Linear Differential Equations and Function Spaces

Linear Differential Equations and Function Spaces

PURE A N D APPLIED MAT H EMAT I C S A Series of Monographs and Textbooks

Edited by

PAULA. SMITHand SAMUEL EILENBERC Columbia University, N e w York 1: ARNOLD SOMMERFELD. Partial Differential Equations in Physics. 1949 (Lectures on Theoretical Physics, Volume V I ) 2: REINHOLD BAEX. Linear Algebra and Projective Geometry. 1952 3 : HERBERT BUSEMANN ANn PAUL KELLY.Projective Geometry and Projective Metrics. 1953 4 : STEFAN BERCMAN A N D M. SCHIFFER. Kernel Functions and Elliptic Differential Equations in Mathematical Physics. 1953 5 : RALPH PHILIP BOAS,JR. Entire Functions. 1954 6: HERBERT BUSEMANN. The Geometry of Geodesics. 1955 7 : CLAUDE CHEVALLEY. Fundamental Concepts of Algebra. 1956 8: SZE-TSENH u . Homotopy Theory. 1959 Solution of Equations and Systems of Equations. Second 9: A. M. OSTROWSKI. Editiqn. 1966 10: J. DIEUWNNE.Foundations of Modern Analysis. 1960 11 : S. I. GOLDBERG. Curvature and Homology. 1962 12: SICUR~UR HELCASON. Differential Geometry and Symmetric Spaces. 1962 Introduction to the Theory of Integration. 1963 13 : T. H. HILDEBRANDT. ABHYANKAR. M Local Analytic Geometry. 1964 14 : S H R E ~ A 15 : RICHARD L. BISHOPAND RICHARD J. CRITTENDEN. Geometry of Manifolds. 1964 16: STEVENA. GAAL.Point Set Topology. 1964 17: BARRYMITCHELL. Theory of Categories. 1965 18: ANTHONY P. MORSE.A Theory of Sets. 1965 19: GUSTAVE CHOQUET. Topology. 1966 20: 2. I. BOREVICH AND I. R. SHAFAREVICH. Number Theory. 1966 AND JUAN JORCE SCHAFFER. Linear Differential Equations 21 : JOSk LUIS MASSE~U and Function Spaces. 1966 22 : RICVARD D. SCHAFER. An Introduction to Nonassociative Algebras. 1966 I n preparation: MARTINEICHLER.Introduction to the Theory of Algebraic Numbers and Functions. FRANCOIS TREVES. Topological Vector Spaces, Distributions, and Kernels. OYSTEINORE.The Four Color Problem.

Linear Differential Equations and Function Spaces JOSE LUIS MASSERA JUAN JORGE SCHAFFER INSTITUTO DE MATEM~TICAY ESTAD~STICA UNIVERSIDAD DE LA REPI~BLICAORIENTAL DEL URUGUAY

MONTEVIDEO, URUGUAY

1966

ACADEMIC PRESS

.

New York and London

COPYRIGHT 0 1966,

BY

ACADEMIC PRESSINC.

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

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

United Kingdom Edition published by ACADEMIC PRESS INC. (LONDON) LTD. Berkeley Square House, London W.l

LIBRARY OF CONGRESS CATALOG CARDNUMBER: 65-26043

PRINTED IN THE UNITED STATES OF AMERICA

This book presents in systematic and detailed form the recent studies of the authors concerning linear ordinary differential equations in the real domain. Rounding off the development of an idea going back to the work of 0. Perron (1930), the theory presented here thoroughly discusses, in the central part of the book, relations between properties of the nonhomogeneous equation-typically, “admissibility” of a pair of function spaces, i.e., the property of the equation having, for each “second member” in one space, at least one solution in the other-and the behavior of the solutions of the homogeneous equation-typically a “dichotomy” or an “exponential dichotomy”, i.e., a kind of uniform conditional stability, ordinary or asymptotic, respectively. There are additional chapters on several connected topics, e.g., almost periodic equations and periodic equations. Considerable emphasis is placed on the methods of functional analysis and on the use of function spaces. T h e theory is developed for equations in a Banach space, but its significance does not depend on this generalization of the usual finite-dimensional setting. T h e book is addressed primarily to readers interested in ordinary differential equations, who will be best prepared to understand its motivation; but no specialized knowledge in this field is required. In functional analysis, a working acquaintance with Banach-space theory, both “soft” and “hard” (but no intimate knowledge of operator theory), is assumed. T h e theory expounded here has wide applications to nonlinear problems, a treatment of which has been omitted for reasons of space.

This Page Intentionally Left Blank

Preface The present book is devoted to the study of certain problems concerning ordinary differential equations, which in a very loose sense pertain to stability theory. Let (1)

k+Ax=O

be a homogeneous linear differential equation and

the corresponding nonhomogeneous equation. Here x represents a function with values in some (real or complex) Banach space X, i = dx/dt, the real independent variable t ranging, say, over R, = [0, 00); A is a mapping of R , into the algebra of (bounded) operators-i.e., endomorphisms-of X, and f is a mapping of R, into X. We do not dwell in this Preface upon the precise statement of the assumptions (of CarathCodory’s type) on the mappings A, f, hor upon the meaning of the term “solution” as applied to Eqs. (1) and (2) under those assumptions ; the essential features of the solutions are however, the same as in the classical (continuous and finite-dimensional) context. The two main classes of properties under investigation are typified by “admissibility” and “dichotomy”, respectively, described as follows.

+

x,

(I) Admissibility. Let B, D be two function spaces, consisting of mappings from R, into X. We say that the pair ( B , D ) is admissible for Eq. (2)-more precisely, for A-if for each f E B there exists a solution x of (2) such that x ED. We consider only the case in which B, D are Banach spaces such that convergence in norm implies convergence in the mean on each compact subinterval of R, . A basic result, obtained by use of the Open-Mapping Theorem, then states (Theorem 51.A) that there exists a solution x E D with its D-norm bounded by a fixed multiple of the B-norm o f f ; thus, in a restricted sense, the concept of admissibility is seen to be related to that of total stability. vii

...

PREFACE

Vlll

(11) Dichotomy. Suppose that we have introduced a suitable concept of “angular apartness” between nonzero elements of X;for instance, the angle itself if X is a Hilbert space or, in general, the angular distance y [ x ,y] = 11 11 y II-ly - 11 x 11-l~11. We say that a (closed) subspace Y of X induces a dichotomy of the solutions of (1) if there exist positive constants y, N , N’, yo such that the following properties (described here in provisional form) hold:


, N’-leY’lt-to)ll

z(to)ll,

with positive constants v and v’, we speak of an exponential dichotomy. I t is clear that these two concepts are closely related to those of uniform conditional stability and conditional exponential (uniform asymptotic) stability, and that in the special case where Y = X they actually coincide with those of uniform stability and exponential stability, respectively.

As far as we know, Perron [2] was the first to point out the importance of these properties and to show that admissibility of the pair ( C , C ) (where C is the space of all bounded continuous functions from R, into X) is under rather general assumptions equivalent to the existence of an exponential dichotomy. After him other authors (e.g., Bellman [I], Kreln [l], KuEer [l], MaIzel’ [l]) endeavored to extend Perron’s results to more general situations. Bellman and KreIn first suggested the use of methods of functional analysis, in the shape of the Banach-Steinhaus Theorem; this is suited to the case of stability ( Y = X; and X finitedimensional), to which they and KuEer restricted themselves. More specific historical references are given in the Notes at the end of Chapters 4-6. The present authors took up the problem in a more general context in 1956-1958, introducing the application of the Open-Mapping Theorem. They published their results in three joint papers, Massera

PREFACE

ix

and Schaffer [l-31, which are the first of a series under the general title, “Linear differential equations and functional analysis”. It was shown later, in a research begun jointly and continued by Schaffer, that much greater generality could be gained, and a kind of “natural context” of the properties of admissibility and dichotomy reached; together with ample developments of other topics and ideas of the preceding papers, this forms the contents of parts IV-IX of the series (Massera and Schaffer [4]; Schaffer [3,5-81). P. Hartman, in addition to contributing to the research reported in Schaffer [5], investigated higher-order equations (Hartman [l]). One of the main instruments for these generalizations is the theory of certain classes of functions spaces, especially of spaces that possess properties of translation invariance ; this theory was developed by Schaffer [ l , 2,4]. This process of step-by-step generalization is responsible for a certain amount of disorder and repetitiveness in the above-mentioned papers, which also obscure the progressive laying bare of the essential structure of the problem. It has thus seemed advisable to write down a clear-cut, systematic account of the theory; this book is the result. We have of course used this opportunity to round out the study of several aspects of the previous research, thus including some new results, and to correct some errors that had passed unnoticed. It was inevitable that our present terminology and notation should deviate somewhat from that used in the original papers. In his very recent book on ordinary differential equation, Hartman gives an account of a part of this theory (Hartman [2], Sections XI1 6-7, Chapter XIII). His exposition describes some abstract methods that promise to be useful for other applications (cf. Section 120). Most of the authors who have dealt with such problems as we consider in this book, notably Perron [2], were motivated less by questions concerning the linear equations themselves than by the application of the theory to weakly nonlinear equations, i.e., equations of the form f+Ax=h

where h: R, x X + X is, in some appropriate sense, “small” as a function of its second argument. Further contributions along these lines, with specific use of the methods introduced in the above-mentioned papers, may be found in the following references: Massera and Schaffer [I], Massera [4], Corduneanu [1, 21, Hartman and Onuchic [I], Hartman [2] (Sections XI1 8-9), to mention a few. The theory in this book may be similarly applied. The main tools for such an application are methods of successive approximation and fixed-point

X

PREFACE

theorems (both Banach’s “trivial” theorem on contractive mappings and Tychonoffs Fixed-Point Theorem). In the general context considered here, the first step in this direction would certainly be a result like Theorem 6.1 of Hartman and Onuchic [l]. Such a study of nonlinear equations would exhibit, incidentally, how close the relationship between admissibility and total stability actually is. We have reluctantly decided to forgo in this book all discussion of nonlinear equations and to remain strictly in the linear domain; we should otherwise have had to face an excessive increase in the already considerable bulk of the work. We should feel that we had missed our aim if the significance, if any, of the theory set forth in this book were taken to reside primarily in the fact that much of it concerns the equations in a general Banach space X instead of, as usual, in a finite-dimensional space (with a distinguished basis thrown in). Indeed, the fundamental structure of the theory, and the power of the functional-analytic approach, are quite sufficiently apparent in the finite-dimensional case. However, the very nature of the methods and results makes the extension to infinite-dimensional spaces so natural that we have gladly paid the price in technical complication that it entails; this price is slight in view of the conceptual complexity involved in any case in the use of function spaces. Our decision has actually produced a considerable benefit in conceptual clarification-as is so often the case with this very kind of extension-which has helped to enrich and simplify the theory even in the finite-dimensional case. We have tried, however, to steer clear of any questions motiwated by the infinite dimensionality of the space, with few exceptions, mainly those required to justify the limitations imposed on certain results (for the signposts to such exceptions, see below); a few of these exceptions are of some intrinsic interest, e.g., Section 66. and the discussion of Floquet’s Theorem in Section 111. On the other hand, strictly finitedimensional methods, such as matrix traces, determinants, subdiagonalization, normal forms, and the like, have been excluded. Surprisingly little in the matter of results seems to be lost in the process (cf. Massera and Schaffer [l], Section 9). Bearing out this point of view, we have entirely disregarded any possible extension of the theory to the case in which the values of A are unbounded operators in X . Such an extension would certainly be of the greatest interest, especially in view of possible applications to partial differential equations. Unfortunately, we have been unable so far to obtain any satisfactory advance in this direction; some fragmentary results, which do not, however, constitute really natural and significant generalizations, have been published by Halilov [ 13 and Domglak [ 1-31.

