Realizability theory for continuous linear systems, Volume 97 (Mathematics in Science and Engineering)

Realizability Theory for Continuous Linear Systems

This is Volume 97 in MATHEMATICS IN SCIENCE AND ENGINEERING A series of monographs and textbooks Edited by RICHARD BELLMAN, University of Southern California The complete listing of books in this series is available from the Publisher upon request.

Realizability Theory for Continuous Linear Systems A. H. Zemanian Department of Applied Mathematics and Statistics State University of New York at Stony Brook Stony Brook, New York

ACADEMIC PRESS

New York and London

1972

COPYRIQHT 0 1972, BY ACADEMIC PRESS, INC. ALL RIGHTS RESERVED. N O PART OF THIS PUBLICATION MAY BE REPRODUCED OR TRANSMITTED IN ANY FORM OR BY ANY MEANS, ELECTRONIC OR MECHANICAL, INCLUDING PHOTOCOPY, RECORDING, OR ANY INFORMATION STORAGE AND RETRIEVAL SYSTEM, WITHOUT PERMISSION IN WRITING FROM THE PUBLISHER.

ACADEMIC PRESS, INC.

I 1 1 Fifth Avenue, New York. New York 10003

United Kingdom Edition published by ACADEMIC PRESS, INC. (LONDON) LTD. 24/28 Oval Road, London N W 1

LIBRARY OF CONGRESS CATALOG CARDNUMBER:12 - 11345 AMS (MOS) 1970 Subject Classification: 93A05 PRINTED IN THE UNITED STATES OF AMERICA

To the Memory of M Y

F A T H E R

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Contents

Preface ................................................................ Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xi xv

CHAPTER 1 . Vector-Valued Functions Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ........... 1.2. Notations and Terminology .............................. ........... 1.3. Continuous Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ........... 1.4. Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5. Repeated Integration and Improper Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. ........... 1.6. Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . I .7. Banach-Space-Valued Analytic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ........... I .8. Contour Integration ...................................... 1.1.

1 2 5

6 9 12 17

20

CHAPTER 2 . Integration with Vector-Valued Functions and Operator-Valued Measures 2.1. lntroduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. Operator-Valued Measures ............................... 2.3. a-Finite Operator-Valued Measures ........................ vii

........... ........... ...........

23 23 29

viii 2.4. 2.5. 2.6.

CONTENTS

Tensor Products and Vector-Valued Functions .......................... Integration of Vector-Valued Functions .................................. Sesquilinear Forms Generated by PO Measures ..........................

34 38 43

CHAPTER 3 . Banach-Space-Valued Testing Functions and Distributions 3.1. Introduction . . . . . . . . . ............................................ 3.2. The Basic Testing-Func Spacearn(A) ............................... 3.3. Distributions . . . . . . . . . . . . . . . . . .................................... 3.4. Local Structure . . . . . . . . . . . . . . . .................................... 3.5. The Correspondence between [ 9 ( A ) ; B ] and [g; [A; B ] ]. . . . . . . . . . . . . . . . . . . 3.6. The p-Type Testing Function Spaces . . .... .. 3.7. Generalized Functions ................................................. 3.8. &-Type Testing Functions and Distributions ............................

49 50 52 57 61 64 67 72

CHAPTER 4. Kernel Operators Introduction ......................................................... Systems and Operators ................................................ 4.3. The Space 8 = 9 ( Y ) ................................................ 4.4. The Kernel Theorem ................................................. 4.5. Kernel Operators ..................................................... 4.6. Causality and Kernel Operators ........................................

4.1. 4.2.

76 77 81 85 89 93

CHAPTER 5 . Convolution Operators

.............. ............. Introduction . . . . . . . . . Convolution . . . . . . . . . ................................... ..................................... Special Cases . . . . . . . . . . . . . . perators with Shifting and 5.4. The Commutativity of Differentiation ..................................................... 5.5. Regularization . . . . . . . . . . . . ........................ 5.6. Primitives....... .................. ........................... 5.7. Direct Products ...................................................... 5.8. Distributions That Are Independent of Certain Coordinates . . . 5.9. A Change-of-Variable Formu ........................................ 5.10. Convolution Operators . . . . . ........................................ 5.1 1. Causality and Convolution Operators .................................. 5.1.

5.2. 5.3.

96 97 98

104 108 110

111 112 115

CHAPTER 6 . The Laplace Transformation 6.1. 6.2.

Introduction ......................................................... The Definition of the Laplace Transformation ............................

117 117

ix

CONTENTS

.

6.3. Analyticity and the Exchange Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . , . . 119 6.4. Inversion and Uniqueness. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 6.5. A Causality Criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

.

.

CHAPTER 7. The Scattering Formulism 7.1. 7.2. 7.3. 7.4. 7.5. 7.6. 7.1. 7.8.

Introduction ......................................................... Preliminary Considerations Concerning L,-Type Distributions , . . . . , . . . . . . . . Scatter-Passivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bounded* Scattering Transforms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Realizability of Bounded* Scattering Transforms . . . . . . . . . . . . . . . . . . . . Bounded*-Real Scattering Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Lossless Hilbert Ports . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Lossless Hilbert n-Port . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

124 125 128 131 134 137 139 143

CHAPTER 8. The Admittance Formulism 8.1. 8.2. 8.3. 8.4. 8.5. 8.6. 8.7. 8.8. 8.9. 8.10. 8.1 1. 8.12. 8.13. 8.14. 8.15. 8.16.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Passivity . .. . . . . . . .. . . .. .. .. . . .. .. .... . . .. .. . Linearity and Semipassivity Imply Continuity . . . . The Fourier Transformation on Y ( H ) . . . . . . . . . . . . . . . ............... Local Mappings.. . . . . . . . . . . . . . . . . . . . . . . . . . . . Positive Sesquilinear Forms on 9 x 9 .................. Positive Sesquilinear Forms on 9 ( H ) x Certain Semipassive Mappings of 9 ( H ) into & H ) . An Extension of the Bochner-Schwartz Theorem . ............. Representations for Certain Causal Semipassive M ........... A Representation for Positive* Transforms. . . . . . . Positive* Admittance Transforms . . . . . . . . . . . . . . . Positive* Real Admittance Transforms . . . . . . . . A Connection between Passivity and Semipassivity ......... A Connection between the Admittance and Scattering Formulisms . . . . . . . . . . The Admittance Transform of a Lossless Hilbert Port . . . . . . . . . . . . . . . . . . . . .

.

149

155 160

174

187 189 192

APPENDIX A. Linear Spaces

194

APPENDIX B. Topological Spaces

198

APPENDIX C. Topological Linear Spaces

201

D. Continuous Linear Mappings APPENDIX

206

APPENDIX E. Inductive-Limit Spaces

21 1

X

CONTENTS

APPENDIX F. Bilinear Mappings and Tensor Products

213

APPENDIX G. The Bochner Integral

216

References

222

.

Index of Symbols.. . . . . . . . . . . . . . . . . . . . . . . . . , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225 Index .................................................................... 221

“Realizability theory” is a part of mathematical systems theory and is concerned with the following ideas. Any physical system defines a relation between the stimuli imposed on the system and the corresponding responses. Moreover, any such system is always causal and may possess other properties such as time-invariance and passivity. Two questions: How are the physical properties of the system reflected in various analytic descriptions of the relation? Conversely, given an analytic description of a relation, does there exist a corresponding physical system possessing certain specified properties ? If the latter is true, the analytic description is said to be realizable. Considerations of this sort arise in a number of physical sciences. For example, see McMillan (1952), Newcomb (1966), or Wohlers (1969) for electrical networks, Toll (1956) or Wu (1954) for scattering phenomena, and Gross (1953), Love (1956), or Meixner (1954) for viscoelasticity. This book is an exposition of realizability theory as applied to the operators generated by physical systems as mappings of stimuli into responses. This constitutes the so-called “ black box ” approach since we do not concern ourselves with the internal structure of the system at hand. Physical characteristics such as linearity, causality, time-invariance, and passivity are defined as mathematical restrictions on a given operator. Then, the two questions are xi

xii

PREFACE

answered by obtaining a description of the operator in the form of a kernel or convolution representation and establishing a variety of necessary and sufficient conditions for that representation to possess the indicated properties. Thus, the present work is an abstraction of classical realizability theory in the following way. A given representation is realized not by a physical system but rather by an operator possessing mathematically defined properties, such as causality and passivity, which have physical significance. We may state this in another way. Our primary concern is the study of physical properties and their mathematical characterizations and not the design of particular systems. Two properties we shall always impose on any operator under consideration are linearity and continuity. They are quite commonly (but by no means always) possessed by physical systems. Of course, continuity only has a meaning with respect to the topologies of the domain and range spaces of the operator. We can in general take into account a wider class of continuous linear operators by choosing a smaller domain space with a stronger topology and a larger range space with a weaker topology. With this as our motivation, we choose the basic testing-function space of distribution theory as the domain for our operators and the space of distributions as the range space. The imposition of other properties upon the operator will in general allow us to extend the operator onto wider domains in a continuous fashion. For example, time-invariance implies that the operator has a convolution representation and can therefore be extended onto the space of all distributions with compact supports. This distributional setting also provides the following facility. It allows us to obtain certain results, such as Schwartz’s kernel theorem, which simply do not hold under any formulation that permits the use of only ordinary functions. Thus, distribution theory provides a natural language for the realizability theory of continuous linear systems. Still another facet of this book should be mentioned. Almost all the realizability theories for electrical systems deal with signals that take their instantaneous values in n-dimensional Euclidean space. However, there are many systems whose signals have instantaneous values in a Hilbert or Banach space. Section 4.2 gives an example of this. For this reason, we assume that the domain and range spaces for the operator at hand consist of Banach-spacevalued distributions. Many of the results of earlier realizability theories readily carry over to this more general setting, other results go over but with difficulty, and some do not generalize at all. Moreover, the theory of Banach-spacevalued distributions is somewhat more complicated than that of scalar distributions; Chapter 3 presents an exposition of it. Still other analytical tools we shall need as a consequence of our use of Banach-space-valued distributions are the elementary calculus of functions taking their values in locally convex spaces, which is given in Chapter 1, and Hackenbroch’s

PREFACE

xiii

theory for the integration of Banach-space-valued functions with respect to operator-valued measures, a subject we discuss in Chapter 2. The systems theory in this book occurs in Chapters4,5,7, and 8. Chapter 4 is a development of Schwartz’s kernel theorem in the present context and ends with a kernel representation for our continuous linear operators. Causality appears as a support condition on the kernel. How time-invariance converts a kernel operator into a convolution operator is indicated in Chapter 5. We digress in Chapter 6 to develop those properties of the Laplace transformation that will be needed in our subsequent frequency-domain discussions. Passivity is a very strong assumption; it is from this that we get the richest realizability theory. Chapter 7 imposes a passivity condition that is appropriate for scattering phenomena, whereas a passivity condition that is suitable for an admittance formulism is exploited in Chapter 8. It is assumed that the reader is familiar with the material found in the customary undergraduate courses on advanced calculus, Lebesgue integration, and functions of a complex variable. Furthermore, a variety of standard results concerning topological linear spaces and the Bochner integral will be used. In order to make this book accessible to readers who may be unfamiliar with either of these topics, a survey of them is given in the appendixes. Although no proofs are presented, enough definitions and discussions are given to make what is presented there understandable, it is hoped, to someone with no knowledge of either subject. Almost every result concerning the aforementioned two topics that is used in this book can be found in the appendixes, and a reference to the particular appendix where it occurs is usually given. For the few remaining results of this nature that are employed, we provide references to the literature. The problems usually ask the reader either to supply the proofs of certain assertions that were made but not proved in the text or to extend the theory in various ways. On occasion, we employ a result that was stated only in a previous problem. For this reason, it is advisable for the reader to pay some attention to the problems. All theorems, corollaries, lemmas, examples, and figures are triplenumbered ; the first two numbers coincide with the corresponding section numbers. On the other hand, equations are single-numbered starting with (1) in each section.

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Acknowledgments

This book was conceived while I was a Research Fellow at the Mathematical Institute of the University of Edinburgh during the 1968-1969 academic year. My tenure in that post was supported by a grant to Professor A. Erdelyi from the Science Research Council of Great Britain. It is a pleasure to express my gratitude for that support. The subsequent development of this book was assisted by Grants GP-7577, GP-18060, and GP-27958 from the Applied Mathematics Division of the National Science Foundation under the administration of Dr. B. R. Agins. I also wish to express my gratitude to R. K. Bose, V. Dolezal, and W. Hackenbroch for various suggestions that have been incorporated into the text. Finally and once again, to my wife, thanks.

xv

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Chapter 1

Vector-Valued Functions

1.1. INTRODUCTION

The purpose of this first chapter is to present a number of results concerning the calculus of functions that map n-dimensional real Euclidean 'space into a locally convex space. In addition, the theory of analytic functions from the complex plane into a Banach space is also introduced. The discussion is quite similar to the elementary calculus and the theory of complex-valued analytic functions. Complications arise at various points, however, because of the more involved structure of locally convex and Banach spaces. Nevertheless, there are no surprises. The primary results of Riemann integration, differentiation, and functions of a complex variable carry over to the present context. The reader may, if he wishes, go directly to Chapter 2 after a perusal of the next section and refer back to the present chapter as the occasion demands. As was mentioned in the Preface, certain facts concerning topological linear spaces will be used. A compilation of them can be found in the appendixes of this book. 1

2

1. VECTOR-VALUED FUNCTIONS

1.2. NOTATIONS AND TERMINOLOGY

The present section is devoted to a description of some of the notations and terminology that are used throughout this book; almost all of them adhere to customary usage. The Index of Symbols at the end of the book lists our frequently used symbols and indicates the pages on which they are defined. Let @ be a set. The notation {4 E 0: P ( 4 ) } , or simply {4: P($)} if 0 is understood, denotes the set of all 4 E @ for which the proposition P ( 4 ) concerning 4 is true. {4i}is,is a collection of indexed elements where the index i traverses the set 1. A sequence is denoted by {+k}F= or {&: k = 1, 2, . . .}, a finite collection by {4k}:=1or {41,.. ., 4"}, and a set containing a single element 4 by {4}. We also use the abbreviated notation (4,) when it is clear what type of set we are dealing with. Let R and A be subsets of @. The notation 4, ll/, 0, . . . E R means that all the elements 4, ll/, 0, . . . are members of R ; 4, ll/, 0, . . . 4 R is the corresponding negation. If 0 is a complex linear space (see Appendix A), R + A (or R - A) denotes the set of elements of the form ll/ + 6 (or respectively ll/ - 0) where Ic/ E R and 0 E A . Similarly, if 4 is a fixed member of @ and a is a complex number, R 4 (or a n ) is the set of all elements of the form II/ 4 (or respectively all/), where I) E 0.On the other hand, R\A is the set of all elements in R that are not in A. Thus, @\A is the complement of A in 0. We also have the customary union u, intersection n, and inclusion symbols c or 3 . Furthermore, 0,and R, denote unions and intersections, respectively, over sequences {Rk} of sets in @. A similar notation is used for finite collections and indexed collections of sets, for example, (JiZlR, and n i E I R i . If @ is a topological space (Appendix B), fi is the interior of 0,and I2 the closure of 0.However, for any complex number a, 5 is the complex conjugate of a. Now, let O k 9 where k = I , . . . , n, be sets. The Cartesiunproduct @, x ... x a,, is the set of all ordered n-tuples {$J~,. . . , 4,,}, where 4, E @k and k = 1. . . . , n . A rulefthat assigns one or more elements ll/ in a set Y to each element 4 i n another set @ is called a relation. Thus, f determines a subset of @ x Y called the graph qf.f and consisting of all ordered pairs {b, I )}, where ll/ is an element assigned to 4 by f. We also use the notation f: 0 c, Y as well as f:4 Hll/ to denote the rule, the sets for which the rule is defined, and the typical elements related by$ The domain off is the set of all elements 4 on which f is defined, in this case, @; 4 is called the independent variable. The range of'fis the subset of Y consisting of all elements ll/ that are assigned by f t o members of 0 ;ll/ is called the dependent rariable.

+

+

1.2

NOTATIONS AND TERMINOLOGY

3

A function f is a relation f : @ -+ Y that assigns precisely one element in Y to each member of @; in other words,fis a function if and only if the equations $ = f 4 and 8 = fi imply that $ = 8. Synonymous with the word “function ” are the terms operator, mapping, and transformation. We say that f:4 H $ maps, carries, or transforms 4 into Y and that f is a function on (orfrom) @ into Y or a mapping of @ into Y. We also say thatfis Y-valued; when Y is the real line or the complex plane, the phrase “Y-valued” is replaced by real-valued or complex-valued, respectively. If R is a subset of Q,, the symbol f(R) denotes the set {$ E Y : $ = f i , 4 E R}. The function g that is defined only on R but coincides withfon R is called the restriction of f to R. On the other hand, f is called an extension of g. A function f is said to be one-to-one or injective and is also called an injection if the equations f 4 = $ and fx = $ imply that 4 = x. In this case, : $ -4, which maps the range off into 0; f is we have the function f called the inverse off. A function f : @ -+ Y is said to be onto or surjective and is also called a surjection if the range o f f coincides with Y . Thus, f is a surjection onto Q, whenever f is injective. I f f : @ c, Y is both injective and surjective, it is said to be bijectice or a bijection. We denote the elements in the range o f f by the alternative notations $ = f& = f (4) = {f,4). On occasion, it will be convenient to violate this symbolism by using f ( 4 ) to denote the function f rather than its range value, as was commonly done in classical mathematics. Whenever we do so, it will be clear from the context what is meant. On still other occasions when the symbol for f is rather complicated, we may use the dot notation f (.) = (f, .) in order to indicate where the independent variable should appear. For example, when t E R , we may denote the function t H cos(sin t) by cos(sin .). When -Y and W are topological linear spaces (see Appendix D and especially Sections D8 and D1 l), the symbol [ Y ;W ] denotes the linear space of all continuous linear mappings of Y into W . Unless the opposite is explicitly stated, it is always understood that [ Y ;W ] is equipped with its bounded topology. When it has the pointwise topology, it is denoted by [ Y ;W],. Thus, if A and B are Banach spaces, [ A ; B] possesses the uniform operator topology, whereas [ A ; B]“ has the strong operator topology. R“ and C” denote, respectively, the real and complex n-dimensional Euclidean spaces. Thus, an arbitrary point t E R” (or t E C“) IS an ordered n-tuple t = {tk}iZ1of real (respectively complex) numbers tk whose magnitude is

-’

We set R = R‘ and C = C ’ . R , denotes the positive half-line {t E R : 0 < t < 03). R,“ is the set of all ordered n-tuples each of whose components

1.

4

VECTOR-VALUED FUNCTIONS

is a real number or co; for example, (2, 03, - 1) is such a triplet. As before, we set Re = R t . We do not allow - 00 as a possible component of any t E R,". Thus, if a E R,", - a is an n-tuple whose components are either real numbers or - 00. If x E R e , the symbol [XI will denote the n-tuple in R," each of whose components is x ; however, the n-tuple [O] is denoted simply by 0. The number n of components in [XI will be implied by the context in which the symbol [ x ] is used. An integer k E R" is an n-tuple all of whose components are integers. A compact set in R" is a closed bounded set in R". Given any set R in R", the diameter of R is denoted by diam R and defined by diam R = sup{ I t - X I : t , x

E

R}.

If x = {xk};=1 and t = {tk};!l are members of R" or R,", the notations x It and x t mean that xk Itk and respectively xk < t k for k = 1, . . . , n. If t 2 0 (or if t > 0), t is said to be nonnegatioe (respectively positive). An interval in R" is the Cartesian product of n intervals in R. As special cases,

-=

we have ( x , y ) = { t E R": x < t < y } , ( x , y ] = { t E R": x

k and all t E [ P , Q ] . )Then,fis also continuous and

Problem 1.4-1. Prove the preceding eight assertions. 1.5. REPEATED INTEGRATION AND IMPROPER INTEGRALS

Under the assumptions of Theorem 1.4-1, the integral

can be written as the repeated integral

1.

10

VECTOR-VALUED FUNCTIONS

To see this, we first rewrite S(A n) as follows:

Set lnkl = max 1 SPkSmk

As

1 ~ " ) +O,

( k Pktk,Pr-I).

the innermost summation in (3) tends to

because, with all the components o f t fixed except for t , , t , H f ( f ) is a continuous mapping of [ P , , Q,] into Y . Now, (4)is a continuous function on [PI,Q,] x * . . x [Pn-l,Qn-l] into Y . Indeed, let At denote an increment in t such that t, remains unchanged. Then, by note IIi of the preceding section,

By the uniform continuity of A the right-hand side tends to zero as I At I + 0, which verifies our assertion. As a consequence, the two innermost summations in (3) tend to

J;;,'

(k-I

pw

dt,

as first Inn[+ O and then In"-'l -0.. Continuing in this way, we see that (3) tends to ( 2 ) as all the Ink/ are taken to zero one at a time in the order of decreasing k. To show that ( 1 ) is equal to ( 2 ) , choose y E r and E E R , arbitrarily. We may write

1.5.

REPEATED INTEGRATION AND IMPROPER INTEGRALS

11

Now, by what we have already shown in this section and by Theorem 1.4-1, we can choose a sufficiently fine rectangular partition n of [ P , Q] such that every term on the right-hand side is less than &/(n+ 1). [Indeed, for one such term, say, the pth term, we can choose a n p such that this term is less than &/(n+ 1) for every refinement of n p . Then, a n can be found that is a refinement of every np.] Hence, the left-hand side of (5) is less than &, which proves the equality between (1) and ( 2 ) . The order of integration in ( 2 ) can be changed in any fashion. This follows from the fact that we can change the order of summation in (3) and then take repeated limits as above to get another repeated integral equal to (1). The next theorem summarizes all this. Theorem 1.5-1. Let f be a continuous function that maps the compact interval [ P , Q] c R", where P I Q, into Y . Then,

IPQj(t) dt

=

1"'

dt,

. . IpQ"j(t) dt,.

PI

Moreover, the order of integration in the right-hand side can be changed in any fashion.

The rest of this section is devoted to a discussion of certain improper integrals. Let Q" E R," be such that one or more of its components are 00. We shall say that Q E R" tends to Q" and shall write Q + Q" if each component of Q tends to the corresponding component of Q". Now, let P E R" be such that P 5 Q" and assume that f is a Y-valued continuous function on [ P , Q ] for every Q E R" such that P I Q 5 Qm. Assume in addition that Q

lim j p f ( t ) d t

Q-Q"

(7)

exists in -Y and is independent of the fashion in which Q -+ Q". [Thus, for example, the components of Q could tend to those of Q" simultaneously or one at a time and in any order without changing (7).] The improper integral

is defined to be the limit (7). In quite the same way, we can define the improper integral (9)

12

1.

VECTOR-VALUED FUNCTIONS

where P-" is an n-tuple with - 00 for some or all of its components and real numbers for the remaining components. When all the components of P-" are -a and those of Qm are co, we write Q"

lp--f ( t ) dt

=

I

f ( t ) dt =

R"

I

fdt.

(10)

R"

It is readily seen that all the assertions of notes I-VI of Section 1.4 continue to hold for (8)-(10). Now, however, the right-hand side of the inequality in note I11 may become co.

Problem 1.5-1. Prove the two statements of the preceding paragraph.

1.6. DIFFERENTIATION

Let Atk E R . In the following, At I k will denote a member of R" all of whose components are zero except for the kth, which is instead A t k . Thus, for any t = {tl, . . ., t,} E R", t -k Atlk = { t i , . . . , t k - 1 ,

fk

-k Atk,

tk+,,

. . ., t,}.

Let f be a Y-valued function on some subset of R". (We remind the reader that throughout this chapter, we are assuming that Y is a separated sequentially complete locally convex space with r as a generating family of seminorms for the topology of Y . ) We say thatfis diflerentiable at the point t with respect to fk iffis defined on some neighborhood o f t and if the quantity f ( t -k At I k ) - f(r) Atk

converges to a limit in Y as I Atkl + 0. That limit is called the deriaative of f w i t h respect to t k evaluated at t and is denoted by a k f ( t )as well as by a , , f ( t ) . Let R be a subset of R".f is said to be differentiable on R with respect to t, iffis differentiable at every t E R; in this case, its derivative a k f = a,,fis also a Y-valued function on R. If a k f is in turn differentiable on R with respect to, say, t i , the second derivative aj akf of f is defined as the derivative of akf on R with respect to t i . Continuing in this way, we obtain the higher derivatives off. Integration and differentiation can be related as follows.

{Qk}z=l

Theorem 1.6-1. Let P = {Pk}i=1E R" and Q = E R" be such that Pk = Qk f o r all components except the j t h component, for which we have

1.6. DIFFERENTIATION

13

instead P j < Qj. Let f be a "Y-valued function having a continuous derivative 0jf on [P, Q]. Then, (1) PROOF. The left-hand side of (I) exists because of the continuity of 0J(t) as a function of t l : Now, let "Y' be the dual of"Y and let FE "Y' be arbitrary. It follows from the definition of oJ that F ojf(t) = OJ Ff(t). So, by note II of Section 1.4 and the fundamental theorem of integral calculus,

But, the weak topology of "Y separates "Y (Appendix D7), and thus (I) is obtained. In regard to a change in the order of differentiation in a second derivative, we have the following result. Theorem 1.6-2. Let f be a "Y-z'alued function on a subset of R 2. If oI!, 02f, and 02 0tf exist and are continuous on a neighborhood n ofa fixed point t E R 2, then 0t 02f exists at t and

(2) Note. We could apply an argument such as that of the preceding proof. But this would only establish that 0t 02f(t) exists as a limit in the weak topology of "Y. To conclude that 01 ol!(t) exists in the initial topology of "Y, we proceed as follows. PROOF. Set t = {xo, Yo}. In the following, hER and IE R are so restricted that {x o + h, Yo + I} remains in a compact interval A contained in nand containing {x o, Yo} in its interior A. We want to show that

(3)

exists and has the value 02 oI!(xo, Yo). Now, (3) may be written as lim lim (ljhl)[f(xo h-Ol-O

+ h, Yo + I) -

f(xo

+ h, Yo) -

f(xo, Yo

+ I) + f(xo, Yo)].

14

1. VECTOR-VALUED FUNCTIONS

We may invoke Theorem 1.6-1 to rewrite this expression as

+ z , yo + I ) - d,f(xo + z , y o ) ] d z

lim lim ( l l h l ) f[dlf(xo h-0

1-0

0 h

=

lim lim ( l l h l ) h-0

But, for any y

E

5 SUP

14 2 I:

140

I I

dz 0

d2 d,f(xo

0

+ z , yo + i)d i .

r, we may employ note 111 of Section

Y P z dlf(X0 +

2 9

Yo

+ C) - 4d,f(xo

7

1.4 to obtain

Yo)].

I n view of the continuity of a, a,f at {xo, y o } , the last expression tends to zero as first I and then h tend to zero. This proves that (3) exists as a limit in Y and is equal to d 2 d l f ( x o ,yo). 0

Let k = { k l , . . . , k,} be a nonnegative integer in R". Dk denotes the differential operator Dk 4 8;: . . . a;.. (4) We also use the notation Dkf=,f(k)as well as Dkf(t) = D,"f(t) = f ' k ' ( t ) . Throughout this book, we adhere to the following conventions. Whenever k is used as in (4), the notation I k I will denote the sum k , + . . . + k,, rather than the magnitude of k as a member of R". Moreover, we shall refer to k as the order of Dk. (This is in contrast to the usual practice of referring to I k 1 as the order of Dk.) Whenever we say that the derivatives of a functionfup to the order k are continuous on a set R, we mean that the functionsf, akf, aj akJ . . . , DkA obtained by successively applying to f the various operators ak to obtain 0% exist and are continuous on R. If R is an open set, it follows from Theorem 1.6-2 that any change in the order of differentiation in Dkfwill not alter Dkfon R. The same is true when R is a compact interval, because the values of Dhf on the boundary on R are continuous extensions of its values on the interior of R. We shall say that a Y-valued function defined on a subset of R" is smooth if the function possesses continuous derivatives of all orders at all points of its domain.

1.6.

15

DIFFERENTIATION

Theorem 1.6-3. Let W be a separated locally conuex space. Also, let F and

f be, respectioely, [Y ; W]-valuedand Y-valuedfunctions having the continuous derivatives ak F and akfon a neighborhood of the point t E R". Then, at t, ak(Ff) =

flf +

(5)

akJ

[Here, Ff denotes the function X H F(x)f (x).] Moreover, f F and f have continuous derivatives up to order k on a neighborhood of apoint t , then, at t, we have

where

Note. The last quantity is the n-dimensional binomial coeficient. For n = 1, we have

where m and q are nonnegative integers in R with q < m. Also, ( 6 ) is called Leibniz's rulefor the differentiation of aproduct. PROOF. In this proof, Atk tends t o zero inside an interval so small that Atk HF(t + At I k ) and Atk H f ( t + At I k ) are continuous and therefore bounded functions on that interval (Theorem 1.3-1). The left-hand side of (5) is the limit of W (if it exists) of

Since [ Y ;W ]possesses the bounded topology, the assumption that F has a derivative at t means that

tends in [ Y ;W ]to G(0) the assumption on f ,

dk

g(Atk)

F(t) uniformly on the bounded sets in Y .By

f( t + At 1 k ) ,

Ark # 0

tends in Y to g(O) 4 f(r). Moreover, {g(Atk)},where Atk traverses its permissible values, is a bounded set in Y . Now, G(Afk).dArk)

- G(o).do) =z [G(Atk)- G ( o ) b ( A t k )

+ G(0)[dArk)- do)].

1.

16

VECTOR-VALUED FUNCTIONS

Thus, both terms on the right-hand side tend to 0 in W , which shows that the first term in (7) tends to (a, F)(t)f(t). The second term in (7) clearly tends to F ( t ) f(t) in W .Thus, ( 5 ) is established. Equation (6) is established by repeated application of (5). 0

a,

As a last consideration, we take up differentiation under the integral sign.

Theorem 1.6-4. Let [ P , Q] be a compact interval in R" and E an open set in R ; t and x will denote variables in [P, Q] and E,respectively. Assume that f is a continuous Y-raluedfuncrion on [P, Q] x E and that a, fexists and is continuous on [P, Q] x E.Set .Q

g ( x ) = J f( t , X) dt,

-

x E .;

(8)

P

Then, a,g exists at each point of E and

PROOF.Fix X E E and restrict A X E R so that x consider

+ A X E S . For

Ax # 0,

By Theorem 1.3-1, d,fis uniformly continuous on [P, Q] x E.Therefore, given any 8 > 0 and any y E r, there exists an q E R + such that

for every t E [P, Q] and all 5 such that 15 - X I < 4. Appealing to note 111 of Section 1.4 and using (lo), we may write

I dt 0

y[hAx(x)l

5

P

1

IlI+AxY[a,f(t, r)

- axf(t.

dr

0 } contained in R. Then, there exists a j n i t e number M = M ( f , E) depending on f and Z such that, for euery (, + LY, and ( + P in the interior k of Z, we have 1 Q(i, a, PJI 5 M , where

PROOF.Let S denote the boundary of 2 . We can choose a circle P in R which encircles E such that

{ I T - (1 : 7 E P , [ E S E I } > 0. Then, upon fixing (, ( + a, and 4' + P as points in 2, we may use Cauchy's d4

LJ

integral formula to write

Theorem 1.7-1. An A-i.alued function on an open set R c C is analytic on R ifand only i f i t is weakly analytic on R. Similarly, an [ A ;B]-ralued,functionis analytic on R ifand only f i t is strongly analytic on R, and this is the case ifand only ifit is weakly analytic on R.

PROOF.We prove only the second sentence, the proof of the first one being quite similar. In view of our previous remarks, we need merely show that the

1.7.

BANACH-SPACE-VALUED ANALYTIC FUNCTIONS

19

weak analyticity of the [ A ; B]-valued function F on R implies its analyticity on Q. Let a E A and b' E B', where B' is the dual of B. Set

Choose Z as in Lemma 1.7-1 and let i,5 + a, and 5 + P be points in 8. By Lemma 1.7-1, there exists a constant M depending on a, b', F, and E but independent of (', a, and P such that

I b"C, By Appendix D12,

a, P)a

llR(c, a,

I

M.

P>aIIB

where M , depends on a, F, and E but is independent of 5, a, and P. An application of the principle of uniform boundedness (Appendix D12) now shows that

llR(5?

