conway funct analysis

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Graduate Texts in Mathematics 96

Editorial Board

F. W. Gehring

P.

R. Halmos (Managing Editor)

C. C. Moore


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Graduate Texts in Mathematics

TAKE Un/ZARING. Introduction to Axiomatic Set Theory. 2nd ed.

2 OXTOBY. Measure and Category. 2nd ed.

3 SCHAEFFER. Topological Vector Spaces.

4

HILTON/STAMM BACH.

A Course in Homological Algebra.

5 MACLANE. Categories for the Working Mathematician.

6 HUGHES/PIPER. Projective Planes.

7

SERRE. A Course in Arithmetic.

8 TAKEcn/ZARING. Axiometic Set Theory.

9 HUMPHREYS. Introduction to Lie Al) ebras and Representation Theory.

10 COHEN. A Course in Simple Homotopy Theory.

11

CONWAY.

Functions

of

One Complex Variable. 2nd ed.

12

BEALS.

Advanced Mathematical Analysis.

13

ANDERSON/FuLI.ER. Rings and Categories of Modules.

14 GOLUBITSKy/GuILLFMIN. Stable Mappings and Their Singularities.

15 BERBERIAN.

Lectures

in

Functional Analysis and Operator Theory.

16

WINTER.

The Structure of Fields.

17 ROSENBLATT.

Random Processes. 2nd ed.

18 HALMos. Measure Theory.

19

HALMos.

A Hilbert Space Problem Book. 2nd ed., revised.

20 HUSEMOLLER. Fibre Bundles. 2nd ed.

21

HUMPHREYS. Linear Algebraic Groups.

22

BARNES/MACK.

An Algebraic Introduction to Mathematical Logic.

23

GREUB.

Linear Algebra. 4th ed.

24 HOLMES. Geometric Functional Analysis and its Applications.

25 HEWITT/STROMBERG. Real and Abstract Analysis.

26

MANES.

Algebraic Theories.

27

KELLEY.

General Topology.

28

ZARISKI/SAMUEL.

Commutative Algebra. Vol.

l.

29

ZARISKUSAMUEL.

Commutative Algebra. Vol. 11.

30

JACOBSON.

Lectures

in

Abstract Algebra

I:

Basic Concepts.

31 JACOBSON.

Lectures in Abstract Algebra

11:

Linear Algebra.

32

JACOBSON.

Lectures in Abstract Algebra Ill: Theory of Fields and Galois Theory.

33 HIRSCH.

Differential Topology.

34 SPITZER. Principles of Random Walk. 2nd ed.

35

WERMER.

Banach Algebras and Several Complex Variables. 2nd ed.

36 KELLEy/NAMIOKA et al. Linear Topological Spaces.

37

MONK.

Mathematical Logic.

38

GRAUERT/FRITZSCHE.

Several Complex Variables.

39

ARYESON.

An Invitation to C*-Algebras.

40 KEMENy/SNELL/KNAPP. Denumerable Markov Chains. 2nd ed.

41 APOSTOL.

Modular Functions and Dirichlet Series in Number Theory.

42 SERRE. Linear Representations of Finite Groups.

43

GILLMAN/JERISON.

Rings

of

Continuous Functions.

44 KENDIG. Elementary Algebraic Geometry.

45

LOEYE.

Probability Theory I. 4th ed.

46 LOEYE. Probability Theory 11. 4th ed.

47

MOISE.

Geometric Topology in Dimensions 2 and 3.

continued

after Index


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John

B.

Conway

A Course

in Functional Analysis

Springer Science+Business Media, LLC


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John B. Conway

Department

of Mathematics

Indiana University

Bloomington, IN 47405

U.S.A.

Editorial Board

P.

R. Halmos

Managing Editor

Department

of

Mathematics

Indiana University

Bloomington, IN 47405

U.S.A

F. W. Gehring

Department of

Mathematics

University

of

Michigan

Ann Arbor,

MI

48109

U.S.A.

AMS Classifications: 46-01, 45B05

Library

of

Congress Cataloging in Publication

Data

Conway, John

B.

A course in functional analysis.

(Graduate texts in mathematics: 96)

Bibliography: p.

Includes index.

1. Functional analysis.

I.

Title.

QA320.C658 1985 515.7

With 1 illustration

II. Series.

84-10568

1985 by Springer Science+Business

Media

New York

Originally published

by Springer-Verlag New York

Inc. in

1985

Softcover reprint of

the

hardcover I

st

edition

1985

c. C. Moore

Department of

Mathematics

University

of

California

at Berkeley

Berkeley, CA 94720

U.S.A.

All rights reserved. No part of this book may be translated or reproduced in any

form without written permission from Springer Science+Business

Media, LLC .

Typeset by Science Typographers, Medford, New York.

9 8 7 6 5 4 3 2 1

ISBN 978-1-4757-3830-8 ISBN 978-1-4757-3828-5 (eBook)

DOI 10.1007/978-1-4757-3828-5


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For Ann (of course)


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Preface

Functional analysis has become a sufficiently large area of mathematics that

it is possible to find two research mathematicians,

both

of whom call

themselves functional analysts, who have great difficulty understanding the

work of the other. The common thread is the existence of a linear space with

a topology or two (or more). Here the paths diverge in the choice of how

that topology is defined and in whether to study

the

geometry of the linear

space, or the linear operators on the space, or both.

In this book I have tried to follow the common thread rather than any

special topic. I have included some topics that a few years ago might have

been thought of as specialized but which impress me as interesting and

basic. Near the end of this work I gave into my natural temptation and

included some operator theory that, though basic for operator theory, might

be

considered specialized by some functional analysts.

The word "course" in the title of this book has two meanings. The first is

obvious. This book was meant as a text for a graduate course in functional

analysis. The second meaning is that the book attempts to take an excursion

through many of the territories that comprise functional analysis. For this

purpose, a choice of several tours is offered the

reader-whether

he is a

tourist or a student looking for a place of residence. The sections marked

with an asterisk are not (strictly speaking) necessary for the rest of the book,

but will offer the reader an opportunity to get more deeply involved in the

subject at hand, or to see some applications to other parts of mathematics,

or, perhaps,

just

to see some local color. Unlike many tours, it is possible to

retrace your steps and cover a starred section after the chapter has been left.

There are some parts of functional analysis that are not on the tour. Most

authors have to make choices due to time and space limitations, to say

nothing of the financial resources of our graduate students. Two areas that


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Vlll

Preface

are only briefly touched here,

but

which constitute entire areas by them

selves, are topological vector spaces and ordered linear spaces. Both are

beautiful theories and both have books which do them justice.

The

prerequisites for this book are a thoroughly good course in measure

and integration-together with some knowledge of point set topology. The

appendices contain some of this material, including a discussion of nets in

Appendix

A. In

addition, the reader should

at

least be taking a course in

analytic function theory at the same time that he is reading this book. From

the beginning, analytic functions are used to furnish some examples, but it

is only in the last half of this text that analytic functions are used in the

proofs of the results.

It has been traditional that a mathematics book begin with the most

general set

of

axioms and develop the theory, with additional axioms added

as the exposition progresses. To a large extent I have abandoned tradition.

Thus

the first two chapters are

on

Hilbert space, the third is on Banach

spaces, and the fourth is on locally convex spaces.

To

be sure, this causes

some repetition (though not as much as I first thought it would) and the

phrase" the proof is

just

like the proof

of . . .

" appears several times. But I

firmly believe that this order

of

things develops a better intuition in the

student. Historically, mathematics has gone from the particular to the

general-not

the reverse. There are many reasons for this, but certainly one

reason is

that

the human mind resists abstraction unless it first sees the need

to abstract.

I have tried to include as many examples as possible, even if this means

introducing without explanation some other branches of mathematics (like

analytic functions, Fourier series, or topological groups). There are,

at

the

end

of

every section, several exercises of varying degrees of difficulty with

different purposes in mind. Some exercises just remind the reader that he is

to supply a proof of a result in the text; others are routine, and seek to fix

some

of

the ideas in the reader's mind; yet others develop more examples;

and

some extend the theory. Examples emphasize

my

idea about the nature

of

mathematics and exercises stress my belief that doing mathematics

is

the

way to learn mathematics.

Chapter

I discusses the geometry of Hilbert spaces and Chapter II begins

the theory of operators on a Hilbert space. In Sections 5-8 of Chapter II,

the complete spectral theory of normal compact operators, together with a

discussion

of

multiplicity, is worked out. This material is presented again in

Chapter

IX, when the Spectral Theorem for bounded normal operators is

proved. The reason for this repetition is twofold. First, I wanted to design

the book to be usable as a text for a one-semester course. Second, if the

reader understands the Spectral Theorem for compact operators, there will

be

less difficulty in understanding the general case and, perhaps, this will

lead to a greater appreciation of the complete theorem.

Chapter I I I

is

on

Banach spaces. I t has become standard to do some of

this material in courses on Real Variables. In particular, the three basic


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Preface

ix

principles,

the

Hahn-Banach Theorem, the Open Mapping Theorem, and

the

Principle

of

Uniform Boundedness, are proved.

For

this reason I

contemplated

not

proving these results here, but in the end decided that

they should

be

proved. I did bring myself to relegate to the appendices the

proofs

of

the representation of the dual of

LP

(Appendix

B)

and the dual

of

C

o

(

X)

(Appendix C).

Chapter

IV hits the bare essentials

of

the theory of locally convex spaces

-enough to rationally discuss weak topologies.

I t

is shown in Section 5 that

the

distributions are the dual

of

a locally convex space.