PREFACE

xi

This book is primarily addressed to readers interested in differential equations, rather than to specialists in functional analysis. Familiarity with the former field is therefore assumed; as far as the latter is concerned, the reader is expected to have a working acquaintance with Banach-space theory, both “soft” and “hard”. More detailed information on this point is contained in the Notes to Chapters 1 and 3. Beyond this, the book is substantially self-contained, without being aggressively so. A very conspicuous exception is Chapter 2, as explained in the Introduction thereof. There are twelve chapters, grouped into three parts. Part I, containing Chapters 1-3, includes preliminary material that does not properly belong to the subject matter of the book as sketched in the preceding discussion. Chapter 1 deals with several topics in the geometry of Banach spaces; Chapter 2 is devoted to the study of the classes of function spaces that are best adapted to the theory of admissibility; and Chapter 3 discusses general properties of linear differential equations and of their solutions. In order that Chapter 2 should not become excessively long, we have limited ourselves to the essential features of the theory and have omitted the proofs of several theorems (to be found in Schaffer [ l , 2,4]). Part I1 is the core of the book, where the main development of the ideas described above takes place. It consists of Chapter 4, on dichotomies; Chapter 5 , on admissibility and important variants thereof; Chapter 6, on the relations between dichotomies and admissibility for equations on R, = [0, +03); Chapter 7, on the dependence of dichotomies and admissibility on the operator-valued function A ; and Chapter 8, where similar problems about equations on R = (- 03, 03) are studied. Part I11 includes complements to, and special cases of, the general theory; viz., Ljapunov’s method (Chapter 9), almost periodic equations (Chapter lo), periodic equations (Chapter 1 I), and higher-order equations (Chapter 12).

+

Chapters are divided into sections: Chapter 6 contains Sections 60 (Introduction) to 67 (Notes); a similar method of numbering applies in each chapter. Sections are divided in turn into unnumbered subsections. The manner of lettering certain items alphabetically (such as theorems, examples, etc.) and of numbering the formulas consecutively throughout each section will be obvious to the reader. It sometimes happens that a theorem is stated in one section and its proof deferred to another. Whenever such a result is used before its proof is given, the fact is brought to the reader’s attention.

xii

PREFACE

A word or phrase in SMALL CAPITALS is the definiendum in a formal definition. Items or whole subsections marked co in the margin are of interest only for an infinite-dimensional space X. Those marked mco concern only a space X containing (closed) subspaces without (closed) complement; i.e., essentially (modulo isomorphisms and a well-known conjecture), a Banach space that is not a Hilbert space. Finally, items marked mmcoare relevant only for a nonreflexive space X. For the convenience of the reader, smaller type has been used for such passages, as well as for certain others that deal with topics that lie outside the mainstream of the theory and may therefore be skipped without disturbing its development. Results obtained in them are not used in the rest of the work. The reader may amuse himself by seeking the two intentional exceptions to this arrangement. Ever since a colleague introduced the thick vertical bar to denote “end of proof”, many have foilowed his example, and many others have felt challenged to design variants and improvements. Inasmuch as our book fairly bristles with thick vertical bars-although they are not that thick-and we do wish to make some significant contribution with this work, we have risen to the challenge with a device that, we hope, will appeal not only to the nautically-minded, but also to those readers who, weary after the crossing of many a vast and uncharted proof, would appreciate the feeling of having reached port safely. The research leading to this book was carried out, as already outlined, by both authors at the Instituto de Matemhica y Estadistica, Universidad de la Rep6blica (Montevideo), and also, during 1960, by Schaffer as a fellow of the John Simon Guggenheim Memorial Foundation at the University of Chicago and at RIAS (Baltimore). The work on the book itself was done by both authors at the above-mentioned Institute, and by Schaffer also at Carnegie Institute of Technology (Pittsburgh) during the academic year 1964-1965; a course given at the latter institution on part of the subject matter discussed here afforded a valuable opportunity for correcting several errors in the manuscript. The first draft of Chapters 2, 3, 7, and 9 was written by Massera, that of the remaining chapters by Schaffer; all chapters were rewritten, revised, and given their final form by both authors jointly. We gratefully acknowledge the valuable assistance of Professor H. A. Antosiewicz, who was kind enough to read the whole manuscript critically. His suggestions have been very helpful; had we found ourselves able to implement more of them, this might indeed have been a better book.

PREFACE

...

XI11

We wish to thank our colleagues at the various institutions mentioned above for comments and suggestions, too numerous to be detailed, during the elaboration of this book. Among them stand out, of course, the contributions of Professor P. Hartman alluded to above. Thanks are also due Mrs. Rebeca E. de Noachas for an arduous task well done. It remains to put on record our appreciation to all at Academic Press, whether or not known to us by name, who spared neither time nor effort to offer valuable editorial advice, to meet our every typographical whim, and to give this book its fine material form.

March, 1966

Joslf LUISMASSERA JUAN JORGE SCHAFFER

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Contents PREFACE

vii

PART I Chapter 1. Geometry of Banach spaces

3

10. Introduction Summary of the chapter selections

Terminology and notation

Continuous

7

11. Angles, splittings, and dihedra Angular distance and related concepts Splittings Dihedra

13

12. Coupled spaces Coupled spaces Subspaces Selections and splittings Reflexive spaces

13. The class of subspaces of a Banach space

18

Two metrics The complemented subspaces The class of closed dihedra A lemma on continuously varying subspaces

26

14. Hilbert space Notation

Angles, splittings, and dihedra

The set of subspaces

31

15. Notes to Chapter 1 Chapter 2. Function spaces

33

20. Introduction Summary of the chapter Further terminology and notation for abstract Functions and function spaces. The relations “stronger than” and < spaces The Lebesgue spaces Lp(X) The space L(X) = R and J = R, . Translation operators

41

21. JY-spaces The lattice N ( X ) Banach spaces in N ( X ) Local closure Completion The relation + xv

CONTENTS

xvi 22. 9-spaces

46

The operators k, , k, f. Lean The lattices 9 and b 9 . Local closure and full spaces The class 9~ Associate spaces Thin spaces The The class S ( X ) domains R, R- , R, . Cutting and splicing at 0

57

23. F-spaces The classes .Y, 9- The operator T- The class FK Associate spaces in Y The functions or(F; Z), B(F; I ) ; the spaces L’,Lm,Lr Thin spaces ,Y+(X) Thick spaces Cutting and splicing at 0 The classes .Y(X),

76

24. Spaces of continuous functions 99-spaces The class S i f ( X ) YP-spaces on ] on ] = R

=

R , and .TV-spaces

25. Notes to Chapter 2

83

Chapter 3. Linear differential equations

84

30. Introduction Summary of the chapter Primitives

31. Solutions

86

Existence, uniqueness, and formulas for the solutions Bounds for the solutions The closedness theorem

32. Associate equations in coupled spaces Associate operator-valued functions Formula

89

Associate equations

Green’s

33. D-solutions of homogeneous equations

92

D-solutions and their initial values Examples and comments on associate equations

34. Notes to Chapter 3

A result

97

PART I1 Chapter 4. Dichotomies

40. Introduction

101

41, Ordinary dichotomies

I02

Definition Dichotomies and solutions in .T-spaces

Examples

CONTENTS

xvii

42. Exponential dichotomies

110

Definition Exponential dichotomies and solutions in Y-spaces

Example

43. Dichotomies for associate equations Dichotomies for associate equations ,X,+ manifolds X,,

117 Ordinary dichotomies and the

44. Finite-dimensional space

120

45. Notes to Chapter 4

122

Chapter 5. Admissibility and related concepts

124

50. Introduction Summary of the chapter Pairs of Banach function spaces

126

51. Admissibility Definition and boundedness theorem Regular admissibility Admissibility and local closure Some remarks on the admissibility of fl-pairs and elated pairs on R , Sets of admissible pairs Inadmissible pairs Equations with scalar A on R ,

D)-manifolds 52. (B,

138

Summary of the chapter (concluded) (B,D)-manifolds (B, D)-manifolds and admissibility (B, D)-subspaces Y-pairs and related pairs Sets of pairs

53. (B,D)-manifolds, admissibility, and the associate equations The polar manifold of a (B,D)-manifold A result on admissible fl-pairs

149

D)-subspaces and the associate equations 54. (B,

155

The polar manifold of a (B, D)-subspace Implications of admissibility for the adjoint equation Sets of (B,D)-manifolds and -subspaces for Y-pairs and related pairs

55. Finite-dimensional space

160

56. Notes to Chapter 5

162

Historical notes

Complemented (B, Lm)-subspaces

Chapter 6. Admissibility and dichotomies

60. Introduction

165

61. The fundamental inequalities

167

xviii

CONTENTS

62. Predichotomy behavior of the solutions of the homogeneous equation

170

Means and slices of solutions Pointwise nonuniform properties of solutions Miscellaneous corollaries

D)-subspaces, and dichotomies: 63. Admissibility, (B, the general case

179

Ordinary dichotomies Exponential dichotomies Sets of pairs

64. Admissibility, (B,D)-subspaces, and dichotomies: the equation with A E M(X)

188

The main theorems Sets of pairs

65. Examples and comments Examples with constant A Counterexamples for the direct theorems

I92

Counterexamples for the converse theorems Examples in infinitedimensional space Estimation of dichotomy parameters

66. Behavior of the solutions of the associate homogeneous equation Implications of the existence of a (B,D)-subspace Implications of the existence of a mere (B,D)-manifold A question about dichotomies

21 1

67. Notes to Chapter 6

22 1

Chapter 7. Dependence on A 70. Introduction

223

D)-subspaces 71. Admissibility classes and (B, Admissibility classes (B,D)-subspaces

224

72. Dichotomy classes

237

Exponential dichotomies Ordinary dichotomies

73. Connection in dichotomy classes: Banach spaces

245

Deformation families Connection by arcs in dichotomy classes

74. Connection in dichotomy classes: Hilbert space

25 I

A bit of motivation Two geometrical lemmas Deformation families Exponential dichotomies: the general case Exponential dichotomies: the exceptional case Ordinary dichotomies Finite-dimensional space

