility problem for Toeplitz operators with continuous symbol was observed in [29]. Proposition 7.24 was proved by Coburn in [16]. The content of Corollary 7.27 is the culmination of several authors including Krein [72], Calderon, Spitzer, and Widom [13], Widom [110], and Devinatz [24]. The proof given here first appears in [29] and independently in [3], where Atiyah used the matrix analog in a proof of the periodicity theorem. A related proof was given by Gohberg and Feldman [45]. The study

Notes

201

of Toeplitz operators With symbol in li c° +C(T) was made in [30] with complements in [77] and [103]. The invertibility criteria stated in Theorem 7.30 and its corollary were given independently by Widom [111] and Devinatz [24] and are based on a study in prediction theory by Helson and Szego [64]. The spectral inclusion theorem given in Theorem 7.32 is based on an extension by Lee and Sarason [77] of a result of the author [31]. The question of the connectedness of the spectrum of a Toeplitz operator was posed by Halmos in [57] and answered by Widom in [113] and [114]. The proof of Theorem 7.45 is a slight adaptation of that in [114] to cover the essential spectrum and avoiding certain measure theoretic considerations as well as the use of the harmonic conjugate. The possibility of the essential spectrum being connected was suggested to the author by Abrahamse. Theorem 7.47 is closely related to various central decompositions in C*-algebra (see [96]) but its application to Toeplitz operators is new, and these results were suggested by the earlier Corollaries 7.51 and 7.52. The first corollary is due to Simonenko [102] and independently to Douglas and Sarason [35] and extends a result of Douglas and Widom [38]. The second corollary is due to Sarason [98]. Several further developments and additional topics should be mentioned. The invertibility problem for symbols in the algebra of piecewise continuous functions has been considered by Widom [110], Devinatz [24], and from the algebra viewpoint by Gohberg and Krupnik [50]. The invertibility problem has also been considered for certain algebras of functions which appear more natural in the context of the line. The algebra of almost periodic functions was considered independently by Coburn and Douglas [19] and by Gohberg and Fel'dman [46], [47]. Actually, the latter authors considered the Fourier transforms of measures having no continuous singular part. The problem for the Fourier transform of an arbitrary measure has been considered by Douglas and Taylor [37] using a deep result [108] of the latter on the cohomology of the maximal ideal space of the convolution algebra of measures. Lastly, Lee and Sarason [77] and Douglas and Sarason [36] have considered the invertibility problem for certain functions of the form n for inner functions 9 and Li. Generalizations of the notion of Toeplitz operator have been considered by many authors: Douglas and Pearcy studied the role of the F. and M. Riesz theorem in [33]; Devinatz studied Toeplitz operators on the H 2 -space of a Dirichlet algebra [24]; Devinatz and Shinbrot [26] studied the invertibility of compressions of operators to subspaces; and Abrahamse studied Toeplitz

202 7 Toeplitz Operators

operators defined on the H 2 -space of a finitely connected region in the plane [I]. A different kind of generalization is obtained by considering Wiener—Hopf operators. A subsemigroup of an abelian group is prescribed, and convolution operators compressed to the corresponding L2 space of functions supported on it are studied. Certain basic results for rather arbitrary semigroups are obtained by Coburn and Douglas in [20]. Earlier work involving half-spaces is due to Goldenstein and Gohberg [52] and [53]. The case of the quarter plane is of particular importance and coincides with Toeplitz operators defined on H 2 of the bidisk. Results on this have been obtained by Simonenko [101], Osher [84], Malygev [78], Strang [106], and Douglas and Howe [32]. In particular, the latter authors obtain necessary and sufficient conditions for such a Toeplitz operator to be a Fredholm operator. They show, moreover, that while such an operator must have index zero, it need not be invertible. In many of the preceding contexts, various topological invariants of the symbol enter into the determination of when the corresponding operator is invertible, and usually these invariants must be zero. In the case of a continuous symbol on the circle, a nonzero invariant corresponds to the operator being a Fredholm operator having a nonzero index. A start at establishing similar results for the other classes of operators has been made by Coburn, Douglas, Schaeffer, and Singer in [21] where a generalized notion of Fredholm operator due to Breuer [8], [9] is utilized. It is expected that questions of this kind will prove important in future developments. Although attention in this book has been confined to the scalar case, the matrix case is perhaps of even greater importance. The function 9 is allowed to have n x n matrices as values and to operate on a C"-valued H 2 -space. Many of the techniques of this chapter can be carried over to this case using the device of the tensor product. More specifically, if 91 is an algebra of scalar functions, 91„ is equal to 91 p M„, where M„ is the C*-algebra of operators on the n-dimensional Hilbert space C", and more importantly, Z(21„) is isometrically isomorphic to Z(91) Q Al,,. In particular, applying this to one of the exact sequences considered in the chapter one obtains the exact sequence (0) —> QE (H 2 ) Al,, -> (C(T)) M„ --) C(E) M„ —> (0) . (The fact that the "scalar" sequence has a continuous cross section is also used.) Since 2E(H 2 ) Q M„ is equal to 5.4..E(H 2 Q C"), one obtains that a matrix Toeplitz operator with continuous symbol is a Fredholm operator if and only if the determinant function does not vanish on the circle. Moreover, the index can be shown to equal minus the winding number of the