Chapter

V treats the weak and weak-star topologies. This is one of my

favorite topics because of the numerous uses these ideas have.

Chapter

VI looks at bounded linear operators on a Banach space.

Chapter

VII introduces the reader to Banach algebras

and

spectral theory

and

applies this to the study

of

operators

on

a Banach space. It is

in

Chapter

VII that the reader needs to know the elements of analytic function

theory, including Liouville's Theorem and Runge's Theorem. (The latter is

proved using the

Hahn-Banach

Theorem in Section IlLS.)

When

in

Chapter VIII the notion

of

a C*-algebra is explored, the

emphasis

of

the book becomes the theory

of

operators

on

a Hilbert space.

Chapter

IX

presents the Spectral Theorem and its ramifications. This is

done

in

the framework of a C*-algebra. Classically, the Spectral Theorem

has been thought

of

as a theorem about a single normal operator. This it is,

but it

is more. This theorem really tells us about the functional calculus for

a normal operator and, hence, about the weakly closed C*-algebra gener

ated by

the normal operator.

In

Section IX.S this approach culminates in

the

complete description

of

the functional calculus for a normal operator. In

Section IX.lO the multiplicity theory

(a

complete set

of

unitary invariants)

for normal operators is worked out. This topic is too often ignored in books

on

operator theory. The ultimate goal

of

any branch

of

mathematics is to

classify and characterize, and multiplicity theory achieves this goal for

normal operators.

In Chapter

X unbounded operators

on

Hilbert space are examined. The

distinction between symmetric

and

self-adjoint operators is carefully delin

eated

and

the Spectral Theorem for unbounded normal operators is ob

tained as a consequence

of

the bounded case. Stone's Theorem

on

one

parameter unitary groups is proved and the role

of

the Fourier transform in

relating differentiation and multiplication is exhibited.

Chapter

XI, which does not depend on Chapter X, proves the basic

properties of the Fredholm index. Though it is possible to do this in the

context

of

unbounded operators between two Banach spaces, this material is

presented for bounded operators

on

a Hilbert space.

There are a few notational oddities. The empty set is denoted by D. A

reference

number

such as (S.lO) means item number 10

in

Section S

of

the

present chapter. The reference (IX.S.lO) is to (S.lO)

in

Chapter IX. The

reference (A.1.l) is to the first item in the first section

of

Appendix A.


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x

Preface

There are many people who deserve my gratitude in connection with

writing this book. In three separate years I gave a course based on an

evolving set of notes that eventually became transfigured into this book. The

students in those courses were a big help. My colleague Grahame Bennett

gave me several pointers in Banach spaces. My ex-student Marc Raphael

read final versions of the manuscript, pointing out mistakes and making

suggestions for improvement. Two current students, Alp Eden and Paul

McGuire, read the galley proofs and were extremely helpful. Elena Fraboschi

typed the final manuscript.

John B. Conway


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Contents

Preface

CHAPTER

I

Hilbert Spaces

l. Elementary Properties and Examples

2. Orthogonality

3. The Riesz Representation Theorem

4. Orthonormal Sets

of

Vectors and Bases

5. Isomorphic Hilbert Spaces and the Fourier Transform

for the Circle

6. The Direct Sum of Hilbert Spaces

CHAPTER II

Operators on Hilbert Space


2. The Adjoint

of

an Operator

3. Projections and Idempotents; Invariant and Reducing

Subspaces

4. Compact Operators

5.* The Diagonalization of Compact Self-Adjoint Operators

6.* An Application: Sturm-Liouville Systems

7.* The Spectral Theorem and Functional Calculus for

Compact Normal Operators

8.* Unitary Equivalence for Compact Normal Operators

CHAPTER III

Banach Spaces


2. Linear Operators on Normed Spaces

vii

1

7

11

14

19

24

26

31

37

41

47

50

55

61

65

70


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xii

3. Finite-Dimensional Normed Spaces

4. Quotients and Products of Normed Spaces

5. Linear Functionals

6. The

Hahn-

Banach Theorem

7.* An Application: Banach Limits

8.* An

Application: Runge's Theorem

9.* An Application: Ordered Vector Spaces

1O. The

Dual

of a Quotient Space and a Subspace

11. Reflexive Spaces

12. The Open Mapping and Closed Graph Theorems

13. Complemented Subspaces of a Banach Space

14. The Principle of Uniform Boundedness

CHAPTER

IV

Locally Convex Spaces


2. Metrizable and Normable Locally Convex Spaces

3. Some Geometric Consequences of the

Hahn-Banach

Theorem

4.* Some Examples of the Dual Space of a Locally

Convex Space

5.

*

Inductive Limits and the Space of Distributions

CHAPTER

V

Weak Topologies

l. Duality

2. The

Dual

of a Subspace and a Quotient Space

3. Alaoglu's Theorem

4. Reflexivity Revisited

5. Separability and Metrizability

6.

*

An Application: The Stone-Cech Compactification

7. The

Krein-Milman

Theorem

8.

An

Application: The Stone-Weierstrass Theorem

9.* The Schauder Fixed-Point Theorem

10.* The Ryll-Nardzewski Fixed-Point Theorem

11.* An Application: Haar Measure on a Compact Group

12.* The Krein-Smulian Theorem

13.* Weak Compactness

CHAPTER VI

Linear Operators on a Banach Space

l.

The Adjoint of a Linear Operator

2.* The Banach-Stone Theorem

3. Compact Operators

4. Invariant Subspaces

5. Weakly Compact Operators

Contents

71

73

76

80

84

86

88

91

92

93

97

98

102

108

111

117

119

127

131

134

135

138

140

145

149

153

155

158

163

167

l70

175

l77

182

187


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Contents

CHAPTER VII

Banach Algebras and Spectral Theory for

Operators on a Banach Space

1. Elementary Properties and Examples

2. Ideals

and

Quotients

3. The Spectrum

4. The Riesz Functional Calculus

5. Dependence

of

the Spectrum on the Algebra

6. The Spectrum of a Linear Operator

7. The Spectral Theory of a Compact Operator

8. Abelian Banach Algebras

9. * The Group Algebra of a Locally Compact Abelian Group

CHAPTER VIII

C*-Algebras


2. Abelian C*-Algebras

and

the Functional Calculus in

C*-Algebras

3. The Positive Elements in a C*-Algebra

4. * Ideals and Quotients for C*-Algebras

5. * Representations

of

C*-Algebras

and

the

Gelfand-Naimark-Segal

Construction

CHAPTER IX

Normal Operators on Hilbert Space

1. Spectral Measures and Representations of Abelian

C*-Algebras

2. The Spectral Theorem

3. Star-Cyclic Normal Operators

4. Some Applications of the Spectral Theorem

5. Topologies

on

J4(.YC')

6. Commuting Operators

7. Abelian von Neumann Algebras

8. The Functional Calculus for Normal Operators:

The Conclusion of the Saga

9. Invariant Subspaces for Normal Operators

1O.

Multiplicity Theory for Normal Operators:

A Complete Set of Unitary Invariants

CHAPTER X

Unbounded Operators

1.

Basic Properties and Examples

2. Symmetric and Self-Adjoint Operators

3. The Cayley Transform

4. Unbounded Normal Operators

and

the Spectral Theorem

XllJ

191

195

199

203

210

213

219

222

228

237

242

245

250

254

261

268

275

278

281

283

288

292

297

299

310

316

323

326


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XIV

5. Stone's Theorem

6. The Fourier Transform and Differentiation

7. Moments

CHAPTER XI

Fredholm Theory

l. The Spectrum Revisited

2. The Essential Spectrum and Semi-Fredholm Operators

3. The Fredholm Index

4. The Components of Yff

5. A Finer Analysis of the Spectrum

APPENDIX A

Preliminaries

l. Linear Algebra

2. Topology

APPENDIX

B

The Dual of LP(fL)

APPENDIX

C

The Dual of Co(X)

Bibliography

List of Symbols

Index

Contents

334

341

349

353

355

361

370

372

375

377

381

384

390

395

399


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CHAPTER

I

Hilbert

Spaces

A Hilbert space is the abstraction of the finite-dimensional Euclidean spaces

of

geometry. Its properties are very regular and contain

few

surprises,

though the presence of an infinity of dimensions guarantees a certain

amount of surprise. Historically, it was the properties of Hilbert spaces that

guided mathematicians when they began to generalize. Some of the proper

ties and results seen in this chapter and the next will be encountered in more

general settings later in this book, or we shall see results that come close to

these but fail to achieve the full power possible in the setting of Hilbert

space.

1.

Elementary Properties and Examples

Throughout this book IF will denote either the real field,

~ ,

or the complex

field, c.

1.1. Definition. I f ' is a vector space over IF, a

semi-inner product

on ' is

a function u: ' X ' IF such that for all

a,

j3 in IF and x, y,

z

in ', the

following are satisfied:

(a)

u(ax +

j3y,

z) = au(x, z) +

j3u(y,

z),

(b)

u(x,

ay

+

j3z)

= iiu(x, y) + jJu(x, z),

(c)

u(

x,

x)

~

0-,,-;------;-

(d)

u(x, y)

=

u(y,

x).

Here, for

a

in IF, i i

= a

if

IF = ~

and i i

is

the complex conjugate of

a

if

IF = c. I f

a

E C, the statement that

a

~ 0 means that

a

E ~ and

a

is

non-negative.


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2

1.

Hilbert Spaces

Note that if

a =

0,

then property (a) implies that

u(O, y)

=

u(a

. 0,

y)

=

au(O,

y)

=

or all

y in 'f.