75. Notes to Chapter 7

269

CONTENTS

xix

Chapter 8. Equations on R

80. Introduction

27 I

8 1. (B, D)-dihedra and admissibility

273

The fundamental theorems Some further results

82. Double dichotomies. Connections with admissibility and (B, D)-dihedra

279

Double dichotomies Examples Connections with admissibility and Predichotomy behavior of the solutions of the homogeneous equation

(B,D)-dihedra

83. Associate equations

293

84. Dependence on A

296

Admissibility classes and closed (B,D)-dihedra classes Connection in double-dichotomy classes

Double dichotomy

PART 111 Chapter 9. Ljapunov’s method

31 1

90. Introduction Summary of the chapter Pointwise properties of the solutions. Exceptional sets

316

91. Ljapunov functions Ljapunov functions Total derivatives

92. Exponential dichotomies

320

93. Ordinary dichotomies

327

94. Notes to Chapter 9

332

Chapter 10. Equations with almost periodic A 100. Introduction

333

Summary of the chapter Spaces of almost periodic functions Almost periodic equations and solutions. Preliminary facts

101. The condition

= (0)

102. Exponential dichotomies

338 34 I

CONTENTS

xx

103. Reflexive and finite-dimensional spaces

343

104. Notes to Chapter 10

345

Equations on R,

The theory of Favard

Chapter 11. Equations with periodic A 110. Introduction

348

Summary of the chapter Spaces of periodic functions Properties of U

111. Floquet representation

35 1

112. Periodic equations and periodic solutions

354

113. The solutions of the homogeneous equation

358

D-solutions Double dichotomies Examples Exponential and double exponential dichotomies

114. Individual periodic equations

369

Chapter 12. Higher-order equations 120. Introduction

373

Summary of the chapter nth primitive functions

121. The (rn

+ 1)st-order equation

376

D)-manifolds 122. Admissibility and (B,

38 1

123. The main theorems

386

REFERENCES

393

INDEX.Author and subject

399

Notation

402

PART I

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CHAPTER 1

Geometry of Banach spaces 10. Introduction

Summary o f the chapter I n this chapter we prepare the geometrical apparatus which we shall use throughout the whole book. Except for some introductory remarks on terminology and notation, we are interested in the geometry of a Banach space or of a pair of Banach spaces in duality. In Section 11 we discuss as thoroughly as will be needed various concepts having to do with “apartness” of elements and subspaces, which replace in a certain sense the concept of angle, proper to euclidean spaces. The technical difficulty inherent in the fact that, in general, a subspace need not be complemented (see below) is overcome in part by the concept of a “splitting”, which replaces the projections associated with a pair of complementary subspaces. T h e properties of pairs of linear manifolds spanning the whole space (“dihedra”) are also examined. Section 12 is devoted to the geometrical properties of pairs of Banach spaces “coupled” by means of a bilinear functional; the simplest and most important case-essentially the only one if the spaces are reflexive -is the evaluation coupling between a Banach space and its dual. The questions treated are in part precisely those discussed in Section 11 for a single Banach space. It is often necessary to decide when two subspaces of a Banach space are “close” to one another; in other words, to define a topology, or even a metric, in the set of all subspaces of a Banach space. This has been done in various ways, all generalizing essentially the same idea in euclidean space. Section 13 gives a summary of some known definitions and of the results which are relevant to our applications. If the Banach space considered is a Hilbert space, the facts in Sections 11, 12, and 13 are expressible in terms of angles, orthogonality, and other “euclidean” geometrical objects and relations. Section 14 3

4

Ch. 1. GEOMETRY OF BANACHSPACES

summarizes this reduction. No further appreciable simplification is obtained by assuming finite dimension, except that, of course, all linear manifolds are then closed. The reader who is not particularly interested in the technical details of the theory in a general Banach space may skip the proofs in Sections 11, 12, and 13. The contents of some subsections (including all of Section 13) are not required until after Chapter 6. There is appropriate warning of this fact, and such passages may be passed over until called for at the appropriate time. Terminology and notation The usual terminology for set theory, vector spaces, and topology is used; we only point out a few items of our specific usage in what follows. The empty set is denoted by 0; set-theoretical differences by \; belonging and not belonging of elements to sets by E and $, respectively. All vector spaces have as scalar field-occasionally denoted by Feither the real or the complex field; the former is denoted by R. Since we usually wish to deal indifferently with real or complex scalars, the terms “real part” and “conjugate” (meaning complex conjugate), etc., and the corresponding notations Re and -, etc., are to be forgiven as harmless redundancies in the real case. The identity mapping in any vector space is denoted by I. A LINEAR MANIFOLD is an additive homogeneous set (vector subspace) in a vector space. An ordered pair (M, N) of linear manifolds in a vector space E is a DIHEDRON if M + N = E. If, in addition, M n N = (0) (so that E is the algebraic direct sum of M, N), the dihedron (M, N) is DISJOINT. An ALGEBRAIC PROJECTION in E is an idempotent linear mapping of E into E. Null-space and range of an algebraic projection, in that order, constitute a disjoint dihedron, and this correspondence between algebraic projections and disjoint dihedra is bijective; the algebraic projection corresponding to a given disjoint dihedron is termed ASSOCIATED WITH the dihedron and described as the algebraic projection ALONG M ONTO N; if P is thus associated with (M, N), then I- P is associated with (N, M). If S is a topological space, the closure of any set W C S is denoted by cl,W or, if no confusion is likely, by clW. Similarly, limits in S, or S-LIMITS, may be denoted by lims. If X is a normed space, the norm is usually denoted by (1 I( or, if absolutely essential, by 11 * Ilx; among the exceptions are the function spaces introduced in Chapter 2. The one-dimensional normed space over the scalar field F is identified with F, and its norm is accordingly written I 1. For every x E X we shall write sgn x = (1 x 11-l~if x # 0,

5

10. INTRODUCTION


0, p > 1 be arbitrary. There exist yi E Yi such that 11 x - yi 11 E , i = 1, 2. But y 1 - y , = z1- z2 , with ziE Y , , 11 zi(1 p~lly, y , I/ ~ P E Ki, = 1, 2. Now Y1 - 2 1 = Y 2 - z2 E Yl n Y , 3 II x - (Yl - 1I),. \< II x - Y1 II II z1 II (1 2 p ~ ) ~Since . was arbitrarily small, x E cl(Y, n Y,); hence cl Y , n cl Y , C c1( Y , n Y,); the reverse inclusion is trivial, The condition is suficient: Assume cl Y , n cl Y , = cl( Y , n Y,). Now (cl Y , ,cl Y2),being a closed dihedron, is gaping by 11.J. Set K = K(c1 Y , , cl Y J , and let x E X be given and p > 1 be arbitrary; then x = u1 u, with ui E cl Yi , Ij ui 11 p ~ l xl 11, i = 1, 2; but ( Y , , Y,) is a dihedron, hence x = z1 z2 with ziE Yi ,i = 1,2. Then u1 - z1= - ( u Z - z,) E cl Y , n cl Y , = cl(Y, n Y2);hence there exists v E Y , n Y, such that 11 u1 - z1- v 11 = 11 u, - z, v 11 ( p - 1)pKll x 11; we set y1 = z1 zi E Y , , y 2 = z, - v E Y,; then x = y1 y , and IIyi 11 I1 ui I1 11 ui - Y i I1 d (1 p - 1)pKIl x )I = p 2 4 ) x 11, i = 1, 2. Hence ( Y , , Y,) is gaping, with K ( Y , , Y,) K = K(c1 Y , , cl Y,). Actually, by 11 .I, equality holds. &


. More generally, if there exists a number s y > 0 such that Z( Y)(O)C s y Z ( X ' / Y o )i.e., , such that (12.4)

d(Yo,x')

< s y sup{I ( y , x') I :y E Z(Y)}

for all x'

E X',

we say that Y has the QUASI-STRICT COUPLING PROPERTY. T h e values of s y verifying the inclusion or (12.4) have a minimum; we shall always assume s y to be this minimum itself. T h e n s y 3 1, and s y = 1 corresponds to the strict coupling property. Since YOo0= Yo, if Y has the sy. [quasi-] strict coupling property, so has Y>'O,and s y o o T h e most important general cases of subspaces having the strict coupling property are described in the following lemma:


0, if and only if Z has characteristic sy' > 0 with respect to Zoo (Dixmier [I], Theoritme 7'); and Dixmier [l] has given an example, for X = lo", of a weakly* dense subspace of X * that has characteristic 0, and therefore does not have the quasi-strict coupling property. Remark 2. 12.A states that for the particular coupled pair X , X * every saturated subspace of either space has the strict coupling property. It is an open question whether this statement, or at least its weaker form involving the quasi-strict coupling property, remains true for every coupled pair of Banach spaces. T h e remainder of this subsection concerns dihedra, and will not be required before Chapter 8.

12.B. Let Y , Z be linear manifolds in X . Then ( Y ,Z ) , (Yo,Zo) are both dihedra, in X , X ' , respectively, i f and only i f there exists a o ( X , XI)-

16

Ch. 1. GEOMETRY OF BANACH SPACES

continuous projection P in X such that P X = 2, ( I - P ) X = Y ; then both dihedra are disjoint, Y , Z are saturated subspaces, and Y , 2, Yo,Z o all have the quasi-strict coupling property. Proof. If ( Y ,Z ) , (Yo,Zo) are dihedra, we have YO0 n Zoo = (YO + 2 O ) O = X'O = {0}, YO0 + 200 3 Y + Z = X ; Y o n Zo = ( Y + Z)O= XO= {0}, so that ( YOo,Zoo),( Y o Zo) , are disjoint closed dihedra. Since Y C YOo,Z C Zoo, we must have equality; i.e., Y , Z are saturated subspaces, and ( Y ,2) is a disjoint closed dihedron. Let P be the projection of X along Y onto 2, and Q the projection of X' along .To onto Yo. For any x E X,x' E X' we have (Px, ( I - Q)x') = ( ( I - P)x, Qx') = 0, whence (Px, x') = ( P x , Qx') = ( x , Qx'); in other words, Q = P', the associate operator of P ; as stated in the preceding subsection, it follows that P is o(X, X')-continuous. Assume, conversely, that P exists as specified in the statement; then ( Y , 2) is obviously a disjoint closed dihedron; the associate operator P' exists and is obviously a projection in X'; and we verify immediately that. = Yo,(I' - P')X' = Zo;then (YO,Zo) is a disjoint closed dihedron. We prove the quasi-strict coupling property for 2, the other cases being analogous. Let x' E X' be given. We have

P'X'

d(Z0, x')

< (1 Px' (1 = sup{( (x, Px') 1 : x E Z ( X ) } = sup{I ( P x , x') I : x E qx)} < (Yl x'> I :Y E II p II Z:(Y)) SUP{I

= II p

so that the property holds, with sz

II S U P 0

< I( €'[I.

