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.
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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:
<
for any solution y of ( 1 ) with y(0) E Y, IIy(t)ll N lly(to)ll, (Di) t 2 to 2 0 ; (Dii) for any solution 2; of (1) such that z(0) has an “angular apartness” 2 y from every nonzero element of Y, 11 z(t)ll N - l 11 z(to)/l, t 2 to 2 0 ; (Diii) for any pair of nonzero solutions y, z of (1) with the properties mentioned in (Di) and (Dii), the “angular apartness” of y(t), z ( t ) is 2 ’yo for all t 2 0.
If the inequalities in (Di) and (Dii) are replaced by
1) y(t)ll < Ne-u(*-*o)ll y(to)ll
and
1) z(t)ll >, 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
<
and sgn 0 = 0. T h e unit sphere (unit ball) (x E X : 11 x 11 l} is denoted by Z ( X ) . Distances between points and sets, and between sets and sets, in X are denoted by d ( . , In a normed space, all topological concepts refer to the norm topology unless the contrary is explicitly stated. A SUBSPACE of a normed space X is a closed linear manifold. A dihedron ( Y , 2) in X is CLOSED if Y , 2 are subspaces; a subspace Y is COMPLEMENTED if there exists a subspace 2 (a COMPLEMENT of Y )such that ( Y ,2) is a disjoint closed dihedron. A PROJECTION is a bounded, i.e., continuous, algebraic projection. It is obvious that null-space and range of a projection constitute a disjoint closed dihedron; if X is a Banach space, the converse statement holds (1O.B). Indeed, a well-known consequence of the Open-Mapping Theorem is: a).
10.A. If ( Y , 2) is a disjoint dihedron in a normed space X , with Y , Z complete, then the algebraic projection along Y onto Z is bounded, i.e., a projection, if and only if X is complete. An immediate consequence is
10.B. In a Banach space, the algebraic projection associated with a disjoint closed dihedron is bounded, i.e., a projection; and a subspace is complemented if and only if it is the null-space [the range] of a projection. If X , Y are Banach spaces, [ X ; Y ] denotes the Banach space of all bounded, i.e., continuous, linear mappings of X into Y . T h e OpenMapping Theorem allows us to call the elements of [ X ; Y ] HOMOMORPHISMS, as we shall frequently do. A homomorphism is an EPIMORPHISM if it is surjective (i.e., onto; it is then open), a MONOMORPHISM if injective (i.e., one-to-one), and an ISOMORPHISM if bijective. If T E [ X ; Y ] is an isometric isomorphism, T is called a CONGRUENCE, and X , Y are said to be CONGRUENT (UNDER T ) . If X is a Banach space over the scalar field F, we may identify [ F ;F ] with F and [F; XI with X ; [ X ;F] is X * , the dual space of X ; and [ X ;X ] is the Banach algebra of ENDOMORPHISMS (OPERATORS) on X ; we denote it by for short. Its bijective elements are the AUTOMORPHISMS of X . In some examples, the Banach space X will be specified to be some space lp of real or complex pth-power-summable sequences, with 1 < p < 00, or the space I", or I," (the subspace of I" of the sequences tending to 0). In all these cases the indices are assumed to run over (1, 2, ...), and we define the elements em = (emJ = (amn), m = 1, 2, ... .
x
6
Ch. 1. GEOMETRY OF BANACH SPACES
Continuous selections
A tool we shall need at several points in this chapter, and also elsewhere in this work, is a part of the theory of continuous selections, as developed by E. Michael. We shall only give the definitions and theorems from that theory which we shall need in the sequel. Let X, Y be topological spaces, and P( Y) the class of subsets of Y. A function q : X + P ( Y ) is a CARRIER. A carrier q is LOWER SEMICONTINUOUS if for each x E X, each y E q ( x ) , and each neighborhood V of y there exists a neighborhood U of x such that for every z E U we have q(z) n V # 0. A CONTINUOUS SELECTION FOR a carrier q is a continuous function f : X -+ Y such that f ( x ) E q ( x ) for all x E X. If Y is a Banach space, let F( Y) be the set of closed convex sets of Y. If X, Y are Banach spaces, let [X; yl, denote the subset of [X; yl consisting of all epimorphisms (homomorphisms onto) of X onto Y. These notations will not be used outside the present section. T h e following theorems are quoted from Michael [l] in a somewhat particularized form and with our terminology.
10.C. (Michael [l], Lemma 7.1). Let X be a metric space, Y a Banach space, and : X -+ F( ?') a lower semicontinuous carrier. For any given X > 1 there exists a continuous selection f for q such that 11 f(x)ll X inf{ll y 11 : y E q ( x ) }for all x E X .
<
T o obtain this form of the statement from the original, we must observe that a metric space is paracompact, and that the real-valued function inf{ll y 11 : y E q ( x ) } is lower semicontinuous on X . 1O.D. (Michael [l], Proposition 7.2). Let X, Y be Banach spaces over the same scalar field, and let T be an epimorphism of X onto Y. For every X > 1 there exists a continuous f : Y --+ X such that, for every y E y: T f Y ) = Y , IIf(Y)ll inf{ll x I1 : T x = y>, and f(4 = .f(r) for all scalars a.
<
10.E. (Michael [l], Theorem 7.4). Let X, Y be Banach spaces over the same scalar field. For any X > 1 there exists a continuous function f : [ A , u], x Y + X such that, for any T E [ X , yl, , y E Y we have T f ( T , y ) = y , Ilf(T, r)ll Wl x II : T x = r>, and f ( a T , FYI = a-l/3f( T , y ) for all scalars a, /3 with a # 0.
<
Observe that 10.D is actually a consequence of 1O.E (but is required for the proof of the latter). Our own use of 10.E is restricted to proving the following lemma:
11.
7
ANGLES, SPLITTINGS, AND DIHEDRA
10.F. If X is a Banach space and A > 1, there exists a continuous A function x0 : X * \ (0) -+ X such that x*(7ro(x*)) = 11 x * 11 and 11 rr0(x*)ll for all x * E X * \ (0). -
<
Proof. Taking Y = F, the scalar field, in lO.E, we have [ X ;Fl0 = X * \ (0). We set T,,(x*) = f ( x * , 11 x * I!). Then no is continuous, A Wll x II : X*(TO(X*)) = x*( f ( x * , II x * 1 ) = II x * Ill and II ro(x*)ll .*(.) < I/ x * 11) = A. &
<
1 1 . Angles, splittings, and dihedra Angular distance and related concepts Throughout this section we shall be dealing with a fixed Banach space X over the scalar field F. We do not. exclude explicitly the case X = {0},but most of what follows is vacuous in that case, and we shall not go out of our way to take it into consideration in statements and proofs. Some precautions are sometimes necessary, here and throughout the book, when the subspace of X that we may be considering happens to be (0) or X itself; the major precautions of this kind are provided by us (for instance, see 1 l.C), but the reader will be trusted to supply the minor ones himself. There are several ways of describing an “angular separation” between nonzero elements of X that could replace the euclidean concept of the angle between vectors; we have chosen the simplest symmetric one: if x, y E X \ {0}, we define the ANGULAR DISTANCE between x and y as y [ x , y] = II sgn x - sgn y II. Thus 0 y[x, y] 2, and Y[X,yl = 0 implies y = ox for some positive real scalar 0. T h e fundamental fact relating this concept to the geometry of X is contained in the following lemma:
<
<
1l.A. For any x, y E X \ {0}, we have y [ x , y] max{ll x I(, (Iy 211 x - Y II.
I\}
<
Because of the symmetry, it is sufficient to consider the case Set y = A x , yl. If II y II ( 1 - b)II x II, the conclusion follows from Proof.
II y II
< II x II.
<
II XI1 = II .r --Y +Y !I < I1 x -Y II If (1 - &)/I x (1 y
< 11 y 11 < 11 x 11,
+ (1 - i-Y) Ilx /I.
it follows from
I/ /I = /I x - I1 II sgn y II = II x - y ~ l l . r - Y l l + ~ l l ~ l l& .
- (11 x
ll - II Y II)sgnY II
Ch. 1. GEOMETRY OF BANACH SPACES
8
The following “Three-Angles Lemma” will be frequently used: 11.B. Let u, v , w E X \ (0) and assume that there exist y’, y” > 0 such that y[u, pv] 2 y’ for any scalar p # 0 and y[hu pv, w ] >, y“ for any scalars A, p not both 0. Then y [ u , av Pw] 2 b’y’’ for any scalars a , /? not both 0; and similarly with u, v interchanged.
+
+
Proof. If a = 0, y [ u , 8.11 = y[P-’u, w ] 2 y” > +y’y”., if P = 0, y [ u , av] >, y’ > t y ’ y ” (since y’, y ” < 2). If a, P # 0, then av /3w # 0, since y[av, -8.11 = y[-a/?-’v, w ] 2 y” > 0. Applying 11 .A twice, we have
+
If Y # (0) is a subspace of X , and x E X \ Y , we may define y [ Y , x ] = inf{y[y, x] : y E Y \ (0)) >, d( Y , sgn x) > 0. This angular distance gives an idea of how far apart from Y the line through 0 and x is. I t would therefore often be met with in such conditions on x as “ x $ Y and y [ Y , x ] >, y” for some y > 0. I t is usually more convenient to use another form of “apartness condition”, namely ‘ 11 x I( X d( Y , x)” for some X > I ; this has the advantage of being meaningful even when Y = (0) or x = 0 (in which cases it is satisfied for any X and, respectively, any x or any Y ) . T h e relationship between the two conditions is as follows:
<
11.C. Let Y # (0) be a subspace of X . I f y > 0, and x E X \ Y satisjes y [ Y , x ] >, y , then (1 x I( 2y-I d ( Y , x). If X > 1 and x E X satisjes /I x 11 X d( Y , x), then either x = 0, or x 4 Y and y [ Y , x ] 3 X-l.
<
<
<
Proof. If y [ Y , x ] y , 1l.A implies, for every y E Y , yII x 11 y [ y , x]11 211 x - y 11, so that the first part of the statement holds. X d ( Y , x), x # 0, then d ( Y , x) # 0, hence x 4 Y and If )I x (1 y [ Y , x] >, d( Y , sgn x) = 11 x 11- * d( Y , x) 2 A-l. &
<
<
If Y , 2 are subspaces of X , Y n 2 = {0}, Y , 2 # {0}, we may define y [ Y , 21 = inf{y[y, 23 : y E Y \ {0},z E 2 \ (0)).
1l.D. If Y , 2 are subspaces of X , Y n 2 = {0}, Y , 2 # {0}, then 2 is a subspace if and only if y [ Y , 21 > 0. I f ( Y ,2) is a disjoint closed dihedron with Y , 2 # {0},and P is the projection along Y onto 2 (cf. l,O.B), then 0 < 11 P 11-l < y [ Y , 21 211 P 11-l.
Y
+
<
1 1. ANGLES,SPLITTINGS, AND
9
DIHEDRA
+
Proof. Let P be the algebraic projection of Y 2 along Y onto 2. Assume y [ Y , 21 > 0, and let y E Y , x E 2 be given, so that P(y z ) = z. z ) = 0; if y = 0, y [ Y , 21II P ( y z)ll ,< 211 z 11; if If x = 0, P ( y 4 Y[Y, -4 II ZII G 211 Y 11. Y , # 0, 11.A gives y [ Y , 21II P(Y Therefore P is bounded, and indeed 11 P 11 2y[Y, 4-l. By lO.A, Y 2 is complete, hence a subspace. Conversely, if Y 2 is a subspace, hence complete, P is bounded, on account of 10.A; and if y E y \ {O}, x E \ {O}, then r [ y ,I. 2 II p 11-l II P(sgn y - sgn )1.I = I/ P 11-l, whence y [ Y , 21 2 11 P 11-l > 0. T h e last part follows by taking Y+Z=X. &
+
+ < <
+
+
+
+
+
Splittings
> 1 a real number. A mapping X if it satisfies, for all x, y E X :
Let Y be a subspace of X , and X q : X + X is a ( Y , +SPLITTING
OF
(a) x - 4(x) E y ; (b) x - y E Y implies q ( x ) = q ( y ) ; ( c ) I1 q(x)ll d ( Y , 4.
<
From (a), (b), and (c) we have, for every x E X , (11.1)
(11.2)
M 4 ) = 4(x), I1 Q(4 I1 d 4 Y , 4
w .
1I.E. For every subspace Y of X and for every A a continuous ( Y , A)-splitting.
>1
there exists
Proof. (The result is trivial if continuity is not required.) Let Q be the canonical epimorphism of X onto the quotient space X l Y . By 10.D, there exists a continuous o : X / Y -,X such that Qo(()= 5, /I o(()ll ,< All ( 1 , and w ( a ( ) = ao(() for every 5 E X / Y and every scalar a. We set q ( x ) = ~ ( Q x )for all x E X ; (a) and (b) are immediately verified, and Ij q(x)II ,< XI1 Qx )I = h d( Y , x); q is continuous; and it satisfies, in addition, q(ax) = aq(x). & We show that continuous ( Y , *)-splittings, which exist for all subspaces Y , are a nonlinear generalization of projections along Y (i.e., with null-space Y ) , which, by 10.B, exist only if Y is complemented.
1 1.F. If Y is a complemented subspace of X and P a projection along Y , then P is a continuous ( Y , A)-splitting foz any h > 1, X >, 1) P 1). Proof. (a) and ( b )are trivially satisfied and P is continuous. For any xE II p x I1 = inf,,, II P(X - All II p II d ( Y , 4. &
x,
<
OF BANACHSPACES Ch. 1. GEOMETRY
10
An element of the range of a (Y, .)-splitting may be prescribed arbitrarily outside Y \ (0):
<
h d(Y, z ) 1l.G. If Y is a subspace of X and z E X satisfies 11 z 11 for some X > 1, there exists for any p > X 2 a continuous ( Y ,p)splitting q of X such that q(z) = z.
+
Proof. On account of 11.E we may assume z # 0, whence z 4 Y; let y* E X * be a bounded linear functional on X that vanishes on Y and satisfies y*(z) = 1, IIy* 11 d(Y, z) = 1. Set v = *(p - A) > 1. By 1 l.E there exists a continuous (Y, u)-splitting, say q'. We compute II z - q'(z)ll II z I1 II q'(.)ll (A 4 d ( Y , 4. For all x E X we define q ( x ) = q'(x) + y*(x)(z - q'(z)). It immediately follows that q is continuous, that it satisfies conditions (a) and (b), and that q(z) = z ; finally, I1 q(x)ll I1 q'(x)ll I\y* 11 d(y,x)ll - 4'(2)1\ (u A v) d(Y, x) = p d(Y, 3). &
+
<
< + <
+ +
+
<
A special result we shall require (for the proof of Theorem 41.F) is the following: 11.H. Let W, Y be subspaces of X such that W C Y ; let qo be a ( W, &)-splitting of Y andq, a ( Y ,A,)-splitting of X . Set X = A, A, XJl and q(x) = q,(x) qo(x - q,(x)). Then q is a ( W ,A)-splitting of X .
+ +
+
Proof. It is a matter of immediate verification that q satisfies conditions (a) and (b) for a (W, .)-splitting of X. Let x E X be given, and set q&) = z, qo(x - ql(x)) = y , so that q(x) = y 2. If lly I1 &(1 AT1) (I z II, we have
+
+
IIq(x)II<(l
+ 4Jl
<
+ y ) ) l l 4 l < ( l +ho(l + y ) ) X , d ( Y , x )
Dihedta The contents of this subsection will be required for the first time in Chapter 8. Dihedra were defined in Section 10, and some properties were mentioned. If (Y, , Y2)is a dihedron in X, and if there exists K > 0 such that for every x E X there are y1 ,yz with yi E Yi, x = y1 y2,
+
1 1. ANGLES,SPLITTINGS,
11
AND DIHEDRA
<
and 11 y, 11 KII x 11, i = 1,2, we say that the dihedron is GAPING; the infimum (not necessarily attained) of the values of K is denoted by K ( Y , , Y,) and called the GAPE of the dihedron; observe that K ( Y , , Y,) 3 i. In 11.K we shall give a neat characterization of gaping dihedra. For the present we note the obvious fact:
1 1 .I. If ( Y , , Y,) is a gaping dihedron and Z , , Z , are linear manifolds in X with Zi 3 Y , , i = I , 2, then (2,, 2,) is a gaping dihedron with K(Z1 , 2,) K ( Y 1 9 Y2).
<
All closed dihedra are gaping, as we shall show; more precisely: 11.J. A dihedron ( Y , , Y2)is closed if and only if it is gaping and Y , A Y , is closed. In particular a disjoint dihedron is closed if and only if it is gaping.
Proof. We may assume, for the proof in either direction, that Yl n Y , is closed. Let Q be the canonical epimorphism of X onto the quotient space X / ( Y , n Y,). Then (QY, , QY,) is a disjoint dihedron in Q X = X / ( Y l n Y,), and Qn-l(QY,)= Y i , i = 1,2. Let Ill be the algebraic projection of Q X along QY, onto QY, , and l7, = I - n,. If the Yi are closed, they are complete, hence so are the QY,. By lO.B, the Ui are bounded. For any given X E Xand any p > 1 there exists y1 E Y , such that Q y , = nlQx, II y1 II d pII Qyl /I PII G II I1 x II; also, if y, = x - y 1 , we have Qy, = n2Qx E QY, , whence y 2 E Y , , I1 Y2 I1 (1 PI1 n,II) II x II. Hence ( Y , , Y2) is gaping, with K ( Y 1 , Y2)
<
1
< + +If II(4Y , II., Y,)
<
is gaping, let p > 1 and ( E Q X be given; there y, , exists x E X such that Qx = 4, 11 x 11 pII 4 11; we may set x = y, with yi E Yi , 11 y i I/ p ~ l xl 11, i = 1,2, where K = K( Y , , Y,). Then Qyi E Q Y , , Qy, Qy, = 4 ; hence Qyi = n,4 and I/ II II yi II d p 2 ~ l 4l 1 , i = 1,2. Thus the IIi are projections and, since p was arbitrarily close to 1, 11 11 K ( Y , ,Y,), i = 1, 2 . By 10.B, the QY, are closed, and so are the Yi = Q-l(QY,), i = 1,2. &
+
<
<
+
<
n, <
Remark. For a disjoint closed dihedron ( Y , , Y 2 )it is obvious that Y , , Y z )= max((1P 11, 11 - P [I}, where P is the projection along Y , onto Y , (cf. 10.B). Therefore, by 1 1.D, y-'[Y, , Y,] < K ( Y 1 , Y,) < 2r-"Y,, y21.
K(
I 1.K. A dihedron ( Y , , Y,) is guping cl(Y, n Y,). Proof.
if
and only
if cl Y , n cl Y 2=
The condition is necessary: ( Y , , Y2) is a gaping dihedron;
Ch. 1. GEOMETRY OF BANACH SPACES
12
set K = K ( Y , , Y,). Let x E cl Y , n cl Y , , Q > 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. &
<
<
<
+
+
+
+
<
+
+ <
+
<
+
+
<
<
We now consider some geometric properties of closed dihedra (which are gaping by 11. J).
1 l.L. If ( Y , , Y,) is a closed dihedron, then d ( Y , n Y, ,y,) < Y , , Y,) d( Y , ,y,) for any y1 E Y,; and similarly with the indices 1, 2 interchanged. K(
Proof. With SZ, LIi as in the proof of 11.J, d ( Y , n Y,, y,) = II QY, I/ = II f l l P Y 1 - Qy,)ll for all yz E y , , whence d ( Y , n y , 9 Yl) < II 171 II d(QY2 QY,) < K ( Y 1 , YJ d ( Y , Y l ) , since II n,II \< K ( Y 1 9 Y,), llQll==l. & 9
9
11 .M. If ( Y , , Y,) is a closed dihedron, A > 1 is given, and q1 is a [continuous] (Y, n.Y,, A)-splitting of Y , , then the mapping q : X + X defined by q ( x ) = ql(yl), where x = y1 y , ,yi E Yi , i = 1, 2 is well defined, and is a [continuous]( Y , , AK)-splitting of X , where K = K ( Y,, Y,); and similarly with the indices 1, 2 interchanged.
+
Proof. Let Q, LIi be as in 11.J. Properties (a) and (b) of q1 imply that there exists a unique mapping w1 : QY, -,Y , such that ql(yl) = wl(Qyl) and s2w,(Qyl) = Qy, for all y1 E Y , . It is easy to see that w1 is continuous if and only if q1 is continuous. It follows that the q defined in the statement is precisely q ( x ) = w,(fl,SZx), which is therefore well defined, and is continuous when q1 is. Furthermore, x - q ( x ) = y , ( y , - ql(yl))E Y,; if x1 - x2 E Y , we have y: - y ; = (x' - xz) -
+
12. COUPLED SPACES
13
12. Coupled spaces
Coupled spaces Let X , X‘ be Banach spaces over the same scalar field F, and let : X x X‘ -+F be a bilinear functional. For any sets V C X , I” C X‘, we define, as usual, the POLAR SETS V o = {x’ E X ’ : Re(x, x’) 1 for all x E V ] , V’O = {x E X : Re(x, x‘) 1 for all x’ E V,}. We shall say that X , X’ are COUPLED (BY (-,.)) if their unit spheres Z ( X ) ,Z(X’) satisfy (a,
a)
<
<
qx)”= Z(X),
(12.1)
Z(x,)” = C ( X ) ;
in other words, if, for every x E X and every x’ E X’, (12.2)
If X , X’ are thus coupled, X ‘ , X are of course coupled by the transposed functional, so that we may, and shall, speak of the unordered pair X , X‘ as coupled by the functional and consider the order as a mere device for labeling the arguments of the functional. From (12.1) and (12.2) it follows at once that I(x, x’)l 11 x 11 11 x’ (1 for all x E X , x’ E X‘; .) is continuous; and X o = (O}, X’O = (0}, so that X , X’ constitute a pair of spaces in duality under (., *), in the sense of Bourbaki [l] (p. 48). If F is the complex field, a standard argument shows that X , X’ are coupled by (., -) if and only if the underlying real spaces are coupled by Re(., T h e most conspicuous instance of a coupled pair is of course that of an arbitrary Banach space X and its dual X * , coupled by the natural or “evaluation” functional (x, x*) = x*(x). Wherever the pair X , X* shall occur in either order, the coupling will be understood to be by means of this functional. If X,X’are Banach spaces coupled by (., .), the mapping @‘ : X’-+X* defined by = (., x’) is, by the first relation in (12.1) or (12.2), a congruence onto a subspace of X*,and the second relation in (12.1) or (12.2) means that W X ‘ is a subspace of characteristic 1 of X * , in the terminology of Dixmier [I] (a duxial subspace of X * , according to Ruston [l]). T h e same remarks apply to the mapping Q, : X -+ X’*
<
(a,
a).
@‘XI
Ch. 1. GEOMETRY OF BANACH SPACES
14
defined by @x = (x, .). If X is reflexive, @‘ and @ are necessarily surjective and, up to a congruence, X’ coincides with X* and X with X’* (Ruston [I], p. 580). We shall return to this special case later. From now on, throughout this section, X , X’ shall always denote Banach spaces coupled by a given bilinear functional .). As usual, we denote by u(X, X’) and u(X’, X ) the weak topologies in X and X’ induced respectively by the other space (Bourbaki [l], p. SO); in particular, if X’= X*, these topologies are the weak topology of X and the weak* topology of X * , respectively. I t is clear that for any set VC X the set V o is u(X’, X)-closed; and if V is convex and balanced (i.e., symmetric or circled, according as F is the real or the complex field) then YO0 is its u(X, X’)-closure (Bourbaki [I], p. 52). Consider a given A EX. It is an immediate consequence of the definition and fundamental properties of the weak topologies(Bourbaki[13 p. 100) that an operator A’ E ( X ) - such that ( A x , x’) = (x, A’x’) for all x E X , x’ E X ’ exists if and only if A is continuous in u ( X , X‘); and then this ASSOCIATE OPERATOR A’ is uniquely defined; equivalently, A‘ exists if and only if @’X’ is invariant under A*, the adjoint operator of A, and then A’x’ = @‘-lA*@’x‘.T h e associate operator satisfies 11 A’ 11 = 11 A 11 and is continuous in u(X’, X ) ; the mapping A 3 A’ is linear and isometric, and A is the associate operator of A‘ in the transposed coupling. If A is invertible, so is A’ (Bourbaki [l], pp. 102-103), and A’-l = (A-l)’. Observe that, if X’= X*, every A E 13 has the associate operator A’ = A*. . (a,
Su bspaces If Y is a linear manifold in X [in X’] its polar set Y o is precisely its annihilator with respect to the coupling functional. If Y = YOo, i.e., if Y is u ( X , XI)-closed [u(X’, X)-closed], we shall say that Y (which is then necessarily closed, i.e., a subspace) is a SATURATED subspace; in particular, YO is always saturated. If X‘ = X*,every subspace of X is saturated. Consider a subspace Y of X, and let 52‘ : X ‘ + X’/Yo be the canonical epimorphism onto the quotient space. T h e bilinear functional <*, : Y x ( X / Y o+ ) F defined by ((y, Q’x’)) = (y, x’) is obviously well defined and, using the notation ( O ) for polar sets with respect to this functional, we have Z(Y) = Z(X’/Y0)(O), Z ( X / Y o ) C Z( Y)c0); the inclusion is equivalent to a>>
(12.3)
d( Yo,x’) 2 sup{ I ( y , x’)
I
:y
E Z( Y ) }
for all
x’
EX’.
15
12. COUPLED SPACES
If this inclusion is an equality, or equivalently, if equality holds in (12.3) for all we say that Y has the STRICT COUPLING PROPERTY; X I ,
another way of describing it is saying that Y and X ' / Y o are coupled by <*, .>>. 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:
<
12.A. Let X be any Banach space. For the coupled spaces X , X * , every subspace of X and every saturated (i.e., weakly* closed) subspace of X * has the strict coupling property.
Proof. For every subspace Y of X , Y * is congruent to X * / Y o , and ( X / Y ) * to Yo, under the natural mappings; thus Y and X * / Y o are coupled, and for a saturated subspace Z C X * , so are 2 = Zoo and X / Z o . &
Remark 1. In general, if X is not reflexive, there may exist nonsaturated subspaces of X * that do not even have the quasi-strict coupling property: indeed, a subspace 2 of X* has this property, with some s y > 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. ,$,
+
+
+
+
Remark. It has been shown (e.g., Schaffer [6], Example 2.1) that for general coupled Banach spaces X, X‘ a disjoint dihedron (Y, Z) of X with Y, Z saturated need not be (X, X’)-disjoint.
Selections and Splittings We give two lemmas concerning the existence of certain continuous mappings.
12.E. For any h > 1 there exists u continuousfunction n : X’ \ (0) + X such that ( n ( x ’ ) , x’) = 11 x‘ (1 and 11 n(x’)ll A.
<
Proof. Consider no : X* \ (O}+X defined by 10.F, set no(Qb’x’), and use the fact that @’ is a congruence. ,$,
~(x’) =
12.F. Let Y be a subspace of X with the quasi-strict coupling property, and p > 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 .
<
XI),
Let q be any continuous (Y, p1/2)-splittingof X (1 1.E). If x’ = 0, this satisfies the conditions of the statement, since p112 < p(I + As,). If x‘ # 0 we must have x‘ $ Yo. Since (12.4) holds, there exists y E Y such that ( y , x’) = 1, II y I1 d( Yo,x’) p1’2~yl(y, 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’) =
set r(x; 0) = q(x);
<
<
Ch. 1. GEOMETRY OF BANACH SPACES
18 Finally,
Remark. A modified form of 12.E would allow us to show that in 12.F y may be chosen as a continuous function of x' $ YO, so that I : X x (X' \ Y o )-+ X is then continuous; we shall not need this additional information.
'
Reflexive spaces
For the convenience of the reader, especially the reader disinclined to pay for utmost generality with additional technical complications, we summarize the simplifications introduced in this section when either member of a coupled pair of Banach spaces is reflexive. We have already stated that if X, X' are coupled and X is reflexive, then @' is a congruence of X' onto X*, so that X' is also reflexive; we therefore may, and shall, restrict ourselves to the coupled pair X, X * , where X and X** are identified. T h e topologies u(X, X * ) , o(X*, X) are simply the weak topologies of X, X', respectively. Any operator A E has the associate operator A' = A*, and similarly for every operator in (X*)". Every subspace of X or X * is saturated; every subspace of X and X * has the strict, a fortiori the quasi-strict, coupling property (12.A). If Y, 2 are linear manifolds in X, then (Y, Z), (YO, Zo) are both dihedra if and only if either the one or the other is a disjoint closed dihedron; and then both are (12.B and 1O.B); hence a dihedron (Y, 2) is (X, X * ) disjoint if and only if it is disjoint and closed.
13. The class of subspaces of a Banach space
Two mettics In this section we give an account of two ways in which a topological and metrical structure may be imposed on the class of all subspaces of a Banach space, and of those properties of these structures which are relevant for our purpose. We shall follow in the main the exposition of Berkson [l], to which we refer the reader for details. T h e contents of this entire section will be applied for the first time in Chapter 7. We consider a fixed Banach space X, and the classes E ( X ) , S , ( X )
13. THECLASS
OF SUBSPACES OF A
BANACHSPACE
19
of all its subspaces and of all its complemented subspaces, respectively. According to our first notion of “closeness” of elements of E ( X ) , two subspaces of X are close to one another if one is mapped onto the other by an automorphism (invertible operator) on X that is close to the identity. More precisely: if Y , Z E E ( X ) , an invertible C E is ACCEPTABLE (FOR Y , 2) if CY = 2. We define the function inf{ 11 C - Ill : C acceptable} yo(
if an acceptable automorphism exists; otherwise.
y,2) =
/ I
A basic neighborhood of Y should then consist of those Z for which , 2) < c. We proceed to show that this defines a topology, indeed a metric topology.
yo( Y
13.A. Forany Y , Z ~ E ( X ) w e h a v e r , ( Y , 2< ) 1, I ro(Y,2 ) - r o ( Z , Y)I < ro(Y , Z)ro(Z,Y ) ;yo( Y , 2) = 0 if and only if Y = 2. For any Y ,2, V E ( x ) we have yo( Y , V ) < ro(Y , z) + r O ( z ,v) + Y , z ) r O ( Z V , ). E
yo(
Proof. T h e cases where any one of the acceptable automorphisms fails to exist are trivial and may be disregarded. If C i s acceptable for Y , Z , so is XC for every scalar X # 0, so that yo( Y , 2) limA+o11 XC - Ill = 1. If yo( Y , 2) < 1, there exists an acceptable C for Y , 2 with 11 C - Ill < 1. taking the Then ro(Z, Y ) 11 C-l - Ill (1 C - Ill ( I - 11 C - I infimum we obtain ro(Z, Y ) - ro(Y , 2) yo( Y , Z)ro(Z, Y ) , which is a fortiori true if ro(Y , 2) = 1. Interchange of Y , 2 completes the proof of the first part. If Y = 2, I is acceptable and r0(Y,2 ) = 0. Conversely, assume r0(Y,2) = 0 ; there exists a sequence (C,) of acceptable automorphisms with limn+, C, = I . For any y E Y , C,y E Z for every n; since 2 is a subspace, y = limn+ooC,y E 2. Thus Y C 2. But ro(Z, Y ) = 0, by the above proof; hence also Z C Y , and equality holds. If C , , C, are acceptable for Y , Z and 2, V respectively, C,C, is acceptable for Y , V , and 11 C,Cl - Ill = Il(Cl - I ) (C, - I ) (C, - I)(Cl - Il1 II Cl - I II II - I II II Cl - I II II - I II. Taking infima, the required inequality follows. &
<
<
<
<
+
<
+
c,
+
+
c,
We set r( Y , 2) = max{ro(Y , Z ) , r o ( 2 , Y)}.I t follows from 13.A that log( 1 + r( Y , 2))is a distance function on E(X). We use an equivalent, more natural distance function, namely (13.1)
6(Y,Z)= infl z r ( V i , Vi+l) :n i=O
=
1,2,
...;
i = o ,..., n;
VieE(X),
1
Vo=Y, V,=Z.
Ch. 1. GEOMETRY OF BANACH SPACES
20
6 is obviously symmetric and satisfies the triangle inequality. Now r( Vf, V,,,) 3 log( 1 r( vi , Vi+,))2 log( 1 t.( y, Z)), so that 6( Y , 2) = 0 implies Y = 2, and 6 is a distance function. Now the function u-’log(l + u ) is decreasing for 0 < u 1. Hence, for any Y , 2 E qx):
c:;
:c:
+
+
<
(13.2)
6(Y,Z)log2 < r ( Y , Z ) l o g 2
<
Since, by 13.A, ro(Y , 2)< E summarize:
<
< 1 implies Y( Y , 2) < E( 1 - ~ ) - l we , may
13.B. 6, defined by (13.1), is a distance function on S ( X ) , equivalent r ) , and satisfies (13.2). A basis of the induced uniformity to log(1 consists of the sets {( Y , 2) : yo( Y , 2) < E } f o i all real E > 0.
+
Remark. If r satisfies the triangle inequality, then 6
= Y.
’The metric space defined on E ( X ) by 6 is denoted S ( X , a), and the metric space defined on S,(X) by E J X , 6) (cf. 13.G). 13.C. S ( X , 6) is complete. Since we shall not require this property, we refer for its proof to Berkson [l], Theorem (4-1).
T h e other notion of “closeness” is defined in terms of the HAUSDORFF of nonempty closed bounded sets (see Kuratowski [l], pp. 106-107), defined, for any such sets M , N , by
DISTANCE
d(M, N ) = max(sup d(M, x), sup d ( N , x)}. SEN
XEM
We define as distance function in S ( X ) the Hausdorff distance between the unit spheres: (13.3) S’(Y,Z )
= d ( Z ( Y ) ,Z(Z))
= max{sup{d(Z(Y),
z) : z E C(Z)}, sup{d(Z(Z),y ) :y E Z ( Y ) ) )
< 1. T h e metric space defined on E ( X ) by 6’ is denoted by E ( X , 6’); the notation E,(X, 6‘) is defined in the same way. 13.D. S ( X , 6‘) is complete: indeed, if (Y,) is a 6’-Cauchy sequence in S ( X ) , and Y = {limn+my , : (y,) a Cauchy sequence, y f LE Y,}, then a’( Y , Y,) = 0. Y E E ( X ) and
13. THECLASS
OF SUBSPACES OF A
BANACHSPACE
21
Proof. Y is obviously a linear manifold. Now (Z(Y,)) is a A-Cauchy sequence. Since X is complete, a fundamental theorem on the Hausdorff distance (Kuratowski [l], p. 314) implies that it has the A-limit M = {lim,4m u, : (un) a Cauchy sequence, u, E Z(Yn)}. Since 11 y n I/ = 11 1im,-=yn 11 for a Cauchy sequence, and since M is closed, we have M = Z( Y ) ,and Y is a subspace; and S’( Y , Y,) = limn+ 4q0,q y , ) ) = 0. &
13.E. If Y , 2 E E ( X ) , Y C Z , Y # 2, then S‘( Y , 2) = 1 . In particular {0}, X are isolated in B(X, 6’). Proof. Let x E Z \ Y be given. For any X > 1 there exists y E Y such that11 x - y I/ X d ( Y , x) = X d ( Y , x - y). Thensgn(x - ~ ) E Z ( Z ) and
<
W V ) ,sgn(x - y ) ) 2 4 Y , sgn(x - Y ) ) 2 I1 - y 11-1 4 K x - Y ) >, A-1.
Since A
> 1 was
arbitrary, the conclusion follows from (13.3).
4
T h e relationship between S and 6’ is described in the following lemma. 13.F. For any Y , Z E B ( X ) , S’( Y , Z ) < 26( Y , Z ) . The uniformity and topology of E ( X , 6) are stronger (finer) than the uniformity and topology, respectively, of S ( X , 6’).
Proof. We claim that (13.4)
6’(Y,2 ) < 2r(Y, Z);
on account of (13.3) it is sufficient to show this for r(Y, Z ) < 1: let CE be acceptable for Y , 2. Now Y = {0} if and only if 2 = {O}, and then (13.4) is trivial; we therefore assume Y , 2 # {O}. For any y E C(Y ) \ (0) we have, using 11 .A, d(V),Y)
< (1 IlY II W ( C Y ) - Y 11 II Y I1 rCQ, rl < 2 /I ( C - or II < 2 II c - 111.
=
Taking the supremum on the left and the infimum over acceptable C on the right, we have sup{d(Z(Z), y ) : y E Z( Y ) } 2r,( Y , 2). Inter-
<
22
Ch. 1. GEOMETRY OF BANACHSPACES
changing Y , Z and using (13.3) we obtain (13.4). But (13.1), and (13.4) applied to V , , Vi+l, plus the fact that 6’ is a distance function, give S‘( Y , 2) < 2 S( Y, 2). &
The complemented subspaces We shall now examine the properties of E,(X) as a subset of E ( X , 6) and S ( X , S’), and compare the structures of I.“,(X,8) and E c ( X ,8’). 13.G. S,(X) is open and closed in S ( X , 6): indeed, 8(Y,2) < 1 implies that both Y , 2 are, or both are not, complemented. If Y , 2 E E,(X), then (13.5)
S(Y, Z )
< r(Y, 2) < inf{ IIP -QII QX
: P,Qprojections, PX = Y ,
= 2).