This

and

similar reasoning shows that for a

semi-inner product u,

(e) u(x,O) = u(O, y) =

for all x, y

in 'f.

In particular,

u(O,

0) =

0.

An

inner product on

'f is a semi-inner product that also satisfies the

following:

(f) I f

u(x,

x) = 0, then x =

0.

An inner product in this book will

be

denoted by

(x, y)

=

u(x,

y).

There is

no

universally accepted notation for an inner product and the

reader will often see

(x, y)

and

(xly)

used in the literature.

1.2. Example. Let

'f be

the collection of all sequences {an: n

~

I}

of

scalars an from IF such that an

=

or all

but

a finite number

of

values

of

n.

I f addition

and

scalar multiplication are defined

on 'f

by

{an} + { P

n

} == {an + P

n

},

a

{

an} == {aa

n

},

then 'f

is a vector space over IF.

I f u({an},{Pn})==L'::=la2n"P2n'

then

u

is a semi-inner product that is

not an inner product. On the other hand,

00

( {an}, {P

n

}) = L an lin ,

n=1

00 1 _

({an}, {P

n

}) = L ;anPn,

n=1

00

({an}, {P

n

})

=

L

n

5

a

n

li

n

,

n=1

all define inner products

on 'f.

1.3. Example. Let (X, J,

p.)

be a measure space consisting of a set

X,

a

a-algebra J of subsets of X, and a countably additive

IR

U {oo} valued

measure p. defined

on J.

I f

f

and

g E L

2(p.)

==

L

2(

X, J,

p.),

then Holder's

inequality implies

f t

E L

1

(p.).

I f

( j ,

g)

=

jftdp.,

then this defines

an

inner product

on

L 2(p.).

Note that Holder's inequality also states that Ifftdp.1 s [flN dp.jl/2 .

[flgl

2

dp.j1/2. This is, in fact, a consequence

of

the following result on

semi-inner products.


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1.1. Elementary Properties and Examples 3

1.4. The Cauchy-Bunyakowsky-Schwarz Inequality. If

( .

.) is

a semi

inner product

on :Y, then

I(x, y)1

2

(x, x)(y,

y)

for all x and y

in

:Y.

PROOF. I f

a E IF and x

and

y E

:Y, then

o (x

-

ay, x

- ay)

=

(x,

x)

-

a(y, x)

- a(x, y)

+ laI

2

(y, y).

Suppose

(y, x)

= be

iO

, b ~

0,

and let a = e-iOt, t in IR.

The

above

inequality

becomes

o

(x,

x)

-

e-iOtbe

iO

-

eiOtbe-

iO

+

t2(y, y)

=

(x,

x)

-

2bt

+ t \ y , y)

= c - 2bt + at

2

== q( t),

where c = (x, x) and a =

(y,

y). Thus q( t) is a quadratic polynomial in

the real variable t and q(t) ~ 0 for all t. This implies that the equation

q(t) = 0 has at

most

one real solution

t.

From the quadratic formula we

find that the discriminant is not positive;

that

is, 0 ~ 4b

2

- 4ac. Hence

o b

2

-

ac

=

I(x, y)1

2

-

(x,

x)(y, y),

proving

the

inequality.

The inequality in (1.4) will

be

referred to as the CBS inequality.

1.5. Corollary. If ( . , .) is a semi-inner product on :Y and

Ilxll

== (x,

X

)1/2

for all x

in :Y,

then

(a)

Ilx

+ yll ~

Ilxll

+

Ilyll for x, yin :Y,

(b)

Ilaxll = lalllxli

for a

in

IF and x

in

:Y.

If

( . ,

.)

is

an

inner product, then

(c)

Ilxll =

0

implies x =

o.

PROOF. The

proofs

of (b) and (c) are left as an exercise. To see (a), note that

for x

and

y

in :Y,

Ilx

+

yll2

= (x +

y,

x + y)

=

IIxl12 +

(y,

x) + (x,

y)

+ IIyl12

=

IIxl12 + 2 Re(x,

y)

+ Ily112.

By

the CBS

inequality, Re(x, y)

~ I(x, y)1

~

IIxIIIIYII.

Hence,

Ilx + yl12 ~ IIxl12 + 211xlillyll +

IIyl12

=

(jlxll + Ilyilf.

The inequality now

follows

by

taking square roots.


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4

1. Hilhert Space,

I f

( . , .)

is a semi-inner product on :r and if x, y E:r, then as was

shown in the preceding proof,

Ilx

+

yl12

=

IIxl12

+

2Re(x ,y )

+

Ily112.

This identity is often called the polar identity.

The quantity

Ilxll

= (x, X

/12

for an inner product ( . , .) is called the

norm of x. I f :r =

IFd (IR

d or C d) and ({ an), {,Bn}) == L ~ ~ l a ) 3 n ' then the

corresponding norm is II {an}

II

= [ L ~ ~ d a n I 2 ] 1 / 2 .

The virtue

of

the norm on a vector space :r is that d(x,

y)

= Ilx - yll

defines a metric on :r [by (1.5)] so that :r becomes a metric space. In fact,

d(x,

y)

= Ilx -

yll

= II(x - z) +

(z -

y)11 ::;; Ilx -

zll

+

liz

-

yll

=

d(x, z)

+ d(z,

y). The other properties of a metric follow similarly.

I f

(

=

IF

d

and

the norm

is

defined as above, this distance function is the usual

Euclidean metric.

1.6. Definition.

A

Hilbert space

is a vector space .Y1' over IF together with

an inner product ( . , .) such that relative to the metric d(x,

y)

= Ilx -

yll

induced

by

the norm, .Y1' is a complete metric space.

I f .Y1'= L 2(fL) and ( I , g) = fft dfL, then the associated norm is Illll =

[flfl2dfLP

/

2.

It

is a standard result of measure theory that

L2(fL)

is a

Hilbert space. I t is also easy to see that IF d is a Hilbert space.

REMARK. The inner products defined on L 2(fL) and IF d are the" usual" ones.

Whenever these spaces are discussed these are the inner products referred

to. The same is true of the next space.

1.7. Example. Let

I

be any set and let [2(1) denote the set of all functions

x:

I

~

IF such that

xU)

= 0 for all but a countable number of

i

and

L

j

E Tlx(i)1

2

f such that (a) 1(0) = 0; (b) for 1

~

k

~

n - 1,

l(k)(t)

exists for all t in [0,1] and I ( k ) is continuous on [0,1]; (c) I ( n

- 1 )

is absolutely

continuous and I(n) E L2(0, 1). For I and

g

in

Yi',

define


21/418

I.2. Orthogonality

7

(b) Show that if

(x + A"',y + A"') == u(x,y)

for all

x

+

A'"

and

y

+

A'"

in

the quotient space

.?r/A"',

then ( . ,

.)

is a

well-defined inner product on

.?r/

A"'.

7. Let Yt' be a Hilbert space over IR and show that there is a Hilbert space : over

C and a map U: Y t ' ~ : such that (a) U is linear; (b) (Uhl'

Uh

2

) =

(hi'

h

2

)

for all hi' h2 in Yt';

(c)

for any k in : there are unique hi'

h2

in Yt' such that

k

=

Uh

l

+

iUh

2

.

(: is called the complexification of

Yt'.)

8. I f

G = {z E C: 0 < Izl < I} show that every J in L ~ ( G ) has a removable

singularity at z =

O.

9. Which functions are in L;(C)?

10. Let G be an open subset of C and show that if a E G, then

{IE

L ~ ( G ) :

J(a) =

O} is closed in L ~ ( G ) .

11. I f {h

n

} is a sequence in a Hilbert space Yt' such that Lnllhnll < 00, then show

that L ' : ~ l h n converges in Yt'.

2. Orthogonality

The greatest advantage of a Hilbert space is its underlying concept of

orthogonality.

2.1. Definition. I f .Yt' is a Hilbert space and I, g E.Yt', then I and g are

orthogonal if U, g)

= O. In symbols,

1.1

g. I f

A, B

~ . Y t ' , then

A .1 B

if

1..1 g for every

I

in

A

and g in B.

I f

.Yt'=

IR

2, this is the correct concept. Two non-zero vectors in

IR

2 are

orthogonal precisely when the angle between them is

7T

/2.

2.2. The

Pythagorean Theorem. II 11,/2"'" In

are pairwise orthogonal

vectors in

.Yt',

then

Ilf1

+

12

+ ... + nl1

2

= 11/1112 +

Ilf2112

+ ... + Il/nl1

2

.

PROOF. I f 11 .1 12' then

Ilf1

+

12112 = U1 + 12'/1

+ 12)

=

Ilfl112 +

2 ReU1'/2)

+

11/2112

by

the

polar

identity. Since

11

.1

12'

this implies the result for

n

=

2.

The

remainder of the proof proceeds by induction and is left to the reader.

Note

that if

I

.1 g, then

I

.1 - g, so Ilf - gl12

=

11.1112

+

Ilg112. The next

result is an easy consequence of the Pythagorean Theorem if

I

and g are

orthogonal,

but

this assumption is not needed for its conclusion.


22/418

8

I. Hilbert Spaces

2.3. Parallelogram Law. If.Yt' is

a Hilbert space

and

f

and g

EO.Yt',

then

Ilf

+ gl12 +

Ilf -

gl12

=

2(llfl1

2

+ IIgI12).

PROOF.