P>Il[A, B ]

M2

I

where M , depends o n F a n d E but is also independent of 5,a, and P. Now, let E E R , be arbitrarily chosen. For all permissible a and B such that I a I and I /3 I are both less than & / 2 M , ,we have that I a - P I < 4 M 2 , so that

By virtue of the completeness of [ A ; B], this implies that, as a + O ,

[m+ 4 - F(0ll. tends to a limit in the uniform operator topology of [A; B]. Hence, F has a derivative at C. Since ican be selected as any point of R by appropriately choosing Z, F is analytic on R. 0 As a consequence of Theorem 1.7-1, many of the results for complexvalued analytic functions can be carried directly over to A-valued or [ A ; B]valued analytic functions. For example, let F be an [A; B]-valued analytic function on an open set R and let a E A and b' E B'. Then, b'F(.)a is a complexvalued analytic function on R, so that Dkb'F([)a exists for every k and every ( E R. This implies that D k Fis weakly analytic and therefore analytic on R for every k, which shows that F i s smooth on R. We finally observe that an adaptation of the proof of Theorem 1.6-3 establishes Liebniz's rule [see (6) of Section 1.61 for the differentiation of Ff, where F is an [ A ;B]-valued analytic function and f is an A-valued analytic function.

1.

20

VECTOR-VALUED FUNCTIONS

1.8. CONTOUR INTEGRATION

Theorem 1.7-1 also allows us to extend contour integration to Banachspace-valued functions. We do so in this section for the integral of an Avalued function f o n a contour P in C. Since [A; B ] is also a Banach space, our results immediately extend to an [ A ; B]-valued function F on P. The integral j p d[ of a continuous A-valued function f on a contour P in C is defined and shown to exist exactly as in the case of a complex-valued function (Copson, 1962, pp. 52-59). We also have, as in the scalar case, the estimate

f(c)

As an immediate consequence of the definition of J p f ( ( )dc as a limit of certain Riemann sums, we can state that, for any a E A, a’ E A’, and b’ E B’,

and

Contour integration is a linear process as in note VI of Section 1.4. Moreover,

where -P is the contour obtained by reversing the orientation of P.

Theorem 1.8-1 (Cauchy’s theorem). Let R be a simply connected open set in C, and let P be a closed contour in R. I f f is analytic on R, then

PROOF.When P is a closed contour in $2,the right-hand side of (2) is zero according to Cauchy’s theorem for complex-valued functions. Equation (4) now follows from the fact that the weak topology of A separates A (Appendix D7). 0 Arguments like that of the preceding proof can be used to extend other standard results from the theory of complex-valued analytic functions.

I .8.

CONTOUR INTEGRATION

21

Theorem 1.8-2 (Cauchy's integral formula). Let R, P, and f be as in Theorem 1.8-1 and let ( E C be apoint inside P. Then,for each nonnegative integer k E R ,

Theorem 1.8-3. Let {f,},"= be a sequence of A-ralued analyticfunctions on an open set R c C such that f, + f i n A uniformly on each compact subset of R. Then, f is also an A-valued analytic function on R and for each nonnegative integer k fik) +f ( k ) in A uniformly on each compact subset of R.

[For this theorem, an argument like the proof of Theorem 1.8-1 would only prove that fik'(() + f (k)(lJ in the weak topology of A . However, an estimation of f,"')(()- f ( k ) ( ( ) using (5) leads to our stronger conclusion.]

{c,

Theorem 1.8-4. Let ?}I+ f([,T) be a continuous A-valued function on R x P, where 4' E R, r E P, R is an open set in C , and P is a contour in C. Assume thatf r ) is an analyticfunction on SZ for each r E P. Set (a,

G(C)

1f ( T , 4 dr. P

Then, G is an A-valued analytic function on R, and Gtk)([)= P

k = I , 2, . . . .

D,"f([, r) dr,

In the next theorem, P is an oriented path in C extending to infinity such that any finite portion P,,obtained by tracing P from one of its points to another is a contour. Let {P,,},"=, be a collection of such finite portions of P with the properties that P,,c P,,,l and Pn= P. We define

u.

I p f ( r ) dr 4 lim n-cc

P,

f(r) dr

if the limit exists. Upon combining Theorems 1.8-3 and 1.8-4, we obtain the following.

Theorem 1.8-5. Assume that f satisfies the hypothesis of Theorem 1.8-4 on R x Pnfor each n. Suppose that, as n + co, J P n f ((, r) dr conoerges uniformly with respect to all 5 in each compact subset of R. Define G by (6). Then, the conclusion of Theorem 1.8-4 holds once again.

Problem 1.8-1. Prove Theorems 1.8-2-1.8-4.

22

1. VECTOR-VALUED FUNCTIONS

Problem 1.8-2. Let P be a contour starting at w E C and ending at z E C. Assume that the [ A ; B]-valued function F and the A-valued function f are analytic on P. Show that

and

Chapter 2

Integration with Vector-Valued Functions and Operator-Valued Measures

2.1. INTRODUCTION

An essential tool needed in our subsequent development of an admittance formulism for time-invariant passive systems is a certain theory of integration due to Hackenbroch (1968), wherein functions taking their values in a given Banach space A are integrated with respect to measures that take their values in a space of operators mapping A into another Banach space B. Chapter 2 is devoted to a presentation of this theory. 2.2. OPERATOR-VALUED MEASURES

Throughout this chapter, T is an arbitrary nonvoid set, 6 is a a-algebra of subsets of T, n = {Ek};=is an arbitrary partition of T, and 1is the collection of all partitions n. (See Appendix G , Sections, GI, G2, and G7.) As 23

24

2.

INTEGRATION WITH VECTOR MEASURES

always, A and B are complex Banach spaces, and H is a complex Hilbert space with the inner product -). The concept of the total variation l p l ( T ) = Varp of p on T, where p is a complex measure, can be extended to any mapping P of 6 into [A; B]. ( a ,

Total variation of P on T !& Var P

Similarly, the semivariation is defined as follows: Semivariation of P on T

War P

These concepts are commonly used in the theory of operator-valued measures (Dinculeanu, 1967). On the other hand, Hackenbroch bases his theory of integration on still another concept of variation, namely, the following: Scalar semivariation of P on T

This definition of SSVar P continues to have a sense when P is replaced by a mapping Non 6 that takes its values in an arbitrary Banach space A, and not merely in the space [A ; B] of operators. The integration of certain complex-valued functions with respect to an A-valued function N on 6 or an [A; B]-valued function P on 6 can be defined as an extension of the integration of the so-called simple functions so long as N and P are additive and have finite scalar semivariations. The process is similar to the usual construction of the Riemann integral of a continuous function. Let us be explicit. As is ;Ind;\tik..in Append;\x G8, a simpre functian f fram T into A is any mapping of the formf= x k a k ,yEr, where a k E A, {Ek)E 9, and zEkdenotes the characteristic function of Ek. We let %,(A) a B,(T, 6 ;A) denote the linear space of all simple functions from T into A. Moreover, G(A) is taken to be the Banach space of all bounded A-valued functionsffrom T into A with the norm I1 * IIG(A) where IlfllG(A)

A I E T Ilf(t)llA*

2.2.

25

OPERATOR-VALUED MEASURES

Hence, Bo(A) c G ( A ) . Finally, % ( A ) will denote the closure of g0(A)in G ( A ) . When A = C , we simplify our notation by setting Yo(C) = 9,,B(C) = 9,and G ( C ) = G. % ( A )is in general a proper subspace of @ A ) . A mapping N of (5 into A is said to be additive if, for any pair of disjoint sets E, F E (5, we have N ( E u F ) = N ( E ) + N ( F ) . In this case, we can define the integral of any complex-valued simple function f = ark xEk, where ak E C , with respect to N by means of the expression

IT

f(?)

dNt

1

N(Ek)ak

a

k

The subscript T on the integral sign is at times dropped when the set T on which the integration is taking place is evident. The subscript t on N signifies that the points in T a r e denoted by t . Here are some immediate results of our definitions. The function

f(0

f+N,

(1)

is a linear mapping of gointo A , and, for all f E Yo,

I sup If(t)I SSVar N . A

t s T

Now, assume that SSVar N < CQ. In accordance with Appendix D, Sections D2 and D5,this implies that the mapping ( I ) has a unique extension that is a continuous linear mapping of B into A . We use the same symbolism for the extended mapping. The inequality (2) continues to hold for all f E 9.Assume still further that N is the mapping P of C into [ A ; B]. For any U E A and b‘ E B’, where B‘ is the dual of B, and for all f E 9,we have

(3) and

[

b‘ jdP‘

4

u = Sd(b’P, 0)

Here, P( .)a is a B-valued additive function on

(5,

f(0.

(4)

and

SSVar P( .)a 5 (lull SSVar P. Also, b’P(.)ais a complex-valued additive function on

(5,

and

SSVar b‘P( .)a 5 11b’II llall SSVar P.

In regard to the functions on (5, our attention will be primarily confined to those functions that take their values in an operator space [ A ; B] and possess the following a-additivity property.

2.

26

INTEGRATION WITH VECTOR MEASURES

Definition 2.2-1. A function P on (5 into [ A ; B ] is said to be a-additive in the strong operator topology of [ A ; B] if it is additive and, given any sequence {Ek}2=lc (5 such that Ek n E j is the void set whenever k Zj,we have that

.( kgl

m

c

= k=

Ek)a

1 P(Ek)a

for every a E A . When this is the case, P is called an operator-valued measure or more explicitly an [ A ; B]-valued measure. If, in addition, A = B = H and the range of P is contained in the space [ H ; H I , of positive operators (see Appendix D I5), P is called a PO measure or apositive-operator-valuedmeasure. Note that P is an [ A ; B]-valued measure if and only if P is additive as a function on (5 into [ A ; B ] and, for every increasing sequence {Ek}F=l in 0 (i.e., Ek c Ek+ for all k ) and for all a E A , P

u Ek a

=

( I

lim P(Ek)a. k-tm

We now investigate in some detail the properties of operator-valued measures and PO measures. Let {Ek}p=,be an increasing sequence in 0 with UkEk = T . Then, for any PO measure P , iiP(Ek)Il

=

sup

(P(Ek)a,0 )

5 sup ( P ( T ) a ,a ) = IIp(T>II* llall = 1

1 1 4= 1

(5)

Moreover, since (P(Ek)a,a) + ( P (T ) a ,a ) for each a E H , it follows that IIp(Ek)II

+

IIp(T)II,

k

+

00.

(6)

Lemma 2.2-1. If M E [ H ; H I + and Mk E [ H ; H I , , where k = 1, 2, . . ., and $ as k + 00, ( M k a ,a) increases monotonically to the limit ( M a , a ) for every a E H , then Mk + M in the strong operator topology of [ H ; HI.

PROOF.If Q E [ H ; H I , and a E H , we may employ Schwarz’s inequality (Appendix A6) to write

IIQa1I4 =

\(eatQaII

I

(pa,a>(QQa,Qa).

But ( Q Q a , Q45 IlQII IIQaII2, and so

IIQal12 5 (Qa,a)IlQll. (7) We may replace Q by M - M kbecause M - h f k E [ H ; H I , according to the hypothesis. Moreover, ( ( M - Mk)a, a) -+ 0 as k + 00. On the other hand, upon invoking the polarization equation (Appendix A7) and making two applications of the principle of uniform boundedness (Appendix D12), we see that IIM - Mkll is bounded for all k . This proves the lemma. 0

2.2.

27

OPERATOR-VALUED MEASURES

Theorem 2.2-1. Let P be an additive mapping of follo wing three assertions are equivalent.

(5

into [ H ; HI,.

The

(i) P is a P O measure on (5. (ii) For all a, b E H , (P( .)a, b) is a complex measure on (5. (iii) For every a E H , ( P ( ' ) a ,a) is a positivejnite measure on (5. Note. As is indicated in Appendix G4, complex and positive measures are by definition a-additive.

PROOF.That (i) implies (ii) and (ii) implies (iii) is clear. To show that (iii) implies (i), we need merely prove that P is a-additive in the strong operator topology of [ H ; H I . Let {&}?=1 be an increasing sequence in (5, and set E = U k E k . Hence, E E (5. Consequently, P(Ek) E [ H ; H I , and P ( E ) E [ H ; H I , . Also, (P(E,)a, a ) increases monotonically to (P(E)a,a) because of the positivity and a-additivity of the measure (P( .)a,a). Hence, by Lemma 2.2-1, P ( E k ) P ( E ) in the strong operator topology of [ H ;HI. 0 --f

Theorem 2.2-2. If P : (5:

-

[ A ; B] is an operator-valued measure, then

SSVar P I 4 sup IIP(E)IIIA;Bl< 00.

(8)

EEO

PROOF.Let F be an arbitrary member of the dual of [ A ; B]. Then, FP 4 FP( .) is a complex measure on (5. Let FP = L + iM,where L is the real part of FP and M the imaginary part. For any 7c = {Ek};= E 1,we may write

1I

L(Ek)

k

I

=

1

+

L(Ek)

-

=L(u+Ek)

1'

-

u' (1- u-)

1

- L(Ek)

L(u

-Ek)?

where and and are taken over those k for which L(Ek)2 0 [respectively L(E,) < 01. Therefore, R E

9

1I

L(Ek)

1

= RE

2

u

[L(

+Ek)

- L(

u-

Ek)]

I 2 s u p IL(E)I. EEO

A similar inequality holds for M , and therefore SSVar F P s Var F P Isup

1 IL(Ek)I + sup 1 I M(&)I

R E 9

R E

9

S 2 s u p IL(E)I + 2 s u p IM(E)I EEO

EEO

I4 sup 1 F P ( E ) I EEO

I4

IlFll SUP IIP(E)II. EEO

2.

28

INTEGRATION WITH VECTOR MEASURES

sup SSVar FP I 4 sup IIP(E)II.

IlFll = 1

EEC

But SUD

SSVar FP

According to Appendix D 13, IlFll = 1

I 1

I=1 1

P(Ek)c(k

P(Ek)ak

1

[ A : B]r

and therefore the right-hand side of (9) is equal to SSVar P.This establishes the first inequality of (8). To obtain the second inequality, let b ' E B' and U E A. According to Appendix G7, the complex measure b'Pa = b'P( * ) asatisfies sup I b'P(E)al I Var b'Pa < a. EE&

Two applications of the principle of uniform boundedness (Appendix D12) complete the proof. 0 Theorem 2.2-3. U P :0: -+ [ H ; HI, is a PO measure, then SSVar P

=

IIP(T)IIIH:Hl.

(10)

PROOF. Choose arbitrarily a partition 7t = {Ek};= E 1,two members a and b in H , and the complex numbers c(k such that I c(kl I1, where k = 1, . .., r. We may write

1(

k= 1

cckP(Ek)a,

b,

1

1

I(P(Ek)a,

b)l.

(11)

By the Schwarz inequality (Appendix A6), the right-hand side is bounded by (P(Ek)a,a)'"(P(Ek)b, b)1/2. But, by the Schwarz inequality for sums, the last expression is bounded by

[c(P(E&?,

(P(&)b, b)]1'2= (P(T)U,a)'I2(P(T)b, b)'"

I llP(T)Il llall llbll. Take a supremum of the left-hand side of (1 1) over all 71 E 9, all clk such that (akl 5 1, and all a, b E H such that ((a(( = I(b(l= 1. This yields SSVar P IIIP(T))I.On the other hand, upon choosing r = I , a1 = 1, and El = Tin the definition of SSVar P , we see that SSVar P 2 IIP(T)II. 0

2.3.

29

6-FINITE OPERATOR-VALUED MEASURES

Problem 2.2-1. A fact we will use later on is the following. Let T = R" and let 0. be the collection of Bore1 subsets of T. For q = {qk};,, E R" and t = { t k } ; = E R", we set qt = x k qk t k . Then, for fixed q, the function tweirlr is a member of 8. Show this. Problem 2.2-2. Let P be a mapping of 0. into [ A ; B]. Show that IIP(T)II I SSVar P I SVar P 5 Var P I co. Problem 2.2-3. With P being an additive mapping of (I into [ A ; B], assume that SVar P < co. Show that an integral J dP,f(t) can be defined for any f E % ( A ) as an extension of the integrals (defined in a natural way) of the functions in %,,(A). Also, show that

and, for any b' E B', b'

1

dP,f(t) = d(b'P,)f(t),

where b'P(.) is an A'-valued additive function on 0. and SVar b'P( .) I Ilb'IIB.SVar P.

[Thus, upon assuming that SVar P < co,we obtain hereby a means of integrating A-valued functions with respect to [ A ; B]-valued measures. However, Hackenbroch's theory encompasses greater generality so far as the measures P are concerned; it takes into account certain measures P for which SSVar P < 03 but SVar P = co. For an example of such a measure, see Hackenbroch (1968, pp. 332-333.)1

2.3. U-FINITE OPERATOR-VALUED MEASURES

Let { Tk}km, be an increasing sequence of sets with TkE (I and Set 0.k

'

{ E E 0.: E

C

Tk},

k

and

It follows that (I, is a a-algebra of subsets in T k .

= 1,

2,

a .

U k

Tk = T.

30

2.

INTEGRATION WITH VECTOR MEASURES

Definition 2.3-1. Let P be a mapping of 0, into [ H : H I , such that the restriction of P to each 0, is a PO measure on 0,. Then, P is called a a-finite PO measure on 0,. As an example of a a-finite PO measure, we may take T = R", Tk= { t E R": I t 1 5 k } , 6,equal to the collection of Borel subsets of Tk,and P equal to the Lebesgue measure on 0 , . P is finite on each bounded Borel subset of R" but is not defined on all the Borel subsets of R".

Throughout this section, P will always be a a-finite PO measure, and h will be a member of 9. It follows that lh(.)l E 93 also. By virtue of Theorem 2.2-3,

exists as a member of [ H ; HI for each k in accordance with the preceding section. Definition 2.3-2. h is said to be integrable with respect to P if, as k

+ CO,

( 1 ) converges in the strong operator topology of [ H ;H I ; the limit is denoted

by

(2) This defines a linear mapping h H J d P , h(t) of Y into [ H ; H I . Upon invoking (4) of the preceding section, we get, for all a, b E H ,

(1

d P , h(t)a, b )

= (lim k - a , JT,

= lim

d(P,a, b)h(t)

JT

k-rm

dP, h(t)u, b )

L

L? j d ( P , a , b)h(t).

(3)

Here, ( P ( .)a, b ) is an additive complex-valued function on 0 , whose restriction to each 0, is a complex measure. Similarly,

Theorem 2.3-1. Let g be a nonnegative.function in 9. g is integrable with respect to P if and only if the sequence { Fk}p=I , where Fk 4

I

dP,g(t) E W; HI,

Tk

is bounded in the strong operator topology.

2.3.

Q-FINITE OPERATOR-VALUED MEASURES

31

Note. By the principle on uniform boundedness, the boundedness of {Fk} in the strong operator topology is equivalent to the boundedness of {Fk}in the uniform operator topology.

PROOF.Since convergent sequences are bounded, the " only if" part of the theorem is clear. Conversely, assume that {Fk} is bounded in the strong operator topology. Clearly, (Fka,a ) 2 0 for every a E H , and therefore FkE [ H ; H I , . Similarly, F,,, - Fk E [ H ; H I , whenever m k. Also, by the principle of uniform boundedness again, llFkll s M , where M is a constant not depending on k . Moreover, upon appealing to (7) of the preceding section, we may write, for each a E H ,

=-

II(Fm

- Fk)a112

5 2 M ( ( F m - Fkb, a)*

The right-hand side tends to zero as k -, co because ( F k a ,a) is an increasing bounded sequence. Therefore, {Fk)is a Cauchy sequence in the strong operator topology and hence must converge because of the sequential completeness of [ H , H ] under that topology (Appendix D l l ) . 0 Theorem 2.3-2. If g E $4 is a nonnegative function and is integrable with respect to P and if h E $4 is such that I h( t ) I _< M g ( t )f o r all t E T, where M is a constant, then h is integrable with respect to P. Moreover,

PROOF.We may decompose h into h = h1

- h, + ih, - ih4,

(6)

where, for each j = 1, 2, 3, 4, hi is a nonnegative function in 3 ' and h,(t) M g ( t ) . Hence, for each k and j ,

(See Appendix 015). This inequality coupled with Theorem 2.3-1 shows that the hi,and therefore h as well, are integrable with respect to P. The estimate (5) follows from (7) and Appendix D l l . 0 The last theorem implies that h is integrable with respect to P whenever lh(*)I is.

32

2. INTEGRA nON WITH VECTOR MEASURES

Theorem 2.3-3. Let g E r§ be a nonnegative function that is integrable with respect to P. Define a mapping Q on (t as follows:

~

Q(E)

JdPtg(t) ~ f dPtxit)g(t), E

EE

(8)

(t,

T

where XE is the characteristic function for E. Then, Q is a PO measure on (t. PROOF. Observe that XEg satisfies the hypothesis of Theorem 2.3-2, so that the right-hand side of (8) exists as a member of [H; H). In fact, Q(E) E [H; H)+ , since

(Q(E)a, a) =

t «r,«.

a) XE(t)g(t) ;;::: 0

for every a E H. It is straightforward to show that Q is additive on (t. Next, we shall show that, for any a E H, (Q(. )a, a) is a positive finite measure on (t. To do this, we have only to show that (Q( . )a, a) is e-additive (Appendix G4). For any J E (t, XJ g is integrable with respect to P according to Theorem 2.3-2. Therefore, (Q(J)a, a) = lim .

m-+oo

f .ur,«, a)XJ(t)g(t),

Now, let {Ed be an increasing sequence in also. Note that

f

Tm

J

E (t.

(9)

Tm

(t

with Uk E k = E. Hence, E

E (t

.ur,« a) XE.(t)g(t)

is an increasing function of both m and k. By virtue of (9), given any s > 0, we can choose an m such that

o~

f d(Pta, a)xit)g(t) - f T

d(Pta, a)XE(t)g(t)
=j(gk

9

ak)*

Therefore,

for every j E g(9,A ) . In particular, choose j such that j ( g , a) = g(to)q(a) where to is an arbitrarily fixed point of T and q E A‘, A’ being the dual of A . The finiteness of (2) implies that gk(t0)ak and hk(to)bkboth converge in A . Consequently, (5) implies that

1

1

4 ( 1 gktt0)ak) = q ( 1 h k ( r O ) b k )

for all q E A’. But the weak topology of A separates A (Appendix D7), and therefore

1gk(tO)ak = 1hk(tO)bk for every to E T. So, truly, Zw does not depend on the representation used for w. Theorem 2.4-1. The mapping Z: Y @ A - t Y ( A ) dejined by (4) is linear, continuous, and injective. PROOF.The linearity of Zfollows easily from the fact that any representation E 9 6 A and its image gkak in S ( A ) converge absolutely under the norms p and 11 * IlCcA, respectively and therefore can be rearranged (Appendix C 12). To show the continuity of Z,let w E Y 6 A and let gk 0 ak be any one of its representations. S e t f = Zw = gkak. Then,

1gk 0 ak of w

1

1

1

IlfIIG(A)

1 llgkllGllakllA.

36

2.

INTEGRATION WITH VECTOR MEASURES

Since this holds for every representation of w, we have that IlfllccA, I p(w), which implies the continuity of I. We now set about proving the injectivity of I. We have to show that, if (gk}r=l C 9 and {Uk}r=l C A are such that ) ) g k ) )I l a k l I < 00 and f 4 gkak is the zero member of 3 ( A ) , then w C gk 0 ak is the zero member of 9 @ A . That w = 0 E Y 6 A means that, given any E > 0, there exists an equivalent representation w = C hk 0 6, , where hk E 3, 6, E A , and llhkII llbkll < E . We shall prove that such an equivalent representation exists. Let {Ei};= be an arbitrary partition of T, let ti be a point in Ei,and let xi be the characteristic function for E i . Therefore, x i ( t ) = 1 for every t E T. Sincef(ti) = 1 kgk(ti)ak = 0, we may write

1

x

xi

r

1

xi

i=l

m

0

k=l

gk(ti)ak = O.

Now, B O A is equipped with the n-topology, and therefore the mapping k gk(ti)ak converges in A ,

{ g , a } H g 0 a is continuous (Appendix FS). Therefore, since x

Upon interchanging the summations on i and k and subtracting the resulting gk 0 ak E 3 6 A , we obtain representation of O E B @ A from w =

Now,

and (2) is finite. Therefore, given any E > 0, there exists an integer s not depending on the choices of the partition { E i } or the points t i E Ei such that

We now state a lemma but postpone its proof.

Lemma 2.4-1. For any given g E B and { E i } ; = ,of Tsuch that

whatever be the choices of t i E E i .

E

> 0, there exists a partition

2.4.

37

TENSOR PRODUCTS AND VECTOR-VALUED FUNCTIONS

According to this lemma, we can choose a partition {Ei}r= such that, for each k = 1, .. . , s,

Hence, for that partition, (7) and (9) imply that ( 6 ) is the equivalent representation for w that we have been seeking. This completes the proof of Theorem 2.4-1. 0

PROOFOF LEMMA2.4-1. By the definition of 9,there exists a partition {Ei}r= of T and a corresponding simple function i C i X i ,

C i E C , i = l ) ...)r,

i= 1

such that, for every i and every t E E , ,

IAt) - ci I < tea Therefore.

for every t E T. The last two inequalities may be combined to yield (8). 0 We now define some additional notations and terminology that we will be using. The image of 9 6A under the mapping I is denoted by 9 6 A . Thus, 9 6 A is the set of all f E % ( A ) having representations of the form(4)suchthat (2) is finite. Furthermore, the image of 9 0 A under I is denoted by 9 0A . Hence, 9 0 A is the set of all . f ~ ’ 3 ( A having ) representations of the form f = gkak. Thus, for any f E ’3 0 A , f ( T ) is contained in a finite-dimensional subspace of A . For this reason, the members of 9 0A will be called fnitedimensionally-ranging functions. Recall that g0(A) is the space of all A-valued simple functions on T. Clearly,

I;=

9JA) c G OA c 9 6 A .

Corollary 2.4-la. Defne the functional

11. Ill on each f E 9 6 A by

(10)

38

2.

INTEGRATION WITH VECTOR MEASURES

Then, I( * 11 is a norm on Y 6 A. Moreover, Y 6 A equipped with the topology generated by 11.11 is a Banach space, and the mapping I dej7ned by (4) is an isomorphism f r o m Y Q A onto Y 6A . PROOF. By Theorem 2.4-1 and the definition of Y 6 A, I is linear, continuous, and bijective. Therefore, its inverse I - I is also linear and continuous (Appendix D14). Also, for w E Y 6A and,f= Iw and for the norm p defined by ( l ) , we have that 11 f 11 = p ( w ) . Thus, I is an isomorphism, I/ is a norm on Y 0A, and Y 6 A must be complete since Y 6A is complete. 0

Henceforth, it will be understood that Y 6 A possesses the topology generated by (1 . / I l . Y o A is the completion of 9 0A with respect to the norm II . I l l .

2.5. INTEGRATION OF VECTOR-VALUED FUNCTIONS

We are now ready to define the integral of an A-valued function with respect to an [A; B]-valued measure. In this section, P will be an [A; B ] valued measure on K. Consequently, SSVar P < co according to Theorem 2.2-2, and dP, g ( t ) exists as a member of [A; B ] for each g E Y.

Definition 2.5-1. Let f E Y 0 A and choose a n y representation f = E Y and ak E A. We define the integral o f f with respect to p by

cL= gk ak, where gk

To justify this definition, we have to show that the right-hand side of (1) does not depend on the choice of the representation forf. Letf= Cf=lh i b i , where h i E Y and b i E A, be another representation. Now, we can find I linearly independent elements e l , . . , , e , E A such that, for each k and i, I

I

ak =

akjej,

bi =

1P i j e j , k= 1

j= 1

where a k j ,/Iii E C . (See Appendix A3.) Upon substituting these sums into the two representations off and invoking the linear independence of the e j , we obtain Ykakj k= I

=

hipij.

i= 1

2.5

INTEGRATION OF VECTOR-VALU ED FUNCTIONS

39

Hence,

This is what we wished to show. Clearly, then, f - j dP,f ( 1 ) is a linear mapping of Y 0 A into B. Still more is true. It is a continuous mapping because of the inequality

which is established as follows. We know that, for any g

E

9,

Consequently, from (I), we have

5

2 llgkll

Ilakll SSVar p *

Since this holds for every representation off, (2) follows. We can now conclude that the mapping f w J d P ,f(r) possesses a unique extension that is a continuous linear mapping of Y 6A into B. This is how we define the integral dP,f ( t ) on any f E Y 0 A . But, since every suchfis the limit under the norm 11. \I1 of a series g k a , , an equivalent definition is the following. Definition 2.5-2. Let f E Y 6A and choose any representation f

=

cp= gka,, where gk E 9 and ak E A . The integral o f f with respect to P is defined by

It follows that this definition is independent of the choice of the representation for f , that the inequality (2) continues to hold for all f E Y 6 A , and that f ~ d Pj , f ( t ) is a continuous linear mapping of 9 6A into B.

40

2.

INTEGRATION WITH VECTOR MEASURES

In the next two theorems, T and X are two nonvoid sets, 6 and a’ are a-algebras of subsets of T and X, respectively, and p is either a complexvalued measure or a positive measure on C’(Appendix G4). Also, a x 6’ denotes the product a-algebra in T x T (Appendix G3). L l ( p ;A) = L , ( X , a’; p ; A ) is the linear space of Bochner-integrable A-valued functions on Xwith respect to the measure p ; as is explained in Appendix (313, L l ( p ;A) is really a space of equivalence classes of functions, but we speak of i t s members as being individual functions. Finally, 9 ( T x X , a x a’; C ) is the Banach space of all complex-valued bounded functions g on T x X that are the limits of sequences of simple functions under the norm [I * IIc, where Ilgllc 4 sup{ 1g(t,

: t E T , x E XI.

We now state a representation theorem for any f

E

(4)

9 6 A.

Theorem 2.5-1. Let F E L , ( X , a‘; p ; A ) and I E 9 ( T x X , 6 x f(t)

J

X

a’;C ) . Then,

dP,4t, x)F(x)

(5)

exists as a Bochner integral for each t E T and defines a function f where 9 = 9 ( T , (5; C ) . Moreover,

Ilf

111 5

E9

IIFIIL,II 11Ic9

6A, (6)

where

and I p I denotes the total-variation measure of p (Appendix G7). Conversely, every f E 9 6 A has a representation of the form (5).

1“

PROOF. We prove the last statement first. Iff E 9 6A, then f = k = l gkak? where gk E 9, ak E A , and llgkll llakll < 00. Let X = {k}F= let a‘ be the set of all subsets of X , and let p ( E ) be the cardinality of E c a’. [This p is called the counting measure; see Rudin (1966, p. 17).] We can choose an I such that I(t, k) = gk(t) and an F such that F(k) = a k . This immediately yields the representation (5) forf. Conversely, the existence of (5) as a Bochner integral is asserted by Appendix G14. To prove the rest of Theorem 2.5-1, let L , O ( p ; A ) 4 L l o ( X , a’; p ; A ) be the space of all simple functions in L l ( p ;A). Thus, L l o ( p ;A ) is the space of simple functions F = EL=,akXEksuch that, if ak # {0}, then p(Ek) is finite. We define a mapping

1

J : G(T x X , 0. x

,,

a’; C ) x L l 0 ( p ;A ) c-t 9 ( T , a; C ) 0A

2.5 by

'1

rJ(', where again F

=

41

INTEGRATION OF VECTOR-VALUED FUNCTIONS

x)F(x) =

d/Lx

X

k=l

j

dflx / ( f ,

Ek

x)ak

9

I;=akzEk.[That dpx K t , x ) E Y(T, 6 ;C ) 'E,

can be seen by taking a sequence of simple functions on T x X that converges to I and then using the estimate in Appendix G13.1 We next observe that 1141,

F)IIG(A)

'

SUP IET

II[J(k

F,l(t>ll"

1141, a

l l

xex

This shows that the linear mapping F H J ( / , F ) is continuous from Llo(p; A ) , supplied with the topology induced by Ll(p; A ) , into 9 6 A . By virtue of the density of Llo(p; A ) in Ll(p; A ) (Appendix G16), that mapping has a unique extension that is a continuous linear mapping of L , ( p ; A ) into Y 6 A . We can conclude that, for any F € L l ( p ;A ) , the f given by ( 5 ) is a member of Y 6A and that the inequality (6) is a consequence of (7). 0

Our next objective is to develop a Fubini-type theorem. Theorem2.5-2. L e t F E L l ( X , 6 ' ; p ; A ) a n d l E Y ( T xX , 6 x 6 ' ; C ) . Then,

where the outer integral on the left-hand side exists in the sense of Definition 2.5-2 and the outer integral on the right-hand side is a Bochner integral. PROOF. That the left-hand side of (8) exists in the sense of Definition 2.5-2 follows from Theorem 2.5-1. On the other hand, for each x E X , l( ., x ) E Y = Y ( T ,6 ;C ) according to Appendix G14. So, dP, l ( t , x) exists in accordance with Section 2.2. We can choose a sequence {I,,},"=, of simple functions in Y ( T x X , 6 x 6';C ) such that 111 - /,I1 + 0. Therefore, for each x E X ,

2.