Exercises 203

determinant. (This latter argument uses the fact that every continuous mapping from T into the invertible nxn matrice is homotopic to a mapping from T to the invertible diagonal matrices.) This latter result is due to Gohberg and Krein [49]. A generalization to certain operator-valued analogs can be found in [31]. Additional results on the matrix and operator-valued case are in [87] and [88]. Lastly, although this book does not comment on it, the study of Toeplitz and Wiener—Hopf operators is important in various areas of physics and probability (see [54], [69], [83]) and in examining the convergence of certain differences schemes for solving partial differential equations (see [83]). Exercises 7.1 If 21 and 93 are C*-algebras and p is an isometric *-linear map from 91 into 93, then o(T) c o(p(T)) for T in 21. (Hint: If T is not left invertible in 21, then there exists {S„},r, 1 in 91 such that iiSn II = 1 and limn . iiSn Tli = 0.) If, in addition, 91 is commutative, then r(p(T)) G h[o (T)] for T in 21. -

7.2 If 9 is a nonconstant real function in U(T), then Tip has no eigenvalues. 7.3 An operator T in Q(H 2 ) is a Toeplitz operator if and only if Tx*i TTx , = T. 7.4 If 9 is a nonzero function in L'(T), then My and Tc, have no eigenvalues in common. 7.5 If 9 is a real function, then Tv is invertible if and only if the function 1 is in its range. 7.6 If 9 is in .ff(T), then W(T4 ) = h[M(9)]. 7.7 If co l , 9 2 , and 9 3 are functions in L'(T) such that Tv ,T4 compact, then 9 1 9 2 = 9 3 .

2

T13 is

Definition If {oc,,}„'_ 0 is a bounded sequence of complex numbers, then the associated Hankel matrix {a u } c, is defined by a u = oc i+j for i,j in Ef . 7.8 (Nehari) If {an }:1 0 is a bounded sequence, then the Hankel operator H defined by the associated Hankel matrix is bounded if and only if there exists a function 9 in L'(T) such that an

=

1

-

2

/ 7'9 (efr)x,,(e h) dt .

27r 0

204 7 ToepUtz Operators

Moreover, the norm of H is equal to the infimum of the norm 11911,,, for all such functions 9.* (Hint: Consider the linear functional L defined on H 2 by L(f) = (Hg, h), where f = gh, and show that L is continuous if H is bounded.) 7.9 (Hartman) If foci }„"_ 0 is a bounded sequence, then the associated

Hankel operator is compact if and only if there exists a continuous function 9 on T such that 1

2n

an = — for n in R + .* 9(ei)x,i (e it) dt 27r 0o (Hint: Use Exercise 1.33 to show that a certain linear functional L is w*continuous if and only if H is compact and apply the analog of Exercise 6.36.) 7.10 An operator H in 2 (H 2) is a Hankel operator if and only if TX H = HTxi • 7.11 If 9 is in L'(T), then n(4) is unitary in Z(L'(T))/2E(H 2 ) if and only if 9 is in QC.* (Hint: Show that the Hankel operator HI, is compact if and only if n(T9 *) is an isometry in Z(INT))/2E(H 2).) 7.12 If 4 is in L'(T), then n(T9) is in the center of Z(E°(T))/2E(T 2 ) if and only if 9 is in QC. Are there any other operators in the center?** 7.13 Show that Tx T1,—Txl, is compact for every x in L'(T) if and only if 9 is in 11"+C(T).* 7.14 If q is in H'+C(T), then o is invertible in L°° (T) if and only if Tq,

has closed range. 7.15 If 9 is in H" + C(T) and A, is in c(T9)\M(9), then either A, is an eigenvalue for Tip or is an eigenvalue for Tv . .