Proof. If S( Y , 2) < 1, there exists a finite sequence, V , = Y, ..., V , = 2, such that Cyzt r ( V i , Vi+l) < 1. Therefore there exists an acceptable automorphism C, for V , , V,+l,and therefore either all V , are complemented, or none is. Let P,Q be projections as stated; they exist by 1O.B. It is sufficient to consider the case in which 11 P - Q 11 < 1. Then C = I - P + Q, D = I P - Q are invertible, and CY = QY C 2. It is easily verified that D(I - P ) = (I - Q)C, so that (I - P)C-lZ = P 1 ( I - Q)Z = {0}, whence C-’ZC Y . Hence CY = 2, and ro(Y,2) < 1) C - I))= IIP-QII. Interchange of Y , Z gives r ( Y , Z ) IIP-QIl. (13.5) follows, using (13.2). &
+
<
13.H. Let Y E Ec(X) be given, Y # {0},X . Let P be any projection along Y. If 2 E E ( X ) ,S’( Y , Z)< min{lI P then Z E E,(X); 11 I - P the range of P is a complement of 2, and the projection Q along Z onto the range of P satisfies
V
PYOO~. Since 1) PI) 1, S’(Y,2) < 1, and by 13.E, Z # (0}, X. Set = PX. We first show that Z n V = (0). If x E (2 n V )\ {0}we have
which is absurd.
13. THECLASS
OF SUBSPACES OF A
We next show that Z be given. Then
+V
BANACHSPACE
23
is closed. Let Z E Z \ { O } , V E V \ (0)
3 dCY, sgn el - sgn z ) 2 d( Y , sgn v) - d( Y , sgn 3) 2 I1 PII-’ - V Y , 2 ) ;
y[z, v ]
taking the infimum, y [ Z , V ] 2 11 P
(13.6)
+
11-1
- S’( Y , Z) > 0;
by 11.D, 2 V is closed. V . Assume that this is not the case. We now claim that Y C Z Since Z V is a subspace, Y n (2 V ) is a proper subspace of Y ; for every X > 1 there exists (cf. proof of 13.E) an element y E Y \ (0) such that 11 y 11 X d( Y n ( Z V ) ,y ) . There exists, then, z E Z such that 11 y - z 11 X 6’( Y , 2)I/ y 11. But (I - P)z E Y , (I - P ) x = z - PZ E Z+ V , so that
+
+
<
+
+
<
+
II Y II < 4 y n (Z V ) , Y ) < IIY - (1 - P)z II < I/ I - p II II Y 3 II < x2 II I - p II a‘( y , Z ) II y II. ~
Since 11 I - P 11 6’( Y , 2 ) < 1, an appropriate choice of X > 1 leads to a contradiction, and our claim is established. Now X = Y V C 2 V C X,so that equality holds, and (2,V ) is a disjoint closed dihedron; let Q be the projection along 2 onto V . By (13.6) and I l . D ,
+
+
(13.7)
I1 Q /I
< 2 11 P I1 (1 - II P I/ S’( Y , Z))F’.
Since V is the common range of P and Q, we have Q P = P . Let x E X be given, Since ( I - P ) x E Y , there exists, for every X > 1, an element z E 2 such that 11 z - ( I - P )x 11 X 6’( Y , Z)ll (I - P ) x 11. Since Qz = 0, we have
<
II (Q - OXII
=
11 Q(1 -
II
= II Q(z - (1 - P>x)II
<
II Q II I1 1 - P II a’(Y, 2)II x 11.
T h e last part of the statement follows, using (13.7), since were arbitrary. & 13.1. B , ( X ) is open in E ( X , 6’). coincide.
> 1 and
x
The topologies of B”,(X,a), E,(X, 6’)
Proof. (0) and X are isolated elements of E ( X , 6’) (13.E) (and a fortiori of E ( X , 6) (13.F)), hence 6’-interior to E,(X). All other
24
Ch. 1. GEOMETRY OF BANACHSPACES
Y EE,(X) are 6’-interior to this set by 13.H; hence the set is open in E ( X , 6’). For any such Y , and any projection P along Y , S’(Y,2) < min(l1 Pll-l, 11 I - PIl-l} implies, by 13.H, 13.G, 6(Y,Z)<2llPII jl I-PII S’( Y , 2) (1 - 11 P 11 6’( Y , Z))-l; therefore the topology of S ,(X , 6’) is stronger than that of E J X , 6); the reverse relation follows from 13.F. & J. Lindenstrauss (private communication) has given a very simple example of a Banach space X for which & ( X ) is not closed in S(X, 8’). On account of 13.G, the topologies of E ( X , S), E ( X , 8‘) do not coincide; further, since &(X, 8) is complete (13.C, 13.G) and & ( X , 8’) is not, the uniform structures of the latter two spaces do not coincide, in spite of 13.1.
The class
of
closed dihedra
T h e closed dihedra in X constitute a subset of the Cartesian product E ( X ) x E ( X ) . Our purpose is to show that this set is open in the product topology when E ( X ) is equipped with either of the metrics 6,s‘. 13.J. The set H ( X ) of all closed dihedra in X and the subset H o ( X ) of all disjoint closed dihedra are both open i n S ( X , 6’) x E ( X , a‘), a fortiori in E ( X , 6 ) x E ( X , 8). The gape K ( * , .) is continuous on H ( X ) , and the angular distance y [ - , -1 is continuous on H o ( X ) minus the dihedra ({0},X ) , ( X , {0}),in either of the induced topologies.
Proof. 1. Since the topology induced by 6 is stronger than that induced by S‘, it is sufficient to consider the latter. We begin with the set H ( X ) . Let ( Y o, 2,) E H ( X ) be given and set K~ = K ( Yo , Zo). Let E be such that 0 < ~ K <~ I ,E and consider any fixed ( Y ,2)E E ( X ) x S ( X ) with 6’( Y o , Y ) + S‘(Zo,2) E . We shall show that ( Y ,2)E H ( X ) , so that H ( X ) is open in S ( X , 6 ’ ) x E(X,S’). Let p > 1 be so small that 2p2K,E < 1. We first claim that for any x E X there exist y E Y , z E Z such that 11 Y 1 , I/ z 11 fKolI X 1 , 11 x - ( y z)ll p2KoEll x 11. Indeed, by the defini, exist yo E Y o , zo E 2, such that y o x0 = x, tion of K ~ there /I yo 11, 11 z0 I/ pKolI x 11. By the definition of 6’ there exist y E Y , x E 2 such that I1 y II II yo I/, II z II II xo II, II y - yo II I1 z - 2 0 II d pE max(l1 y o 11, 11 xo l } p2KoEllx 11; and these y, 2: satisfy our claim.
<
< <
+ <
<
<
<
+
+
2. We next show that ( Y , 2) is a (closed) dihedron and estimate its gape. Let X E Xbe given. We construct inductively sequences ( x n ) in X , (yn)in Y , (z,) in 2 as follows: xo = x; if xnP1is given, the preceding part of the proof warrants our choosing y n E Y , z, E Z and x, = xnP1 --
13. THECLASS
OF SUBSPACES OF A
BANACHSPACE
25
( ~ n + z n )in such a way that I1yn 11, I1 z n I1 ,< pKolI xn-111, I1 xn I1 6 p2KoEll xn-1 I/. BY induction, II xn II d (f2Kot)nll < I 1 9 I/ Y n 11, II zn I1 d fKo(f2Ko‘)n-111 x /I. Since p2Ko€ < < 1, (x:yj), zj) are Cauchy sequences in the subspaces Y , 2, respectively; let y E Y , z E Z be their limits. By the (yj zj)/l = /I x,+~ 11 ( p 2 ~ o ~ ) nx+ 1 definition of ( x n ) we have 11 x 1) x /I -+ 0 as n oc), so that y z = x; hence ( Y ,2) is a (closed) dihedron. ~ 1 ~II y0 11,, II z 11 P K , (p2Kor)nIIx II = pKo(l - p2KoE)-’l/ x /I. Since p was arbitrarily near 1, we have K = K ( Y , 2) ~ , ( l- K ~ E ) - ~ , K~KE. or K - K~ But K E K , E ( ~- K ~ E )< - ~8/&= 1; with p > 1 such that p 2 K E < 1, the above argument may be repeated with ( Y o ,2,) and ( Y ,Z ) interchanged. We thus obtain K, - K ,< K K ~ E , and combining with the previous result,
(1,
x:+ + x,“
--f
<
<
<
< <
I K-l This shows that the gape ogy induced by 6’.
K(*,
-’I < . .
- Kg
E.
*) is continuous on H ( X )under the topol-
3. To prove the statements about H,(X) we actually show a little more. Let H ’ ( X ) be the subset of S ( X ) x S ( X ) consisting of all pairs ( Y ,Z ) of subspaces such that Y n 2 = {0}, Y , Z # {0}, and Y 2 is closed; by 11.D, the last condition may be replaced by y [ Y , Z ] > 0. We show that H ’ ( X ) is open and y [ * , -1 continuous on it under the topology induced by 6’. Let ( Y o ,Z,) E H ’ ( X ) be given and set yo = y [ Y o, Z,]. Let E be such that 0 < 4r < yo < 2, and consider any fixed ( Y , 2)E B ( X ) x E ( X ) with S’( Y o, Y ) S’(2,, Z ) E . We shall show that ( Y ,2)E H’(X). Since E < 1, 13.E implies that Y , Z # (0). Choose any p > 1 such yo . L e t y E Y \ (01, z E Z \ (0)be given. There exist yoE Z( Yo), that 4 p ~i zo E Z(Z,) such that 11 yo - sgn y 11 11 zo - sgn z 11 pr. It. follows that II y o II 11 zo!I 2 2 - pE, whence /I sgn yo - sgn y I1 II sgn zo sgn x 1) ,< 2pt. Therefore
+
+
<
+
+
Y [ Y , 4 = I1S
P Y
- sgnz/l>,
<
+
rbo , Z”1 - 2P‘ 2 Y o - 2PC > 0;
hence Y n Z = (0)and y = y [ Y , Z ] 3 yo - 2~> 0 so that ( Y ,Z ) E H ’ ( X ) . Since y >, yo - 2~ > 2r, we can repeat the argument with ( Y o ,Z,), ( Y , 2) interchanged, and conclude that 1 y - yo I 2 ~ S,O that y [ * , -1 is continuous on H ‘ ( X ) under the topology induced by 8‘.
<
4. By 13.E, ({O}, X ) , ( X , (0)) are isolated points of E ( X , 8’) x B ( X , 6’); and obviously H o ( X ) = ( H ( X )n H ’ ( X ) )u {({0},X ) , ( X , {0})),so that H o ( X )has the properties in the statement. &
26
Ch. 1. GEOMETRY OF BANACHSPACES
Remark. Since the projection of H o ( X )on either factor of E ( X ) x E ( X ) is precisely E,(X), 13.J yields a new proof of the fact (13.1) that E , ( X ) is open in E(X, 6’).
A lemma on continuously varying subspaces The following lemma is a simple consequence of the properties of the topologies which we have introduced:
13.K. Let U , V be continuous functions on the real interval J with such that U(to)= V(to)for some toE J. If Y is a invertible values in subspace of X such that for each t E J either U(t)Y C V ( t ) Yor U ( t ) Y 3 V ( t ) Y , then U ( t ) Y = V ( t ) Y for all t~ J.
x,
Proof. We may assume, without loss of generality, that V = I. T h e function t -+ U ( t ) Y : J + E ( X , 6) is continuous, since U(t”)Y = U(t”)U-l(t’)(U(t’)Y)and liml*+l*11 U(t“)U-l(t‘)- Ill = 0. By 13.F, the function t -+ U ( t ) Y : J -+ E ( X , 6‘) is a fortiori continuous; hence S‘(Y, U ( t ) Y )is a continuous function of t; but by the assumption and 13.E it can only take the values O,l, and actually equals 0 for t = to. Therefore 6’(Y, U ( t ) Y )= 0, t E J, which proves our assertion. & 14. Hilbert space
Notation We shall show in this section in what way the geometrical concepts and properties introduced in the preceding sections may be expressed in “euclidean” terms when the Banach space X is actually a Hilbert space. We deal in this section with a Hilbert space X;the inner product is denoted by (., .). If the scalars are the complex numbers, the underlying real space is a Hilbert space with inner product Re(-, .). The set of elements orthogonal to all the elements of a set V C X is denoted by VL = {x E X : (x, v ) = 0 for all v E V>. Occasionally we shall use Z ( X ) = {x E X : (1 x (1 = 1). If Y is a subspace of X, the orthogonal projection with range Y is denoted by P,. The abbreviations Dim and Codim stand for orthogonal dimension, and codimension, respectively. We denote the natural isometric conjugate isomorphism from X onto X* by VJ; i.e., VJX = (., x). If Y is a subspace of X, then m y L = Yo. If A EX,we denote its Hilbert adjoint (Hermitian transpose) by A’;
27
14. HILBERT SPACE
i.e., A t = m-’A*w. For most purposes, we shall speak as if UJ performed an identification of X and X * ; the evaluation coupling between X and X * is then replaced by the inner product.
Angles, splittings, and dihedra T h e fundamental “euclidean” geometric concept we need is that of angle between vectors, between subspace and vector, and between subspaces. If x , y E X \ (0), then the ANGLE +(x, y ) is uniquely defined by the conditions
I1 x /I IIY II cos a ( x , Y ) = Re(x,y),
0
If Y # (0) is a subspace of X and x E X inf{+(y, x) : Y E y \ {O}), or
< a(x,Y) <
7r.
\ Y , we define *( Y ,x)
=
T h e definition of the angle between subspaces Y , 2 is slightly more complicated, and we shall use this angle only if neither Y C Z nor Z C Y (whence Y , Z # (0)). If Y n Z = (0), we set + ( Y , 2) = inf{ M y , 2) : y E y \ (O}, z E z \ (0)); or (14.2)
I (Y 4 I cos + ( Y , Z ) = s u p --!--:yyE IIY !I I, 27 II
I
Y\{O}, Z E Z \ { O } / ,
0
< +(Y,Z) d &.
Otherwise, we set V = ( Y n Z)* and observe that Y n V # {0}, since for anyy E Y \ 2 we have P,y E ( Y n V)\ (0); likewise Z n V # (0); and ( Y n V ) n (2 n V ) = (0). It is therefore meaningful (and corresponds to the usual concept of “dihedral” angle) to set +(Y, 2) = +(Y n V , 2 n V ) ; this is also consistent with the former definition, since Y n Z = (0) implies Y n V = Y , Z n V = 2. We observe that angles involving subspaces lie between 0 and &r, and are therefore uniquely defined by their cosine or their sine. T h e relationship between angles and angular distance is given by the obvious formula (14.3)
Y[X,rl = 2 sin
+a(.,Y ) ,
x, Y E
x \ (0).
14.A. If Y # (0) is a subspace and x E X \ Y , then y [ Y , x] = 2 sin &+( Y , x), and 11 x 11-l d( Y , x) = sin +( Y , x) (thus 11 x 11 < h d( Y , x),
28
Ch. 1. GEOMETRY OF BANACH SPACES
>
x # 0 is equivalent to sin +(Y, x ) k l ) . If Y, Z # (0) are subspaces with Y n Z = (0}, then y[Y, 21 = 2 sin 8 +( Y, 2).
Proof. Immediate from the definitions. 9, 14.B. If ( Y , 2) is a disjoint closed dihedron, not ( X , (0))nor ((01, X ) , and P is the projection along Y onto 2, then sin Y, 2) = 11 P 11-l.
a(
Proof. I f x E Y u 2, either P x = 0 or P x = x . If x $ Y u 2, consider the triangle with vertices 0, Px, x ; by the Sine Law,
11 Px 11 sin 3 ( Y , 2 ) < 11 Px I/ sin +((I =
1) x 1) sin
*(-3c,
-
P)x, -Px)
-(I - P)x)
< 11 3c I);
<
therefore (1 PI( sin +(Y, 2 ) 1. Now let y E Y , z E 2 be unit vectors, and set x = z - ( y , z)y. Then Px = z, II Px II = 1, and II x 112 = 1 - 2(y, z)2 ( y , z)2 = sin2 +(y, 2). Therefore 11 P 11 sin +(y, z) 1. Taking the infimum, I/ P 11 sin Y , 2) 2 1, and equality holds. &
+
>
a(
14.C. If Y , Z are subspaces of X , Y either Y C Z or Z C Y or +(Y, 2) > 0.
+Z
is closed
if
and only
if
Proof. Assume that neither Y C 2 nor 2 C Y. Set V = ( Y n Z ) l . If Y + Z is closed, so is ( Y 2) n V ; if the latter set is closed, so is Y + Z = P;l((Y 2) n V).But(Y + 2) n V = ( Y n V ) ( Z n V ) ; and (Y n V)n ( Z n V ) = (0). By l l . D and 14.A, ( Y Z ) n V is closed if and only if sin 8 +(Y n V , Z n V ) = sin 8 <(Y, 2) > 0. &
+
+
+ +
As far as splittings are concerned, all that need be said is that, since every subspace is complemented, we may restrict ourselves to the “projection splittings” considered in l l . F ; it does not appear to be convenient to limit ourselves exclusively to those involving orthogonal projections. Observe however, that I - P y may be spoken of as the unique ( Y , 1)-splitting (with h = l!) if Y # X. T o complete this survey of geometrical objects introduced in Section 11, we give the precise value of the gape of a closed dihedron. 14.D. If (Y, 2) is a closed dihedron, Y, Z # X , then K(Y,Z)sin + ( Y , Z )
= 1.
Proof. T h e assumption implies that neither subspace is contained in the other. Set V = (Y n 2)l.Then ( Y n V , Z n V ) is a disjoint closed dihedron in V (cf. proof of 14.C) and the norms of the projections
29
14. HILBERT SPACE
of V along Y n V onto Z n V and along 2 n V onto Y n V are = l/sin +(Y n V , 2 n V) = l/sin <(Y, 2) (by 14.B). z, with For each x, we have a unique decomposition x = u y u = Py,,x E Y n 2, y E Y n V , z E Z n V. T h e above-mentioned x = P,x --t x, y x ---f y. projections are the mappings y Take any x E V, so that u = 0, and assume x = y' z f , y f E Y, zf E 2. T h e n y' - y = - ( z f - z ) E Y n 2. Therefore 11 y f 11 = 11 y -k (Yf - Y>ll 2 II y 1 , 11 zf II 2 II II; since x = y x E V was arbitrary, K ( Y, Z) 2 l/sin +( Y, Z). Take any x E X. If x E Y n Z, then x = x 0, x E Y, 0 E Z. Otherwise, set
+ +
+
+
+
+
+
y'=y+
IIY II IIYII +Ilzll"'
here y f E Y; zf E Z, y'
z'=z+
II z II IIY I1
+ I1 z II u;
+ xf = x; and
' 1 sin <(Y,Z).Therefore K( Y,Z) and similarly 11 x 11 2 11 21 and equality holds. .&
< l/sin a(Y,Z),
Remark. For any subspace Y # X, K ( X ,Y) = 1; and K ( X ,X)= 4, as the reader may verify at once. We define a special type of closed dihedron: a closed dihedron ( Y ,2) is PERPENDICULAR if either Y = X or Z = X or <(Y, 2) = &r. 14.E. For given subspaces Y , 2, the following conditions are equivalent: (a) ( Y , 2 ) is a perpendicular closed dihedron; (b) Y * C Z ; (c) ( I - P,)(I - Py) = 0. Proof. The equivalence of (b), (c) is trivial. Set V = ( Y n Z)l. As in the preceding proof, (Y, 2 )is a closed dihedron if and only if (Y n V, Z n V) is a disjoint closed dihedron in V . Thus (a) is equivalent to the statement that Z n V is the orthogonal complement of Y n V in V, i.e.,
Z n V = ( Y n V ) l n V = ( ( Y n V ) + ( Y n Z ) ) * =Y l . Since Y LC V always, this is again equivalent to Y l C 2, i.e., (b). &
Ch. 1. GEOMETRY OF BANACHSPACES
30
The
set o f
subspaces
In the Hilbert space X the theory of Section 13 becomes greatly simplified. For one thing, E,(X) = E(X). For the other, the following identities hold. 14.F. Fm any Y, Z E E ( X )we have r(Y, 2) = 6(Y,Z)= 6'(Y, Z)=
II Pr - P z IIProof. We first show that
w,2) < r(Y, 2)
(14.4)
by an adaptation of the proof of 13.F. As in that proof, we may assume r( Y, Z)< 1; let C be an acceptable automorphism for Y, 2. Then, for any y E C(Y),we have P d E C(2), Cy E 2, so that 4 W ) , Y ) = II (1 - Pzlr II < II (1 - ClY II Q II 1 - c II. Continuing as in the proof of 13.F, sup{d(Z(Z), y) : y E Z(Y)} Interchanging Y, 2 we obtain (14.4). We next show that
< ro(Y,2).
We follow the proof in Achieser and Glasmann [l], p. 75. For any x E X we have ( P y- Pz)x = Py(I - Pz)x - ( I - Py)Pzx, and the two summands are orthogonal. Now "(1- Pr)PzX 11 = 11 Pzx - P,PZx 1) S'( Y , 2) II Pzx )I since (IPYPZx II II Pzx II. Further, using at the end the preceding argument with Y, 2 interchanged,
<
<
IIPy(1- Pz)xlla= (Py(1- P z ) ~(1, - p z ) ~= )( ( I - Pz)Pr(l- P.z)x, (1-Pz)x)
< II (1- Pz)Pr(I - P Z )II~II (1- pz)x II < 8'(Y,2)I1f'r(1- P Z )It~II (1- Pz)x II,
1
whence 11 P d I - Pz)x 11 mentioned above,
< S'( Y ,Z)ll(I-
Pz)x 11. Using the orthogonality
+ II (1 - Pr)pzx t12 < 8'a(Y,Z)(IIPZxI!2+ ll(1-Pz)xl12)
II (Py - p z ) IIp~ = II Pr(l- Pz)x Ila and (14.5) follows.
= ~'2(y,~)11x112,
IS. NOTESTO CHAPTER 1
31
Now (14.4), ( 1 4 4 , and 13.G imply r( Y , 2) = 8’( Y , 2) = 11 P, - Pz 11. Since r now obviously satisfies the triangle inequality, 8( Y , Z ) = r( Y , 2) (Remark t o 13.B). J,
Remark. I n view of 14.E, we shall simply write %(X) instead of Jrn( X , 8) = E ( X , 8‘).
< 1, then D im Y = D im 2, - P,I( = 8’(Y,Z ) = Codim 2. For any cardinals d, c such that d c = Dim X , the set ;E(X) = { Y EE ( X ) : Dim Y = d , Codim Y = c} is open and 14.G. Codim Y
If 11 P ,
+
=
closed in S ( X ) .
Prooj. Since 6(Y,2) < 1 (by 14.F), there exists a n acceptable automorphism C for Y,2 ; it preserves orthogonal dimensions and codimensions. &,
15. Notes to Chapter 1 We assume in the reader a certain familiarity with the theory of Banach spaces. As works of general reference for this theory we mention Bourbaki [l], Dunford and Schwartz [l], Hille and Phillips [I], Riesz and Sz.-Nagy [l], Day [I], and Taylor [l]. Our notation and terminology, while in general agreement with current usage (one innovation: the concept of dihedron) does not adhere strictly to the language of any one of these books. Concepts related to the “angular apartness” of elements and subspaces, dealt with in Section 1 I , have been proposed by several authors. Our angular the name angle-by distance y [ x , y] of elements was introduced-under Clarkson [l]. In this connection we also mention Del Pasqua [l], [2], and Sundaresan [11. Concerning the topic discussed in Section 13, we mention that a first generalization (cf. 14.F) of the distance 11 P , - Pz 11 between subspaces of a Hilbert space was given for Banach spaces by Krein, Krasnosel’skii, and Mil’man [l] with the concept of opening e(Y, 2 ) of the subspaces Y , 2, defined as
The properties of the opening were further studied by Gohberg and Krein [l]. The closely related distance function S’( Y , 2 ) was introduced by Gohberg and Markus [l], and independently by Berkson [l] who compared with it the distance function log(1 Y ( Y ,Z ) ) , defined by Massera and Schaffer [l], pp. 562-563.
+
32
Ch. 1. GEOMETRY OF BANACH SPACES
We add an important postscript to Chapter 1. Throughout almost the whole book, the concepts introduced and the results obtained, as well as the proofs, remain unaffected if the norm of the fundamental space X is replaced by an equivalent norm (for coupled spaces X , X ’ we must add that the new norms should preserve the coupling assumption, otherwise we should have to deal with “quasi-coupled” spaces, which we prefer to avoid); the verification of this assertion is left to the reader, and exceptions to it will be specially noted. In particular, if X is finite-dimensional, there is no loss in assuming the norm to be euclidean, so that Hilbert-space terminology applies; although there is usually no substantial gain in this assumption, we shall in general abide by it whenever questions specific to finite-dimensional spaces are discussed.
CHAPTER 2
Function spaces 20. Introduction
Summary o f the chapter In this chapter we introduce several classes of function spaces that will play a fundamental role throughout the study of differential equations that follows. T h e elements of these spaces appear in two distinct capacities: (i) as parts of the equations themselves, and (ii) as prospective solutions of the equations. Since we restrict ourselves to the case of ordinary differential equations with a real independent variable, we only consider spaces of functions whose domain is an interval J of the real line R ; the values of the functions, however, may belong to any Banach space X . As far as situation (i) above is concerned, it seems natural to place ourselves in “Carathiodory’s conditions”: that is, to assume that the functions are (strongly) measurable and (Bochner) integrable on every compact subinterval J ‘ C J ; and to use, as a basic topology, that of convergence in the mean on every such J‘. All such functions with the same domain J and the same range-spaceX constitute, with this topology, the space ,L(X), or L(X) for short; this space occupies a central position in our work. Provision is made, however, for the more traditional requirement of continuity by means of the special consideration of spaces of continuous functions. In (ii) it may seem natural to consider only spaces of continuous functions with a topology related to uniform convergence on each compact subinterval; this point of view has proved to be inconveniently restrictive; since these functions are in L(X) and on compact intervals uniform convergence implies convergence in the mean, it is possible to adopt the more convenient point of view that subsumes the spaces used in situation (ii) under the more general context suggested by (i). After some introductory matter on terminology, notation, etc., 33
34
Ch. 2. FUNCTION SPACES
included in this Introduction, Section 21 discusses the class M ( X ) of function spaces, consisting of normed spaces F that are algebraically contained in L(X) and whose norm-topology is stronger than the topology induced by L(X). A lattice structure is defined in M ( X ) , and questions such as completeness and local closure are discussed (the unit sphere of the “local closure” of F E M ( X )is the L(X)-closure of the unit sphere of F). Section 22 concerns the subclass S(X) of M(X).A space F E M ( X ) belongs to 9(X) when, for any f in the unit sphere of F,any measurable function g : J + X with 11 g 11 l l f l l (pointwise a.e.) is also in the unit sphere of F. The spaces in this class have a richer structure. In particular, a kind of dual of an 9-space (the associate space or Kothe dual) is defined under certain mild restrictions. Section 23 deals with subclasses of 9 ( X ) with still richer structure: they consist of spaces that are, roughly speaking, invariant under translations t ---+ t - T of the domain J , which must in this case be the whole line or a half-line. These spaces include the Orlicz spaces and, more particularly, the Lebesgue spaces Lp for 1 p co. Section 24 deals with similar subclasses of N ( X ) consisting, however, of spaces of continuous functions. In this chapter we are forced to deviate from our purpose of making the main developments reasonably self-contained. In order to keep the chapter within bounds commensurate with its role in the book, and to avoid a lengthy excursion into a field that is perhaps not of principal interest to the reader, we shall refer for many of the proofs to previously published work, especially to SchSer [l]. The fact of a proof’s being given in extenso is no indication of the importance of the result: it may simply mean that it is new or that there is no convenient reference.
<
< <
Further terminology and notation for abstract spaces. The relations “stronger than” and
<
The following items supplement the terminology and notation introduced in Section 10. A convex set K in a vector space is BALANCED if x E K, h E Z(F) implies Ax E K (i.e., K is symmetric or circled, depending on whether F is the real or the complex field). If K is a nonempty balanced convex set, its intersection with any ray {Ax : h 2 0} is either the whole ray, or a half-open segment, or a closed segment. The set obtained from K by including the endpoints of all half-open segments obtained in this way is another balanced convex set, the RADIAL CLOSURE of K,denoted
20. INTRODUCTION
35
by rad K. If rad K = K, K is RADIALLY CLOSED;for any nonempty balanced convex K , rad K is obviously radially closed. I n a topological vector space obviously rad K C cl K; therefore rad K is bounded if K is bounded. If X is any normed space with norm 11 / I x , and a > 0, we define the normed space aX by Z ( a X ) = aZ(X); i.e., X and aX coincide algebraically, and 11 x llJx = a-lll x [ I x for all x E X. Let 2 be a locally convex H a u s d o d topological vector space (normable or not) and let Y be a normed space (with norm I( * / I y ) contained algebraically as a linear manifold in 2; we say that Y is STRONGER THAN 2 if the norm-topology of Y is stronger than the topology induced by 2. This relation can be expressed in any of the following equivalent ways: for every continuous seminorm 7r of 2 there exists a number y 11 for all y E Y; (2) Z( Y) is 2-bounded; (3) the trivial injection Y -+2 is continuous. (1) a,
> 0 such that n ( y ) < a,II
-
If, in particular, 2 is a normed space (with norm 11 I[=), (1) and (2) may be formulated as follows: (1’) there exists a = a( Y , 2) > 0 such that 11 y
y; (2’) Z(Y)C aZ(2) (with the same
llz
< ally 11
for all
Y E
a
as in (1’)).
T h e relation “stronger than” is transitive. If Y is stronger than 2, we also say that 2 is WEAKER THAN Y. T h e relation “stronger than’’ depends on the algebraical and topological structure of Y,not on its particular norm. If Y, 2 are normed spaces, Y stronger than 2 and 2 stronger than Y hold simultaneously if and only if Y and 2 coincide as topological vector spaces, though not necessarily as normed spaces; their norms are equivalent, and we say that Y, 2 are NORM-EQUIVALENT: this relation between normed spaces is of course an equivalence. We require a more precise order relation for normed spaces, one that does take into account the norms themselves. If Y, 2 are normed spaces, we write Y 2 if Y is stronger than 2 and a( Y, 2) 1 in (1’), i.e., if Y is algebraically contained in 2 (hence the notation) and 11 y llz I1 y I1 for all y E Y; or, equivalently, Z( Y) C 42). If Y 2 we also write 2 2 Y. T h e relation is clearly an order relation. T h e statement “ Y is stronger than 2” may now be formulated in still another way:
<
<
<
(2”) Y
<
< a 2 (with the same a as in (1’)).
<
Ch. 2. FUNCTION SPACES
36
Functions and function spaces We consider the real line R, and denote by p Lebesgue measure on R; all measure-theoretic terminology shall refer to p unless otherwise noted, An INTERVAL is any connected set J C R containing more than one point; the usual parentheses-and-square-bracketsnotation is used for intervals with given endpoints, finite or infinite. Special instances in constant use are R- = (- 03,0], R, = [0, +a). Our function spaces will consist of strongly measurable functions on a given interval J with values in a given Banach space X; or rather, of equivalence classes of such functions modulo null sets in J ; we shall, however, usually stick to the less accurate terminology and identify such equivalent functions. The individual functions will usually be denoted by letters of the same type as those used for the elements of their range-space, e.g., f,g for a general Banach space X, but 'p, $ for the scalar field F. When, as is generally the case, there is no danger of confusion, a function assuming a constant value on J will be denoted by the same symbol as its value. If q,f,g,f *, U,V are strongly measurable functions on J with values in F,X, X, X*, [ X ; yl, [Y; 4, respectively, the functions Re 'p, q, I 'p I, Ilfll, sgnf, f g, 'pf, (f,f*>,Uf,?ware defined by the corresponding pointwise operations on their values; they are strongly measurable, and their equivalence classes indeed depend unambiguously on the equivalence classes of the component functions. If the latter are continuous, so are the composite functions, except sgnf; and this function is continuous too if, in addition,f(t) # 0, t E J. Obviously, 'p = I 'p I sgn 'p, f = l l f l l sgnf. The (measurable) characteristic function of any measurable set E C J is denoted by x,. The set of (equivalence classes of) real-valued measurable functions on J is a (conditionally o-complete) lattice under the order relation where 'pl 'p2 means ' p l ( t ) tp2(t)a.e. (almost everywhere-i.e., except on a null set) on J. 'p is POSITIVE if 'p 2 0; the sequence (pn)is INCREASING if 'pn+l 2 'pn , n = 1, 2, ...; expressions like "strictly positive" or "strictly increasing" are ambiguous and will not be used for such functions. The lattice operations are denoted by sup and inf, If 'pl < 'pa, we define the INTERVAL ['pl, 'pa] = {$ : $ is measurable, 'pl $ 'pa}. If E C J is measurable, p ( E ) > 0, and 'p is a real-valued measurable function on J, the ESSENTIAL SUPREMUM of 'p on E is ess supE'p = inf(A E R : AxE >, x e } ; and similarly for ess inf, 'p. If to is a point or an endpoint of J,we have ess lim s u p f + ,'p(t) =lirnd+, ess supJn(f,e.fo+a, 'p, and similarly for ess lim inff+lo'p(t). If both coincide, their common
+
<
<,
<
< <
20. INTRODUCTION
37
value is ess limt,io p(t).. T h e obvious modifications are made in defining co. the essential limits for t -+ - co or t -+ Function spaces will be denoted by bold-faced capitals such as F, G , and classes of function spaces by script capitals, such as N, 9; either standing alone or with subscripts, etc. When the same letter of either type is required for different range-spaces, the range-space may be indicated as an argument, e.g., L(X), N ( X ) .I n the case of L(X) (defined in the preceding subsection, and more precisely in the next subsection but one), as well as in the cases discussed in Sections 22, 23, and 24, this notational convention reflects a deeper underlying structure: as the precise definitions will show, in these cases the symbol without the argument, say F, indicates a space of real-valued functions, and F(X) is a related space of functions with values in X . Examples of this situation are the spaces Lp(X), 1 < p 00 (see next subsection). I n the few instances when the domain of the functions has to be mentioned, it is indicated as a subscript on the left, as, e.g., in ,L(X); a different convention is used in the special case in which spaces of functions defined on R, R,, and R- must be used in the same context; it is introduced in Section 22 (p. 54), where it is first needed. Thick bars are used to denote the norm in our function spaces, as, e.g., I in order to distinguish it from the norm in the range-space, which is denoted, as usual, by 11 11 or, in case it happens to be the scalar field, by I * I.
+
<
- IF,
The Lebesgue spaces Lp(X) Particularly important instances of the function spaces we are to consider are the “Lebesgue spaces” defined as follows. Let J C R be a given interval and let X be a Banach space. If 1 < p < co, ,Lp(X), or Lp(X) for short, is the space of all strongly measurable functions f : J - X with. $,Ilf(t)llP dt < co, provided with the norm 1, = ( J , ~ ~ f ( t )dt)’/p; ~ ~ ~ and ,L”(X), or L“(X) for short, is the space of all strongly measurable functionsf: J-+X with ess sup,l(f((<00 (“essentially bounded” functions), provided with the norm I = ess sup, 11 f 11. T h e notation for the norms illustrates the convention that a subscript Lp(X) is invariably replaced by p , 1 < p co; and that in the special case of the norm, I is further abbreviated to I * I; this convention is in force throughout the book. We also adhere to the notational convention announced in the preceding subsection, and set LP(R) = LP,1 p 00 (the classical real Lebesgue spaces); this will be further justified in Section 22 (p. 55).
If
If
-
<
< <
Ch. 2. FUNCTION SPACES
38
Holder's Inequality may be written
p-1
+ q-1
= 1,
where co-l = 0, as always in this context. We quote the following classical results (cf. Taylor [l], pp. 376-386).
< <
00, are complete. If q-l = 1 - p-l, 20.A. The spaces LP(F), 1 p the mapping I/ I/* : L*(F)-+(LP(F))*dejned by ( y , #*) = J,y(t)I/(t)dt is an isometrical monomorphism, and indeed a congruence ;f p < co; in particular, LP(F) is rejexive if 1 < p < co; more in particular, L2(F) is a Hilbert space with the inner product ( y , I/) = J J y ( t )$(t) d t . ---f
The space L(X) Let J C R be a given interval and X a Banach space. T h e set of all (equivalence classes of) strongly measurable functions f : J --t X that are (Bochner) integrable on every compact subinterval J' C J becomes, with the topology of convergence in the mean on every such J' - i.e., locally convex topoconvergence in pL1(X)of the restrictions to J'-a logical vector space, which we denote by ,L(X) or, more usually, L(X). We write in particular L(R)= L. A set of seminorms defining the topology of L(X) is given by {J,, IIf(t)ll d t } , where J' runs over all compact subintervals of J , or at least over a class whose union is J. Since J is the union of a finite or countable set {Ju} of abutting compact subintervals, L(X) may be represented as the direct product X, ,"V(A'); the projection nuon . the 14h direct factor is simply the restriction of the functions to Since ,uL1(X)is a Banach space, L(X) is thus a Frechet space, i.e., complete and metrizable (and, in particular, a Hausdorf€ space). L is obviously a lattice under the ordering and it is "orderconvex"; i.e., if y1 , y z E L, cpl F~ , then [yl , y 2 ] C L. Jy
<
20.B. For any rpl , rp2 E L, y1 sequentially compact in L.