For

any

f

and

g in

.Yt'

the

polar

identity

implies

Now

add. P:

Ilf

+ gl12

=

Ilfll2

+

2 Re(j, g)

+

Ilg11

2

,

Ilf -

gl12

=

Ilfll2

- 2 Re(j, g)

+

Ilgll

2

.

The

next

property of a Hilbert space is truly pivotal. But first we need a

geometric

concept

valid for any vector space over IF.

2.4.

Definition. I f r

is

any

vector space over

IF

and

A

s:;;

r, then

A

is a

convex set

if

for any

x

and

y

in

A

and

$

t

:$

1,

tx

+

(1 - t)y EO

A.

Note

that {tx + (1 - t)

y:

$

t :$

I} is the straight-line segment joining

x and

y. So

a convex set is a set A such that

if

x

and

y

EO A, the entire line

segment joining x and

y is

contained

in

A.

I f r is a vector space, then any linear subspace in r is a convex set. A

singleton set is convex.

The

intersection of any collection of convex sets is

convex. I f .Yt' is a Hilbert space, then every

open

ball B(f;

r)

= {g EO .Yt':

Ilf - gil

h

o

.

By assumption, L(h

o

)

= lim[L(h

n

- h + h

o

)] =

lim[L(h

n

) - L(h) + L(h

o

)] = lim L(h

n

)

- L(h) + L(h

o

).

Hence

L(h) =

limL(h

n

) .


26/418

12

1. Hilbert Spaces

(b)

=

(d): The definition of continuity at 0 implies that L - I ({a ElF:

lal

0 such that

B(O; 8) ~ L -I({a ElF: lal 0: IL(h)1 s cllhll,h

in

X} .

Also, IL(h)1 s IILllllhll for

every

h in X .

PROOF.

Let a

=

inf{ c >

0:

IIL(h)1I s cllhll, h in

X}.

It will be shown that

IILII

=

a;

the remaining equalities are left as an exercise.

I f

f

>

0,

then the

definition of IILII shows that ILlIhll + f)-lh)1 s

IILII.

Hence IL(h)1 s

IILII(lIhll + f). Letting f

~

0 shows that IL(h)1 s IILllllhll for all h. So the

definition of a shows that a s IILII. On the other hand, if IL( h)1 s cllh II

for all

h,

then

IILII

s c. Hence

IILII

s a.

Fix an ho in X and define L:

X ~ IF

by

L(h) = (h, h

o

>.

It is

easy to

see that L is linear. Also, the CBS inequality gives that IL(h)1

= I(h,

ho>1

s IIhllllholi. So L is bounded and IILII s

IIholl.

In fact, L(ho/liholl)

=

(ho/liholl,

ho>

=

IIholl,

so that

IILII

=

IIholi.

The main result of this section

provides a converse to these observations.

3.4. The Riesz Representation Theorem. I f

L:

X ~ IF is a

bounded

linear

functional, then

there

is a

unique

vector ho

in

X such that L(h) = (h, h

o

>

for

every h in X. Moreover,

IILII

=

IIhali.


27/418

I.3. The Riesz Representation Theorem

13

PROOF. Let ,A = ker L. Because

L

is continuous, ,A is a closed linear

subspace of

.:It'.

Since we may assume that ,A

* :It',

,A

1. *

(0). Hence there

is a vector fo in ,A

1.

such that

L(fo)

= 1. Now if h E.:It' and a = L( h),

then

L(h

-

afo)

=

L(h)

-

a

=

0;

so

h

-

L(h)fo

E , A .

Thus

0= (h -

L{h)fo,Jo>

=

(h,Jo> -

L{h)llfoIl

2

.

So if ho =

Ilfoll-%,

L(h) = (h, h

o

> for all h in

.:It'.

I f ~ E.:It' such that (h, h

o

> = (h, h

o

> for all h, then ho - h ~

..l.:lt'.

In

particular,

ho -

h ~

..1

ho

-

h ~ and so ho

=

h

o

. The fact that

IILII =

IIh

o

l

was shown in the discussion preceding the theorem.

3.5. Corollary.

I f

(X,

D,

p.)

is a measure space

and F:

L2(p.)

-> IF

is a

bounded

linear functional, then there is a unique h

0

in L

\ p.)

such that

for every h in L 2(p.).

Of course the preceding corollary

is

a special case of the theorem on

representing bounded linear functionals on

LP(p.),

1

:::;

p

for

every

h in .Jf'. (c) What is the norm

of

the linear functional L defined in (b)?

4.

With the notation as in Exercise 3, define L: .Jf'-'> C by L({ a,,}) =

L ~ ~ l n a " A n - l ,

where IAI < l.

Find

a vector ho in .Jf' such that

L(h)

=

(h,h

o

>

for every h

in .Jf'.

5.

Let .Jf' be the Hilbert space described in Example l.8. I f 0 < t

s

1, define L:

.Jf'-'>

IF

by L(h) = h(t). Show that L is a bounded linear functional, find liLli,

and

find the vector ho in .Jf' such that L( h) = (h, ho

>

or all h in .Jf'.

6. Let .Jf'= L2(0, 1) and let e(l) be the set of all continuous functions on [0,1] that

have a continuous derivative. Let t E [0,1] and define L: e(l)

-'>

IF by L(h) =

h'(t). Show that there

is

no bounded linear functional

on

.Jf' that agrees with

L

on eel).


28/418

14

1. Hilbert Spaces

4. Orthonormal Sets of Vectors and Bases

I t will be shown in this section that, as in Euclidean space, each Hilbert

space can be coordinatized. The vehicle for introducing the coordinates is

an orthonormal basis. The corresponding vectors in IF d are the vectors

{el,eZ, . . . ,e

d

} , where

e

k

is the d-tuple having a 1 in the kth place and

zeros elsewhere.

4.1. Definition. An

orthonormal

subset of a Hilbert space X is a subset

Iff

having the properties: (a) for

e

in Iff, Ilell = 1; (b) if

e

l

,

e

2

E Iff and

e

1

=1= e

2

,

then e

l

..1

e

2

A

basis

for X is a maximal orthonormal set.

Every vector space has a Hamel basis (a maximal linearly independent

set).

The

term

"basis"

for a Hilbert space

is

defined as above and it relates

to the inner product on

X . For

an infinite-dimensional Hilbert space, a

basis is never a Hamel basis. This is not obvious, but the reader will be able

to see this after understanding several facts about bases.

4.2. Proposition. I f

Iff is an orthonormal set in X , then there is a basis for

X

that contains

Iff.

The

proof

of

this proposition is a straightforward application of Zorn's

Lemma and is left to the reader.

4.3. Example. Let

X =

L ~ [0,2'17] and for

n

in

71..

define

en

in X by

en(t) =

(2'17)-I/Zexp(int). Then

{en: n E

7I..} is an orthonormal set in

X .

(Here L ~

[0, 2'17]

is the space of complex-valued square integrable functions.)

It is also true that the set in (4.3) is a basis, but this is best proved after a

bit

of

theory.

4.4. Example.

I f

X = IFd and for 1 ~ k ~ d, e

k

= the d-tuple with 1 in the

kth place and zeros elsewhere, then

{e

l

,

...

,

e

d

}

is a basis for

X .

4.5. Example. Let

X =

12(1) as in Example 1.7.

For

each i in I define e

i

in X by ei(i) = 1 and ei(J) = or j

=1=

i. Then {e

i

: i

E I } is a basis.

The proof of the next result is left as an exercise (see Exercise

5).

It is very

useful

but

the proof is not difficult.

4.6. The Gram-Schmidt Orthogonalization Process. If X

is a Hilbert

space and

{h

n

:

n

E N}

is a linearly independent subset

of X ,

then there

is

an orthonormal set {en: n

EN}

such that for every n, the linear span

of

{e

l

, . ,

en} equals the linear span

of

{hi" '" h

n

}


29/418

IA.

Orthonormal

Sets of

Vectors

and Bases 15

Remember that VA is the closed linear span of

A

(Exercise 2.4).

4.7. Proposition.

Let {e1, . . . ,e

n

} be an orthonormal set in

and

let

A

=

V{

e

l' ...

en}.

I f

P is the orthogonal projection

of

onto

A ,

then

for all

h in

.

n

Ph = ~

(h,ek)e

k

k ~ l

PROOF.

Let Qh = r,Z=l(h, ek)e

k

. I f

1:s;

j :s; n, then (Qh, e

j

) =

r ,Z=/h,ek)(ek ,e)

=

(h ,e ) since ek.L

e

j

for k =F j. Thus (h - Qh,e)

=

0 for 1 :s;

j

:s;

n.

That is,

h

-

Qh

.L A for every

h

in . Since

Qh

is

clearly a vector in A , Qh is the unique vector h 0 in A such that

h

- ho . LA

(2.6). Hence Qh

=

Ph for every h in .

4.8. Bessel's Inequality.

I f

{en:

n

EN}

is an orthonormal set

and

h E

,

then

00

~ I(h, e

n

)12 :s;

IIhll

2

.

n=l

PROOF. Let

h

n

=

h

- r,Z=l(h,ek)e

k

.

Then hn.L

e

k

for 1 :s; k:s;

n

(Why?).

By the Pythagorean Theorem,

IIhll

2

=

IIh n

ll

2

+ ~ l

(h, en)ekr

n

n

;::::

~ I(h, e

k

)1

2

k=l

Since n was arbitrary, the result is proved.

4.9. Corollary.

I f

C

is an orthonormal set in

and

hE

,

then

(h,

e)

=F 0

for

at

most

a countable number

of

vectors e in

C.

PROOF.