7.16 (Widom—Devinatz) If 9 is an invertible functions in Lf (T), then T4, is invertible if and only if there exists an outer function g such that

I arg g 91 < n/2, — 6 for 6 > O. 7.17 Show that there exists a natural homomorphism y of Aut[Z(C(T))1

onto the group Hom + (T) of orientation preserving homomorphisms of T. (Hint: If 9 is in Hom + (T), then there exists K in 2E (H 2 ) such that 74 K is a unilateral shift.) -

7.18 Show that the connected component Aut o [Z(C(T))1 of the identity

in Aut[Z(C(T))] is contained in the kernel of y; Is it equal?**

Exercises 205

7.19 If 9 is a unimodular function in QC and K is appropriately chosen in 2C(1-1 2 ), then a (T) = (T,p + K)*T(Tc,+ K) defines an automorphism on rrIMITI‘

Lt 11/4 )1 III €,111, 111,11.1%.01 v1 y. 1,711 ,0 W CuaC. €11' CLUC viiivi

Fill aino

tit

(111J W1111 WO 11 ,4_, c

exhaust leery.* Is a in Aut o [Z(C(T))I?** 7.20 If V is a pure isometry on fe and En is the projection onto r , then Ut = 0 en (E„ r En+i ) is a unitary operator on .ff for e it in T. If we set 13,(T) = 14* TU, for T in the C*-algebra Cy generated by V, then the mapping F (e a ) = 13, is a homomorphism from the circle group into Aut [C y] such that F(eit) = 13,(T) is continuous for T in C v . Is it norm continuous? Moreover, for V = TX ,, the unitary operator U, coincides with that induced by rotation by e1 ' on H 2 . 7.21 Identify the fixed points a v for the fl, as a maximal abelian subalgebra of Cy . (Hint: Consider the case of V= Tx „ acting on 1 2 (1 + ).) 7.22 Show that the mapping p defined by inn

P(T) =

for T in Cv

0 fit (T) dt

a

is a contractive positive map from C v onto v which satisfies p (TF) p(T) F for T in C v and F in a y . are pure isometrics on /e l and Jr2, respectively, such that there exists a *-homomorphism (1) from C vi to E ve with (1)(1/1 ) = V2, then (I) is an isomorphism. (Hint: Show that po(I) = (1)o p and that dola vi is an isomorphism.) 7.23 If V 1 and

V2

7.24 If A and B are operators on le and X', respectively, and (I) is a *-homomorphism from EA to CB with (I)(A) = B, then there exists *-isomorphism

'I' from CA e B to EA such that W(AEDB) = A. 7.25 If V is a pure isometry on if and W is a unitary operator on Yl , then there exists a *-isomorphism 4 from C y , onto Eve such that Iii(Vi ) = V2.

are nonunitary isometrics on H 1 and H2, respectively, then there exists a *-isomorphism '1 1 from C v , onto C v2 such that T(V1 ) = V2. 7.26 (Coburn) If V1 and

V2

Definition A C*-algebra 91 is said to be an extension of 2C by C(T), if

there exists a *-isomorphism (1) from 2C(0) into 91 and a *-homomorphism from 21 onto C(T) such that the sequence (0)---> 2C (()

C (T)

(0)

206 7 Toeplitz Operators

is exact. Two extensions 91 1 and 21 2 will be said to be equivalent if there exists a *-isomorphism 0 from 21 1 onto 21 2 and an automorphism a in Aut[QC(X)] such that the diagram ot QEGyo--->(0) commutes.

ce

C (0)

e

QE(10 -2-÷ 91 2

412

7.27 If N is a positive integer and Z is, is the C*-algebra generated by Tx ,,, and 2E(H 2 ), then ZN is an extension of £ by C(T) with IF (TIN ) = Moreover, Z N and Zilf are isomorphic C*-algebras if and only if N = M.

(Hint: Consider index in ZN .) K2 = Ew e H2, Q be the orthogonal projection onto K 2 , and define Sc, = QM4 IK 2 . If N is a negative integer and S N is the C*-algebra 7.28 Let

generated by Sxr, and 2C(K 2 ), then SN is an extension of 2C, by C(T) with 11/(Sx,) x i . Moreover, the extensions Z it,r and S N are all inequivalent despite the fact that the algebras Z N and e jv are isomorphic. 7.29 Let E be a closed perfect subset of T and p be a probability measure on T such that the closed support of p is T and the closed support of the restriction pE of p to E is also E. If 21E is the C*-algebra generated by MX1