42

INTEGRATION WITH VECTOR MEASURES

in [ A ; B]. But, for each n, the left-hand side of (9) is a simple [ A ; B]-valued function of x . Therefore, the right-hand side is a measurable function of x (Appendix G9). Since F E L l ( p ;A ) , we can choose a sequence {F,,}such that F, E L I o ( p ;A ) and F, F in L l ( p ;A ) as well as almost everywhere on X (Appendix G16). Then, for each 11, J T dPl /,,(t, x)F,,(x)is a simple B-valued function of x , and, as n co,it converges almost everywhere on X to -+

-+

Therefore, (10) is a measurable function of x . Moreover, llF(*)llAE L l ( p ;R ) , and

Consequently, by Appendix G15, the right-hand side of (8) truly exists as a Bochner integral. Next, we observe that

according to the inequality (2) of Section 2.2 and Appendix G11. On the other hand, forfdefined by (9,we have t h a t f e S o A. So, by (2) and (6),

5 liFliLlsup I / ( t , .)I

SSVar P.

(1 3)

1, x

The inequalities (12) and (13) and the density of L l o ( p ;A ) in L , ( p ;A ) imply that we need merely establish the equality in (8) for every F e L l 0 ( p ;A ) . So, let F = I akzEI. E L I o ( p ;A ) and let 6‘ E B‘ be arbitrary. We may write

I;=

according to Appendix GI7 and (4) of Section 2.2. But, when # 0, p and b’P( .)akhave finite total variations on Ek and T, respectively (Theorem 2.2-1 and Appendix G7), and therefore we may apply the scalar Fubini theorem to the right-hand side of (14) to interchange the integrations. Then, upon extracting 6‘ and the u k ,we obtain b’ j d P I j d p x I ( t , x)F(x).

2.6.

43

SESQUILINEAR FORMS GENERATED BY PO MEASURES

Since b' E B' is arbitrary, (8) has been established for every F EL I o ( p ;A ) . This completes the proof. 0 Problem 2.5-1. Show that Definition 2.5-1 is consistent with the definition of

f dP,f ( t ) indicated in Problem 2.2-3, for the case where SVar P < 03 and f E 9 0A . [The last two conditions imply that SSVar P < co, according to Problem 2.2-2, and that f

E

%(A).]

2.6. SESQUILINEAR FORMS GENERATED BY PO MEASURES

We end this chapter with a discussion of certain positive sesquilinear forms generated by PO measures. In the first part of this section, P is restricted to being a PO measure on 6. Hence, P maps 6 into [ H ; H I , , where H is a complex Hilbert space with the inner product Moreover, SSVar P = [IP(T)II < co according to Theorem 2.2-3. As usual, E denotes the complex conjugate of any number, function, or measure u. ( a ,

.)a

Lemma 2.6-1. Let f and v be any two members of Y 0 H . Dejne a function 23, on 9 0H x 9 0H by choosing any two representations

where gk , hj E Y and a k , bj E H , and then setting

Then, 8,is a positive sesquilinearform on the space Y 0 H x Y 0 H, and I23P(f,

011

s IIp(m IlflllIlvlll.

(2)

PROOF. An argument like the one following Definition 2.5-1 shows that the right-hand side of (1) is independent of the choices of the representations for f and v. Moreover, 8, is clearly a sesquilinear form on Y 0 H x Y 0 H. To prove the inequality (2), we write

IIP(T)II 2 llgkII k

llhj\l llbjlI.

Ilakll

i

(3)

Since the left-hand side does not depend on the representations for f and v, we may take the infimum over all such representations to get (2).

2.

44

INTEGRATION WITH VECTOR MEASURES

Finally, we show that Bp is a positive form. Let g k , be a simple function that approximates g k , and set r

Then, r

'P(f,f)

-'P(fO,fO)

r

c 1(

=k = l

j=l

IdpIISk(t)G)

-gk,O(t)gj,o(t)lak,

aj).

Through an estimate similar to (3), we see that the right-hand side can be made arbitrarily small by choosing the g k , appropriately. Moreover, functions of the form (4) are themselves simple functions. Thus, to complete the proof, we need merely establish the positivity of 23, on functions of the form fo = a i z E I ,where the Ei comprise a partition of T and are therefore pairwise disjoint. Whence,

and therefore n

SAY0 fo) 9

=

C1 (P(Ei)ai

3

ai) 2 0.

i=

Lemma 2.6-1 has been completely established. 0 In view of Lemma 2.6-1, we can extend 23, continuously onto the Cartesian product 96 H x 96 H supplied with the product topology (Appendix D5).The resulting mapping, which we also denote by B p , will be a positive sesquilinear form on 9 6 H x 9 6 H that satisfies the inequality (2) for all f, u E Y 6 H . We use this result to define still another kind of integral.

Definition 2.6-1. For anyf, u E 4 6 H , we set

I

d(PIf(0, 40)a 2 3 P M v).

(5)

The next theorem presents an explicit formula for ( 5 ) that can be used when representations forfand v are given in accordance with Theorem 2.5-1. The following notation is used. T, X , and Yare three nonvoid sets, &, &', and 6'' are o-algebras of subsets of T, X , and Y, respectively, and 1-1and v are complexvalued measures or positive measures on &' and &", respectively. We know from Theorem 2.5-1 thatf'e 9 6 H if and only if it has the representation

2.6.

SESQUILINEAR FORMS GENERATED BY PO MEASURES

45

where F e L I ( X ,6’; p ; H ) and 1 E Q(T x X,6 x 6’; C ) . Similarly, g E Q 6 H if and only if v(t) =

f dv, m ( t ,Y ) V ( Y ) , Y

(7)

where V € L 1 (Y , 6”;v ; H ) and m E S(T x Y , 6 x 6 ” ; C ) .

Theorem 2.6-1. Let f and u have the representations ( 6 ) and ( 7 ) . Then,

PROOF.We first note that the mapping ( 4 x , Y ) H l ( t ,x ) m ( t ,A is a member of Y(Tx X x Y , 6 x 6’ x 6 ” ; C ) . This fact and a straightforward estimate using ( 2 ) of Section 2.2 show that the right-hand side of (8) exists as a scalar integral. For the integrable simple functions r

=

1 akX& E L pcx,

k= 1

6 ‘ ;p ; H ,

and S

V

=

1 bjXI, E Llo(Y, 6 ” ; V ; H ) , j= 1

the right-hand side of (8) is equal to

Since p , V, and ( P ( ‘ ) a k ,b j ) have finite total variations on E k r I j , and T, respectively, whenever ak # 0 and b j # 0, we may apply the scalar Fubini theorem and then extract the ak and bj to obtain

Thus, (8) is true for any integrable simple functions F a n d V.

46

2.

INTEGRATION WITH VECTOR MEASURES

To show that (8) remains true for all F E & ( X , 0'; p ; H ) and

v E L,( Y , (5"; v ; H ) , we need merely show that both of its sides depend continuously on F and V with respect to the L , norms (Appendix D5). For the left-hand side, this follows from

5 IIP(T)II IIFIIL, SUP I l ( t ,

x>l II

f , X

w,

SUP f,

I m ( t , Y)l.

(9)

Y

It is easily seen that the right-hand side of (8) is also bounded by the righthand side of (9). 0

In the remainder of this chapter, P : Km -+ [ H ; H I , is a a-finite PO measure. Also, we let g E Y, g(t) > 0 for all t E T, and g be integrable with respect to P (Definition 2.3-2). Moreover, we set

Q ( E )4

I

E

dPfg(t>,

By Theorem 2.3-3, Q is a PO measure on Yg(H)4 Y,(T, (5; H )

(5.

E E 0.

Finally, we set

{ f e Y(H):f/g E 9 6H } .

(10)

Lemma 2.6-2. Y g ( H )c 9 d , ( H ) c Y 6 H .

PROOF.We will use the obvious fact that, if u E 9 and q E Y 6 H , then uq E 9 6 H . Since g E 9, Jg E 9 also. By definition, for any f E Y,(H), we have thatflg E 9 6 H . Therefore, Jgflg =f / J g E Y 6 H , which implies that Yg(H)c Yd;(H).Another multiplication by Jg shows that F?dq(H) c Q 6 H. 0 Lemma 2.6-3. Iff E 9 , ( H ) , thenf has the representation ( 6 ) where, in addition, thefunction { t , x } ~ l ( tx ), / g ( i ) is a member o f Y ( T x X , (5 x (5'; C). This lemma follows directly from Theorem 2.5-1. Definition 2.6-2. For any f E Y g ( H ) we , set

Note that the right-hand side has a sense according to (10) and Definition 2.5-2. Clearly, f~ d P , f ( t ) is a linear mapping of Yg(H)into H. Moreover,

2.6.

47

SESQUILINEAR FORMS GENERATED BY PO MEASURES

this definition of dPtf ( t ) is independent of the choice of g. To see this, first note that, for every f E Q,(H) n Q 0H , we may choose a representation, f= gka,, where a, E H, g k E 3,and lgk(t)I < Ckg(t) for all t E T, the ck being constants. [Indeed, sinceflg E Y 0H , we can writeflg = qkak and choose ck supt Iqk(t)l.] Therefore,

I;=

According to Theorem 2.3-2, the right-hand side does not depend on the choice of g. Next, it can be seen as before that every f E Q g ( H ) has a representation of the form f = g k ak, where ak E H , g k E 9,1gk.t) I < Ckg(t) for all t E T, the ck are constants, and Cp=lCkllakII < 00. Set f , = gkak. As r + 00,

I;=l

in H because

Since the left-hand side of (1 1) does not depend on the choice of g, neither does the right-hand side.

Definition 2.6-3. For anyf, v E Q,(H), we set

In view of Definition 2.6-1 and Lemma 2.6-2, the right-hand side has a sense and defines a sesquilinear mapping on Y,(H) x %,(If). Moreover, (12) is independent of the choice of g, as can be seen through an argument similar to the one used for Definition 2.6-2.

Theorem 2.6-2. Let f E 9 , ( H ) and v E Yg(H) have the representations ( 6 ) and (7). Then, (8) of Section 2.5 and (8) of this section still hold true in the present situation, where P is a a-jinite P O measure.

PROOF.By Definition 2.6-2,

48

2.

INTEGRATION WITH VECTOR MEASURES

By virtue of Lemma 2.6-3, Theorem 2.5-2, and Theorem 2.3-4, the righthand side is equal to

and this justifies (8) of Section 2.5. A similar manipulation establishes the other equation. 0

Problem 2.6-1. Show that the definition of b, given by (1) does not depend on the choices of the representations forfand u. Problem 2.6-2. Show that (12) does not depend on the choice of g. Problem 2.6-3. Prove the other part of Theorem 2.6-2. Problem 2.6-4. Let P be a a-finite measure and let g E 9 be a positive function [i.e., g(t) > 0 for all t ] that is integrable with respect to P. Furthermore, IetS, u E gg(H)and let {Ek}:=l be an increasing sequence in 0 with UEk = T. Show that

and

Chapter 3

Banach-Space-ValuedTesting, Functions and Distributions

3.1. INTRODUCTION

As was mentioned in the Preface, the natural framework for a realizability theory of continuous linear systems is distribution theory. Since the signals in the systems of concern to us take their values in Banach spaces, the properties of Banach-space-valued distributions are essential to our purposes. The present chapter is devoted to a discussion of such distributions; they constitute a special case of the vector-valued distributions of Schwartz (1957). We start with a description of the primary testing-function space in the theory of distributions, namely 9 ” ( A ) . As always, A and B denote complex Banach spaces.

49

3.

50

BANACH-SPACE-VALUED DISTRIBUTIONS

3.2. THE BASIC TESTING-FUNCTION SPACE 9 " ' ( A )

Let nz be an n-tuple each of whose components is either a nonnegative ) the integer in R' or co. Also, let K be a compact set in R".9 K m ( A denotes linear space of all functions 4 from R" into A such that supp 4 c K and, for every integer k E R" with 0 2 k m, 4(k)is continuous. We assign to g K m ( A )the topology generated by the collection { y k : 0 I k I m} of seminorms, where

Since y o is a norm, g K m ( A is ) separated. Moreover, it is metrizable because the collection { y k } is countable (Appendix C7). Lemma 3.2-1.

9Km(A)

is complete and therefore a FrPcliet space.

PROOF.Since g K m ( A )is metrizable, we need only establish its sequential completeness (see Appendix CI1). Let {4i}?=, be a Cauchy sequence in g K m ( A )In . view of (1) and the completeness of A , we have that, for every k as restricted above, there exists an A-valued function $k on R" for which 4!k)--t $ k uniformly on R". By note VlII of Section 1.4, $ k is continuous. Also, by Problem 1.6-3, $bk) = $k . Clearly, supp t,b0 c K. Hence, $o is the ) {+i}. O limit in g K m ( A of

Note that, if every component of m is finite, g K m ( A )is a Banach space because its topology is the same as that generated by the single norm p , where

P(4)

max

Yk(4).

O$k$m

When all the components of m are

00

(i.e., when m

=

[a]), we denote

g K m ( Aby ) a K ( A ) .Moreover, we set g K m ( C= ) gKm and 9 K ( C )= g K .

Now, let { K j } Y = , be a sequence of compact sets in R" such that Kl c K2 .* K j = R",and every compact set J c R" is contained in some K j . We define g m ( A )= 9:,,(A) as the inductive-limit space generated by the 9 E j ( A ) . That is, c K3 c

9,

uj

u9 g j ( A ) , m

9"(A) = 9 l " ( A )=

k= 1

and this space possesses the inductive-limit topology (see Appendix El). As before, we set 9[m1(A) = B(A),Bm(C)= 9", and 9 ( C ) = 9. This definition of g m ( A )does not depend on the choice of { K j } .Indeed, for any other sequence {Hi}zl of compact sets with the required properties, 9 E j ( A ) and 9 G i ( A ) are identical as linear spaces because every 9 F j

u

u

3.2.

THE BASIC TESTING-FUNCTION SPACE

9"'(A)

51

is contained in some 9zi and conversely. To show that the topologies are 9z,(A). Given any K j , the same, let A be a convex neighborhood of 0 in we can find an H i containing K j . By definition of the inductive-limit topology, A n 9 E i ( A )is a neighborhood of 0 in 9 z , ( A ) .Moreover, 9 F j ( A )is a subspace of 9 i i ( A ) , and its topology is the same as that induced on it by 9;,(A) because both topologies are generated by the same seminorms y k . Hence, An is a neighborhood of 0 in 9E,(A). Consequently, A is a convex 9 F j ( A ) . Similarly, every convex neighborhood of neighborhood of 0 in 0 in 9 E j ( A ) is a convex neighborhood of 0 in 9 i , ( A ) . Consequently, the two inductive limit topologies are identical 9"'(A) is clearly a strict inductive-limit space (Appendix E3). Moreover, it possesses the closure property defined in Appendix E4 because each 9g4(A) is complete. As a consequence, the following assertions hold. 9"'(A) IS a complete separated locally convex space. A linear mapping f of 9"'(A) into another locally convex space W is continuous if and only if its restriction to each 9 E j ( A ) is either sequentially continuous or bounded. A set is bounded in 9"'(A) if and only if it is contained and bounded in some 9 $ ( A ) . Similarly, a sequence { 4 j }converges in W ( A ) if and only if it is contained and converges in some 9 g j ( A ) .Thus, a linear mapping on W ( A )is continuous if and only if it is sequentially continuous. (See Appendix E.)

ui

uj

u

ui

Lemma 3.2-2. Let J and Kbe two compact intervals in R" such that J contains a neighborhood of K . Then, given any 4 E gK"'(A),there exists a sequence {4j}y= c g J ( A )such that $ j + 4 in 9,"'(A).

PROOF.Set

and 9

P

= 1,

2, . . * .

(In this proof, all integrations are over R".) It follows that q, is a smooth nonnegative function, diam supp q p = 2/p, and j q,(t) dt = 1. Next, set

For all sufficiently large p , supp 4, c J . Moreover, we may differentiate under the integral sign (Theorem 1.6-4) and integrate by parts (Problem 1.6-1) to obtain, for any fixed k E R" such that 0 Ik I m,

4r)(t)= J$(k)(x)vp(t - x) d x .

3.

52

BANACH-SPACE-VALUED DISTRIBUTIONS

Hence,

sup

l14(k'(t)- 4 ' k ' ( X ) I I A *

lr-xl 0 such that for all 4 E 9K"'(A),

11 (f, 4) 11 B 5 M p p ( 4 ) ,

(1)

where

Pp(4) 4L Omax

II(f,

(4)

4)IIB.

To indicate that the pointwise topology is being used, we employ the notation [ 9 " ' ( A ) ;B]". Both the bounded and pointwise topologies are separating.

Lemma 3.3-2. Every f through the equation

E

[ P ( A ) ;B] uniquely defines a g

(9, 0)a

(f, Oa),

E

[9"'; [ A ;B ] ]

0 E grn,a E A.

(5)

This assertion remains true when 9"'(A) is replaced by 9 K " ' ( A ) and 9"'by gK"' for any given compact set K.

PROOF.Having fixed upon some 0 E B"', we define a mapping,je of A into B by j,a A (f, Oa) for all a E A . It readily follows that j e is linear. That it is also continuous follows from Lemma 3.3-1. Indeed, for any compact set K that contains supp 6' and for all a E A , we have Iljeallr, =

II(fe9

0a)IIB 5 MpAOa) 5 M l l a l l A ~ ~ ' ( O h

54

3.

BANACH-SPACE-VALUED DISTRIBUTIONS

where

~ ~ ' (=0 )max sup 1 O c k ) ' ( t ) l . O s k s p tsR"

Hence, IljOll[A: B ]

(6)

Mpp'(0).

Next, set ( g , 0 ) j,. This uniquely defines g as a mapping from 9"into [ A ; B ] . g is linear because, for any a E A , a, p E C, and 0, $ E grn,

+ P$>a = (.A

( 9 , a0

+ P$a>

= a(ft =

ea> + K.A $4 + P ( g , $>)a.

( 4 g ,0)

Moreover, (6) implies that g is continuous. The second statement of the lemma is established in the same way. 0

For any nonnegative integer k

E

R", we define the generalizeddrflerentiation

Dk on [ 9 " ( A ) ;B] as follows. First, note that Dk is a continuous linear mapping of 9 m + kinto ( ~g), " ( ~ since ) its restriction to every 9 z + k ( A )is a continuous linear mapping of 9 : + k ( A ) into a K r n ( A )This . allows us to define Dk on any f E [ g r n ( A )Bl ; by

( 0 %4)

A (-

l ) l k l ( . A Dkb),

6 E 9rn+k(A)

(7)

because the right-hand side has a sense. Theorem 3.3-1. Dk is a continuous linear mapping of [ g r n ( A ) B] ; into [ 9 r n ' k ( A ) B], ; as wellasof [ 9 " ( A ) ;B]" into [ 9 r n ' k ( A ) B]". ; PROOF.Observe that the right-hand side of (7) is a member of B, so that ) B. It follows readily that D Y E [ 9 m + k ( A )B; ] and that Dkfmaps 9 r n + k ( Ainto Dk is a linear mapping of [ g r n ( A ) B] ; into [ 9 m + k ( A )B; ] . To show that Dk is continuous, let @ be a bounded set in 9 " ' + k ( A ) .As 4 traverses @, Dk4 traverses a bounded set in 9 " ( A ) ; call it 0.So,

supll(D% + € @

4>llB

= suPll(f,

Dkb)llB

4EO

= supll(f,

O>llB*

So, truly, Dk is continuous from [ 9 " ( A ) ;B] into [ 9 m ' k ( A ) ;B ] . By restricting @ to sets of one element each, we obtain the same conclusion with the pointwise topologies. 0 Another operator of importance to us is the shifting operator ur, which is also called the translation operator. Let T E Rn be fixed. 6, is defined on any

3.3.

DISTRIBUTIONS

55

4 E 9 " ' ( A ) by (a,4)(t)= 4(t - z). It is an automorphism on 9"'(A). On the other hand, a, is defined on any f E [9"(A);B] by (arf,

4)

'( f ,

0-r

4),

4 E g"(A)*

(8)

As a consequence, a, is an automorphism on [ 9 " ( A ) ; B], as well as on

[.9"(A); B]". (Show this.) Two distributions f, g E [ 9 ( A ) ; B] are said to be equal on an open set SZ c R" if (f, 4 ) = ( g , 4 ) for every 4 E 9 ( A ) such that supp 4 c a. The null set o f f is the union of all open sets on which f is equal to zero (i.e., is equal to the zero distribution). The complement of the null set is called the support off and is denoted by suppJ Thus, suppf is a closed set. A property of distributions that we occasionally use is the following. If f E [ 9 ( A ) ;B] is equal to zero on every set in a collection of open sets, then it is equal to zero on the union of these sets. The proof of this assertion is precisely the same as it is for complex-valued distributions on 9 and can be found, e.g., in the work of Zemanian (1965, Section 1.8). One consequence of this result is that supp f is the smallest closed set outside of which f is equal to zero. Another direct result is the following. (f, 4 ) depends only on the values that 4 E 9 ( A ) assumes on any arbitrarily small neighborhood of suppJ Indeed, if $ E 9 ( A ) is equal to 4 on a neighborhood of supp f, then supp(4 - @) is contained in the null set off, so that (f, 4 - $) = 0. We conclude this section with two examples of [ A ;B]-valued distributions. Example 3.3-1. Let F be a fixed member of [ A ; B] and let S be the delta function defined on any 4 E .9'(A) by (6, 4 ) 4(0). We define FS as a mapping on G'(A) by (Fa, 4 )

e F4(0),

4 E 9'(4.

(9)

Consequently, FS E [ 9 ' ( A ) ; B]. As is indicated in Lemma 3.3-2, F6 generates a member of [ g o ;[ A ; B]], which we will also denote by FS. Thus, in accordance with ( 5 ) and (9), we write, for any d E 9' and a E A , (FS, O)a = (Fa, Oa)

= F[O(O)a]= [Fd(O)]a

or (FS, 0)

= Fd(0).

Upon applying the generalized differentiation Dk to FS, we obtain, for any

4 E .9k(A), ( Dk(FS),4 ) = (- l)'k'F4'k'(0) = F( Dk6, 4).

(10)

We define FDkS as the distribution that assigns to 4 the value indicated on the right-hand side of (10). Thus, we have Dk(FS) = FDkS. 0

56

3.

BANACH-SPACE-VALUED DISTRIBUTIONS

Example 3.3-2. Let h E L , ( [ A ;B]); that is, the [ A ; B]-valued function h is Bochner integrable with respect to Lebesgue measure on the Borel subsets of R". We define a mapping f of 9'(A) into B by setting

By Appendix G11,

II(L 4)llB

I141Ll SUP II4(t)ll,. 1

It follows that h H f is a continuous linear mapping of & ( [ A ; B]) into [go@); B]. This mapping is injective; indeed, if / I generates the zero member of [ 9 ' ( A ) ; B], then JRn

h(t)O(t)dt a

=0

for all 0 E 9 and all a E A , which by Appendix G12 implies that h ( t ) = 0 for almost all t . Thus, L , ( [ A ; B]) can be identified as a subspace of [9 '( A ) ; B]. Any member of [ 9 ' ( A ) ; B] that can be generated in this way from a member of L,( [ A ; B]) will be called a regular [ A;B]-valued distribution. 0

Problem 3.3-1. Show that the shifting operator oris an automorphism on [ 9 Y A ) ;BI. Problem3.3-2. Let h be an [H; HI-valued function on R with the following two properties. It is strongly measurable with respect to Lebesgue measure on the Borel sets; that is, h(*)ais a measurable H-valued function for every a E H. It is locally essentially bounded; that is, for every compact set K c R , there exists a constant MK such that Ilh(t)II 5 MKfor almost all t E K. Define a mapping f on any 4 E 9' by

(f, 4)a

a / h ( t ) U 4 ( l ) dt.

Show that f E [9';[H; HI].

Problem 3.3-3. Let P be a a-finite [ A ; B]-valued measure on the bounded Borel subsets of R". For every 4 E 9 ( A ) ,define f by

(L4) a p p 4 . Show that f E [ D ( A ) ;B]. f is called the [ A ;B]-valued distribution generated by P.

3.4.

57

LOCAL STRUCTURE

3.4. LOCAL STRUCTURE

The objective of this section is to show that every Banach-space-valued distribution can be represented on any compact interval as a finite-order derivative of a continuous Banach-space-valued function. We do this by means of a method employed by Sebastiao e Silva (1960). As before, K is a compact interval in R", and m = {mi}:= denotes an n-tuple each component of which is either a nonnegative integer or co. %?,"(A) is the linear space of all functions 4 from K into A such that, for each integer k E R" with 0 I k 2 M , dCk)is continuous. %?,"(A) is equipped with the topology generated by the collection { Y k : 0 5 k 2 m > of seminorms, where again Yk(4)

suPl14'k'(t)llA. 1 E K

%?Kn'(A) is a FrCchet space, its completeness being established as was Lemma 3.2-1. When every component of m is finite, %?,"(A) is a Banach space. By identifying each t,b E 9,"(A) with its restriction to K , we' can and will view 9,"(A) as a subspace of %?,"(A). Now, let p = {pi):=l be a nonnegative integer in R", and let [L, Q] be a compact interval in R", where L = { L i } : = ]< Q = {Q,}Y=,. It is a fact in interpolation theory that, for each i, there exist 2(pi + 1) polynomials on [ L i , Qi], which we denote by g x i , v i , where X i = L i , Qi and vi = 0, . . . , p i , such that df'g,,, v,(Li)

= dvi.pi

a f i S l , ,v i ( Q i ) = 0,

9

afigQi, v,(Li)

= 0,

d;'gQi, vi(Qi)

= dvi,pi

*

Here, p i = 0 , . . . , p i and d v i , p i is the Kroriecker delta (i.e., dvi,,li = 1 for v i = p i and d v i , p i = 0 for vi # pi). Now, consider the function d b obtained from some 4 E gKP(A) by means of the following formula:

A summation on the X's means the sum of all possible terms obtained by is a summation over setting each X , equal to either L, or Q,. Also, 1s i s n ,

0 1 vi_ 0.

For each fixed t E R", we denote the function X H J,(f - s)on K by J,(t - .), Also, we remind the reader that the symbol [2] denotes the it-tuple each of whose components is equal to 2.

3.4.

59

LOCAL STRUCTURE

Lemma 3.4-3. Let 4 E g f ; + [ 2 1 ( A )Then, . t H J,(t

- .)DP+[Z]4 ( t >

is a continuous mapping of R" into %Kp(A).Similarly, r HJ,(t - .) is a continuous mapping of R" into VKp. PROOF.Let the integer k E R" be such that 0 I k d p . Also, fix t and let I be any compact interval in R" containing t in its interior. As a function of { t , 4,

D,kJ,( t - X)D,P + [2lf$( t ) is a continuous function from I x K into A , and therefore it is uniformly continuous on I x K. So, as At -+0 in R",

D,kJ,(t

+ At - X ) D P ' " ~ ~+( ~At) + D,kJ,(t

- x)DP"']q5(t)

in A uniformly with respect to all x E K. This proves our assertion for the function (1). The proof for the other function is the same. 0

Lemma 3.4-4. For any

4 E 9i+[z1(A),

Note. I n the right-hand side, we have the Riemann integral of the continuous %'KP(A)-valued function (1).

PROOF.We may repeatedly integrate by parts to establish that (- l)IpI

s

J,(t - x ) D p + " l ~ ( tdt ) =

K

s

Jo(t - x)DcZ1q5(t) dt

= q5(x).

K

Equation (2) follows directly. 0 We now appeal to note 11 of Section 1.4 and to Lemma 3.4-2 in order to apply 7 c p to (2) under the integral sign. Thus, for any E 9i+[21(A),

where G,(t, *) = n,J,(t - -).

The integral in (3) is a continuous gKP(A)-valued function of t E K because it is the result of first applying the function (1) and then applying x p , both of which are continuous mappings. Similarly, G,(t, -) is a continuous gKPvalued function o f t E K.

60

3.

BANACH-SPACE-VALUED DISTRIBUTIONS

Lemma 3.4-5. Let f E [9Y'(A); B], let K and J be compact intervals in R" such that K c 3, and let p be the integer in R" corresponding to f and J in accordance with Lemma 3.3-1. Then, there exists a unique 1 E [ 9 K p ( A ) ;B] that satisjies the relation (1, O )

=

lim ( f , O j ) j-+m

(4)

for all O E .9KP(A) and all sequences {e,} c .9j"'(A) that tend to 0 in 9 j p ( A ) . Thus, (1, $) = ( f , $)for all $ E 9 K m ( A ) .

PROOF.Choose any 8 E 9 K p ( A ) .By Lemma 3.2-2, we can choose a sequence {O,}j", c gJ'"(A)such that O j + 0 in 9 j p ( A ) . We infer from the inequality in Lemma 3.3-1 that {( f , 0 , ) ) is a Cauchy sequence in B. Its limit is taken to be the value (1, O), and this value also satisfies the inequality in Lemma 3.3-1. It follows that 1 E [.9KP(A);B]. Moreover, this definition of 1 is independent of the choice of the sequence {O,}, and 1 is uniquely determined by the values thatfassigns to the 4 E 9j"'(A). 0 Theorem 3.4-1. Let f E [ W ( A ) ; B] and let K be a compact interval in R". Then, there exists an integer p E R" with 0 p _< m and a continuous [ A ;B]valuedfunction h on Ksuch that,for'all4 E .9:'[21(A),

I

(f,4 ) = h(t)DP+['l4(t) dt. K

(5)

In general, p and It depend on,f and K. PROOF.Let J and p be chosen as in Lemma 3.4-5. Then, 1 can be applied to (3), where now 4 E 9F'[21(A), and note 11 of Section 1.4 can be invoked to get

Let g E [gKP; [ A; B]] be related to 1 in accordance with Lemma 3.3-2. Then, (6) can be rewritten as

Upon setting h(t) = (- l)Ip1(g(*), G,(t, we arrive at (5). h is a continuous [ A ; B]-valued function because it is the composite of the two continuous mappings G,(t, .): K - +Q K P and 9: 9 K p -+ [ A ; B]. a))

That p and h depend on f and K can be shown by examples. 0

3.5. [ B ( A ) ;B] AND [9; [A; B ] ] CORRESPONDENCE

61

Corollary 3.4-la. Let f E [gm; A ] and let K be a compact interval in R". Then, there exists an integer p E R" with 0 p I m and a continuous A-oa1uedfiitictio.n h on Ksuch that, for all 4 E BE'[21,

(f,4 ) =

1

h ( t ) D P + [ 2 1 4 ( dt )t .

(7)

K

Here, too, p and h depend in general onf and K.

PROOF.By Theorem 3.4-1, (7) holds true for a continuous [ C ; A]-valued function h on K. But [ C ;A ] is identical to A. 0 Problem 3.4-1. Supply the details of the proof of Lemma 3.4-1. Problem 3.4-2. Show that the quantities p and I? in Theorem 3.4-1 depend in general on f and K.

3.5. THE CORRESPONDENCE BETWEEN [ 9 ( A ) ; B], AND 19; [ A ; Bl1

The natural identification between [ 9 ( A ) ;B ] and [9; [ A ; B ] ] ,which we alluded to at the beginning of Section 3.3, will now be established. In the following, B O A (or B K O A ) denotes the linear space of all 4 ~ 9 ( A ) [respectively 4 E g K ( A ) ]having representations of the form 4 = 0, a,, where 8, E 9 (respectively 0, E 9,), ak E A , and the summation is over a finite number of terms.

1

Lemma 3.5-1. Let K and N be compact interoals in R" such that K c fi. Given any 4 E 9 K ( A ) ,there exists a sequence {4j}T= c BN0A that converges in B N ( A )to 4. Thus, 9 0A is dense in 9 ( A ) . PROOF.