< cpz, the
<,
interval [yl, y2] is weakly
Proof. See Schaffer [l], Lemma 3.2. T h e required fact that intervals are weakly sequentially compact in each ,uL1(X) was proved by Kakutani [l]. &
20. INTRODUCTION
J
=R
and J
39
= R+ . Translation operators
We collect in this subsection various facts that specially concern functions defined on R and R, . 20.C. Assume that J = R or J = R, . Consider a function p : J --t R,. If there exist positive numbers H , T , 7, with q > 1 , such that p(to + r ) 2 to) and ~ ( t2) H ~ ( t 0for ) all t o E J ) to < t < to 7 , then p ( t ) 2 H 9 -1 e p ( l - l o ) p(to)for all to E J , t 2 t o , where p = 7-1 log?,
+
Proof. For any such t o , t set n = [(t - t , ) ~ - ~(greatest ] integer). so thatp(t) 3 Hp(to nr) 2 Hqnp(tO)> Then?”+’ = elr(n+1)72 e/d(f-fo), H ?-le/d 1- l o ) ?(to)* A A function p E L (on R or R,) is said to be STIFF if p 0 and there exists a number A > 0 such that
+
(20.2)
tsJ t infJ’ttdv(u) du
> 0.
Obviously, stiffness depends on the behavior of p for large I t I only.
20.D. Assume that J = R or J = R, . A function p E L,p 2 0, is stiff i f and only if there exist positive numbers p, M such that for all to E J , t 3 to , (20.3)
Proof. Assume that p is stiff, and let K > 0 be the value of the infimum in (20.2) for an appropriate A > 0. T h e positive-valued, given by + ( t ) = exp (f, p(u) du) satisfies nondecreasing function p(t, A ) 3 eKp(to),to E J. Then 20.C implies (20.3) with M = e-“, p = K A - ~ .Conversely, if p satisfies (20.3), it satisfies (20.2) with any A > -p-Ilog M > 0. &
+
+
We also introduce the following useful notation. For any f E R+L(X) (where X is any Banach space), f # 0, we define the “numbers” so(f), s( f),0 so( f ) < s( f) m in such a way that (s0(f), s( f))is the smallest open interval in R, such that f vanishes a.e. outside it; that is, = sup{t : = 0}, s( f ) = inf{i : x [ ~ ,f~ = ) 0). If F is any set or space consisting of such functions (plus, perhaps, 0), we set s0(F) = inf{s0(a : f E F \ (0)) and s( F) = sup{s( f) : f E F \ (0)).
<
<
s0(a
Ch. 2. FUNCTION SPACES
40
xc-m,rlf= 0},
s+Jf) = inf{t E R, : ~ ~ ~ = , + 0); ~and , f if F is a set or space consisting of such functions, sJF) = inf{s-(f) :f E F} and s+(F) = sup{s+(f) :JE F}. All these “numbers” are ordinarily used only if they are finite. T h e additive group or semigroup properties of R and R, , respectively, are reflected in the definitions of certain TRANSLATION OPERATORS applied to functions on that domain. We begin with functions on R, with values in any Banach space X . If f : R -+ X and T E R, we define the “translate” TTf : R --t X by
T,f(t) = f ( t - T ) ,
t
E
R.
If f is strongly measurable, so is T,f, and TT preserves equivalence modulo null sets. Obviously each T , is a continuous linear bijective mapping of L ( X ) onto L ( X ) for every T . We note the following trivial relations among these operators:
(20.4) (20.5)
To = identity; I1 TTfll =
T,l l f l l
TTItT2 = T,,T,, for all
T~
,
T~
E
H;
for all T E R and all strongly measurablef: R -+ X .
For the domain R , the definitions are slightly more complicated. --t X and T E R, , we define T:f, T;f as
I f f : R,
It is often convenient to introduce T, , defined for all T E R as T, = T,f if T 2 0 and T, = TI, if T 0 (no confusion will arise with the T, defined above for functions on R). We also define a TRUNCATION OPERATOR 0,for each T E R+ as 0, = T , + C , so that 0 , f = xLs,.,,f. Again iff is strongly measurable, so are all TTfand 0,f, and these operators preserve equivalence modulo null sets and are continuous linear mappings of L(X) into L ( X ) . We have
<
(20.6)
(20.7)
T;T,f = T+ = T0
= identity
Z ‘ ~ + r 2 = T:lT:2, TGfrz= Tl,TT,
(20.8) II T,’fII =
q llflll II T;fII for all
T
E
=
; y ll.fll9
for all
T E
T,
R,;
, T~ E R,;
II qfll = or llfll < llfll
R, and all strongly measurable f : R ,
Hence for every strongly measurable f : R , (20%
for all
II TT,T,fll = TTlTT211flld T,,,,, l l f l l
+X,
for all
,
T~ T~ E
R.
-+
X.
21. M-SPACES
41
Finally, we state the principal relations between the translation operators defined for functions on R and those defined for functions on R, . Iff : R -+ X is given and f + is its restriction to R , , (20.10)
7’.TjL == O , ( T f ) +
T;f+
=
(T-f),
for all
T E
R,
,
21. .”-spaces
The lattice . 1 “(X) Throughout this section we let J be a fixed interval, which will be the domain of all functions in our function spaces unless otherwise noted. We consider a Banach space X , the corresponding space L(X), and the following partially ordered classes: full, J’.(X)-of all normed spaces that (a) the class ,t’(X)-in are stronger than L(X), with the order relation defined in the Introduction (p. 35); (b) the class r ( X ) of all nonempty, radially closed, L(X)-bounded, balanced convex sets in L(X), with the order induced by set-theoretical inclusion. Following our usual notational convention, we write N ( R ) = JV. An order-preserving bijective correspondence may be established between . A ’ ( X )and r ( X ) by associating with each space F E N ( X ) its unit sphere Z(F). Indeed, Z(F) is obviously convex, balanced, and radially closed; it is L(X)-bounded since F is stronger than L(X). Conversely, if K E r ( X ) , let F be the linear manifold spanned by K in L(X), and define a norm I IF in F by I f IF = inf{X : X > 0, k ’ f K}-the ~ “Minkowski functional” of K . This definition is meaningful, since K “absorbs” every element of F; it defines a seminorm because K is convex and balanced, and this seminorm is a norm since K , being bounded in the Hausdorff space L(X), does not contain any complete ray; and obviously Z(F) = rad K = K. Since distinct normed spaces have distinct unit spheres, the mapping F -+ Z(F) : .A ‘ ( X )-+ T ( X ) is indeed bijective; it is order-preserving by the definition of the order relation Now T ( X ) is a lattice, with K , A K , = K , n K , and K , v K , defined as the radial closure of the convex hull of K , u K,; the crucial fact is that, L(X) being locally convex, the convex hull of an L(X)bounded set is L(X)-bounded. T h e lattice r ( X )is actually conditionally complete: if S is a nonempty index set and K , E F ( X ) , 5 E S, we have A K , = n K E , and if K , C KO for some fixed KOE r ( X ) and all
<
<.
42
Ch. 2. FUNCTIONSPACES
5 E 8 then VK, is the radial closure of the convex hull of UK,. If in particular 9 is a totally ordered set and K, C K,. whenever 4’ follows 4, we simply have VK, = rad UK,. T h e correspondence established above transfers this conditionallycomplete-lattice structure from r ( X ) to N ( X ) ,and it remains to interpret the lattice operations in terms of N ( X ) . A little reflection shows that this interpretation is as follows: if F , , Fz E N ( X ) , F, A F, is, algebraically, the intersection F, r\ F , , and is provided with the norm 1 f IF,,jF, = max{lf , I f Fl v F, is, algebraically, F, , and is normed with the norm I f IF, V F, = the sum F, inf{lfl IF, IF, : f = f i + f i ; f i E Fi , i = 12}. If F, E N ( X ) for all 5 € 9 ,then AF, is that (possibly proper) submanifold of nF, on which sup{l f IF[ : 4 E S} < co, with this supremum as norm; and if in addition F, F, for some fixed F, E N(X) and all 4 E 9, then VF, is the sum F, (set of finite sums f,,f t E F,) with the norm I f I”Ft = inf{C I f , IP, : f = E f t , f t E F, 9 f , = 0 except for finitely many 4 E 9).If in particular E is a totally ordered set and F, F,. whenever 4’ follows 4, then VF, is UF, with the norm I f lVF, = inf{l f IF[ : 5‘ E 8 such that f E FE). I t is important to note that norm-equivalence between spaces is preserved under the taking of finite meets and joins in JV(X): if, e.g., F, and F, are norm-equivalent and G, and G, are also normequivalent, then so are F, A G, and F, A G, , F, v G,and F, v G , . This is no longer necessarily true for infinite meets. Formulation (1) of the relation “stronger than” in the Introduction (p. 35) may be specialized, in the case of “&”-spaces” to the following necessary and sufficient condition for a normed space F that is algebraically a linear manifold in L(X) to be an element of Jtr(X):
+ If,
IF.$
IF1
+
< 1
1
<
,
(N) for each compact subinterval J’ C J there exists a number >, 0 such that SJ, 11 f(t)ll dt < a,, I f IF for all f E F .
a,,
We denote by a ( F ; J’) the least value of a,, that satisfies (N); clearly, a(F; J’) = sup{J,, IIf(t)ll dt : f E Z(F)).
<
<
21.A. If F, G E &“(X), F G, then a ( F ; J’) a(G; J’) f o r every compact subinterval J’ C J. For a given F E . N ( X ) , a(F; J’) is a nondecreasing subadditive function of J’.
Proof. Trivial.
9,
<
Consider the space L*(X), 1 p \< co. By taking cp = I(f 1) E Lp, x,. E Lq in Holder’s inequality (20.1) we find that Lp(X) satisfies (N) with aJ, = I x,. 1, , so that L*(X) E . N ( X ) ; by taking f to be a
t,h
=
21. N-SPACES constant multiple of xP it follows at once that, indeed, a(LP(X);J’)
I X J , I,
43 =
= (tL(J’>)’-p-’.
Banach spaces in . N ( X )
We consider the class of complete spaces, i.e., Banach spaces, in
. N ( X ) , and denote it by b.N(X). This class obviously contains the spaces Lp(X), I < p < 00. and only if the L(X)-limit of every 21.B. F E .A ‘ ( X )is complete F-Cauchy sequence in Z(F) belongs to Z(F).
Proof. T h e condition is meaningful, since every F-Cauchy sequence is an L(X)-Cauchy sequence and therefore has an L(X)-limit (which coincides with the F-limit if the latter exists). T h e “only if” part is trivial. T o prove the “if” part, let (fn) be any F-Cauchy sequence in F; since this sequence is F-bounded, the assumption implies that its L(X)-limit, say g, is in F . For given E > 0 there exists m, such that - f,, IF E for all m , n 2 m, . For any m 3 m, , (fn - f n J is then an F-Cauchy sequence in EZI(F), hence its L(X)-limit g - fm is in Z(F); in other words, lg - f,,L IF E for all m 2 m, . Hence (f,) F-converges to g. &
<
If,
<
An important consequence of 21.B is: 21.C. b.K(X) is a sublattice of . N ( X ) and contains the lattice meet of any subclass.
Proof. See Schaffer [I], Theorem 2.1. $, Among spaces in bM(X), the relation “stronger than” has a particularly simple characterization:
21.D. If F , G E b./V’(X), F is stronger than G i f and only if F is contained as a linear manifold in G. I n particular, all spaces in b N ( X ) that are supported by one and the same linear manifold in L(X) are normequivalent. Proof. A corollary of 21.C via the Open-Mapping Theorem; see Schaffer [ I], Corollary 2.1. & Local closure
K
E
r ( X ) obviously implies clLo,K E r ( X ) . For every F E N ( X ) the relation ZI(1cF) = C ~ ~ ( ~ , Zdefines ( F ) a space
it follows that
44
Ch. 2. FUNCTION SPACES
IcF E .,t’’(X),which we call the LOCAL CLOSURE of F. T h e operator Ic is obviously a closure operator (i.e., increasing, order-preserving, idempotent) on the lattice . N ( X ) . If F = IcF, i.e., if 2 ( F ) is L(X)-closed, we say that F is LOCALLY CLOSED. Since local closedness is not necessarily invariant under norm-equivalence, we say that F E M ( X ) is QUASI LOCALLY CLOSED if it is norm-equivalent to a locally closed space; equivalently, if F and 1cF are norm-equivalent. T h e class r ( X ) of convex sets, having done its duty, now disappears from our considerations.
21 .E. Assume that F E .N(X). A function f E L ( X ) belongs t o IcF i f and only if there exists a sequence ( f n ) in F with limL(x)fm= f and lim infl f, IF < 00, and then I f llcF \< lim infl f m I F . For every compact subinterval J’ C J , we have a(F; J ’ ) = a(Ic F ; 1’). Remark. T h e “only if” part may be refined thus: i f f E lcF, there exists a sequence (f,) in F such that lim,,,, f, = f, liml f, IF = If l l c F . Proof. All except the equality a(F; J’) = or(1cF; J’) follows from the definition and the metrizability of L(X); the equality, from 21.A and from Fatou’s Lemma applied to the sequence ( f n ) in the Remark. T h e meet of any class of locally closed spaces in J r ( X ) is obviously locally closed; since norm-equivalence is preserved under finite lattice operations, the meet of a $finite class of quasi locally closed spaces in A r ( X ) is quasi locally closed. It is not true in general that the join of even two locally closed spaces in .A”(X) is locally closed; see however 22.F.
21 .F. A quasi locally closed space F E .A”(X) is complete. Conversely, if F E b./Y‘(X) and i f F and 1cF consist of the same elements, then F is quasi locally closed. Proof. An immediate consequence of 21.B and 21.D; see Schaffer [l], Theorem 2.2. &
It is easy t o show that a reflexive space in b.N’(X) must be locally closed (see Schiiffer [I], Theorem 2.3). I t will be shown later (although a direct proof here would not be difficult) that the spaces Lp(X), 1 p GO, are locally closed (see 22.D, 22.S).
< <
Completion For any given F E N ( X ) the class {G : G E b N ( X ) , G >, F} is not empty, since it contains 1cF by 21.F; its infimum b F E b.h“(X)
21. M-SPACES
45
(by 21.C) is therefore well defined. b is a closure operator on the lattice bF lcF, whence, by 21.A and 21.E,
N ( X ) ,and F (21.1)
<
<
a(bF; J ' )
= a(F;J ' )
for every compact interval J' C J.
I t is important to describe a construction of b F in terms of F-Cauchy sequences: 21.G. If F E N ( X ) , b F consists of all L(X)-limits of F-Cauchy sequences, with the norm (21.2)
If
(bF
inf{lim Ifn IF : (fn)
an F-Cauchy sequence, limL(x,f, = f } .
<
Since F b F and b F is complete, every F-Cauchy sequence is a bF-Cauchy sequence and has a bF-limit, which is also the L(X)limit. Let F, be the linear manifold of all such limits, with the seminorm given by the right-hand side of (21.2) (all the properties of a norm except definiteness are trivial). Iff E F, and (f,) is any F-Cauchy JbF = limJf, JbF sequence with bF-limit (or L(X)-limit) f, then limlfn taking the infimum over such sequences, we conclude that IbF I f IF1, which proves that F, is normed and that F F, bF. We claim that F, is complete; since then F, bF, equality holds, as required by the conclusion. I t remains to establish our claim. Let (fn) be an F,-Cauchy sequence; we choose a subsequence, if necessary-and relabel it (f,)-in such a way that IftIr+,- f i n IF1 2-", m = 1, 2, ...; it is sufficient to show that this subsequence is F,-convergent. Since fnL+, - f, E F, , there exists for each m an F-Cauchy sequence (g,,) with Ignr,,+, gmn IF 2-"-". I gtn, IF ,:::2limn-tmL(x)gmn =fm+l -fm , m, n = 1, 2, ... . We set h, = gi,7L-iE F. NOW 1 h,+, - h, IF Proof.
If
< <
If
IF; <
<
<
<
<
C::: I gi.n+i-i - gi.n-i
+
xi=,
<
+
<
<
IF I g,, I F (n - 1)2-" 2-,+1 2-n+ln. Therefore (h,) is an F-Cauchy sequence; let h be its L(X)-limit; then h E F, . For any fixed m we consider the F-Cauchy sequence (h,+, m m . gi,m+n-i), whose L(X)-limit is h llmn+mL(x,gin=
xi=l h I hm+n Ci=,gi.m+n-i x:+, 2P+'-11). 2 Now P + l ; therefore I h + f,
xi=,
m
- (fm
-
IF
=I
m+n-1
gi.m+n-i
IF, < 2-m+l.
IF
G
Since this holds for all m, the F,-Cauchy sequence (fi,,) F,-converges to h fi, and F, is complete. 9, =
Remark.
- fnb
+
b F is a kind of completion of F, but the trivial injection
F + b F need not be isometrical.
Ch. 2. FUNCTION SPACES
46
The relation
*
Up to now we have considered N ( X ) for a fixed X . Let now X , Y , 2 be Banach spaces over the same scalar field, and consider three spaces, F E N ( [ Y ; Z ] ) ,G E N ( [ X ;yl), H E M ( [ X ;Z ] ) . We write F . G s H if U E F, V E G imply U V E H, I U V IH < I U IF1 V IG.
-
21.H. If F - G H, F, < F, G, < G, H, >, H, then F1 G, * H, . If F * G H, the mapping ( U , V ) UP': F x G + H is contin--f
uous.
Proof. Trivial.
&
Remark. Applications of the final statement of 21.H are obtained if X is a Banach space over the scalar field F, and we consider functions v, f, f*, U , V with values in F, X , X * , respectively; if these functibns are in the appropriate function spaces, the mappings (v, f) -+ vf, (f*,f ) - < f , f*>,(fJ f*)-<*,f*>fh(u,f)-uj;( U , UVare continuous. An example of the relation * is the classical result that LP( [ Y ; 21) Lg([Xi Y ] ) L'([X; 4) if 1 + rll = p-' + q-'. A similar result for Orlicz spaces is given by O'Neil [l]. More general instances of this relation will be pointed out later (22.V, 24. J, IoO.C, Section 112, p. 357).
x,x,
v-+
-
22. 9-spaces
The lattices 9 and b9. Local closure We again let J be a fixed interval throughout this section (further specialization will be made in the next-to-last subsection). For the moment we consider spaces of real-valued functions. T h e class 9, or, in full, ,9consists , of all spaces F E N that satisfy the condition
(F) if that I $
I
v E F and
< 1 v 1,
+ is a real-valued measurable function on J such
then
4 F and I$ IF < Iv IF
*
Thus a normed space F of (equivalence classes of) real-valued measurable functions belongs to F if and only if it satisfies conditions ( N ) and (F). 22.A. 9 is a sublattice of N ,and indeed contains the meet and join of any nonempty subclass of 9. (provided the latter exists in .N)
22. 9-SPACES Proof. See Schaffer [l], Theorem 3.2.
47
&
We denote by b 9 the class of all complete spaces i n s , i.e., b 9 = F n b N . We obviously have L P E b 9 , 1 p 00.
< <
22.B. b 9 is a sublattice of 9 (hence of b M ) and contains the meet of a n arbitrary nonempty subclass. If F E 9, then b F E b F . Proof. The first part is obvious from 21.C and 22.A. For the last, see Schaffer [l], Theorem 3.4, with the proof simplified on account of 21.G. 9,
We turn to the question of the local closure of “9-spaces”. Here the first part of 2i.E may be refined as follows: 22.C. Assume that F c 9. If tp E lcF, there exists an F-bounded increasing sequence ( y n ) in F , 0 vn I tp I, such that I tp 1 = lim, y n and I tp llcF = lim( tpa IF . Conversely, zf (tp,) is an F-bounded increasing sequence in F , q,L>, 0, then tp = lim,cp,& exists, and ‘p E lcF, I tp J l c F liml q n IF Proof. See Schaffer [l], Theorem 3.5. 9,
<
5
<
As an almost immediate corollary, we have, in view of 21.F: 22.D. If F
E
9, then 1cF E b F .
Proof. See Schaffer [I], Corollary 3.2.
9,
22.E. A space F E .F is quasi locally closed if and only if, f o r every F-bounded increasing sequence (tp,) in F , tpn >, 0, its L-limit tp belongs to F ; it is locally closed if and only if, in addition, I y IF = limJ vn IF , or, equivalently, I tp IF ,< lim) pn IF. A space F E F is quasi locally closed if and only if F and 1cF consist of the same elements. Proof. See Schaffer [I], Theorem 3.6 and Corollaries 3.3, 3.4. &
22.F. The class of quasi locally closed spaces i n 3 is a sublattice of b 9 ; the subclass of locally closed spaces is a sublattice of the former, and contains the meet of any subclass. Proof. See Schaffer [I], Corollary 3.6. T h e essential step is the proof of the fact that if F , G E are~locally closed, then F v G is locally closed. This result (Schaffer [I], Theorem 3.7) lies deeper than most of the others, and depends essentially on 20.B. &
Ch. 2. FUNCTION SPACES
48
The operators k,, k, f. Lean and full spaces If F E 9 we define k,F as the linear manifold of all 97 E F with compact (essential) support, i.e., vanishing a.e. outside a compact subinterval of J , and k F as the F-closure of koF; both k,F and k F are provided with the norm of F. Obviously, k F consists of exactly those functions q~ E F for which liml xJ\f m ~ = 0 for any nondecreasing sequence (I,) of compact intervals with UJ, = J.
IF
22. G. k, , k are dual closure operators (decreasing, order-preserving, idempotent) on the lattice 9, and k is a dual closure operator on b 9 . For any F E F,k,F kF F, lck,F = lckF = 1cF; and a(koF; 1’)= a(kF; J’) = a(F; J’) for all compact subintervals 1’ C J . For any
<
F E b 9 , bk,F
=
<
kF.
Proof. Trivial, using 22.C for the properties concerning local closure.
&
k b F is complete, Remark. Since bF is complete for any F E 9, whence b k F bkbF = kbF. Actually, equality holds, but we shall not require this fact.
<
A space F E 9 such that k F = F is termed a LEAN space; leanness is invariant under norm-equivalence. Instances of lean spaces are the Lp, 1 p < 00; on the other hand, L“ is not lean if J is not compact; we shall always denote the important lean space kL” by Lg (this space belongs to b 9 ) : it consists of all measurable essentially bounded functions with essential limit 0 at those endpoints of J that do not belong to 1. E L such that x f y E F If F E 9 we define f F as the set of all for all compact subintervals J’ C J, and supJ.I x f , y IF < co,provided with this supremum as norm. It is immediate that f F E S,and indeed fF = V{G E 9: koG = koF}. A space F E S such that fF = F is termed a FULL space; if F is norm-equivalent to a full space or, equivalently, is norm-equivalent to fF, then it is termed QUASI FULL. T h e spaces Lp, 1 p a,are full; and fL: = L“.
<
< <
22.H. f is a closure operator on 9 and on b 9 . For any F E F , < fF < lcF, k,fF = koF, kfF = kF, fk,F = f k F = f F ; and a(fF; J’) = a(F; J’) for all compact subintervals J’ C J. Every [quasz) locally closed space in 9is [quasa) full. The meet of any class of full spaces is full.
F
22. 5-SPACES Proof. Trivial, except for the fact that F this follows from 21.B.
E
49
b 9 implies fF E b 9 ;
The class YK T h e class 9, consists of all those spaces F E 9that contain all essentially bounded functions with compact support, i.e., that contain k,Lm algebraically. It is obvious that F E 9,if and only if F E 9 and xJ. E F for all compact intervals J' C J. We denote by b S K the subclass of all complete spaces in F,, i.e., b 9 , = b . F n 9 , . T h e spaces Lp, 1 p 00, belong to b 9 , . We state a trivial lemma which is very frequently used in the book.
< <
22.1. If F E 9,and g,, IJJ are measurable real-valued functions on J that coincide outside a compact interval, and are essentially bounded (in particular, continuous) on it, then g, E F if and only if IJJ E F.
22. J. If F E F and there exists a continuous function g, E F such that g,(t) # 0, t E J , then F E 9,. If F E b 9 , this suficient condition is also necessary. Proof. T o prove the sufficiency, let J' C J be a compact subinterval, and CT = min,,,, 1 g,(t) I > 0. Then xJ' u-l I g, I, whence x,. E F. T o prove the necessity for F E b 9 , , we observe that there exists a nondecreasing sequence (J,) of compact intervals such that U J , = J , and a sequence (g,,) of continuous functions such that xJ, g,, xJ,+, , n = 1, 2, ... . Therefore (g,J is an increasing sequence in F. Thus 2-i I pi lilyi) is an F-Cauchy sequence, which converges to some g, E F. But 0 I g,. l;tqn I v1 so that the sequence converges uniformly to its L-limit, which is g,. Therefore 9) is continuous, and g, 3 2-"1 g,, Ii1xJ, for each n. Since U J , = J , g, vanishes nowhere. &
<
<
<
(xr
<
<
9, is obviously a sublattice of 9, and b9,asublatticeof 9 , ( a n d of b 9 ) . Neither need contain the meet of an infinite subclass; but if F E 9,and G E 9, G 2 F, then obviously G E 9,; therefore SK contains the join of any subclass, provided it exists in 9, i.e., in A'-. If F E 9,, then k,F, kF, fF E 9,, and b F , 1cF E b T K ;if F E b F K , also kF, fF E b 9 , . T h e quasi locally closed spaces and the locally closed spaces in 9,constitute sublattices of b s , , on account of 22.F.
50
Ch. 2. FUNCTION SPACES
Associate spaces
Let F €9be given. We consider the set F' of all measurable realvalued functions $ on J such that
where k depends on $ alone. Since q E Z(F) implies, by condition (F), I v 1 sgn $ E Z(F), (22.1) implies, and hence is equivalent to, the apparently stronger condition (22.2)
I,I p(t)+(t)I dt < A
for all
v E z(F).
For complete spaces, these conditions are equivalent to an apparently weaker one: 22.K. If F E b F and $ is a measurable real-valued function such that p,h E L1for all v E F, then $ E F .
Proof. If (22.2) failed to hold, there would exist a sequence (9")in F, vn 2 0, such that I vn IF d 2-", JJv,&(t) I $(t) I dt 2 1. Then (C:vi) is an increasing F-Cauchy sequence; if v is its F-limit, hence its L-limit, we have cp 2 v c ,n = 1, 2, ..., whence the contradiction < Z: J,vn(t) I $(t) I dt < J J p ) ( t ) I $(t) I dt < & F' is obviously a linear manifold and may be provided with the seminorm
x:
(22.3)
I $ IF'
= = SUP
11s dt)+(t)I v 1s
22.L. If F E then ($F)' (as seminormed spaces).
di :
J
I
z(F)
I d t W ( t ) I dt : T€Z(F)I < m.
=
(kF)' = (fF)'
=
(bF)' = (1cF)' = F'
Proof. Since k,F, kF, fF, bF, 1cF €9 and lck,F = lckF = lcfF = lcbF = lcF, it is sufficient to prove (1cF)' = F'. Assume that 4 E F, and consider any q~ E Z(1cF). By 22.C there exists an increasing sequence (rpn) in Z(F), q n 2 0, with lim, vn = I rp 1. By (22.3) and B. Levi's Theorem, J, I p)(t)$(t) I dt = lim J,vn(t)I $(t) I dt < I $ Ip,. Therefore $ E (IcF)', whence F' C (IcF)', and 1 $ I(loF). < I $ IF, . The reverse inclusion and inequality follow at once from F < 1cF. ,&
22.
51
%-SPACES
If we merely assume F E F or even F E b 9 , it may well happen that I * IF, fails to be a norm (if all functions in F vanish a.e. on some fixed nonnull measurable set E C J ) or that, even if it is one, $ E F ’ does not imply $ E L. We have the following theorem, however:
, is a normed space with the norm given by (22.3), 22.M. If F E S KF’ and is a locally closed space in b F K ; I xJ. IF, = a(F; 1‘) for every compact subinterval J‘ C J . The mapping $ --+ $*: F’ -+ F* defined by ( q ~ ,+*) = JJqJ(t)$(t)dt is an isometrical monomorphism, and the following “Holder’s Inequality” holds: (22.4) I I
< 1J I d t ) # ( t )I dt < I?’’ IF
I # IF’
for all
P ‘ E F,
/J
E F‘.
<
If F,G E FK, F G, then F’ 3 G’. For any F E F Kwe have F” = k F ; the mapping F -+ F’ is an involutory lattice antiautomorphism of the lattice of locally closed spaces in F K . Proof. Let $ E F’ be given; for every compact subinterval J‘ C J , we have x,, E F and therefore JJ, 1 #(t) I dt = J,x,,(t) I $ ( t ) I dt I x,. IF I $ IF,; it follows that I $ IF, = 0 if and only if = 0. T h u s I * IF, is indeed a norm; also F ‘ satisfies (N) and ol(F’; J’) I x,, I F ; F’ obviously satisfies (F) also. T h e isometrical monomorphism of F’ into F*, as well as formula (22.4), follow at once from the definitions. For every compact subinterval J’ C J we further have:
+
<
<
Therefore xJ. E F’, whence F‘ E F K ,and I x,. IF, = a ( F ; 1’). If (I&) is an F’-bounded increasing sequence of positive elements in F’ and if $ = lim, $ n , we have, by (22.4), JJ I v ( t ) I &(t) dt I $, IF, for all q~ E Z(F), n = 1,2, ... . By B. Levi’s Theorem, JJ I p)(t) I $ ( t ) dt liml IF, for all q~ E C(F). Therefore $ E F’, I $ IF, liml $, IF,; by 22.E, F’ is locally closed. T h e fact that F , G E FK,F G implies F’ G’ is obvious. I t (F’)’= F” for any F E S K . is equally obvious from (22.4) that F Since F” is locally closed, we obtain IcF F ” ; we must prove that F” 1cF in order that equality hold; since F” = (IcF)” by 22.L, this is equivalent to proving that F” F when F is locally closed. For this purpose we use essentially the method of Luxemburg and Zaanen [l], Theorem 1: We assume, then, that F is locally closed. We choose a nondecreasing
<
<
<
<
<
<
<
<
52
Ch. 2. FUNCTION SPACES
(In)
sequence of compact intervals such that U Jn = J. For each n the set Zn= x,~,Z(F) is convex, and nonempty and L-closed (since F contains I x,, JilxJ, and is locally closed). Let g, E F" \ (0) and p > 1 be given, and set g,, = min{I cp 1, nx,,} E F. If we identify functions vanishing outside J , with their restrictions to Jn , the set Znis convex and closed in JnL1and does not contain p( g,n Iilg,n , In view of the canonical congruence between (,nL1)* and .L" (see 20.A), the Hahn-Banach Theorem yields the existence of a function w, E L", vanishing a.e. outside J,, , such that .f,fwn(t)g)n(t) dt > I q n IF and I JJwn(t)#(t) dt I 1 for # Zn 9 hence for all # E Z(F). This last fact implies wn E ,Z(F'), and therefore pi g, IF" 2 p I v n IF! > I vn IF for all p > 1 and all n. Since (vn) is an increasing sequence of positive elements in F, with lim,g,, = I q~ I, and the sequence is now seen to be F-bounded, 22.E implies g, E F, Ig, IF I g, IF., so that indeed F" F. On the lattice of locally closed spaces in SK, F -+F' is order-reversing and involutory (since F" = 1cF = F), hence bijective: it must therefore be a lattice antiautomorphism. &
<
<
<
Remark. On account of 22.L, the assumption be weakened to F E 9, 1cF E S K .
F E S Kin 22.M may
Under either assumption in the preceding Remark, F' is called the ASSOCIATE SPACE OF
F.
+
< <
22.N. (LP)' = L*,p-' q-' = 1, 1 p 00, and (L:)' =L '. Therefore LP is locally closed, 1 p CO. If F ES,, then F' is weaker than L" [than L'] if and only if F is stronger than L' [than L"]. If F E .FK is quasi locally closed, then F' is stronger than La [than L'] if and oniy if F is weaker than L' [than L"].
< <
Proof. From 20.A and 22.M it follows that (LP)' = Lq, p-'+ 4-1 = 1, 1 < p < 00, and that L' is a subspace of (L")' = (L;)'; that this subspace is not proper follows from the fact that 1 E Z(Lm). T h e remainder of the statement follows from 22.M. 9,
Thin spaces If B, D E 9, we say that D is THIN WITH RESPECT TO B if for every continuous function q ED, g,(t) # 0, t E J, there exists # E B such that #g,-' 4 L'. (The definition does not involve the algebraic or the topological structure of B, D, and would be meaningful for any two sets of measurable functions, but such generality does not interest us; see, however, Section 24, p. 78.) Obviously, if B, B1,
53
22. 9-SPACES
E F, B, is weaker than B, D, is stronger than D, and D is thin with respect to B , then D, is thin with respect t o B,. From 22. J, D is trivially thin with respect to any B if D 4 S K ; on the other hand, if J is compact and D E T Kthen , 1 E D and D is trivially not thin with respect to any B. This motivates some restrictions in the following lemma; we are also not interested in the complications that arise when B is not complete.
D, D,
The 22.0. Assume that J is not compact and that B E b e D E FK. following statements are equivalent: (a) D is thin with respect t o B ; (b) i f g, E D is continuous and g,(t) # 0, t E J , then v-l 4 B’; ( c ) i f g, E D , $ E B’ are continuous, lim infl V(t)$(t)l = 0 as t tends to the set of endpoints of J that are not in J ; (d) D is thin with respect t o 1cB. If B E b F K (or at least 1cB E bTK), the preceding conditions are implied by, and ifD is also quasi locally closed, are equivalent to: (e) B’ is thin with respect t o D’. Proof. T h e implication (a) 4 (b) is trivial by the definition of B’; (b) 4(a) follows from 22.K. (c) + (b) is trivial (take $ = v-l); (b) 4 (c) is clear for g,, $ that vanish nowhere, and is reduced to this case by the fact that every continuous function with compact support belongs to D and to B’ (the latter by direct verification). T h e equivalence of (a) and (d) follows via (b) and the fact that (1cB)‘ = B’ (by 22.L). Assume 1cB E b S K . In (c), D and B’ = (1cB)’ enter symmetrically; therefore (a)-(d) imply that B’ is thin with respect to D’ provided D = D”, i.e., by 22.M, when D is locally closed (and D‘ E b e which always holds); hence also when D is quasi locally closed. Conversely, since B’ is locally closed, (e) implies that D” = lcD, a fortiori D, is thin with respect to B” = lcB, so that (d) holds. 9,
Remark. T h e restriction of local closedness for the implication (a) + (e) in 2 2 . 0 is not superfluous: LT is thin with respect to L’, but (Ll)’= L” is not thin with respect to (L:)’ = L’. 22.P. If p(J) = co and B E b F K , D then D is thin with respect to B.
E
sK, D stronger
than B,
Proof. If a continuous QI E D , q ( t ) # 0, t E J satisfies y5q1-l for all $ E B, it does so in particular for $ = q ~ ,and hence 1 which is absurd. &
E E
L1 L1,
Ch. 2. FUNCTION SPACES
54
.
The domains R, R - , R+ Cutting and splicing at 0 This subsection, which will not be required until Chapter 8, is devoted to the study of the relations between %spaces of functions defined on R on the one hand and on R- and R+ on the other. There is no difficulty in extending all considerations to the case of any interval J that is cut into two at an interior point; but we prefer to consider only the particular case. We use the following conventions in this connection: the range of t is understood to be R, unless the contrary is explicitly stated or is implied by this agreement: a subscript - or attached to a function defined on R denotes its restriction to R- or R + , respectively; and attached to other symbols, refers to R- or R, as domain, in a manner to be explained from case to case. A superscript - or is merely a convenient index: in particular, we write x- = xR , x+ = xR+ (defined on all R). Concerning the use of the signs f, in either position, we agree that all upper signs (i.e., those in both subscripts and superscripts) are to be taken together throughout the formula or sentence, and likewise for the lower signs; and the formula applies to both cases or to either, as the context may require. Thus, while L = RL,we set L, = RkL.T h e mapping v --t y - 0v+ is obviously a FrCchet-space isomorphism of L onto the outer direct and likesum L- @ L, . Again, while 9=RRwe set 9?*= Rt9, wise for b z PK, etc. we denote by F* the normed space that is algebraically If F E { y :~y E~ F) and has the norm Iyt IF+ = Ix*y I F . W e observe that the norm is well defined and that x*v-+v, is a congruence between the subspace xfF = (9,E F : xTy = 0} of F (which is indeed a subspace by condition (N)) and F,; all this makes sense on account of condition (F). I n the sequel we shall write I q* instead of I cp+ IF*. T h e proof of the following results is straightforward and is left to the reader (cf. Schaffer [6], Lemmas 3.1, 3.2, 3.3, and 3.4).
+
+
z
IF
22.Q. (1) Assume that F E 9.Then F+ E 95,and cp E F if and only if vk E F*; more precisely, the mapping y -+y - 0v+ is an isoI v IF I v- IF -t morphism of F onto F- 0 F+ and max-,+l y + IF
<
i v + IF. ( 2 ) The mappings F
<
F* : 9-+9*are surjective complete lattice homomorphisms, and commute with the operators lc, b, k,, k, f, and (for F E gKor at least 1cF E FK) with the operation of taking the associate space. If F is locally closed, or full, F* are both locally closed, or full, respectively. ---f
22. 9-SPACES
55
( 3 ) The mapping F --f (F- , F,) : 9+F-x Ftis surjective; if F, G E F, = G * ,then F and G are norm-equtvalent. F E 9 is complete, oris lean, or belongs to .FK, i f and only i f both F* are complete, or are lean, or belong to SKk, respectively. Remark 1. On account of 22.Q,(2), the notations IcFh, bF*, etc., are unambiguous, and parentheses may be omitted.