on L2 (//) and QE(L2 (pE)), then 91 E is an extension of £ by C(T) with (Mxi ) = x i . Moreover, two of these extensions are equivalent if and only if the set E is the same. 7.30 If E is a closed subset of T, then E E0 u (ehn: n 1), where E0 is a

closed perfect subset of T. Let p be a probability measure on T such that the closed support of p is T and the closed support of the restriction pEo of p to E0 is also E0 . If 21E is the C*-algebra generated by W = Mx , 0 I.?, CI Me lt. e P (7)), then 21 E is an on L2 (1u) e 12 (7) and QC(L2 (40 ) extension of QC by C(T) with '11 (W) = x i . Moreover, two of these extensions are equivalent if and only if the set E is the same. 7.31 Every extension 21 of 2E by C(T) is equivalent to exactly one exten-

sion of the form ZN, SN, or 21E .* (Hint: Assume 21 is contained in Q(.1e) and decompose the representation of QE on Jr. Use Exercises 5.18-5.19.) 7.32 If T is an operator on Jr such that cre (T) = T and T* T — T T* is compact, then the C*-algebra generated by T is an extension of QE by C(1)

with T(T) = x i . Which one is it?

Exercises 207

7.33 If B 2 is the closure of the polynomials in L2 (D) with planar Lebesgue measure and PB is the projection of L2 (D) onto B 2 , then the C*-algebra e Rq, = MV1B 2 , is an — )}, wher generated by the operators {F p : 9 e C(D extension of C(T) by .2E with tP(R2 1 ) --,--- x i . Which one is it? 7.34 If 91 is an extension of .2E by C(T), determine the range of the mapping from Aut (91) to Aut(C(T)).* 7.35 If A. is in T and 9 is in If (1), then 0 (FA) c o-(0 ). (T9)) h(0(FA)), where 0 is the harmonic extension of 9 to the gilov boundary of H'. 7.36 (Widow) If 9 is in PC, 9* is the curve obtained from the range of 9 by filling in the line segments joining 9(e'`) to 9(e + ) for each discontinuity, then Tv is a Fredholm operator if and only if 9 # does not contain the origin. Mores?ver, in this case the index of Tv is minus the winding number of 9* .* (Hint: Use Section 7.51 to show that Tv is a Fredholm operator in this case and that the index is minus the winding number of 9 # . If 9' passes through the origin, then small perturbations of 9 produce Fredholm operators of different indexes.) 7.37 (Gohberg—Krupnik) The quotient algebra (PC)/QC (H 2 ) is a commutative C*-algebra. Show that its maximal ideal space can be identified as a cylinder with an exotic topology.* 7.38 If X is an operator on H 2 such that Tv * XTv — X is compact for each inner function 9, then is X = K for some tji in H" and compact operator K?** 7.39 If for each z in C, 9z is a function in L'(T) such that Me``) is an entire function in z having at most N zeros for each eh in T, then the set of z for which Tc„ fails to be invertible is a closed subset of C having at most N components.* (Repeat the whole proof of Theorem 7.45.)

References

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ceedings of Conference on Operator Theory, Hungary, 1970. 38. R. G. Douglas and H. Widom, Toeplitz operators with locally sectorial symbol, Indiana U. Math. J. 20, 385 388 (1970). 39. P. Duren, HP spaces. Academic Press, New York, 1970. 40. T. W. Gamelin, Uniform algebras. Prentice Hall, Englewood Cliffs, New Jersey, 1969. 41. I. M. Gelfand, Normierte rings, Mat. Sb. (N.S.) 9 (51), 3-24 (1941). -