(Yl

x')

I

:Y E ~ ( Y > } ,

9,

If ( Y , 2) is a dihedron satisfying the conditions of 12.B, we say that it is (X, XI)-DISJOINT;the dihedron (YO,Zo) is then (X', X)-DISJOINT. An immediate consequence of 12.B (of interest pending an answer to the open question in Remark 2 above) is: 12.C. If Y is a subspace of X that is either the null-space or the range of a u ( X , XI)-continuous projection, then Y has the quasi-strict coupling

property. In the most important cases, the definition of an dihedron can be simplified:

(X, X')-disjoint

X*)-disjoint if and only if it 12.D. A dihedron ( Y , 2) of X is (X, is disjoint and closed. A dihedron ( V , W ) of X* is (X*, X)-disjoint i f and only i f it is disjoint and V , W are saturated, i.e., weakly* closed.

12. COUPLED SPACES

17

Proof. T h e “only if” parts follow from 12.B. Assume that (Y, Z) is disjoint and closed: then the projection P along Y onto Z is, as is every operator, weakly continuous, i.e., u(X, X*)-continuous. Assume that (V, W) is disjoint and V, W are saturated. Set Y = Vo, Z = Wo; then Y, Z are subspaces of X with Yo= V, Zo = W, Y n 2 = (V W ) O = X*O = (0). Let Q be the projection of X* along Zo onto Yo. On the other hand, consider the linear manifold U = Y Z C X and let R be the algebraic projection of U along Y onto 2. Under the canonical monomorphism Y : X + X**, R becomes the restriction to Y U of Q*, the adjoint projection of Q; therefore R is bounded. Since Y, Z are complete, it follows from 1O.A that U is complete, hence a subspace of X. But then Y Z = U = Uoo= ( Y 2 ) O 0 = ( Y on Z0)O = (O}O = X, so that Q = R* is weakly* continuous, i.e., o(X*, X)-continuous. ,$,

+

+

+

+

Remark. It has been shown (e.g., Schaffer [6], Example 2.1) that for general coupled Banach spaces X, X‘ a disjoint dihedron (Y, Z) of X with Y, Z saturated need not be (X, X’)-disjoint.

Selections and Splittings We give two lemmas concerning the existence of certain continuous mappings.

12.E. For any h > 1 there exists u continuousfunction n : X’ \ (0) + X such that ( n ( x ’ ) , x’) = 11 x‘ (1 and 11 n(x’)ll A.


1 a real number. For any h > 1 and any x‘ E X‘, II x’ II h d( Yo,x’), there exists a continuous ( Y ,p(l + hs,))-splitting Y ( * ; x’) of X such that (~(x; x’) = 0 for all x E X .


l = p14,; then (1 y (1 (1 x‘ I( Ap1/2sY.We then set r(x; x’) = q(x) - (q(x), x’)y. Obviously, Y ( - ; x‘) is continuous and satisfies conditions (a) and (b) for a (Y, -)-splitting; also, (r(x; x’), x’) = ,) n, n = 1, 2 , ... , which is absurd; the set of continuous functions in F is therefore not closed, contradicting (c). (b) implies (a). If /3(F; 0) = 0 there exists a sequence (l,), 0 < 1, < 1, such that B(F; I,) r3. Define the nonoverlapping compact inter&], n = 1, 2, ... . There exists a continuous vals ], = [ n - 81, , n rp(t)nr< (n) = n function 'p on ] s o that 'p(t) = 0 for t 4 U 1,, and 0 ' for t E In,n = 1, 2, ... . Then rp is unbounded, but rp = xJnyE F 1 since I xJ,'p IF n/3(F;1,) < co; this contradicts (b). &


0 i f and only ;f F is weaker than L’. (2) Assume that F ditions are equivalent:

E

b y + and set so = so(F). The following con-

(a) a(F; 0) > 0; (b) F is weaker than O,oL1; (c) F is weaker than 0,L’for some

7

E

R,

Proof. See Schaffer [l], Lemma 4.14 and Theorem 4.21 for the proof of (2). If J = R , , (1) is a particular case of (2); if J = R, the proof of (1) is entirely similar and simpler. 9, Similar results hold for 1 --+ co: 23.S. Assume that F E b y K . The following conditions are equivalent: (a) P(F; 00) < 00; (b) F is weaker than LF; (c) lcF, is weaker than L“ (equivalently, than L:). The preceding conditions are implied by, and i f F is quasi locally closed are equivalent to: (d) F is weaker than L“; (e) F contains a sti8 continuous function.

Prooj. The equivalence of (a), (b) is proved in Schaffer [l], Theorem 4.22. By (23.11), &?(F; co) < P(lcF,; co) < P(F; co), so that (a) holds if and pnly if p(lcF,; co) < co; the preceding equivalence applied to the locally closed space lcF, instead of F yields the equivalence of (a), (c). (d) implies (e), since 1 E L “ and 1 is stiff and continuous. Assume that (e) holds, ‘p E F is stiff and continuous, and sytp(u) du 3 K, t E J , for certain d, K > 0. By 23.M,

< 2 ~ - dl Iq~ IF < co, and (e) implies (a). If F is quasi locally closed, (b) obviously implies (d). 9,

so that P(F; co)

Remark 1. If F is quasi locally closed, the word “continuous” may be deleted in (e), on account of the remark preceding 23.M.

23. 9-SPACES

67

Remark 2. T h e corresponding question about a ( F ; co) is answered by the trivial observation that, for any F E 9 or y+,a(F; CO) < 00 i f and only if F is stronger than L'. We can now use these results to improve on the second part of 22.N. 23.T. Assume that F E b y . Then: (1) F' is weaker than L" or , : L equivalently, if and only if F is stronger than L'; ( 2 ) F' is weaker than L1i f and only i f F is stronger than L"; (3) F' is stronger than L" i f and only i f F is weaker than L1; (4) i f F E b y K , (F,)' is stronger than L' if and only i f F is weaker than L,".

Proof. ( I ) , (2) follow from 23.0 and 22.N (first part). (3) follows from 23.Q, 23.R, and (23.10), taking into account that F' is, by 23.0, locally closed. T o prove (4), we observe that 23.0 implies that the locally closed space (F,)' is stronger than Lf if and only if (F,)"= IcF, is weaker than (L')' = L", and this occurs, by 23.S, if and only if F is weaker than L;. &

Thin spaces We take up, for F-spaces, the concept of a space D being thin with respect to a space B,introduced in Section 22 (p. 52). T h e first part of the following lemma allows us to restrict our attention to the case in which B E b y , even when J = R, . 23.U. (1) If J = R , , B E b y + , D E y K ,then D is thin with respect to B i f and only i f D is thin with respect to b T-B E b y . (2) Assume that B E b y and D E YK. Each of the following conditions is suficient for D to be thin with respect to B: (a) (b) (c) (d) (23.12)

B = L' and D is not weaker than L"; B is not stronger than L' or D E b y K is not weaker than L ; : D is stronger than M,;in particular, D is lean; each continuous 'p E D satisfies inf IA-2Sf+d

I~

( u I )du : t E

J, 0 < A

t

(e) each continuous 'p E B' satisfies (23.12).

< 1 1 = 0;

Ch. 2.

68

FUNCTIONSPACES

The following condition is necessary, and also suficient: (f)

if D

is quasi locally closed

B is not stronger than L' or D is not weaker than L".

Proof. Proof of (1). The "only if" part is trivial, since bT-B 2 B. Assume that D is thin with respect to bT-B, a fortiori with respect to IcT-B = T-1cB (by 23.1). We claim that D is thin with respect to 1cB; by 22.0 this completes the proof of the "if" part. To establish our claim, let E D be continuous with p)(t) # 0, t E J , and let I,/I E T-lcB be such that I,/Iv-l+L1. Now OsoI,/IE 1cB (where so = s,(B)) by 23.1, and obviously x [ o , s , l ~ ~E- lL', for q~ is bounded away from 0 on [0, so]; therefore (@so$)qrl 6 L'. Proof of (2).

1. If

q~ E

D is continuous and vanishes nowhere,

(Ll)'implies 1 < I rp-l I I rp I, whence 1 E D, which contradicts (a). If (a) holds, therefore, 9-l + (Ll)',and 22.0 implies that D is thin with respect to L'.

9-l

EL"

=

2. We prove the sufficiency of (d): if 9 E D is continuous and vanishes nowhere, 'p-' E B' would imply, by 23.0, 23.G, q~-l E M; by Schwarz's Inequality we should find, for 0 < A 1,


0 we define the functions A,?, (lip, by A,rp(t) = J:',rp(u) du, AArp(t) = J 1~ e - ~ ( ~ - ~ ) du; r p ( uwe ) also write A,"rp for the function given by A:rp(t) = J;e-"("-%p(u) du provided the integral exists (and is finite) for all t E R, (for this it is necessary and sufficient that it occur for any single value of t ) . For reasons of symmetry-as little as there is -we also consider the function A,T;rp, and observe that A,Td+rp(t)= 1 ~ ( udu; ) here I+ = max(0, r} for r E R (this notation will not 1-A)+ signifying restriction to R,: interfere with the use of the subscript the two will not occur together). Thus A, , A,T,+,A : , are linear positive mappings of the vector lattice L into itself; A: has the same properties, but its domain is a linear manifold in L; the values of these mappings are continuous functions, and the positivity implies

',

+

When J = R, matters are simpler. For rp E L,d, u > 0 we set A,y(t) = 1 st-,l+A rp(u) du, A;rp(t) = J-me-O(l-u)rp(u)du, Azrp(t) = du, the last two subject to existence (and finiteness) of the integrals. A , , A ; , A,+ then behave like the above-defined mappings on J = R, , and satisfy (23.13). T h e fundamental relations between these mappings, in connection with F-spaces, are described in the following lemma.

Jp-u(u-L)rp(

23.V. (1) Assume that J = R+ , F E b y K and rp E L, rp 2 0. Then the following statements are equivalent : (a) (c) (e) (g)

[(b)] [(d)] [(f)] [(h)]

AArp E F for every [some] d > 0; A,T;rp E F for every [some] A > 0; AArp E F for every [some] u > 0 ; AZrp exists and E F for every [some] u > 0.

I n particular, T E L,rp 2 0, satisfies rp E M if and only if each or any one of A,rp, A,Tirp, Airp, Atrp is bounded.

( 2 ) Assume that J = R, F E b y K and rp E L, rp >, 0. Then the following statements are equivalent:

70

Ch. 2. FUNCTION SPACES

(a) [(b)] Ad'p E F for every [some] d > 0 ; (c) [(d)] A;tp exists and E F for every [some] u > 0 ; (e) [(f)] A,+tp exists and E F for every [some] u > 0.