Let L be a compact interval in R" such that K c L and L c fi. Let

d, = inf{ I t - X I : t E K , x E R"\L}, d2 = inf{ 1 t - XI : t E L , x E R"\N). Thus, d , and d2 are greater than zero. We choose a sequence {QP},"=,, where each Ep is a collection { Q p , i } i of open sets in R" with the following three properties. Only a finite number of the Q p , intersect any bounded set in R". For every p and i, diam QP, < d l . Finally, sup diam QP, 1

+ 0,

p

00.

3.

62

BANACH-SPACE-VALUED DISTRIBUTIONS

Next step: For each p , we can choose a collection {I),, i } i c 9 such that, for every i and 1, we have supp I),, c Q,, i , 0 I I),,,i ( t ) I 1, and I),, i ( t ) = 1. [See, for example, Zemanian (1965, Section 1.8).] For each p , we define the mapping J , on 9K(A)into gL0A by

xi

where each t i is chosen as some member of 0,. i . It follows that, as p J p 4 tends to # in g L O ( A ) .Indeed, su~II+(t)- C 4 ( t i ) I ) p , i ( t ) I I A I SUP i

t

t

4(ri)III)p,i(f)

i

5 SUP i

1 II+(t> -

+ co,

SUP

II4(t) - +(ti)Il>

t€Rp,i

and the right-hand side tends to zero as p + co because of the uniform continuity of # E 9K(A). Next, let q,, where q = 1 , 2 , . . . , be the function defined in the proof of Lemma 3.2-2, and let R , be the linear mapping on gO (A)defined by

9-

1

9(x)q,(* - X) dx.

R"

If q-' < d,, then supp R,9 c N whenever supp 0 c L. Moreover, R , is continuous from g L O ( A ) into BN(A). Indeed, for any O€BL0(A) and any nonnegative integer k E R",

supllDtkR,O(r)ll =

e(x)D,kq,(r - X) d x

t

Now, for any 4 E g K ( A ) , consider R , J p 4 E gN 0A. By what we have already shown, for fixed q > d; ' and as p --+ co, we have that R , J,, # --t R , 4 in BN(A).Furthermore, by the proof of Lemma 3.2-2, R 4 4 + 4 in 9N(A) as q + co. The fact that 9 N ( A ) has a countable base of neighborhoods of 0 (Appendix C7) can now be exploited to extract a sequence { R ( , , J p j $ } 7 = , that converges in 2 , , , ( A ) to 4. This establishes the first conclusion. The second conclusion follows from the definition of 9 ( A ) as an inductivelimit space.

Lemma 3.5-2. 9 0A is dense iii 9'"(A) wliatewr be m.

3.5. [ g ( A ) ;B] AND [9;[ A ; B]]CORRESPONDENCE

63

PROOF.9 ( A ) is a dense subspace of 9"'(A), and the topology of 9 ( A ) is stronger than that induced on it by 9 " ( A ) . Consequently, this lemma is an immediate result of the preceding one. 0

Theorem 3.5-1. There is a bijection from [ 9 ( A ) ;B] onto [9;[ A ; B ] ] defined bY

(9,O a = ( f , @a>,

(1)

where 0 E 9, a E A , g E [9; [ A ; B ] ] ,and f E [ g ( A ) ;B]. Note. Because of this result, we will subsequently denote g and .f by the same symbol, sayf, and will replace (1) by

(2)

( f , @>a= ( f , ea>.

PROOF.Lemma 3.3-2 shows that f uniquely determines g though (1). So, let us consider the converse. By Corollary 3.4-la, for any given g E [9;[ A ; B ] ] and compact interval K c R", there exists a nonnegative integer q E R" and a continuous [ A ; B]valued function / I on K such that, for all O E g K , (9,8 ) =

j

h(t)Dq8(t)dt.

K

(3)

We let 9 k ( A ) be the space of all 4 E 9 ( A ) whose supports are contained in and supply with the topology induced by g K ( A ) .Also, 9~ 9 g ( C ) . We define a mapping,fX on g i e ( A )into B by

ICI)

(fie,

j h(W41CI(t)d t , K

ICI E %(A).

(4)

It follows that,f, E [ 9 , ( A ) ; B ] . Moreover, if we set = 8a with 8 E and we see from (3) and (4) that (9,%)a = (f k , Oa), in agreement with (1). This procedure determines an f g E [Qk(A); B] for every compact interval K. We now assert that, if the compact interval J contains K,then the restriction of,fi to 9 k ( A ) coincides withf, . Indeed, for 8 E g kand a E A , (fJ, Oa) = (g,8)a = ( f g , %a), and therefore coincides with fk on 9 k 0A . But, as a consequence of Lemma 3.5-1, gg0A is dense in 9&), and this implies our assertion. It now follows that there exists a unique f~ [ 9 ( A ) ;B] whose restriction to each 9 k ( A ) coincides with,fk. Thus, every g E [9;[ A ; B ] ] uniquely determines anfE [ 9 ( A ) ;B] by means of (1). 0 a E A,

fJ

64

3.

BANACH-SPACE-VALUED DISTRIBUTIONS

3.6. THE p-TYPE TESTING FUNCTION SPACES

The rest of this chapter discusses various generalized-function spaces, most (but not all) of which are subspaces of [B"(A);B ] . It is possible to formulate our discussion in such a general fashion that each space of interest to us becomes a special case obtained by making particular choices for certain parameters. The only exceptions to this are the spaces of L,-type distributions, which are discussed at the end of this chapter. Just as with distributions, generalized functions are continuous linear mappings on certain testingfunction spaces. Thus, our first objective is to develop a general formulation for the needed testing-function spaces. Throughout the rest of this chapter p, v, i , , j , f, and p denote nonnegative integers in R , whereas k = { k v } t = lis a nonnegative integer in R". As before, m = {m,}:=1, where each m, is either a nonnegative integer or 03. A sequence {Kj}T= will be called a nested closed cover of R" if each K j is a closed subset of R", K j c K j + l for every j , K j = R", and each compact subset of R" is contained in one of the K j . Let {Kj}j"=oand {Ip}Pm,o be two nested closed covers of R". For each j and 1, let there be given a continuous function t j ,f(t) from R" into R such that t j ;f(t) > 0 for every t E R". Also, assume that t j ,f(t) 2 tj+l,f ( t )for all j , f, and t. For every j and p , we define the functional pi, on suitably restricted functions 4(t)from R" into A by

uj

,

Pj,p(4)

4

max

O =

max

SUP

O ~ k , s m i n ( p ,m,) tsl,

II 4'k'(t>IIA

&"(A) is a Frichet space and is normal. For fixed m, &"(A) is the largest of the p-type testing-function spaces because 9 " ' ( A )c &'"(A) for any other space 9 " ( A ) . Indeed, every A-valued function 4 on R" such that 4(k)is

3.

66

BANACH-SPACE-VALUED DISTRIBUTIONS

continuous whenever 0 5 k 5 m is a member of €"(A). Thus, l l ~ ( k ) ( t ) /has l no restriction on its growth as I t ( co. Furthermore, the canonical injection of 9 m ( A )into &"(A) is continuous. I l l . Y m ( A ) . Now, K j = R", I , = R", and -+

tj, ~

t =) (1

+ I t I '1'

for everyj, q, I, and t . Thus, Y m ( A )is degenerate, and Pj,p ( 4 ) = ~

SUPII(~ +

max

p ( 4 = )

Oqkv4min(p,mv)teR"

I I 2>p4'k'(t>IIA.

Here, too, Y m ( A )is Frechet and normal. The members of Y P [ " ' ( A )= Y ( A ) are called A-valued testing functions of rapid descent. Now, l\+(k)(t)l/ tends to zero faster than any negative power of [ t I as 1 t I 00. IV. Y Z d ( A ) . Let c = {c,}:, E R" and d = {d,,}:= E R". Set

n n

Kc,d ( l ) =

Kc,. dy(fv)?

v= 1

where

Also, let K j = R", I , = R", and P j ,p ( 4 )

t i ,, ( t ) = K c , , ( ? ) for allj, q, 1, and t . So, max

= Pc, d, p ( 4 ) =

Ock.crnin(p,m,)

"P 11 ' c ,

d(t>4'k'(t)ll A

'

feR"

The resulting p-type testing-function space 2 ' : d ( A ) is degenerate and therefore Frtchet. It is not normal because g m ( A )is not dense in Y E d ( A ) .This space and the next one arise naturally in the study of the generalized Laplace transformation (Zemanian, 1968a, Chapter 3). The members of these spaces have exponential bounds on their behavior as 1 t I -+ co. where each w, is either a real number V. 9 " ( w , z ; A ) . Let w = { w,,}:' or - co, and let z = {z,};=1, where each z,, is either a real number or 00. Let {cj}T= and {dj}T= be two sequences in R" with the following properties: cj+l < c j and d j + l > dj for every j . Also, upon denoting the components of c j by c j ,,and of dj by d j ,,, where v = 1, . . . , n, assume that, for each v and as j - + co, c j , ,+ w, and d j ,,-+ z,. Now, set K j = R", I, = R", and ti,, ( t ) = 2 tj+l, as is required. ~ , ~ , d , ( t )for every j , p , f , and t . Thus ti,,(?) Moreover P j ,p ( 4 ) = P c j , d,,

p(4)

=

max

S U P l l K c j , d,(t)4'k'(t)llA O < k v < m i n ( p . m,) feR"

Thus, Y m ( w ,z ; A ) is the inductive-limit space not strict. On the other hand, it is normal.

.

ujY ; , d j ( A ) . However, it is

3.7.

GENERALIZED FUNCTIONS

67

VI. 9-"'(A). In this and the next special case, n = 1, so that we will be dealing with functions on the real line R. Also, k is now a nonnegative integer in R , and m is either a nonnegative integer in R or 00. We set K j = ( - co,j ] ,I , = [ -p, a),and ti,I ( t ) = 1 for allj, p , 1, and t . Thus, the functions 4 in 9 - " ( A ) have their supports bounded on the right, and Pj,p(4)

=

max ~ ~ P I I ~ ' ~ ' ( ~ ) I I A *

osksp

rErp

However, l14(k'(t)llhas no restriction on its growth as t + - 00. The space 9-'"(A) is a normal space as well as a strict inductive-limit space having the closure property. VII. 9+"'(A). Here again, n = 1. Also k and m are as in the preceding case. We set K j = [ - j , a), I , = (- 00, p ] , and t,, I ( t ) = 1 for all j , p , 1, and t . The functions in 9 + " ' ( A )have their supports bounded on the left, and

D+"'(A)is both normal and strict and has the closure property. Problem 3.6-1. If I , 3 gi,then p i , is a norm on $,(.A) for every p 2 q. Conversely, if pi, is a norm on $,(A), then 1, 2 g j .Prove these assertions.

,

Problem 3.6-2. Prove that Ij"(A) is complete. Problem 3.6-3. Show that the differentiation operator Dk is a continuous linear mapping of P ' + k ( A )into . F ( A ) . Problem 3.6-4. Show that &"(A), Y " ( A ) , Y"(w, z; A), .9-"'(A), and 9 + " ' ( A )are all normal. Also, show that the shifting operator or is an automorphism on each of these spaces.

3.7. GENERALIZED FUNCTIONS

Given any p-type testing-function space $"'(A), a continuous linear mapping of F ( A ) into B is said to be an [ A ; B]-valuedgeneralizedfunctionon R". As we done with distributions, we will subsequently identify [$"; [ A ; B]] with [$"'(A); B], and thereby [P'; [C; B]] with the space [P; B] of B-valued generalizedfunctions on R".Since $"'(A) contains 9 ( A )and induces a topology on 9 ( A ) weaker than that of 9 ( A ) , the restriction of any f E [$"'(A); B] to 9 ( A ) is a member of [ 9 ( A ) ;B] (Le., is an [ A ; B]-valued distribution on R"). If P ( A ) happens to be normal, [$"'(A); B] can be treated as a subspace of @ ( A ) ; B] by identifying each f~ [$'"(A); B] with its restriction to .9(A).

68

3.

BANACH-SPACE-VALUED DISTRIBUTIONS

This is because the said restriction determines f uniquely on all of $"'(A) by virtue of the density of 9 ( A ) in $"'(A). The topology we will usually employ for [$'"(A); BJ is the topology of uniform convergence on the 6 - s e t s in $"'(A), which we will simply call the 6-topology. This is the topology generated by the collection {ye}ess of seminorms defined by

ydf)

suPll.

(8) We end this section by noting certain support conditions possessed by the members of [&"'(A); B] and [ 9 + " ' ( A ) ;B ] . These members are distributions since both &"'(A) and 9+"'(A)are normal.

Theorem 3.7-3. f E [&"'(A); B] ifand only iff compact set.

E

[9"'(A);B] and supp f is a

PROOF.Given any f E [&"'(A);B ] , there exists a neighborhood A of 0 in €"'(A) such that II(f, 4)llB2 1 for all 4 E A. Upon referring to Case I1 of the preceding section, we see that A contains a set SZ consisting of all 4 E &"'(A) such that max supI14'k)(t)IlAIE , 0 s k y $ m i n ( p , m,) f

EI,

where I , is a compact set. This implies that supp f c I , . Indeed, iff is not equal to the zero distribution on R"\I, , then we can find a.O E 9 ( A ) such that supp 0 c R"\I, and ( f , 0) + 0. But then, M0 E R for all real numbers M , and I I ( f , MO)I/ can be made larger than 1 by choosing M appropriately. This is a contradiction. Conversely, iff E [ W ( A ) ; B ] and supp f is a compact set, then, by virtue of the paragraph just before Example 3.3-1, f has a unique extension onto any 4 E €"'(A) defined by ( f , 4 ) = ( f , A4),where A E 9 is equal to 1 on a neighborhood of suppf. This extension is clearly linear and is continuous because the convergence of {&,,} to 0 in &"'(A) implies the convergence of {h$"} to 0 in 9 " ' ( A ) . 0 A similar argument establishes the following result.

Theorem 3.7-4. . f [ ~ 9 - " ( A ) ; B] (or f E [ 9 + " ' ( A ) ;B ] ) fi and only sf f E [ W ( A ) ;B] and suppf is bounded on the left (or respectively on the right). Problem 3.7-1. Show that, when 9"'(A) is normal, the restriction of any , f ~[Y"'(A);B ] to 9 ( A )uniquely determinesfon all of 9"(A).

Problem 3.7-2. Prove Theorem 3.7-4. Problem 3.7-3. Let .P"'(A) be any one of the spaces &"'(A), 9'"'(A), z ; A ) , &"'(A), and 9+"'(A). Show that the shifting operator is an automorphism on [9"'(A);B]. ~""(MJ,

72

3.

BANACH-SPACE-VALUED DISTRIBUTIONS

Problem 3.7-4. Every continuous A-valued function is a regular A-valued distribution (see Example 3.3-2). The canonical injections of &''(A) into [go; A ] and of 9 ' ( A ) into [ g o ;A ] are continuous. Verify these assertions. Problem 3.7-5. Show that 9 ' ( A ) is a subspace of [Yo; A ] for every p-type testing-function space 9'. Problem 3.7-6. Let % be a continuous linear mapping of 9 ( A ) into [9;B]. Define 5Y.l as a mapping on 9 by

('JJzf, $ ) a

(%(fa), 4 ) ,

(9)

where,f E 9, 4 E 9, and a E A. Show that llJl is a continuous linear mapping of 9 into [9;[A; B]].

uj

uj

Problem 3.7-7. Let Y" = .fj"and f' = f ; be p-type testingfunction spaces consisting of complex-valued functions. Let % be a continuous linear mapping of [Ym;A]" into [f'; B]'. Define a mapping 91 on [Y"; C ] by (9), where now f E [Y"; C ] , 4 E y ,and a E A. Show that 9.Ji is a continuous linear mapping of [Y";C]" into [f'; [A; B]]". Also, show that this result is again valid when the'G-topologies of [Y"; A ] , [ f ' ;B ] , [Y"; C ] , and [ f ' ; [A; B ] ]are replaced by the pointwise topologies. 3.8. L,-TYPE TESTING FUNCTIONS AND DISTRIBUTIONS

The &-type testing functions do not fit the general formulation for the p-type testing functions. They are instead defined as follows. Let p E R be fixed with 1 I p < co. The space QLP(A)is the linear space of all smooth A-valued functions 4 on R" such that, for each integer k E R" with k 2 0,

We set g L P ( C = ) g L PThe . members of g L P ( A are ) said to be L,-type testing functions. Minkowski's inequality shows that y p , k is a seminorm on gLP(A). We assign to gL,(A)the topology generated by {y,, k } k > O . This separates the space because, if y,, ,,(4) = 0, it follows from the nonnegativity and continuity of III$(.)II that 4(t) = 0 for every t. Thus, 9,,(A) is a metrizable space. The shifting operator 0,: 4 ( t ) w $ ( t- r ) is an automorphism on gLP(A), and differentiation is a continuous linear mapping of QLP(A)into 9,,(A). The space gLP(A) is a subspace of the space L,(A) of all (equivalence classes of) A-valued Bochner-integrable functions with respect to Lebesgue measure on the Bore1 subsets of R" (see Appendix G19). Moreover, the canonical injection of 9&4) into L,(A) is continuous.

3.8. L p - TESTING ~ ~ FUNCTIONS ~ ~

A N D DISTRIBUTIONS

73

In turn, &(A) becomes a subspace of [9; A ] when everyf E L J A ) is identified with a mapping g of 9 into A through the equation

Indeed, g is clearly linear on 9. Moreover, for p > 1 and q = p / ( p - l ) , we have from Holder's inequality (Appendix G20) that

for all

4 E Q K , and hence g E [Q; A ] . When p

=

1 , we may write

for all q5 E QK and thereby conclude once again that g E [9;A ] . These estiA ] is continmates also show that the canonical injection of &(A) into [9; uous. We now develop some inequalities that we shall subsequently need. Let 1 + denote the n-dimensional unit-stepfunction defined by I+(?) = l + ( t l ) ... l + ( t n ) , where in the right-hand side 1 + is the unit-step function on the real line (see Section 3.4). As usual, D[ll 4 dl a,, denotes the differential operator of order [l]. Also, let 4 E €["(A) and let A E 9 be such that A(t) = 1 for It I < 1 and A(?) =,0 for I t I > 2. For any fixed t E R", we may write

A(?

- x ) ~ ( x )= ( - 1)"

Upon setting f

=x

JR" 1 +(T)D:''[A(~- x +

T)&X

- T)] dT.

and then estimating the result, we get

where M is a constant and E = { T : I t - T I < 2). More generally, if k is a nonnegative integer in R" and if q5 E gk+[ll(A), then

Also, upon applying Holder's inequality (Appendix G20), we obtain, for I,v(t)) = 2 uj(t)vjo,

where (., .) is the inner product for H . Actually, u is a fictitious quantity, which is uniquely related to Fa. Indeed, given any Fa = u j f , , one need merely replace.6 by e j to obtain u. Upon borrowing some concepts from electrical network theory, we can consider u to be an H-valued voltage signal and u an H-valued current signal on the system of Figure 4.2-1. Thus, 91 can be taken to be an admittance operator. Were H n-dimensional Euclidean space, our system would be called an n-port by network theorists. In analogy to this, we shall call the present system a Hilbert port.

1

4.2.

SYSTEMS AND OPERATORS

79

We can view our system in a somewhat different way to obtain a closer analogy to the electrical n-port. We exploit the isomorphism between H and the Hilbert space e,; that is, we identify u with its sequence {uj} of Fourier coefficients and u with {uj}.The coefficient u j is considered to be the voltage signal at thejth port of an electrical system, and u i the current at that port, as indicated in Figure 4.2-2. The polarity of uj andthe direction of uj are so chosen that uj(t)vj(t) denotes the instantaneous power entering the jth port. Thus, we have arrived at an n-port, where now n = co. One might call this an co-port (Zemanian, 1972, Section 8). This ends our discussion of Example 4.2-1. 0

co-port

I I

I Figure 4.2-2

It is only when considerations of power flow arise that an inner product is needed for the space in which the signals at hand take their values. In this book, this will occur only when we impose the assumption of passivity, which states in effect that the system does not contain energy sources that can transmit energy to the exterior of the system. Thus, in the absence of passivity, we can and shall assume that the signals take their values in Banach spaces. In general, therefore, our analyses will involve signals that are Banachspace-valued functions or distributions. We shall refer to a system having such signals as a Banach system. It should also be pointed out that it is usually

80

4. KERNEL OPERATORS

to transient phenomena that the realizability theories of this book are applied. This means that time is the independent variable for the signals, which are therefore functions or distributions on the real line R. However, there do occur physical phenomena involving signals on R”that are amenable to some of the subsequent theories. An example of this is the optical system discussed by Meidan (1970). For this reason, we shall allow, at least initially, signals defined on R”. A Banach system may have many different Banach spaces associated with it. For example, the signal u representing some physical variable at one location x within the system may be an A-valued distribution, whereas the signal u for another physical variable at a different location y may be a B-valued distribution. Moreover, the system defines the relation %: U H U . % need not be an operator; that is, more than one u may be assigned to some particular u. (An example of this is the ideal transformer of electrical network theory when u is taken as the current vector and u as the voltage vector.) Howeuer, a basic assumption imposed throughout this book is that every relation with which we shall be concerned is truly an operator. Furthermore, a Banach system may define many different relations depending on the choices of the locations x and y and the physical variables. The term ‘‘ Banach system ” refers to the entire system and not to any particular relation generated by it. Moreover, it is possible for certain operators generated by a given Banach system to exhibit various properties such as linearity, time invariance, and passivity, while other operators generated by the same system do not. For this reason, the postulates in our subsequent realizability theory will be imposed on particular operators and not on the system as a whole. Indeed, it is such operators and not entire systems that comprise the main concern of this book. Let us now define the concept of a “ Hilbert port.” Assume that in a given Banach system we have singled out two physical variables u and u that are complementary in the following sense: When both u and u are ordinary functions taking their values in a (not necessarily separable) complex Hilbert space H, the inner product (u(t), u(t)) represents the net complex power entering the Banach system at the instant t . Then, the Banach system with these two variables so singled out is called a Hilbert port. We shall borrow some terminology from electrical network theory by referring to the relation %: U H U , when it is truly an operator, as the admittance operator ofthe Hilbertport, and to the relation 2B:$(u + U)H+(U - u), when it too is an operator, as the scattering operator of the Hilbert port. This agrees with the usual terminology for electrical n-ports when u is identified with the voltage vector and u with the current vector. For the scattering operator, certain normalizations of u and u are also implied by this (Carlin and Giordano, 1964, p. 225). It is the admittance and scattering operators with which we shall be concerned when the passivity hypothesis is imposed.

4.3. THE SPACE 2 = g(Y)

81

Still another operator we could consider is the impedance operator 3 :U H U , where again u is current and v is voltage. However, everything we shall say about ' i l lcan be applied equally well to 3 by interchanging the roles of u and u.

4.3.

THE SPACE 2 = 9 ( Y )

Schwartz's kernel theorem (Schwartz, 1957, p. 93) characterizes separately x g R Sinto C in terms of complexcontinuous bilinear mappings of gRn valued distributions on R" x R". There now exist a number of alternative proofs for it (Bogdanowicz, 1961 ; Ehrenpreis, 1956; Gask, 1960; Gelfand and Vilenkin, 1964, Section 1.3). For our purposes, we need an extension of this theorem to separately continuous bilinear mappings of g R nx gR.(A) into B. Actually, Bogdanowicz's proof establishes the kernel theorem for mappings of gRnx 9 R s ( A ) into C, and with some obvious modifications, we can replace C by B. His proof is the subject of this and the next section. The present section is devoted to a generalization of the space g R , ( A ) resulting from the replacement of A by a more general type of space V . In the following, we let Y be the strict inductive limit of a sequence {Yj}y=of FrCchet spaces. Since every FrCchet space is separated and complete, we can conclude that Y possesses the closure property (Appendix E4). For each j , we let Z j denote a sequence { [ j , y } ~ = oof seminorms that generates the topology of Y j. We can always choose the multinorm Z j such I(,,z I . this we do. As a consequence, a base of that [ j , o I neighborhoods of 0 in Y jconsists of all sets of the form

cj,,

* a ;

{v

E Y j :5 j , q ( ~ ) < E } ,

where E E R + and q are arbitrary. Now, let { K j ) y = l be a sequence of compact intervals in R" such that K j c Rj+l for every j and K j = R". We let 2 4 gRn(Y) denote the linear space of all smooth Y-valued functions on R" having compact supports. It follows from Theorem 1.3-1 that, for any h E Z , h(R") is a bounded subset of Y . Consequently, according to Appendix E4, h(R") c V j for some j depending on h. We now let Z j A gK,(Yj) be the linear space of all h E 2 such that h(R") c V j and supp h c K j . Thus, Z j c X j + , for every j , and Z=UZj. Fix j and consider Z j . For any h E Z j and any nonnegative integer p E R", h ( p ) is a continuous function from R" into V and its range is contained in " Y j . Since the topology of Y j is identical to the topology induced on "Yj by "Y, h ( p ) is also continuous from R" into V j .This means that, for any

u

82

4. KERNEL OPERATORS

given 5 E Z j and p , we can define a finite-valued functional xp, on X i by means of x p , c ( h )Li sup [[h'P'(t)], h e S j . 1E

R"

Each xp, is a seminorm on S j .We equip i?Vj with the topology generated by the collection Ti4 {xp, H DxP&t, x) is a uniformly continuous function on R" x R" and has a compact support therein. Moreover, (Dkh)(t)= D,"r$(t, .). So, the same considerations show that h is a smooth function from R" into g R S ( A ) .Clearly, supph is bounded. Thus, to each 4 E g R n + . ( Athere ) corresponds a unique h E &'. Conversely, let EX be given. We define an A-valued function 4 on R"+"by setting 4(?,x) = [h(r)](x).The function 4 has a compact support in R"+' because h has a compact support in R" and the range of h is contained in some D,,(A). That 4 is smooth can be shown in the following way. The fact that h is continuous from R" into g R , ( A )implies that, for any nonnegative integer k E R", for a fixed t E R", and as At + 0 in R", D,"#(t + At, x) tends to D,k$(r, x) uniformly for all x E R". But D,"$(r, x) is a continuous function of x for each fixed t . We can conclude therefore that D,k4(t, x) is a continuous function of { t , x} whatever be k .

4.4.

85

THE KERNEL THEOREM

Next, let di 4 d,, denote differentiation with respect to the ith component of r. By setting up the incremental definition of d i h and taking the limit, we see that ai$(t, x ) = [dih(t)](x).After proceeding exactly as in the preceding paragraph, we can conclude that D,k a,, 4 is a continuous function on Rn+S , whatever be k. Moreover, Theorem 1.6-2 implies that the order of differentiation in DXka,, 4 can be changed in any fashion without altering the result. These arguments can be applied to all the derivatives of 4, which leads to the conclusion that 4 is a smooth A-valued function on R"". Since supp is bounded, 4 E 9Rn+s(A). We have so far shown that the equation ti

h(4 = 4(t, * > (8) sets up a bijection from X onto aRn+.(A).Clearly, this bijection and its inverse are linear. On the other hand, we have from (7) that Xp,c,(h)= max SUP lI4%Wt, x>llA. Osks[ql

f E

R";

XE

R'

This shows that, given the compact intervals Kjc R" and L j c R', the bijecConsequently, ,. tion defined by (8) is an isomorphism from X j onto g K J Y Z the bijection is also an isomorphism from 2f Onto gRn+s(A). Problem 4.3-1. Prove (1). Problem 4.3-2. Show that X possesses the closure property; i.e., for each j , Z j is a closed subspace of 2fj+l. 4.4. THE KERNEL THEOREM

We continue to use the notation defined in the last section. Moreover, we can let { p l } z 0 ,where Pl(4)

max

SUP

Osk 0 and Consider any h E S ; h will be in let M , and x p , c be as in Lemma 4.4-1. By virtue of Lemma 4.3-2, we can select an ho of the form (4) in Section 4.3 such that

llZi(h) - 2dh)II 5 llZi(h - h0)ll

+ II2i(h0) - 2dh0)ll + II2dh0 - h)ll*

For all i and I, the first and third terms on the right-hand side are both bounded by .c1/3 by virtue of Lemma 4.4-1 and (8). Moreover, there exists a constant N > 0 such that, for all i , 1 N , the second term is bounded by ~ ~ because 1 3 of (6). Hence, by the completeness of B, 2 , ( h ) converges to a limit, say 2(h). This defines a mapping 2 from Sj - into B, which is linear since each 2,is linear. Moreover,

=-

11 2(h) 11 B 5

[(h) for all h E X?j-l.But, Lemma 4.3-1 implies that x p , { is a continuous seminorm on S j- Thus, 2 is continuous on every .#',- 1. This being true for every j , we have that 2 E [&; B ] . xp.

4. KERNEL OPERATORS

88

Equation (6) now shows that (1) holds true. Clearly, given 2,331is uniquely determined by (1). On the other hand, Lemma 4.3-2 coupled with (1) of Section 4.3 states in effect that the set 0 of all elements of the form $ v , where 4 E 9 and u E V ,is total in 2.Hence, any member of [&; B] that coincides with 2 on 0 must be identical to 2 on &. This completes the proof of Theorem 4.4-1 except for the proof of Lemma 4.4-1. In order to establish that, we shall need still another lemma.

Lemma 4.4-2. Given any nonnegative integer p valued function ri on R" by

E R",

define the complex-

where qi is thefunction definedin theprecedingproof. L e t j > 1 andlet $ E b e s u c h t h a t $ ( t ) = 1 on K j + l . Then, f o r a l l i > j a n d a l l h E & j - l ,

PROOF.For any therefore

4 E g K ,and

(qi

i

gKj+2

>j , we have that supp q i * 4 c Kj+l and

* 4)(t)= JR" qi(t - T M T )

d~

= /Rn$(t)ri(f - T ) D ~ + " ] ~ (dr. T)

Here, we have used repeated integration by parts. Consider the function T H $(*)ri(*- T ) D ~ + [ ' ] + ( T ) .

It is smooth on R" with values in gKj+2, and its support is contained in K j . By appealing to (9,we may write, for any 4 E g K 1any , u E V j ,and all i>j,

zi(40)= m(qi *

4 9

0)

= lK1%l($(*)ri(. - T),

Dp+'*'4(T)U)dT.

In the last step, we have used Note I1 of Section 1.4 and the separate continuity and bilinearity of +m.

4.5. KERNEL OPERATORS

With i > j still, define the linear operator

89

lion X i by

k i ( h ) 4 J" 'iUl(+(-)ri(. - T), D P + [ l l h ( t ) )ds,

h EX j.

&

(10)

KI

Thus, from (2), we have where s = p + [I], o = [ j , V for some integer v, and N is a constant. This shows that % i is continuous. Obviously, 2, and 2 , coincide on the set { $ u : 4 E g K Iu, E V j }and therefore on its span S. Now, let h E X i - , and ho E S. We may write

II2i(h) - %i(h)IIB 5 IIXi(h - h0)II

+ Il%i(ho - hll.

Upon referring to ( I ) of Section 4.3, to (4) and ( I l), and to Lemma 4.3-2, we see that 2,and %i coincide on X j - Finally, note that the interval K j of integration in (10) may be replaced by K j - when h E X j . This proves (9). 0

-,

PROOF OF LEMMA 4.4-1. We may apply the estimate (2) (with g K j +replaced by g K j fand 2 V iby V j- I ) to (9), where p is chosen equal to [I]. This yields, for all i > j ,

where s = p + [I ] and p = ( j - l ,4. It is not difficult to show that the quantity within the braces is bounded by a constant not depending on i. (This is a result of the fact that diam supp q i is bounded for all i.) Thus, { 2 i } i ,isj an Moreover, for 0 5 i S j , each 2,is equicontinuous set of mappings on Consequently, {Z,},'is ",also equicontinuous on continuous on & ' j - l . X j - I . 0 4.5. KERNEL OPERATORS

Theorem 4.4-1 provides a characterization of the continuous linear operators from g R S ( A )into [ 5 B R , , ; B]"; this is the content of the present section. We start with a special case of Theorem 4.4-1.

Theorem 4.5-1. Corresporzding to every separately continuous bilinear mapping 911 of 9 R t l x g R s ( A ) info B there exists one and only one distribution f~ [ Q R n f S ( A )B; ] such that

9x(4>u,

= ( f ( t %x),

4(r)v(x>>, 4 E g R n

9

QRS(A).

(1)

4. KERNEL OPERATORS

90

PROOF.We choose -Y- = gRs(A), as was done in Example 4.3-1, and then employ (8) of Section 4.3 to set up an isomorphism 3 from &' = 9 R n ( 9 R s ( A ) ) onto g,,+.(A). This induces a bijection from [ Z ;B ] onto [ g R n + = ( A B ) ;] by means of the equation

2(44 = ( f ( h x > , 4 ( t ) v ( x ) > , where 2 E [&'; B ] . (See Figure 4.5-1). But, according to Theorem 4.4-1, every = 2(&v).