210 References 42. 1. M. Gelfand, D. A. Raikov and G. E. Silov, Commutative norrned rings, Usp. Mat. Nauk 1, 48-146 (1946); Amer. Math. Soc. Transl. 5 (2), 115-220 (1951). 43. A. Gleason and H. Whitney, The extension of linear functionals defined on H', Pacific J. Math. 12, 163-182 (1962). 44. C. Goffman and G. Pedrick, First course in functional analysis. Prentice-Hall, Englewood Cliffs, New Jersey, 1965. 45. I. C. Gohberg and I. A. Fel'dman, Projection methods for solving Wiener-Hopf equations, Moldavian Academy of Sciences, Kishinev, 1967 (Russian). 46. T. C. Gohberg and I. A. Fel'dman, On Wiener-Hopf integral difference equations, Dokl. Akad. Nauk SSSR 183, 25-28 (1968) (Russian); Soviet Math. Dold. 9, 1312-1316 (1968). 47. I. C. Gohberg and I. A. Fel'dman, Wiener-Hopf integral-difference equations, Acta. Sc!. Math. 30, 199-224 (1969) (Russian). 48. 1. C. Gohberg and M. G. Krein, Fundamental theorems on deficiency numbers, root number, and indices of linear operators, Usp. Mat. Nauk 12, 43-118 (1957) (Russian); Amer. Math. Soc. Transl. 13 (2), 185-265 (1960). 49. I. C. Gohberg and M. G. Krein, Systems of integral equations on a half line with kernels depending on the difference of arguments, Usp. Mat. Nauk 13, 3-72 (1958) (Russian); Amer. Math. Soc. Transl. 14 (2), 217-287 (1960). 50. 1. C. Gohberg and N. Ya. Krupnik, On an algebra generated by Toeplitz matrices, Functional Anal Prilozen. 3, 46-56 (1969) (Russian); Functional Anal. Appl. 3, 119-127 (1969). 51. Seymour Goldberg, Unbounded linear operators. McGraw-Hill, New York, 1966. 52. L. S. Goldenstein, Multi-dimensional integral equations of Wiener-Hopf type, Bull. Akad. Stiince RSS Mold. no. 6, 27-38 (1964) (Russian). 53. L. S. Goldenstein and I. C. Gohberg, On a multi-dimensional integral equation on a half-space whose kernel is a function of the difference of the arguments and on a discrete analogue of this equation, Dokl. Akad. Nauk SSSR 131, 9-12 (1960) (Russian); Soviet Math. Dokl. 1, 173-176 (1960). 54. U. Grenander and G. Szegii, Toeplitz forms and their applications. University of California Press, 1958. 55. P. R. Halmos, Introduction to Hilbert space and the theory of spectral multiplicity. Chelsea, New York, 1951. 56. P. R. Halmos, Shifts on Hilbert spaces, I. Reine Angew. Math. 208, 102-112 (1961). 57. P. R. Halmos, A glimpse into Hilbert space. Lectures on modern mathematics. Vol. 1, 1-22. Wiley, New York, 1963. 58. P. R. Halmos, A Hilbert space problem book. van Nostrand-Reinhold, Princeton, New Jersey, 1967. 59. P. Hartman and A. Wintner, On the spectra of Toeplitz's matrices, Amer. J. Math. 72, 359-366 (1950). 60. P. Hartman and A. Wintner, The spectra of Toeplitz's matrices, Amer. J. Math. 76, 867-882 (1954). 61. H. Helson, Lectures on invariant subspaces. Academic Press, New York, 1964. 62. H. Helson and D. Lowdenslager, Prediction theory and Fourier series in several variables, Acta Math. 99, 165-202 (1958). 63. H. Belson and D. E. Sarason, Past and future, Math. Scand. 21, 5-16 (1967). 64. H. Helson and G. Szego, A problem in prediction theory, Am. Mat. Pura AppL 51, 107-138 (1960). 65. E. Hewitt and K. Stromberg, Real and abstract analysis. Springer-Verlag, New York, 1965.