< <
Remark 2. (Lp)k = R,L’, 1 p co, and (LT)* = R*LF. T h e notations L$ , LF’ are therefore unambiguous.
The class 9 ( X ) We again consider any interval J as the range of t. I n this section we have been dealing with real-valued functions only; we now return to the consideration of spaces with values in a given Banach space X . For every space F E .Fwe denote by F(X) the normed space of the (equivalence classes of) strongly measurable functions f : J + X such in the subthat 11 f 11 E‘F,provided with the norm f IF(X)= 111 f 11 script of the norm we drop the argument X whenever practicable. We see that F(R) = F, which agrees with our notational convention; the p co, introduced earlier, is consistent with this notation Lp(X), I new definition. T h e class {F(X) : F E F}for one and the same X is denoted by F ( X ) . This notation is given fuller meaning by the following results, which show that the properties of the spaces F(X) faithfully reflect the properties of the corresponding spaces F.
IF;
< <
22.R. If G is a normed space of strongly measurable functions from J into X , then G E F ( X ) ij and only if G satisfies ( N ) (i.e., G E N ( X ) ) and (Fx): If f E G and g : J - X is strongly measurable and satisfies Ilgll
&
22.S. . F ( X ) is a sublattice of N ( X ) . The mapping F
+ F(X) :
S+ F ( X ) is a complete lattice isomorphism that commutes with the
operators lc and b. F(X) is complete, or locally closed, or quasi locally closed, if and only if F is complete, or locally closed, or quasi locally closed, respectively. For every F E 9, a(F(X); 1’)= a(F; 1’)for all compact subintervals 1‘ C J .
Ch. 2. FUNCTION SPACES
56
Proof. See Schaffer [l], Theorems 3.2, 3.3, 3.4, 3.8, and Corollaries 3.1, 3.6; the last part follows at once from the proof of 22.R. 91. On account of 22.S, such notations as VF,(X), lcF(X), and bF(X) are unambiguous and do not require additional parentheses. If we define k,(F(X)), k(F(X)), and f(F(X)) in the obvious ways, it is a quite easy consequence of 22.R and 22.S that they are respectively equal to (k,F)(X), (kF)(X), and (fF)(X), and we may omit parentheses. With the obvious definitions for spaces in F ( X ) , F(X) is LEAN or [QUASI] FULL if and only if F is lean or [quasi] full, respectively. In the same way, if J = R, the notation F*(X), etc., is unambiguous in the language of the preceding subsection, and requires no additional parentheses. We may similarly write b S ( X ) , S+(X), etc. 22.T.
If F E 9, F(X) is reflexive i f and only i f F and X
are reflexive.
Proof. See Schaffer [l], Theorem 3.9. This result is essentially due to Halperin [l]. 9, For coupled spaces X, X’ and a space F E SK(or a slightly more general case) we examine the relations between F(X) and F’(X’), where F’ is the associate space of F.
22.U. Assume that X , X’ are Banach spaces coupled by ( - , * ), and that F E 9with 1cF E gi (in particular, F E FK). I f f’ E L(X’), the following statements are equivalent: (a) f’ E F’(X’); IF f o r all (b) there exists k 2 0 such that I JJ(.f(t),f ’ ( t ) ) dt I ,< k f E F(X); (c) there exists k, 2 0 such that I J,(f(t), f ’ ( t ) ) dt 1 k, I f IF f o r all f E k,F(X);
If
<
and then If’ IF, is the common infimum of the values of k and k, satisfying (b) and (c), respectively. Proof. (a) implies (b). I f f E F(X), f ’ E F’(X’),the strong measurability of f , f ’ and the continuity of ( , * ) imply that ( f , f ’ ) is measurable. Since 1) f 1) E F , 11 f ’ 1) E F’, 22.M with the Remark yields, by formula (22.4), e
1s
J
(f(t),f’(t)>
dt
I < J Ilf(t) I/ IIfVJ II J
<
so that (b) holds with inf k 1.f’ IF‘. (b) implies (c). Trivial, with inf k,
dt
< If‘lF I f ’
< inf k.
IF, *
57
23. F-SPACES
(c) implies (a). Let h > 1 be given and let IT be the corresponding function defined by 12.E. For any y E k,F we set
Since the set { t : f ' ( t ) f 0} is measurable, and since IT is continuous, f is strongly measurable. By 12.E, 11 f I( G XI y I, hence , f k,F(X), ~ Again by and, using (c), 12*E, < f , f'> = I v I [If' 11 I f IF < hl q I F . I y ( t ) I Ilf'(t) II dt hk, I v I F . Since y E k,F was arbitrary, we have ilf' 11 E (k,F)' = F' (by 22.L), i.e., f ' E F'(X'),with If' IF. ,< hk, . Since h > I was arbitrary, 1.f' IF. < inf k, . &
sJ
<
T h e F-spaces provide important instances of the relation duced at the end of Section 21 :
3
intro-
22.V. Assume that X , Y, Z are Banach spaces over the same scalar field; and that F E .F and G E . N ( [ X ;Y]). I f G L"([X; yl), then F([Y; Z]) . G 3 F([X; Z]) and G * F([Z; XI) 3 F([Z; Yl); if G L:([X; Y)], then F([Y; Z]) G * kF([X; 21) and G .F ( [ Z ; q )
<
<
=>
kF([Z; Y]). Proof. Trivial, using 21.H.
9,
A further example, for any F E 9 with 1cF E FKand any Banach space X , is F'(X*) * F(X) 3 L'; this follows from 22.U.
23. .T-spaces
The classes .F,.T+ Throughout this section, J is either R or R , , unless otherwise noted, and the translation and truncation operators defined in the Introduction are extensively used. We again deal for the moment with spaces of real-valued functions. T h e class .T consists of all spaces F E ,Fthat satisfy the following additional conditions: ( 2 ) F f YO}; (T) if q E F,
7E
R, then
TT9) E
F, I T+
IF < I y IF
*
If J = R, it follows from (20.4) that the inequality in condition ( T ) may be replaced by equality. If J = R , , condition (T) implies, via (20.6),
(T +)
if y
E
F,
7
E
R, , then T,+vE F,
I T : y IF = I v
IF;
Ch. 2. FUNCTION SPACES
58
and indeed ( T ) is equivalent to (T-) together with “if q E F, T E R, , then T,p, E F”, on account of condition (F): for if (T+)and the additional condition hold, (20.6), (20.8), and (F) yield I TFp, IF = I TfTYv IF =
I@ r q
IF
< Iv
IF
*
For ] = R, we define the class F-’as the class of all spaces F that satisfy (2) and (T-); it thus contains .Fas a subclass.
E
9
General Remark 1. In this section, all results concerning 9 are proved for ] = R, , but are valid also for J = R (with the exceptions specially noted), the proofs being either the same except for insignificant alterations, or even simpler; save for an occasional indication, they are left to the reader. An exception is of course the subsection on cutting and splicing at 0 (p. 74), where both ranges of t occur together. We do not use the notation fl,Rt.Fto distinguish the two classes F, because they are not strictly analogous. General Remark 2. In using the numerous references to Schaffer [ 11, it must be kept in mind that our present classes .7,F+ for J = R , are there denoted by F#, F,respectively. For spaces F E F+the number s,(F) defined in the Introduction (p. 39) exists on account of (2) and plays an important role. It is obvious that s0(F) = 0 for F E Y (on J = R+). For any F E Y - ’ and T E R , we define the spaces T f F , O,F as the manifold { T:v : v E F} and the subspace ( 0 ,: v~E F} of F, respectively, both provided with the norm of F.
<
<
T,fF O,F F 23.A. If 5; E F-,T E R, , then TfF, O,F E F+, (with equality at the second sign i f 7 s,(F)), s,(T:F) = 7 s,(F), s,(O,F) = max(7, s,(F)}.
<
Proof. Trivial, using O,Tf
=
T,f (from (20.6)).
+
9,
23.B. (1) 9 is a sublattice of S,and contains the meet of any infinite subcluss unless this meet is {0},and the join of any infinite subclass, provided it exists in 9, i.e., in M . (2) The meet of a n y j n i t e subclass of ,F+belongs to F+;so does the meet of any inJinite subcluss, unless it is (0); and so(AFt)3 sup so(F,), with equality i f the subclass is Jinite. (3) If the index set E is totally ordered, and i f Fc E F+, 5‘ E E and F,, whenever $. precedes t’, then VF, E .F+,provided the join exists in 9, i.e., in A’.
F,
<
23. T-SPACES
59
Proof. See Schaffer [ l ] , Theorem 4.2 and Corollary 4.1. We point out that the fact that F A G # (0) for F, G E F or F+ is not at all trivial. $,, ‘ is not a sublattice of 9, it is a conditionally Remark. Although 5 complete lattice under the order relation 6 :the meet operation is the same as i n 9 (i.e., in N); if 8 is a nonempty index set, F, E .FA, and F, E.P, F, < F, for all 6 E 5, the join V’F, E .FAcoincides algebraically with the join V F, in 9, and if 9 E V F, then Iy Iv+ F E . = inf,,,+ I T,fv there are other equivalent ways of describing this join. We omit the proof, but point out that the required fact that for any F, , F, E Y+there exists F, E Y-’, F, , F, F, , will follow from 23.G.
<
The operators b, lc, k, , k, f map 7 into F and 7- into
23.C.
Y-’.
Proof. For b, lc, see Schaffer [l], Theorems 4.3, 4.4. For k,, k, f the result is trivial. & We use the notations b F = b2F n F, b y + = b 9 n F+, YK= FKn F, b F K = bRKn 7 .
23.D. b y ,
r Kb ,- 7 are ~ sublattices of 7 ;the first
contains the meet
of any subclass, provided this meet is # (0).
b S K are sublattices Proof. 22.B, 23.B,(1), and the fact that SK, O f S .
,$,
T h e translation invariance expressed by condition (T) or (T+)implies a certain “homogeneity” of the T-spaces with ‘respect to the domain J. A typical and important illustration is the following property of the function a ( F ; 1’). 23.E. (1) If F E Y ,and 1’ C J is a compact interval with p(J‘) = I, then a ( F ; 1‘)= a(F; [0, 4). (2) I f F E F+,and J’ C R, is a compact interval with p( J’) = I , then a(F; J’) < a ( F ; [s,(F),s,(F) + I ] ) , with equality if J’ C [s,(F), a).
J
Proof. For the proof of (2), see Schaffer [I], Corollary 4.3. For R , , (1) follows from (2). For J = R, (1) is trivial. &
=
As a. consequence of 23.E, we define the function a ( F ; I ) for 1 > 0 as equal to a(F; [s,(F),s,(F) 1]) if F E F+,and equal to a ( F ; [0,1]) if F E .Y. This function will be further studied in a later subsection ( P a 63)-
+
60
Ch. 2. FUNCTION SPACES
< <
Obviously, Lp E b y K , 1 p co, with a(Lp, I ) = 1 l - P - l (p. 43). We introduce two further important spaces in b y K . T h e space M consists of all p) E L such that supIcJJ:+I I p)(u) I du < co, with this supremum as norm. We set M, = kM,and easily find that M, is characterized as the subspace of M consisting of those p) E M for which 1+1 lim,t,+wJl I p)(u) I du = 0. We remark that the choice of any length of the interval of integration other than 1 would yield spaces that are norm-equivalent to M,M, .
23.F. M, M,
byK;
E
M is locally closed; M, contains no s t i f
functions. Proof.
Trivial, except for the local closedness of &
M,for which use
22.E and B. Levi’s Theorem.
T h e extreme importance of M follows from the following immediate consequence of 23.E (and the fact that k is order-preserving):
23.G. M is the weakest space, and M, the weakest lean space, in ,T and in Y+:more precisely, if F E F or F E F+,then F a(F; 1)M; if F is also lean, then F a ( F ; 1)M, , and F contains no stifJ functions.
<
<
The Operator TIn this subsection we examine relations between the class ,T+ and the subclass Y, so that we assume J = R , . Our purpose is to extend each F E Y+ as narrowly as possible to a space in ,T: in this way we hope to be able to reduce questions concerning spaces in 9to the corresponding questions about the more amenable spaces in Y. T o this end, we define T-F as the linear manifold {T;? : p) E F; 7 E R+}, provided with the norm I T;p) = I0,p) IF; this definition is meaningful on account of (20.8) and condition (F), and unambiguous, for if Tcp), = T;q2 with, say, T~ 3 T ~ then , (20.7) yields OT2p)z= TTfgT;v2 = T,+z-,lT:T;yl = TTfg--71@71p)1, and hence, by (T-), I O,lp)l IF = I it also clearly defines a norm. This definition is adequate for our purpose:
IT-F
IF;
23.H. If F E Y+, then T-F E Y, T-F = A{G : G E Y, G 3 F}; a(T-F; I) = a(F; 1) for all 1 > 0, and the trivial injection F + T-F (in fact, F -+ ay0T-F, where so = s,(F)) is isometrical. If F E b y + , then the trivial injection T-F -+ bT-F is isometrical, hence so is the trivial injection F -+ bT-F. T- is increasing, order-preserving, and idem-
23. F-SPACES
61
potent on F+, and its restriction to 9-is the identity. T- commutes with the operators k,, k, and f. Proof. See Schaffer [l], p.237. T h e last statement is trivial.
&
I n general, the completeness of F does not imply the completeness of T-F. For quasi locally closed F, however, much more is true-such a space is indeed exactly a truncated T-F: 23.1. If F E F+,then T-lcF = 1cT-F. If F is [quasi] locally closed, T-F is also [quasi] locally closed and F = t3,"T-F = T,+!T-F, T-F = {T;y : y E F}, where so = s,(F).
Proof. See Schaffer [l], Theorem 4.7.
&
The class YK We consider in some detail the class FKfor J = R and J = R, . We already know that if F E F then F E YKif and only if the characteristic function of every compact subinterval of J belongs to F ; it follows at once from ( T ) and (F), however, that it is necessary and sufficient to this end that F contain the characteristic function of a single arbitrary compact subinterval of J , e.g. xro,ll. It will be useful to introduce, for F E F', the notation P(F; 1) = I IF = I x J . IF, where J' is any compact subinterval of J with p( J ' ) = . I . Obviously, P(F; 1) is a nondecreasing and subadditive function of 1 for a given F, and satisfies, for all 1 > 0, (23.1) (23.2)
4 7 1)/3(F;1) 3 I /3(F;1) 3 /3(G;I ) /3(koF;1) = P(kF; I)
if F
= /3(fF; 1) = P(F; 1 ) .
T h u s B(F; 1) plays a kind of counterpart t o a(F; I ) , as will be seen more clearly in a later subsection (p. 63). , its translates, Any F E F K must contain, together with x [ , , ~ ] all all finite linear combinations thereof, and all functions bounded in absolute value by such linear combinations. These remarks motivate the introduction of two special spaces, of which the first is by far the more important. T h e space T consists of all y E L such that (23.3)
Ch. 2. FUNCTION SPACES
62
xy
for some sequence (T,) in R and some sequence (a,) in R, with a, < co, provided with inf E r a , as norm, where the infimum is taken over all pairs of sequences satisfying (23.3). T h e space S is similarly defined, except that the sequences are assumed to be finite.
23.J. T E b y K , and T is locally closed and lean; k,T = S and b S = T. T is the strongest space in b y K ; more precisely: if F E b Y , then F E b y K if and only if F is weaker than T, and then F >, P-'(F; l)T.The same conclusion holds with b y , b Y K replaced by Y, T Kand , T replaced by S = k,T. I f y E L satisfies j ~ ( t I ) pe-"ltt for all t, where p, u > 0, then E T.
<
Proof. See Schaffer [l], Section 4.4 (pp. 238-247). S = k,T follows from the fact that the trivial injection S -+ T is isometrical, and the leanness from this and from bS = T. T h e last statement is obvious from the definition of T. 9, A very important result concerns the locally closed spaces in F.
23.K. If F E Y is locally closed, then F 21 for all 1 > 0. Proof. See Schaffer [2], Theorem 2.
E
FKand a ( F ; l)P(F;1)
<
&
23.L. The class of quasi locally closed spaces in F is a sublattice of b y Kand the class of locally closed spaces in 9-i s a sublattice of the former and has M as weakest and T as strongest element. Proof. 22.F, 23.K, and 23.F, 23.G) 23.5.
9,
From 23.K it follows that if F E F is locally closed, and if y
E
J' is a compact subinterval of J, and 1 >, p(J'), then JJ. I v(t)I dt
<
F,
<
a ( F ; 1) I v IF 2@-l(F; I ) I y I F . T h e inequality between the first and last members subsists even when F is not locally closed (but is in YK), provided cp is continuous on J':
23.M. Assume that F E FK. If v E F is continuous on the compact interval J' C J , and 1 >, p ( J ' ) , then P ( F ; l)J,. I v(t)I dt < 21 I v I F . Proof. See Schaffer [2], Theorem 1; the adjustment of the proof > p ( J ' ) is obvious. 9,
to include the case I
If F E b Y K we are often primarily interested in the continuous functions contained in F ; it turns out that among all the spaces in b y K that have the same set of continuous functions with the same norms
63
23. F-SPACES
as F the “smallest” has some very convenient properties, as the following lemma shows. In all questions concerning the continuous functions only there is of course no loss in replacing F by this “smallest” space. A similar construction is valid for noncomplete F E F K, but we do not consider it. 23.N. If F E b y K , set F, = A{G E b y K: ‘p E F continuous implies y E G, I ‘p IG = I q Then F, E b y K , F, F, and:
<
IF}.
if
(I)
I v IF,
y
E
L is continuous, y
E
F ;f only and ;f
I F IF; if I’ > 1 > 0, B(F; I) < /3(Fc; 1) if 1 > 0, g(lcF,; 1) 3 $(F; 1).
‘p
E
F,, and then
=
(2) (3)
< /3(F; 1’);
Proof. See Schaffer [4]. 9, Associate spaces in F
T h e most important facts about associate spaces of F-spaces are summarized in the following lemma. 23.0. If F E 7 , F‘ is a normed space with the norm given by (22.3), and is a locally closed space in b y K ; /3(F‘; I ) = m(F; 1) for all 1 > 0. If F , G E F , F G , then F’ >, G’. For any F E F , F” = 1cF; the F‘ is an involutory antiautomorphism of the lattice of mapping F locally closed spaces in y.
<
--j
Proof. By 23.C and 23.K, 1cF ~ 7 this ~ makes ; the Remark to 22.M applicable, and the conclusion follows, once the obvious fact that F‘ satisfies condition (T) is noted. 9,
We thus see that, for 7-spaces, no further assumption need be made in order that F’ be a normed space and F’ E M . I t follows from 23.0 and 23.1, that M’ must be norm-equivalent to T, and T’ to M; actually a simpler and stronger result holds: 23.P. Proof.
M‘
=
T and T’= M.
See Schaffer [l], Lemma 4.10.
9,
The functions a(F; I ) , P(F; I ) ; the spaces L’, L“,
L :
It will appear in the body of the book that one of the things that mainly matter about a 7 - s p a c e F is the behavior of the functions
Ch. 2. FUNCTION SPACES
64
a ( F ; I>, B(F; I) (the latter if F E Y K ) for small and large 1. We show in this subsection that this behavior is intimately connected with the relationship of F with certain “pivotal” spaces, namely L’, L“, L:. Although some conclusions can be drawn if F is not complete, our main results will be stated only for F E b y . To begin with, we recall certain properties of a ( F ; I) (defined for F E Y and F E Y+) and P(F; 1) (defined for F E F K ) :both are positive-valued nondecreasing subadditive functions of I > 0. We may therefore extend their definition to the extended half-line by setting a ( F ; 0) = liml+,, a ( F ; l), a(F; 00) = liml+m a ( F ; I), and similarly for B(F; 0), B(F; 0 0 ) ; the monotonicity and subadditivity of both functions is preserved. We remark that if a ( F ; 0) = 0 then a ( F ; 1) is continuous, and similarly for B(F; I ) . For the sake of reference, we collect here several properties of these functions that are either trivial or have been proved before, save the obvious extensions to 1 = 0, 1 = co (21.A, 21.E, (21.1), 22.G, 22.H, 22.M, 23.E, 23.H, (23.l), (23.2), 23.K, 23.N); they are valid for
O
+F; 1) = Xa(F; I ) , a(F; 1) < a(G;1)
for h
> 0,
F,GEForF-’, F G G ; (23.5)
(23.6) (23.7)
a($F; 1) = a(kF; I)
a(T-F; I )
= a(fF; 1) = a(bF; 1) = a(lcF; 1)
= a(F; 1)
for F E Y or F-;
= a(F; I )
for F E F+;
B(hF; 1) = h-’B(F; I ) , B(F; I ) 3 B(G;1)
for h
> 0,
F, G E Y K , F < G ; (23.8)
(23.9)
B(lcF; 1) < p(bF; 1) < B(k,,F; 1) = &kF; 1) = B(fF; 1) = B(F; I ) for F E Y K ; a(F; @(F; 1) 2 1
for F E .TK;
a(F; I)B(F; 1) < 21
for locally closed F E F;
(23.10)
a(F; 1) = P ( F ; 1)
(23.11)
$B(F; 1) < B(lcF,; I )
for F E F;
< B(F,; 0 < B(F; U
FEbFK, We come to our main theorems.
I t > 1.
for
23. T-SPACES Assume that F
23.Q.
E
65
b y , . The following, conditions are equivalent:
(a) 18(F;O) > 0; (b) every continuous function in F is bounded; (c) the set of continuous functions in F is F-closed; (d) F, is stronger than L". If F is quasi locally closed, the preceding conditions are equivalent to: (e) F is stronger than L". Proof. We prove the implications (a) + (d) + (c) -+ (b) -+ (a). F ; the implication T h e implication (e) -+ (d) is trivial, since F, (a) .+ (e) when F is locally closed is proved in Schaffer [l], Theorem 4.20.
<
(a) implies (d). Set P(F; 0) = 18, > 0. Let y E F be continuous. For any t E J , p > 1 there exists a compact interval J' C J such that P I 'p I b I 'p(t)l x J , ; therefore P I P I F b I d t ) I B(F; PcL(J')) 2 18, I d t ) I. Since this holds for all t and p > I , we conclude that 'p is bounded, I v I ',81 I 'p I F . Therefore I 'p IF = max{I 'p J F , I 'p I}, S O that F A &'L" is one of the spaces G in 23.N whose infimum is F, . Therefore F, F A &'L" /3;'L". (d) implies (c). Assume that (d) holds and that (rp,) is an F-convergent sequence of continuous functions in F, with F-limit rp. Then (y,) is an F-Cauchy sequence, hence (isometrically) an F,-Cauchy sequence, hence, by (d), an L"-Cauchy sequence, hence uniformly convergent to a continuous limit; this is also the L-limit, hence coincides with 'p. Thus y is continuous. (c) implies (b). Assume that 'p E F is an unbounded continuous function; there exists a sequence (7,) in J such that I'p(7,) 1 >, 2", n = 1, 2, ... . Consider the sequence ($,) defined by $, = x : 2 - i T - 7 i 1 'p 1; by (T),(F), it is an F-Cauchy sequence; let $ be its F-limit. It is also an increasing sequence of continuous functions, so that $, being its n = 1, 2 , ... . If L-limit, is also its pointwise limit, and $ 3 were continuous, $(o)2 $,(o) = 2-f 1 ~ ( 7 1 ~>,) 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). &
<
<
<
+
1'
<
1;
+
< 1;
<
1'
66
Ch. 2. FUNCTION SPACES
23.R. (1) Assume that F E b y . Then a(F; 0) > 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,
<
< II-' IM j
t+A
91
(4I du,
J,
t
+
contradicting (23.12); therefore 9-l B'; use of 22.0 completes the proof. Similarly, (e) implies that B' is thin with respect to D', and therefore (e) is sufficient, on account of 22.0. 3. If a continuous function tp does not satisfy (23.12)' I 9 I is stiff (take d = I). Now (b) implies that either D or-by 23.T,(l)-B' is not weaker than L:; by 23.S, either D or B' contains no stiff continuous function; hence (b) implies (d) or (e), and is therefore sufficient. By 23.F, 23.G, condition (c) implies that D contains no stiff function, so that (d) holds; thus (c) is also sufficient. 4. If D is thin with respect to B, and if B were stronger than L1 and D weaker than L", we could conclude that L" is thin with respect to L', which is absurd; therefore (f) is necessary. If D is quasi locally closed, (f) implies (b), and is therefore sufficient. $,
Query. Is condition (f) (necessary and) sufficient even if D E b y K is not quasi locally closed?
23. Y-SPACES
69
Thick spaces We introduce several operations on functions in L; we must distinguish the cases J = R, and J = R, and begin by taking up the former, more complicated one. Assume then that J = R,. For every rp E L and any numbers d, u > 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)
<
and
(b) implies (g). In analogy to the proof of (d) + (e) we have, for the d > 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
<
(23.1 7)
Proof of (2). Similar to, but simpler than, the preceding proof; the details are left to the reader. Instead of (23.14)-(23.17) we have
From now on we again consider J = R, and J = R together as best we can. If B E b y or b y - , and D E b y , , we say that D IS THICK WITH RESPECT TO B if A, maps B into D; on account of (23.13), which is also valid for J = R, it is sufficient to assume that cp E B, y 2 0 imply Aly ED. (Although the definition makes sense if B, D are not complete, we are not interested in this case.) Obviously, 23.V implies that A, may be replaced in this definition by any A, , or by any A : , A : , A,Td+ (if J = R,) or A ; , A: (if J = R). Just as obviously, if B, B, E b y or b y - , D, D , E b y K , B1 is stronger than B, D1 is weaker than D, and D is thick with respect to B, then D, is thick with respect to B, . Further properties of thick spaces are given in the following lemma.
23.W. Assume that B E b y or b y + and D E b y K . Then: (1) If D is thick with respect to B, the restrictions to B of A, , A, T i , A : , A: ( i f J = R,) and A,, 11: ( i f J = R) are in [ B ;D ] . ( 2 ) If D is thick with respect to B E b y , then B' is thick with respect to D'; and conversely i f D is quasi locally closed. ( 3 ) D is thick with respect to M [to M,] if and only i f D is weaker than L" [than L:]. (4) T [L:] is thick with respect to B if and only i f B is stronger than L' [than M,]. (5) If D is quasi locally closed and weaker than B , then D is thick with respect to B. ( 6 ) If also B,E b y or b y - , and D, E b y , , and if D, D, are thick with respect to B,B, , respectively; then D AD^ is thick with respect to B A B, , and D vDl is thick with respect to B vB, (provided this space is in 9-+, when B, BI E b y + ) . I n particular, L" A D , L: A D are thick with respect to L1 v B, (L' v B) A M o , respectiwely.
72
Ch. 2. FUNCTION SPACES
Proof. Proof of (1). We denote by A,, etc., the restrictions to B. If A, : B --t D were not bounded, there would exist a sequence (rpn) in B such that Irpn In < 2-" and I AH" ID >, PI. On account of (23.13) we may assume rpn 2 0. Then (C:rpi) is an increasing B-Cauchy sequence; let cp be its B-limit, so that 0 < rpn < rp, n = 1, 2, ... . Since A, is positive, 0 < A,rp, < AArp, whence I A,rp ID 2 n, n = 1, 2, ..., which is absurd; therefore A,, E [B; D]. The proof for the other mappings could be similar, but we may use (23.14)-(23.18), the fact that 11 Td+1; = 1, and the fact that, for fixed A > 0, ead(l - e--O')--l takes the minimum 4, and obtain the following relations (all norms in [B; D1):
Proof of (2). We carry out the proof for J = R,, the other case being simpler. Let rp' ED', cp' 2 0, be given. For any rp E B, 9 2 0, we have A,T:rp E D ; using Fubini's Theorem, we obtain
Irn
p)(t)Alp),'(t) dt
=
0
Srn I 0
[
1+1
p)(t) dt
p)'(u) du
m
=
0
=
t
P'(u)A~T:~(u)du
Srnp)'(u)
du
0
< II
p)(t)dt (U-l)+
II[B; D ~ I
I,I v'
ID,,
whence Alrpf E B',and B' is thick with respect to D'. Conversely, the same argument implies that if B' is thick with respect t o D', then D" = 1cD and, if D is quasi locally closed, the norm-equivalent D itself, is thick with respect to B" = IcB, a fortiori to B. Proof of ( 3 ) . The "if" parts are trivial. If D is thick with respect to M, then All = 1 E D (if J = R + ; if J = R, All = 2 E D ) so that D is weaker than L". If D is thick with respect to M o , then 0 XLO, I1 G 4 X r o . z+11, 50 that B(D; I ) II (11 ll[M,,;D] whence /?@; 00) < 00; by 23S, D is weaker than L,".
<
<
9
Proof of (4). The part not in square brackets follows from (3) and (2) by 22.N and 23.P. The part is square brackets is trivial. cp
Proof of ( 5 ) . We again carry out the proof for J 2 0, be given; hence rp E D.
=
R,
. Let rp E B,
23 F-SPACES
73
I
For any y’ ED’, y’
>, 0, we have, using Fubini’s Theorem,
Smn p ’ ( t )dt 1
t+l
/ m p ’ ( t ) A l p ( t dt )
=
= J dv
whence A,?
=
t
0
E
J
0
1
1 1
m
p(u)dii
~ ’ ( l ) d t T;T(t)du
n
0
q’(t)T;p)(t)dt G I P‘
ID.
I P ID1
D“ = 1cD. Since D is quasi locally closed, A l y
E
D.
Proof of (6). Trivial, since A, is linear. T h e last part follows by (3) and (4): e.g., L” A D, L’vB are norm-equivalent to (LaAD)v T, (M A B) v L’, respectively (by 23.5, 23.G), and of these last spaces the former is thick with respect t o the latter.
9,
Remark. If J = R , and B E b y - , it can be shown, using 23.H and 23.W,(1), that D E b y K is thick with respect to B if and only if it is thick with respect t o bT-B E b y . For any q~ E L , Adv is a kind of smoothed-out version of tp. We now show that if D is thick with respect to B and 9 E B is continuous and satisfies a weak smoothness condition, then not only AApbut p itself is in D. A continuous real-valued function is NONCOLLAPSING TO THE RIGHT [TO THE LEFT] if there exist numbers 6, k > 0 such that t , t’ E 1 ,t’ 2 t[t’ t ] and I t’ -t I S imply I rp(t’)l k I cp(t)l.
<
<
23.X. Assume that B E b.7 or b y + and D E b Fp If D is thick with respect to B and if q~ E B is continuous and noncollapsing to the right w to the left, then q~ E D. Conversely, if B E b y is quasi locally closed, and every continuous P E B that is noncollapsing to the right, or each that is noncollapsing to the left, satisfies p E D , then D is thick with respect to B. Proof. We give the proof for J = R , . Assume that D is thick with respect to B. Let P E B be noncollapsing to the right, with constants 6, It; we may assume P 2 0. Then Sp < kA,P, whence p ED. If p is noncollapsing to the left, STFp kA,p, so that T ; ~ ED, I whence @# = T,fT;v ED; but this differs from P only on [0, 61, and both functions are continuous there; by 22.1, P E D. T o prove the converse, we observe that, for any rp E B,(1; I P I is noncollapsing to the right, and A: I p 1 is noncollapsing to the left, with 6 > 0 arbitrary and k = e-a; both functions are in B,since B E b y K (by 23.K) is quasi locally closed, and 23.W,(5) is applicable with B instead of D. The conclusion is then immediate. &
<
74
Ch. 2. FUNCTION SPACES
Cutting and splicing at 0 We now investigate the matter of cutting and splicing functions and function spaces at 0, introduced in the corresponding subsection of Section 22 (p. 54), as it applies to 7-spaces. (This will be required in Chapter 8.) T h e notational conventions introduced there are in force. F denotes the class F corresponding to ] = R, and Y+the class 9 corresponding to ] = R, . But now 7- denotes the subclass of 9-that is obtained by SYMMETRIZATION (i.e., mirror-reflection -t) from F+. We assume the whole at 0, or change of variables t theory of .T+-spaces, developed in this section, to be applied, under symmetrization, to F-. Obvious changes in notation make T; , T,‘ the translation operators corresponding under symmetrization to T,f, T;, respectively; and so on. T h e details are left to the reader. --f
23.Y. ( I ) The mappings F 4 F-l-: F + are surjective complete lattice homomorphisms, and commute with the operators lc, b, k, , k, f, and with the operation of taking the associate space. (2) If F E b y , then F is weaker than L’, stronger than L’, weaker : , stronger than L“, if and only if both [either one of] F* are [is] than L respectively weaker than L : , stronger than Li , weaker than L:*, stronger than Lz. ( 3 ) If B E b z D E FK,then D is thin with respect to B i f and only ;f either one of D+ is thin with respect t o B* . (4) If B E b< D E b y K , then D is thick with respect t o B i f and only if both D+ are thick with respect to B* . Proof. Proof of (1). This follows at once from 22.Q, (2) and (20.10), except for the surjectivity of the mappings. If Ff E Y+ , we define F as the set of those cp E L for which (T,v)+E F+ for all 7 E R and SUP,^^ I (T,v)+IF+ < co, provided with this supremum as norm. It is easy to see that F E 7 , F+ = F+;and indeed this is the “largest” space with this property. For this last reason, F is complete, full, locally closed, if and only if F+ has the respective property.
Proof of (2). T h e “only if” parts are trivial; it remains to prove the “if” parts in square brackets, which we do for F+ . Since a(F; I> = a(F+;I ) , P(F; I) = /3(F+;I), the “if” parts of the statements concerning L’ and Lr follow from 23.R, 23.S, and Remark 2 to 23.S. Assume finally that F+ is stronger than LT, say F+ < a :. For any q~ E F we have ess supR I v I = supn ess supR+I (T,v)+ I = SUPn I (Tmv)+I d A “Pn I (TfLv)+ IF d A SUPn I Trip IF = A I v IF whence v E L“, I I < A1 cp IF, so that F is stronger than L“. 9
23. .T-SPACES
75
Proof of ( 3 ) . Trivial.
<
Proofof (4). For any F E B, A1y+ (Alv)+ and the latter function differs from Al(y+ T:rp+) only on [0,1], both being continuous there. Therefore, if either of the functions A , F + , (App)+ belongs to D, , so does the other, by 22.1 and 23.V,( 1) (observe the two different uses of A , , one for J = R , , the other for J = R ! ) . T h e corresponding relation holds for D-, and the conclusion follows from 22.Q,(1). &
+
Remark 1. It is clear that M, , M,, , T, are precisely the spaces M, M , , T defined on J = R + , so that there is no ambiguity in the former notations.
Remark 2. I t is almost obvious that, in contrast to 22.Q, (3), the mapping F -+(F- , F,) : F --t 9x F+is not surjective.
The classes .F(X), F-+(X) We finally take u p the theory of spaces of functions with values in a given Banach space X,developed at the end of Section 22 (p. 5 3 , and apply to it the study of F-spaces that was made in this section. We define the classes F ( X ) = {F(X) : F EF}for J = R and J = R , , and F’(X) = {F(X) : F E F’}. These classes are subclasses of F ( X ) ; in analogy to 22.R we have the following direct characterization.
23.2. If G is a normed space of strongly measurable functions from = R or J = R , into X, then G E F ( X ) if and only if G satisfies (N) and (F,) (i.e., G E S ( X ) ) ,and also
J
(2,)
(T,)
G # {0} (unless X = (0)); if f E G , 7 E R , then TTfE G , I T,f IG< I
f IG.
R , , then G E F + ( X )if and only ifG satisfies (N), (Fx),(Z,), and (T;) i f f E G , ~ E R + , t h e nT , f f E G , I T T f f I C = I f l G .
If J
=
Proof. Immediate from 22.R and the definitions of F,.F+.& In addition to the instances of “dropped parentheses” given in Section 22 (p. 56), we mention that s,(F(X)) = s,(F), etc.; and under the obvious definitions for Tf(F(X)),etc., we may write unambiguously TfF(X), O,F(X), T-F(X), bF(X), b.F+(X), F+(X), etc. We also call attention to the fact that 22.U is applicable to any F E Y on account of 23.C and 23.K.
Ch. 2. FUNCTION SPACES
76
24. Spaces of continuous functions
9% spaces ’As was mentioned in the Introduction, we wish to consider, in addition to the spaces of locally integrable functions studied in the preceding sections, spaces with an analogous structure, but consisting entirely of continuous functions. We do not develop the theory with any degree of completeness, but only describe as much of it as is strictly necessary for the sequel. Since we occasionally wish to treat spaces of continuous functions as special cases of spaces of measurable functions, we should identify with each continuous function all functions coinciding with it a.e.; we agree, however, to use the term “continuous function’’ in its usual meaning unless the context requires this identification. We again consider a given interval J , and real-valued functions defined on J . An important role is played by the space C,or, in full, JC,the space of all bounded continuous real-valued functions on J with the supremum norm. C is a subspace of L” under the abovementioned identification, so that C E b N . W e set Co= C A L : , the subspace of C consisting of those v E C for which lim y ( t ) = 0 when t tends t o an endpoint of ] that is not in J . We define F V , or, in full, , F W , as the class of normed spaces F of continuous real-valued functions defined on J that satisfy (N) (so that F EN) and (FC) If cp E F and # is a continuous real-valued function on J such that I # I d I v 1 9 then # and I # IF 19) I F .