For each n;:::: 1 let C

n

= {e E

C: I(h,

e)1 ;::::

l in} . By

Bessel's

Inequality,

C

n

is finite. But U::'=lC

n

= {e E C: (h,e

n

)

=F O}.

4.10. Corollary. I f C is an orthonormal set and h E , then

~ l(h,e)1

2

:s; IIhll

2

.

ef", f

This last corollary is just Bessel's Inequality together with the fact (4.9)

that

at

most a countable number of the terms in the sum differ from zero.

Actually, the sum that appears in (4.10) can be given a better interpreta

t ion-a

mathematically precise one that will be useful later. The question is,


30/418

16

I.

Hilbert Spaces

what

is meant

by

I:{ hi:

i

E I } if hi E.Yf' and I is an infinite, possibly

uncountable, set? Let

:7 be the collection of all finite subsets of

I and

order

:7

by inclusion, so :7 becomes a directed set.

For

each F in :7, define

hF= L { h

i

:

i E F}.

Since this is a finite sum, hF is a well-defined element of

.Yf'.

Now {h F:

FE:7} is a net in .Yf'.

4.11. Definition. With the notation above, the sum I:{ hi: i

E

I} converges

if the net {h

F: F E

:7} converges; the value of the sum is the limit of the

net.

I f

.Yf'=

IF,

the definition above gives meaning to an uncountable sum of

scalars. Now Corollary 4.10 can be given its precise meaning; namely,

I:{I(h,e)1

2

: e E

6"} converges and the value::; IIhl1

2

(Exercise 9).

I f the set

I

in Definition

4.11

is countable, then this definition of

convergent

sum

is

not

the usual one. That is, if {h

n}

is a sequence in .Yf',

then the convergence of I:{ h n:

n EN}

is not equivalent to the convergence

of

: r : ~ l h

n

The former concept of convergence is that defined in (4.11) while

the latter means that the sequence { I : k ~ l h d r : ~ l converges. Even if .Yf'= IF,

these concepts do not coincide (see Exercise 12).

If,

however, I:{ h

n:

n

EN}

converges,

then

I : r : ~ l h n converges (Exercise 10). Also see Exercise 11.

4.12. Lemma. I f C is an orthonormal set and h E.Yf', then

L{(h ,e )e :

e E C }

converges in

.Yf'.

PROOF. By (4.9), there are vectors

e

1

,

e

2

,

. . . in C such that {e E

C:

(h,e) *" O} = {e

1

,e

2

, . . .

}.

W e a l s o k n o w t h a t I : r : ~ 1 1 ( h , e n ) 1 2 : : ; IIhl1

2

0, there is

an

N such that I : r : ~ N I ( h , e n ) 1 2 < ,,2. Let

Fa =

{e

1

,

...

, eN-d

and

let :7= all the finite subsets of C. For F in

:7

define

h

F

==

L{

(h, e)e:

e

E F}. I f

F

and

G

E:7

and both

contain

Fa,

then

IIh

F

- hGII2 = L{I(h,e)1

2

:

e

E (F\G)

U(G\F)}

00

< e

2

.

So {h

F: F E :7} is

a Cauchy net in

.Yf'.

Because .Yf' is complete, this net

converges. In fact, it converges to

I : r : ~ l ( h , en)e

n

4.13. Theorem.

I f

C

is an orthonormal set in

.Yf',

then the following

statements

are equivalent.

(a) C is a basis for .Yf'.

(b)

I f

h E.Yf' and h

.1

C, then h =

0.

(c) V C= .Yf'.


31/418

1.4. Orthonormal Sets of Vectors and Bases

(d) I f

h

E.Yl',

then h

=

L{

(h, e)e:

e

E O"}.

(e) I f g and h E.Yl', then

(g,h)

=

L{(g ,e ) (e ,h) :

eEO"}.

17

(f) I f h E.Yl', then IIhl1

2

= L{I(h, e)12: e E O"} (Parseval's Identity).

PROOF.

(a) =* (b): Suppose h .1. 0" and h =f- 0; then

O"U{

h/llhll} IS an

orthonormal set that properly contains 0", contradicting maximality.

(b)

=

(c): By Corollary 2.11,

VO"=.Yl'

if and only if

0".1

= (0).

(b)

=*

(d):

I f h

E.Yl', then

f= h

- L{(h,e)e:

e

E

O"}

is a well-defined

vector by Lemma 4.12. I f e

1

E

0",

then

(I ,

e

1

) =

(h,

e

1

) - E{ (h,

e)(

e, e

1

) :

e EO"} = (h, e

1

)

-

(h, e

1

) = O. That is, f

EO".1

. Hence f = O. (Is every

thing legitimate in that string of equalities? We don't want any illegitimate

equalities.)

(d) =* (e): This is left as an exercise for the reader.

(e) =*

(f):

Since

IIhl1

2

=

(h,

h), this is immediate.

(f) =* (a): I f 0" is not a basis, then there is a unit vector eo (Ileoll =

1)

in

.Yl' such that eo.l.. 0". Hence, 0 = E{I(e

o

,e)1

2

: e E O"}, contradicting (f) .

Just as in finite-dimensional spaces, a basis in Hilbert space can be used

to define a concept of dimension.

For

this purpose the next result is pivotal.

4.14. Proposition. If .Yf' is a Hilbert space, any two bases have the same

cardinality.

PROOF. Let 0" and

.f7

be two bases for

.Yf'

and put e = the cardinality of

0",

TJ = the cardinality of .f7. I f e or TJ is finite, then e =

TJ

(Exercise 15).

Suppose both e and

TJ

are infinite.

For

e in

0",

let

~ ={ f

E.f7: (e, f )

=f

O}; so ~ is countable. By (4.13b), each f in .f7 belongs to at least one set

~ , e

in

0".

That

is, .f7=

U { ~ :

e

E

O"}.

Hence

TJ

~

e .

~ o

=

e.

Similarly,

e

~

TJ.

4.15. Definition.

The

dimension

of a Hilbert space is the cardinality of a

basis and is denoted by dim.Yf'.

I f (X,

d) is a metric space that is separable and

{B;

=

B(x

i

; e;):

i

E I} is

a collection

of

pairwise disjoint open balls in X, then

I

must be countable.

Indeed, if D is a countable dense subset of

X,

Bi () D =f- 0 for each i in I.

Thus there is a point Xi in Bi ()

D.

So {Xi: i E I} is a subset of

D

having

the cardinality of

I;

thus

I

must be countable.

4.16. Proposition.

If.Yf'

is an infinite-dimensional Hilbert space, then .Yl' is

separable if

and

only if dim.Yf'= ~ o'


32/418

18 I. Hilbert Spaces

PROOF. Let g be a basis for

.Yf'. I f e

l

,

e

2

E g, then lie

l

- e211

2

=

IIedl

2

+

IIe

2

2

= 2. Hence

{B(e;

1/v1): e

E g}

is a collection of pairwise disjoint

open balls in .Yf'. From the discussion preceding this proposition, the

assumption that

.Yf'

is

separable implies g is countable. The converse is an

exercise.

EXERCISES

1.

Verify the statements in Example

4.3.

2. Verify the statements in Example 4.4.

3. Verify the statements in Example 4.5.

4.

Find an

infinite orthonormal set in the Hilbert space of Example 1.8.

5. Using the notation of the Gram-Schmidt Orthogonalization Process, show that

up

to scalar multiple e

l

=

hl/llhdl

and for n

~ 2, en

=

Ilh

n

-

Inll-l(h

n

- /,,),

where

In

is the vector defined formally by

-1 l

h l ~ h l )

In

= de t

det[

(hi '

) r j ~ l (hI' h

n

-

l

)

hI

In

the next three exercises, the reader is asked to apply the Gram-Schmidt

Orthogonalization Process to a given sequence in a Hilbert space. A reference for

this material is pp.

82-96

of Courant and Hilbert

[1953].

6. I f

the sequence

1, x, X 2, .

. . is orthogonalized in L

2

( - 1, 1),

the sequence

en(x) = [t(2n + 1)]1/2P

n

(x)

is

obtained, where

Px) =

_ l _ ( ~ ) n

(x

2

-

1)"

n 2nn dx .

The functions P

n

(x)

are called Legendre polynomials.

7.

I f

the sequence

e-

x

'

/2, xe-

x

2

/2,

x

2

e-

x2

/2, ...

is orthogonalized in

L2(

-

00,

(0),

the sequence en (x) = [2n

n y';

]-1/2

H"

(x )e-

X2

/2

is

obtained,

where

Hn(x) = (-1)" ex2( ~ ) ne-x2.

The functions Hn are Hermite polynomials and satisfy

H;(x) =

2nH,,_

1(x).

8.

I f the sequence e-

x

/

2

, xe-

x

/

2

, x

2

e-

x

/

2

, . . . is orthogonalized in L

2

(O, (0), the

sequence en(x) =

e-

x

/

2

L

n

(x)/n

is

obtained, where

Ln(x)

=

e

x

( ~ r

(xne


33/418

1.5.

Isomorphic Hilbert Spaces and the Fourier Transform for the Circle

19

11. I f

{h

n

}

is a sequence in a Hilbert space and

L ~ ~ J i l h n l l

< 00, show that

L{ h

n

:

n

EN}

converges in the sense of Definition 4.11.

12. Let {an} be a sequence in

f

and prove that the following statements are

equivalent: (a) L{ an: n

E

N} converges in the sense of Definition 4.11. (b) I f 'IT

is any permutation of N, then L ~ ~ l a , , ( n ) converges (unconditional convergence).