References 2II 66. K. Hoffman, Banach spaces of analytic functions. Prentice-Hall, Englewood Cliffs, New Jersey, 1962. 67. Sze To Y. Homotopy theory. Academic Press, New York 1959. 68. R. S. Ismagilov, The spectrum of Toeplitz matrices, Dokl. Akad. Nauk SSSR 149, 769-772 (1963); Soviet Math. Dokl. 4, 462-465 (1963). 69. M. Kac, Theory and applications of Toeplitz forms, in Summer institute on spectral theory and statistical mechanics. Brookhaven National Laboratory, 1965. 70. T. Kato, Perturbation theory for linear operators. Springer-Verlag, New York, 1966. 71. J. L. Kelley, General topology. van Nostrand-Reinhold, Princeton, New Jersey, 1955. 72. M. G. Kreln, Integral equations on half line with kernel depending upon the difference of the arguments, Usp. Mat. Nauk 13, 3-120 (1958) (Russian); Amer. Math. Soc. TransL 22 (2), 163-288 (1962). 73. N. H. Kuiper, The homotopy type of the unitary group of Hilbert space, Topology 3, 19-30 (1965). 74. Serge Lang, Analysis 11. Addison-Wesley, Reading, Massachusetts, 1969. 75. P. Lax, Translation invariant subspaces, Acta. Math. 101, 163-178 (1959). 76. P. Lax, Translation invariant spaces, Proc. Internat. Symp. Linear Spaces, Jerusalem, 1960, pp. 251-262. Macmillan, New York (1961). 77. M. Lee and D. E. Sarason, The spectra of some Toeplitz operators, J. Math. Anal. Appl. 33, 529-543 (1971). 78. V. A. Malygev, On the solution of discrete Wiener-Hopf equations in a quarter-plane, Dokl. Akad. Nauk SSSR 187, 1243-1246 (1969) (Russian); Soviet Math. Dokl. 10, 1032-1036 (1969). 79. K. Maurin, Methods of Hilbert spaces. Polish Scientific Publishers, Warsaw, 1967. 80. M. A. Naimark , Normed rings. Noordhoff, Groningen, 1959. 81. J. von Neumann, Eigenwerttheorie Hermitescher Funktionaloperatoren, Math. Ann. 102, 49-131 (1929). 82. J. von Neumann, Zur Algebra der Funktionaloperationen and Theorie der Normalen Operatoren, Math. Ann. 102, 370-427 (1929). 83. S. J. Osher, Systems of difference equations with general homogeneous boundary conditions, Trans. Amer. Math. Soc. 137, 177-201 (1969). 84. S. J. Osher, On certain Toeplitz operators in two variables, Pacific J. Math. 37, 123-129 (1970). 85. R. Palais, Seminar on the Atiyah-Singer index theorem. Annals of Math. Studies, Princeton, 1965. 86. J. D. Pincus, The spectral theory of self-adjoint Wiener-Hopf operators, Bull. Amer. Math. Soc. 72, 882-887 (1966). 87. H. R. Pousson, Systems of Toeplitz operators on H 2 II, Trans. Amer. Math. Soc. 133, 527-536 (1968). 88. M. Rabindranathan, On the inversion of Toeplitz operators, J. Math. Mech. 19, 195-206 (1969). 89. C. E. Rickart, Banach algebras. van Nostrand-Reinhold, Princeton, New Jersey, 1960. 90. F. Riesz, fiber die Randwert einer analytischen Funktion, Math. Z. 18 (1922). 91. F. Riesz and M. Riesz, fiber die Randwert einer analytischen Funtion 4e Congr. des Math. Scand. 27-44 (1916). 92. F. Riesz and B. Sz.-Nagy, Functional analysis. Ungar, New York, 1955. 93. M. Rosenblum, On a theorem of Fuglede and Putnam, J. London Math. Soc. 33, 376-377 (1958). 94. M. Rosenblum, A concrete spectral theory for self-adjoint Toeplitz operators, Amer. J. Math. 87, 709-718 (1965).

212 References 95. W. Rudin, Real and complex analysis. McGraw-Hill, New York, 1966. 96. David Ruelle, Integral representations of states on a C*-algebra, J. Functional Analysis 6, 116-151 (1970). , Trans. Amer. Math. Soc. 127, 179-203 97. D. E. Sarason, Generalized interpolation on (1967). 98. D. E. Sarason, On products of Toeplitz operators (to be published). 99. D. E. Sarason, Invariant subspaces, Studies in operator theory, Math. Assoc. Amer., Prentice-Hall, Englewood Cliffs, New Jersey (in press). 100. L J. Schark, The maximal ideals in an algebra of bounded analytic functions, J. Math. Mech. 10, 735-746 (1961). 101. 1. B. Simonenko, Operators of convolution type in cones, Mat. Sb. 74 (116) (1967) (Russian); Math. USSR Sb. 3, 279-293 (1967). 102. 1. B. Simonenko, Some general questions in the theory of the Riemann boundary problem, Izv. Akad. Nauk SSSR 32 (1968) (Russian); Math USSR fry. 2, 1091-1099 (1968). 103. J. G. Stampfli, On hyponormal and Toeplitz operators, Math. Ann. 183, 328-336 (1969). 104. M. H. Stone, Linear transformations in Hilbert space and their applications to analysis. Amer. Math. Soc., New York, 1932. 105. M. H. Stone, Applications to the theory of Boolean rings to general topology, Trans. Amer. Math. Soc. 41, 375-481 (1937). 106. G. Strang, Toeplitz operators in the quarter-plane, Bull. Amer. Math. Soc. 76, 13031307 (1970). 107. B. Sz.-Nagy and C. Foia, Harmonic analysis of operators on Hilbert space. Akademiai Kiado, Budapest, 1970. 108. J. L. Taylor, The cohomology of the spectrum of a measure algebra, Acta Math. 126, 195-225 (1971). 109. 0. Toeplitz, Zur theorie der quadratischen Formen von unendlichvielen VerAnderlichen, Math. Ann. 70, 351-376 (1911). 110. H. Widom, Inversion of Toeplitz matrices II, Illinois J. Math. 4, 88-99 (1960). 111. H. Widom, Inversion of Toeplitz matrices III, Notices Amer. Math. Soc. 7, 63 (1960). 112. H. Widom, Toeplitz matrices, Studies in real and complex analysis, Math. Assoc. Amer., Prentice-Hall, Englewood Cliffs, New Jersey, 1965. 113. H. Widom, On the spectrum of Toeplitz operators, Pacific.1. Math. 14, 365-375 (1964). 114. H. Widom, Toeplitz operators on H p , Pacific J. Math. 19, 573-582 (1966). 115. N. Wiener and E. Hopf, Uber eine Klasse singulären Integral-gleichungen, S.-B. Preuss Akad. Wiss. Berlin, Phys.-Math. Kl. 30/32, 696-706 (1931). 116. A. Wintner, Zur theorie der beschrfinkten Bilinear forrnen, Math. Z. 30, 228-282 (1929). 117. K. Yoshida, Functional analysis. Springer-Verlag, Berlin, 1966.