9-N can be identified with one and only one such 2 by setting !Ill(&, v) Thus, this theorem is established. 0

Figure 4.5-1

We shall use the right-hand side of ( I ) to define an operatorf., which we call a kernel operator or alternatively a composition operator. GivenfE [ 9 R n + s ( A;) B ] and any v E g R S ( A )we , define the composition productf. v as a mapping on by all 9 E gR,,

(f. 0,4) A

( f ( t , x), 4(t)v(x>>,

t E R",

x E R".

(2)

Thus, f . v maps g R ninto B linearly and continuously. Hence, the kernel operatorf: v ~ f maps . ~g R S ( A )into [ g R m B ];. Clearly, f. is linear. To show its continuity, let @ be a bounded set in g R nThere . exists a compact interval K in R" such that @ c g K .Also, let J be any compact interval in R'. Then, for all v E g J ( A ) ,the mapping{t, x} H 4(t)v(x) is a member of g Kx J ( A ) .Therefore, there exists a constant Q and a nonnegative integer r = { r , , r 2 } E R"" such that

s u ~ l l ( f .v, 4>11 5 SUP Q max su~lI~~C4(t)4x)lll. +E@

O s k s r r,x

4.5.

91

KERNEL OPERATORS

But the right-hand side is bounded by

P max sup((Dk2u(x)(( = f'pr,(u), OSkZ4r2

x

where P does not depend on u. Thus, f.is continuous on Q,(A) for every J . This implies that f.is continuous on g R I ( A ) . We summarize these results as follows.

Theorem 4.5-2. For any given f E [ 9 R n + . ( A ) ; B], the kernel operatorf. is a continuous linear mapping of 9 , , ( A ) into L[a;, B]. Our next objective is to develop a converse to Theorem 4.5-2. We first state a lemnia whose proof is quite straightforward.

Lemma 4.5-1. Let %be a continuous linear mapping of QRs(A) into [Q,"; Define 9'R . from '3 by 9)31(#, u)

'

ll".

Condition E3 will be verified once we show that, as 4 traverses @ and t traverses R, (1 -!- f2)14(j)(t+ traverses an 6-set in U ( w , z). To prove the latter, let w < c < 0 < d < z, so that Y c .Yc,d.By Liebniz's rule for the differentiation of a product and by the argument following (9), we have that, for every k, a )

S U P IK,,d(X)D,k[(l xcR

+ t 2 )I 4 ( i ()t + x)ll

is uniformly bounded for all t E R and all 4 €0, which is what we want. The next theorem has hereby been established.

Theorem 5.3-3. If y E [ Y ( A ) ;B ] , then v w y * u is a continuous linear B ] when M J < 0 < z. mapping of [U(w,z ) ; A]" into [Y; Problem 5.3-1. Prove Lemma 5.3-2. Problem 5.3-2. Show that, if y E [ & ( A ) ;B ] , then linear mapping of [Q; A ] into [Q; B ] .

u

UHY

* v is

a continuous

Problem 5.3-3. Show that, if y E [ U ( w , z ; A ) ; B ] , where w < z, then ~ * uy is a continuous linear mapping of [U(M', z ) ; AIs into [U(w,z ) ; B]".

5.4. THE COMMUTATIVITY OF CONVOLUTION OPERATORS WITH SHIFTING AND DIFFERENTIATION

In each of the three cases considered in the preceding section, the convolution operator y * commutes with the shifting operator and differentiation. For instance, in case I, we have the following result.

5.5. REGULARIZATION

Theorem 5.4-1. If y integer in R", then

E

105

[ 9 ( A ) ;B ] , L' E [b;A ] , t E R",and k is a nonnegative

and

in the sense of equality in [9;B ] . PROOF.Since 6, H or6, is an isomorphism on 9, we may write = (At),( ~ ( x )6,(t , (ar(y * u),6,) = (Y * 0, = (Y(t>, (v(x - z), 6,(t + 4)) = (Y

+ x + ?>>) * (arv), 6,).

This establishes (1). A similar argument with c, replaced by Dkestablishes (2). 0

is an important fact. As was indicated in the That y * commutes with introduction, it means that y * is a translation-invariant operator. Moreover, 9 ( A ) can be identified as a subspace of [d;A ] in accordance with Example 3.3-2, and the canonical injection of 9 ( A ) into [ b ;A ] is continuous. Consequently, the restriction of y * to 9 ( A ) is a translation-invariant kernel operator. A major objective of this chapter is to demonstrate the converse; namely, every translation-invariant kernel operator is a convolution operator. For Cases I1 and 111 of the preceding section, we have the next theorem. Its proof is quite similar to that of Theorem 5.4-1. Theorem 5.4-2. Let t E R and let k be a nonnegative integer in R . Then, (1) and ( 2 ) are equalities in [ X ;B ] under either one of the following conditions: (i) y E [ 9 - ( A ) ; B ] , v E [9-; A ] , X = 9- . (ii) y E [ Y ( A ) ;B ] , c E [ Z ( w , z ) ; A ] , where 11'

-= 0 < z, X = 9'.

Problem 5.41. Establish Theorem 5.4-2. Do the same for the convolutions of Problems 5.3-2 and 5.3-3.

5.5. REGULARIZATION

When the convolution operator y * , where y E [ 9 ( A ) ;B ] ,is applied to any * u is a smooth B-valued function. This is called the regularization of y by v, a process we shall now investigate. u E 9 ( A ) , the result y

5. CONVOLUTION OPERATORS

106

Lemma 5.5-1. Let y

E

[ 9 ( A ) ;B ] and u E 9 ( A ) . Set 4 ( Y ( t ) , u(x - 0 ) .

(1)

Then, UH u is a continuous linear mapping of 9 ( A ) into d ( B ) . PROOF.Clearly, u maps R" into B. The argument that was applied to (2) of Section 5.3 can again be used to show that u is smooth [i.e., u E b(B)]and that U'k'(X)

= ( y ( t ) ,u ( k ) ( X - t ) )

(2)

for every nonnegative integer k E R". To show the continuity of the linear mapping D W U , let K and N be arbitrary compact sets in R".Then, { Z ~X

.): x E K , u E 9 , ( A ) } c 9 J ( A ) ,

where J is some other compact set in R". Since the restriction of y to g J ( A )is continuous and linear, S U P l I U ( k ) ( X ) l I B = SUP XEK

II,

which is what we wished to establish. Finally, we demonstrate that any primitive g off differs from the primitive f(-” defined by (2) by a constant member of [9;A ] . For any $ E 9,

So, upon applying g to the decomposition (1) of any r-0

= lim 0 = 0.

Now, one primitive of zero is zero. Since g is also a primitive of zero, it must differ from zero by a constant member of [9;A ] . In other words, (3) holds. Obviously, there cannot be two different values for a satisfying (3) for all 4. 0 Problem 5.6-1. Prove thatf(-’) as defined by (2) is a member of [9;A ] .

5.

110

CONVOLUTION OPERATORS

5.7. DIRECT PRODUCTS

In this section, t E R" and x E R". Moreover, 4(t, x ) will now denote a function on R"" (and not the value of 4 at { t , x } , which is our usual interpretation). Let f E [ g R n A ;] and g E [ g R S C]. ; We define the direct product f ( t ) x g(x) as a mapping on any 4(t, x ) E g R n by fS

( f ( 0 x g(x), 4(4 4 )A! (fW,( g ( x ) , 4(4 4 ) ) .

(1)

The right-hand side has a meaning since, as a function o f t , ( g ( x ) , 4(t, x ) is a member of gR,, . This can be shown through the same argument as that used in regard to (2)-(4) of Section 5.3. That argument also establishes the following equation : Qk(g(x>, 4(4 X I > = ( g ( x ) , Q W t ,

XI>.

(2)

It follows from (1) that f ( t ) x g(x) is a linear mapping of gRnfS into A . Moreover, a direct estimate of the right-hand side of ( I ) with the use of (2) shows that f ( t ) x g(x) is continuous on .CBJ for every compact set J c R"". Thus, f ( ? > g(x) i 9 R " f S ; A useful fact is the following. The restriction off(t) x g(x) to the set SZ of all testing functions of the form e(t)lC/(x),where 0 E g R n and II/ E g R Suniquely , determinesf(t) x g(x) on all of g R 8 , + This * . is because SZ is total in gfln+*. (See Schwartz, 1966, pp. 108-109.) For this restriction, we have

(f(t>x g(xh W)lC/(X)) = (5 e x g , II/>. 5.8. DISTRIBUTIONS THAT ARE INDEPENDENT OF CERTAIN COORDINATES

Let j

E [9R"+";

t = {tl,

., t n > E R",

* *

r] = {Vl,

{t,I'}

=

A],

* * * 9

?I, E> R",

{t1 , . . ., t n v . . ., ~

We say that j is independent of

9

r]

19

s E >

Rn+"*

if, for every z E R"" with

Z = (0,.

. . , 0, T I , . . . , Zs},

we have that o r j = j [that is, if j ( 5 , r]) is independent of shifts through the r] coordinates]. It will now be shown that such a distribution can be written as

5.9.

A CHANGE-OF-VARIABLE FORMULA

111

the direct product y ( t ) x l(q), where y(t) denotes a member of [9,,; A ] and l(q) denotes the regular distribution corresponding to the function that equals 1 everywhere on R". Let = e(tl, . . . t., ql,. . . , q s - l ) m s ) ,

m,

where 8 E 9 R n + s - 1 and + e E R 1Now, . j is independent of qs. Therefore, $ H ( j , 8 $ ) is a member of [aRl ; A ] ,which is also independent of qs . We may therefore invoke Lemma 5.6-1 to write (j9

ell/> = a(@

I*(V")

4% = a(w1(%),

$(qs))7

where a(8) is a member of A depending on 0 but not on $. Upon fixing $ such that J $(qs) dqs = 1, we see that a(8) = ( q , e), where q E [QR,,+$- I ; A ] is uniquely determined by j . In view of the last paragraph of the preceding section, we can conclude that j is equal to the following direct product: j ( 5 , '1) = q(t1,.

. * , t",q1, . . ., V s - 1 )

x I(%).

Next, we observe that, since j is independent of shifts through qs-l, so, too, is q. Indeed, let ur be a shift through the coordinate qs-l only and let 8 and $ be as above. Then, ( 4 , e>(l, $) = ( j , ell/> = ( U r j ,

e$> = (q, a-re>(l, $)*

Since this holds for all such 8 and I), q = u,q. Therefore, by applying the argument of the preceding paragraph to q, we see that there exists a unique p E [ 9 R n + = - 2 ; A ] such that i(t,?)=P(t1,...,t,,11,...,rls-2) x 1(qs-A x l(qJ

We may also write l ( ~ " - ~x ) l(qJ = l ( ~ " - ~q,). , Continuing in this way, we arrive at the following result.

Theorem 5.8-1. Let j E [ E R n +A=];be independent of q . Then, there exists a unique y E [ 9 R n ; A ] such that

At, 4 = Y ( t ) x

w

5.9. A CHANGE-OF-VARIABLE FORMULA

In this section, 9 = QR,, . Let z, 5 E R" and set z = U c , where U is a nonsingular linear transformation on R". Thus, U can be represented by a nonsingular n x n matrix of real numbers and has an inverse U - '. Moreover, IU I = I U 1, where I U I denotes the magnitude of the determinant of U.

-'

-'

5. CONVOLUTION OPERATORS

112

Given any f E [9; A ] , we let f ( U [ ) denote the linear mapping from 9 into A defined by ( f ( 4 ,I u I -Wu -'z>>, 0 E 9. (1) f(UC) is continuous. Indeed, if the sequence (0,) tends to zero in D, then, clearly, { 1 U I -'B,(U -'z)} tends to zero uniformly for all z E R" and the supports of these functions remain contained within a fixed compact set. Moreover, the chain rule for differentiation (Kaplan, 1952, p. 86) shows that the same is true for each fixed-order differentiation of these functions. Thus, I Ul -'0,(U-'z) + O in 9 as v 3 oc). The continuity of f ( U [ ) now follows A]. from (1). We conclude that f(UC) E [9; An alternative form for (1) can be obtained by setting $(z) = O(V-'z):

= ,I u I$(UC)>.

(2)

5.10. CONVOLUTION OPERATORS

We are at last ready to establish the condition (namely, translation invariance) under which a kernel operator from 9 ( A ) into [9; B ] becomes a convolution operator. Once again, it is understood that 9 = BR7and 9 ( A ) = 9 R 4 4

Dejinition 5.10-1. Let S and I be spaces of functions or distributions on R" and assume that S and I are closed under the shifting operator 0,. A mapping % from 5F into I is called translation invariant if u,%f = %u, f for every f E S and every z E R". In certain subsequent discussions, we shall call the members of a given family of operators translation varying to indicate that the condition of translation invariance is not imposed even though certain members of that family may be translation invariant. Thus, translation-invariant operators are considered to be a special case of translation-varying operators. When n = 1 and the space R' on which the members of S and I are given is interpreted as the time axis, the adjective translation-invariant is commonly replaced by time-invariant and translation-varying by time-varying. Assume now that % is a continuous linear mapping of 9 ( A ) into [9; B]". By Theorem 4.5-3, % is a kernel operator; that is, 'ill=f on 9 ( A ) , where f E [9RIm(A); B ] . If, in addition, % is translation invariant, we may write for every 4 E gYv E 9 ( A ) , and z E R",

< f ( 4 XI, 4(t + Mx))= <mu, a-74) = II

IM sup

sup

OSkSr t t

- 1/ICI

I e-"'Dk[A( 1 C 1 t)edC']I.

(2)

6.

122

THE LAPLACE TRANSFORMATION

Note that the right-hand side is independent of d and that (2) holds whenever Re [ > u. Moreover, remains bounded for all t and [ such that Re [ > cr > 0 and t 2 - 1/ I [ 1. As a result, the right-hand side of (2) is bounded by a polynomial in 1 [ I. Sufficiency. We first establish a lemma concerning classical Laplace transforms.

Lemma 6.5-1. Assume that, on the halfplane C,, A {[ E C: Re [ > cr}, G is an [ A ; B]-valued analytic function and

IIG(C)II

MI51- 2 ,

(3)

where M is a constant. Sef

I

00

g(r)

(1/2n)

-m

G(c + iw)e(c+iw)t do,

c

> u.

(4)

Then, g is a continuous [ A ;B]-valuedfunction for all t and is independent of the choice of c > u. Moreover, g(t) = 0 for t < 0 , and m

.

g(t)e-5' dt,

G([)=

[ E C,.

0

Infact, G is the Laplace transform of the regular distribution f generated by g.

E [9(u,

co ;A ) ;B ]

PROOF.That g does not depend on the choice of c > u follows from (3) and Cauchy's theorem (Theorem 1.8-1). Its continuity follows from the continuity of tHe-ctg(t), which in turn can be shown by writing

1 I 2n -

"

1

-m

IIG(c

+ iw)ll

and noting that the right-hand side tends to zero as x -P 0. Similarly, it can be seen that e-"g(t) is bounded on the domain { { t ,c}: t E R, c > o}. This implies that g(t) = 0 for t < 0. Indeed, if g(t) # 0 at some f < 0, then e - c t ~ ~ g (can t ) ~be ~ made arbitrarily large by choosing c large enough. We now observe that o ~ G ( + c iw) is a smooth [ A ;B]-valued function. Hence, by virtue of (3), the standard proof for the Fourier-inversion formula (see, for example, Zemanian, 1965, Section 7.2) can be applied to obtain

6.5. A CAUSALITY CRITERION

123

(5) from (4).Finally, g generates a regular distribution f E [Y(a, co ;A); B] by means of the definition

jm s(tW(t>dt,

(f,4 )

-m

4 E %T

00 ;

4

because, for any a, b E R with a < a < b < 00, we can choose c with a < c < a and then write, for all 4 E Y,,,b(A),

IIU, 4)llB

)I

= 1000e-'fg(t)e'c-a)feaf4(t) d t

/I

1 e(c-a)'dt. 00

Isup Ile-cfg(t)ll sup Il&,,b(t)4(t)ll t

f

0

Thus, f E [Y,,, ,,(A) ; B ] , and therefore f E [&'(a, co ;A); B]. Hence, (5) is equivalent to G ( [ ) = (i?f)([) for [ E C, . Our lemma is established. The sufficiency part of Theorem 6.5-1now follows readily. By the condition (l), there exists a positive integer m such that G([) 4 Y([)/["'satisfies the hypothesis of the lemma. Therefore, G = !i?Jwhere f E [L?(o, co ;A ) ; B ] . According to Problem 6.3-2, Y([)= ("G([) = (i?f ("))([) for [ E C, . We conclude by noting that c suppf = supp g c [O,

a).0 Since the condition supp y c [0, co) is equivalent to the causality of y * (Theorem 5.1 1-1), the last theorem provides the following causality criterion promised in the title of this section. suppf

Corollary 6.5-la. Let y E [ 9 ( A ) ;B ] and Y([)= (i?y)([)for [ E ay.The convolution operator y * is causal on 9 ( A ) if and only i f there exists a havplane {( E C : Re [ > a} on which Y is analytic and 11 Y([)II is bounded by a polynomial in I [ I. In this case, y is causal on [9; A ] as well. For the last sentence, see Theorem 5.11-2.

Chapter 7

The Scattering Formulism

7.1. INTRODUCTION

So far, we have seen that the linearity and continuity of an operator W from 9(H) into [Q; HI is equivalent to a kernel representation for W (Theorems 4.5-2and 4.5-3),that W is in addition translation invariant if and only if it has a convolution representation (Theorem 5.10-1),and that the causality of W is characterized by a support condition on the kernel or unit-impulse response of these representations (Theorems 4.6-1 and 5.11-1). However, we have not as yet developed any results arising from energy considerations. This is our last objective. A Hilbert space is the natural framework in which to examine questions concerning energy and power flow, and consequently we will be concerned henceforth with Hilbert ports. The net energy e(Z) absorbed by a Hilbert port over some time interval I c R is described in two distinct ways, de124

7.2. L

P -DISTRIBUTIONS ~ ~ ~ ~

125

pending on whether the scattering formulism or the admittance formulism is chosen. In the former case,

where 2B is the scattering operator for the Hilbert port. In the latter case, e ( l ) = Re

J’I (%u(t), u(t)) dt,

(2)

where % is the admittance operator. If the Hilbert port has no energy sources within it, it cannot impart more energy to its surroundings during the time interval IT { t : - co < t T}, where T I co, than it has received, and thus, e(lT)2 0. This property is called passivity. However, one should (and we will) distinguish between the passivity conditions arising from (1) and (2) because they affect the representations for 2B and % in very different ways. Also, the cases where T = 03 and Tis finite but arbitrary will also be treated separately, the former one being a weaker assumption than the latter. We will take up the scattering formulism in this chapter and the admittance formulism in the next. Passivity is a strong assumption. For example, linearity and passivity imply causality and continuity. This is fairly obvious in the scattering formulism (see Section 7.3) but is by no means obvious in the admittance formulism (see Sections 8.2 and 8.3). Similarly, linearity, continuity, time invariance, and causality do not ensure that the unit-impulse response g of the operator at hand is Laplace-transformable. However, g is indeed Laplace-transformable when the assumption of passivity is added. Much of the subsequent discussion will be directed toward the Laplace transform of g and will result in the so-called frequency-domain formulation for our realizability theory. Throughout our discussion of Hilbert ports, we adopt the natural assumption that the signals at hand are functions or distributions on the real time axis. Thus, in this and the next chapter, t E R and 9 = Q R 1 .

-=

7.2. PRELIMINARY CONSIDERATIONS CONCERNING L,-TYPE DISTRIBUTIONS

We start by establishing certain properties of the distributions in [ 9 , , ( A ) ; B] and [ 9 , * ( A ) ; B]. These two spaces were discussed in Section 3.8. Lemma 7.2-1. I f f € [ g L 1 ( A )B] ; and if s u p p f c [0, co), t h e n f e [ Y ( O , co; A ) ; BI.

7. THE SCATTERING FORMULISM

126

Note. According to Lemma 3.8-3 [ g L 2 ( A )B; ] c [ g L , ( A ) ;B ] , and hence this lemma also holds for all f E [9&l); BI.

PROOF.Let c, d E R with 0 c c < d c co.Let 4 E Zc, d ( A ) . Finally, let 1 be a smooth, real-valued function on R such that A(t) = 0 for - 00 c t < - 1 and A(t) = 1 for c t < 03. Then, A+ egL1(A). Indeed

-+

and the right-hand side is finite because Il&')(t)(j I Ne-" on - 1 c t < co for some constant N . As was indicated in Section 3.3, (f, 4) depends only on the values that 4 assumes on some arbitrarily small neighborhood of suppf. Therefore, we can extend the definition off onto Z c , d(A) by means of the equation (f, 4)

n4>,

= (f,

4

sc,d(A)*

Sincef E [ g , , ( A ) ; B ] ,there exists a constant M > 0 and a nonnegative integer r such that

In view of (I), the right-hand side is bounded by

Hence,f E [.Yc,d ( A ) ; B ] .Since this is so for every c, dsuch that 0 < c c d < co, we have f E [Z(O,co ;A ) ; B ] . 0

Lemma 7.2-2. Let f E [ 9 ( A ) ;B ] andp E R with 1 < p c co. Zf

:/

Il(f*

$)(t)llBP

dt

W

for every $ E 9 ( A ) , then f E [QL,(A); B ] , where q = p / ( p - 1). PROOF.Set

4

Let $ E 9 ( A ) and 4 E 8. In the following, $ ( t ) 4 $(-t). Then, f * E &([A;B ] ) and f * $ E b ( B ) according to Theorems 3.5-1 and 5.5-1. Moreover, = ,

7.2. L p - DISTRIBUTIONS ~ ~ ~ ~

127

This inequality is Holder's (Appendix G20). Because of (2), the left-hand side traverses a bounded set of real numbers when 4 traverses B and t,b andfare held fixed. Thus,f* 6 traverses a bounded set in [ 9 ( A ) ;B]'. Now, let K and N be any two compact intervals in R such that K cfi.By Lemma 3.7-1, there exists a constant M > 0 and an integer m 2 0 such that, for all I E 9&4), supll(f* 6, I>ll* < max suPllA(k'(t)llA. (3) 918

Osksm toR

Also, by Lemma 3.2-2, for any 8 E giR"'(A),we can find a sequence {I,}T= c 9&4) such that I j -,8 in g N m ( A ) This . fact and (3) imply that f * 6; has a unique extension as a member of the space [gK"'(A);B ] . (See Lemma 3.4-5.) This extension, which we also denote by f * 6, satisfies (3) with I now allowed to be any member of QKm(A).Therefore, as q5 traverses p, f * 4 traverses a bounded set in [ g K m ( A )B]". ; Moreover, for any 8 E gK"'(A),we may write

(f* 8, 4 ) = , where the usual definition of distributional convolution is used. Thus, f * 8, 4 ) traverses a bounded set in B as #I traverses B. Next, set 8 = xu, where x E gK"'and a E A. We obtain


a, where in the right-hand side, f E [9;[ A ; B ] ] and (f* x, 4 ) E [ A ; B ] . Since the left-hand side traverses a bounded set in B as #I traverses B, we have, from the principle of uniform boundedness (Appendix D12), that (f* x, q5) traverses a bounded set in [A ; B ] . Hencef * x is a continuous linear mapping of 9, supplied with the topology induced by g L qinto , [A; B ] . [See Appendices C8 and D2(iii).] But 9 is dense in g L qand , thereforef * x has a unique extension as a continuous linear mapping of gLQ into [A ;B ] . Upon denoting this extension by f* x, we obtain f * x E [aL,; [ A ; B ] ] for every x E 9,"'. Finally, let y E 9 be such that y = 1 on a neighborhood of 0 in R.Let K be a compact interval in R such that supp y c K. Set n = m + 2, where m is the integer corresponding to N =I k as above. Set

7. THE SCATTERING FORMULISM

128

+ c, where 5 E g K .Thus, f =f * 6 =f * D"(yJ,) + f * c = D " ( f * YJ,)

Then, S = D"(yJ,)

+f* 5.

According to the preceding paragraph, bothf* yJ, and f * { are members of [gLq; [ A ; B ] ] . But then, so too is S, because [gL,;[ A ; B ] ] is closed under D". By Theorem 3 . 8 - 1 , f ~ [QLa(A);B]. 0 It is worth mentioning here that the converse to Lemma 7.2-2 is not true, as can be seen through the following counterexample due to L. Schwartz. Let p = 2, A = C,and B = L , . Also, let f be the identity operator on L , . Then, clearly, the restriction o f f t o gL2is a member of [gL2; L , ] . But, for any E 9,

*

(f**)(t>

= ( f ( x ) , * ( t - 4) = * ( t

- .). The right-hand side is a mapping of the real line into L , . However, it does not satisfy (2) because II$(t - * ) l l L 2 = I I $ I I L 2 , so that the integrand of (2) is in this case a constant with respect to t . This counterexample has still another implication. It is a fact that any f E [ Q L 2 ; C] has a representation as a finite sum of derivatives of functions in L,; i.e.,

f

=

2hy,

hk E L , .

k= 1

(See Schwartz, 1966, p. 201.) This is no longer true for every f E [gL2; B], where B is an arbitrary Banach space. Indeed, if it were true, we could choose B = L , as above and then write, for any E 9,

c

* $ = 1 h p * If5 = hk * * ( k ) , where h k E L,(B). It can be shown that h k * $(k) E L,(B) so th a tf * I,$E L,(B). But we have already noted in the preceding paragraph that f * Ic/ 4 L,(B) = f

L,(L,) for a properly chosenf. Problem 7.2-2. Let h EL,(B), where B is a Banach space, and let Show that h * II/ E g L 2 ( B ) .

7.3.

I// E 9.

SCATTER-PASSIVITY

Let u be the input signal on a Hilbert port under the scattering formulism. If q is an ordinary function at some instant of time t , then Ilq(t)llH2is the instantaneous power injected by the input signal into the Hilbert port.

7.3.

SCATTER-PASSIVITY

129

Similarly, if 2B is the scattering operator and if 2Bq is also an ordinary function at the instant t, then ll(2Bq)(t)llH2is the instantaneous power carried out of the Hilbert port by the output signal. Therefore, the net power absorbed by the Hilbert port at time t is Ilq(t)llZ- II(2Bos)(t>l12.

(1)

In general, neither q nor ‘2Bq need be an ordinary function at t and indeed they may be singular distributions throughout some neighborhood of t. In this case, ( 1 ) will not possess a sense as the net power absorption. Nevertheless, our intent is to allow distributional inputs and outputs and at the same time make use of a passivity assumption. This is accomplished by first assuming that 2B is passive on a domain of ordinary functions [namely 9 ( H ) ] ,then developing certain representations for 2B, and finally extending 2B onto wider domains by means of those representations. (In this regard, see also the discussion at the beginning of Section 4.6.) Definition 7.3-1. Let 2B be an operator whose domain contains a set 3 c L , ( H ) . 2B is said to be scatter-semipassive on X or, alternatively, contractivefrom X into L , ( H ) if, for all q E X and for r a 2Bq, we have that r E L,(H) and

If, in addition, 3 = L2(H), then we simply say that 2B is contractive on LZW. A stronger condition than scatter-semipassivity is stated by the next definition, where the following notation is used. Given any T E R , we define the function 1, on R by l T ( t )= 1 for t IT and l T ( t )= 0 for t > T. If h is any function on R,the function tw l,(t)h(t) is denoted by 1,h. Definition 7.3-2. Let 2B be an operator whose domain contains a set 9 with the property that 1,q E L,(H) for every q E 9 and every T E R . 2B is said to be scatter-passive on g if, for all q E 9 and all T E R and for r A 2Bq, we have that 1 r E L 2 ( H )and

,

d -

m

If 2B is scatter-passive on X c L,(H), then it is also scatter-semipassive on X, as can be seen by letting T + 03. However, the converse is not true in general. For example, a pure predictor defined by (!Dq)(t)4 q(t x > 0, is scatter-semipassive but not scatter-passive on 9 ( H ) .

+ x), where

130

7.

THE SCATTERING FORMULISM

Lemma 7.3-1. Let 2B be a linear scatter-semipassive operator on .9(H). Then, ll3 is continuous from 9 ( H ) into L 2 ( H ) and therefore inlo [ 9 ;HI" as well. Moreover, 2B has a unique linear contractive extension onto L2(H). PROOF. The inequality (2) states in effect that, when 9 ( H ) is equipped with the topology induced by L,(H), 2B is continuous from 9 ( H ) into L2(H).But, the canonical injections of 9 ( H ) into L 2 ( H )and of L 2 ( H )into [ 9 ;HI" are both continuous, and this implies the first sentence. The second sentence follows from Appendix D5 and the fact that 9 ( H ) is dense in L2(H).0

Not only does the scatter-passivity of a linear operator imply continuity, it also implies causality. In fact, we have the following result originally pointed out by Wohlers and Beltrami (1965). Theorem 7.3-1. Let 2B be a linear operator on 9 ( H ) . Then, 2B is causal and scatter-semipassive on 9 ( H ) ifand only i f 2 B is scatter-passive on 9 ( H ) . PROOF. Let 2B be scatter-passive on 9 ( H ) and, as before, set r 2Bq, where q E 9 ( H ) . We have already noted that (2) can be obtained from ( 3 ) by taking T - t co,and therefore 2B is scatter-semipassive. Next, assume that 2B is not causal. This means that, for some T E R, we have q(t) = 0 for - 03 < t < T and r(t) # 0 for all t in some set of positive Lebesgue measure contained in (- 03, T ) . Then, (3) cannot hold, and this contradicts the scatter-passivity of $11.Hence, 2B must be causal on 9 ( H ) . Conversely, assume that '123 is causal and scatter-semipassive on 9 ( H ) . Given any T E R , choose an X E R such that T < X . Let 8 E d be real-valued and such that e(t) = 1 for - 00 < t 5 0, e(t) = 0 for 1 5 t < co, and 8 is monotonic decreasing for 0 < t < 1. Set

~ ( t ) l Tx,( t ) 4

e(-).Xt -- TT

Given q E 9 ( H ) , set g = !lB(lq) and r = 2Bq. By scatter-semipassivity, both g and r are members of L 2 ( H )and by causality, g = r almost everywhere on ( - 00, T ) . Thus, we may write

The second inequality is due to the scatter-semipassivity of $11.By choosing X sufficiently close to T, 1 ; IIlq1I2 dt can be made arbitrarily small because

This establishes (3) and thereby the scatter-passivity of 2B. 0

7.4.

BOUNDED* SCATTERING TRANSFORMS

131

Theorem 7.3-1 shows that the following two assumptions on an operator

2B are equivalent: (i) 2B is linear, causal, and scatter-semipassive on 9(H). (ii) 2B is linear and scatter-passive on 9 ( H ) . Under a postulational approach to realizability theory, (i) may be considered a better form for a hypothesis because causality and scatter-semipassivity are independent conditions (see Wohlers and Beltrami, 1965 or Zemanian, 1968b). However, statement (ii) recommends itself by virtue of its conciseness. Later on, the reader should bear in mind that causality is a consequence of (ii).

7.4. BOUNDED* SCATTEIUNG TRANSFORMS

Definition 7.4-1. A function S of the complex variable ( is said to be a bounded* mapping of H into H (or simply bounded*) if, on the half-plane C , 4 {(:Re > 0}, S is an [H; HI-valued analytic function such that II S(C)II[H ;H] 5 1.

Our aim in this section is to show that the Laplace transform of the unitimpulse response of a linear translation-invariant scatter-passive operator on 9 ( H ) is bounded*. The converse assertion (namely, every bounded* mapping is such a Laplace transform) will be established in the next section. Theorem 7.4-1. If the operator 2B is linear, translation-invariant, and scatter-semipassive on 9 ( H ) , then 2B = s * on 9 ( H ) , where s E [9,,(H); HI.

PROOF.By Lemma 7.3-1, $113is continuous from 9 ( H ) into [9; H]”. There-

= s *, where s E [ 9 ( H ) ;HI, according to Theorem 5.10-1. Let denote the norm for L,(H). The scatter-semipassivity of 2B implies that, for all $ E 9 ( H ) ,

fore,

11

*

[ILz

Lemma 7.2-2 now shows that s E [9,,(H); HI. 0 Theorem 7.4-2. If the operator 2B is linear, translation-invariant, and scatter-passive on 9 ( H ) , then its unit-impulse response s possesses a Laplace transform whose strip of definition contains the halfplane C , 4 {C: Re C > 0).