In particular, 'p E L, tp 2 0, satisjies any one of Ad'p, A;'p, A,+'pis bounded.

'p E

M

if and only if each or

-

Proof. Proof of (1). We prove the implications (a) 3 (c) --+ (d) 5 (e) + (f) (a) and (a) + ( b ) 3 (g) + (h) * (a); the implications marked by simple arrows are trivial. (a) implies (c). For any d > 0, the functions A,Titp and c A d ' p E F differ only on [0, A ] and are continuous there. By 22.1, A,Titp E F. (d) implies (e). For the d > 0 of (d), and for any u > 0 and all tER+,

m

=

C e-jU*Tj+dAdTiv(t); i=O

now e-juAT&AdT2'pE F for all j , and C:3-0 I e-fuAT&AdTi'pIF ~ ~ = o e - i IuAdTd+Q) d IF < 00; since F is complete, AAtp E F, and

=

( f ) implies (a). For the u > 0 of (f), and any d > 0 and all t E R, , eud.ro l+Ll e-u(t+d-u) 'p(u) du = eudEAA'p(t), so that Ad'p E F,

Adtp(t)


0 of (b), and any u > 0 and all t E R+ ,

the completeness of

F then implies A,"'pE F, and

23. F-SPACES

71

(h) implies (a). For the CT > 0 of (h), and any d > 0 and all t E R, , A,y(t) e"Jf-'(u-%p(u) du = eodA:y(t), so that A,cp E F, and


,

%(t)I

4 2Y[Xl(t"), X,(tO)l exp(2jJ ,I1 4)1I du) .

Proof. We take norms in (31.1), with xo = x(to),and apply Gronwall's Lemma between to and t ; we obtain

and (31.5) follows immediately. Formula (31.6) follows by integrating both members of (31.5) over J' with respect to t o . Formula (31.9) follows from (31.7) applied to the solution UU-l(t,) of (30.3), which takes the value I at t = t o . T o prove (31.10), consider the solution of (30.1) given by x = 1) xl(to) Il-lxl - 1) x2(t0)I)-1x2. Applying (31.7), 11.A, and again (31.7), we have

The closedness theorem

T h e following theorem is fundamental for future applications. Its main import is that the set of triples (A,f, x), where x is a solution of (30.2), is closed in L ( 8 ) x L(X) x L(X).

32. ASSOCIATE EQUATIONS

89

I N COUPLED SPACES

31.D. T H E O R E M . Let (A,), (f,) be sequences in L(R), L(X), respectively, and let x, be some solution of 5, + A,x, = f n , n = 1, 2, ... . If the limits lim LAn = A, lim, f n = f , lim, x, = x exist, then the function x is (except f o r equivalence modulo a null set) a solution of (30.2) and x, -+ x unijormly on every compact subinterval of J as n -+ 00.

Proof. Since ( x n ) converges in L(X), there exists a subsequence (x,(~))which converges pointwise a.e.; we may refer the latter convergence to the continuous functions x, themselves, instead of to their

equivalence classes; it is therefore meaningful to infer that there exist to E J , xo E X such that limj+mxncj,(t0)= xo , Consider the solution x‘ of (30.2) such that x’(to) = xo . Let J’ C J be any compact subinterval, which we may assume to A,(j,y = f n ( j ,contain t o . Since y = x,(~) - x’ is a solution of j f ( A - An(j))x‘,and since the second member converges to 0 in L(X) as j + m, it follows immediately from (31.5) that x,(~)+ x’ uniformly in J’; hence limj+mLxn(j) = x’, so that x’ = x (i.e., x’(t) = x ( t ) a.e.) and, under this identification, x is indeed a solution of (30.2). For each n, z = x, - x is a solution o f i: A,z = f n - f ( A - A,)x, and since the second member converges to 0 in L(X) as n + co, and lim, x, = x , it follows from (31.6) that x, -+ x uniformly on J’. &

+

+

+

+

32. Associate equations in coupled spaces Associate operator-valued functions

32.A. Let X, X’ be a pair of coupled spaces, and assume A given. The following conditions are equivalent:

E

L(X)

(a) For every representative of A , A ( t ) is continuous in o(X, X’) for almost all t E J. (b) [(c)] There exists A‘ E L((X’)“)such that any continuous [constant] functions f , f ’ from J into X, X‘, respectively, verify ( A f , f ’ ) = ( f , A”).

Proof. (a) implies (b). Under (a) there exists, for a chosen representative A and almost every t E J , the associate operator (A(t))’; if we define A‘ : J -+ X’ by A’(t) = (A(t))‘ for such t (and = 0, say, elsewhere), the isometry of the operator-to-associate-operator mapping implies A‘ E L((X’)”);and ( A ( t )f ( t ) ,f ’ ( t ) ) = ( f ( t ) , A’(t)f’(t)) for those t , so that ( A f , f’) = ( f , A”). ( b ) implies (c). Trivial.

90

Ch. 3. LINEAR DIFFERENTIAL

EQUATIONS

(c) implies (a). For y y t o , t E J , t > t o , we set V ( t o ,t) = A(u)du, V ‘ ( t o ,t ) = JI, ?’(u) du. For any x E 1X , x‘ E X’ we have, by (c), ( V ( t o ,t ) ~x’) , = Jf, (A(u)x, x’) du = J1, (x, A’(u)x’) du = (x, V ‘ ( t o ,t ) x ’ ) . Therefore V ’ ( t o ,t) = (V(to, t))’, the associate operator of V ( t o , t). If A, A’ are any representatives of their respective classes, then liml,lo(t - to)-’V(t0,t) = A(to),liml,fo((t - to)-lV(to,t))’ = lim,,lu(t - to)-lV’(to,t) = A’(t,) for almost all to E J (Hille and Phillips [l], p. 87). On account of the isometry of the operator-toassociate-operator mapping, we conclude that A(to) has the associate operator (A(to))’= A’(to), and therefore is continuous in u ( X , X’), for almost all to E J. (A simpler proof of (c) --f (a) is possible if X is separable). & 1

J1,

If the conditions of 32.A are satisfied, we call A‘ the ASSOCIATE of A, and say that A HAS AN ASSOCIATE; A is the associate of A’ in the transposed coupling. In the special case X‘ = X * - which occurs, in particular, whenever y is reflexive - every A E L(R) has an associate, namely A*, defined by A *( t ) = (A(t ) ) * .

Associate equations Let X, X‘ be a pair of coupled spaces, and assume that A E L(X) is given and has an associate. There exists, then, the associate element A’ E L ( ( X ) “ ) .We consider the equations (32.1)

2’

- A’x’

= 0,

where f’E L ( X ’ ) and the solutions x’ have values in X‘. Equation (32.1) is called the ASSOCIATE EQUATION of (30.1), which is in ‘ t u r n the associate equation of (32.1) in the transposed coupling. It is a loose but convenient figure of speech to call (32.2) an ASSOCIATE EQUATION of (30.2). In the special case X‘ = X* (which occurs, in particular, whenever X is reflexive), A has the associate A * and the associate equations become the ADJOINT EQUATIONS (32.3)

X*

- A*%*= 0,

(32.4)

**

-

A*%* = f*.

9

32. ASSOCIATE EQUATIONS

IN COUPLED SPACES

if X is a Hilbert space, they are replaced (under the mapping by the (Hermitian) adjoint equations in X itself :

91 m-’)

Green’s Formula

32.B. Assume that A E L ( x ) has an associate. Let f E L(X),f‘ E L(X’) be given. A n y solutions x, x‘ of (30.2), (32.2), respectively, satisfy, for any t o , t E J:

+

(f(u),

X ‘ W

du.

to

Proof. Using 30.A, formulas (30.2) and (32.2) yield (x, x’)’ = (x,

A‘x‘) - (AX, x’)

+ (x,f‘) + (f,x’);

the first two terms cancel and, by 32. A, integration between to and t gives (32.5). d, Formula (32.5) is frequently referred to as Green’s Formula. I n particular, taking f = f’ = 0, we have: 32.C. Assume that A E L(x)has an associate. For any solutions x, (30.l), (32.1), respectively, (x, x’) is constant on J.

x’ of

32.D. Assume that A E L(x)has an associate. If U is an invertiblevalued solution of (30.3) and ;f U(to) is continuous in u ( X , X’) for some to E J (in particular, if U is the solution with U(to)= I ) , then so is U ( t ) for every t E J; hence U‘(t) = (U(t))’ exists for all t E J ; and U’-l = (U-1)’ is a solution of (32.6)

vf

-

A’V’

=

0

(where the notation V’ is not meant to imply that V’ is an associate in general). Proof. Let V’ be the solution of (32.6) with V‘(to)= (U-l(t0))’. Let x E X, x’ E X‘, t , E J be given. Applying 32.C to the solutions UU-’(t,)x, V‘x’ of (30.1), (32.1), respectively, we have (x, V‘(tl)x‘) =

DIFFERENTIAL Ch. 3. LINEAR

92

EQUATIONS

(U(t,>U-l(t,)x, V‘(t,)x’) = (U-l(t,)x, x’); thus U-l(t,) has the associate operator V‘(t,); hence U-l(t,) and U(t,) are continuous in o(X, X’), and V’ (U-l)’ = U‘-l. 9, Remark. Let U be in particular the solution with U(to)= I. Let U , be any other solution of (30.3) with Ul(to) continuous in o(X, X‘). By (31.4), U , = UU,(t,), so that, by 32.D, U ; = (Ut(t,,))’U’ exists, i.e., U,(t) is continuous in o ( X , X’) for all t; and if V , is any solution of (32.6) we have, by 32.D and by (31.4) applied to the solutions of (32.6), V ; = U’-lV;(t,). Therefore U;V; = ( Ul(to))’V’(to) is a constant on /.

33. D-solutions of homogeneous equations D-solutions a n d their initial values Consider a space D E b N ( X ) . A solution of (30.1) or (30.2) that belongs to D is termed a D-SOLUTION. For spaces D(X) with D E b 9 we follow the rule established in Chapter 2 and omit the argument X in subscripts, in the expression “D-solution”, and the like; in case J = R, or J = R, and D E b y K , the fact that any solution is a continuous function allows us to replace the space D by D, (by 23.N): every D-solution is a D,-solution and conversely, and its D-norm and D,-norm coincide. Assume that B E b J ( X ) and that f E B, x is a D-solution of (30.2), J’ is a compact subinterval of length I.(/’)= I of J, and t E J’. I t follows from (31.6) that (33.1)

11 x ( t ) 11

< (z-la(D;j ’ )I

in particular, iff (33.2)

=

ID

+ or(B;1’)

IflB)

‘“p(/

I

,I1 A ( f r 11) d@);

0, so that x is a D-solution of (30.1), then

11 x ( t ) 11 d

j ’ )I

ID

exp(l,, 11 A(u) 11 du) *

While we are juggling the bounds given in 31.C, we may note, for

/ = R , or J = R, the “smoothing” effect of the assumption that A E M(x)on the solutions of (30.1). Let x be such a solution; then

(31.7) yields 11 x ( t ) 11 < [ j x ( t ’ ) 11 exp(1 A IM) for all t, t’ E J with I t’ - t I < 1, so that 11 x 11 is noncollapsing to both left and right (cf. Section 23, p. 73); on the other hand, (31.8) gives

33. D-SOLUTIONS OF HOMOGENEOUS

EQUATIONS

93

for all t E 1, and this shows that every M-solution of (30.1) is bounded, i.e., an Lm-solution, and every M,-solution an L:-solution. We now return to the study of D-solutions of (30.1) in general. I t will be convenient to assume, as we do without further mention, that O E J ; this can be obtained, without loss, by means of a shift in the independent variable. 33.A. Assume that D E bJlr(X). The set X , of all D-solutions of (30.1) is a subspace of D. The mapping 17 defined by l 7 x = x(0) is a monomorphism from X, into X.