Index

A Absolute continuity, of measures, 104 Absolutely convergent Fourier series, 56, 61 Abstract index, 36, 62, 133, 138 Abstract index group, 36, 38 Adjoint of bounded linear transformation, 30 of closed, densely defined linear transformation, 119 of Hilbert space operator, 81-82 Alaoglu theorem, 10 Algebra of bounded holomorphic functions, 26, 150 Annihilator, of subspace, 30, 80 Asymptotically multiplicative functions, 169 Atkinson theorem, 128 Automorphism of Banach algebra, 61 of C*-algebra inner, 143 weak, 144 B Banach algebra, 34, 32-62 Banach space, 2, 1-31 Banach theorem, 13, 31 Beurling theorem, 45, 153

Bilateral shift, 89 Boundary, for function algebra, 61 Bounded below, operator, 83 Bounded variation, function, 14 Brown-Halmos theorem, 182 C C*-algebra, 91 Calkin algebra, 127 Cauchy-Schwarz inequality, 64 tech cohomology group, 38 Center, of algebra, 196 Characteristic function, 18 Circle group, 51 Classical index, 130, 138 Closed convex hull, 182 Closed graph theorem, 29 Coburn theorem, 184, 185, 205 Cohomotopy group, 39 Commutant, of algebra, 108 Commutator, 180 ideal, 145, 179 Compact operator, 121, 145 Complete metric space, 2 Conjugate space, 6 Convolution product, 54 Corona theorem, 156, 173 Cross-section map, 178 Cyclic vector, 105, 157

213

Index

2 14 D

Dimension, Hilbert space, 76, 79 Tli rert v1m of Banach spaces, 29 of Hilbert spaces, 78 Directed set, 3 Dirichlet problem, 171 Disk algebra, 51-53 Division algebra, 42 E

Eigenspace, generalized, 131 Essential range, 57 Essential resolvent, 191 Essential spectrum, 190, 191 Essentially bounded, 24 Exponential map, 36 Extensions, of C*-algebras, 205 Extreme point, 28, 78 F

Fatou theorem, 174 Fiber, of Alcc , 172 Finite rank operator, 59, 121 Fourier coefficient, 55 Fourier series, 55 Fredholm alternative, 131 Fredholm integral operator, 125 Fredholm operator, 127 Fuglede theorem, 114, 120 Function algebra, 51 Functional calculus, 93, 99 extended, 112-114

H

Hahn-Banach theorem, 11, 12, 78 "Inflict-1 mntrix, 2n3 Hankel operator, 203 Hardy space, 26, 70, 149-176 Harmonic extension, 168 Hartman theorem, 183, 204 Hartman-Wintner theorem, 179, 183 Hausdorff-Toeplitz theorem, 115 Hellinger-Toeplitz theorem, 116 Helson-SzEgo theorem, 187 Hermitian operator, 84 Hilbert space, 63-80, 66 I

Idempotent operator, 116-117 Indicator function, see Characteristic function Initial space, of partial isometry, 95 Inner function, 152 Inner-outer factorization, 158 Inner product, 63 Inner product space, 64 Integral operator, 125, 132, 145 Invariant subspace, 98 simply, 151-152 Involution, of Banach algebras, 82, 91 Irreducible subset, 141 isometry, 95, 117, 146 Isomorphism, of Hilbert spaces, 76 3