(c)

L ~ ~ l l a n l

Yf'.

Similar such arguments show that the concept

of

"isomorphic" is

an equivalence relation

on

Hilbert spaces.

I t

is also certain that this is the


34/418

20

1.

Hilbert Spaces

correct equivalence relation since an inner product is the essential ingredient

for a Hilbert space and isomorphic Hilbert spaces have the

"same"

inner

product.

One

might object that completeness is another essential ingredient

in

the definition of a Hilbert space.

So

it

is

However, this too is preserved

by

an isomorphism. An

isometry

between metric spaces

is

a map that

preserves distance.

5.2. Proposition. I f V: ~ X

is

a linear map between Hilbert spaces, then

V is an isometry

if

and only if

(Vh,

Vg)

= (h,

g) for all h,

g in .

PROOF. Assume (Vh, Vg) =

(h,

g) for all h, g in . Then II Vhl1

2

=

(Vh, Vh) = (h, h) =

IIhl1

2

and V is an isometry.

Now

assume that

V

is an isometry.

I f

h,

g

E

and

;\.

E

IF,

then

Ilh + ;\.g112 = II Vh +

;\.VgI1

2

. Using the polar identity on both sides of this

equation gives

IIhl1

2

+ 2ReX(h,g) + 1;\.1

2

11g11

2

= IIVhl1

2

+ 2ReX(Vh,Vg) + 1;\.1

2

1IVgI1

2

.

But

II

Vhll =

Ilhll and II Vgll

=

Ilgll, so this equation becomes

ReX(h, g)

=

ReX(Vh, Vg)

for any ;\. in IF.

I f

IF = IR, take ;\. = 1. I f IF = C, first take ;\. = 1 and then

take ;\. = i to find that (h, g) and (Vh, Vg) have the same real and

imaginary parts.

Note

that an isometry between metric spaces maps Cauchy sequences into

Cauchy sequences. Thus an isomorphism also preserves completeness.

That

is, if an inner product space is isomorphic to a Hilbert space, then it must be

complete.

5.3. Example. Define S: 12 ~ 12 by

S(

0'1,0'2' . . . ) = (0,0'1,0'2' ... ). Then

S is an isometry that is not surjective.

The

preceding example shows that isometries need not be isomorphisms.

A word about terminology. Many call what

we

call an isomorphism a

unitary operator.

We shall define a unitary operator as a linear transforma

tion U: Y l ' ~ Yl' that is a surjective isometry. That is, a unitary operator is

an isomorphism whose range coincides with its domain. This may seem to

be a minor distinction, and in many ways it is. But experience has taught me

that there is some benefit in making such a distinction, or at least in being

aware of it.

5.4. Theorem. Two Hilbert spaces are isomorphic

if

and only

if

they have the

same dimension.

PROOF. I f

U:

Y l ' ~

X is

an isomorphism and I

is

a basis for

,

then it

is

easy to see that UI == {Ue:

eEl }

is a basis for X . Hence, dim

=

dim X .


35/418

I.5. Isomorphic Hilbert Spaces and the Fourier Transform for the Circle

21

Let Yt' be a Hilbert space and let

g

be a basis for Yt'. Consider the

Hilbert space

l2(g).

I f h

EYt',

define h:

g ~ IF

by h(e) =

(h,e). By

Parseval's Identity h E 12(

g)

and

Ilhll

=

IIhll.

Define

U: Y t ' ~ 12(

g) by

Uh

=

h.

Thus

U

is linear and an isometry.

It

is easy to see that

ranU

contains all the functions f in

l2(

g) such that f( e) =

or all but a finite

number of e; that is, ranU is dense. But

U,

being an isometry, must have

closed range. Hence

U: Y t ' ~ 12( g)

is an isomorphism.

I f f is a Hilbert space with a basis %, f is isomorphic to 12(%).

I f

dim Yt'= dim

f, g

and

%

have the same cardinality; it is easy to see that

12( g) and 12( % ) must be isomorphic. Therefore Yt' and f are isomorphic

5.5.

Corollary.

All

separable infinite dimensional Hilbert spaces are isomor

phic.

This section concludes with a rather important example of an isomor

phism, the Fourier transform on the circle.

The

proof of the next result can be found as an Exercise on p.

263

of

Conway [1978]. Another proof will be given later in this book after the

Stone-Weierstrass Theorem is proved.

So

the reader can choose to assume

this for the moment. Let U} = {z

E

C:

Izi

< I}.

5.6. Theorem. I f f:

aU} ~

C

is

a continuous function, then there

is

a

sequence {Pn(z,

i )} of

polynomials

in

z and

z such

that Pn(z,

z) ~ fez)

uniformly

on aU}.

Note that if

z

E aU},

z

= z-l. Thus a polynomial in

z

and

z

on

aU}

becomes a function of the form

I f

we put

z

=

e

iO

,

this becomes a function of the form

Such functions are called trigonometric polynomials.

We can now show that the orthonormal set in Example 4.3 is a basis for

L ~ [ O ,

27T]. This is a rather important result.

5.7. Theorem.

If

for each n

in

71..,

en(t)

==

(27T)-1/2

exp(int), then {en:

nElL} is

a basis for

L ~ [ O , 27T].

PROOF. Let .07= O : : : Z ~ - n < X k e k :


36/418

22

I. Hilbert Spaces

is

'??= {IE

Cdo,

277"): f(O) = f(277")}.

To do this, let

fE Cf/

and define

F:

o[)) ~ C by F(e

it

)

= f(t).

F is continuous. (Why?) By (5.6) there is a

sequence

of

polynomials in z and z, {Pn(z, Z)}, such that Pn(z, z) ~ F(z)

uniformly

on

O[)).

Thus Pn(eil,e-

it

)

-->

f(t)

uniformly on

[0,2'1T].

But

Pn(

e

it

, e-

it

) E:Y.

Now

the closure of Cf/ in L ~ [0,277") is all of L ~ [0,277") (Exercise 6). Hence

V{e

n

: n E

Z}

= L ~ [ 0 , 2 7 7 " )

and

{en}

is thus a basis (4.13).

Actually, it is usually preferred to normalize the measure on [0,277"). That

is, replace

dt

by (277")

-1 dt, so

that the total measure of

[0,277") is 1.

Now

define

en(t)

= exp(int). Hence {en:

n E Z}

is a basis for .Yl'=

L ~ ( [ 0 , 2 7 7 " ) , ( 2 7 7 " ) - l d t ) . I f

fE.Yl', then

5.S

is called the nth Fourier coefficient of f, n in Z. By (5.7) and (4.13d),

00

5.9

n= - 00

where this infinite series converges to f in the metric defined by the norm of

.Yl'. This is called the Fourier series

of

f.

This terminology is classical and

has been adopted for a general Hilbert space.

I f .Yl'

is any Hilbert space and

iff is

a basis, the scalars

{< h, e); e

E

iff}

are called the Fourier coefficients of h (relative to iff) and the series in

(4.13d) is called the Fourier expansion

of

h (relative to iff).

Note

that

Parseval's Identity applied to (5.9) gives that L : ; ' ~

_ool/(n)1

2

L2(X, a, ""1) EB

L

2

(

X, a, lL2) defined by VI

=

11

EB

12' where

fj

is the equivalence class of

L

2

(X,

a,,.,,)

corresponding to

I,

is well defined, linear, and injective. Show that

U

is an isomorphism iff

""1

and ""2 are mutually singular.


40/418

CHAPTER

II

Operators on Hilbert Space

A large area

of

current research interest is centered around the theory

of

operators on Hilbert space. Several other chapters in this book will be

devoted to this topic.

There

is a marked contrast here between Hilbert spaces and the Banach

spaces that are studied in the next chapter. Essentially all of the information

about the geometry of Hilbert space is contained in the preceding chapter.

The geometry of Banach space lies in darkness

and

has attracted the

attention

of

many

talented research mathematicians. However, the theory of

linear operators (linear transformations) on a Banach space has very few

general results, whereas Hilbert space operators have an elegant and well

developed general theory. Indeed, the reason for this dichotomy is related to

the opposite status of the geometric considerations. Questions concerning

operators on Hilbert space

don't

necessitate

or

imply any geometric difficul

ties.

In addition to the fundamentals of operators, this chapter will also

present an interesting application to differential equations in Section

6.


The

proof of

the next proposition is similar to that of Proposition 1.3.1 and

is left

to

the reader.

1.1. Proposition. Let yt> and f be Hilbert spaces and A: yt> f a linear

transformation. The following statements are equivalent.

(a) A is continuous.

(b) A is continuous at O.


41/418

ILL

Elementary Properties and Examples

27

(c)

A is continuous at some point.

(d) There is a constant

c

>

0

such that

IIAhl1 ~ cllhll

for all h

in .Yf'.

As

in

(1.3.3),

if

IIA

II = sup{IIAhll: h E

.Yf',

Ilhll

~ I},

then

IIAII = sup{IIAhll:

IIhll

= I}

= sup{IIAhll/llhll:

h

*

O}

= inf{ c

> 0: IIAhl1 ~

cllhll, h in

.Yf'}.

Also,

IIAhl1

~

IIAllllhll.

IIAII

is called the

norm

of

A

and

a linear transfor

mation with

finite norm is called

bounded. Let

Jj(.Yf', g) be

the

set of

bounded linear transformations from .Yf' into g. For.Yf'=

g, Jj(.Yf',.Yf')

== Jj(.Yf'). Note that Jj(.Yf',

0=)

= all the bounded linear functionals on .Yf'.