7. THE SCATTERING FORMULISM

132

PROOF. By Theorems 7.3-1 and 7.4-1, B3 = s *, where s E [9L2(H);HI, and B3 is causal. Hence, supp s c [0, 00) according to Theorem 5.1 1-1. We now

invoke Lemma 7.2-1 to conclude that s E [ Y ( O , and has a strip of definition containing C + . 0

00;

A ) ; B] so that 2 s exists

Lemma 7.4-1. Let a, b E R be such that b < 0 < a. Assume that s E [9L2(H); b(H), s * 4 exists and is a H ] and supp s c [O, a)).Then, for each 4 E Pa, smooth H-valued .function. Moreover, there exists a constant L > 0 and an integer I2 0 such that ~l(s* +)(t)ll I Le-b' max supllebr4(k)(t) 11. OSksl rER

(1)

PROOF.Since [9L2(H);HI c [9L,(H); HI, we have from Theorem 5.5-2 that s * 4 is a smooth H-valued function and (8

* 4)(0= M X ) , 40 - x)>.

-+

E be~ such that 1(x) = 1 for < x < 00 and 1(x)= 0 for -00 < x < - 1 . Let Bp LA sup, 1 l ( p ) ( xI.) Then, there exists a constant M > 0 and an

Let A

integer 12 0 such that Il(s * 4>(t>ll= II<s(x), X x M t - x)>Il 03

<M

max OSkSl p = O

(i)Bk-pe-b'

ebxd x -1

suplleb'4(P)(t)ll. roR

This implies (1). 0 Lemma 7.4-2. Let ?lB be a linear translation-invariant scatter-passive operator on 9 ( H ) and let q E Y c , d ( H )where , d < 0 < c. Then, for r a 'Dqand for every T E R, we have that

PROOF.Let a, b E R be such that d < b < 0 < a < c. Given anyq E Y cd(H), , c 9 ( H ) that tends to q in Ya,b(H). we can find a sequence {4j}T=1 Indeed, let 1 E 9 be such that 1(t) = 1 for I t I < 1 and A(t) = 0 for I t I > 2. We may write

7.4.

133

BOUNDED* SCATTERING TRANSFORMS

Now, Dk-”[A(t/j)= I ] is equal to zero for I t I I j ; and for I t J > j , it is bounded by a constant that does not depend on j . Moreover, for each p , Ka b(t)

- 0

IIKa,b(t)DPq(t)llIsupllKc,d(t)DPq(t)II

(4)

~,,d(t) as j - , 00. Consequently, the right-hand side of (3) converges uniformly to 0 on R. Thus, upon setting $ j ( t ) 4 A(t/j)q(t),we obtain the sequence we seek. Next, note that 2l3 = s *, where s satisfies the hypothesis of Lemma 7.4-1 according to Theorems 5.11-1, 7.3-1, and 7.4-1. Set $ j A 2134j = s * + j . Since supp s c [0, m), $ j E 9 + ( H ) . Also, set r 2Bq = s * q. By Theorem 5.5-2, r E I ( H ) . So, 11q1I2, llr112, I14j112,and Il$j112 are all continuous functions on R. Furthermore, we may invoke Lemma 7.4-1 and the fact that I ebr/Ka, b(t)I I1 to write Irl>j

rsR

111>j

e*‘IIr(t)- $j(t)lI I L max supllebrDk[q(t)- 4j(t)lII -,0,

j -, co. ( 5 )

Osksl IER

Now, given any T E R,consider

1

T J-m

T

[11q112 -

I I ~ I dI t~ -I J- m [ll4j1I2- II$~II~I dt

sm + ST I I ~ I I I I ~

I

IIqII 114 - 4jll dt

-m

-

’r

+

I

dt

+ J’- m I I -~

T

-m

114 - 4jll

I

IIdjII dt

T

II$jll

dt.

(6)

-=

Since b < 0 a and since $ j -,q in 9“. b(H), there exists a constant K not depending o n j such that Ilq(t)ll I Ke-br and I14j(t)II I Ke-b’. Also, sup ebrllq(t)- 4j(t)ll -,0,

j -, 00.

IeR

These facts imply that the first and second integrals on the right-hand side of (6) tend to zero as j + co. Similarly, by virtue of Lemma 7.4-1 and (9, Ilr(t)II, II$j(t)II, and ebrIlr(t)- $j(t)ll satisfy similar conditions, and, as a result, the third and fourth integrals on the right-hand side of (6) tend to zero as j - , co. Condition (3) now follows from the scatter-passivity of 2B on 9 ( H ) and the consequent nonnegativity of the second integral on the left-hand side of (6). 0 The main theorem of this section is the following.

Theorem 7.4-3. VlB is a linear translation-invariant scatter-passive operator = s *, where s E [QL2(H); on 9 ( H ) , then 2l3 has a convolution representation HI and supp s c [0, m). Moreover, the Laplace transform S of s exists with a strip of deJinition containing C + {t; E C : Re t; > 0) and is bounded*.

7.

134

THE SCATTERING FORMULISM

PROOF.We have already established everything except for the condition

c

IIS(c)ll[H:Hl I 1 for all E C+ . (See Theorems 5.11-1, 7.3-1, 7.4-1, and 7.4-2.) Let a E H and T ER. Choose z E R such that 7 > T. Also, let 1 E d be such that A(t) = 1 for -co < t < 7 and A(t) = 0 for z 1 < t < 00. For any E C+ , set q(t) = aertA(t).Thus, with -Re ( = d < 0, we have that q E Y cd, ( H )for every c E R. So, we may use Lemma 7.4-2 and ( 2 ) for r = 2Bq.

+

c

We now invoke Theorem 5.5-2 to write r(t) = (s * qNt) = M x ) ,q(t - x ) ) . For any fixed t < T , q(t - x ) = aer('-x)when x is restricted to a sufficiently small neighborhood of [0, 03). But supp s c [0, co), and therefore r(t) = (s(x), aec(t-x))= ec'S(c)a. Thus, J -

m

a,

Since this holds for all a E H and since the integral on the right-hand side is positive, this implies that IIS(c)IICH;Hl 1 for any E C+ . 0

c

7.5. THE REALIZABILITY OF BOUNDED* SCATTERING TRANSFORMS

We will now show that every bounded* mapping of H into H i s the Laplace transform of the unit-impulse response of a linear translation-invariant scatter-passive operator on 9 ( N ) . We start with two lemmas.

Lemma 7.5-1. Let 4 E ~ ( Hwith ) s u p p 4 c [0, co). Then, 0 and for each fixed t E R, h(t)e-"'

= (1 /2n)

S(o

I O U

+ i o ) @ ( a + io)e'"' d o .

-m

In view of (2), IIh(t)e-"'II I (1/2n) Jm Q(o)do < co.

-

7.

136

THE SCATTERING FORMULISM

Since this holds for all o > 0 and since supp h c [0, a),it follows that, for any fixed a > 0, Ilh(t)e-"'ll E L , . Now, set H(c) S(c)cD(()for [ E C + and consider

rm

Ilh(t)e-"'IIZ dt

=

jm(h(t)e-"', (1/2n) -m

Jm -m

H(a + iw)eiw' dw dt

)

(3)

where again o > 0. By a judicious use of Lemma 7.5-2 and Fubini's theorem, we may first bring the integration on w outside the inner product in the righthand side of (3), then reverse the order of integration on w and ?, and finally bring the integration on t inside the inner product t o obtain Parseval's equation :

r*

1

00

Ilh(t)e-"'J12dt = (1/2n)

Since IIS([)ll I I for

-m

IIH(o + iw)l12do.

(4)

c E C , , the right-hand side of (4) is bounded by m

By (2) and Lesbesgue's theorem of dominated convergence, (5) tends t o

On the other hand, as a -+ 0 + , llh(t)e-"'112 increases monotonically t o the limit Ilh(t)l12 at each t . So, by the theorem of B. Levi (Williamson, 1 9 6 2 , ~62), . the left-hand side of (4) tends t o JZm Ilh(t)112dt. We conclude that

rm 11(s

* 4 ) ( t ) l l 2 dt

(1/2n)

jm

Il@(iw)ll2 dw

0 ) , the restriction of S(a) to H , is a member of [ H , , H , ] .

The next theorem is actually a corollary to Theorem 7.4-3. Theorem 7.6-1. If is a linear translation-inrariant scatter-passive operator on 9 ( H ) and if 21 maps Q(H,) into [ 9 ( R ) ;H,], then 2U = s *, where s E [ 9 , , ( H , ) ; H,], and S 2 s is bounded*-real. PROOF.By Theorem 7.4-3, 2B = s *, where s E [ 9 L z ( H ) ;HI, and S is bounded*. Choose a sequence {$j}j"=o c 9 ( R ) which tends to 6 in [ E ( R ) ;R ] . Then, for any a E H , and any 4 E 9 ( R ) , ('rO($j a), 4 ) E H , . But <m($ja)q

4 ) = (s(t)t

($j(x)a, $(t

+ XI>) 40) +

(

~

3

Therefore, (s, $a> E H , . But 9 ( R ) 0H , is total in 9 ( H r ) ,and so s E [ 9 ( H r ) ; H , ] . Hence, by the density of 9 ( H , ) in 9,,(H,), we have s E [ g L Z ( H r )H; , ] .

7.7.

139

LOSSLESS HILBERT PORTS

It now follows as in the complex case that s E [L?(O, 00); [H,; H,]] since supp s c [0, co) according to Theorem 7.4-3. Thus, for any a > 0, S(a) = ( s ( t ) ,e-"') E [H,; H,]. 0 As a corollary to Theorem 7.5-1, we have the following result. Theorem 7.6-2. If S is bounded*-real, then, in addition to the conclusion of Theorem 7.5-1, we haoe that s E [ 9 L z ( H r )H,] ; and s * maps 9 ( H , ) into [ g ( R ) ;Hr1. PROOF. We know from Theorem 7.5-1 that s E [ g L Z ( H )HI. ; The conclusion s E [ g L 2 ( H r )H,] ; will follow as in the preceding proof once we show that s

maps 9 ( H , ) into H,. This will also imply that s * maps g ( H , ) into [ 9 ( R ) ;H,]. Let 4 E 9 ( H r ) .According to the inversion formula for the Laplace transformation (Theorem 6.4-I), (s,

4)

=

lim w-m

(U/W

lW

--w

~ e c ' d w4,( t ) ) ,

where Re 4' > 0 and w = Im 4'. We have to show that ( s , 4) E H,. But this will be established when we prove that

Jyw~ ( w c '

dw E [ H r ; ~

r

1

(2)

for each w > 0 and t . Let a, b E H,. Then, (S(a)a,b ) is real for 0 real and positive. By the reflection principle and the decomposition S(c) = S , ( c ) + is,([),where S , , S2 E [H,; Hr1, we have ( S , t b , b) = (sl(Oa,b) and (S,(%)a,b) = - (S2(l)a,b) for E C , . Since this is so for every a, b E H,, S,(a + iw) and S,(a + iw) are respectively even and odd functions of w . We now obtain ( 2 ) by noting that the imaginary part of the integral there is the zero member of [H,; H,]. 0 Problem 7.6-2. Verify that the complexification of a real Hilbert space is a complex Hilbert space. Problem 7.6-2. Assume that 2J.I is a linear translation-invariant scatterpassive operator on 9 ( N , ) with range in [B(R);H,]. Show that 2J.I has a unique extension as a linear translation-invariant scatter-passive operator on %H). 7.7. LOSSLESS HILBERT PORTS

There are physical systems with the property that their net energy absorptions for all time from signals of compact support are always zero. Any electrical network consisting exclusively of a finite number of inductors and

7.

140

THE SCATTERING FORMULISM

capacitors has this property. Passive systems of this sort are usually called lossless. In regard to Hilbert ports, we shall use the following definition. Definition 7.7-1. Let (m be an operator whose domain contains a set (m is said to be lossless on X if 2J.l is scatter-passive on X and, for every q E X and r 4 (mq,

X c L,(H).

When X supplied with the topology induced by L,(H) is a normed linear space and when 2J.l is linear, condition (1) can be restated by saying that '2B is isometric from X into L,(H). It should be borne in mind, however, that losslessness requires in addition that 2J.l be scatter-passive on fa(H). The pure 4 , t < 0, is isometric from 9 ( H ) into predictor, defined by 4 ~ 0 ~ where L , ( H ) but not scatter-passive on 9 ( H ) an d hence not lossless on 9 ( H )according to our definition. It should also be noted that a linear operator 2J.l is lossless on 9 ( H ) if and only if (1) is satisfied and (113 is causal on 9 ( H ) ; this is a consequence of Theorem 7.3-1. When the Hilbert space H is separable, a lossless convolution operator can be characterized by the fact .that its scattering transform S(o io) is bounded* and, as o -+ 0 + , S(o i o ) converges for almost all w to an operatorS(io)suchthat IIS(iw)all = llall foralla E H.AnylinearoperatorT~[ H ;HI having this property (namely llTall = llall for all a E H ) is said to be isometric on H. As our first result along these lines, we have the following assertion.

+

+

Theorem 7.7-1. Let H be a (not necessarily separable) complex Hilbert space. Assume that S is an [ H ;HI-ralued analytic function on C+ such that, for all E C, , IIS(l)II I P( 1 (I), where P is a polynomial. Also, assume that, as o -+ 0 + and for almost all o,S(a + iw) -+ S(iw) in the strong operator topology, where S ( i o ) is a linear isometric operator on H. Set s = i?-'S (i.e., s is the unique member of [ 9 ( H ) ;HI whose Laplace transform coincides with S on C+). Then, s * is lossless on 9 ( H ) , and therefore S is bounded*.

c

PROOF. We first observe that, by Theorem 6.5-1, s E [ 9 ( H ) ;HI and supp s c [0, 00). Next, we let 4 E 9 ( H ) with supp 4 c [0, 00) and let @([) = 4(t)e-{' dt as before. @ is analytic on C. By the estimate of Lemma 7.5-1, we have, for 0 < a < 1,

IIS(a

+ iw)@(o+ io)l125 Q(w),

where Q E L , . Moreover, for any fixed o where S(o + io)converges strongly as o -+ 0 + , we may write IIS(a

+ io)@(o+ i o ) - S(io)@(iw)llIIIS(o + iw)II II@(o+ i o ) - @(io)II + II[S(a + iw) - S(io)]@(io)ll.

7.7.

141

LOSSLESS HILBERT PORTS

+

By the principle of uniform boundedness, IIS(a iw)lJ is bounded for 0 c a < 1. We can conclude that the left-hand side tends to zero as a -+ O + . Thus, by Lebesgue’s theorem of dominated convergence and Parseval’s equation,

1

1 “ lim IIS(a + iw)@(a + iw)I(’ dw a - ~ +27~ - m

1

1 IIS(iw)@(iw)11’ dw 271

=-

Moreover, Parseval’s equation [see (4) of Section 7.51 and the theorem of B. Levi show that the left-hand side is equal to lim a-O+

j Il(s * 4)(t)e-a‘112dt = s l l ( s * 4)(t)I12dt.

Thus, s * satisfies (1) for every q = 4 E 9 ( H ) with supp 4 c [0, co).The translation invariance of s * shows that s * satisfies (1) for all 9 E 9 ( H ) . Since supp s c [0, a),s * is causal and therefore lossless on 9 ( H ) . Theorem 7.4-3 now implies that S is bounded*. 0 Note. A small modification of the foregoing proof shows that, if we weaken the hypothesis of Theorem 7.7-1 by replacing the assumption that S(iw) is a linear isometric operator on H by the condition IlS(iwll I 1 for almost all w , then we can still conclude that S is bounded*. The next theorem is a sort of converse to Theorem 7.7-1. Its proof makes use of the following known fact. Let ff and J be separable complex Hilbert spaces. If F i s an [H; J]-valued function that is analytic and bounded on C , , then there is a set R c R whose complement R\R has measure zero such that, for each w E R and as a -+ O+, F(a + iw) converges in the strong operator topology. This is proven by Sz.-Nagy and Foias (1970, pp. 185-187) for [ H ; J]-valued bounded analytic functions on the unit disc {z E C: IzI < I } and strong convergence along radial lines. The present version can be obtained by mapping the unit disc into C + through = ( I + z)(l - z)-’. The radial lines becomes circles centered on the imaginary axis and passing through 5 = 1. That strong convergence for C(c) A F(z) also occurs along horizontal lines in the [ plane follows from the inequality

c

IIG(0

+ iw) - C(0 + iq)II I 4 ( 1 w - ql/a)SUP

SEC+

IIG(i)ll,

0 0 for all 4 E 9#). Now, consider

8(49 $1 a8,(4, $1 4 “(4, $1 + 5G31. By virtue of the semipassivity of W, 8 is a positive sesquilinear form on 9,(H) x 9 , ( H ) . The Schwarz inequality (Appendix A6) yields

since

$1 I I [8(4, 4)8($9 $)11/2. I 4) I Is(+,$1I , we get I $1 I I I W$,$1 I + m 4 , 4)8($, $ ) P 2 I f w 4 9

W($)Z(4),

(3)

154

8. THE ADMITTANCE FORMULISM

Now, for every 4 E g K ( H )other than the zero function, the mapping

is continuous and linear from g K ( H ) into C. Thus, f4 E [ g K ( H ) ;C]. In view of (3), the collection of all suchf4 is a bounded set in [ g K ( H ) ;C]". Therefore, that collection is equicontinuous on g K ( H ) (Appendix D9). As a result of this and the continuity of 11 ]IL, on g K ( H ) , we can find a continuous seminorm p on g , ( H ) such that

Iw4,$1 I

5 P($)Z(4)

(6) $ E g K ( H ) . Combining this with

and simultaneously 1) $[IL2 5 p($) for all 4, (5), we now obtain Z(4)/P(4) 5 1 + [z(4>/P(4>11'2. Consequently, there exists a constant M > 0 such that Z ( 4 ) I Mp(4). By (6) again, I w49 $11 5 M P ( 4 ) P ( $ ) , (7) which verifies the continuity of 23 on g K ( H ) x g , ( H ) . Next, set JI = Ba, where 0 E g Kand a E H.By (7),

so,

II(W @llH =

SUP

llall = 1

I((W4, (0,a l l I SUP

llall = 1

MP(4)P(ae).

Let 0 be a bounded set in 9.Then, 0 is a bounded set in g N for some compact interval N. Moreover, given any other compact interval J , we can choose K to contain N and J. Then, p will be a continuous seminorm on g , ( H ) , and 0 will be a bounded set in g K .Consequently,for every 4 E g , ( H ) , SUPll(~4,OllH I PP(4),

eEe

where P is a constant. This proves that '9l is continuous from g , ( H ) into [g; HI. Since J can be chosen arbitrarily, '3 is continuous from 9(H)into [9;Hl. 0 Problem 8.3-1. Assign to L?(H) the topology generated by the collection

{q.}g of seminorms defined by Wk(f)

=

I

Kk

Ilf(t)llH

dt,

.fEL:"'(H)~

8.4.

THE FOURIER TRANSFORMATION ON

Y(H)

155

where {&};= is a nested closed cover of R. Then, Lp(H) becomes a Frkchet space. Use the closed-graph theorem (Appendix DlO) to prove that every linear semipassive mapping on 9 ( ~ is continuous ) from 9 ( ~ into ) LP(H).

8.4. THE FOURIER TRANSFORMATION ON Y ( H )

Subsequently, we shall need a variety of results concerning the Fourier transformation 8. We gather them in this section. The Fourier transform of any 4 E Y ( H ) is denoted by $ 4 84 and defined by

Some differentiations under the integral sign and integrations by parts show that

1

m

(io>~$(~)(o> = e-'mrD?[( - itlk+(t)l tit, -m

and this implies that 8 is a continuous linear mapping of Y ( H )into Y ( H ) . The same argument as that which holds for complex-valued functions (Zemanian, 1965, Section 7.2) establishes the following inversion formula for 8: m

+(t) = ( ~ - 1 $ ) ( t ) = (1/21t)

-m

(2)

$(w)e'a' dw.

Here, 3-l denotes the inverse Fourier transfbrmation. By the same manipulations as above, we see that S-', too, is a continuous linear mapping of Y ( H ) into Y ( H ) . It also follows that 3 must be a bijection. Thus, 8 is an automorphism on Y ( H ) . We take note of the following version of Parseval's equation: 2n J m (4, $W)dw,

4 9

+

E Y(H).

(3)

This can be obtained by using Fubini's theorem and Lemma 7.5-2. We now present three lemmas concerning complex-valued functions.

Lemma 8.41. Let 9 * 9 denote the set of allfunctions 4 with the representation q5 = 9 * $, where 8, $ E 9. Then, 9 * 9 is dense in 9.

156

8. THE ADMITTANCE FORMULISM

This is established by choosing a sequence {8k}z=l 8, dt = 1, and supp 8, c { t : 1 t I 5 k-'} and then showing that 8, * 4 +

x

c 9 such that 8, 2 0,

in 9.

Lemma 8.4-2. Let I a B(9). Also, let I * E denote the set of all functions with the representation x = #$, where 8, $ E 9.Then, I * 9 'is dense in 9.

PROOF.Choose any $ E Y and any Y-neighborhood fi of $. Then, B-'$ E Y, and R 6 S-'(fi) is an Y-neighborhood of 4. By the density of 9 in Y, there exists a E 9 n h. But, since the topology of 9 is stronger than that induced on it by Y and since h is also an Y-neighborhood of c, there exists a %neighborhood of ( such that E c h. By the preceding lemma, wecan find a 8 * $ E 9 * 9 in Z. Thus, 8 * $ E R. But B(8 * $) = #$. Therefore, #$ E 6.0

4

Lemma 8.4-3. Given any bounded Bore1 set E c R, there exists a function 8E9suchthat 2 1forallqEE.

PROOF. We can choose a compact interval K 3 E and a function E such that $(q) > 2 for all q E K. Therefore, $ E Y. We can also choose a sequence {$k}?= c 9 that converges in Y to $. Therefore, $I, -+ $ in Y . This implies that $k tends to $ uniformly on K. Thus, the sequence ( 4 k ) contains an element 8 with the required properties. 0 We finally take note of a standard result concerning complex-valued distributions, namely the Bochner-Schwartz theorem. In the following, 6 denotes the collection of all Bore1 subsets of R. A temperedpositive measure p is a positive measure on 6 such that (1 t 2 ) - p E Ll(R, 6; p ; C) for some integerp. (See Appendix G, Sections G4, G10, and G13.) On the other hand, f E [9; C] is called positive-definite if, for every 4 E 9,

+

Theorem 8.4-1. f E [g; C ] is positive-definite if and only if there exists a temperedpositive measure p such that, for every 4 E 9,

Proofs of this theorem are given by Schwartz (1966, pp. 276-277) and Gelfand and Vilenkin (1964, Section 3.3). According to the standard definition of the Fourier transformation on [CB;C] (see, for example, Zemanian,

8.5. LOCAL MAPPINGS

157

1965, Section 7.8), (4) means that f is the Fourier transform of the distribution generated by p. Later on, we shall meet a tempered complex measure. This is a a-finite complex measure v (see Appendix G12) for which

J-r.tldlvl (1 + t 2 ) - p converges as k --t co if p is chosen sufficiently large.

8.5. LOCAL MAPPINGS

Let W be a linear translation-invariant semipassive mapping of 9 ( H ) into [9; HI. By Theorem 8.3-1, W is also continuous. Therefore, 92 = y *, where y E [ 9 ( H ) ; HI, according to Theorem 5.10-1. It now follows from Theorem 5.5-1 that W is a continuous linear mapping of 9 ( H ) into b(H). We define the sesquilinear mappings 8 and b on 9 ( H ) x 9 ( H ) as before. For instance,

w4,$1 %l(4, *)

Srn(W4(0, W )

dt,

-m

4,

*

E9 w .

(1)

By the proof of Theorem 8.3-1, 8 is a continuous sesquilinear form on gK(H)x 9#) for every compact interval K , and therefore so too is 8 because

B(49 $1 a @9?(49*)

f

i&J f m

-m

w 4 , $1 + (4, %*)I

dt.

(2)

Another result from Hackenbroch (1969) is a characterization of those causal operators W for which b = 0. One property such operators have is that they respond only to present values of input signals, but not to past values nor, by causality, to future values. A precise definition of this property is the following.

Definition 8.5-1. A mapping W of 9 ( H ) into [9; HI is said to be local if supp % ' $ c supp 4 for all 4 E 9 ( H ) . For linear mappings, local operators are the same thing as the so-called memoryless operators of systems theory (Willems, 1971, p. 14). Theorem 8.5-1. Let W be a linear translation-invariant causal mapping of g ( H ) into [9;HI andsuch that b = 0. Then, W is local. Note. The condition b = 0 implies that W is semipassive on 9 ( H ) .

8.

158

THE ADMITTANCE FORMULISM

PROOF. That W is local is equivalent to the following assertion. If 4 E 9(H) vanishes on a neighborhood A of T ER, then W 4 also vanishes on A. Now, 4 can be written in the form 4 = 4, + &, where 4,, 42E 9(H), supp 41 c [t,, t 2 ] ,and supp 4zc [ f 3 , t,] for certain t,, t 2 , t 3 , t, E R with t, < t z < T < t 3 < t 4 . So we need merely prove that, if 4 E 9(H) and supp 4 c [c, d ] , then supp %q!~ c [c, d]. Since W is causal by hypothesis, W+ vanishes for t < c. Again by hypothesis, @(4,J / )= 0 for all E 9(H),so that

+

J(W4, J/) dt = - J(4, WJ/) dt.

(3)

Choose J/ such that J/(t)= 0 for t < d. Therefore, (W+)(t)= 0 for t < d by causality. Hence, the right-hand side of (3) is equal to zero. But the behavior of J/ for t > d is unrestricted, and we know that 8 4 E b ( H ) . Therefore, (W+)(t) = 0 for t > d as well. 0 In order to prove Hackenbroch's characterization of W (see Theorem 8.5-2), we shall need the following result.

Lemma 8.5-1. Let A and B . be complex Banach spaces. Assume that c [ A ;B],'where Lk # 0 for every k. Then, there exists an a E A such that Lk a # 0for every k.

{Lk}km,

,

PROOF.Let F k 4 {a E A : Lka = O}. Our lemma will be proved when we show that A # up= Fk . First, note that each Fk is a closed linear subspace of A and that Fk # A because L k # 0.We now show that F , is nowhere dense; that is, its closure F k has no interior points. Suppose that b is an interior point of F k . Since F k = F k , this means that there exists an E E R, such that O(b, &) c F k , where O(b, 6) {a E A : IJa- bll < E}. Also, since Lk # 0, there exists a nonzero c E A such that Lkc # 0. Then, d = b + ~ l l ~ l-'cl E O(b, E). Upon applying Lk to d and using the fact that Lk b = 0,we obtain Lkd=3EllCll-1LkC #o. This is a contradiction, and therefore F , is truly nowhere dense. Baire's category theorem (Appendix C13) now implies that A # F , . 0

,

+

u

Theorem 8.5-2. W is a linear translation-invariant causal mapping of 9(H) into [9;HI such that % = 0 ifand only if,for all 4 E 9(H),

8.5.

LOCAL MAPPINGS

159

where Po , PI, .. .,P,, E [ H ; H ] and Pk'

k = 0,1, . ..,n.

= (-l)k"Pk,

(5)

Note. As before, the prime denotes the adjoint operator.

PROOF.If % is defined by (4), then it is clearly a linear translation-invariant causal mapping on 9 ( H ) into 9 ( H ) . Some integrations by parts show that 8 = 0. Conversely, assume that % is a linear translation-invariant causal mapping of 9 ( H ) into [9; HI and such that 8 = 0. For any fixed T E R and a, b E H , consider the mapping

e b ((wa)(T), b),

e E D.

(6)

Since % is a continuous linear mapping of 9 ( H ) into &(H), (6) is a complexvalued distribution. Its support is {T} because % is local. Therefore, by a standard result for complex-valued distributions (Zemanian, 1965, Theorem 3.5-2), ( w e a m b) =

i

k=O

a,(a, b ) e w ) ,

(7)

where the ak(a, b) denote complex numbers and satisfy the following condition. For each fixed choice of the pair a, b E H, all but a finite number of the tlk(a, b) are equal to zero. This finite set will change in general for different choices of a and b. However, the ak(a,b) do not depend on T, because of the translation invariance of %. Moreover, it is not difficult to show that each is continuous and sesquilinear on H x H supplied with the product topology. Therefore, according to Appendix D15,there exists a Pk E [H; HI such that 6 ) = (pk a , b). Thus, (7) becomes

We now show that there exists a finite integer n such that (Pka, b) = 0 for all k > n and all a, b E H. If this is not true, there exists a subsequence {Q,}Y=, of {pk}& such that Q j # 0 for all j . By Lemma 8.5-1, there is an a E H such that Q j a # 0 for all j . But ( Q j a , -)E [H; C], and so, by Lemma 8.5-1 again, there is a b E H such that ( Q j a ,6) # 0 for all j . This contradicts the fact that only a finite number of coefficients in the right-hand side of (8) can be nonzero. It now follows that, for any E 9 0 H ,

+

8.

160

THE ADMITTANCE FORMULISM

Since 9 0 H is dense in 9 ( H ) , (9) remains true for all 4 E 9 ( H ) . This establishes (4) since b and Tare arbitrary. We turn now to the proof of (5). Set Qk Pk + (- l)kpk’.Some integrations by parts show that, for any 4, $ E 9 ( H ) ,

The right-hand side is equal to zero by hypothesis. Thus, for any a, b E H and any 8, A E 9,

11

(Qka, b)O(”(t)yt)dt = 0.

We may apply Parseval’s equation [see (3) of Section 8.41 to this and employ the fact that (gO(”)(q)= (iq)k&q)to get

1

(Qk

4 b)(irl)k’%ll)X(rl)drl = 0

(1 1)

for all 8, 1 E 2. According to Lemma 8.4-2, 2 . 9’ is dense in 9. Therefore, (1 1) continues to hold with %I replaced by any x E 9. Consequently,

2 (&a,

b)(ir])k= 0

k=O

for all q. Whence, ( Q k a ,b) = 0 for all a, b E H , which implies that Qk = 0. This proves (5). 0 Problem 8.5-1. Verify that the quantities c(k occurring in the proof of Theorem 8.5-2 are continuous sesquilinear forms on H x H and do not depend on T. Problem 8.5-2. Let !Jl be a continuous linear translation-invariant local mapping of 9 ( H ) into [ 9 ;HI. Show that

where Q k E [ H ; HI.

8.6. POSITIVE SESQUILINEAR FORMS ON 9 X 9

We shall use the Bochner-Schwartz theorem to obtain a representation for positive, separately continuous, translation-invariant sesquilinear forms on 9 x 9. See Appendices A6 and F1 for the definition of a positive, separ-

8.6.

POSITIVE SESQUILINEAR FORMS ON

9

X

9

161

ately continuous, sesquilinear form on 9 x 9. As for translation invariance, we have the following. Definition 8.6-1. Let X be a space of functions from R into H that is closed under translations (i.e., if 4 EX, then or+E X for every T E R). A sesquilinear form 23 on X x X is said to be translation invariant if 23(0,4,

for all

4, tb,

0 7

ICI)

= 23(4?$1

E X.

Given any y

E

[9;C ] , define the mapping 23 on 9 x 9 by

-

B(e, A) A ( y , e * A T ) ,

e, A E 9,

(1)

where At(t) A( - t). Then, !3 is a separately continuous translation-invariant sesquilinear form on 9 x 9. Indeed, the sesquilinearity is clear. The separate continuity follows from the fact that both O H 0 * It and AH 0 * At are continuous mappings of 9 into 9. Finally, the translation invariance follows from the easily established identity

o * V,

=

* (o,~)f

Conversely, every separately continuous translation-invariant sesquilinear form on 9 x 9 has the representation (1). Indeed, by Theorem 4.5-1, there exists a unique f E [ g R 2 C ;] such that

w 0 ,4= ( f ( t ,

4,e ( t ) W >

for all 8, 1 E 9.From the translation invariance of 23 and the totality in gR2 of the set of functions of the form O ( t ) A a , we see that f ( t , X) = f ( t - 7,X

- T)

for every T E R. We may now proceed exactly as in Section 5.10 to show that

w e ,4= ( Y * A, e>,

(2)

where y E [ 9 ;C ] is uniquely determined by 23. A simple manipulation converts the right-hand side into ( y , 0 * At). Thus, we have established the following.