<
We denote by b F V the subclass of complete spaces in 5%‘ (this is slightly misleading, since F E 9%need not imply b F E 9%). It should be noted that 9%is not a subclass of 9, so that the theory of Section 22 is not immediately applicable. However, a connection may be established between the two classes by means of the concept of “envelope”. For any F ~9%‘ we consider the set e F of all measurable real-valued functions # defined on J such that I # I I v I for some q~ E F, and we put I # = inf{l v : v E F, 1 (CI 1 1 cp I). We call e F the
IeF
ENVELOPE O f
IF
< <
F.
24.A. Assume that F ~9%‘. Then e F is a normed space with the norm I * IeF; e F €9, and indeed eF = I\{G E fl : G 2 F}. a(eF; J’) = a ( F ; J’) for every compact interval J‘ C J . The trivial injection F + eF is isometrical, and F consists of all continuous functions in eF. If F is complete, e F is complete.
24. SPACES OF
77
CONTINUOUS FUNCTIONS
Proof. See Schaffer [l], Theorem 6.1 and Corollary 6.1. We observe that
C,C,E b 9 % , e C
=
L", eC,
=
Lz
IE,
.
24.B. 9%is a sublattice of N ,and b F % is a sublattice of 9%. The operator e is an injective lattice homomorphism ,from 9%into 9 and from b 9 % into b 9 ,
Proof. T h e fact that 9%is a sublattice of JV follows as in Schaffer [l], Theorem 3.2; the statement about b F V , from 21.C. Let F, G E %% be given. Since F eF, G eG, we have F v G e F v e G E %; by 24.A, e ( F v G ) e F v eG; similarly, e ( F A G ) e F A eG. Cone(F v G). If versely, eF, e G e ( F v G), whence, in %, e F v e G $ E eFAeG, p > 1, there exist v1 E F, vz E G such that I $ 1 I yi I, but then I ICI I d v = i = 1, 2, and 1 ~ I F1 9 I V Z IG P I ICI I e F A e G ; i n f { l ~I)i I q z I}. F AG,IT lF,+,C d PId~ I e F A e G ; therefore $ E e(F A G ) and, since p > 1 was arbitrary, I # le(FAG) d I $J leFAeC, so that eFAeG FAG). Thus e : 9% 9 is a lattice homomorphism; that it is injective and maps b F % into b 9 follows from 24.A. &
<
< <
<
< < <
<
<
-
<
We shall show later (24.1) that 1cF = lc(eF) E b 9 for any F E FW. For any F E F V , we define k,F, kF exactly as we did for 9-spaces in Section 22 (p. 48) and we again call F LEAN if k F = F. (The characterization of kF given there is of course meaningless for spaces of continuous functions.) We have:
and 24.C. k, and k are dual closure operators on the lattice 9%, k is one on the lattice b%%; for all F E b9$?, bk,F = kF. k, commutes with e and ekF keF for all F E F%.
<
Proof. Trivial; the last statement follows from k F by the use of 24.A. &
< keF, keF
E
S,
An important property of complete spaces F E %% that is not shared by the rest is the fact that F-convergence implies uniform convergence on every compact subinterval of J: 24.D. r(F; J')
Assume that F E b F % . For every compact interval J' C J , =sup(l~r~ :9 ~ E1 z(F)} <
Proof. Assume that for some J' the supremum were not finite. There would then exist a sequence (yn) in F, vn >, 0, such that I v n IF 2-", I xPvnI 3 n. T h e increasing sequence is an F-Cauchy sequence; let y be its F-limit, hence its L-limit, so that
<
(c:vi)
Ch. 2. FUNCTION SPACES
78 0 n
< vn < v, n = 1, 2, ...; but y F is < I x , q n I < I x , q I < co for all n, which E
continuous; therefore is absurd. 9,
T h e definition of THINNESS of a space D with respect to a space B, given in Section 22 (p. 52) for F-spaces extends at once to the case D E 9, B E%%, but is immediately reduced to the previous case by the following trivial lemma.
Then D is thin with respect to 24.E. Assume that D E ~B ,E 9%. B if and only if D is thin with respect to eB. We turn to the question of cutting and splicing at 0 for the ranges J = R, J = R+ , required in Chapter 8. I n this paragraph and the following lemma the notational conventions appropriate to this topic (p. 54) are in force. Thus we write S%= R9%,9%* = Ri9%‘. For any F E 9%, we again define F+ as the set {Ti : E F}, but in order = I leF, since x*y is to define the norm we must put I general not continuous (as for9-spaces, we use the subscript F instead of Fi); it is obvious that I ‘p+ = inf{l t,h : t,h E F, t,h+ = ?+}, and similarly for IvWe observe that Ck = RiC, coi= RkCO.
IF
IF
IF.
IF
24.F. (1) Assume that F
E 9%. Then Fk E 9%+, and e(Fi) = is a continuous function on R, then E F if and only if (eF)*. If v* F*; then max-,+l ?* IF I IF I v- IF f I v+ IF * are surjective complete (2) The mappings F -+F* : 9%+ 9%+ lattice homomorphisms, and commute with k, , k. (3) F E 9%is complete or lean if and only if both F i are complete or lean, respectively.
<
<
Proof. T h e first statement in (1) follows at once from the definition of Fi . T h e remainder of (1) and (2) is in part trivial, and in part follows easily from 24.A and from 22.Q applied to the envelopes. T h e statement (3) concerning lean spaces is trivial. T h e “only if” part of the statement (3) concerning complete spaces is almost obvious and is left to the reader. T o prove the “if” part, let Fi be both complete, and let (tpn) be an F-Cauchy sequence. Then (qn*)are Fi-Cauchy sequences, with Fi-limits y*, say. Since yn-(0) = yn+(0)= v,(O), n = I , 2, ..., and Fi-convergence implies pointwise convergence by 24.D, we have q-(O) = cp+(O), and there exists a continuous function rp on R such that v+ = @. By (l), cp E F, and liml 9) - v n IF lim(I I- - yn- IF I @ - v n + IF) = O* 9,
<
+
On account of 24.F,(1), we may write e F + unambiguously.
24. SPACES OF
79
CONTINUOUS FUNCTIONS
The class 9 % ( X ) We assume J to be a given interval and X a given Banach space. For any F E .F%,F(X) denotes the normed space of all continuous = functions f : J 4 X such that 1) f 1) E F, with the norm 111f 11 IF; in the sequel we omit the argument X in subscripts, etc. Thus F = F(R). Since the only space common to F and 9%is {0}, the notation F ( X ) will introduce no confusion with the corresponding one for 9-spaces, and is more convenient to use than the distinctive notation F[X] introduced in Schaffer [I], p. 261. We define the class F%(X) = {F(X) : F E SW}, so that 9%? = FW(R).
If
24.G. (1) If G is a normed space of continuous functions from J into X , then G E 9 U ( X ) if and only if G satisfies condition (N) (i.e., G E J V ( X ) ) and (FC,) I f f E G and g is a continuous function from J into X such that Ilgll , < I l f l l , t h e n g E G a n d l g I c , < I f l c . (2) For any F E P i ? F(X) ? , is complete
if
and only if F is complete.
Proof. T h e proof of (1) is exactly as that of 22.R (see Schaffer [l], Theorem 3.1). T h e “only if” part of (2) is an immediate consequence. T o prove the “if” part, let (f,) be an F(X)-Cauchy sequence in Z(F(X)), and let f be its L(X)-limit. For every compact interval J’ C J, 24.D imp1ies I X J , ( f m - f n ) I = I X J , 11 f m - f n 11 I r ( F ; 1’)I f , - f n IF 9 so that ( f n ) converges uniformly.on every such J’, and f is continuous. On the other hand, ( 1 1 f n 11) is an F-Cauchy sequence, and its F-limit must be its L-limit 11 f (I. Therefore 11 f (1 E F , f E F(X), and I f =’ liml (1 f , 11 I F I , so that f~ Z(F(X)). By 21.B, F(X) is complete. &
<
<
IF
On account of 24.G,(2), we may write bSW(X) unambiguously for {F(X) : F E b 9 % } = F U ( X ) n b M ( X ) . 24.H. 9 U ( X ) is a sublattice of N ( X ) . The mapping F -+ F(X) : 9%49 W ( X ) is a lattice isomorphism, and maps b 9 W onto bFW(X).
Proof. Same as for 24.B (see Schaffer [I], Theorem 3.2); the last statement follows from 24.G,(2). & On account of 24.H, such notations as VF,(X) are unambiguous.
If we define k,(F(X)), k(F(X)) in the obvious way, it follows at once from 24.G, 24.H that they are equal to ($F)(X), (kF)(X), and that parentheses may be omitted. With the obvious meanings, F(X) is then
Ch. 2. FUNCTION SPACES
80
if and only if F is lean. I n the same way, if J = R, the notation is unambiguous, as is also S % ? * ( X )etc. , F*(X), for F E S%?,
LEAN
We turn to the fundamental question of local closure of *%-spaces.
24.1. If F E 9% then lc(F(X)) = Ic(eF)(X), whence lc(F(X))
=
(lcF)(X) E b*(X).
<
<
Proof. Since F e F , we obviously have F(X) (eF)(X) (the latter lc(eF)(X). T o prove space being in F(X)) and hence lc(F(X)) the converse inequality it is sufficient to show that lc(F(X)) E F(X): for by 24.A and 22.S it follows that (eF) (X)= I\(G E S ( X ) : G >, F(X)}, and since lc(F(X)) 2 F(X) is locally closed we obtain lc(F(X)) >, lc(eF) (X), as required. Since specialization to X = R yields 1cF = lc(eF), we indeed also have lc(F(X)) = (1cF) (X). I t remains to prove that lc(F(X)) E S ( X ) , i.e., by 22.R, that this space satisfies (FX). Assume that f E Ic(F(X)), g : J -+ X is strongly I l f l l . There exists a sequence (f,) in F(X), measurable, and 11 g 11 Ifn'lF Ifllc(F(X)) such that liml,(X)fn = f * Now g E L ( X ) , and the continuous functions are dense in L(X) (see Hille and Phillips [l], p. 86); so there exists a sequence (h,) of continuous functions such that limLo,h, = g. Set g, = inf{(If, 11, 11 h, I\} sgn h,; these are contin[If, 11, so that g, E F(X), Ig, IF uous functions, and 11 g, 11 But Ilg - gn II d I1 g - hn II SUP{O, II hn II If, IF If I l c ( F ( X ) ) * 11 f, II} -+ 0 sup{O, (1 g 11 - I l f l l } = 0, so that g,-tgin L(X). Therefore g lc(F(X)), Ig Ilc(F(X)) liml g, IF If l l c ( F ( X ) ) * &
<
<
<
<
9
<
<
+
+
<
On account of 24.1 we may write lcF(X) unambiguously (notice how the use of the same notation for spaces in F ( X ) and S U ( X ) helps). We conclude by picking up a few loose threads. 24.J. Assume that X , Y , 2 are Banach spaces over the same scalar % G E N ( [ X ; Y]). If G d C ( [ X ;yl), field; and that F ~ 9 and then F([Y; 21) G * F([X; 4) and G * F([Z; XI)=> F([Z; Y ] ) ; if G C , ( [ X ;Y ] ) , then F([Y; 4) G kF([X; 21) and G F([Z; XI) * kF([Z; ul).
<
-
-
-
Proof. Trivial, using 21.H and the fact that an (operator) product of continuous functions is continuous. & We again consider &hegiven space X. We denote by C(X)lnVthe set of those U E C ( x )that are invertible-valued, i.e., for which U ( t ) E is invertible for every t E I, and that further satisfy U-l E C ( X ) .
24. SPACES OF
CONTINUOUS FUNCTIONS
81
24.K. C(X)fnvis open in C(x), and the mapping U +. U-' of C(x')lnv onto itself is continuous. Proof. Assume that U, E C(X)lnv. If U E C(X), I U - U, I < we have 11 UU,' - I 11 < 11 U - U,iI 11 U;lII < I U -- U , I I Uil I < I ; therefore UUcl is invertible-valued, hence so is U. Since taking the inverse is continuous in X, U-' is continuous, and 11 u-1 - U,' jl < /I U,' 11 11 U,U-l - Ill < I u,1 I I( UU,' - 111" I U - U , 1% I U;' In+' < 00. Thus U-' - Utl E C(x), whence U-' E C(X), U E C(X)jnv, SO that C(X)jn, is open; also, I u-1 - U,' I < I U,'IZ I u - u, I(1 - I U,' I I u - u, p,so that U + U-' is continuous in C(X)jnv. &
I U;l I-',
x;
x:
FV+-spaces on J
=R,
and FV-spaces on J
<
=R
I n order to deal with translation-invariant spaces of continuous functions, we must consider the cases J = R, , J = R separately. the subclass of 9% Assume that J = R,. We denote by 9%+ consisting of all spaces F E 9%that satisfy (Z) and (T-). I t is obvious that the last-named condition requires that y(0) = 0 for all y E F. This shows, incidentally, why there is no class of spaces of continuous functions analogous to F on R, . Just as F+with respect to FV+ is not a sublattice of 9 V ,but contains all finite meets. Also, if F E F%+, then k,F, k F E T%+. Assume on the other hand that ] = R. We denote by F% the subclass of 9%consisting of all spaces F E 9%that satisfy(Z) and (T). This class is easily seen to be a sublattice of 9%. If F EF V , then
k,F, kF E F$?. 24.L. (1) If J = R, and F E F%*, then e F E F+. If J = R and F E F % then , eF E F ~ . (2) Proof. Proof of (1).
See Schaffer [I], Theorem 6.2.
Since F satisfies ( T ) and (Z), e F Proof of (2). By 24.A, e F E 9. satisfies ( T ) and contains ,yJ. for some compact interval J' C R. Hence the conclusion. tp, We denote the subclasses of complete spaces in F V + , F% by b y % + , by%?, respectively. For spaces in these classes we have a considerable improvement on 24.D. T o formulate it for J = R,, we require the space Co E b F V + , the one-codimensional subspace of C consisting of those y E C that have ~ ( 0 = ) 0.
82
Gh. 2. FUNCTION SPACES
24.M. (1) I f J = R, , Cois the weakest space in b y % * . (2) I f J = R, C is the weakest space in b y % . Proof. We carry out the proof of (1); that of (2) is even simpler. Let F E b y % - + be given; there exists v0 E F, vo 0, and .toE R, such that cpo(to)= 1. Set y = y(F; [0, to])< co, as in 24.D. We claim that F < yC. Let q~ E F be given. For 0 < T < to we indeed have I Y ( T ) I < y I v IF, by definition of y. For r >, t o , we define # = inf{l ?(T) I y o , T;+ I y I}; this function is continuous, and 0 < # < I ~ ( 7 I )9 0 , SO that # E F- But 0 < T:-,,,# < I I < @7-t0
< I v 1, so that 14 I F d I IF Now I d r )I = I dT)1 v O ( t O ) = y I # IF < y I v I F , and our claim is established. Since, however, ~ ( 0= ) 0 for all
v E F, as noted above, we actually have F
< yCo.
&
Rkmark. 24.M,(1) may be reformulated, in view of 24.A, as follows: = R,, a space F E b M is in b9-W if and only if it is the subspace of all continuous functions vanishing at 0 of a space G E b y + that contains the characteristicfunction of some interval and is stronger than L" (for the "only if" part set G =eF). Correspondingly, 24.M, (2) implies: If J =R, a space F E b M is in b 9 Y if and only if it is the subspace of all continuous functions of a space G E by-, that i s stronger than L".
If J
It is possible to extend the definition of THICKNESS of a Banach space D, with respect to another B, given in Section 23 (p. 71) for D E b y , and B E b y or b y * , unchanged to the case where, instead, B E b y % + (if J = R,),or B E b y % ?(if J = R);however, this new case is reduced to the former, on account of 24.A, 24.L, and the following trivial lemma.
24.N. Assume that D E b y K , and B E by%'+ (if J = R,) or B E b y % (if J = R). Then D is thick with respect to B if and only ifD is thick with respect to eB. Assume that X is a given Banach space. If J = R, , we define the class F%+(X)= {F(X) : F E Y%+}; if J = R, we define y % ( X )= {F(X) : F E TU}. On account of 24.G,(2), we may write bT%+(X), by%(X) unambiguously for the respective subclasses of complete spaces. We have the usual characterization lemma (cf. 23.2).
24.0. (1) I f J = R, , and if G is a normed space of continuous functions from R , into X, then G E T%?+(X) if and only ifG satisfies ( N ) , (FC,) (so that G EF%(X)),and (Z,), (T;). (2) I f J
=
R, and ifG is a normed space of continuous functions f r o m
25. NOTESTO
CHAPTER
2
83
R into X , then G E F % ( X )if and only ifG satisfies (N), (FC,) (so that G E ~ % ( X )and ) , (Zx), (Tx). Proof. Immediate from 24.G, (1) and t h e definition of YW, F%.&
25. Notes to Chapter 2 We continue assuming in the reader a certain familiarity with functional analysis, and mention as works of general reference those enumerated in the first paragraph of the Notes to Chapter 1 (p. 31). For the theory of measurability and integration of functions with values in a Banach space we may refer to Hille and Phillips [l] and Bourbaki [2]. Our theory of 9-spaces is closely related to the work of DieudonnC [I], Kothe [I], Lorentz and Wertheim [I], Ellis and Halperin [I], Luxemburg [l], Luxemburg and Zaanen [I], [3], [4], [5], [6], [7], among other authors and papers. The spaces considered by Luxemburg and Zaanen are essentially what we call locally closed spaces in SK(but the domain is a rather general measure space and compact intervals are replaced by ad hoc “bounded sets”); the theory of associate spaces is also in its main lines contained in these papers. The intended Lpplication of these functional-analytic concepts and tools to the theory of linear differential equations has suggested, however, extending the research in two opposite directions: (i) the study of more general structures, arising from the partial elimination of certain assumptions (the theory of the class N ( X ) , the consideration of spaces that are not quasi locally closed, etc.); (ii) the study of richer structures, namely those satisfying certain properties of translation-invariance (the classes F ( X ) , .F4(X), etc.). We have refrained, however, from developments pretending to utmost generality: Section 21, for instance, could have been set up in a much more abstract form-not concerning function spaces in particular-as was done in part in Schaffer [l], Section 2; instead of restricting the domain of our functions to be an interval of the real line, a general measure space might have been used, provided in some cases with a group or a semigroup of measure-preserving transformations; and so forth. As was pointed out in the Introduction, we have preferred to stick to the moderate degree of generality that best fits the application to differential equations. An important instance of function spaces-to be precise, of locally closed for J = R or J = R+-is given spaces in 9- for any interval J , and in 9, by the Orlicz spaces. Their place in the theory of this chapter is described in Schaffer [I], Section 5, supplemented by Massera and Schaffer [4], Lemma 3.4. As general references for the theory of Orlicz spaces we mention, among many others, Krasnosel’skii and Rutickii [I] (for a domain of finite measure only), Luxemburg [I], Luxemburg and Zaanen [2].
CHAPTER 3
Linear differential equations 30. Introduction Summary o f the chapter T h e main purpose of the book is the study of certain differential equations, namely, the linear equations
+ AX = 0,
(30.1)
X
(30.2)
A+Ax=f,
which we introduce in this chapter. T h e independent variable, always called t , ranges over a subinterval J (i:e., a connected set containing at least two points) of R ;the domain J , whether left arbitrary or further specified, remains fixed and generally unmentioned. T h e terms A , f are elements of L(x),L(X), respectively, X being a given Banach space. A SOLUTION x is a primitive function (in a sense that will be made precise below) from J into X such that, if i is its derivative, Eq. (30.1) or (30.2), as the case may be, becomes an equality in L(X). Besides Eqs. (30.1) and (30.2), it is useful to consider the operator equation (30.3)
O+AU=O,
x.
where a solution U is a primitive function from J into Mul_iplication (on the left) in 8 by a given element defines an element of X,and this mapping is isometrical; therefore (30.3) becomes a special case of (30.1), with X replaced by T h e same remark applies to other equations in similar to (30.3) that will occur later on. In the chapters dealing with special topics (beginning with Chapter 9), additional introductory material on differential equations will be included as required in each case.
x.
84
30. INTRODUCTION
85
Section 31 includes the fundamental properties of the above equations: existence and uniqueness of solutions, formulas that connect the solutions of (30.l), (30.2), (30.3), and bounds for these solutions; one subsection is devoted to the proof of a theorem on convergent sequences of solutions of a sequence of equations of the form of (30.2). T h e theory of associate equations in coupled spaces, including that of the adjoint equation, and the corresponding Green's Formula, is developed in Section 32. Section 33 contains some preparatory material on the set of those solutions of the equations that belong to a given function space.
Primitives
A function f from J into a Banach space X is called a PRIMITIVE (FUNCif there exists an element of L ( X ) ,always denoted, correspondingly, by f, such that f ( t ) = f ( t o ) J : , f ( u )du for all t E J , to being any fixed number in J. T h e function f,which is uniquely determined by f, is called the DERIVATIVE off; as an element of L ( X ) , it is actually an equivalence class of functions modulo null sets. A primitive is always absolutely continuous on every compact subinterval of J ; the converse is true if X is uniformly convex-in particular, a Hilbert space, and more particularly, finite-dimensional- or under certain other assumptions (Clarkson [l], Dunford and Morse [I]), but not for arbitrary X (Bochner [ 13). TION)
+
30.A. Let X , Y , Z be Banach spaces over the same scalar field. If U , V are primitives with values in [ Y ;Z], [ X ; Y ] , respectively, then U V is a primitive i n d ( U V ) ' = UV U6'. In particular, if X is a Banach , f *, U , V are primitives with values space over the scalar field F and if q ~ f, in F , X , X * , 3,3,respectively, then qIf, ( f , f * > , ( * , f * > f , Uf,U V are primitives; if X , X' are coupled spaces and f,f' are primitives with is a primitive. If U is a primitive values in X , X ' , respectively, then (f,,f'> with invertible values in -f, then U-' is a primitive and (U-l). = - U-l/-JU-l.
+
Proof. T h e first result is trivial; the particular cases follow by replacing X , Y , 2 by X and F in some arrangement and, for the coupled spaces X , X ' , by using the congruence @' : X ' + @'XI C X * (Section 12). T o prove the last statement: since U is continuous, so is U-l, hence - 7J-lUU-l E L ( 8 ) . Choose a fixed to E J and define a primitive V by V ( t ) = U-l(t0) U - l ( ~ ) O ( u ) U - l ( du, u ) t E J. Then U V , V U are primitives, and ( U V ) ' = UV U P = U(V - U-l) =
s"l,
+
Ch. 3. LINEAR DIFFERENTIAL
86
EQUATIONS
- I ) , and similarly ( V U ) ’ = ( V U - I ) U - l o ; here Ofu-l, E L(X). Thus UV - I, V U - I are, respectively, solutions W - OU-’W = 0, W - WU-10 = 0, satisfying “(to) = 0;
OW1(UV U-lO
of this solution is unique (see Section 31), and therefore coincides with the trivial solution 0. Therefore UV = V U = I , 17-1 = V . J,
30.B. Assume that X is a Hilbert space, and that f is a primitive with values in X \ (0). Then 11 f I(, sgn f are primitives, and I I(f 1). I Ilfll, II (Sgn II llf ll / Ilfll. Roof. By 30.A, Ilfl12 = U,f) is a primitive; hence so is 11 f (I. Again by 30.A, ~ ~ fis ~ a primitive, ~ - l hence so is sgn f = 11 f 1 1-F and f = 11f11’ sgn f l l f l l (sgn f)’ Taking into account the identity obtained by differentiating (sgnf, sgn f ) = 1, we find llf1I2 = Ilfll‘2 + llf1(2 1) (sgn.f)‘ )I2, whence the conclusion. &
<
n. <
+
.
3 1. Solutions Existence, uniqueness, and formulas for the solutions
T h e definitions of primitive and of solution given in Section 30 imply that a continuous function x from J into X is a solution of (30.2) with x(t,) = xo for some to E J if and only if it satisfies the integral equation
+ 1 f(u) du t
(31.1)
x ( t ) = xo
to
t
s,
A(u)x(u)du.
Since any compact subinterval of J is covered by a finite union of intervals J’ C J such that J’,,(( A ( u ) (( du < 1, the (Banach) Fixed-Point Theorem for contractive mappings, applied in ,,C(X) for each J’, shows, as usual, the existence on all J of a solution of (31.1) and its uniqueness. Therefore the existence and uniqueness of the solution of (30.2) with given “initial” value at to is established, and the same is true in particular of (30.1) and (30.3). It is obvious that the solutions depend only on A, f as equivalence classes and not on the particular representative chosen (cf. 90.A). 31.A. If U is a solution of (30.3) and U(to)is invertible for some to E J , then U ( t ) is invertible for all t E J. of P - V A = 0 Proof. Let V be the solution (with values in that satisfies V(to)= U-l(t0); the existence and uniqueness of V
3 1. SOLUTIONS
87
follow as for the solutions of (30.3). Set W , = U V , W , = V U ; by 30.A, W, , W, are primitives, and direct computation shows that they are the solutions of I@, AW, - W I A = 0, = 0, respectively, with Wl(to)= W z ( t o )= I . Again existence and uniqueness hold for these equations; therefore W , = W , = I; and V = U-l. & I n view of 31.A it is permissible to speak of INVERTIBLE-VALUED SOLUTIONS of (30.3).
+
m,
31.B. (a) Let x, y , U , U , be solutions of (30.l), (30.2), (30.3), (30.3), respectively, with U invertible-valued. Then, for all t o , t E J , (31.2)
(31.3)
4)= W ) W t , ) x ( t o > , At) = W ) ( W t , l Y ( t O ) + j t
U-’(ulf(u)
to
(31.4)
Ul(t)
w,
= w)~-l(to)~l(t”).
(b) If x is a solution of (30.1) and 9 a scalar-valued primitive, then q ~ xis a solution of (30.2) with f = $x.
Proof. Direct verification, taking into account uniqueness and 30.A. Q Formulas (31.2), (31.3), (31.4) show that the values of the solutions of (30.1), (30.2), (30.3) at each t E J, as well as the solutions themselves as elements of L(X) or L(x),depend linearly and continuously on the “initial” values at t o , and on f. If t o , t , E J are given, and U is an invertible-valued solution of (30.3), U(tl)U-l(t0) is an automorphism of X which, by (31.4), is independent of the choice of U ; we refer to it as the MAPPING ALONG THE SOLUTIONS OF (30.1) FROM to TO t,; for any xo E X, xo and x1 = U(t,)U-l(to)xo are indeed, by (31.2), the values, at to and t, , respectively, of one and the same solution of (30.1). Bounds for the solutions T h e elementary bounds of the solutions, given by the following lemma, will be widely used in the sequel.
3 1 .C. Let J‘ C J be a compact subinterval of length p(J’) (a) If x is a solution of (30.2) and t o , t (31 3)
II 4 t ) II d (I1 *%(to)ll
E
=
J’, then
+ J ,IIf(4 II dtr) exp(jJ,II 4 4 II ^.) J
1.
9
Ch. 3. LINEAR DIFFERENTIAL
88
EQUATIONS
I n particular, if x is a solution of (30.1) and to , t
E
J', then
(c) If xl, x2 are nonzero solutions of (30.1) and t o , t E J ' , then (31.10)
Y[Xl(t>,
%(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.
<
Proof. Assume first that clY C X,, . Since X,, is the range of the monomorphism I7 described in 33.A, n-l(clY) is a subspace of X,. T h e restriction of I7 to this subspace is thus an isomorphism of
94
Ch. 3. LINEAR DIFFERENTIAL
EQUATIONS
the Banach space I7-’(clY) onto the Banach space clY; its inverse, the restriction of IT-l to clY, is therefore bounded; the “only if” part is therefore proved, with S, taken as the norm of this inverse. Assume, conversely, that S , exists. The restriction of l7-l to Y is then a bounded linear mapping from Y into the Banach space X, , and has a unique bounded linear extension, say Y :clY + X,. But then both ZW: clY -+ X and the identity injection clY X are bounded linear extensions of the identity injection Y -+ X, and therefore coincide; hence clY = RY(clY) C ITX, = X,, . Both parts of the proof break down if Y = {0}, but in this case the statement is trivially true. & ---f
An immediate corollary of this theorem is Assume that D € b M ( X ) . X,, is closed i f and 33.C. THEOREM. only if there exists a number S 2 0 such that every D-solution of (30.1) satisfies I x ID S 11 x(0) 11.
<
Remark. If X is finite-dimensional, the existence of S follows immediately from the fact that the linear manifold of all solutions of (30.1) is finite-dimensional, and that therefore the same is true of X, . We agree to denote by S,, S the minimum value of the numbers satisfying the conclusions of Theorems 33.B, 33.C, respectively; or, in full, by Sy,(A), SD(A), respectively; when no confusion is likely, the subscript D and/or the argument A will be dropped.
33.D. Assume that Y E b N ( X ) and that E is a subspace of D. S, . In paiticular, if X, is
If X,, is closed, X,, is closed and S , closed, X, is closed.
<
R o o f . Apply Theorem 33.C twice. & In this connection we state some special results, which will not be used in the sequel; for the proof we refer to ScMffer [3], Theorems 7.1, 7.2, with the obvious adjustements. 33.E. If X , = X , , then X , is closed. If J = R , or J XoM, = X,, , then XoM is closed.
For the important case of $-spaces,
J
= R,
and if
the following remark is useful:
33.F. If D E b 9 and X,, # {0}, then D E b S K . In particular, R , or J = R, and D E b y , then X,, # (0) implies D E b y K .
=
if
33. D-SOLUTIONS OF HOMOGENEOUS
95
EQUATIONS
Proof. Let x be a non-trivial D-solution of (30.1); then 11 x 11 E D is a continuous function that vanishes nowhere on J. By 22. J, D E FK. &
co
Examples and comments The following two examples illustrate cases in which X,, is not closed. 33.G. EXAMPLE.Let J be R,, and let X be the real or complex separable Hilbert space 12. Let (30.1) be the system Xn
+ tanh(t - n)
*
x,, = 0,
n
=
1,2,
...;
here A is given by a diagonal matrix-hence its values are symmetric or Hermitian operators-and A e C ( 8 ) with 11 A 11 = 1. Any solution x of this system is given by its components xn(t) =
cosh n cosh(t - n) Xn(O),
n = 1,2, ...,
<
so that each I x,, I E T, since I x,,(t)I 21 xn(n)l e-lt-,,l. Let D be any space in b y K (cf. 33.F); then D is weaker than T (by 23.J). X,,,then contains all points of X with only finitely many nonzero components and is therefore dense in X. Consider, however, the solution x with x(0) = (n-l): we find, for all t 2 1,
therefore x does not belong to M(X), let alone to the stronger space D(X). Hence X,, # X, and XoDis not closed. Q* 33.H. EXAMPLE.If in Example 33.G we change variables, setting y = efx and changing y back into x for the sake of consistency in notation, (30.1) becomes the system X,,
+ (tanh(t - n) - 1
) =~ 0, ~
tt =
1,2, ...;
here A is still given by a diagonal matrix (and therefore has symmetric or Hermitian values) and A E C ( 8 ) ;now (1 A 11 = 2. The components of a solution x are now given by et cosh n Xn(t)
= cosh(t - n)
X,,(O)
=
1
+ tanh(t - n)1 - tanh n
X,,(O),
n
=
1, 2,
...;
96
Ch. 3. LINEAR DIFFERENTIAL
EQUATIONS
every 1 x, I is bounded and nondecreasing-hence 1) x 1) is nondecreasingand 1imt+, x,(t) = (1 eZn)x,(0).Hence X, contains all points with only finitely many nonzero components, but does not contain the point (n-I), so that X , is a dense linear manifold that is not closed. &
+
Remark. We have assumed J = R,, because we require these examples for future reference; however, they also subsist for J = R.
0000
In these examples, A was not constant; cases with constant (but not symmetric or Hermitian) A have been described by Massera and Sckffer [I], Example 4.1, and [4], Example 2.1. If X is not a Hilbert space, it may also happen that X,, is a subspace, but is not complemented; instances are given by Massera and Schiffer [4], Examples 2.2 and 4.1 (the latter reproduced in Example 44.B); in the former, X is any l p , p # 2 (and thus reflexive, indeed uniformly convex, A is constant, X , is a subspace that and separable, provided p # I, a), is not complemented, and all bounded solutions are actually constants; in the latter, X = l“, A is given by a constant diagonal matrix, X , = X, and X,,,, = l:, which is not complemented. The question may be raised, whether every linear manifold Y in a Banach space X may be an X,, , i.e., Y = XoD(A)for some A E L(8) and some D E bst, , say (cf. 33.F). If J = R, and X is infinite-dimensional, the answer is no: 33.1. If A,, A, E L(Q, D,, D,E b*K and Xq(Ai) 4- XODB(AJ = X, then the dihedron (XoD,(Al), XOD2(Az)) is gaping. Proof. See Section 81 (p. 276). & Now, if X is infinite-dimensional, there certainly exist dihedra that are not gaping; by 33.1, at most one member of such a dihedron can be XoD(A) for some A E L(8), D E bSK; at least one, therefore is “forbidden.” However, if Y is a complemented subspace, then Y = XoD(I- 2P), where P is any projection along Y and D is any space in b y K (hence weaker than T); indeed, the solutions of (30.1) are then given by x ( t ) = e-t(I - P)x(O) etPx(0). Therefore, if X is finite-dimensional the answer to the above question becomes yes.
+
0000
Query. Is a subspace that is not complemented ever “forbidden” in this sense?
A result on associate equations W e prove a special result concerning a pair of associate equations on J = R, . We let X,X‘ be a pair of coupled spaces. W e also recall
34. NOTESTO CHAPTER 3 the definition of thinness of one *-space (Section 22, p. 52).
97
with respect to another
33.5. Assume that J = R , and that A E L(x)has an associate. Consider spaces D , E E b S K . If D is thin with respect to E‘, or E with respect to D’ (in particular, if D, E E b.35 and either D or E is stronger where XiE = than M, or not weaker than L:) then XAEC (XOD)O, X&A’). In particular, X(,C (X,)O and X i C (X,,),. Proof. Assume the contrary to be the case; there exist then a D-solution x of (30.1) and an E-solution x’ of (32.1) such that (x(O), x’(0)) = u # 0. From 32.C, 11 x 11 11 x’ 11 3 I (x, x’) I = I u I > 0. Since 11 x 11 E D C D” and 11 x’ 11 E E C E”, 22.0,(c) shows that the thinness assumption yields a contradiction. T h e particular cases follow from 23.U,(2); (b),(c). &
Remark. If the answer to the Query following 23.U were yes, the thinness assumption in 33.5 could be replaced by “if either D or E is not weaker than L””. T h e result would be best possible, since for A = 0 and any D and E weaker than L“, we have X,, = X, = X , XhE = X i = X’. We therefore formulate the question and give a partial affirmative answer, namely when A E M(W)and D, E E b F K . Query. Can the assumption “if D is thin with respect to E‘, or E with respect to D’ ” be replaced in 33. J by “if either D or E is not weaker than L”” ? 33.K. Assume that J = R , and that A E M(X) has an associate. Consider spaces D, E E bcFK.If either D or E is not weaker than L“, then XhEC (X,,)’.
Proof. Assume the conclusion to be false, and let x, x‘, u be as in the proof of 33.5. We remarked, at the beginning of the section, that if A E M(X), every M-solution, hence every D-solution for D E b y K , of (30.1) is bounded; the same applies to the associate equation. ThereI o I I x’ I-l, 11 x‘ 11 > 1 u I I x I-l, so that D and E contain fore 11 x [ ) nonzero constants and are therefore weaker than L“, contradicting the assumption. 9.
>
34. Notes to Chapter 3 In the treatment of linear differential equations we have placed ourselves in the case of CarathCodory’s assumptions, in which the usual continuity
98
Ch. 3. LINEAR DIFFERENTIAL
EQUATIONS
requirements are replaced by mere integrability conditions. This is essential for a satisfactory and fruitful application of the classes of function spaces and related concepts studied in Chapter 2 to the properties of differential equations -an enterprise that constitutes one of the central purposes of this book. The discussion of existence problems, etc., under these general hypotheses even in the nonlinear case, but in finite-dimensional space only, may be found, for instance, in the book of Coddington and Levinson [l] (especially Chapter 2). We also mention the book of Sansone and Conti [ 13 (especially Cap. l), where, in particular, a general form of Grmwall’s Lemma may be found. As a work of general reference for functions with values in a Banach space, their integrals, etc., we may indicate Hille and Phillips [I].