Jensen's inequality, 174 K

G

Gelfand-Mazur theorem, 42 Gelfand-Naimark theorem, 93 Gelfand theorem, 44, 45 Gelfand transform, 40 Gleason-Whitney theorem, 164 Gohberg theorem, 146 Gohberg-Krupnik theorem, 207 Gram-Schmidt orthogonalization process, 79 Grothendieck theorem, 30

Kernel of integral operator, 126 of operator, 83 Kolmogrov-Krein theorem, 175 Krein-Mil'man theorem, 28 Krein-Smuliyan theorem, 30 Krein theorem, 147 L

Lebesgue space, 7, 23-26, 68-70 Linear functional, 5, 72

215

Index Linear transformation bounded, 21 closable, 118 closure of, 118 self-adjoint, 119 symmetric, 119 Localization, of C*-algebras, 196 M Matrix operator entries, 116-117 Maximal abelian subalgebra, 88 Maximal ideal space, 44 Metric, 2 Metric space, 2 t Multiplication operator, 87 Multiplicative linear functional, 32, 39 N Nehari theorem, 203 Net, 3 Cauchy, 3 Newman theorem, 173 Norm, 1 Hilbert space, 64 Normal element, in C*-algebra, 91 Normal operator, 84 Normed linear space, 4, 27 Numerical radius, 115 Numerical range, 115 0 Open mapping theorem, 22, 23 Operator, 81 Orthogonal, 65 Orthogonal complement, 71 Orthonormal, 65, 79 Orthonormal basis, 74-75 Outer function in H l , 172, 174 in H 2 , 157 P Parallelogram law, 66 Partial isometry, 95 Piecewise continuous function, 176 Polar decomposition, 97, 98 Polar form, 95 Polarization identity, 64

Polynomials analytic trigonometric, 51 trigonometric, 51 Positive element, of C*-algebra, 94, 117 Positive operator, 84 Projection, 86 Pure isometry, 173 multiplicity, 173 Putnam theorem, 120 Pythagorean theorem, 66, 73 Q Quasicontinuous function, 176 Quasinilpotent operator, 133, 145 Quotient algebra, 43 Quotient C*-algebra, 139 Quotient space, 20 R Radical ideal, of Banach algebra, 60 Radon-Nikodym theorem, 25, 80 Range, of operator, 83 Reducing subspace, 98 Reflexive Banach spaces, 28-30 Resolvent set, 41 Riemann-Stieltjes integral, 15 Riesz theorem, 18 Riesz (F. and M.) theorem, 154 Riesz functional calculus, 60 Riesz-Markov representation theorem, 20 Riesz representation theorem, 72 S Self-adjoins element, of C*-algebra, 91 Self-adjoint operator, 84 Self-adjoint subset, 46 Semisimple algebra, 60 Separating vector, 105 ilov boundary, 61, 164, 173 ilov theorem, 53 Similarity, of operators, 117 Spectral inclusion theorem, 179, 188 Spectral mapping theorem, 45 Spectral measure, scalar, 113 Spectral projection, 113 Spectral radius, 41 Spectral subspace, 113

2 16

Index

Spectral theorem, 93 Spectrum, 41 operator, 85 Square root, of operator, 95 *-Homomorphism, 88, 110 State, 147 pure, 147 Stone-tech compactification, 58, 59 Stone-Weierstrass theorem, 46 generalized, 50 Strong operator topology, 100 Subharmonic function, 174 Sublinear functional, 11 Subnormal operator, 183 Summability, in Banach space, 4 Symbol, of Toeplitz operator, 184 Symmetric linear transformation, see Linear transformation, symmetric Szégo theorem, 175 Sz.-Nagy theorem, 173

T Tensor product of Banach spaces, 31 of Banach algebras, 62 of Hilbert spaces, 79 of operators on Hilbert spaces, 118 Toeplitz matrix, 177 Toeplitz operator, 177

U Uniform boundedness theorem, 29 Unilateral shift, 96, 117, 136 Unit ball, of Banach space, 10 Unitary element, of C*-algebra, 91 Unitary equivalence, of operators, 117 Unitary operator, 84, 117 V von Neumann theorem, 173 von Neumann double commutant theorem, 118 von Neumann-Wold decomposition, 173 Volterra integral operator, 132

W W*-algebra, 101 w-topology, 29 w*-topology, 9, 30, 78 Weak topology, 9 Weak operator topology, 100 Weierstrass theorem, 48 Weyl theorem, 145 Weyl-von Neumann theorem, 146 Widom-Devinatz theorem, 204 Widom theorem, 195, 207 Wiener-Hopf operator, 200 Wiener theorem, 56 Wintner theorem, 183