Proof. By Theorem 31.D, the linear manifold of all solutions of (30.1) is closed in L(X); its intersection XD with D is therefore closed in the stronger topology of D. I7 is obviously linear and injective; by (33.2) it is bounded, with

(1 I?(\< inf[l-la(D; J') exp(l ,)I A(u) 11 du) : J'

a compact interval,

J

00

We proceed to study the linear manifold l 7 X , = (x(0) : x a D-solution of (30.1)}, which we denote by X,, or, in full, X,,(A). T h e fact that the solutions are examined at t = 0 plays no essential role: the set of values of the D-solutions of (30. I ) at any fixed to E J is simply the image of X,, under the (automorphic) mapping along the solutions of (30.1) from 0 to t o . If D = L"(X), D = LT(X) (equivalently, D = C ( X ) , D = C,(X)), we abbreviate the notation X,, to X , , X,, , respectively. T h e linear manifold X,, need not be closed, as examples 33.G, 33.H in separable Hilbert space will show. T h e following results will be used copiously in our further work; among other uses, they are relevant to the question of the closedness of X O D . Assume that D E b N ( X ) and let Y be a linear 33.B. THEOREM. manifold, Y C X,, . Then cl Y C X,, if and only if there exists a number S y 3 0 such that every solution x of (30.1) with x(0) E Y satisfies I x ID S Y I I 40) 11.


0, and for every X > 1 there exist N' = " ( A ) > 0, yo = yo@) > 0, such that any solutions y , x of (40.1) with y(0) E Y, 11 x ( 0 ) 11 M(Y,x ( 0 ) ) satisfy:


0, and for some h > 1 there exist N' > 0, yo > 0, such that any solutions y , z of (40.1) with y(0) E Y , 11 z(0) 11 hd(Y, z(0))satisfy (Di), (Dii), (Diii).


0, and for every [some] h > 1 and any [some] ( Y ,A)-splitting q of X there exist N' = N'(q) > 0, y o = yo(q) > 0, such that any solutions y , z of (40.1) with y(0) E Y , q(z(0)) = z(0) satisfy (Di), (Dii), (Diii). (e) [(f)] For every [some] h > 1 there exists D = D(h) > 0 such that any solutions x, y , z of (40.1) with x = y z, y(0) E Y , 11 z(0) 11 Ad( Y , z(0))satisfy:

+

(D'i) (D'ii)

IIr(t> II < D II x ( t 0 ) I1 11 x(t) (1 < D 11 x(to) 11

f o r all for all


1 and any [some] ( Y , A)-splitting q of X there exists D = D(q) > 0 such that any solutions x, y, z of (40.1) with z(0) = q(x(O)), y = x - z satisfy (D'i), (D'ii), or, equivalently, that for every u E X we have (D"i) (D"ii)

11 U(t)(U-l(t,)u-qq(U-'(t,)u))II~DII u I [ for all t > t o > O ; 11 U(t)q(U-l(t0)u)1) < D 1) u 1) for all to >, t >, 0.

Proof. The equivalence within conditions (g), (h) is obtained by setting x = UU-'(t,)u, whence u = x(to), z = Uq(U-l(t,)u). We consider the following diagram of implications:

(4 + J

L

L

J

(b)

1

(4

(4

(4

(f) L

'(h)

(g)

J

All the implications are trivial if we use the definition of a splitting and (1 l.l), (1 1.2), except those marked with heavy arrows, and of these only (h) + (e) is not quite straightforward. (d) implies (h). Let (d) hold for the ( Y ,A)-splitting q and let x, y, z be as assumed in (h); we have y(0) E Y and, by ( l l . l ) , q(z(0)) = z(O), so that y , z satisfy (Di), (Dii), (Diii) by (d). Assume that y, z # 0; taking t = to in (Diii) and applying l l . A t o y(to), -z(to), we have max{lly(to) 10 I1 ~ ( t , 11)) 2y0' I1 x(to) 11. (D'i), (D'ii) then follow, using (Di), (Dii), respectively (the latter with t , to interchanged); they hold with D = 270' max{N, N ' } ; the same conclusion may be reached 2. if y or z is 0, from (Dii) or (Di), respectively, since yo (e) implies (a). Let y be a solution of (40.1) with y(0) E Y . Application of (D'i) to x1 = y 1 = y , z1= 0 with any h yields (Di) with N = inf,,, D(h). Let h > 1 be given and let z be a solution of (40.1)


0 such that condition 41.A,(d) is satisfied for X,, and this splitting. 2. For any y E Yo\ {0} we have by assumption inf,>, 11 U ( t ) y Ij 2 N;l lim sup,,, 11 U ( t ) y [I > 0. Also, if y l , y 2 E Yo\ {0}, then we get I inf,,, II U(tlY1 II - in620 II U ( t b 2 II I G SUPf>O II U(t)(Yl - Y2) II N , 11 y1 - y z 11. T h e continuous function inf,,, 11 U ( t ) y 11 has a positive minimum, say fl, on the compact set { y :y E Y o , 11 y 11 = l}. Any solution y of (40.1) with y ( 0 )E Yo therefore satisfies, for any to , t 3 0,

+

+

+


0 unless x = 0; therefore X,, = (0). But (0) does not induce a dichotomy, for if it did we should have, by 41.C, I/ x ( t ) 11 N'-l )I x(0) (1 for an appropriate N' > 0 and all solutions x and all t 3 0; however, if m is an integer so large that e2m > N' - 1, and if x is the solution with x(0) = em , we should find limt+mI/ x ( t ) I/ = ( I eZm)-l < N'-I = N'-' 11 x(0) I), a contradiction. &

+

+

Ch. 4. DICHOTOMIES

110

42, Exponential dichotomies

Definition We propose to discuss a kind of behavior of the solutions of (40.1), termed exponential dichotomy, which differs from an ordinary dichotomy in the additional requirement that the solutions starting from the “inducing” subspace Y decay in a uniformly exponential manner, while those that start far from Y grow in a similar way. This type of behavior is both simpler in structure and richer in properties than its ordinary counterpart, as the analysis in this section and later chapters will show. T h e precise definition runs as follows: T h e subspace Y of X INDUCES AN EXPONENTIAL DICHOTOMY OF THE SOLUTIONS OF (40.l), or AN EXPONENTIAL DICHOTOMY FOR A , if there exist v, u’, N > 0, and for every h > 1 there exist N‘ = ” ( A ) > 0, yo = yo(A) > 0, such that any solutions y , z of (40.1) with y(0) E Y , II z(0)11 Ad( Y , z(0))satisfy:

< I1Y ( t ) I1 < Ne-”(l-I0)I1Y(t0) I1

>

for all t >, to 0; for all t to >, 0; Y[Y(t),4 t ) l 2 Yo for all t 0 if y , z # O (same as for ordinary dichotomy).

(Ei) (Eii)

>

11 z ( t ) I( 2 N‘-leY’(f-fO)11 z ( t o )11

Obviously, Y then induces an ordinary dichotomy for A. Since all solutions starting from Y tend to 0 and all others are unbounded, the subspace inducing the exponential dichotomy is unique (in contrast to the case of an ordinary dichotomy: if A = 0, every subspace induces one); it is therefore unambiguous, and often convenient, to say that A POSSESSES AN EXPONENTIAL DICHOTOMY.

We have, in analogy with Theorem 41 .A, several equivalent conditions for an exponential dichotomy; they include one condition, (am), of slightly different form:

Let Y be a subspace of X . The following statements 42.A. THEOREM. are equivalent: Y induces an exponential dichotomy for A; (a) There exist Y, v’, N , Nk) y, > 0, and for every h > 1 there (a,) exists T = T(h) 2 0, such that any solutions y , z of (40.I ) with y(0) E Y , 11 z(0)11 Ad( Y , z(0))satisfy (Ei) and


Nk-leY’(t-lO)II 4 t o ) II y [ y ( t ) ,z ( t ) ] > ym for all 11 z ( t ) 11

for all t >, T

if

t >, t o 2 T; y, z # 0.

42. EXPONENTIAL DICHOTOMIES

111

(b) There exist v, v’, N > 0, and for some h > 1 there exist N’ > 0, > 0, such that any solutions y , z of (40.1) with y ( 0 ) E Y , 11 z(0) 11 h d( Y , z(0))satisfy (Ei), (Eii), (Diii). (c) [(d)] There exist v, v’, N > 0, and for every [some] h > 1 and any [some] ( Y ,h)-splitting q of X there exist N’ = N’(q) > 0, yo = yo(q) > 0, such that any solutions y , z of (40.1) with y(0) E Y , q(z(0)) = z(0) satisfy (Ei), (Eii), (Diii). (e) [ ( f ) ] There exist v, v‘ > 0, and for every [some] h > 1 there exists D = D(h) > 0, such that any solutions x, y , z of (40.1) with x =y z , y ( 0 ) E Y , I/ z(0)11 h d( Y , z(0)) satisfy:


0, and for every [some] X > 1 and any [some] ( Y ,A)-splitting q of X there exists D = D(q) > 0, such that any solutions x, y , z of (40.1) with z(0) = q(x(O)), y = z - x satisfy (E’i), (E’ii), or, equivalently, that for every u E X we have


, 0 ; (E”ii) 11 U(t)q(U-l(t,)u)II< De-’’(fo-t)IluII for all to 2 t 2 0.