P A R T I1
This Page Intentionally Left Blank
CHAPTER 4
Dichotomies 40. Introduction
In this chapter we embark on the main line of development of this book with a study of the homogeneous equation (30.1), i.e., (40.1)
* + Ax = 0,
where the range of the independent variable t is J = R , (a fact that shall be understood throughout the chapter), and A E L(x)is given, X being a fixed Banach space. We describe two types of behavior of the solutions of (40.l), which might be loosely termed uniform conditional stability, and uniform asymptotic (or exponential) conditional stability, respectively: uniform, because independent of the “initial” value to; conditional, because some solutions remain (or become) small, while others remain (or become) large. There is in addition an “apartness” condition on the two sets of solutions. We have coined the generic term dichotomy for this kind of behavior; more precisely, (ordinary) dichotomy, and exponential dichotomy, respectively. Ordinary and exponential dichotomies will be the most “perfect” types of behavior of the solutions of (40.1) that we shall consider, and they will serve as a standard in discussing other types. Sections 41 and 42 introduce the concepts of an ordinary and an exponential dichotomy, respectively, give several equivalent formulations, and describe their properties and the relations with the classes of solutions belonging to certain function spaces. Ordinary dichotomies are definitely trickier to handle than the apparently more elaborate exponential dichotomies. In Section 43 we discuss the relation between dichotomies, ordinary or exponential, of the solutions of (40.1) and of the associate equation, in a pair of coupled Banach spaces. 101
102
Ch. 4. DICHOTOMIES
T h e simplifications which occur in the statements when X is a Hilbert space are minor; they are therefore dealt with in passing at the appropriate points. They a_re some special results, however, if X is finite-dimensional; they are given in Section 44, We have included the statement of some theorems (e.g., Theorem 41.E) which are meaningful in the terms of this chapter, but for which we have to, or choose to, defer the proof until later. Sometimes it is only one proof out of two, both of interest, that is given now, the other being deferred (e.g., Theorem 41 .F). Throughout this chapter, U denotes the solution of (30.3) that satisfies U(0) = I .
4 1. Ordinary dichotomies Definition Speaking informally and very loosely, a subspace Y of X is said to induce a dichotomy of the solutions of (40.1) if the solutions starting from Y at t = 0 remain uniformly bounded, the solutions starting far from Y remain uniformly bounded away from 0, and both sets of solutions remain uniformly apart. Now for a precise definition: T h e subspace Y of X INDUCES A DICHOTOMY OF THE SOLUTIONS OF (40.1), or A DICHOTOMY FOR A, if there exists N > 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:
<
(W I1 Y ( t ) I1 (Dii)
< N I1Y(t0)I1
11 z ( t ) 1) 3 N'-l 11 z(t,,) Y b ( t ) , 441 3 Yo
11
for all t >, to >, 0; for all t 2 to >, 0 ; for all t >, 0, if y , z # 0.
For the sake of precision, we may describe such a dichotomy as ORDINARY, in contrast to the exponential dichotomy to be introduced in Section 42. It is not necessary, in fact, to assume the existence of N', yo for eoery A ; this and similar apparent weakenings and reformulations of the definition are taken care of by the next theorem:
41.A.
THEOREM. Let Y be a subspace of X . The following state-
ments are equivalent : (a) Y induces a dichotomy for A. (b) There exists N > 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).
<
41. ORDINARY DICHOTOMIES
103
(c) [(d)] There exists N > 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
<
t 3 to 2 0; to 3 t 0.
(g) [(h)] For every [some] h > 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)
<
<
Ch. 4. DICHOTOMIES
104
<
h d ( Y , z(0)). Application of (D'ii) to x1 = z1= z, with 11 z(0) 11 y1 = 0 yields (Dii) with N' = D ( t o , t are interchanged). If y, z # 0, we apply (D'i) to y1 = IIy(to)Il-ly, z1 = - I1 4 t O ) II-lz, x1 = y1 z1, t = to , and obtain r[y(to),z(t0)l = II %(to) II 2 D-l 11 yl(to)11 = D-l, so that (Diii) holds with yo = D-l. (h) implies (e). Assume that (h) holds for the ( Y ,A,)-splitting q, with D(q) = D o . Let h > 1 be given and x , y, z as assumed in (e). Define the solutions y 1 , z1 of (40.1) by z,(O) = q(x(O)), y , = x - z1. The solution w = y - y1 = z1 - z has w ( 0 ) E Y, whence q(w(0)) = 0. Now 1) z(0) 11 h d( Y , z(0)) = h d( Y , zl(0)) h 11 z,(O) 11, so that applying (D'ii) of condition (h) with z1instead of z, and t = 0,
+
<
II 4 0 ) !I d (1
(41.1)
<
+ 4 I1 4 0 )II < 2hDo II 4 t O ) II .
Now let t 2 to be given. Applying (D'i) to y 1 instead of to y , and again to w , w , 0, 0 instead of x, y, z, t o , adding, and using (41.1), we have I1 r(t)II < II Y l ( 4 !I II w w II d Do(l!xV0) II II 4 0 ) II)
+
+
d Do(1 4- 2XDO) II 4 t o ) II '
<
Let t to be given; in the same way, but using (D'ii) for z1 instead of x (yet (D'i) for w as before) we find (1 z(t) (1 Do( 1 UD,)1) x(to) 1). We conclude that (e) holds with D = D(X) = Do(l 2XD0). &
<
+ +
Additional equivalent conditions for a dichotomy are available for complemented subspaces: 41.B. A complemented subspace Y of X induces a dichotomy for A i f and only i f any one of the following equivalent conditions holds: (a) [(b)] There exists N > 0, and for every [some] complement 2 of Y there exist N' = " ( 2 ) > 0, yo = yo(Z) > 0, such that any solutions y , z of (40.1) with y(0) E Y , z(0) E 2 satisfy (Di), (Dii), (Diii). ( c ) [ ( d ) ] For every [some] projection P along Y there exists D = D(P)> 0 such that (Dii) (1 U(t)(Z - P)U-'(t0) 11 (Diii) )I U(t)PU-'(t,) 1)
for all t 2 to 2 0; for all to >, t >, 0.
Proof. If we let P be the projection along Y onto 2, and use 11 .F, we have the chains of implications: 41.A, (c)+ (a)+(b)+41.A,(d) and41.A,(g)+(c)+(d) Theorem 41.A completes the proof. 9,
+41.A,(h).
41. ORDINARY DICHOTOMIES
105
If X is a Hilbert space, every subspace is complemented; in 41.B, (b), (d) one might (but should not necessarily) choose 2 = Y l , P = I - P , . Of Theorem 41.A, the only condition that remains of interest, besides the definition of dichotomy itself, is (b). We reformulate the definition and this condition, together with 41.B, (a), (b), for this case:
41 .C. If X is a Hilbert space, the subspace Y induces a dichotomy for A if and only if any one of the following reformulations of the de$nition, of 41.A,(b), of 41.B,(a), of 41.B,(b), respectively, holds: (a) [(b)] Either: Y # (0) and there exists N > 0, and for every [some] w , 0 < w ,< 4p, there exist N’ = N’(w) > 0, wo = w o ( w ) , 0 < wo $rr, such that any solutions y , z of (40.1) with y , z # 0, y ( 0 ) E Y , +(Y, z(0))3 w satisfy (Di), (Dii), and (Diii,) a ( y ( t ) , z ( t ) ) 2 wo for all t 3 0;
<
or: Y = (0) and there exists N’ > 0 such that every solution z of (40.1) satisjies (Dii). (c) [(d)] There exists N > 0, and for every [some] complement 2 of Y there exist N ‘ = “ ( 2 ) > 0, wo = wo(Z), 0 < wo b, such that any solutions y , z of (40.1) with y(0) E Y , z(0)E 2 satisfy (Di), (Dii), (Diii”).
<
Proof. T h e reformulations are carried out using 14.A. T h e possibility w = &r in (a), (b) follows from (c) by taking 2 = Y l . &
of taking
Remark 1. In the definition of dichotomy and in conditions 41.A, (b), (e), (f) we may, if Y # {0}, replace the condition ‘ 1 1 z(O)[l X d( Y , z(0))” by “ y [ Y , z(O)] 2 y” for some y > 0, by virtue of I1.C; then N ’ , y o , D become functions of y , and the case Y = (0) must be stated separately. We omit the details.
<
N’(q), D(X), D(q) may be taken, Remark 2. T h e functions ”(A), as the proof of Theorem 41.A (especially the implication (h) -+ (e)) shows, to be nondecreasing, and indeed linear, functions of X only, and yo(X),yo(q) as the reciprocals of such functions. Use of 11. D and 1 l . F shows that, in 41.B, ” ( 2 ) may be taken as a nondecreasing linear function of y [ Y , Z]-l, y o ( Z ) as the reciprocal of such a function, and D ( P ) as a nondecreasing linear function of [I P[I. T h e corresponding facts for 41.C follow from 14.A. Remark 3 . In view of later developments (see in particular 42.D) it is pertinent to ask whether (Diii) is redundant in the definition of dichotomy or in conditions 41.A, (b), (c), (d) and 41.B,(a), (b), and whether this is the case for (Diii,) in conditions 41.C, (a), (b), (c), (d).
Ch. 4. DICHOTOMIES
106
T h e answer is no, even for two-dimensional X (for one-dimensional X, (Diii), (DiiiH) are vacuously satisfied) and constant A, and for the most stringent of those conditions, namely the definition of dichotomy and 41 .C, (a): this is illustrated in Example 41 .G. Remark 4. The question as to which subspaces of X induce a dichotomy for some A E L ( 8 ) is trivially answered by the observation that if A = 0 the solutions of (40. I ) are the constants, and therefore every subspace of X induces a dichotomy for 0. Some related questions will be taken up in Section 72 (p. 241).
Dichotomies and solutions in 9'-spaces We shall now examine the relationship between subspaces inducing dichotomies and the linear manifolds X,, introduced in Section 33. We shall restrict ourselves to considering spaces D(X) with D E b y ; on account of 33.F we may assume without loss that D is weaker than T, i.e., D € b y K . T h e most important of those linear manifolds in the present connection are X, and X,, .
If the subspace Y of X induces a dichotomy for 41.D. THEOREM. A, then X,, C Y C X , , and X,, is a subspace. If D E b y K , then X,, = X, or X,, C X,, C Y, according as D is, or is not, weaker than
L".
Proof. We refer to the definition of dichotomy. Condition (Di) implies Y C X,, and condition (Dii) implies X,, C Y. T h e linear mapping y -+ Uy : Y + L"(X) is bounded since (Di) implies 11 Uy 11 N 11 y 11 (or by Theorem 33.B); the inverse image X,, of the subspace L:(X) is therefore closed in Y, hence a subspace of X. Let D E b y K be given, and consider any D-solution x of (40.1) with x(0) 4 Y; condition (Dii) implies, for an appropriate N' > 0, 11 x ( t ) 11 N'lf+'(Ix(u) 11 du for all t 0. Since x E M(X) (by 23.G), it follows that x is bounded, hence x(0) E X, . Since Y C X, anyway, we have X,, C X , . If D is weaker than L", the reverse inclusion holds, and equality follows. Assume that there exists a D-solution x of (40.1) with x(0) 4 X , . If x(0) E Y, we have u = inf,,, 11 x ( t ) 11 2 N-l lim SUP^+^ 11 x ( t ) 11 > 0; if x(0) 4 Y, (Dii) a fortiori implies u = inf,,, 11 x ( t ) 11 > 0. T h e n 1 0-l 11 x 11, and we must have 1 ED. Therefore D is weaker than L". &
<
<
<
41.E. THEOREM. If the subspace Y of X induces a dichotomy for A , so does every subspace Z with Y C Z C X,
.
41. ORDINARY DICHOTOMIES
107
Proof. See Section 63 (p. 181). & 03
Remark. Example 41 .H below illustrates, in separable Hilbert space, the fact that there may exist a subspace Y that induces a dichotomy although X , is not closed and therefore does not induce one itself.
41.F. THEOREM. If the subspace Y of X induces a dichotomy for A, and if X,,has finite codimension with respect to Y , then X,, also induces a dichotomy for A. Proof I (for Proof I1 see Section 63 (p. 182)). 1. By Theorem 41.D, X,, is a subspace contained in Y . We shall assume that X,, # Y , since otherwise there is nothing to prove. By the assumption there exists some ( Y , A,)-splitting q1 of X and constants Nl , N ; , yol > 0 such that condition 41.A,(d) is satisfied for Y . Let Yo be any (finite-dimensional) complement of X,, with respect to Y ; observe that Y o # {O}. Let Q be the projection of Y along X,, onto Y o . Set A = A, II Q II A, II Q II, and q ( 4 = q1(4 Q(x - q,(x)) for every x E X . By 1 l . F and ll.H, q is an (X,,, A)splitting of X . We claim that there exist N , N', yo > 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,
+
+
+
<
If y # 0 and u is a nonzero solution of (40.1) with u(0) E X,, , and if to 3 0 is given, we apply (Di) to 11 y ( t o ) 1 1 - l ~ - 11 u(to)11-l~ and obtain
3. Since X,, C Y , condition (Di) holds for X,,, with N = N l . Let z be a solution of (40. I ) such that q(z(0)) = z(0).We define the solutions yi , ~1 of (40-1) by zl(0) = ql(z(O)),yl(0)= Q(z(0)- zl(0))E Yo ,
108
Ch. 4. DICHOTOMIES
+
so that z = y1 z1 and ql(zl(0)) = xl(0). Assume for the moment that y l , z1 # 0. By (Diii) for Y , ybl(t), - zl(t)] 3 yol for all t. We apply 1 l.A and use (Dii) for z1and (41.2) for y1 , and obtain for any t 2 to 2 0
I1 dt)II 2 b 0 l max{llx(t) II Z l ( t ) Ill 2 5 0 1 max{BW IIYl(t0) IL
(41.4)
119
w1II %(to) Ill
II 4 2 0 ) II
2 "-l
9
where N' = 4yi: max{/3-1Nl, N i } . T h e inequality between extreme terms in (41.4) subsists if y1 or z1 is 0, since yol 2. Thus (Dii) holds for X,, with this. N'. Assume that z # 0, and let u be as in part 2 of the proof. Assume for the moment that y l , z1# 0, and let t 3 0 be given. By (41.3), y[u(t), pyl(t)] >, for all scalars p # 0; by (Diii) for Y, y[pu(t) vyl(t), zl(t)] 2 yol for all scalars p, v not both 0. By the Three-Angles Lemma l l . B we conclude that y [ u ( t ) , z(t)] 3 bO1/3Ny2. T h e same conclusion subsists if y1 or z1= 0, since yol , pNy2 2 (the latter by (41.3)). Therefore (Diii) holds for X,, with yo = $yol/3N~2.&
<
+
<
00
Remark. The conclusion of Theorem 41.F does not remain true in general if the codimension of X,,,, with respect to Y is not finite: Example 41.1 illustrates this fact for a symmetric- or Hermitian-valued bounded continuous A in separable Hilbert space.
Examples
41 .G. EXAMPLE. Let X be two-dimensional real or complex euclidean space, with Cartesian coordinates xl, x 2 . Let (40.1) be the system $l $2
- x2 = 0 = 0.
Every solution of the system is given by x ( t ) = ( x l ( t ) , x 2 ( t ) ) = (xl(0) x,(O)t, ~ ~ ( 0 ) )We . shall show that X , = {x : x2 = O} satisfies (Di), (Dii) of the definition of dichotomy or, equivalently, of condition 41.C, (a), but not (Diii), or equivalently (Diii,) (even for w = T&I only). Solutions y with y(0) E X , are constants; hence (Di) holds with N = 1. Let w , 0 < w an, be given and let z # 0 be a solution 11 z(0) 11 cos w . We set such that + ( X , , z(0)) 3 w , i s . , I zl(0) I tm = - 1 z2(0) Re(z,(O)f,(O)). An easy computation shows that,
+
<
<
I09
41. ORDINARY DICHOTOMIES
if t,, < 0, )I z ( t )11 increases as t goes from 0 t o co, whereas, if t, > 0, I/ z ( t )11 decreases as t goes from 0 t o t, and then increases as t goes from t,, t o 00. I n the latter case, therefore, if t t , 3 0, we have
Therefore (Dii) holds with N' = N ' ( w ) = I/sin w in any case. However, consider the solutions y ( t ) = ( I , 0), z ( t ) = ( t , l), with y(0) E X , , + ( X , , z(0))= in: they satisfy tan *(y(t), z ( t ) )= t-l, so that limi+m +(y(t),z ( t ) )= 0, and (Diii,) does not hold. W e remark that, on account of Theorem 41.E or a direct simple proof, no subspace at all induces a dichotomy. 03
41.H. EXAMPLE (Example 33.H continued). In Example 33.H, IJxIIis nondecreasing for every solution x of (40.1); hence X,, = (0) induces a dichotomy for A , with N' = 1 (e.g., by 41.C); but X , is not closed. &
O3
41.1. EXAMPLE.Let X be the real or complex Hilbert space P. Let (40.1) be the adjoint equation to the equation in Example 33.H: namely, the system k, - (tanh(t - n)
-
l)x, = 0,
n
=
I , 2, ...;
A is again given by a diagonal matrix, has therefore symmetric or Hermitian values, and satisfies A E C ( x ) ,/I A (1 = 2. Any solution x is given by its components cosh(t - n) % ( t ) = et cosh n
1 tanh n + tanh(t n) x,(O), -
"(O)
=
1
-
n = 1, 2,
...;
every I x, 1 is nonincreasing, hence /I x /I is nonincreasing, and X,, = X induces a dichotomy for A , with N = 1 . Also, 1imt+=x,(t) = (1 eZn)-lx,(0), so that limt+z11 x ( t ) 11 > 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
<
(Eii,)
(Eiii,)
> 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:
<
yo
+
<
(E’i) IIy(t) 11 (E’ii) 11 z(t)11
< De-’(f-fo) 11 x ( t o ) 11
< De-””lo-f)11 x(to)11
for all t 2 to 2 0; for all to 2 t >, 0.
(g) [(h)] There exist v, v‘ > 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
<
(E”i) /I U(t)(U-’(t,)u- q(U-l(to)u))ll De-’(f-fo)lluII for all t 2 to >, 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),
+
<
<
(42.1)
II to) II
+
<
< II ~ i ( t o )II + II Zl(tn)II < (A + l)Ne-Yto II zi(0) II + II zi(to)II < ( ( A + I)NN‘e-(‘+V’)to+ 1) II II < p I1 Zl(tn) II . zl(t0)
Applying 1 l.A when y 1 # 0 (then z # 0, z1 # 0), and (42.1) in any case, we have, for all t 2 to 3 T ,
II 4 t ) I1 3
b”II zdt) II >, bo”-’eY’(t-tO)II
z&o) II
2 +p-lyoN’-lev’(t--to)I( z(to)11;
Ch. 4. DICHOTOMIES
112
hence (Eii,) holds with N i = 2pr;’N’ . If y , z # 0, application of (42.1) and, if necessary, of 1 1.A, yield for any t T
>
3W Y O so that (Eiii,) holds with y , = *P-?~.
(a,) implies (a). Let h > 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,
<
<
<
+
< <
For complemented subspaces we have a result analogous to 41.B:
42.B. A complemented subspace Y of X induces an exponential dichotomy for A if and only i f any one of the following equivalent conditions holds: (a) [(b)] There exist v, v’, N > 0, and for every [some] complement 2 of Y there exist N’ = ”(2) > 0, yo = y o ( Z ) > 0, such that any solutions y , z of (40.1) with y ( 0 ) E Y , z(0) E 2 satisfy (Ei), (Eii), (Diii). (c) [(d)] There exist v, v‘ > 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)
<
for all t 2 to 2 0; for all to 2 t 2 0.
Proof. Same as for 41.B (using Theorem 42.A instead of Theorem 41.A). 9,
If X is a Hilbert space, we make the same remarks as for ordinary dichotomies and obtain, in the same manner, a result corresponding to 41.C: 42.C. If X is a Hilbert space, the subspace Y induces an exponential dichotomy for A i f and only i f any one of the following reformulations of the definition, of 42.Al(b), of 42.A,(a,), of 42.B,(a), of 42.B1(b), respectively, holds:
42. EXPONENTIAL DICHOTOMIES
113
(a) [(b)] Either: Y # (0) and there exist v, v', N > 0, and for every [some]w , 0 < w &r, there exist N' = N ' ( w ) > 0, w 0 = w 0 ( w ) , 0 < wo &r, such that any solutions y , x of (40.1) with y ( 0 )E Y , a ( Y ,z(0)) 3 w satisfy (Ei), (Eii), and (Diii,,); or: Y = (0) and there exist v', N' > 0 such that every solution z of (40.1) satisjies (Eii).
<
<
Either: Y # {0} and there exist v, v', N , NL > 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,)
<
< w,
a ( y ( t ) ,z ( t ) ) 3
(Eiii,,,) or:
Y
<
=
w,
for all t
>
> 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").
<
Remark 1. Condition 42.A, (a,) is more interesting as a necessary than as a sufficient condition for an exponential dichotomy; for this reason we have not formulated equivalent conditions which would correspond to 42.A, (b), (c), (d) as 42.A,(a,) does to 42.A,(a) (the definition), for they are obviously implied by 42.A,(a,). These remarks apply correspondingly to 42.C,(a,). An analogous condition for ordinary dichotomies would have been a sufficient condition (same proof as in Theorem 42.A), but not a necessary one, as the case A = 0 in any X of dimension > 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
<
large t shows that v’ a, so that, replacing v’ by any smaller value, if necessary, we indeed have v’ < a , p‘ > 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
<
II~-”(T+~)
> 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
+
<
(42.2)
1 1< 1
+ (1 + + (1 -
E)%2
1
+
*
E ) W
Let z # 0 be any solution with Q ( X , , z(0))2 w , i.e.,
I z,(O) I cot w. For any t >, T ,
I zl(0) 1
<
43. DICHOTOMIES FOR
ASSOCIATE EQUATIONS
117
Therefore, for any t b to 2 0,
+
so that (Eii,) holds with Y’ = 1, Nk = (1 e)/(1 - E ) , T = T(w). However, consider the solutions y ( t ) = (e-I, 0), z ( t ) = (eft,h(t),e f ) , with y ( 0 ) EX,, Q ( X , , z(0))= $T: they satisfy tan Q (y(t),z ( t ) )= l/t,h(t), so that limt+, Q ( y ( t ) ,z(t)) = 0, and (Diii,) does not hold. &
Remark. We cannot choose q~ E M , on account of 42.D, but a slightly more elaborate argument shows that it is sufficient to assume q~ E L and ftl ~ ( udu ) 7 co to obtain an example with the same properties.
43. Dichotomies for associate equations Dichotomies for associate equations
L e t ’ X , X‘ be a pair of coupled Banach spaces. We shall relate the existence of a dichotomy, ordinary or exponential, of the solutions of (40.1) to the existence of a similar dichotomy of the solutions of the associate equation (32.1). T h e fundamental result is as follows:
Assume that A E L ( 2 ) has an associate. If the 43.A. THEOREM. subspace Y of X has the quasi-strict coupling property and induces a n ordinary [exponentiall dichotomy for A , then Yo induces an ordinary [exponential] dichotomy for -A’. Proof I (for Proof I1 see Section 66, p. 212). We carry out the proof for exponential dichotomies; the argument for ordinary dichotomies is the same. Before giving this “geometrical” proof in the general case, we wish to exhibit an “instant” proof in the special case in which Y is the nullspace of a a(X, X’)-continuous projection P (equivalently, Y is a member of an ( X , X’)-disjoint dihedron; by 12.C, Y automatically has the quasi-strict coupling property, and this situation certainly obtains if X is a Hilbert space). For this purpose, we observe that ( I - P)’is a projection along Yo; that, by 32.D, U‘-’ is the solution of the opera-
Ch. 4. DICHOTOMIES
118
tor equation (32.6) with value I‘ at t = 0; and that, applying condition 42.B,(d), we obtain, for appropriate v, v’, D > 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
<
45. NOTESTO CHAPTER 4
123
(40.1) with y(0) E X o , z(0) 4 X o satisfy (Ei) and (Eiia), (Eiiim), the latter two to T,(z(O));i f the codimension of X o is jinite (in particular, i f X is finite-dimensional), X,, induces an exponential dichotomy.
for all t
It is shown in Massera and SchafTer [3], Examples 3.3,3.4, that the assumption that A EM(X) in the converse part cannot be omitted, and that neither can the condition on the codimension of X o The former example is a variant of Examples 65.D-65.G (real scalar equations) ; the latter consists in the adjoint of the equation of Example 33.G.
.
CHAPTER 5
Admissibility and related concepts 50. Introduction Summary of the chapter
In this chapter we study not only the homogeneous equation (30.1), i.e., (50.1)
x
+ Ax = 0,
but most especially the nonhomogeneous equation (30.2), i.e., (50.2)
*+Ax=f
for given A E L ( x ) and varying f E L(X). The fundamental theme is the relation between certain “test functions” f and “nice” solutions of (50.2) for these f. T h e crudest expression of this theme is the notion of admissibility of a pair of classes of functions, both in L(X), namely, the class of “test functions” and the class of “nice” functions: the pair is admissible if for every “test function” f equation (50.2) has a “nice” solution. Section 51 deals with this concept of admissibility in the case (the only one developed in this book) in which both classes are Banach function spaces of the type described in Chapter 2. Such pairs of spaces are discussed later in this introduction. T h e fundamental result is a boundedness theorem, Theorem 51 .A. In order to understand the relevance of the concepts introduced in Section 52 and grasp fully their relationship to the main theme it is preferable to defer even a preliminary description until after Section 51. Sections 53 and 54 deal with the associate equations and will be more fully described then. Section 55 contains some results for finitedimensional X. 124
50. INTRODUCTION
125
In the remainder of this introduction and in Section 51 the domain of t is any interval J ; in Sections 52, 53, 54, 55 we shall specify it to be R , , a fact of which the reader will be reminded.
Pairs o f Banach function spaces
We shall be concerned with pairs (B, D), where B, D E b M ( X ) ; the term PAIR,or .,+'"-PAIR, without further specification, shall always have this meaning in the sequel. T h e pair (B, ,D1) is STRONGER THAN the pair (B, , D,), and the latter pair is WEAKER THAN the former, if B, is weaker than B, and D1 is stronger than D, (recall that these relations are not strict); by virtue of 21 .D, the relation is equivalent to the algebraic inclusions B, C B, , D, C D, . If (B, , D,) is both stronger and weaker than (B, , D,), then B, , B, are norm-equivalent, and so are D, ,D, (equivalently, B, , B, consist of the same elements, and D, , D, also consist of the same elements); we may say in this case that the pairs are EQUIVALENT.
Important special cases of pairs occur when the constituents belong to the narrower classes of function spaces described in Chapter 2. Typical of these are 9 - P A I R Si.e., , pairs (B(X),D(X))-which we shall always abbreviate to (B, D)-with B E b F , D E bSK; since we shall expect D(X) to contain nonzero solutions of (50.2), and perhaps of (50. l), it is preferable to require, from the outset, that D E bFK rather than D E b S . If also B E b F K, we say that (B, D ) is an S K - ~ ~ ~ ~ . Parallel to F-pairs we have ~ V - P A I R Spairs : (B(X),D(X))-also abbreviated to (B, D)-with B E b F % , D E b S K . It may seem paradoxical to make this requirement on D instead of taking it to be a space of continuous functions, but our choice turns out to present the least complications. If J = R, , we have some special classes of pairs involving translationinvariant spaces: F-PAIRS and F+-PAIRS (the former included in the latter) are F - p a i r s (B, D) with B E b y , B E b y + , respectively, and D E byK; F%+-PAIRSare 9%-pairs (B, D ) with B E bcT%+, D E b y K . If J = R,we similarly have 9-PAIRS,that is, 9 - p a i r s with B E b y , D E b.&, and .Y%-PAIRS,that is 9-%-pairs with B E b y % ? , D E b y K . Summarizing: Pairs (B(X),D(X)), abbreviated to (B, D): 9-pairs F--pairs .f%?-pairs
B Eb S BEbFK B E b.9W
i
D E bFK
Ch. 5. ADMISSIBILITY AND
126
RELATED CONCEPTS
For J = R,:
For J
=
7-pairs .P-pairs 3-W-pairs
BEh 9 B E b3-+ BEb S P
I
D E b&
9-pairs ,TV-pairs
BEb y B E bykp
1
DEbrK.
R:
5 1. Admissibility
Definition and boundedness theorem We consider Eqs. (50.1) and (50.2) for given A E L(x), the range of t being an interval J C R. Let ( B ,D ) be a pair (of Banach function spaces, in the sense defined in the Introduction); we say that (B, D ) is ADMISSIBLE, or, in full, ADMISSIBLE FOR A (sometimes, loosely, FOR THE EQUATION (50.2)), if for every f E B Eq. (50.2) has a D-solution. An immediate consequence of the definitions is that, if a pair is admissible, every weaker pair is also admissible. Our main result is then: 51 .A. THEOREM. If the pair ( B ,D ) is admissible for A , there exists a number K > 0 such that for every f E B and every number p > 1 there is a D-solution x of (50.2) with I x I,, pKI f I,, .
<
Proof. Let Y be the linear manifold of all D-solutions of Eq. (50.2) for all f E B, i.e., of all primitives x E D that satisfy i A x E B. T h e mapping l7:Y 4B defined by l7x = i Ax is linear and, by assumption, surjective. Since B, D are stronger that L(X),Theorem 31.D implies that the graph of l7 is closed in D x B. Since B, D are Banach spaces, the Open-Mapping Theorem implies (cf. Hille and Phillips [l]. Theorem 2.12.1) the existence of a number k > 0 such that for every f E B there exists x E n-lw)with I x I D kl f T h e conclusion then holds with K the infimum of all possible values of k. &
+
+
<
IB.
T h e value of K obtained in the proof is the least value that satisfies the statement. When necessary, it shall be denoted in full by K B , D ( A ) , but the subscripts and/or argument will be omitted when no confusion is likely.
Remark. It is a consequence of the selection theorem 10.D that, for given p > 1, the solution x in Theorem 51 .A may be chosen so that
5 1.
I27
ADMISStBILtTY
the mappingf-t x : B 3 D is continuous (and preserves multiplication by scalars). A more comprehensive result, involving the dependence on A , will be given in Theorem 71.B. We have not defined admissibility for function spaces that are not complete; however, the following lemma is sometimes useful:
5 I .B. Let the spaces F , G E . M ( X ) be given. Assume that there exists a number k > 0 such that for every f E F equation (50.2) has a solution x E G satisfying I x Ic k Jf IF . Then the pair (bF, bG) is admissible and K , , F . I , G d k.
<
P Y O O ~Let . f~ bF, p > 1 be given. By 21.G f is the L(X) limit of an F-Cauchy sequence; using also the formula (21.2) for JbF , we conclude that there exists a sequence (f,) in F such that IF ~ ( f l l , and ~ = f. Let x , be a solution of i8LAx, =f, such that x,EG, I x, Ic k l f , I F . Then 21.G implies the existence of x = lim7,+z 1 ;xi and the fact that x E bG, I x I,,c C: Ixn IG k Ifn IF p k l f lbF.By Theorem 31.D, x is indeed a solution of (50.2). S i n c e f c b F and p > 1 were arbitrary, the conclusion follows. 9,
If
xy+If,,
<
xyh
xr
<
<
<
<
We assume from now on, without loss, that 0 E J. We then have an addendum to Theorem 51.A:
51 .C. If the pair (B, D ) is admissible, there exists a number C 3 0 such that iff E B, p > 1, and x is a D-solution of (50.2) with I x ID < pKlf
IB >
Proof.
then
11 x(o)ll d pClf
IB
.
Let J‘ be any compact subinterval of J with O E J’, and (33.1) we obtain 11 x(O)(l p C l f where
<
p( J’) = 1 its length. From
C
=
IB,
+
(Kl-la(D; 1’) a(B; j ’ ) )exp(S, [I A(u) jl du). &
We shall always take C to be the smallest number satisfying the statement of 51.C; we denote it in full by CB,,(A), but shall again omit subscripts and/or argument when convenient.
Regular admissibility
In many situations, mere admissibility is not enough to derive further results. We therefore define a slightly more restricted concept: the pair (B, D) is REGULARLY ADMISSIBLE (FOR A ) if it is admissible and X,, is closed. For finite-dimensional X , admissibility and regular admissibility
AND Ch. 5. ADMISSIBILITY
128
RELATED CONCEPTS
coincide; this is no longer true even for a separable Hilbert space X, as Example 65.S will show. For regular admissibility we have a boundedness theorem that is sharper than Theorem 51.A. We now use, in addition to the numbers K , C defined above, the number S = SDintroduced by Theorem 33.C.
Assume that the pair (B, D ) is regularly admissible 51.D. THEOREM. f o r A. Then for every f E B and every A > 1 [and every ( X o D A)-splitting , q of XI there exists a D-solution x of (50.2) with (1 x(O)(( hd(X,,, ,x(0)) [with q(x(0)) = x(O)]; every D-solution x with these properties satisfies I x ID AK’l f 1s , 11 x(O)/l ACI f I B , where K‘ = K 2SC. Proof. T o prove the first part, we let q be an (XoD,A)-splitting of X . By the assumption, there exists a D-solution x’ of (50.2); let x be the solution of (50.2) with x(0) = q(x‘(0)). T h e n q(x(0)) = x(O), and x - x’ is a solution of (50.1) with x(0) - x‘(0) E X,, , so that x - x’ E D ; hence x E D. For the proof of the second part, let x be a D-solution of (50.2) with 11 x(0)ll < A d(X,,, x(O)), and let p > 1 be given. By Theorem 51.A there exists another D-solution x’ with I x’ 1, p K l f IB , 11 x’(0)ll pC(f . Now x - x’ is a D-solution of (50.1); therefore x(0) - x’(0) E X,, , so that (1 X(O)ll A d ( X m 3 X(0)) = A ~ ( X O X’(0)) D AII x’(0)II f IB By Theorem 33.C, I x - X’ ID < SJIx(0) - x’(0)ll < ( 1 A)pSCI f Is. Thus I I D < I ID I ID < P ( K (I h ) s c ) ( f I B < pXK’lf(B* Since p > 1 was arbitrary, the conclusion follows. &
<
< +
<
<
<
9
+ ’-
<
+ +
Remark. If x(0) = 0, the proof shows that I x
+
< <
I D < ( K + S C ) I f IB.
51.E. Assume that the pair (B,D) is regularly admissible and that X,,, is complemented; let Z be a complement of X,, . For every f E B there exists exactly one D-solution x of (50.2) with x(0) E Z , and this solution satisfies I I D < max{ P 11 ?I 11) ’ K’l f )B 11 x(o)lI < 11 11 f ) B , where I? is the projection along X,, onto Z and K’ is as in Theorem 51.D. 9
Proof.
ll.F. and Theorem 51.D.
&
Admissibility and local closure We shall meet in the course of our work several results of the general tenor: “if such-and-such a pair is admissible, and ..., then such-and-such another pair is also admissible”. A first, and trivial, instance was the remark that the admissibility of a pair implies the admissibility of all weaker pairs. We prove here some less trivial theorems of this kind.
51. ADMISSIBILITY
129
51 .F. THEOREM. Assume that the pair (B, D) is admissible. I f there exists a subspace Y of X such that Y C X,, and such that the quotient space X / Y is reflexive-in particular, if X itself is repexive, or if X,, is closed and X/X,, is reflexive-then (lcB, IcD) is also admissible. Proof. Set K = K B , D , C = C B , D , let S y= S y , be the number defined by Theorem 33.B, and choose an arbitrary p > 1. Let f E 1cB be given and set u = I f llcB . There exists a sequence (f,), f,, E B, If , u, such that limnjm f , = f . By Theorem 51.A and 51.C there exists a D-solution x, of f, Ax, = f , such that I x, 1, pKI f , pKu, 11 x,(O)II pCu. It follows that, in the quotient space XjY, 11 x,(O) Y (1 pCu. Since X / Y is reflexive, we may select a subsequence of (f,) (which we relabel (f,)) such that ( ~ ~ ( 0Y)) has Y, where x, may be chosen so that a weak X/Y-limit, say x, (1 x , (( p2Co. Let x be the solution of (50.2) with x ( 0 ) = x,; we claim that x E 1cD. Since (x,(O) Y) converges weakly to x(0) Y in X/Y, there Y) converges (strongly) in exists a sequence (y,) such that (y,(O) X / Y to x(0) Y, where y, is, for each n, a finite convex combination of the xj ,j >, n; thus y, E D ; Iyn ID pKo, II yn(o)ll pcu; and j, Ay, = g, ,whereg, is the corresponding finite convex combination of the f j ,j >, n; therefore g, E B, Ig, la u, and 1imnjm Lgn= f . Since (y,(O) Y) converges to x(0) Y, there exists a sequence (z,,) in X, z,, - y,(O) E Y, such that 1imnjm z,, = x(0). Let z, be the solution of S, Az, = g, that satisfies z,(O) = z,,. Since z,(O) - y,(O) E Y C X,, , z, - y, is a D-solution of (50.1) for each n. Therefore z, E D and, applying Theorem 33.B,
IB < Is <
+
+
<
<
<
+
+
<
+
+ < < +
+
+
+
+
<
+
I zn ID d I Y ,
+I
+
d PKO SYII zn(0) - Y J O ) II < P(K + S Y C b + S Y II zn(0)II ID
~n
- Yn ID
1
whence
On the other hand, applying (31.5) to z,, - x, g, - f , and to = 0 for every compact interval J’ C J containing 0, we conclude that z, --t x uniformly on each such interval; a fortiori, 1imnjm z, = x. By 21.E, it follows that x E lcD, as claimed, and that
< P(K -t- (1 + p ) s Y c ) If t I r B . Since p > 1 was arbitrary, KlcB,lcD < K + 2SJ. & I
llcD
Ch. 5. ADMISSIBILITY AND
130
RELATED CONCEPTS
If D E bSK(X),we can prove a sharper result, which is then obviously applicable to 9 - p a i r s and 9%-pairs: Assume that the pair (B,D(X)) is admissible, where 51.G. THEOREM. D E b F K . If there exists a subspace Y of Xsuch that Y C X,, and such that the quotient space Xi Y is rejlexive-in particular, if X itself is rejlexive or if X,, is closed and XjX,, is reflkxive-then (lcB, fD(X))is also admissible. In particular, if the $-pair or S V - p a i r (B, D) is admissible and there exists a subspace Y as described, then the %-pair (lcB, fD) is also admissible.