Theorem 8.6-1. 23 is a separately continuous translation-invariant sesquilinear form on 9 x 9 $and only if 23 has the representation (l), where y E [ 9 ;C ] . y is uniquely determined b y 23, and conversely. If, in addition, 23 is positive, we have the next result.

8.

162

THE ADMITTANCEFORMULISM

Theorem 8.6-2. 23 is a positive, separately continuous, translation-invariant sesquilinearform on 9 x 9 ifand only if23 has the representation

B(0, A) =

J

R

dpt O(t)X(t),

0, A E 9,

(3)

where p is a temperedpositive measure on the Borel subsets of R . PROOF. The “ i f ” part of this theorem follows quite readily. Indeed, the sesquilinearity is again clear. The separate continuity follows from the continuity of the Fourier transformation from 9 into Y and the fact that p is a tempered positive measure. The translation invariance is an immediate result of the identity (@,8)(w) = e-iw‘O(w). Finally, the positivity of 23 follows from the positivity of p because

s(e,0) = j d p t J 8 ( r ) I ’ 2 0. For the “only if” part of this theorem, we first invoke Theorem 8.6-1 to obtain (2). Upon setting 0 = X E 9, we may write

1( y * 8)O dt

=(y

* 2, A)

= B(A,

A) 2 0.

Thus, y E [9; C] is positive-definite. By the Bochner-Schwartz theorem (Theorem 8.4-l), there exists a tempered positive measure p on the Borel subsets of R such that

B(0, A) = ( y , 0 * At) But G(0 * A t )

=

=

J dp g(0 * A’).

03(At) = Oj.This completes the proof.

0

8.7. POSITIVE SESQUILINEAR FORMS O N 9 ( H ) x 9 ( H )

Our purpose in this section is to extend Theorem 8.6-2 to certain sesquilinear forms on 9(H)x 9 ( H ) . 0 will denote the collection of all Borel subsets of R,and 0, will denote the set of all bounded Borel subsets of R. In Definition 2.3-1, we choose 0,as the collection of all Borel subsets of { t : I t I I k } , where k = 1, 2, . . . . Definition 8.7-1. A tempered PO measure P is a a-finite PO measure on

0, such that g&t)

(I

+

lt12q)-’

is integrable with respect to P for some nonnegative integer q.

8.7.

POSITIVE SESQUILINEAR FORMS ON

9(H)x 9(H)

163

As in Section 2.6 [see Equation (10) there], we set

Bgq(H) { f

E

B ( H ) :g;

YE9 6H } .

Lemma 8.7-1. For each nonnegative integer q, Y ( H ) c Bg4(H).Moreouer,

4 H g ; ' 4 is a continuous linear mapping of Y ( H ) into B 0 H . PROOF. Let

4 E Y ( H ) and consider the Fourier inversion formula:

4(t) = (1/2n)

I &w)eiW' d o .

We know that 6 E Y ( H ) c L,(H). By Theorem 2.5-1, 4 E 9 6 H . Also, by (6) of Section 2.5, 1141115 I I $ l l L , , where 11 . is the norm for ' 3 6 H (see Corollary 2.4-la). Hence, 1 1 4 1 1 15

1 II

$(W)llH

R

dw

Since the Fourier transformation is an automorphism on Y ( H ) , this shows that the canonical injection of Y ( H ) into '3 6 H is continuous. Our lemma now follows from the fact that multiplication by g; is a continuous linear mapping of Y ( H ) into Y ( H ) . 0 By virtue of this lemma and Definitions 2.6-2 and 2.6-3, the integrals

have meaning for any 4, t+b

E

Y ( H ) and any tempered PO measure P

Theorem 8.7-1. If P is a tempered PO measure, then

4

-1

dP, 4 ( t )

(1)

is a continuous linear mapping of Y ( H ) into H . Moreover,

(2) is a positive continuous sesquilinear form on Y ( H ) x Y ( H ) . Note. The mapping (1) is called the tempered [ H ; HI-valued distribution generated by P.

8.

164 PROOF.

THE ADMITTANCE FORMULISM

As in Section 2.6, we set

Q(E) 4

I

E

df',

s,(t),

where E is any Bore1 subset of R and q is chosen large enough to ensure that co.We may then write, for any r$ E Y ( H ) , 11 Q(R)IllH;

-=

II Q ~ ~ ~ I I ~ ~ ; ~ ~. I I ~ ~ ~ ~ ~ ~ I ~ ' O

5

[See (2) of Section 2.5.1 This result combined with Lemma 8.7-1 implies the first conclusion. With respect to the second conclusion, we already know that (2) is a positive sesquilinear form on ggs(H)x 9 J H ) (see the paragraph before Definition 2.6-1) and therefore on Y ( H ) x Y ( H ) . To show its continuity, we invoke (2) of Section 2.6 to write, for any 4, cc/ E Y ( H ) ,

I

1I

jd(pfNO7cc/(t)) = /d(Q,[s,(t)l- ''2r$(0, [ g & W '/2$(t>) 5 IIQ(R)II Il(1 + t2q)'/24(t>ll 1 Il(1

I

+ t2')'/2$(t>ll

1.

(3)

We now appeal again to Lemma 8.7-1 to complete the proof. 0 Theorem 8.7-2. With P again being a tempered PO measure, set

$1 4

/ d(P,dXt), W)?91 $ R

EY(H).

(4)

Then, 'illis a positive, continuous, translation-invariant, sesquilinear form on Y ( H ) x Y(H). PROOF. Sesquilinearity is again clear. The translation invariance follows readily from the identity

(80,r$)(t) = e-"'c$(t). For the continuity, combine an estimate like (3) with the facts that &-+g,-'+ is a continuous linear mapping of Y ( H ) into 9 6H and the Fourier transformation is an automorphism on Y ( H ) . The positivity is asserted by Theorem 8.7-1 since $ E Y ( H ) . 0

4,

The principal theorem of this section is the following converse to Theorem 8.7-2 due to Hackenbroch.

8.7.

POSITIVE SESQUILINEAR FORMS O N

9(H)X 9(H)

165

Theorem 8.7-3. Every positive, separately continuous, translation-invariant, sesquilinear form % on 9 ( H ) x 9 ( H ) has the representation

~ 4II/) ,= J’ d(Pr

4, II/ E WH),

$(t>>,

R

(5)

where P is a tempered PO measure. P is uniquely determined by PI.

PROOF. (i) Assuming for the moment that the representation ( 5 ) holds true, we first show that the tempered PO measure P is uniquely determined by ‘u. Let 4 = Ba and II/ = Ab, where 8 , I E 9 and a, b E H . By virtue of (1) of Section 2.6. Definition 2.6-3, and Theorem 2.3-4, we may write

BI(Bu, Ab) = J’d(P, 8(it)~, X(t)b)

By (3) of Section 2.3, this becomes

BI(Ba, Ib) = d(P, a, b)O(f)X(t),

(6)

where 8, E 2’. Now, 2’ * 2’ is dense in 9’ according to Lemma 8.4-2. Moreover, it follows from Theorem 8.7-1 that

c j d(Pra, b ) ~ ( t ) ++

is a continuous linear mapping of Y into C. Hence, given any a, b E H , 2t determines d(P,a, b)c(t) for all C E 9’.This means that (P,a, b) is uniquely determined as a a-finite complex measure on the bounded Bore1 sets in R (Appendix (312). Since this is true for every a, b E H , % uniquely determines P on those sets. A) A a(&, Ab), where 0, A E 9. (ii) Let a, b E H as before. Set Then, BI,, b is a separately continuous translation-invariant sesquilinear form on 9 x 9. Upon setting ’u, = a,,, and using the polarization equation (Appendix A7), we may write %‘u,,b

= &[PI,+, - % a _ ,

+

i%,+ib

- i%,- ib] .

Note that PI, is positive on 9 x 9. So, by Theorem 8.6-2, for each a E H , there exists a tempered positive measure p a such that

Upon setting

166

8.

THE ADMITTANCE FORMULISM

we have that p a , b is a tempered complex measure and

Note that B I uniquely determines Pl,, b for any given a, b E H . Hence, by the argument in part (i) of this proof, p a , b is uniquely determined by the given Iu, a, and 6 . (iii) We now show that, for each a, b E H , po,b has the representation p a , b ( E ) = (P(E)at

b),

(8)

where E is any bounded Bore1 set in R and P ( E ) E [ H ; H I , . First, note Bla, b(O, A) is sesquilinear in its dependence on a that, for any fixed 0, A E 9, and b. It follows from (7) that {a, b}Hp a , b(E) is a positive sesquilinear form on H x H because p a is a tempered positive measure. Furthermore, by Lemma 8.4-3, there exists a 8 E 9such that I g ( t ) I 2 1 for all t E E . So,

Now, '21 is separately continuous on 9 ( H ) x 9 ( H ) and therefore, according to Appendix F2, continuous. Hence, p,(E) IB1(0a, 0a) I Mllall', where M does not depend on a. Therefore, by Appendix D15, (8) holds, and P ( E ) E [ H ; HI,. (iv) Our next objective is to demonstrate that P is tempered. Since p a , b is a a-finite complex measure, it follows from Theorem 2.2-1 that P is a a-finite PO measure on 0,. We have to prove that gq is integrable with respect to P for some q. We start by showing that the mapping

is continuous from Y x H x H into C , as well as linear with respect to O and a and antilinear with respect to b. (It is understood that 9 'x H x H is supplied with the product topology.) Since both Y and H are Frtchet spaces, we need merely establish that the mapping (9) is separately continuous (Appendix F2). Since p a , b is a tempered complex measure, the mapping is continuous and linear on Y for fixed a and b. Now, assume that 0 and b are fixed and define the mapping Fe, b : H C by

-

For the moment, assume in addition that

8.7.

POSITIVE SESQUILINEAR FORMS ON

9 ( H ) x Q(H)

167

Then, by (7)) FO,

1

= dpo, b , I

= %a,

= %(la, lb)*

b(c,

This implies that F 0 . b is continuous and linear because % is separately continuous and sesquilinear. We now invoke Lemma 8.4-2; thus, for arbitrary 8 E 9, we can choose a sequence {6k}p=, such that each Ok has the form (10) and O k + 8 in 9. By what we have already shown, F e k , b E [H; C] and Fek,b(a) + Fe, b(a) for each a E H . It follows that Fg, b E [ H ; c]also (Appendix 011). A similar argument shows that the mapping (9) is continuous and antilinear with respect to b. We can summarize our results so far as follows. There exist two nonnegative integers q and k and a constant M > 0 such that, for all a, b E H with I(al(I 1 and (Ib((5 1 and for all 8 E Y with max sup 1(1 OspSk I e R

we have that

Ij

+ I t12q)e(p)(t)lI 1,

dkz.b, t e ( t )

1

(11)

M-

(See Appendix F2.) Now, let 8 E 9 be nonnegative and such that O(t) ej(t) = ( I

+ p12q)-18(j-1t),

=

1 for I tI I 1. Set

j = 1 , 2 , ...

.

Then, sup{l(l+ I t 1 2 q ) 8 y ) ( t ) : t E R , O I p S k , j = 1 , 2 , ...} P C < C O . So, by (1 l), for all a E H with llall I 1 ,

1

oI where again p a , I

dpo,

p a ,a ,

e j ( t ) IC M ,

j

=

i , 2 , ...,

So,

This proves that, for all j ,

/IS,

tlsj

d P 1 ( 1 + I t l 2 q ) - 1 [~H~; HI

ICM.

(See Appendix 015.) By Theorem 2.3-1, g, is integrable with respect to P. Thus, P is tempered.

8.

168

THE ADMITTANCE FORMULISM

(v) So far, we have shown that

a(&,I b ) = / d ( P , a , b)&t)X(t>

for all 0 , A E 9 and all a, b E H . From the sesquilinearity of 91, (1) of Section 2.6, and Definition 2.6-1, we see that (5) holds for all 4, $ E 9 0H . By the hypothesis and Appendix F2, 2l is continuous on B(H) x 9 ( H ) . Also, by Theorem 8.7-2, the right-hand side of (5) defines a continuous mapping on Y ( H ) x Y ( H ) and therefore on 9 ( H ) x .9(H). But, since 9 0H is dense in 9 ( H ) , 9 0H x 9 0H is dense in 9 ( H ) x 9 ( H ) . Therefore, ( 5 ) holds for all 4, $ E 9(H). 0

8.8. CERTAIN SEMIPASSIVE MAPPINGS OF 9 ( H ) INTO b ( H )

In this section, we study the properties of certain mappings of 9(H)into 6 ( H ) , which occur in subsequent realizability theorems. For any H-valued function 4 on R , we set

4AX) 4 40 - x) and

Lemma 8.8-1. For any

E

9 ( H ) andanypositivep,

m

qP$,(q) = i p

4(p)(t- x)(e-IXq - 1) d x , -m

This lemma can be established by integrating by parts.

Lemma 8.8-2. Let P be a PO measure and set j(x)

Sm -m

dP,,e'Xq.

8.8.

Also.for cjJ

E ~(H),

OF

SEMIPASSIVE MAPPINGS

~(H)

INTO cff(H)

169

set

(9Jl l cjJ)(t)

~ f~J(X)cjJ(t

(9Jl 2 cjJ)(t)

~ foo

- x) dx

(1)

j(x)cjJ(t - x) dx.

(2)

and

o

These equations are equivalent to

= foo

(9Jl I cjJ)(t)

dP,J>t(-,,)

(3)

-00

and

= foo dP~$[tl(-").

(9Jl 2cjJ)(t)

(4)

-00

Moreover,

fOO

(9Jl I cjJ(t), t{J(t)) dt

= foo d(P~$(, ),

-00

$(,,)),

cjJ, t{J

E

~(H),

(5)

-00

and

Re

r

(9Jl 2 cjJ(t), cjJ(t)) dt =

t foo d(P~ $(,,), $(,,)),

-00

cjJ

E ~(H),

r E R, (6)

-00

where in the last equation, t{J(t) ~ {cjJ(t), 0,

t s; r, t > 0.

(7)

PROOF: Equations (3) and (4) follow directly from Theorem 2.5-2, which also asserts that the integrals in (I) and (2) exist as Bochner integrals. It can be shown that x H j(x)cjJ(t - x), roll cjJ, and rol2 cjJ are all continuous H-valued functions. (See Problem 8.8-1.) Consequently, the left-hand sides of (5) and (6) exist. The right-hand sides of (5) and (6) also exist by virtue of Theorem 2.6-1. Moreover,

J

(rollcjJ(t), t{J(t)) dt

=

R

J (J R

=f =

dt

j(x)cjJ(t - x) dx, t{J(t)) dt

R

J(j(t - x)cjJ(x), t{J(t») dx

J J(J dP~ dt

ei(l-x)~cjJ(x),

t{J(t)) dx.

8.

170

THE ADMITTANCE FORMULISM

By Theorem 2.6-1 again, the right-hand side is equal to

1d(P, dxd,

&I>).

This verifies (5). Next, consider

f

Ir(4)

4(t),4(t))d t

(9~12

-m

=

1' f ( A t dt

-m

- x)4(x),

4 0 ) dx.

-m

Since the values of P, are positive operators and therefore self-adjoint, it follows that the adjoint ofjix) isj( -x). By using this fact, we see that

Upon reversing the roles of t and x in the right-hand side, adding I,(+) to -

Zr(4),and

using (7), we get

2 Re I,(+)

=

1'

dt

-m

j' ( A t - x > 4 ( x ) , 4(0) dx -m

An application of Theorem 2.6-1 as before yields (6). 0

In the rest of this section, Q is a tempered PO measure. Also, p where q = I , 3, 5, . . . ; hence, ip = - 1 . Set P(E) =

E

dQ,U

= 2q,

+ vP)-',

+

where E is any Bore1 set in R and p is chosen so large that (1 vP)-' is integrable with respect to Q. Thus, P is a PO measure according to Theorem 2.3-3. We define j as in the preceding lemma. Also, 23 and 9 denote the sesquilinear forms defined in Section 8.3; namely, for any operator ill from g ( H ) into L?'(H),

8.8.

SEMIPASSIVE MAPPINGS OF

Theorem 8.8-1. For any

9 ( H ) INTO € ( H )

171

4 E Q ( H ) , set

J-m

J-m

and

Then, !TIl and W, are linear translation-invariant semipassive mappings of 9 ( H ) into € ( H ) . In addition, !TI2 is causal on 9 ( H ) . Finally, for iN = W, or %=!TI2,

for all

4, $ E 9 ( H ) .

PROOF. As before, the integrals in (8) and (9) exist as Bochner integrals, and x w j ( x ) b ( t - x), W,4, and !TI2 4 are continuous H-valued functions. Clearly, !TI1 and W 2 are linear and translation invariant, and in addition %, is causal. If we can show that W land !TI2 are semipassive on 9 ( H ) , then it will follow from Theorem 8.3-1 that W, and !TI2 are continuous from 9 ( H ) into [ 9 ;HI. This will imply in turn that !TIl and ‘92, are convolution operators (Theorem 5.10-1) and therefore map 9 ( H ) into € ( H ) (Theorem 5.5-1). First, consider the case of W = !TIl. An application of Theorem 2.5-2 yields

(W14)(t)=

eiqX4(t- x ) dx

dP, R

+

R

I I dP,

R

For each k

=

(1 - einx)+(P)( t - x ) dx.

(11)

(1 - eiqx)@p)(t - x ) dx.

(12

R

1 , 2, . . . , set

+ 1- k dP, k

R

By using Theorems 2.2-3 and 2.5-1 and the extension of (2) of Section 2.5 onto 93 6H 3 !3(H), we obtain the estimate

I(%,, k 4(t),$(t)) 1 5 llP(R)ll[H;HI jR [II$(X)IlH + 2 ~ ~ ~ ( p ) ( x ) ~ / H l x dxll $(t)llH.

where

4, $ E 9 ( H ) .

(1 3)

8. THE ADMITTANCE FORMULISM

172

By Theorem 2.5-2 again, we may also write

It is not difficult to show that the quantities in both pairs of brackets on the right-hand side converge in the strong operator topology as k + 03 uniformly k $ ) ( t ) -, (g1$)(f) in H for each for all x E R.We can infer from this that (g1, t E R. This result in conjunction with (13) allows us to invoke Lebesgue's theorem of dominated convergence (Appendix G18) to conclude that, as

k-,

03,

I

I

$(t)) dt

(gl,k$(r),

R

(gl$(r),

R

$(l)) dt.

(15)

On the other hand, by Lemma 8.8-1, Definition 2.6-2, and Theorem 2.5-2 again, k (%I,

k

$)(t) =

-k

k

dP,(l 4- vIp)6t( - 'I)=

-k

dQ, 6t( - ?).

Therefore, by Lemma 8.8-2,

Im(fll,k4(0,NO)dt

=

-m

-k

d(Q,6('1)~$)-,J m d(Q, ~ ( ' I L$).

This implies (10). Indeed, the values of Q are positive operators and therefore self-adjoint. Hence, in view of Definition 2.6-3 and (1) of Section 2.6, we may write in effect

BB,($, 4) = /(%$, $1 dt = Sd(Q6,6> = 146, Q6) = Sd(Q$,

$1 =

I(%$,

$) dt

=

!%I~($, $1.

But, for 4 = $, the right-hand side of (10) is nonnegative. Hence, Rl is semipassive on 9(H).

8.8.

SEMIPASSIVE MAPPINGS OF

9 ( H ) INTO b ( H )

173

We now take up the case where % = !R2. Again by Theorem 2.5-2, (%2

4)(t) = 2 jRdP,

W

+2

eiVx4(t- x ) d x 0

sR

m

dP, 0

(1 - eiqX)@p) ( t - x ) d x .

Following our previous procedure, we set

+2

1

k

d P , s W ( l - e hX) + ( P ) ( t- X) d x 0

-k

and conclude again that

ask+co. As the next step, we shall show that

We now set 2, i r j ! k dP,( -q)"-'. Since the values of P are positive operators and therefore self-adjoint, it follows that 2,' = (- l)Tr, where r = 1, . . . , p - 1. Upon integrating by parts, we see that Re

/ (Zr

"(t), $(t)) d t = 0.

$J(~-

R

We infer from this result and (20) that

s

-t Re R ('%, k d'(th

cb(t>) d t = Re

/ ( f dQ, R

-k

&tl(

- q), $ ( t ) ) dt.

Since Q is a PO measure on [ -k, k], we may invoke (6) and take than any support point of 4 to equate the right-hand side to

-t

-k

T

larger

d(Q, 6(4,6Cd).

This establishes (19). Observe that (17) still holds in the present case. The combination of (17)-(19) establishes

%d4,4)

Re

R

(% 4(t), 4(t)) dt

=

/

R

d(Q, &v), ~ ( Y I ) ) .

(21)

174

8.

THE ADMITTANCE FORMULISM

Since the right-hand side is nonnegative, we have hereby proved the semipassivity of %2 on 9 ( H ) . Finally, we need merely apply the polarization equation (Appendix A7) to both sides of (21) in order to get (10). 0

’-

Problem 8.8-1. With P being a PO measure, prove that k

d P , eiqx--+

k

1

m

d P , e’“”

-m

in the strong operator topology uniformly for all x E R. Then, prove the assertion made in the proof of Lemma 8.8-2 that x H j ( x ) 4 ( t - x), !lJ331,$, and %N24are all continuous H-valued functions. Also, prove the assertion made in the proof of Theorem 8.8-1 that (%,, k 4)(t)+ (gz,4)(t) in H for every t E R.

8.9. AN EXTENSION OF THE BOCHNER-SCHWARTZ THEOREM

The Bochner-Schwartz theorem (Theorem 8.4-1) gives a representation of any complex-valued positive-definite distribution as the Fourier transform of a distribution generated by a tempered positive measure. Hackenbroch’s extension (Hackenbroch, 1969, Corollary 3.5) of this representation to operator-valued distributions is the subject of the present section. A similar extension is given by Kritt (1968). Definition 8.9-1. An f E [ 9 ( H ) ;HI is said to be positive-dejinite if, for all

4 EW H ) ,

Theorem 8.9-1. Corresponding to each positive-dejinite f E [ 9 ( H ) ;HI, there exists a tempered PO measure M such that,for every 4 E 9 ( H ) ,

Note. The customary definition of the Fourier transform of a complexvalued distribution (see, for example, Zemanian, 1965, p. 203) can be extended to the members of [ 9 ( H ) ;HI. By virtue of this, this theorem can be restated by saying that every positive-definite distribution in [ 9 ( H ) ;H ] is the Fourier transform of a tempered [ H ; HI-valued distribution generated by a tempered PO measure. In this regard, see also Theorem 8.7-1.

8.10.

CERTAIN CAUSAL SEMIPASSIVE MAPPINGS

175

PROOF.Set 3 = f *. Then, 3 is a continuous, linear, translation-invariant mapping of 9 ( H ) into b ( H ) . As usual, set

Then, 23 is a positive, separately continuous, translation-invariant, sesquilinear form on 9 ( H ) x 9 ( H ) . So, by Theorem 8.7-3, there exists a unique tempered PO measure Q such that

W 4 ,$1 =

1 R

d(Q, &I), $),

4, @ E g ( H ) .

(3)

Now, starting with Q , we proceed as in Section 8.8 to define P and j . We then define by (8) of Section 8.8. As was shown in the proof of Theorem 8.8-1, BR,($, $) is equal to the right-hand side of (3). Therefore,

j$W, *) dt = J (WJ, *) d t . R

Upon replacing $ by Oja, where a E H , j = 1, 2, . . . , and O j d a,S for an arbitrarily chosen z E R , we see that '3 = S l .Furthermore, (1 1) of Section 8.8 and Lemma 8.8-1 show that

(%4)(0 = JR dQrl&(-?I. Since 4Jx)

4(f - x) and

$(x)

+(-x),

we may write

( f , 4 >= (fn6)(0) = (%6)(0)

=

J

R

6,= 4 and

dQd%-r~).

This yields (2) when we set M(E) A Q( - E ) for any bounded Bore1 set E . 0 Problem 8.9-2. Show that, i f f (2), then f is positive-definite.

E

[ 9 ( H ) ;HI possesses the representation

8.10. REPRESENTATIONS FOR CERTAIN CAUSAL SEMIPASSIVE MAPPINGS

We now develop Hackenbroch's representations for linear, translationinvariant, causal, semipassive mappings on 9 ( H ) . The first result is a timedomain characterization.

176

8.

THE ADMITTANCE FORMULISM

Theorem 8.10-1. % is a linear, translation-itioariant, causal, semipassioe mapping on 9 ( H ) ifand only if,f o r every 4 E 9 ( H ) ,

+ rm[j(0) - j ( ~ ) ] # ~ ) (-t x ) d x , JO

where the following conditions are satisjied: PkE [ H ; HI and P k ' = (- I ) k " P k . (As before, Pk' is the adjoint of Pk .) Also, p = 2q, where q is an odd positive integer. Finally,

J

j(x) =

dP, e"jX,

(2)

R

where P , is a PO nieasure on the Bore1 subsets of R.

PROOF.We first derive (1) from the stated properties of 91. As in the proof of Theorem 8.9-1, 53% is a positive, separately continuous, translationinvariant, sesquilinear form on 9 ( H ) x 9 ( H ) . By Theorem 8.7-3, there exists a unique tempered PO measure Q such that

%(4, ICI) = 4

/

R

d ( Q , 6 ( ~ )$ (~! I ) ) .

(3)

We define P from Q as in Section 8.8, j by (2), and 912 by (9) of Section 8.8. We also set 934 fS2. Upon comparing (10) of Section 8.8 with (3), we see that 8, = 8 %Theorem . 8.8-1 also states that 91 is a linear, translationinvariant, causal, semipassive mapping on 9 ( H ) . Therefore, so, too, is % - 91.Since !&-, = 8n- Bg1= 0, it follows from Theorem 8.5-2 that

[(a- 9JI)@](f)=

f pk 4(k)(t),

k=O

4 E g(H).

This combined with the definition of 9JI yields (1). That any operator satisfying (1) possesses the stated properties follows from Theorem 8.8-1 and the identity Re

IR

(Pk4(k)(t),4(t))dt

which is a result of the condition pk'

= (-

= 0,

l ) k + ' P k .0

We turn to a frequency-domain characterization for the Laplace transform of the unit-impulse response of the operator %. With j defined by (2), we have that

8.10.

CERTAIN CAUSAL SEMIPASSIVE MAPPINGS

177

Moreover, for any q5 E E ( H ) , xHj(x)q5(x) is a continuous H-valued function according to Problem 8.8-1. As a result, the mapping

4 HJom.i(x)+(-v)dx + J m[i(o>- j ( x ~ p ) ( xdx ) 0

is a member of [9Ll(H);H I . Indeed,

and a similar estimate holds for the second integral. With 'illdefined by (I), we have from Theorem 5.5-2 that 91 = y y E [9Ll(H); H ] is defined on any q5 E L f a , b ( H ) with b < 0 < a by

+ /"[j(0) - j ( ~ ) ] q 5 ( ~ ) (dx. x) 0

*, where

(4)

By continuous extension, this equation is seen to hold for all 4 E gL,(H). Clearly, supp y c [ 0, a). By Lemma 7.2-1, we now have that y E [ Y ( O , 00 ;H ) ; HI. Consequently, the Laplace transform Y of y exists and, for every c E C , and a E H , we have ~ ( i > a [(L'y)(i)]a = (y(x), ae-i")

Theorem 2.5-2 allows us to reverse the order of integration. We then integrate with respect to x and note that a E H is arbitrary, to get

These results combined with Theorem 8.10-1 and the uniqueness property of the Laplace transformation yield the following frequency-domain realizability theorem. Theorem 8.10-2. If 91 is a linear, translation-invariant, causal, semipassive mapping on 9 ( H ) , then 92 is a convolution operator y * where y E [QL,(H);H ] satisfies (4). Moreover, Y ( c ) ( S y ) ( [ )existsfor at least all c E C, andsatisfies (5). Here, Pk , P, , and p satisfy the conditions stated in Tlieoren?8.10-1. Conversely, for every function Y having the representation (9,there exists a

178

8.

THE ADMITTANCE FORMULISM

unique convolution operator 91 = J * suclt that (2jj)(() = Y(4') f o r 4' E C, . Moreover, J E [BL,(H); HI, and y is gicen on any 4 E 9L,(H) by (4). Finally, '9l is a linear, tratislatioti-invariant,causal, seniipassive mapping on 9 ( H ) .

8.11. A REPRESENTATION FOR POSITIVE* TRANSFORMS

If in Theorem 8.10-2 the condition of semipassivity on 9 ( H ) is replaced by the stronger requirement of passivity on 9 ( H ) , then the representation (5) in the preceding section takes on a stricter form. In particular, the following additional conditions occur: n = 1, P , E [H; HI,, and p = 2. The proof of this result, which is given in the next section, is based upon a representation due to Schwindt (1965) for certain [H; HI-valued functions, which we call positive*. Dejnition 8.21-2. A function Y of the complex variable 4' is said to be a positive* niapping of H into H (or simply positive") if Y is an [H; HI-valued analytic function on C , such that R e ( Y ( l ) a ,a ) 2 0 for every 4' E C, and a E H.

When H = C, [C; C] may be identified with C, in which case the following classical result is in force. Theorem 8.11-1. F is a coniplex-valued positive* function for all 4' E C , , F adinits the representation

if and

only if,

Mhre X E R , X 2 0, arid 9l is a j n i t e positive measure on the Bore1 subsets of R . F uniquel), determines X aniong the cornplex numbers and 91 among the complex nieasures. A proof of this is given by Akhiezer and Glazman (1963, Section 59). Actually, in their version, N is a bounded nondecreasing function o n R and the integral is interpreted in the Stieltjes sense. However, this is entirely equivalent to the present statement; see Zaanen, 1967, pp. 33-34, 63-64. Schwindt's representation is a generalization of ( I ) to the case where F is [ H ; HI-valued, as follows.

8.1 1 .

A REPRESENTATION FOR

POSITIVE* TRANSFORMS

Theorem 8.11-2. Y is a positive* inupping of H into H [ E C , , Y can be representedby

179

if and only if, for all

wl?ereP , E [ H ; H I , , Po is a skew-adjoint member of [ H ;HI, and P,, is a PO measure on the Borel subsets of R. PROOF.Assume that Y is defined by (2). Since rl

(1 - irl5M5 - irl)

is a member of 9,it follows from (4) of Section 2.2 and Rudin (1966, p. 201) that the integral on the right-hand side of (2) is weakly analytic and therefore analytic on C , . Therefore, so, too, is Y. Furthermore, for all a E H , we have that (P,a, a) 2 0, ( P ( - ) a ,a ) is a finite positive measure, and (Po a, a) is imaginary. Also,

for all ( E C , . It follows that Re(Y(()a, a ) 2 0 for [ E C , and a E H . So, truly, Y is positive*. Conversely, assume that Y is positive* and set

F&a, 6 ) = ( Y ( i ) a , b),

a, b E H .

(3)

Thus, for each fixed 5 E C , , F , is a continuous sesquilinear form on H x H . Also, for fixed a E H , 5 H Fc(a,a ) is a complex-valued positive* function. So, by Theorem 8.1 1-1,

where, for each fixed a E H , X ( a ) is a real nonnegative number and N,(a) is a finite positive measure. The notation N,,(a) denotes the measure E H [N,,(a)](E),where E is any Borel subset of R. Upon setting Po 4 f [ Y ( l )- Y(l)’],we see immediately that Po is skew-adjoint and that

(Poa, a ) = i Im F,(a, a).

(5)

Now, from (3) and (4) and the fact that ,%‘(a)( is uniquely determined by FS(a,a), we see that X satisfies the following two identities. For all a, b E H and B E C, X(B4 =

IB I ’ x ( 4

(6)

8.

180

THE ADMITTANCE FORMULISM

and

X(a

+ 6) + X ( a - 6) = 2X(a) + 2X(b).

(7)

Similarly, N,(Pa) =

I B I",(a>

(8)

and

N,(a

+ b) + N,(a - b) = 2N,(a) + 2N,(b).

(9)

Next, set x(a, 6)

+

# [ X ( a 6) - ~ ( -a6) + iX(a + ib) - iX(a - ib)].

(10)

Any functional on H that takes on only nonnegative values and satisfies (6) and (7) defines through (10) a positive sesquilinear form x on H x H such that x(a, a) = X ( a ) 2 0.

(1 1)

(See, for example, Kurepa, 1965.) Similarly, we define the complex measure Q,(a, b): E H [Q,(a, b)](E)on the Borel subsets E of R by Q,(u, b)

$[N,,(a+ b) - N;(a - b) + iN,,(a + ib) - iN,(u - ib)]. (12)

Again (see Kurepa, 1965), for every E , {a, b}t+ [Q,(a, b)](E) is a positive sesquilinear form on H x H such that

[P,(a, a>l(E>= [N,(a)l(E) 2 0.