Proof. T h e equivalence within conditions (g), (h), as well as the equivalence o f all the conditions except (a,) follows precisely as in the proof o f Theorem 41 .A. (a) implies (aw). Let A, > 1 be fixed, and set N’ = N’(X,), yo = y,(h,). Choose a fixed p > 1 and set, for every h > 1, T = T(X) = (v + v’)-’ log(NN’(h + l)(p - l)-l). For any X > 1, let y , z be solutions o f (40.1) with y ( 0 ) E Y, 11 z(0) 11 < h d ( Y , z(0)).y satisfies (Ei). By means o f a (Y, A,)-splitting or otherwise, we have z = y1 zl, where y l , z1 are solutions of (40.1) with ~ ~ (E Y, 0 )11 ~ ~ ((1 0 ) A, d(Y, z(0))= A0 d ( Y , .do)). We have II Yl(0)II II z(0) II II 4 0 )II (A + 1) 11 zl(0) 11. For any to 2 T we have, using (a),

+


1 be given, and set T = T(h). Let y , z be solutions of (40.1) with y ( 0 ) E Y , I( z(0) 11 h d ( Y , z(0)).y satisfies (Ei). If 0 to ,t T, (31.7) yields 11 z(t)11 >, 11 z(to)11 eY’(l-fu)exp(-v’T +(u) 11 du). Combining this with (Eii,), we obtain (Eii) with N’ = N , exp(v’T JrIl A(u) 1) du). I f y , z # 0 and 0 t T , (31.10) and (Eiii,) applied at T imply y [ y ( t ) , z(t)] > b l y ( T ) , z ( T ) ] exp(-2Jr11 A(u) (1 du) 2 #y, exp(-2JiII A(u) (1 du). Using (Eiii,) for t > T , we conclude that (Eiii) holds with yo = #y, exp( -2s; 11 A(u)II du), and (a) is proved. 9,


0, and for every [some] projection P along Y there exists D = D(P) > 0, such that (E,”i) 11 U(t)(I- P)U-’(t,) 11 < De-”(l-lo) (E,“ii) 11 ,U(t)PU-’(t,) )I De-y’(lo-l)


0 and W , , &r, and for every w , 0 < w &r, there exists T = T ( w ) 3 0, such that any solutions y , z of (40.1) with y(0) E Y , + ( Y , z(0)) w satisfy (Ei), (Eii,), and

0

(a,)


T if

y , z # 0;

(0) and condition (a) (or (b)) holds.

(c) [(d)] There exist v, v', N > 0, and for every [some] complement 2 of Y there exist N' = " ( 2 ) > 0, wo = w o ( Z ) , 0 < w0 b, such that any solutions y , z of (40.1) with y ( 0 ) E Y , z(0)E 2 satisfy (Ei), (Eii), and (Diii").


1 illustrates at once. Remark 2. T h e contents of Remarks 1 and 2 in the preceding section apply verbatim to Theorems 42.A, 42.B, and 42.C. We add the important observation that the parameters v, v' have the same values for all conditions. It is therefore meaningful to speak of ALLOWABLE v, v' for A , without specifying the condition in which they appear. We denote by ;= ;(A), ;' = G'(A) the suprema of the allowable values of v, v', respectively. Clearly v, v' are allowable, with matching values of the other parameters, whenever they are less than these suprema. T h e suprema may be attained, or finite but unattained, or infinite: the reader may verify that these alternatives obtain, for V , for instance with the scalar equations P x = 0, P t(1 t)-'x = 0, P t x = 0, respectively; and for ;', with the corresponding adjoint equations.

+

+

+

+

Remark 3. As for ordinary dichotomies, (Diii) in the definition of exponential dichotomy and in conditions 42.A,(b),(c),(d) and 42.B,

Ch. 4. DICHOTOMIES

114

(a),(b); (Eiii,) in condition 42.A,(am); and (Diii,), (Eiii,,) in the conditions of 42.C, are not redundant. This is shown in Example 42.F for two-dimensiona1.X (for one-dimensional X the question is vacuous) and for the most stringent of those conditions, namely the definition of exponential dichotomy and 42.C,(a). However, we now have an important case in which these conditions are redundant: 42.D. Assume that A E M(X),and that y , z are nonzero solutions of (40.1) that satisfy (Ei), (Eii)for certain v, v’, N , N‘ > 0. Then they satisfy (Diii) with y o = e-ap-Pp’*’N-fi’”-e > 0, where

(where we suppose, as we may, that v‘ < a). The assumption of (Diii), (Eiii,), (DiiiJ, (EiiimH)is therefore redundant in the definition and respective conditions for a n exponential dichotomy when A E M(2). Proof. For a fixed to 2 0, the solutions y1 = II $(to)II-ly, z1= 11 z(to)ll-lz satisfy (Ei), (Eii), respectively. Using Jt, 11 A(u) 11 du < a(t - to + 1) for t 2 t o , a comparison o f (Eii) and (31.7) for z1and


0. We set r = (v v’)-l log(pp’-’NN’) > 0 fsince N, N’ 2 1) and find 1) yl(to T ) 11 Ne-”: 11 zl(to r ) 1) 2 Applying (31.7) to the solution z1- yl, we obtain

+

+

+

yry(to),z(t,)i = II z,(t,) - yl(to)II

+

+

3 II zl(to -Y ~ O , >/ ( N ’ - l e u ’ T - Ne-VT)e-a(T+l) = yo


0.

T h e redundancy of (Diii), (Diii,) follows; that of (Eiii,), (Eiii,,). is proved by taking to >, T and assuming that z merely satisfies (Eli,) in the above argument, in which N‘ is replaced by Nk . &

0000

Remark 4. In contrast to the case of ordinary dichotomies, it is not known whether every subspace Y of X induces an exponential dichotomy for some A. If Y is complemented (hence always if X is a Hilbert space) the answer is yes: if P is any projection along Y, Y induces an exponential dichotomy for the constant Z - 2P, since then U(t) = e-l(Z - P ) + etP, so that (E,“i),(E,”ii)hold. If Y is not complemented, it cannot induce an exponential dichotomy for periodic A (including constant A ; see 113.L), nor for almost periodic A (102.B). We therefore formulate (cf. also Query following 33.1):

42. EXPONENTIAL DICHOTOMIES ~~

115

Quuy. (a) Does there exist a Banach space X and a noncomplemented subspace Y such that Y induces an exponential dichotomy for some A E L(X)?

(b) If so, is this true for all X and all noncomplemented Y ? ( c ) What are the answers if we require A E M ( 2 ) ? We conclude this subsection by giving a simple necessary condition for A to possess an exponential dichotomy: 42.E. THEOREM. If A then (1 A 11 is a s t i f function.

E

L ( x ) possesses an exponential dichotomy,

Proof. There exists either a nonzero solution y satisfying (Ei) for appropriate v, N or a nonzero solution z satisfying (Eii) for appropriate v', N ' . Using ( 3 1.7) we respectively obtain, for all t 2 to 2 0,

T h e conclusion follows by 20.D.

&

Exponential dichotomies and solutions in .%spaces In analogy to the case of ordinary dichotomies, we examine the relationship between subspaces inducing exponential dichotomies and the linear manifolds X,, for D E b y K . In contrast to the elaborate results in Theorems 41.D, 41.E, 41.F, with their lengthy proofs, the situation here is as simple as might be wished: If the subspace Y of X induces an exponential 42.F. THEOREM. dichotomy f o r A , then X,,,= Y f o r every D E b y K . Proof. Since every decreasing exponential function belongs to T, (Ei) of the definition of exponential dichotomy implies Y CXOT; since no increasing exponential function belongs to M, (Eii) of the same definition implies X,, C Y . Since D E b y K is stronger than M and weaker than T (by 23.G, 23. J), XoTC X,, C X,, . &

116

Ch. 4. DICHOTOMIES

We observe, in particular, that if A possesses an exponential dichotomy, Theorem 42.F tells us that the subspace inducing it is precisely X,,(A) (or X,,(A) with any D E b y K , for that matter).

Example 42.G. EXAMPLE.Let X be two-dimensional real or complex euclidean space, with Cartesian coordinates xl, x 2 . Let v be any continuous, real, nonnegative, nondecreasing function on R, with 1 limf-,m~ ( t=) 00. We define the function by +(t)= e-21 J eZUF(u)du =

+

0

1

JIoe-2vq(t - v) dv; this function is nondecreasing, and limf-,m$(t) 2 1+1 liml-,we-2 J, v(u) du = 03. Let (40.1) be the system R,

+ x1 - px2 = 0 = 0.

R, - X?

Every solution of the system is given by x ( t ) = ( x l ( t ) , x 2 ( t ) ) = (e-'xl(0) e$!(t)x2(0), e1x2(0)). We shall show that X o = {x : xz = O} satisfies (Ei), (Eii) of the definition of dichotomy; or rather, equivalently (cf. proof of Theorem 42.A), (Ei), (Eii,) of condition 42.C,(a,); however, X, does not satisfy (Diii) or, equivalently, (DiiiH), even for w = &7r only, as we shall see. If y is a solution with y(0) E X , , then 1) y ( t )1) eL = 1 yl(0) I is a constant, so that (Ei) holds with v = 1, N = 1. Let w , 0 < w &7r be given, and choose any E , 0 < E < 1. Choose T = T ( w ) in such a way that Ee24,b(t) 2 cot w for all t >, T ; this is possible since e 2 V ( t ) t a.We remark that, for all D >, 0, we have

+


, T ,

I zl(0) 1


0,

11 ~

< De-v’(t-to) for all P)l/-l(t)ll < De-v(to-t)

- 1 ( t ) ~ ~ ’11 (=t 11~ U(~,)PU-’(~) ) 11

11 u‘-l(t)(l - P)’U’(to)\l= 11 U(to)(Z-

t 2 to 2 0,

for all to 2 t 2 0,

so that condition 42.B,(d) holds for t h e associate equation, with v , v‘ interchanged. The proof for the general case is patterned on the preceding argument, but the projections must be replaced by an appropriate splitting. We intend to use condition 42.A,(g) for the given equation and to show that condition 42.A,(e) holds for the associate equation, with v , v’ again inter-

changed. Let then h x’ = y’

+ z’,

> I be given, and let x’, y’, z’ be solutions of (32.1) with y’(0) E Yo, I1 z’(0)ll Q h d(Yo,~’(0)).Let p > 1 be fixed

arbitrarily, and let r be the function defined by 12.F, so that Y(*; z’(0)) is a continuous ( Y ,p( 1 h s,))-splitting of X . We apply condition 42.A,(g) ) for this particular splitting and observe that D = D(r(.;~ ’ ( 0 ) )depends only on p(1 Asy), and not further on z‘ (Remark 2 to Theorem 42.A and its corollaries). Let x be an arbitrary solution of (40.1); let y , z be the solutions defined by z(O)=r(x(O); z’(O)),y=x-z, so that y ( 0 ) E Y , whence (y(O),y‘(O))=O; by 12.F, also (z(O), z’(0)) = 0. Therefore 32.C implies that (x, y ’ ) = (z, x’) and (x, z’) = ( y , x‘) are constants. Using (E’ii), (E’i) we find

+

+

I(W>Y’(~))l = I ( 4 t O ) l

x’(tll)>l

< De-”‘t-t”’ll x(t)ll II x’(t0)Il for all t

I(x(t), z’(t)>l = I(r(to), x‘P0))l

to 2 0,

< De-”(to--l)llx(t)lI II x’(t0)Il for all t , 3 t 2 0.