Proof. I n the proof of Theorem 51.F we found that z, -+x uniformly on each compact interval J’ C J. For each such J‘ we have, for every n,
I XJIx
ID
< I XJ’%
ID
f
I XJ’(l;n
-
+I we find Ix p x ID < lim “1 ID
taking the limit superior as n 4 00 (1 p)SyC) lloB for each
p(K
+ +
If
ID
X J ’ ID
I XJ’(’%
I;
-
I z,
J’. Therefore x E fD(X). &
ID <
00
Remark 1. If X,, is closed, a subspace Y CXoDsuch that X / Y is reflexive exists if and only if X/X,, itself is reflexive: the “if” part is obvious; and if such a Y exists, the quotient spaces ( X / Y ) / ( X , , / Y )and X/X,, are congruent, and the former is reflexive. If in addition X,, is complemented, any complement is naturally isomorphic to X/X,, , so that the reflexivity of this quotient space is equivalent to the reflexivity of the complement.
00
Remark2. Under the assumptions of Theorem 51.F [51.G] it does not follow that the regular admissibility of (B, D) implies the regular admissibility of (IcB, 1cD) [(lcB, fD(X)); or (IcB, fD)], not even if X is a separable Hilbert space (SO that X,, is complemented): this is illustrated by Example 65.P.
00 0000
We have not been able to remove the reflexivity assumptions in the preceding results, although there are some very special situations in which they can be omitted (e.g., Theorem 51.1, Theorem 62.J). We therefore formulate:
00 0000
Query. Is the assumption of the existence of Y with the stated reflexivity properties redundant in Theorem 51.F? In Theorem 51.G? In the case of %-pairs or %%-pairs 7
coca
Under the assumptions of Theorem 51.F or Theorem 51.G it does not follow that (lcB, D ) is admissible (see Example 65.B); for 9 - p a i r s and 9%-pairs, Theorem 51.G suggests that this failure is due to behavior at the “open ends” of J : it is necessary t o “fill” D to f D to restore
51. ADMISSIBILITY
131
admissibility; we might try instead to “thin out” 1cB at those open ends; and we indeed have the following theorem, in which no reflexivity condition is required:
51.H.THEOREM.Assume that the *-pair or *%-pair (B,D) is [regularly] admissible; then the F - p a i r (klcB, D) is also [regularly] admissible.
, Proof. T h e result on regular admissibility is a trivial consequence of that on admissibility. T h e proof of the latter for J = R, is given in Section 52 (p. 143), for J = R in Section 81 (p. 278); the case of any noncompact J reduces to one of those by an appropriate change of independent variable; the details are left to the reader. If J is compact, every -9-pair or *%-pair (B, D) is regularly admissible, with X,,, = X, since D E b F K and therefore D(X) contains all bounded continuous functions on J, in particular all solutions of (50.1) and of (50.2) for any feL(X); in this case the theorem is trivial. & a2 ma2
03 0003
To conclude this subsection, we state a theorem of the same type as Theorems 51.F, 51.G, without reflexivity assumptions, but only for an adjoint equation, and for .F-pairs or 9tV-pairs. For reflexive spaces this result is superseded by Theorem 51.G. The theorem will not be used in the sequel. Assume that X is separable and that the *-pair or 51.1. THEOREM. 9%-pair (B, D) is admissible for -A* in X*. Then (IcB, IcD) is also admissible for -A*.
Proof. For J = R, , in Section 52 (p. 143); for 1 = R, in Section 81 (p. 278); the proof is completed as for Theorem 51.H. & a2
Query. Is the assumption that X be separable redundant in Theorem 51.1 I
0303
Some remarks on the admissibility o f .%pairs and related pairs on R , Due to the peculiar one-sided nature of the classes Y+,9-V+and their awkward properties (e.g., their not being sublattices of S,.F%) it is desirable to find cases in which the admissibility of a F + - p a i r or F%+-pair can be replaced by that of a .F-pair, ideally a stronger one. No new ideas are involved in the following results; in all of them we assume J = R, .
Ch. 5. ADMISSIBILITY AND
132
RELATED CONCEPTS
51. J. Let (B, D ) be a given F+-pair. I f the .F+-pair (O,B, D ) is [regularly] admissible for some T >, 0, then (B, D) itself is [regularly] admissible.
+
Proof. For given f E B(X), let y be a D-solution of j Ay = @,f. Let x be the solution of (50.2) that satisfies X(T) = ~ ( 7 ) T. h e n x, y are continuous functions that coincide outside [0, T ] . By 22.1, x E D(X). &
In motivating the following result, we recall that, if B E b 9+, the infimum of all spaces F E b.F such that F 2 B is given by bT-B (cf. 23.H); for B E b9-V+, the corresponding space is bT-eB. 51 .K. If the F+-pair or Y V + - p a i r (B, D) is [regularly] admissible, then the F - p a i r (kT-lcB, D ) is [regularly]admissible. I f B is lean, or ;f D is quasi full and there exists a subspace Y of X s u c h that Y C X,, and X / Y is reJEexive, the 3 - p a i r (bT-B, D ) or (bT-eB, D) is [regularly] admissible. Proof. (This depends on Theorem 51.H, the proof of which is still incomplete). By Theorem 51.M, the F + - p a i r (klcB, D ) (cf. 24.1, 24.L) is admissible. But by 23.1, if so = so(B) we have klcB = kOSoT-lcB = @.+kT-lcB, and, since T-lcB is locally closed, kT-lcB E b y . By 51.5, (kT-lcB, D ) is admissible. If B is lean, we have, for B E Y+,bT-B = bT-kB = bkT-B < kbT-B klcT-B = kT-lcB (23.H, Remark to 22.G, 23.1); if B E .FW, similarly, bT-eB = bT-ekB bT-keB < kbT-eB klcT-eB = kT-lceB = kT-lcB (24.C plus the previous argument, and 24.1). T h e conclusion follows. If D is quasi full and the reflexivity assumption holds, (1ckT-lcB, D ) = (T-lcB, D ) is admissible (22.G, 23.1, and Theorem 51.G) since D, fD are norm-equivalent, and bT-B < 1cT-B = T-lcB, or bT-eB < 1cT-eB = T-lceB = T-lcB. &
<
Sets
bf
<
<
admissible pairs
We collect a few results concerning the structure of the sets of admissible and of regularly admissible pairs for a given A EL@); further results will appear in later sections and chapters (e.g., 52.P, 52.Q, 52.S, 63.0, 63.P, 64.D,64.E). The first theorem is a generalization of the obvious fact that, if (B,, D), (B,,D) are both admissible pairs, then (B,v B,,D) is also admissible.
51.L. THEOREM. Let D E b.N(X) be given, and let 9? C bN(X) be a non-empty class of spaces such that (B,D ) is [regularly] admissible fm e w ~ y
51. ADMISSIBILITY
I33
B E a. Then there exists B, E b.M(X), weaker than each B E a, such that (B, , D ) is also [regularly] admissible, provided there exists some space E E X ( X ) weaker than each B E 99; more precisely, there exist numbers uB > 0, B E 9, such that F = V{u,B : B E .@} exists and B, = bF.
<
Let uB be so small that uBKB 1 Proof. Set KB = KB,Dfor each B €9. and uBB E for each B E B ;the last condition ensures the existence of F as defined in the statement (and F E). Then B, = bF E bX(X) and uBB F B, , so that B, is weaker than each B E 9.It remains to prove that (B, , D ) is admissible. T h e statement concerning regular admissibility follows trivially. Let f E F and p > 1 be given. For some n there exist spaces B,, ..., B, and functions f i , ...,h, with f , E B, , i = 1 , ..., n, such that f = f, and Ifi let plf IF . Since (BE, D) is admissible, there exiits, by Theorem 51.A, a D-solution x, of k, A x , = f t with Ix, ID pKB,lfzI, p o i f l f , l B t , Z' = I, ..., n. But then x = X, isaD-solution IX, ID P u i ; I f t lBtl d $1 f IF T h e of (50.2) with I x ID d admissibility of (B, , D) follows from 51.B, where G = D = bD. &
< < <
<
c"
<
1:
<
+
C:
<
c:
En
<
-
51 .M. Assume that J = R , or J = R . Let D E b y K be given. There exists B, E b c T such that a ,T-pair (B, D ) is [regularly] admissible and only if B is stronger than B, , provided some such [regularly] admissible .Fpair exists, namely, provided (T, D) is [regularly] admissible.
Proof. We apply Theorem 51.L with E = M(X) (an account of 23.G) to the class of all B(X), B E by-, such that (B,D ) is admissible. We use the fact that F ( X ) is a sublattice of N ( X ) , and the properties of b for 9-spaces (22.A, 23.B, 22.S, 23.C); the final remark will follow from Theorem 52.K. & Remark. T h e corresponding statement holds, if J = R, for .FV-pairs; we merely use E = C ( X ) (24.M,(2)), and replace B, by B, A C ( X ) . However, the proof fails for J = R, and .F+-pairs or .TW-pairs, since .T+,.FW are not sublattices of N ;and the argument of the proof of Theorem 51.L does not go through if we use the special supremum (Remark to 23.B) instead of Results may be obtained, however, via 51.K, either if D is quasi-full and there exists a subspace Y C X,, as specified in 5I.K, or if we restrict ourselves to lean spaces B; their formulation is left to the reader.
v+
v.
Query. Does 51.M remain true with J = R, , B, pairs? With B, E by%?+ and P P - p a i r s ?
E b.T+,
and
Y+-
Theorem 51.L suggests the following question: suppose we are given
B,
I34
Ch. 5. ADMISSIBILITY AND
RELATED CONCEPTS
and a class of D’s such that all the pairs (B, D) are admissible; is there a D o , stronger than all the D’s such that (B, Do) is also admissible? T h e situation here is not so simple as for the opposite problem dealt with above. For one thing, we have to restrict our attention to regular admissibility (no loss if X is finite-dimensional); for another, even then the answer is not an unqualified yes. The most general result we have is: Let B E b N ( X ) be given, and let 9 C b.M(X) be a 51.N. THEOREM. nonempty class of spaces such that (B,D) is regularly admissible for every D E 9.If X,, Dl D,) = XoD,for any D,, D, E 9, there exists DoE b.N(X), stronger than each D EB, with XoD0= XoD, and such that (B, Do) is regularly admissible. More precisely, there exist numbers uD > 0, D €9, such that Do = ~ { u D D :D ~ 9 } .
Proof. On account of the assumption there exists a subspace Y of X such that X,, = Y for all D €9. For any f E B and any D, ,D, € 9 ,let x1 , x2 be solutions of (50.2) with x i ED^ , i = 1, 2. Then x1 - x, is a D, v D,-solution of (50.1), so that x,(O) - x2(0) E XO(D,VD,) = XoD,= Y ; therefore x, - x, ED, , whence x2 E D,; and similarly x, ED, We conclude that for given f E B the set of D-solutions of (50.2) is the same for every D €29; we call these solutions 9-solutions. Set Kb = KB,D 2SDCB,, and uD = max{S, , KA} for each D €9. We then define Do as in the statement, so that DoE b N ( X ) (by 21.C) and Do is stronger than each D € 9 . On account of this last property, XoD,C Y. Conversely, let y be any solution of (50.1) with y(0) E Y (a %solution). By Theorem 33.C, uG1ly 10 ~~;~&IIy(o)11 < IIy(o)ll for every D €9, SO that y ED, ; hence Y C XoDo , and equality holds. For a given f E B and h > 1 let x be a %solution with 11 x(0)ll h d( Y , x(0)); such a solution exists by Theorem 51.D, and by the same theorem it satisfies u;’I x ID huG’KbIf la A1 f lB for all D €29, so that x ED,. Since f was arbitrary and XoD,,= Y, the pair (B, Do) is regularly admissible. &
.
+
<
<
<
<
The following example shows that the assumption “XOIDI D,) may not be weakened to “XOD, = XOD2”.
9,
= XoD,
51.0. EXAMPLE.Let J = R , or J = R. Let p, be any nonnegative, continuous, continuously differentiable, unbounded function in L’, with ~(0) = 0. Let (50.2) be the equation in F (the scalar field) 2 - (1
+ p,)-lfjx= f.
51. ADMISSIBILITY
135
+
The general solution of (50.1) is x = (1 p)x(O). We consider D1= L“(F), D, = L1(F); then XoD,= XODe = {0}, XO(DIVD,) = F = X. For B we take the set of continuous functionsf = u( 1 y~)-lqj, u E F, with If Ir, = lul. For suchf, (50.2) has the general solution x =up (1 p)x(O). For x(0) =0, x E D,; for x(0) = -u, x ED,; thus both (B, D1) and (B,D,) are (regularly) admissible; but if u # 0 there is clearly no solution in D, AD, &
+
+ +
.
In the preceding example, D, and D, are (locally closed) spaces in b.FK-actually in b y K . This gives rise to the following question: is the conclusion of Theorem 51.N true for 9-pairs or 9‘V-pairs if the assumption “XO(DIVDp) = XOD,” is replaced by “XoDI= XoD, ”. ? The answer depends very much on the nature of J: if J is compact, it is a trivial yes, since every *-pair or SV-pair (B, D) is regularly admissible with X,, = X (cf. proof of Theorem 51.H); if J contains one endpoint (so that, under a change of variable, we may suppose that J = R+), there are partial affirmative answers (e.g., 52.P, 52.Q; see also the Query following 52.S); if J contains no endpoint (so that we may suppose J = R), the answer is an emphatic no: indeed it remains negative if we restrict ourselves to 9--spaces, to a set 9 of two elements, to dimension 1 and to A = 0: 51.P. EXAMPLE.Assume J = R, X = F (the scalar field) and A = 0. Let D-, Df be the subspaces of L“ consisting of those p E L” that satisfy lirn,+-“ y(t) = 0, lirn,,,, p(t) = 0, respectively. We have XoD-= Xon+= (0); the F-pair (Li, D-) is admissible, since for any f~ L1(F) t f(u) du of (50.2) is in D-(F); similarly, the the solution x ( t ) = but D- A Df = L:; and (Ll, L:) is not F-pair (Ll, D+) is a&&sible; admissible, since for anyfcL1(F) and any solution x of (50.2) we have m lirn,,,, .$t) - 1imt+-, x ( t ) = s-,f(u) du, and this is # 0 for some fEL’(F). 9,
s
Inadmissible pairs The question may be raised, whether a given pair is admissible for some A E L(x). This question is not very interesting for a compact J (we have already noted that in this case every 9 - p a i r or 9%’-pair is always admissible-see proof of Theorem 51.H). We mention here only one interesting negative result for J = R , or J = R; there are other pairs, even .F-pairs, that are similarly “absolutely inadmissible,” but we have not been able to give a complete characterization. L): is not admissible for any A 51.Q. The .F-pair (M, F-pair (L“, L): is not admissible for any A E M(8).
E L(x);
the
Ch. 5. ADMISSIBILITY AND
136
RELATED CONCEPTS
Proof. Let A E L(X) be given, and let U denote the solution of (30.3) with U(0) = I. For a given constant K > 0 and every n = 1,2, ... we 1 so small that, if Jn = [n,n l,], then choose a number I , , 0 < 1, jJn11 A(u)ll K , n = 1, 2, ... For some fixed xo E X , 1) x,, 11 = 1 we set
f(t)=
+
< . I;lU(t)U-l(n + fn)xo,
<
tE
In,
71
= 1, 2,
...
00
lo,
t$$Jln.
By (31.9), t E Jn implies
Ilf(0ll< l;'ek.
(51.1)
Jyl
<
Therefore IIf(t)ll dt = JJn Ilf(t)l[dt ek,n = 1, 2, ..., so thatfE M(X). If A EM(^), we might have taken 1, = 1, k = I A l M , and (51.1) then implies f E L"(X). Let y be any solution of (50.2) for thisf. By (31.3) we have, for every n, y(n
+ In)
= =
+ +/ U(n' + /n)U-l(n)y(n) + ~ ( n fn)U-l(n)y(n)
~0
J*
u(n + ~ n ) ~ - l ( u ) f (duu )
9
whence, using (31.9), 1 = II xo I1
< II Y(n + 4l)Il + II Y(4ll ek.
Since this holds for all n, y $ L:(X).
&
We shall give later certain large classes of F-pairs, Y+-pairs, and YW-pairs (including all pairs formed by Orlicz spaces) that are always admissible for some A E L(x)(Examples 65.N, 65.0, with their presentation).
Equations with scalar A on
R,
We prove a simple result, useful in the construction of examples, for the case when J = R, , A = T I , g, E L.
51.R. Assume that J = R, , A
=
TI,g, E L. Set
51. ADMISSIBILITY The 9 - p a i r or 9 % - p a i r (B, D ) is admissible for A ( 1 ) [ E D, X,, given by
=
(51.2)
X , and for every f .r(t) =
E
137
if and only if either
B(X) the solution x of (50.2)
5 ( t ) jt E-'(ulf(u)
du
0
belongs to D(X); or (2) [ $ D , X,, given by (51.3)
=
{0}, and f o r every f~ B(X) the solution x of (50.2) x(t) =
jm E-'(ulf(u) du
-EP)
exists and belongs to D(X). Proof. By direct verification, U = [I is the solution of (30.3) with U ( 0 ) = I . Therefore X,, = X or = {0}, according as 4 E D or E $0,by (31.2). I t follows from (31.3) that (51.2) and (51.3) (the latter if meaningful) define solutions of (50.2). T h e "if" part of the statement then holds trivially, and we proceed to prove the "only if" part. We assume that (B, D ) is admissible. If 4 E D , X,, = X, every solution of (50.2) is a D-solution, in particular the one given by (51.2); this disposes of case (1). We therefore assume that 5 I$ D, X,, = (0). Let f~ B(X) be given, and choose some x, E X , 11 x, I( = 1. Since llfll x, E B(X), there exists a D-solution y of j A y = l l f l l x, which, by (31.3), satisfies
+
A t ) = &)jr(O)+ (jt0 5-Yu) IImII q x , l
(51.4)
.
Now J/ t-'(u)llf(u)]l du, being nondecreasing, has a finite or infinite limit as t -+ co; therefore 11 y(0) (J: [-l(u)llf(u)l\du)x, 11 also has afinite or infinite limit as t + co.If this limit is > 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). &
+
<
Jy" <
51 .S. Under the assumptions of 5 1 .R, the 9 - p a i r or F%-pair (B, D) is admissible for A if and only $, in R instead of X , (B, D ) is admissible for v.
Proof. We apply 51.R to both A = T I and v, and use the fact that if x, E X , 11 x, 11 = 1, then E B if and only if #x0 E B(X); and similarly for D and D(X). &
+
138
Ch. 5. ADMISSIBILITY AND
52.
RELATED CONCEPTS
(B,D)-manifolds
Summary of the chapter (concluded) We are now in a position to resume our interrupted sketch of the contents of the chapter. We specify once and for all that in the remainder of this chapter the range of t is R, . All pairs will be 9-pairs or 9%-pairs. In attempting to apply the concept of admissibility of an 9 - p a i r or 5%-pair (B,D) to the investigation of the structure of solutions of (50.1) and (50.2), as well as of the associate equations, it happens that often only those frsB(X) are needed that have compact support (i.e., f E k,B(X)); the full admissibility of the pair is used only to provide, via Theorem 51.A, a bound for D-solutions of (50.2) for such f. Since k,B(X) is not complete, the use of the whole space B(X) can be avoided only at the price of additional assumptions. Incidentally, if X is infinite-dimensional, it often happens that, even assuming the admissibility of (B, D), nothing can be done because X,, is not closedan additional reason for seeking a subtler approach. It is desirable to construct a concept that will embody those elements of the described situation that are relevant to the intended analysis of the structure of the equations. As one of the relevant technical details, we observe that, iff has compact support, the D-solutions of (50.2) are characterized precisely by their coinciding, for large t, with D-solutions of (50.1) (cf. 22.I), i.e., solutions of (50.1) that start from X,,, at t = 0. The concept we seek will be that of a (B, D)-manifold (a (B, D)-subspace is a closed (B,D)-manifold), which replaces X,, in the role it plays above. Briefly, a (B, D)-manifold is a linear manifold Y contained in X,, with the following property: for every f E k,B(X) there exists a solution x of (50.2) which coincides for large t with a solution of (50.1) that starts from Y at t = 0, and is such that I x 1, k l f , where k > 0 does not depend on f (the fact that x is a D-solution follows from Y C XoD). The artificiality involved in the definition is amply compensated by the way in which the new concept clarifies and organizes the subsequent developments. The present section gives a precise definition, the main properties (one of the most important being that a (B,D)-manifold is also an (lcB, D)-manifold) and the connection with admissibility; we note, in particular, that the admissibility of (B, D) implies that X,, is a (B, D)manifold. Interest will later center on (B, D)-subspaces, however. (B,D)-manifolds and -subspaces are a very keen tool (as contrasted with admissibility) for studying the relations between Eqs. (50.l), (50.2), and the associate equations in a space X' coupled to X. In section 53 we
<
IB
52. (B, D)-MANIFOLDS
139
consider the implications for the associate equations of the existence of a (B, D)-manifold for (50.2).Section 54 deals with the more important case of (B, D)-subspaces. A typical result is: Let (B, D ) be an TK-pair or a 9 - p a i r ; i f Y is a (B, D)-subspace, then Yo is a (D’, B')-subspace for the adjoint equation; the converse holds i f D is quasi locally closed. T h e use of the adjoint equation also yields some interesting facts about Eq. (50.2) itself; these results are collected in a subsection of Section 54. Special results for finite-dimensional X, or special forms of the previous results for this case, are collected in Section 55; the most conspicuous is: Let ( B , D ) be an 9 , - p a i r or a Y - p a i r ; if ( B , D ) is admissible, then (D‘, B’) is admissible for the adjoint equation; the converse holds if D is quasi locally closed.
(B, D)-manifolds We are working with a given A E L(x). Assume that f E L(X) has a compact support, and set s = s ( f ) (cf. Section 20, p. 39). If x is any solution of (50.2),we shall always denote by x, , or, in full, xmA, that solution of (50.1)that satisfies x,(s) = x(s); obviously, xm(t) = x ( t ) for all t s. If D E b S K , then x E D ( X ) if and only if x, ED(X), i.e., xm(0) E X O D (by 22.1). All pairs occuring in this section shall be 9 - p a i r s or 9 g - p a i r s . We consider a given one, (B, D). A linear manifold Y C X is called a (B, D)-MANIFOLD (FORA , or, speaking loosely, FOR THEEQUATION (50.2))if Y C XoDand there exists a number k > 0 with the following property: for every f E k,B(X) there exists a solution x of (50.2)such that xm(0) E Y (whence x E D(X)) and I x ID kl f I B . We denote the infimum of all possible values of k by K,, or, in full, K Y B , D ( A ) , with the usual convention about dropping subscripts and arguments; clearly, if B # {0}, K , > 0. A closed (B,D)-manifold is called a (B, D)-SUBSPACE.
<
52.A. If Y is a (B, D)-manifold, there exists a number C , >, 0 such that i f f € koB(X), p > I , and x is a solution of (50.2)with x,(O) E Y and pcYlf IB ‘ I I D P K Y l f l B then 11 x(0)ll
<
3
<
Proof. Same as for 51.C. & We shall let C,, or, in full, CyB.D(A),denote the smallest number satisfying the statement of 52.A. There are several results of the general form: “if Y is a (B,D)manifold, and ..., then 2 is a (B,, D,)-manifold”; some rather trivial
Ch. 5. ADMISSIBILITY AND
140
RELATED CONCEPTS
instances are collected in the following lemma; others will be given in Theorem 53.D, 53.1, Theorems 54.F, 54.G, 54.1, etc.
52.B. Assume that Y is a (B, D)-manifold. Then: (a) Y is a (B, , D,)-manifold for any 9 - p a i r or F F - p a i r (B, , D1) (such that (kB, , D,) is) weaker than (B, D ) ; (b) any linear manifold 2 such that Y C 2 C X,, is also a (B, D)manifold; in particular, so is X,, itself; (c) if Y C X,,,, , Y is a (B,kD)-manifold. Assume in addition that Y is closed, i.e., a ( B ,D)-subspace, and that 2 is a linear manifold, Z C Y. Then: (d) $ 2 is dense in Y , 2 is also a (B,D)-manifold; D A D1)(e) i f 2 is a ( B ,D,)-manifoldfor some D , E b g K ,then 2 is a (B, manifold.
Proof of (a). There exist numbers jl,6 > 0 such that kB, PB, D 6 D l . I f f E kpBl(X)= k,kB,(X), then f E k,B(X); for given p > 1 there exists a solution x of (50.2) with xm(0)E Y C X,, C XoD, ID psrcYlf IB d p P G K Y 1 f l B l * and I ID, d K,. Proof of (b). Trivial; obviously K , Proof.
<
<
<
Proof of (c).
<
Trivial.
.
Proof of (d). Firstly, 2 C Y C X,, Further, l e t f e k,B(X) and p > 1, and let x be a solution of (50.2) with x,(O) E Y , I x ID pKyIf IB. Since 2 is dense in Y , there exists a solution y of (50.1) such that xm(0) y(0) E 2 (whence y(0) E Y ) and 11 y(0)ll (p - 1) If IB NOWz = x y is a solution of (50.2) with z, = x, y, so that zm(0)E 2,and I z ID I x ID
<
+
1-Y
ID
d fKY
If
IS
f 'YIIy(0)ll<(PKY
+
.
<
+
+ <
+
(f-l)SY)IfIB(whereS,=SYD
is the constant of Theorem 33.B). We find K , equality holds.
< K y ,but by part (b)
Proof of (e): Set K,, = KZB,*,, Czl = CZB,,,. With f, p, x as in the proof of (d), let z be a solution of (50.2) with zm(0)E 2, I z ID, PKzllf lB so that 11 z(o)/l Pczllf IB * Then y = - * = zco - x, is a solution of (50.1) with y(0) E Y and IIy(0)ll 11 x(0)ll 11 z(O)11 p(Czl Cy)If I B . It then follows as above that
<
1
I
ID
hence z E ( D A D l ) ( X ) ,
< d K Y +1Z'(Y'
+
<
+
+
'Y))lflB
;
< <
52. (B, D)-MANIFOLDS
141
A fundamental result is the following, which in particular often allows us to disregard the case of an 9%‘-pair in dealing with (B, D)manifolds. 52.C. THEOREM. A linear manifold Y C X is a (B, D)-manifold and only if it is an (lcB, D)-manifold.
if
Proof. T h e “if” part is trivial by 52.B,(a). Assume that Y is a (B, D)manifold, and recall that (lcB,D) is an 3 - p a i r (by 22.S, 24.1). Let f E kolcB(X) and p > 1 be given and set s = scf). By 22.C and 22.S, or the proof of 24.1, there exists g E B(X) such that 11 g 11 11 f I( (whence g E KoB(X)with s(g) s) and Ig IB If I I ~ Band
<
<
<
+
By the assumption, there exists a solutiony o f j Ay = g withy,(O) E Y and I y ID pKyl g IB pKyl f llcB. Let x be the solution of (50.2) that satisfies x(s) = y(s), so that x, = y, , xm(0)E Y. We apply (31.5) to x - y , f - g in [0, s], with to = s; using (52.1) and the fact that x and y coincide outside [0, s] we find
<
<
+
therefore I - y ID < ( p - l ) l f ( I C E ; thus I ID Iy ID I - y ID ( p K , ( p -. 1))lf llcB . T h e conclusion follows with KYlcB\< K , = K,; the reverse inequality follows from the proof of 52.B,(a), so that equality holds. &
<
+
We investigate cases where all
<
(B,D)-manifolds are dense in X o , .
52.D. Assume that Y C X is a (B, D)-manifold. If x is a solution of (50.1) with x(0) 6 clY, then ~ 1 x1 11-l E L’for every q~ E B (i.e., 11 x 11-l E B’). Proof. On account of 52.C there is no loss in assuming that B E b 9 and is locally closed. Set 6 = d(clY, x(0)). Let q~ E B , p > 1, 7 3 0 be given. Set f = xr0,.]l q~ 1 sgn x E koB(X). By the assumption there exists a solution y of (50.2) such that ym(0)E Y, 11 y(0)ll pC,lf IB pCylp,lB. by 31.B,(b), another solution z of (50.2) is given by z ( t ) = x ( t ) Jo xto,71(u)l d U ) I II 4u)Il-l du; and z m = x J; I T ( 4 l II Nu)Il-l du.
Tt
<
<
142
Ch. 5. ADMISSIBILITY AND
RELATED CONCEPTS
Thereforey - z is a solution of (50.1), and we have y so that
-
z = Y,
- Z, ,
since T >, 0, p > 1 are arbitrary, we conclude that J: I v(u)J1) x(u)ll-' du d 6 - 1 C , I ~I,, so that ~ I I I ~ I I - ~ EIfL B ~ .E b F K or B Eb y , we find 111 I[-' 1,' * *e
<
Assume that D is thin with respect to B (inparticular, 52.E. THEOREM. that (B,D) is a Y - p a i r , OY F + - p a i r , or Y V + - p a i r , not weaker than (L1, L:))! If Y C X is a (B, D)-manifold, then Y C X,, C clY. Proof. If x is a solution of (50.1) with x(O)$clY, 52.D and the assumed thinness of D imply that 11 x 1) 4 D , whence x(0) $ X,, . & For 9-pairs, etc., see also Theorem 52.N. 00
Remark. We cannot require Y = X,,, = clY in Theorem 52.E unless X is finite-dimensional: indeed, if X is infinite-dimensional and there exists any (B,D)-manifold, there always exists one that is not closed, by 52.B,(d); we can then always choose Y in such a way as to make one of the equalities fail. The fact that X,, need not be closed will be illustrated in Example 65s.
(B, D)-manifoZds a n d admissibility We mentioned at the beginning of this section that there is a close connection between the existence of a (B, D)-manifold and the admissibility of ( B , D ) or of some related pair. T h e central result in this connection is the following:
52.F. THEOREM. If(B, D ) is admissible, then X , , is a (B, D)-manifold. Conversely, if there exists a (B, D)-manifold, then (kB, D) is admissible. Proof. Assume that (B,D ) is admissible and set K = K B , , . For givenfE k,B(X) and p > 1 there exists, by Theorem 5 1 .A, a D-solution xof (50.2)with Ix ID p K J f I,; it follows that x, ED(X), i.e., x m ( 0 ) ~ X o D ; K. thus Y = X,, is a (B, D)-manifold with K , Assume conversely that Y is a (B, D)-manifold with the constant K , . For anyf E k,B(X) and p > 1 there exists by the definition a D-solution x of (50.2) with I x 1, pK,I f . Since bkoB(X) = kB(X) (22.G or 24.C), the admissibility of (kB, D) follows from 51 .B, with F = k,B(X)
<
<
<
IB
52. (B, D)-MANIFOLDS and G = D ( X ) = bD(X); and since K Y B . D = KYkB.D)* &
KkB,D
143
(actually, equality holds,
Remark. It is not possible in general to strengthen the conclusion of the second part of Theorem 52.F to the admissibility of (B,D) itself, even for a scalar equation with constant A ; cf. Example 65.B. However, certain additional conditions do make this strengthened condition true, as the following corollaries show. 52.G. I f B is lean, then (B, D ) is admissible ( B , D)-manifold.
[if and only i f X,, is] a
if and only if
Proof. Theorem 52.F, 52.B,(b), and the fact that k B
there exists
= B.
&
52.H. If D is quasi full-in particular, quasi locally closed-and if there exists a subspace Y of X such that Y C X,,, and such that the quotient space X / Y is rejlexive-in particular, if X itself is rejlexive or X,, is closed and XjX,, is reflexive-then (B, D ) is admissible if and only if there exists [if and only if X,, is] a (B, D)-manifold.
Proof. T h e “only if” part follows from Theorem 52.F. If a (B, D)manifold exists, the same theorem implies that (kB, D ) is admissible; by Theorem 51.G, so is (lckB, f D ) = (lcB, f D ) ; since D , fD are norm-equivalent, (B, D ) is weaker, hence also admissible. & We are now in a position to supply the pending proofs, for J=R+, of two theorems of Section 5 I .
Proof of Theorem 51.H for J = R, . By Theorem 52.F there exists a (B, D)-manifold; by Theorem 52.C this is also an (lcB, D)-manifold; again by Theorem 52.F, (klcB, D ) is admissible. &
Proof of Theorem 51.1 for J
= R,
.
By Theorems 52.F, 52.C, Y * = X&--A*) is an (lcB,D)-manifold for -A*, with constants K,* , C y *. Let f * E lcB(X*) and p > I be given, and let ( ~ ( n )be ) an increasing sequence in R, with limn+,T(n) = 00. For each n, set
f,* = X [ O . r ( n ) ] f *
kO1cB(X*)~
There exists a solution x,* of x:
<
-
<
If,*
IlCB
< If* I l c B ‘
A* x,.’ - f,* with x&(O)E
Y* and
I ., ID*p K Y ‘ If* IlCB > 11 x,*(0)ll pcY*lf* llCB ’ Therefore (x,*(o)) is a bounded sequence in X*. Now X is separable, so that Z ( X * ) is metrizable and compact in the weak* topology (Day [l], p. 43), hence weakly* sequentially compact; thus, selecting a subsequence of ( T ( n ) ) if necessary, we may assume that (x,*(O)) converges weakly* to, say, X: E X * . Denote by x*
Ch. 5. ADMISSIBILITY AND RELATED CONCEPTS
144
the solution of (32.4) with x*(O) = xz. We claim that, for every t E R , , (x,'(t)) converges weakly* to x * ( t ) : indeed, for any n so large that T ( n ) >, t , xc - x* is a solution of (32.3) in the interval [0, t ] , and therefore satisfies, by 32.D, x*,(t) - x * ( t ) = (U(t))-l*(x,*(O) - x*(O)), where U is the solution of (30.3) with U(0) = I ; and (U(t))-l* is continuous in the weak* topology. It follows from the weak* sequential compactness of Z ( X * ) that, for every tER,,
< li;$f II < II x,* II.
II x*(t)II
(52.2)
II x;(t) II = y P Pn(t>
where 9, = inf,,, II x; Thus Pn ED, I Pn ID,G pKylf* I i c ~ But (vn)is an increasing sequence of positive elements of D; by 22.C, sup, pn = l i q + m Lcp, E 1cD; by (52.2), x* E lcD(X*); and the computaKB,D. & tion shows that KlcB,lcD K,*
<
<
(B, D)-subspaces For (B,D)-subspaces we have a theorem that is the analogue of Theorem 51.D on regular admissibility. This theorem plays a central role in the sequel, but we shall not develop any major consequences of it at present.
Assume that Y C X is a (B, D)-subspace. For every 52.1. THEOREM. f E k,B(X) and every A > 1 [and every ( Y ,A)-splitting q of XI there exists a solution x of (50.2) with xm(0) E Y and 11 x(0)ll h d( Y , x(0)) [q(x(O)) = x(O)]; eoery solution x with these properties satisjes I x ID AKLlf ( B 11 x(o)ll x c Y l f IB 9 where Kb = K Y f 2sYcY *
<
<
9
<
Proof. Same as for Theorem 51.D (using Theorem 33.B instead of Theorem 33.C). & Remark. If, in the last part of the statement, x(0) = 0, the proof shows that
I x ID < ( K Y
+
sYcY)lf
IB
*
D)-subspace; let Z be 52.J. Assume that Y C X is a complemented (B, a complement of Y , and let P be the projection along Y onto Z.For every f~ k,B(X), the unique solution x of (50.2) with xm(0)E Y and x(0) E 2 is given by
1 U(t)(I t
(52.3)
x(t) =
- P)U-'(u)f(u)du -
Jm
U(t)PU-l(u)f(u)du,
t
0
t E R,,
(where U is the solution of (30.3) with U(0) = I ) and satisfies I x I < D" max{l, 11 PII}'KilfIB 11 x(0)lI 11 cI'lflB where Kb is as I n Theorem 52.1. 9
<
9
52. (B, D)-MANIFOLDS Proof.
145
From (52.3),
so that x is a solution of (50.2), by (31.3), and x(0) E 2; also, x, = U(I - P ) J"'" U-*(u)f(u)du, whence ~ ~ (E 0Y .)T h e remainder follows from Theorem 52.1 and 1 I .F. &
%pairs and related p a i r s
F--pairs and .FV+-pairs are no less disagreable to work with in connection with (B, D)-manifolds than in questions concerning admissibility (Section 51); the remedy, however, is more radical here than in the previous context (cf. 51.K): any (B, D)-manifold for such a pairs is a (B, , D)-manifold for a stronger Y-pair: 52.K. Assume that (B, D ) is a F+-pair or a Y P - p a i r . If Y is a (B, D)-manifold, then Y is a (T-lcB, D)-manifold (where T-lcB E b F is locally closed and weaker than B), a fortiori'a (bT-B, D)- or (bT-eB, D)manifold.