1.2. Proposition.

(a) If A and

B E

Jj(.Yf',

g) ,

then A + B

E

Jj(.Yf',

g) ,

and

IIA + BII ~ IIAII + IIBII

(b)

I f

a

E

0= and A

E Jj(.Yf',

g ) , then aA

E Jj(.Yf',

g) and IlaA11

=

lalllAII

(c)

I f A

E

Jj(.Yf',

g)

and

B E

Jj(g,

l ') ,

then

BA

E

Jj(.Yf', l')

and

IIBAII

~

IIBIIIIAII

PROOF. Only (c) will be proved; the rest of the proof is left

to

the reader.

I f

kEf, then IIBkl1 ~ IIBllllkll. Hence, if h E.Yf', k = Ah E f and so

IIBAhl1

~ IIBllllAhl1 ~ IIBIIIIAllllhll

By

virtue

of the preceding proposition,

dCA,

B) = IIA -

BII

defines a

metric on Jj(.Yf',

g) .

So it makes sense to consider Jj(.Yf',

g)

as a metric

space. This will

not

be examined closely until

later

in

the

book, but

later

in

this

chapter the idea

of

the

convergence

of

a sequence

of

operators

will

be

used.

1.3. Example.Ifdim.Yf'=n< o o a n d d i m g = m < oo,let{el, . . . ,en}be

an

orthonormal

basis for .Yf' and let {f

l

, . . . , f

m

} be an

orthonormal

basis

for f .

It

can

be shown

that

every linear

transformation

from

.Yf'

into g is

bounded

(Exercise 3).

I f

1

~ j

~ n , 1 ~

i

~

m ,

let aij =

(Ae

i

, f

i

).

Then

the

m X n

matrix (a

i

) represents

A

and every such matrix represents an

element

of Jj(.Yf',

g) .

1.4. Example.

Let

[2 ==

[2(1\1)

and

let

e

l

, e

2

, ...

be its usual basis.

I f

A

E Jj(l2),

form

aij =

(Ae

j

, e

i

) .

The

infinite

matrix ( a

i

)

represents

A

as

finite

matrices

represent

operators on

finite dimensional spaces. However,

this representation has limited value unless the matrix has a special form.

One difficulty is

that

it is

unknown how to

find the

norm

of

A

in terms of


42/418

28

II. Operators

on

Hilbert Space

the entries in the matrix. In fact, if 2 < n < 00, there is no known formula

for the norm of a matrix in terms of its entries. A sufficient condition that is

useful is known, however (see Exercise 11).

1.5. Theorem. Let (X,

D, JL)

be a a-finite measure space and put

:if '=

L 2( X, D, JL) == L 2(JL). I f

0, the a-finiteness of the measure space implies that there is a

set .::1 in

D,

0

f

is a linear

transformation such that L:IIAenll < 00. Show that A is bounded.

4. Proposition 1.2 says that d(A,B) = IIA

- BII

is a metric on ?I(Ji ', f). Show

that ?I(Ji',

f)

s complete relative to this metric.

S.

Show that a multiplication operator

M.p

(1.S) satisfies

M;

=

M.p

if and only if


45/418

II.2. The Adjoint of an Operator 31

2. The Adjoint of an Operator

2.1. Definition. If X and :f are Hilbert spaces, a function

u: Xx:f IF

is a

sesquilinear form

if for

h,

g in X , k,

f

in : f , and

a,

f3 in IF,

(a)

u(ah +

f3g, k)

=

au(h,

k) + f3u(g, k);

(b)

u(h,ak + f3f)

=

au(h, k) + 13u(h,f).

The prefix "sesqui" is used because the function is linear in one variable

but

(for IF = C) only conjugate linear in the other. ("Sesqui" means

" one-and-a-half.")

A sesquilinear form is bounded if there is a constant M such that

lu(h,

k)1

.::;;

Mllhllllkil

for all

h

in X and

k

in

: f .

The constant

M

is

called a bound for

u.

Sesquilinear forms are used to study operators. I f A

E

l(

X ,

X"), then

u(h, k) == (Ah,

k)

is a bounded sesquilinear form. Also, if

B E

leX", X),

u( h, k) ==

(h, Bk)

is a bounded sesquilinear form. Are there any more? Are

these two forms related?

2.2. Theorem. I f u:

Xx:f

IF is

a bounded sesquilinear form with bound

M,

then there are unique operators A in l(

X ,

X") and B

in

l( X",

X )

such

that

2.3

u(h,k) = (Ah,k) = (h,Bk)

for all h in

X and

k

in

:f and IIA

II, IIBII .::;;

M.

PROOF.

Only the existence of

A

will be shown.

For

each

h

in

X ,

define

L

h

:

% ~ IF by Lh(k) = u(h,

k).

Then Lh is linear and ILh(k)1 .::;; Mllhllllkil.

By

the Riesz Representation Theorem there is a unique vector f in X" such

that

(k , j )

=

Lh(k)

= u(h,k) and 11Il1.::;; Mllhll. Let Ah =f .

It

is left as

an exercise to show that

A

is

linear (use the uniqueness part of the Riesz

Theorem). Also, (Ah, k) = (k, Ah) = (k, f) =

u(h, k).

I f Al

E

leX,

% ) and u(h, k) =

(AIh,

k), then

(Ah

- AIh, k) = 0 for

all k; thus Ah - AIh = 0 for all h. Thus, A is unique.

2.4. Definition. If

A

E

l(

X ,

X"),

then the unique operator

B

in

leX",

X )

satisfying (2.3) is called the adjoint of A and is denoted by

B

= A*.

The adjoint of an operator will usually be used for operators in l( X),

rather than

l(

X ,

X"). There is one notable exception.

2.5. Proposition. I f U E l( X ,

: f ) ,

then U is an isomorphism if and only

if

U is invertible and U-

I

= U

*.

PROOF. Exercise.


46/418

32

II. Operators on Hilbert Space

From now

on

we will examine and prove results for the adjoint

of

operators in d( ").

Often, as

in

the next proposition, there are analogous

results for the adjoint

of

operators

in

d( ", .Jf"). This simplification is

justified, however, by the cleaner statements

that

result. Also, the interested

reader will have no trouble formulating the more general statement when it

is needed.

2.6. Proposition.

If A, B E d(") and a E

IF,

then:

(a) (aA + B)*

= aA*

+ B*.

(b)

(AB)* = B*A*.

(c) A** == (A*)*

=

A.

(d)

I f

A is invertible in

d(")

and

A

-1

is its inverse, then

A*

is invertible

and

(A*)

-1 = (A -1)*.

The proof of

the preceding proposition is left as

an

exercise,

but

a word

about part

(d) might

be

helpful. The hypothesis that

A

is invertible

in

d( ") means that there is an operator A -1 in d( ") such that

AA -1 =

A -lA = I. I t is a remarkable fact that if A is only assumed to

be

bijective,

then A

is invertible

in

d(

").

This is a consequence

of

the Open Mapping

Theorem, which will be proved later.

2.7. Proposition.

If A

E

d("), IIAII

=

IIA*II

=

IIA*Alll/2.

PROOF. For h

in ",

Ilhll:s;; 1, IIAhl1

2

= (Ah, Ah) = (A*Ah, h) :s;;

IIA*Ahllllhll :s;; IIA*AII :s;; IIA*llllAII Hence

IIAI12:s;;

IIA*AII :s;;

IIA*IIIIAII

Using the two ends of this string of inequalities gives IIAII :s;; IIA*II when

IIAII is cancelled. But A

=

A** and so if

A*

is substituted for

A,

we get

IIA*II

:s;;

IIA**II =

IIAII Hence

IIAII = IIA*II. Thus

the string

of

inequalities

becomes a string

of

equalities

and

the

proof

is complete.

2.8. Example. Let

(X,

g,

Jl)

be a a-finite measure space

and

let

M

be

the

multiplication operator with

~ m b o l

cp (1.5).

Then M*

is M;p, the multipli

cation operator

with symbol

cpo

I f an

operator

on

IF d is represented by a matrix, then its adjoint IS

represented by the conjugate transpose of the matrix.

2.9. Example. I f

K

is the integral operator with kernel

k

as in (1.6), then

K * is the integral operator with kernel k *(x, y) == k (y, x).

2.10. Proposition.

I f

S: /2 ---> /2

is

defined by S(a

1

, a

2

,

) =

(0,a

1

,a

2

, ), then S

is

an isometry and S*(a

1

,a

2

, )

=

(a

2

,a

3

, .. . ) .

PROOF.

It has

already been mentioned that S is an isometry (1.5.3).

For (a,,)

and (/3

n

) in /2, (S*(a

n

),(/3n

=

a

n

,S(/3n =

a

1

,a

2

, ... ),(0,/31,/32'


47/418

II.2. The Adjoint of an Operator

33

...

= a

2

P

l

+ a

3

P2 +

...

= a

2

,a

3

,

),(f3l,/3

2

,

. Since this holds

for every (f3

n

), the result is proved.

The

operator S in (2.10) is called the

unilateral shift

and the operator S

*

is called the backward shift.

The operation of taking the adjoint of an operator is, as the reader may

have seen from the examples above, analogous to taking the conjugate of a

complex number. I t

is

good to keep the analogy in mind, but do

not

become

too religious about it.

2.11. Definition. I f A

E

JB( .Yt'), then: (a) A is hermitian or self-adjoint if

A* = A; (b) A is normal if AA* = A*A.

In the analogy between the adjoint and the complex conjugate, hermitian

operators become the analogues of real numbers and, by (2.5), unitaries are

the analogues of complex numbers of modulus

1.