Since, for fixed t , x ( t ) is an arbitrary element of X , we conclude that x‘, y’, z’ satisfy (E’i), (E’ii). Therefore the associate equation and the subspace Y osatisfy condition 42.A,(e) for the given A, with v, v’ interchanged and with the above-defined D , which depended, for fixed p, on h alone. & , T h e most important special case is obtained by taking X ‘ = X * :

43.B. THEOREM. Let A E L(R) be given. The subspace Y of X induces an ordinary [exponential] dichotomy for A ;f and only if Yo in X* induces an ordinary [exponential] dichotomy for -A*. Proof. Theorem 43.A and 12.A. &

43. DICHOTOMIES FOR ASSOCIATE

EQUATIONS

119

43.C. Assume that X is a Hilbert space and that A E L ( 2 ) is giwen. The subspace Y of X induces a n ordinary [exponential] dichotomy f o r A if and only if Y'- induces a n ordinary [exponentiall dichotomy f o r -A+. 00 0000

00 0000

Returning for a moment to the case of a general coupled pair X,X , we mention one further corollary of Theorem 43.A: 43.D. Assume that A E L ( 2 ) has an associate. Assume that both the subspace Y of X and Yo have the quasi-strict coupling property. If Y induces a dichotomy for A , so does Yw; if Y induces an exponential dichotomy for A, then Y is saturated.

Proof. Apply Theorem 43.A twice and recall the uniqueness of the subspace inducing an exponential dichotomy. & , ca

Ordinary dichotomies and the manifolds Xo,X,* The remarks we shall make are closely connected with Theorems 41.D, 41.E, 41.F; since they will be superseded, for finite-dimensional X , by Theorem 44.A, the present subsection is of interest for the infinite-dimensional case only. All references to polar sets shall be understood with respect to the coupled pair X , X * . Assume that the subspace Y induces a dichotomy for A E L(2). By Theorem 43.B, Yo induces a dichotomy for - A * ; thenYO C X,* (where X,* = X,*(-A*); by Theorem 41.D), whence (X,*)OC Y. If X,* is closed, Theorem 41.E implies that it also induces a dichotomy for -A*; if, in addition, it has the quasi-strict coupling property-in particular, if it is saturated-then Theorem 43.A implies that (X,*)O induces a dichotomy for A. Therefore, in this case, the subspaces inducing dichotomies for A are precisely those satisfying (X,*)OC Y C X o (cf. Theorem 41.E). However, X,* need not be closed, and then (X,*)O need not induce a dichotomy: witness Example 43.F below. We now state a theorem (to be proved later) which shows that a small part of the preceding argument can still be salvaged. In order to understand the relation between this theorem, Theorem 41 .F, and the results in the next section, we recall that X , C (X,*)O (33.J). Assume that the subspace Y induces a dichotomy for 43.E. THEOREM. A. If W* is a subspace of X * such that Yo C W* C X,* and such that Y o has finite codimension with .respect to W*,then W*O (which satisjes W*O C Y ) induces a dichotomy for A. The subspace (X,*)O is the intersection of all subspaces that induce dichotomies for A.

120

Ch. 4. DICHOTOMIES Proof. See Section 63 (p. 181). &

Remark. Theorem 43.E cannot be strengthened to state: “if Z is a subspace of X , (X,*)OC Z C Y , codimension of Z with respect to I’ finite, then Z also induces a dichotomy for A”, as the following example shows.

43.F. EXAMPLE (Examples 33.H, 41.H, 41.1 continued). Consider A as in Example 41.1, where X is separable Hilbert space and A is symmetricor Hermitian-valued, and A E C(2). Here X = Xo(A) induces a dichotomy. The adjoint equation is given by Examples 33.H, 41.H, and here Xo(-At) is dense in X but not equal to X ; also, (X,(-At))l = {0}does not induce a dichotomy for A. If u E X \ Xo(-At) (e.g., u = (n-I)), then the one-dimensional subspace spanned by u obviously does not induce a dichotomy for -At; by 43.C, the hyperplane {u}’ does not induce a dichotomy for A , but does satisfy the condition in the preceding Remark. & 44. Finite-dimensional space

Let X be finite-dimensional; according to our general agreement (cf. Section’ 15) we assume the norm to be euclidean. Most of the analysis in the preceding sections remains significant, although some of the proofs could be simplified somewhat. In particular, the relevant conditions for ordinary and exponential dichotomies are those given in 41 .C, 42.C, respectively, plus the important conditions 41 .B,(c),(d) and 42.B,(c),(d), which deal with bounds for I( U ( t ) ( I - P)U-’(t0) 11, II U(t>PU-l(t,>II * A definite simplification specific to the finite-dimensional case occurs in connection with Theorems 41.E, 41.F, 43.E, which relate the subspaces inducing dichotomies to the manifolds X , , X,, , (X,*)O.

44.A. THEOREM. The class of all subspaces of X that induce a dichotomy for given A E L ( 2 ) is either empty or coincident with the class of all subspaces Y such that X,, C Y C X , (this class includes X,, , X,). In the latter case, (Xo0(A))l= X,( -At) and (X,(A))l= X,,(-A+). Proof. (This depends on Theorem 41.E, the proof of which is still outstanding.) T h e first part follows immediately from Theorems 41.D, 41.E, 41.F. For the second part, X,,(A) induces a dichotomy for A , hence (Xo0(A))linduces one for --At (43.C); but the first part of the statement, applied to - A t , implies (Xo0(A))lC Xo(- A t ) ;

44. FINITE-DIMENSIONAL SPACE the reverse inclusion follows from 33.J. Interchanging A , -At find ( X o O ( - A t ) ) l= X o ( A ) , whence X,,,(-A+) = (Xo(A))'-. a3

a3

121 we

Remark. Theorem 44.A does not hold in general for infinite-dimensional X.Example 43.F in separable Hilbert space violates the first part, since X0,(,4) = (0) does not induce a dichotomy for A , and Xo(-A+) # X is not even closed; for this reason it also violates (X,(A))I = Xo(-At). However, for finite-dimensional X, this condition is equivalent to (X,(--At))l = X,(A), and this, together with (X,(A))I = X,( -At) does hold for Example 43.F. This raises the following question for general coupled X,X ' and an A E L(x) that has an associate: if there exists a subspace Y (perhaps having the quasi-strict coupling property) that induces a dichotomy for A , does it follow that (X,,(A))O= X,(-A')? The answer is no: a counterexample with constant A in X = Z1 and with X ' = X* can he given: see Example 44.B below. We have not been able to construct a counterexample in a reflexive space, let alone in Hilbert space.

44.B.

EXAMPLE.Let X be the separable space 1'. Let (40.1) be the

system 2, - n-lx* = 0,

n

=

1,2, ...,

so that A is constant and given by a diagonal matrix. Every solution x of this system is given by its components x,(t) = et"tc,(0),

n

=

1 , 2, ... .

Every I x, j is nondecreasing-hence /I x Ij is nondecreasing-and if x # 0 we have, for some n, x, # 0 and limt+m)I x(t)JJ>, limt+wet/nl x,(O)J = 00. Therefore X, = X , = (0) induces a dichotomy, with N' = 1. We identify X* with l", so that the evaluation functional becomes m ( x , y ) = XI x n y n .The adjoint equation is then the system j,

+ n-ly,

= 0,

n

=

1,2, ...;

every solution y of this system is given by its components

y,(t)

= e-t'nyn(0),

n

=

1,2, ..,

Every I y, j is nonincreasing, hence /I y 11 is nonincreasing; and X,*= X*=lm induces a dichotomy for -A*, with N = 1. We claim that X,*, =.:I Obviously, e, E X& for every n ; since the set {e,} spans a dense manifold in and since X;, is a subspace (Theorem 41.D), we have 1; C X&.

fr,

122

Ch. 4. DICHOTOMIES

Assume conversely that y is a solution with y(0) $1;. lim sup 11 y ( n ) 11 n-m

> lim sup e-l n-m

Then

1 m(0) I > 0,

.

so that y(0) $ X,$, Hence X& C lo“, and equality is proved. However, (X0)O = (0)O = X* = l m # 1”0 = Incidentally, it can be shown that the (noncomplemented) subspace X & = 1: induces a dichotomy for -A* (Massera and Schaffer [4], Example 4.1). &

45. Notes to Chapter 4 The kind of behavior of the solutions of the homogeneous equation that we call exponential dichotomy was essentially considered by Perron [2] (Satz l), although he restricts himself to finite-dimensional X and A E C(R), and his conditions apply after A has been brought by an appropriate transformation (Perron [l]) into triangular matrix form; it was Maizel’ [l] who showed that Perron’s conditions are equivalent to the exponential growth conditions of the solutions given by (Ei), (Eii). Neither Perron not Maitel’ mention condition (Diii), which is indeed redundant in the case considered by them, on account of 42.D; however, the condition imposed by Maizel’ that a certain determinant be bounded away from zero is essentially nothing but condition (Diii). The case where (Eii) and (Diii) are vacuously satisfied, i.e., X itself induces an exponential dichotomy, is more commonly found in the literature on stability of solutions of differential equations; apparently Persidskii [l] and Malkin [l] used it for the first time, the latter even in the case where A . i s continuous but not bounded. A closely related concept, which we render as “uniformly noncritical behavior” of the solutions of a (generally nonlinear) differential equation was introduced by Krasovskii [l]. In the case of our homogeneous linear equations, the definition of this concept may be rephrased as follows: the behavior of the solutions of (40.1) iS UNIFORMLY NONCRITICAL if for every number k > 1 there exists a number T = T(k) > 0 such that for any solution x of (40.1) and any to 3 T we have maxlt+, 11 x(t)ll 3 kII x(to)ll. The following theorem may then be proved (cf. Massera and Schaffer [3], Theorem 3.5): 45.A. If there exist a subspace Y of X andpositive numbers h > 1, V , v‘, N , N’, such that any solutions y , z of (40.1) with y(0) E Y , 1) z(0)ll h d ( Y , 40))satisfy (Ei) and (Eii), then Y = Xo and the behavior of the solutions of (40.1) is unifortnly noncritical. Conoersely, if A E M(R) and the behavior of the solutions of (40.1) is uniformly noncritical, then X o is closed, and there exist numbers v , v’, N , N,, ym, and a function T,(x) 3 0 defined in X \ X o , such that any solutions y, z of


k > 0, it follows from (51.4) that &t) kill y(t)ll for all sufficiently large t , whence f E D (by 22.I), contradicting the asumption. Therefore (J: t-'(u)Il f(u)lldu)xo= -y(O), and (51.4) yields y ( t ) = { - f ( t ) [-'(u)lIf(u)lI du}x,. Therefore the integral in (51.3) exists, and 11 x 11 11 y 11, whence x E D(X). &

+