(13)

With these results, we see from (4), (1 I ) , and (13) that Re F,(a,

4 = x ( a , 4 + [Q&,

a)l(R>.

Thus, x(a,a> 5 IF,(a,

41 5

IIY(l>ll llal12.

We may now conclude from Appendix D15 that there exists a unique PI E [ H ; H I , such that (P,a, b) = x(a, b) for all a , b E H . In the same way, we see that, for every Borel set E c R, there exists a P,,(E)E [ H ; H I , such that (P,(E)a, b ) = [Q,(a, b)](E). In fact, Theorem 2.2-1 asserts that P, is a PO measure. Upon combining these results, we see from (4) that, for all a E H ,

where P o , PI, and P, satisfy the conditions stated in the theorem. Upon appealing to the polarization equation and Equation (4) of Section 2.2, we finally obtain (2). 0

8.1 1.

A REPRESENTATION FOR POSITIVE* TRANSFORMS

For any a E H and any

f~

E R, , (2)

181

yields

Both the real and imaginary parts of the quantity within the brackets are bounded on the domain {{a, q } : I < a < co,

-00

< q < 0O}

and tend uniformly to zero as f~ + 00 on any compact subset of the q axis. As a consequence, the integral tends to zero and a-'(Y(a)a, a ) tends to (P,a, a). By the polarization equation, therefore, lim a-'(Y(a)a, b ) = ( f l u ,b) a+ m

(14)

for all a, b E H . We shall make use of this result in a moment. When studying real passive operators in Section 8.13, we will meet a special type of positive* functions, namely, the positive*-real functions. In anticipation of this, we determine the special form that Schwindt's theorem assumes for these functions. Assume throughout the rest of this section that H is the complexification of a real Hilbert space H,. Definition 8.11-2. A function Y is said to be a positive*-real mapping of H into H (or simply positiue*-real) if Y is positive* and, for each real positive number a, the restriction of Y(a) to H , is a member of [H,; H,].

Lemma 8.11-1. Let Q be a skew-adjoint member of [ H ; H I . Then, Q is real ifand only if( Qa, a) = 0for all a E H , .

PROOF.If Q is real, (pa,a) is real for all a E H,. But, (Qa, a) is imaginary because Q is skew-adjoint. Therefore, (pa, a) = 0. Conversely, assume that (pa,a) = 0 for all a E H , and set Q = Q , + iQ, , where Q , and Q2 are real. Then, Q' = Q,' - iQ2'. Since Q is skew-adjoint, Q, = - P I ' and Q 2 = Q2'. By the preceding paragraph, ( Q l a ,a) = 0 for all a E H,, so that (Q2a, a) = 0. Since Q , is self-adjoint,

+ ib), a + ib): a, b E H,, /la + ibl/ = I ) , and, moreover, ( Q2(a+ ib), a + ib) is real. Therefore, ( Q2(a+ ib), a + ib) = ( Q 2 a, a) + b, b) = 0. lIQzll

= sup((Q2(a

(Q2

Hence, Q2 = 0, so that Q is real. 0

8.

182

THE ADMITTANCE FORMULISM

Theorem 8.11-3. Y is a positiiv*-real niappiiig of H into H ifarid onlj. fi tlie representation ( 2 ) possesses tile fidlowitig additional properties: Po and PI are real mappings, and P, = P - , . Note. The measure E , P - , ( E ) Li P,( - E ) .

P - , is defined as follows. For any Bore1 set

PROOF.Assume that P o , P , , and Pll have the stated properties. We may set P, = L, iM,, where the measure L, and M, take their values in [H,; H,]. It follows that L , = L - , and M, = - M - , . For i= c E R , , the imaginary part of the integral in (2) is

+

and this is equal to zero by the oddness and evenness of the integrands. Thus, Y ( c ) is real, so that Y is positive*-real. On the other hand, when Y is positive*-real, (14) shows that PI is real. Moreover, for any a E H,, we have ( Y ( l ) a ,a) = ( P l a ,a)

+ (Pea, a> + j" d(P,a, a). R

The left-hand side as well as the first and last terms on the right-hand side are real numbers. Therefore, so, too, is (Pea, a). But Po is skew-adjoint, which requires that (Pea, a) be imaginary. Hence, (Pea, a ) = 0. By Lemma 8.11-1, Po is real. We can now conclude that G(a)

/R

1 - irp LI(P,,0 , 6) 7 = ([ Y(a) - Pla - Po]a, b ) a - 1q

is a real number for all a, b E H , and all a E R , . It follows from the reflection principle that G(i) = G(5) for all iE C, . We infer from this that a,bEH,,

CEC,.

By the uniqueness assertion of -Theorem 8.1 1-1, ( ( P , - P-,)a, 6) = 0 for all a , b E H,, and therefore P, P -,. 0 2

Problem & I / - I . An [H; HI-valued analytic function Y is said to have a pole at a point i E C if, for some neighborhood R of z, for all iE R\(i}, and for some positive integer 11,

8.12.

POSITIVE* ADMITTANCE TRANSFORMS

183

where F, E [ H ; HI and the series converges in the uniform operator topology. The pole is called simple if n = 1. Assume that Y is positive* and has a pole at a point z = iq of the imaginary axis. Show that the pole is simple and F - , E [ H ;H I , .

8.12. POSITIVE* ADMITTANCE TRANSFORMS

We are at last ready to establish the realizability conditions for a linear translation-invariant passive admittance operator. Consider the distribution y E [ 9 ( H ) ;HI defined by

+ Cm[j(0)- ~ ( X ) I ~ ( ~ ) d( Xx ,) ' 0

where P ,

E

[ H ; H I , , Po is a skew-adjoint member of [ H ; H I , j(x)

1 d P , eiqx, R

and P , is a PO measure on the Bore1 subsets of R. As was shown in Section 8.10, y E [ g L , ( H ) ;H I , and ( I ) continues to hold for all 4 E g L , ( H ) .Moreover, the Laplace transform of y is precisely Schwindt's representation for a positive* mapping of H into H . One of the things we shall show is that (1) characterizes the unit-impulse response J' of a passive convolution operator. Lemma 8.12-1. Assume that y E [gL,(H); HI and that J' * is passive on 9 ( H ) . Choose c, d E R sucli that d < 0 < c. Then, y * is passive on L?c.d(H). PROOF.Choose a, b E R such that d < h < 0 < a < c. As was shown in the proof of Lemma 7.4-2, given any 4 E L?c,d(H), we can find a sequence {+j}y= c 9 ( H ) that converges in L?,,b(H) to 4. Upon setting 11/ = y * 4 and $ j = y * c $ ~we , obtain from Theorem 5.5-2 that t,hj 11/ in B ( H ) . The lemma now follows from the estimate --f

-T

"7

184

8.

THE ADMITTANCE FORMULISM

Theorem 8.12-1. If % is a linear translation-invariant passive operator on 9 ( H ) , then % is a convolution operator y * ,where y E [ g L , ( H ) ;H ] satisfies (1). Moreover, Y !i?y exists and is positive.* Conversely,for every positive* mapping Y of H into H , there exists a unique convolutionoperator% = y *such that ( 2 y ) ( [ )= Y ( [ )forall[ E C, . Moreover, y is represented by (l), so that y E [ g L , ( H ) ;HI and supp y c [0, GO). Also, '9l is a linear translation-invariant passive mapping on 9 ( H ) .

PROOF.Assume that the operator '3 is linear, translation-invariant, and passive on L@(H). By Theorem 8.2-1, % is causal on L@(H). So, by Theorem 8.10-2, '3 = y *, where y E [ Q L I ( H ) ; HI and supp y c [0, GO). By Lemma 7.2-1, y E [ Y ( O , G O ; H ) ; HI so that Y 4 i?y exists and has a strip of definition containing C+ . We will now show that Re(Y([)a,a ) 2 0 for all [ E C , and a E H . To this end, we first invoke Theorem 5.5-2 to write ( f l d ) ( t )= ( Y ( 4 , 40 - 4)

for all 4 E Y C , d ( H ) ,where d < 0

-= c. By Lemma 8.12-1, for any T ER,

Fix upon a [ E C , and choose X > T.Also, choose O E Ep such that O(t) = err on -a < t < X and O(t) = 0 on X + 1 < t < GO. Finally, fix c, d E R such that -Re [ < d < 0 c c. Then, 0 E Y cd,. Upon putting 4 = Oa in (2) and noting that 4 ( t - x ) = aei(' - x ) for - 00 < t < T and for all x in some neighborhood of supp y , we can manipulate (2) into

I

T

Re( Y ( [ ) u ,u )

e2

dt 2 0.

- W

This implies that Re( Y([)a,a) 2 0. Thus, Y is positive*. We now appeal to Schwindt's representation and the uniqueness property of the Laplace transformation to conclude that y satisfies (I). This establishes the first half of our theorem. Next, assume that Y is positive*. By Schwindt's representation, Y = Qy, where y has the representation (1). In view of Theorem 8.10-2, the remaining statements of Theorem 8.12-1 are all clear except for the assertion that 3 4 y * is passive on 9 ( H ) . To show this, let T ER , 4 E . 9 ( H ) , and a E H . Since Po is skew-adjoint, (Poa, a) is imaginary, so that

8.12.

POSITIVE* ADMITTANCE TRANSFORMS

185

so that

Now, set

(%,4)(t)

+/

/m.i(x)q5(t - x) dx 0

m

[ j ( O ) - j ( ~ ) I @ ~ ) (t x> dx.

0

The substitution of the definition ofj(x) and an application of Theorem 2.5-2 converts this into (%14)(t) =

J

dP,&t,(-V)

+ JR dP,[-M#J(t) + q2&1(-v)1.

We are using here the notation defined at the beginning of Section 8.8. For any positive integer k , we define !JIl,k by k (%i,k+>(f) -k

q 2 ) - iQk#’(f),

dPq&t](-q)(l

(5)

where Q k = J!-k dP,q. Note that (&a, a) = J!-k d ( ~ , aa)q, , which is a real number. Hence,

By the argument given in the proof of Theorem 8.8-1 [see, in particular, (18) of Section 8.81,

Re

lT

T

(ml,k4(t>,

-m

4(t))d t

Re J-:%l

6(f>,4([>)d t

(7)

a s k -+ 00. We define the measure M on any Borel subset E of [ - k , k ] by

M(E)

J

E

dP,(1

+ q2>

and set M ( J ) = 0 if J is a Borel set that does not intersect [ - k , k]. Upon identifying M with the PO measure P given in Lemma 8.8-2, we may invoke (6) of Section 8.8 in conjunction with ( 5 ) and (6) to write

8.

186

THE ADMITTANCE FORMULISM

In view of (7),

Upon combining (3), (4), and (8), we see that 9 is passive on 9 ( H ) . 0 Theorem 8.12-1 states the fundamental realizability conditions relating to positive* functions. A similar proof for it, which, however, is not based on Theorem 8.10-2, is given by Zemanian (1970a). As is indicated there, we need merely assume that 9 is linear, translation-invariant, and passive on 9 0H in order to establish the first half of Theorem 8.10-2. It should also be noted that Theorem 8.12-1 possesses an extension to n dimensions (that is, to the case where 9 = g R ,is replaced by BR,,)due to Vladimirov (1969a, b).

8.13. POSITIVE*-REAL ADMITTANCE TRANSFORMS

Here we show how the realizability conditions for an admittance operator are sharpened when that operator is real. Once again, we assume that H is the complexification of a real Hilbert space H , . Theorem 8.13-1. If 9 is a linear translation-invariant passive operator on 9 ( H ) and Y'iTl maps 9 ( H , ) into [ 9 ( R ) ;H , ] , then 9 = y *, where y E [aLI(Hr); H , ] satisjies (1) of Section 8.12 with PI, P o , and P , restricted as in Theorem 8.1 1-3. Moreover, Y g f!y is positive*-real. Conversely, if Y is a positive*-real mapping on H into H , then Y = f!y on C , , where y is a member of [ g L , ( H , ) ;H , ] and satisfies ( I ) of Section 8.12 with P , , P o , and P, restricted as in Theorem 8.1 1-3. Moreover, 9 g y * maps 9 ( H , ) into [ 9 ( R ) ;H , ] and is a linear translation-invariant passive operator on 9 ( H ) .

PROOF.We have already seen that every positive* function is the Laplace transform of a unique y E [ g L , ( H ) ;HI given by ( I ) of Section 8.12. We now invoke Theorems 8.1 1-2 and 8.11-3, which give the necessary and sufficient conditions on P,, P o , and P, in order for Y to be positive*-real. In this case, y E [ g L , ( H r ) ;H , ] because the imaginary part of dP,ei'fxequals zero by virtue of the condition P , = P - , . The second half of our theorem now follows from the second half of Theorem 8.12-1. Under the hypothesis of the first half of the theorem, we can prove that y E [ 9 ( H r ) ; H , ] and Y ( o )E [ H , ; H , ] for every o E R, exactly as in the proof of Theorem 7.6-1. Hence, Y is positive*-real, and, by Theorem 8.12-1 again, 9 = y *, where y has the stated properties. 0

8.14.

PASSIVITY AND SEMIPASSIVITY CONNECTION

187

Problem 8.23-2. Assume that % is a linear translation-invariant passive operator on 9 ( H , ) with range in [ 9 ( R ) ;H,]. Show that '3 has a unique extension as a linear translation-invariant passive operator on 9 ( H ) .

8.14. A CONNECTION BETWEEN PASSIVITY AND SEMIPASSIVITY

Some conditions under which a semipassive convolution operator is passive is given by the next theorem, the scalar version of which was given by Konig and Zemanian (1965). Its proof is based upon the following observations. The function j in the representation ( I ) of Section 8.12 is a strongly continuous [ H ; HI-valued function by virtue of Problem 8.8-1. Therefore, j is strongly measurable, and, by the principle of uniform boundedness, it is bounded in the uniform operator topology on every compact set. Thus, in accordance with Problem 3.3-2, j defines a member of [ 9 ;[ H ; HI] (which we also denote b y j ) by means of the equation ( j , o)a

l j ( x ) a e ( x ) dx,

aEH,

e E 9,.

Now,

converges in the strong operator topology and defines a strongly continuous function of x. Therefore, (1) is also a member of [ 9 ;[ H ; HI]. Through an integration by parts, (j:j(t)l+(i)

) 1:

dt, --e(')(x) a =

j(x)aO(x) d x = ( j , 0 ) a .

By our usual identification between [ 9 ( H ) ; HI and [a;[ H ; HI], j F j ( x ) $(x) dx is the value assigned to any q5 E 9 ( H ) by the generalized derivative of (1). In the same way, [ j ( O ) - j ] l + is strongly continuous and therefore a member of [ 9 ;[ H ; HI]. Thus

Jorn[j(o)- ~ ( X > I ~ ( ~dYx X ) is the value assigned to any q5 E 9 ( H ) by the second generalized derivative of [ j ( O ) - j l l + * However, for p 2 4,

8.

188

THE ADMITTANCEFORMULISM

is not the value assigned to 4 by the second generalized derivative of a strongly continuous [ H ; HI-valued function. This is because f 4 D’{[j(O) - jJl+} and therefore D”-’{[j(O) - j J l + }are not strongly continuous. To show this, let m

ek(x)

e - k 2 x 2 / / - m e - k 2 xd2x ,

k

= 1,

2, . . . .

Since Ok E 9’c QL, and [ j ( O ) - j ] l + E [QL,(H);H I , we may write, for any aEH,

((fa,

ek>,

a) = ( ( W [ j ( O ) - j I l

+I, e k > a ,

a ) = ( ( [ j ( o )- j N + , e:’))a, a )

An application of Theorem 2.5-2 and two integrations by parts convert the right-hand side into

1

m

d(P, a, a)q2

Ok(x)eiqxd x .

0

This is equal to

41 d(P,,a, a)q2[exp(- q’/4k2)] erfc( - iq/2k), where erfc denotes the complementary error function (see Zemanian, 1965, pp. 350, 357). But Re erfc( - iq/2k) = 1, and thus (2) clearly does not tend to zero as k + 00. It would have to do so if f A D’{[j(O) - j ] l + } were strongly continuous at the origin because Ok -,6 and supp f c [ 0, 03). Finally, we note again that Y 4& !i?y is positive* if and only if y is represented by ( 1 ) of Section 8.12. In view of these results, we need merely compare Theorems 8.10-1 and 8.12-1 in order to conclude the following. Theorem 8.14-1. Let y E [Q(N); HI. Then, y if y satisfies the following conditions.

+

* is passive on Q ( H ) ifand only

(i) y = PI6(” w ,where PI E [ H ; HI+ , is as usual theJirst generalized derivative of the delta functionaI, w is the second generalized derivative of an [ H ; HI-valued function on R that is continuous with respect to the strong operator topology, and supp w c [0, 03). (ii) y * is semipassive on Q(H).

8.15

189

ADMITTANCEAND SCATTERING FORMULISMS

8.15. A CONNECTION BETWEEN THE ADMITTANCE AND SCATTERING FORMULISMS

With %: U H U and !XI: q H r denoting respectively the admittance and scattering operators for a Hilbert port, the variables u, u, q, r E [ 9 ;HI satisfy v=q+r,

u=q-r.

(1)

If a particular Hilbert port has an admittance operator, need it have a scattering operator as well? No. For, a substitution of (1) into %v = u yields %r

+ r = %q - q.

Upon setting % = - 6 *, we see that only q = 0 will satisfy this relation, whereas any r will do. This means that 2B:q H r does not exist as an operator. A similar manipulation shows that % will not exist as an operator when '123 = -6 *. However, when % is a linear translation-invariant passive operator on Q ( H ) , !XI exists and is a linear translation-invariant scatter-passive operator on Q ( H ) . The converse is not true in general, but it will be true if we assume in addition that I + S(() possesses an inverse in [ H ; HI for each ( E C , , where I is the identity operator on H and S is the scattering transform. To show these results, we first establish a connection between S and the admittance transform Y.

Lemma 8.15-1. Let Y be positive*. Then, for every ( E C , , I possesses an inverse in [ H i HI. Moreover, S ( . ) P [ I +Y(.)]-"I-

is bounded *.

+ Y(() (2)

Y(.)]

PROOF.Fix ( E C, and set F A I + Y ( ( ) . We wish to show that F-' exists. Set b = Fa, where a E H . Then, 2(a,a) I 2 ( a , 4

+ ( Y o , 4 + (0, Y a ) = (6, a) + ( a , b) 5 21(a, 611 S211all llbll,

This implies that F is injective. F ( H ) is a closed linear subspace of H . Indeed, let b be a limit point of F ( H ) and choose a sequence {b,} c F(H) that converges to 6. Then, b, = Fa,, where a, E H . By (3), llan - a m l l

IIF(an - am)II

=

llbn - bmll

-+

0.

8.

190

T H E ADMITTANCE FORMULISM

So, {a,} is a Cauchy sequence and converges therefore to an a E H. Thus, Fa, + Fa by the continuity of F, whereas Fa,, = b, -+ b. Hence, b = Fa E F(H). If F ( H ) were a proper subspace of H, there would be an a E H with a # 0 such that (Fa, a) = 0 (Berberian, 1961, p. 71). That is, ( [ I + Y(c)]a, a) = 0. But this cannot be since Re( Y([)a, a ) 2 0. Hence, F is surjective. Thus, F - ' exists on H. Moreover, F - ' E [H; HI according to Appendix D14. In the same way as in the scalar case, it can be shown that [ I + Y( . )I-' is analytic at every point [ where I + Y([) has an inverse and Y is analytic. (In this regard, see Problem 8.15-1.) Thus, [ I + Y( . )I-' and therefore S are [H; HI-valued analytic functions on C+ . Upon premultiplying ( 2 ) by I + Y and rearranging the result, we obtain Y(Z + S ) = I - S. Thus, for any a E H, Ilal12 - IIS(0all2 = Re([[ - S(i)la, [ I + S(0la) =

Re( Y(O[l

+ S(C)Ia, [ I + S(0la) 2 0.

This implies that IlS([)II I1 for all ( E C+ . We have hereby shown that S is bounded*. 0 Now, assume that '!Jl is a linear translation-invariant passive operator on 9(H). Therefore, 91 = y *, where Y A 2 y is positive*. Moreover, for any t, E 9 ( H ) , u A y * u is a Laplace-transformable member of $B+(H) whose strip of definition contains C, . Consequently, q = t(v + u) and r = +(u - u ) have the same properties. Upon substituting ( I ) into u = y * u, taking Laplace transforms, and rearranging the result, we obtain [I

+ Y(OIN0 = [I - Y(OlQ(5),

iE C+ .

(4)

In view of Lemma 8.15-1, this may be rewritten as

44')= [ I + Y(Ol-l [I -

Y(i)lQ(O

= S(4')Q(09

iE C+ ,

where S is bounded*. Applying the inverse Laplace transformation, we get

r=s*q,

(5)

where s A S - ' S and r and q correspond to the given u E $B(H) as above. We use (5) to define the scattering operator '2u s * on other q and in particular on 9 ( H ) . Since S is bounded*, the next theorem follows immediately from Theorem 7.5-1. Theorem 8.15-1. LcJt 91 br a linear translation-iiii~ariarit passioe operator on Q(H) and let Y be the corresponding admittance transform. Define S by ( 2 ) andset s Q - ' S . Then, XI A s * is a linear translation-iiioariant scatter-passive

operator on 9(H).

8.15.

191

ADMITTANCE AND SCATTERING FORMULISMS

Let us now go into the opposite direction, starting with a given scattering operator 2B and deriving from it an admittance operator '3. We first note that this may not always be possible even when 1113 is linear, translation-invariant, and scatter-passive on 9 ( H ) . A counterexample is 1113 = -6 *. Lemma 8.15-2. Assume that S is bounded* and that, for every [ E C,,

I

+ S([)possesses an inverse in [ H ; HI. Then, Y( . )

[I

+ S( . )]-"I

- S( * )]

(6)

is positive* and satisjies (2). PROOF. We again have that [ I + S( )]-I is an [ H ; HI-valued analytic function on C , . Therefore, so, too, is Y. Since I + S commutes with I - S, ( 6 ) yields

Y(Z + S ) = I - s.

(7)

Now, let b E H be arbitrary and fix [ E C , . Since [ I an a E H such that [ I + S([)]a = b. Therefore,

+ S(4')I-l

exists, there is

Re( W b , b ) = Re( Y[Z + S([>la, [ I + S(0la) = ReW - m > l a ?[ I + S([)Ia> = llallZ - IIS(5)al12 2 0. Thus, Y is positive*. Lemma 8.15-1 now shows that [ I This allows us t o solve (7) to get (2). 0

+ Y([)]-'

exists.

To obtain an admittance operator from a given scattering operator, we proceed in just about the same way as for Theorem 8.15-1 except that now we rely on Lemma 8.15-2. Theorem 8.15-2. Let B ' 3 be a linear transfation-invariant scatter-passive operator on 9 ( H ) , and let S be the corresponding scattering transform. Assume that I + S ( ( ) possesses an inverse in [ H ; HI for every [ E C , . Also, define Y by ( 6 ) and set y 2-l Y. Then, '3 4 y * is a linear translation-invariant passive operator on 9 ( H ) , and the scattering operator generated by '3 in accordance with Theorem 8.15-1 coincides with 2B. Problem 8.15-1. Let G be an [ H ; HI-valued analytic function on C , such that G([)-' exists for all [ E C, . Choose any z E C , . Show that, for all [ E C , such that

II [G(4

- G(r)lG(z)-

II
0 and a finite collection {yl, . . . , y,} c I' such that

p(f(4)) 5

f €I

max Yk(4)

1SkSm

CONTINUOUS LINEAR MAPPINGS

209

for all ~ E YIf .Y is a FrCchet space, the following three assertions are equivalent : (i) E is bounded in [ Y ;W ] . (ii) E is bounded in [ Y ;WIu. (iii) Z is equicontinuous. D10. The following is one version of the closedgraph theorem: Let Y and W be FrCchet spaces and f a linear mapping of Y into W . Assume that, for every sequence { c $ ~ which } converges to zero in Y and for whichf(4,) converges to t,b in W , we always have that I) = 0. Then, f is continuous.

D11. Let A and B be normed linear spaces with norms I I . I I A and l l * l l B , respectively. The bounded topology of [ A ; B ] is equal to the topology generawhere ted by the single norm 11. I([A;

(The supremum notation means that the supremum is taken over all a E A for which llallA = I.) This topology is called the uniform operator topology. When B is a Banach space, [ A ; B ] under the uniform operator topology is a Banach space, too. For any fixed a E A , the mapping y o : f H 11 f(a)llB is a seminorm on [ A ; B ] . The topology generated by thecollection { y a : a E A } of seminorms is called the strong operator topology of [ A ; B ] . It is the same as the pointwise topology of [ A ; B ] , defined in Appendix D8. In this case, if A is a Banach space, [ A ; B ] under the strong operator topology is a sequentially complete separated space; if, in addition, {fk}km, converges in this topology tof, then llfll I h~+mllhll. Finally, for any given a E A and b‘ E B’ (8’ is the dual of B), Y ~ , ~ , : ~ H Ib‘(f(a))I is also a seminorm on [ A ; B ] . The topology generated by the collection { y o , 6 , : a E A , 6‘ E B’} of seminorms is called the nieak operator topology of [ A ; B ] . Each of these topologies separates [ A ; B ] . D12. Upon applying Appendix D9 in the context of Appendix DI I , we obtain the principle of uniform boundedness: If E is a subset of [ A ; B ] and if ~~p,,,llf(~ ~~ ll dlpl llg(.>lld l p l . We define F d p A s f d p , where f is any member of F. The notation L,(T, (5; p ; A) = Ll(p; A) denotes the linear space of all equivalence classes F, and this space is a Banach space under the norm 11 * where

s

=s

s

IIFIIL,

'J

T

ll.f(*N4~lPl9

f E F .

220

APPENDIX G

When T is R",6 the collection of all Bore1 subsets of R",and p Lebesgue measure, we denote L,(p, A) simply by &(A), and L,(C) by L,. It is customary to represent F by any member f E F and to replace F by f in all manipulations on the members F of L l ( p ; A ) . In fact, we shall say that the members of L l ( p ; A ) are the functions f and will maintain the tacit understanding that f should really be replaced by the equivalence class F to which f belongs. 614. Let 9(T,6 ;C ) be the space of all complex-valued functions g on T that are the limits under the norm ( 1 . JIG, where llgllG supfETIg(r)I,of sequences of simple functions on T. Thus, every member of 9(T, 6 ;C ) is a bounded function. Iff E L,(T, 6 ;p ; A ) and g E Q(T,6 ;C ) , then f g E L,(T, 6 ; p ;A ) . Furthermore, if T, X , 6,and 6' are as in Appendix G3 and if I E G(T x X , 6 x 6';C ) , then, for each fixed x E X , we have I(-, x) E G(T, 6 ;C). G15. I f f is a p-measurable function on T into A and if )If(?)[IA where g E L,(T, 6 ;p ; R),then f E L,(T, 6 ;p ; A ) .

Ig(t),

616. Let Llo(T,6 ;p ; A ) = L l 0 ( p ;A ) denote the linear space of all integrable A-valued simple functions on T. [Here again, it is understood that the members of L l 0 ( p ;A) are really equivalence classes of functions differing ffom the simple functions on no more than p-null sets.J Given anyf E L l ( p ; A), we can choose a sequence {hk}km,1c L I o ( p ;A ) such that hk + f i n Ll(p; A ) and hk(t) -+ f ( t ) almost everywhere on T. (Any one of the subsequences mentioned in the first paragraph of Appendix G I 0 will do.) Consequently, L l 0 ( p ;A ) is dense in L l ( p ; A ) . Moreover, L , ( p ; A ) is the completion of Ll0( p ;A ) under the norm 11 [I L,.

G17. Let A4 be a continuous linear mapping of thecomplexBanach spaceA into another such space B. Iff € L 1 ( T ,6 ;p ; A ) , then M f ( * )E L ~ ( T6,;p ; B) and

j f& In particular, i f f dual of B, then

E L,(T, 6 ;p ;

=

JWv) 4.4

*

[ A ; B J ) , a E A , and 6'

E

B', where B' is the

,

G18. The following is the theorem of dominated convergence. If {fk}p= c L l ( p ; A ) , if Ilfk(t)ll Ig(t) for each k,where g E L 1 ( p ;R ) , and iffk(t) +f ( t ) for

221

THE BOCHNER INTEGRAL

-

almost all t E T, then f E L , ( p ; A) and, as k + and JT h dp $T f dp.

(319. Let p E R and p 2 1. Let {fk}:= simple functions on T such that slIfk(')

03,

JT

Ilh(t)-f(t)II

dlptl - 0

be a sequence of integrable A-valued

+o

-fj(')IIAp

as k and j tend to 00 independently. Then, there exists an equivalence class F consisting of all A-valued measurable functions f on T such that any two members of F are equal to each other for almost all t E F and, for any f E F, the integral

s

Ilfk(

exists and tends to zero as k + and

s Ilf(

')//Ap

'

)-f(

00.

=

*

)IIAp

dl p1

Moreover,

lim

k-tm

1

Ilfk(

11 f ( ' ) l l A P ')IIAp

E

Ll(T, 6 ;lpl ;R),

dlpl.

The space of all such equivalence classes F is denoted by L,(T, 6 ;p ; A ) = L p ( p ;A ) , and it is a Banach space under the norm 11. [IL,, where

As before, we shall represent F by any one of its membersf. Moreover, we shall speak of L p ( p ;A ) as consisting of all f in all F, the partitioning of the space of all such f into equivalence classes being understood. It is a fact that f E L p ( p ;A) if and only i f f is an A-valued measurable function of R" and IIf ( . ) I I A E Lp(p;R). In the special case where is the a-algebra of Bore1 subsets of R" and p is Lebesgue measure, we denote Lp(p;A) by L,(A) and Lp(C)by L,.

G20. The following is an extension of Holder's inequality. Let p , q E R be such that p > 1 and p - ' + 4 - l = 1. Moreover, let f E L,(T, 6 ;p ; A) and g EL&T,6 ;p ; C ) . Then,fg E L , ( T ,6 ;p ; A ) , and A

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Index of Symbols

A.4 W,l64 B,4 B('lI, j"; tr), 196 91(B),10 7 18= 18lJl' 153 ~=~lJl' 153 18p,43

Q:.(A),5 0 ~+

meA), 67

e: m(A), 67
15,

G(A),24 25 Q:; C),220 ~(T x X, Q: x '1:'; C),4O ~,(H) = fI,(T, Q:; H),46 ~o(A) = ~o(T, Q:; A), 24 ~(A), ~(T,

en,3 C+,131 ~Km(A), 57 '1:,23 '1:"",29 diam, 4 D"14 ~(A), 50 ~J[(A), 50 ~(A), 65 ~J[m(A), 50 ~Lp (A), 72

H,4 .;tf',81 inf,4 1,145 J(A),65 J"'(A),6 5 JJ"'(A), 64

lim, 4 lim, 4 225

226

INDEX OF SYMBOLS

L 1(JL ; A) = L 1(T, Q:; JL; A), 219 L 1 0 (JL ; A) = L 1 O(T, Q:; JL; A), 220 L~oC (H), 150 L,(JL; A) = Lp(T, Q:; JL; A), 221 !l'c. d(A), 1I8 !l'~ a(A), 66 !l'(w, z; A), 1I8 !l'm(w, z; A), 66 2, 1I8 max, 4 min,4 %[T], 147 9/,78 .P,23 R+,3 R",3 Re",3 9l[Tj, 147

SSVar,24 sup, 4 supp,4,55 SVar,24

"p.,,82 pJ,p,64

a,,54

0,2

0.,2

Some Additional Special Symbols

1+, 58, 73

MI., 12 ~,

4

0,4 .,90

*,97 -,155 0,37,61,69

6,37 (8),214 @,215 ,." 10, 14, 24 [·j,4

11'11 = II'IIA, 204 11'11.. 37

.'I'm(A),66

(., '),4, 205

Var,24 vol, 4 :!E, 156

(', 'j, 4 [-, '),4 'j, 4 [.; 'j, 3, 208 [.; 'j., 68

Yc,d.t.

r.

1I8

yo, 50 Yp. 0, 72

[.; -r. 3, 68, 208

Ye,53,68 SJ. 0, 57 0.,12 K c • d, 66, 1I8

'1-,2 ·.....·,2 'H',2