Normal operators, as we

shall see, are the true analogues of complex numbers. Notice that hermitian

and unitary operators are normal.

In light of (2.8), every multiplication operator M is normal; M is

hermitian if and only if cf> is real-valued;

M

is unitary if and only if

1cf>1

==

1 a.e. [ILl. By (2.9), an integral operator K with kernel k is hermitian

if

and

only if

k(x,

y)

=

key,

x)

a.e.

[IL

X

ILl.

The unilateral shift is not

normal (Exercise

6).

2.12. Proposition.

I f

.Yt' is a C-Hilbert space and A

E

JB(.Yt'), then A is

hermitian

if

and only if

(Ah, h)

E

R

for all h in

.Yt'.

PROOF. I f A = A*,

then

(Ah, h) = (h,

Ah)

=

(Ah, h); hence

(Ah, h) E

R.

For

the converse, assume

(Ah, h) is

real for every h in .Yt'. I f a E C and

h, g E.Yt', then

(A(h +

ag), h

+

ag)

= (Ah, h) +

ii(Ah,

g) + a(Ag, h)

+

laI

2

(Ag,

g)

E

R. So this expression equals its complex conjugate. Using

the fact that

(Ah, h)

and (Ag,

g) E

R yields

a(Ag, h)

+

ii(Ah,

g) = ii(h, Ag)

+

a(g, Ah)

= ii(A*h, g)

+ a(A*g,

h).

By first taking

a

= 1 and then

a

= i, we obtain the two equations

(Ag, h)

+

(Ah, g) = (A*h, g)

+

(A*g, h),

i(Ag, h)

-

i(Ah, g) =

-i(A*h, g)

+ i(A*g, h).

A little arithmetic implies

(Ag,

h) = (A*g, h), so A = A*.

The

preceding proposition

is

false if it is only assumed that .Yt' is an

R-Hilbert space. For example, if A = [ _ on R 2, then (Ah, h) = 0


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34


for all h in 1R2. However,

A*

is the transpose of A

and

so

A* =1=

A. Indeed,

for

any operator

A on an IR-Hilbert space,

(Ah, g)

E

IR.

2.13. Proposition.

If

A

=

A*, then

IIAII

= sup{I(Ah, h)l: Ilhll = I}.

PROOF. Put

M

=

sup{I(Ah,

h)l: Ilhll = I}.

I f

Ilhll =

1,

then I(Ah,

h)1 ::::;

IIAII; hence

M::::; IIAII. On

the

other

hand, if

Ilhll

= Ilgll = 1, then

( A

(h

g),

h g) =

(Ah,

h)

(Ah,

g)

(Ag,

h) + (Ag, g)

= (Ah,

h)

(Ah,

g) (g, A*h) +

(Ag, g).

Since A = A*, this implies

(A{h g),h g) =

(Ah,h)

2Re(Ah,g) + (Ag,g).

Subtracting

one

of these two equations from the other gives

4Re(Ah, g) =

(A{h

+ g), h + g) - (A(h - g), h - g).

Now

it is easy

to

verify that

I(Af,f)1 ::::; MIlf112

for

any

f in . Hence

using the parallelogram law we get

4Re(Ah, g)

::::; M(llh +

gl12

+ Ilh _

g112)

= 2M(llh112 + Ilg11

2

)

=4M

since h a n d g are unit vectors.

Now

suppose

(Ah, g)

= eiol(Ah, g)l.

Replacing

h in the inequality above with e-ioh gives

I(Ah,

g)1 ::::; M if

Ilhll

= Ilgll = 1.

Taking the

supremum

over all g gives IIAhl1

::::;

M when

Ilhll

= 1.

Thus IIAII ::::; M.

2.14. Corollary. If A

=

A* and (Ah, h)

=

0 for all h, then A

= o.

The preceding

corollary is

not

true unless A = A *, as

the

example given

after Proposition

2.12 shows. However, if a complex Hilbert space is

present, this hypothesis

can

be deleted.

2.15.

Proposition. If is

a C-Hilbert space and A

E .'14()

such that

(Ah, h)

=

0

for all h

in

, then A

=

o.

The

proof of

(2.15) is left to

the

reader.

I f

is a

C-Hilbert

space

and

A

E

~ ( ) ,

then B =

(A + A*)j2 and

C

= (A -

A*)j2i are self-adjoint

and

A

=

B

+

iC.

The operators Band

C are called, respectively, the real and imaginary parts of A.


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n.2.

The Adjoint of an Operator

35

2.16. Proposition. I f A E 88() , the following statements are equivalent.

(a)

A is normal.

(b) JJAhJJ

=

IIA*hll

for all h.

I f

is a C-Hilbert space, then these statements are also equivalent to:

(c) The real and imaginary parts

of

A commute.

PROOF. I f h

E

, then JJAhJJ2 - JJA*hJJ2 = (Ah, Ah) -

(A*h,

A*h) =

A*A -

AA*)h, h).

Since

A*A

-

AA*

is hermitian, the equivalence of (a)

and (b) follows from Corollary 2.14.

I f

B,

C are the real and imaginary parts of A, then a calculation yields

A*A

=

B2

-

iCB

+ iBC +

C

2

,

AA*

=

B2

+

iCB

-

iBC

+

C

2

.

Hence

A*A =

AA* if and only if

CB

=

BC,

and so (a) and (c) are

equivalent.

2.17. Proposition. I f A

E

88(

) ,

the following statements are equivalent.

(a) A is an isometry.

(b) A*A = I.

(c)

(Ah, Ag)

=

(h, g)

for all h,

gin .

PROOF. The proof that (a) and (c) are equivalent was seen in Proposition

1.5.2. Note that if

h,

g

E

, then

(A*Ah, g)

= (Ah,

Ag).

Hence (b) and

( c) are easily seen to be equivalent.

2.1S. Proposition.

If

A

E

88(), then the following statements are equiv

alent.

(a) A is unitary.

(b)

A is a surjective isometry.

(c)

A is a normal isometry.

PROOF. (a) = (b): Proposition 1.5.2.

(b) = (c):

By

(2.17),

A*A

=

I.

But it is easy to see that the fact that

A

is

a surjective isometry implies that

A -1

is also. Hence by (2.17) 1=

(A -l)*A -1 = (A*)-lA -1 = (AA*)-\ this implies that A*A = AA* = I.

(c) = (a): By (2.17), A*A = I. Since A is also normal, AA* = A*A = I

and

so

A

is surjective.

We conclude with a very important, though easily proved, result.

2.19. Theorem. I f A E 88(), then kerA = (ranA*)-L.

PROOF.

I f

h

E kerA and g E , then

(h, A*g) = (Ah, g) = 0,

so kerA

c:::;;

(ran

A*)

-L .

On

the other hand, if

h ..1

ran

A*

and g

E

, then

(Ah, g)

=

(h,

A*g) = 0; so (ran

A*) -L

c:::;; ker A.


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36


Two facts should be noted. Since A* * = A, it also holds that ker A* =

(ran A) 1- Second, it is not true that (ker

A)

1- = ran A* since ran A* may

not

be

closed. All that can be said is that (kerA)1-= cl(ranA*) and

(ker

A*)

1-

= cl(ran

A).

EXERCISES

1.

Prove Proposition 2.5.

2.

Prove Proposition

2.6.

3. Verify the statement in Example 2.8.

4. Verify the statement in Example 2.9.

5.

Find the adjoint of a diagonal operator (Exercise 1.8).

6. Let S be the unilateral shift and compute

SS*

and S*S. Also compute S"S*"

and S*"S".

7.

Compute the adjoint of the Volterra operator V (1.7) and V + V*. What is

ran(V

+

V*)?

8.

Where was the hypothesis that '

is

a Hilbert space over C used in the proof of

Proposition 2.12?

9.

Suppose

A = B

+

iC,

where

B

and C are hermitian and prove that

B = (A

+

A*)/2,

C

= (A - A*)/2i.

10. Prove Proposition 2.15.

11.

I f

A and B are self-adjoint, show that A B IS self-adjoint if and only if

AB = BA.

12. Let L ~ ~ o a " z " be a power series with radius of convergence

R,

0 < R :0:; 00. I f

A

E

8l(

' )

and

IIA

II

0 as

n

---> 00. I f

BA

= AB, show that BT =

TB.

14. I f fez) =

expz

=

L ~ ~ o z " / n

and A

is

hermitian, show that f(iA) is unitary.

15. I f A is a normal operator on

' ,

show that A is injective if and only if A has

dense range. Give an example of an operator

B

such that ker

B = (0) but

ran

B

is

not dense. Give an example of an operator C such that C

is

sUIjective but

kerC * (0).

16. Let Mq, be a multiplication operator (1.5) and show that kerMq,

=

(0) if and

only if /L({x: CP(x)

=

OJ)

=

O. Give necessary and sufficient conditions on cp

that ran Mq, is closed.


51/418

11.3. Projections and Idempotents; Invariant and Reducing Subspaces

3. Projections and Idempotents; Invariant and

Reducing Subspaces

37

3.1. Definition. An idempotent on is a bounded linear operator E on

such that

E2

= E. A projection is an idempotent P such that ker P

=

(ran P) -L

I f .A :s; , then PJ I is a projection (Theorem 1.2.7).

It

is not difficult to

construct an idempotent that

is not

a projection (Exercise 1).

Let E be any idempotent and set .A = ran E and JV= ker E. Since E is

continuous, JV is a closed subspace of

.

conway funct analysis

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