Proof. Set so = so(B). By Theorem 52.C, Y is an (IcB, D)-manifold, with constant K,. Now by 23.1, 1cB = OsoT-lcB. Let f E koT-lcB(X) and p > 1 be given; then O s , f ~O,qokoT-lcB(X)= k,lcB(X), and there exists a solution y of j Ay = Osof with ym(0)E Y and Iy ID pK,( Oso f llcB pKYlf IT-lcB. Let x be the solution of (50.2) with x(so) = y(sn); then x ( t ) = y(t) for t 2 s o , so that x, = ym, x,(O) E Y . Applying (31.5) to x - y and x[o,s,l f for all t E [0, so], with to = so, we find
+
<
<
so that IxID
< (pKY +
Ix-YID $ 0 ) I x [ O . , s o ] ID
exp(/soll A ( u )11 d'))
If'
IT-ICB
.
Therefore Y is a (T-lcB, D)-manifold. T h e last part of the statement follows from 52.B,(a) since bT-B 1cT-B = T-lcB (by 23.1) or, T-lc(eB) = T-lcB (by 24.1). & similarly, bT-eB
<
<
In dealing with 9 - p a i r s , etc., the specific space B is often irrelevant; it therefore becomes useful to have a condition for a manifold to be
146
Ch. 5. ADMISSIBILITY AND
RELATED CONCEPTS
a (., D)-manifold, and a corresponding one for admissibility (which completes the proof of 51.M).
Assume that D E b y K is given. The linear manifold 52.L. THEOREM. Y is a (B, D)-manifold for some F - p a i r or F + - p a i r or F P - p a i r (B, D) if and only i f it is a (T, D)-manifold. Proof. T h e “if” part is trivial. T o prove the “only if” part we may assume, by 52.K, that B E b F and B is locally closed. But then (T, D) is weaker than ( B , D ) (by 23.K, 23.J), and 52.B,(a) completes the proof.
.
52.M. Assume that D E b FKis given. Some F - p a i r or F + - p a i r or F W - p a i r (B, D ) is [regularly] admissible i f and only if(T,D ) is [regularly] admissible.
Proof. By Theorem 52.F, X , , is a (B,D)-manifold; hence by D)-manifold. But T is lean (by 23. J); by 52.G, Theorem 52.L, a (T, (T, D ) is admissible. T h e “if” part is trivial. & We take another look at Theorem 52.E. In the case of 7-pairs, etc., an affirmative answer to the Query following 23.U would allow us to obtain the conclusion of Theorem 52.E under the assumption that (B,D) is merely not weaker than (L1, L“). There is one case in which this implication does hold; for (B, D)-subspaces (and consequently in the finite-dimensional case) the following result will be superseded by Theorems 64.A, 64.B. E M(R), that (B,D ) is a Y - p a i r a F + - p a i r or a F V + - p a i r , and that Y C X is a (B, D)-manifold. If x is a solution of (50.1) with x(0) 4 clY, then inf, 11 x(t)ll > 0. If (B, D) is not weaker than (L1, L“), then Y C X,, C clY.
Assume that A 52.N. THEOREM.
OT
Proof. By Theorem 52.L, Y is a (T,D)-manifold. By 52.D,
11 x 11-l E T’= M. By (31.7),
Therefore inf, 11 x(t)ll 2 I 11 x ll-llG exp( - I A IM) > 0. T h e last part follows from Theorem 52.E if B is not stronger than L1,and from the first part of the present theorem if D is not weaker than L“ and consequently contains no function bounded away from 0. &
52. (B,D)-MANIFOLDS Oo
Remark I . rem 52.N.
147
The Remark to Theorem 52.E is equally valid for Theo-
Remark 2. T h e last part of Theorem 52.N is best possible, since, if A = 0 E M(8),(Ll, L") is admissible and every linear manifold is an (Ll, L")-manifold (Example 65.A). Sets o f pairs
We obtain an analogue of Theorem 51.N for (B,D)-subspaces; and some improvements on that theorem for 9-pairs, etc. Let B E b S or b S % and the subspace Y of X be 52.0. THEOREM. given. Let 9 C b S K be a nonempty class of spaces such that Y is a (B, D)subspace for every D E 9. There exists Do E b S K , stronger than each D E 9, such that Y is a (B, Do)-subspace (more precisely, there exist numbers uD > 0, D E 9, such that Do = A {uDD: D E 9}), provided any one of the following conditions is satisfied: (1) (2) (3) (4)
there exists G E FK, stronger than each D E 9; 9 is countable; y f (0);
s(B) =
00.
+
Proof. Set uh = max{S,, , K;,}, where K;, = Kn,D 2SYDCYBeD for each D € 9 .Under conditions (3), (4) we set uD = u; ; under condition ( I ) we choose each U, so large that uD 2 ub , G uDD; under condition (2) we order 9 as a sequence (DJ and set uDn= max{u&, I In any case we then define Do as in the statement, so that DoE b F . The proof that DoE b S K depends on the specific condition assumed and will be deferred. For any solution x of (50.1) with x(0) E Y and every D EBwe have x ID O;~S,,II x(0)Il I( x(O)/l, so that x ED,(X). Thus Y C XoD,. Let f E $B(X) and A > 1 be given, and let q be a ( Y , A)-splitting of X . Let z be the solution of (50.2) with z(s( f ) ) = 0, so that z, = 0. Define x as the solution of (50.2) with x(0) = q(z(O)), whence x(0) - z(0) E Y , q(x(0)) = x(0); further, x, = x, - z, = x - z, so that x,(O) E Y. For every D € 9 ,Theorem 52.1 gives I.;' x ID Au;'K;,lf In A1 f In. We conclude that x ED,(X), Ix ID, A( f In . To complete the proof it remains to show that DoE FK. Under condition (I), the choice of uD implies Do G, so that Do E FK. Under condition (2), I X[~.LII~, IX L ~ . ~ I ~uDn ~ , for each 1 > 0 and every integer
<
<
<
<
<
<
<
<
148
Ch. 5. ADMISSIBILITY AND
RELATED CONCEPTS
2 1. Therefore uitIx[O.l]lD, = maXl 0. Thus Do E S ~ . Under condition (3), there exists a nonzero Do-solution x of (50.1), since Y C XoDo;since 11 x 11 E Do is a continuous function vanishing nowhere, DoE SK Assume, finally, that condition (4) holds, but condition (3) does not. Let to > 0 be given. There exists p E B with s(p) > to; we may assume without loss that to 6 so < s < co (where so = s,(Q), s = s(Q)), so that Q E kJ3; further, that Q 2 0, and that j:oy(u) du = 1. We choose any xo E X \ {0} and define f = T U X , ,where U is the solution of (30.3) satisfying U(0) = I; thus f E k,B(X). Since Y = {0}, the solutions x, x in the t to above construction coincide, and are a Do-solution; but for 0 (<so) we have
.
< <
x ( t ) = x ( t ) = -U ( t )
1' U-'(u)f(u) du t
= -U ( t ) x ,
S8
Q(U)
du = -U(t)x, .
t
Thus 11 x I( is a continuous function in Do that does not vanish anywhere in [0, to]; therefore x ~ ~ . ~E,Do, ] and DoE.SE;C. Q
52.P. Let the lean space B E b S or b9'V be given. Let 9 C b S K be a nonempty class of spaces such that (B,D) is regularly admissible for every D E 9.If XoDl= XODlfor any D, ,D, E 9,there exists DoE bSK ,stronger than each D € 9 ,with X,,, = X,, for all D E 9, and such that (B, Do) is regularly admissible, provided any one of conditions (l), (2), (4)of Theorem 52.0 holds or XoD# {0}, D € 9 , Proof. With Y = X,, for all D € 9 , Y is a (B, D)-subspace (Theorem 52.F); with Do as in Theorem 52.0, Y is a (B,D,)-subspace, hence (B, Do) is admissible, by 52.G; also, Y C XoDo; but Do is stronger than every D € 9 ,hence X,,, C Y, and equality holds. Q
52.Q. Let B E b 9 or b S W be given. Let 9 C bSKbe a nonempty class of D E 9. If XoD,= XoD,for any D, ,D, ~ 9 ' a n X/XoDl d is refexive, there exists a full space DoE bSK , stronger than each D € 9 ,with XoDo= XoDfor all D-E9, and such that (B, Do) is regularly admissible provided any one of the conditions (l), (2), (4)of Theorem 52.0 hold or XoD# {0},D E 9. quasi full spaces such that (B, D) is regularly admissible for every
Proof. We may assume without loss that every D E 9 is full. The proof is then the same as for 52.P (22.H implies that Do is full), except that 52.H is used instead of 52.G to prove the admissibility of (B,Do). Q
53. (B, D)-MANIFOLDS, ASSOCIATE
EQUATIONS
149
Remark. Conditions (l), (2), (3), (4)cannot be entirely omitted in Theorem 52.0, 52.P, 52.Q: indeed, condition ( 1 ) is a necessary condition (with G = Do); but consider, e.g., the case X = F, A = 0: then condition (1) is violated, for instance, if 9 is the class of all quasi-full D E b.FK that are not weaker than L"'; however, the remaining assumptions of all three propositions are satisfied (with Y = X,, = (0)) if B is any space in b F or b 9 V that violates condition (4). The verification is left to the reader. We observe that condition (1) is satisfied if 9 C b y K (with G = T; by 23. J), and condition (4),if B E b Y or bY+ or by%+. This points to the following refinements of Theorem 52.0 and 52.P for Y-pairs and related pairs; instead of the corresponding corollary of 52.Q we have a much stronger result, 62.G (without reflexivity conditions). Other related results will be obtained later (62.H). We call attention to 51.M for a strong statement in the opposite direction. 52.R. Let B E b Y OY bY+ or by% + and the subspace Y be given. Let 9 C b YKbe a nonempty class of spaces such that Y is a (B,D)-subspace for every D € 9 .Then there exists Do E b T K , stronger than each D €9, such that Y is a (B, Do)-subspace.
Proof. Theorem 52.0withG = T, using the fact that Do =A O,DEZ 9, 52,s. Let the lean space B E b y or b Y + or b y % + begiven. Let 9 C bYK be a nonempty class of spaces such that ( B , D ) is regularly admissible for every D € 9 .If XoD,= XoD,for any D , ,D , € 9 ,there exists Do E b y K , stronger than each D € 9 ,with XoD,= X,, for all D E 9, and such that ( B ,Do) is regularly admissible. Proof. Proof of 52.P, using 52.R. &
Query. Does 52.P remain true if the assumption that B is lean is dropped ? Does 52.S ?
53. (B, D)-manifolds, admissibility,
and the associate equations The polar manifold o f
a
(B,D)-manithold
In this section and the next we consider not only Eqs. (50.1) and (50.2) but also the associate equations: i.e., we start out with coupled Banach
150
Ch. 5. ADMISSIBILITY AND
RELATED CONCEPTS
spaces X , X’ and an A E L(2) that has an associate A’, and consider Eqs. (32.1) and (32.2), namely, (53.1)
&’ - A‘x’ = 0,
(53.2)
&’ - A’x’ =f’,
in X . We shall sometimes specialize X ’ to X* (and the associate equations to the adjoint equations (32.3), (32.4))-especially in order to prove certain facts about (50.1), (50.2) which can be stated without reference to associate equations (e.g., Theorems 54.F, 54.G); in all these cases the assumption that A have an associate is redundant; we therefore agree to assume in Sections 53 and 54 that A has an associate, i f and only if this associate, or the associate equations, are mentioned in a statement and X ’ = X * is not expressly assumed. The notations and terminology shall refer to the original equations or to the associate equations, as the context may require: e.g., X o = Xo(A), but X i = Xi(--A’). Since the spaces B , D involved will now be required to have associate spaces (Sections 22, 23, especially 22.M, 23.0), every pair (B, D) in this section and the next shall be an %&air or a .7-pair (unless the contrary is explicitly stated). Then (D’, B‘) (this notation is an abbreviation for (D’(X’),B’(X’)))is also an SK-pair or a Y-pair, respectively. In this section we consider the implications of the existence of a (B, D)-manifold, with no assumption of closedness. The reader who is mainly interested in finite-dimensional theory is requested, however, to be patient: the present arrangement seems to be best for an orderly exposition, but the results in this section and the next are summarized for the finite-dimensional case in Section 55. In our developments we are guided by two main principles, an ideal formulation of which (that turns out to be correct for finite-dimensional X : see Theorems 55.A and 55.C) is: “if Y is a (B, D)-manifold for A, then Y o is a @’, B‘)-manifold for -A’ ”, and “if (B, D) is admissible for A then (D’, B’) is admissible for -A‘ ”. We begin with an attempt at following the former of these guiding ideas; the theorem we prove is fundamental to all our work with associate equations. 53.A. THEOREM. Assume that Y C X is a (B,D)-manifold for A . I f f ’ E k,,D’(X) and i f x‘ is any solution of (53.2) such that xk(0) E Yo, then X’ E B‘(X’) and I X’ IB, < KYIfl I, CyIIx’(0)lI.
+
Proof. Consider arbitrary f E koB(X), p > 1 and set s = max{s(f),s(fl)>. By the assumption there exists a solution x of (50.2) with xm(o) y , I ID d p K Y l f IB 11 x(o)ll d p c Y l f lB 32*c 9
53. ( B , D)-MANIFOLDS, ASSOCIATE
151
EQUATIONS
we have (x(s), x‘(s)) = (x,(s), x: (s)) = (x,(O), xk(0)) = 0. Using 32.B (Green’s Formula) and 22.U (cf. the last sentence of Section 23, p. 75),
22.U (as before) then implies the conclusion. & An immediate and important corollary is 53.B. THEOREM. If Y C X i s a (B,D)-manifoldfor A , then YoC Xi,,
Proof. Set f’ = 0 in Theorem 53.A and observe that x;
.
= x’.
& We next turn to the question, raised by Theorem 53.B, of how much of Xi,, is occupied by Yo;since Yo is a subspace and Xiw need not be (as Example 65.Q will illustrate), we cannot always have Y o = X i w ; we shall describe cases where equality does hold (Theorem 53.E), but in the meantime we show that, in a sense, Yo can be made large: e.g., Y can be so modified that any specific element of XiB,belongs to YO (Theorem 53.D). 53.C. If Y C X is a (B, D)-manifold for A and x i E Xi,, is given, then Z = Y n (Fxi)O is also a (B, D)-manifold for A (where FxA is the at most one-dimensional subspace spanned by xi).
Proof. Let x’ be the solution of (53.1) that satisfies x‘(0) = xi; in particular, x’ E B’(X’). If Y C (Fxi)O then 2 = Y, and there is nothing to prove. We assume, therefore, that there exists a solution x of (50.1) with x(0) E Y and (x(O), x’(0)) = u # 0. Of course x E D ( X ) . Choose any f~ k,B(X), p > 1, and set s = s( f ) . By the assumption there exists a solution y of (50.2) with y,(O) E Y and I y ID < pKYlf , IIy(o)ll < p C Y l f IB - By 32*c, < ~ ~ ( o ) , ~ ’=(y,(s),x’(s))=(y(s), (o)) x’(s))* Using 32.B (Green’s Formula) and 22.U,
152
Ch. 5. ADMISSIBILITY AND
RELATED CONCEPTS
so that zm(0)E 2. Finally,
thus Z is also a ( B ,D)-manifold, with K ,
< the expression in braces.
&
53.D. THEOREM. Assume that Y C X is a (B,D)-manifold for A. If W C X is a linear manifold such that Yo C W C X & , and the codimension of Yo in W isfinite, then Z = Y n W ois also a (B, D)-manifold for A (and Zo= W ) .I n particular, eaery element of XhB,is contained in Zo for some ( B ,D)-manifold Z C Y . Proof. We proceed by induction on d = dim(W'/Yo), the codimension of Yo in w'.If d = 0, W' = Yo, Y n W o= Y n YO0 = Y, and there is nothing to prove. Assume the conclusion true whenever d = n, and let the assumption hold with d = n 1; then W = V' Fxi for some x i E X', where V' is a linear manifold with YOC V', dim(V'/Yo) = n. By the induction assumption, U = Y n V'O is a (B,D)-manifold. By 53.C, 2 7 Y n W o= Y n V'O n (Fx;)O = U n (FxA)O is also a (B, D)-manifold. &
+
+
We now describe cases where Yo = Xi;w.
53.E. THEOREM. Assume either (1) that D is thin with respect to B (in particular, that ( B ,D) is a Y - p a i r not weaker than (L1, Lc));or (2) that A E M(X)and that ( B ,D ) is a Y - p a i r not weaker than (Ll,L"). If Y C X is a ( B ,D)-manifold f o r A , then Yo = (XOD)O = XiB,. Proof. By Theorem 52.E or Theorem 52.N and Theorem 53.B we have (XOD)O = Yo C XiB,. It remains to show that Xi,,, C (XOD)O or, equivalently, XoDC(Xi,.)o. Assume that x is a solution of (50.1) with x(0) 4 (XAB,)O;there exists then a solution x' of (53.1) with x'(0) E XhB, (i.e., a B'-solution) such that (x(O), x'(0)) = u # 0. By 32.C we have I (J I = I(x, x'>l d II x II II x' II, SO that II x 11-l < I I-llI x' II, and II x 11-l E B'. T h e conclusion that x ( 0 ) $ X o Dthen follows as in the proofs of Theorem 52.E or Theorem 52.N, respectively. &
53. (B, D)-MANIFOLDS, ASSOCIATE
* *
EQUATIONS
153
We present a corollary showing that in certain cases a (B,D)-manifold must be closed.
53.F. Assume that D is quasi locally closed; assume also that D is thin with respect to B (in particular, that (B,D) is a F-pair not weaker than (Ll,L")). If there exists a (B,D)-manifold for A and a (D', B')-manifold for -A', then X,, is a saturated rubspace; if (B, D) is admissible for A and (D', B') is admissible for -A', both admissibilities are regular. Proof. There is no loss in assuming that D is locally closed, so that D" = D. By Theorem 53.E,X i B , is a saturated subspace; and by that theorem and Theorem 53.B, X,, C (XoD)OO= (XiB,)OC X,,,, = X,, , so that X,, = (XiB,)Ois a saturated subspace. The last part follows by the use of Theorem 52.F. &
A result on admissible F-pairs T h e theorem we are about to prove (by an application of Green's Formula, similar t o that in the preceding proofs) is an attempt to bear out the guiding principle "(B, D) admissible for A implies (D', B') admissible for -A' "; it concerns 5 - p a i r s only.
53.G. THEOREM. Assume that (B,D) is a 5 - p a i r . If (B,D) is admissible, or i f there exists a (B, D)-manifold, f o r A , there exist numbers k, , k, > 0 such that, if x' is any M-solution of (53.2)f o r somef' E D'(X'), then X' E B'(X') and I X' In, 6 k,l X' lM k21f' ID, .
+
Proof. Let Y be a (B, D)-manifold (the assumption that (B,D) is admissible implies the existence of Y by Theorem 52.D). Choose any f~ k,B(X), p > 1, and set s = s ( f ) . There exists a solution x of (50.2) with ~ ~ (E Y 0 , )I x I D p K y 1 . f IB 9 II x(0)ll &,If IB . Using (33.1) and the fact that 11 -A' 11 = I( A 11 we have
<
By Schwarz's Inequality we have
<
154
Ch. 5. ADMISSIBILITY AND
If this minimum is attained at T , say, s 32.B (Green’s Formula) and (53.3))
RELATED CONCEPTS
< < s + 1, we obtain, using 7
the conclusion follows from 22.U (cf. the last sentence of Section 23, p. 75); we find
53.H. Assume that (B, D ) is a Y-pair. If (B, D ) is admissible, or i f there exists a (B, D)-manifold, for A, then Xi,,, = XiM. Proof. By Theorem 53.G with f’ = 0, XiMC Xip; the reverse inclusion holds because B‘ is stronger than M (by 23.G). &
53.1. Assume that (B, D ) is a F-pair. If (B, D ) is admissible, or i f there exists a (B, D)-manifold, for A, and if Z’C X is a (D’, G)-manifold for -A’ for some G E WK , then Z’ is a (D’, B’)-manifold for -A’. Proof. Since G is stronger than M, 53.H implies Z’C XiG C XAM= Xhp . For anyf’ E k@’(X’) and any p > 1 there exists a solution x‘ of (53.2) with xL(0) E 2‘ and I x’ IG pKZ.If’ ID, . Now x’ E M(X’), with I x’ lM < a(G; 1)1 x’ IG; by Theorem 53.G we have I x’ IB’ (pK,.k,a(G; 1) + k 2 ) l f ’ ID., so that Z’is a (D‘, B’)-manifold. &
<
<
Remark. The proof of 53.1 shows that Z’is actually a (D’, B’ A G)manifold.
54. ( B , D)-SUBSPACES, ASSOCIATE
155
EQUATIONS
54. ( B ,D ) - s u b s p a c e s a n d the associate equations
The polar manifold o f a (B,D)-subspace In this section we show how the analysis begun in the previous section is developed and deepened if the ( B ,D)-manifolds involved are assumed to be ( B ,D)-subspaces (with the quasi-strict coupling property-a
redundant assumption if X = X*). The remarks and agreed assumptions at the beginning of Section 53 remain in force until further notice.
54.A. Let Y C X be a ( B ,D)-subspace f o r A with the quasi-strict coupling property. I f f ’ E k,D‘(X‘) and i f x‘ is a solution of (53.2) with xL(0) E Yo,then d( Yo,x’(0)) < s,S,If’ ID. . Proof. Let x be any solution of (50.1) with x(0) E Y. Setting s = s(f’) and applying 32.C we have (x(s), x’(s)) = (x(s), x&(s)) = (X(O), XL(0)) = 0.
Using 32.B (Green’s Formula), Theorem 33.B, and 22.U,
By the definition of the quasi-strict coupling property, d( yo,X‘(0))
d SY ‘ sup{ I (x(o), x‘(0)) I : X(o) 2(y ) } < sYsY
If’
ID’
*
&
We come now to the main result of this section. Let Y C X be a ( B ,D)-subspace f o r A with the 54.B. THEOREM. quasi-strict coupling property. Then Y o is a (D‘, B‘)-subspace f o r -A’. Proof. Yo is a subspace and, by Theorem 53.B, Yo C XiB,. Let and some (Yo,A)-splitting q‘ of X’be given. For an arbitrary f ’ koD’(X’) ~ set s = scf’), and let z’ be the solution of (53.2) with z’(s) = 0, so that &, = 0. Define x’ as the solution of (53.2) with x’(0) = q’(z’(0)).Then x’ - z’ is a solution of (53.1) and x: = x: - zk = x‘ - z‘, so that xL(0) = x’(0) - z’(0) E Yo. By (11.2) and 54.A, 11 x’(0)ll A d( YO, x’(0)) hs,S,If’ ID, . From Theorem 53.A it then follows that I x’ IB. < ( K , + hSySyCy)lf’ID,, so that Yo is a (D’, B‘)-subspace for -A’ with K,o < K, s,S,C,. &
A
>1
<
<
+
54.C. Let Y be a saturated subspace of X such that Y and Y o both have the quasi-strict coupling property. I f D is quasi locally closed, Y is a ( B ,D)-subspace f o r A i f and only i f Yo is a (D’, B’)-subspace for -A‘.
156
AND Ch. 5. ADMISSIBILITY
RELATED CONCEPTS
Proof. Theorem 54.B applied to (50.2) and (53.2); the relations YO0 = Y, B 1cB = B”, D” = 1cD norm-equivalent to D ; and 52.B,(a). 4, We summarize Theorem 54.B and 54.C for X‘ = X * (we omit the special formulation for a Hilbert space):
<
54.D. Let Y be a subspace of X . I f Y is a (B, D)-subspace f o r A , then Y o in X * is a (D’,B‘)-subspace for -A*; if D is quasi locally closed, the converse is also true. Proof.
Theorem 54.B and 54.C, with 12.A.
4,
Rermark. T h e assumption that D be quasi locally closed in the converse parts of 54.C, 54.D cannot be omitted in general: see the Remark to Theorem 55.A. Implications o f admissibility for the adjoint equation Theorems 54.B and 52.F imply that if there exists a (B, D)-subspace for A with the quasi-strict coupling property, then (kD‘, B’) is admissible for -A‘. Is it possible to conclude that (D’, B’) itself is admissible? Since D’, B’ are locally closed, this is surely the case, by 52.H, if X’ is reflexive, or, equivalently, if X is reflexive (so that X’ = X * up to a congruence). We shall show that the conclusion remains true if X‘ = X*, without reflexivity assumptions (Theorem 54.E); and we formulate the following question: 00
coo3 00
coo0
Query. If X’ # X* and there exists a (B, D)-subspace for A with the quasi-strict coupling property, is (D’, B’) admissible for --A’ ?
Remark. For the applicability of 52.H in this case it is sufficient, since YO C XiB,(by Theorem 53.B; Y is the (B, D)-subspace), that X’/Yo be reflexive; it is a simple consequence of the quasi-strict coupling property of Y that this reflexivity is equivalent to the reflexivity of Y. Therefore if the assumed (B,D)-subspace is reflexive, the answer to the Query is affirmative. I f there exists a (B, D)-subspace f o r A (in particular, 54.E. THEOREM. if (B, D ) is regularly admissible f o r A), then (D’, B’) is admissible f o r -A* in X*. Proof. See above for a proof for reflexive X.
oc, mco
Let Y be the given (B, D)-subspace. Let f * E D’(X*) be given. For any x, E Y set y(xo) = 0 < x ( t ) , f * ( t ) ) dt, where x is the solution of
-s“
54. (B, D)-SUBSPACES, ASSOCIATE
EQUATIONS
157
(50.1) with x(0) = xo; this is meaningful, by 22.U, since x E D ( X ) ; thus 9 is a bounded linear functional on Y : linearity is trivial, and I q(xo) I I x ID If* ID, S y ( f *ID, (1 x, (I. By the Hahn-Banach Theorem there exists an element x t E X * , (1 xz 11 S y l f * ID, , such that (xo , x,*) = 7(x0) for all xo E Y . Let x* be the solution of (32.4) with x*(O) = xz. Summarizing,
<
<
<
for all solutions x of (50.1) with x(0) E Y. Let f E $B(X), p > 1 be given, and set s = s ( f ) . By the assumption there exists a solution z of (50.2) with ~ ~ (E 0Y ,) I z ID d p K y I f Ir > II Z(0)II d &',If Ir * Applying 32-B (Green's Formula) and using (54.1) with x = z, ,
Applying Green's Formula once more and using (54.2), /' dt = (z(s), x*(s)> 0
- ( d o ) , x*(O))
-
1'
( z ( t ) , f * ( t ) )dt
0
hence, using 22.U and (54.1),
22.U then implies x *
E B'(X*);and
indeed KD,,B,(-A*)< K , + S y C y . 4p1
Sets of (B,D)-manifolds and -subspaces for F-pairs and related pairs In this subsection we shall always assume that (B, D ) is a Y - p a i r , or a F+-pair,or a Y W - p a i r , thus upsetting the agreement made at the beginning of Section 53. We intend to show that, roughly speaking, the class of (B, D)-
158
Ch. 5. ADMISSIBILITY AND
RELATED CONCEPTS
subspaces for a given A does not depend very closely on the spaces constituting the pair; this is particularly the case for quasi locally closed D, which we take u p first. There are some very fragmentary results on (B, D)manifolds. The location of this subsection is motivated by the methods employed, which use the developments in Sections 53 and 54. It is sufficient, however, to consider only the adjoint equations, i.e., work with X' = X*. This exempts us, incidentally, from additional assumptions on A (existence of an associate); all polar sets, etc., are taken in the coupled spaces X , X*. Let D be quasi locally closed. If there exists a ( B ,D)54.F. THEOREM. subspace, then X,, = X,, ,and every (B, D)-manifold is a ( B , M)-manifold and conversely.
Proof. On account of 52.K there is no loss of generality in assuming that B E b y and is locally closed; and we may also assume that D is locally closed, so that B" = B, D" = D. ( B , D)-manifolds are trivially (B,M)-manifolds (52.B,(a)). By Theorem 54.B there exists a (D', B')subspace for -A*; by 53.H applied to the adjoint equation, X,, = X,,. = XoD. By 53.1 applied to the adjoint equation, every (B,M)-manifold for A is a ( B , D")-, i.e., a ( B ,D)-manifold for A. & Remark. We shall show that X,, = X,, holds under the weaker assumption that D is quasi full (62.C); without any assumption on D this is no longer true, even for X = F and A = 0 (Example 65.A). Let D be quasi locally closed. If there exists a ( B ,D)54.G. THEOREM. manifold (in particular, if ( B , D) is admissible), then every (B,D)-subspace is a (T, D)-subspace and conversely.
Proof. A ( B , D)-subspace is always a (T, D)-subspace (Theorem 52.L). To prove the converse, there is again no loss in assuming that B E b y and that B , D are locally closed. Let Y be a (T,D)-subspace; by 54.D and the fact that T' = M, Y o is a (D', M)-subspace for -A*. By 53.1, YO is a (D', B')-subspace for -A*. Again by 54.D, Y is a (B,D)-subspace for A. & 54.H. Let D be quasi locally closed. In each of the following inclusions either equality holds or the included class is empty:
{(B,D)-subspaces}
C
1{(T,
1
{ ( B , M)-subspaces} C {(T, M)-subspaces}. D)-subspaces}
Proof. Theorems 54.F and 54.G. 9,
54. (B, D)-SUBSPACES, ASSOCIATE
EQUATIONS
159
Remark. All six possible combinations, except that in which all four classes are empty, can be illustrated with a scalar equation with constant A (Examples 65.A, 65.B); the remaining case can be illustrated either with an equation in two-dimensional X with constant A (Example 65.C) or with a scalar equation with bounded continuous A (Examples 65.K, 65.L). The conclusion of Theorem 54.Gremains true under different conditions for (B,D) (cf. Query 2 below): Assume that (B, D) is not weaker than (Ll, Lorn). If 54.1. THEOREM. there exists a (B, D)-manifold (in particular, if (B, D) is admissible), there exists a (T,D)-subspace Y if and only if X,, is closed, and thm Y = X,, and this is a (B, D)-dspace. Proof. Let Y be a (T, D)-subspace. We now use the space D, , defined by 23.N, which has the same continuous functions, with the same norms, as D. Then Y is a (T, D,)-subspace, a fortiori a (T, IcD,)-subspace; but there exists a (B, D)-manifold, hence a (B, lcD,)-manifold. By Theorem 54.G, Y is a (B, lcD,)-subspace. By 23.S, (B, lcD,) is not weaker than (L1, Lf). By Theorem 52.E applied to (B, IcD,), and by the fact that Y is a (T, D)-subspace, we have, Y C X,, = XoDo C XOLCDo C cl Y = Y, so that Y = X,, and X,, is closed, hence a (B,D)-subspace. If X,, is closed, it is a (B, D)-subspace, by 52.B,(b), and hence a (T, D)-subspace, by Theorem 52.L. & We now return to an arbitraryy-pair or Y+-pair or Y W - p a i r (B, D). D)-subspaces, we begin In attempting a description of the class of all (B, by observing that, by definition and by Theorem 53.B, every (B,D)manifold satisfies Y C X,, , Y oC X,*, ; we should like to show that every subspace satisfying these conditions is a (B, D)-subspace, provided any exists. This is indeed the case for finite-dimensional X (55.B), and we have the following general result on these lines: Assume that there exists a (B, D)-dspace Z. Let Y 54.J. THEOREM. be a subspace of X such that either Y C X,, and ( Y n Z)O C X,*, cl( Y Z ) C X,, and Y oC XzM , then Y is a (B, D)-subspace.
+
, or
Proof. We may assume without loss that B E b Y and is locally closed. By 53.H, X,*, = X& . Assume the first alternative condition: by 54.D,Zo is a (D’, B‘)-subspace for -A*. Since Zo C . ( Y n Z)O C X&, , 52.B,(b) implies that (Y n Z)O is a (D’, B’)-subspace for -A*; again by 54.D,Y n 2 is a (B,1cD)-subspace for A, since (IcD)’ = D’. Now Y n Z C Z, so that 52.B,(e) implies
Ch. 5. ADMISSIBILITY AND
160
RELATED CONCEPTS
that Y n Z is also a (B, D)-subspace. Since Y n Z C Y C X,, , 52.B,(b) D)-subspace. implies that Y is a (B, Assume the second alternative condition: since Z C cl(Y Z) C X,, , cl(Y + Z) is a (B,D)-subspace. By 54.D, (cl(Y Z))O = (Y Z)O = YO n ZO is a (D', B')-subspace for -A*. Since Y o nZo C Y oC X&, , YO is a (D', B')-subspace for -A*. Again by 54.D, Y is a (B, 1cD)-subD)-subspace. Since Y C cl(Y Z), 52.B,(e) implies that Y is also a (B, space. &
+
+
+
+
54.K. Assume that either X,, is closed or X,*, is weakly* closed (i.e., a saturated subspace). The class of (B, D)-subspaces is empty or equal to the class of (T, D)-subspaces: the latter is empty or consists exactly of those subspaces Y that satisfy Y C X,, , YO C X &
.
Proof. Let Z be a (B, D)-subspace, Y a subspace with Y C X,, , Yo C X,jM.We have Z C X,, , whence Y 2 C Xo, ; if X,, is closed, cl(Y Z) C X,, We also haveZO C XzM(Theorem53.B), Yo z ° C X & ; if X,*, is a saturated subspace, then (Y n Z)O = (YO" n ZO")O = ( Y o Z0)OoC X2M. Theorem 54.1 implies that in either case Y is a (B, D)subspace. Together with Theorem 52.L this implies the conclusion. &
+ +
.
+
+
Remark. The inclusion relations for Y, Yo may be written equivalently (X,,), C Yo C XoM, when Xo, is closed; and (X&)O C Y C X,, , when X,*, is weakly* closed. oc)
a3
a3
Query 1. Does the conclusion of 54.K remain true if the closedness assumptions on X,, , X,*, are removed ? Or at least the first part of the conclusion ?
In case of an affirmativeanswer to the last part of Query 1, Theorems 54.G and 54.1 suggest (and give partial answers to) the following question (for infinite-dimensional X): Query 2. If there exists a (B, D)-manifold, is every (T,D)-subspace also a (B, D)-subspace ?
55. Finite-dimensional space Let X be finite-dimensional, with euclidean norm (cf. Section 15). All the developments in Sections 51 and 52 remain significant, and indeed the fundamental proofs do not become much simpler. Of course the concepts of admissibility and regular admissibility are merged, as are those of (B, D)-manifold and (B, D)-subspace; and the reflexivity of
55. FINITE-DIMENSIONAL SPACE
161
X simplifies the statement of Theorem 51.F and all its consequences; but we offer no significant special results for the finite-dimensional case. T h e situation is slightly different for the contents of Sections 53 and 54. At the cost of a minor blurring of details, all the results of these sections are summarized and superseded by the following propositions: 55.A. THEOREM. Let (B,D) be an FK-pair or a 9 - p a i r . If Y is a ( B ,D)-subspace for A, then Y* is a (D', B')-subspace for -At; the converse holds ifD is quasi locally closed. If the set of all (B,D)-subspaces for A is not empty, it consists of all subspaces Y such that (XoBt(-At))* C Y C XoD(A),where XOB( -At) = XoM( -At) if (B, D) is a F - p a i r . If D is thin with respect to B,there is at most one (B,D)-subspace, and if there is, then XoB*( -At) = (XoD(A))*. Proof. The first statement follows from 54.D. A (B,D)-subspace for A, say Y, satisfies Y C XoD(A),Y* C XoBe(-A+)(54.D or Theorem -At))l C Y. Assume conversely that there exists a 53.B), hence (XoB,( (B, D)-subspace for A, say Y o , and that Y is a subspace satisfying (XoB,( -At))l C Y C XoD(A).Then Y+C ( Y n Yo),C XoB,( -A+); by Theorem 53.D, Y n Yo is a ( B , D)-subspace for A ; but Y n YoC Y C XoD(A);by 52.B,(b), Y is a (B, D)-subspace for A. The special result for F-pairs follows from 53.H. The last statement follows from Theorem 52.E and the first part of the proof or from Theorem 53.E. 4,
Remark. The assumption that D is quasi locally closed cannot be entirely omitted in the converse part of the first statement .of Theorem %.A, even for X = F and A = 0 (Example 65.A). 55.B. Let (B,D) be a F-pair of a F+-pair or a F P - p a i r . The set of (B,D)-subspaces is either empty or equal to the set of all subspaces Y with (xOM(--At))*
XOD(A).
Proof. Theorem 55.A using 52.K; or 54.K (including remark). 4,
Remark 1. Theorem 55.A applied to the adjoint equation, or Theorem 54.F, yields the information that XoD(A)= XoM(A)if D is quasi locally closed; but it will be shown in 62.D that the same conclusion holds under the weaker assumption that D is quasi full. Remark 2. It may well happen that (XOM( --At))* C XoD(A)and yet the set of all (T,D)-subspaces(which contains that of all (B, D)-subspaces) is empty: see Remark to Examples 65.K, 65.L, with X = R, bounded continuous A, and any D. Finally, and most important:
162
Ch. 5. ADMISSIBILITY AND
RELATED CONCEPTS
Let (B, D ) be an FK-pair or a Y - p a i r . If (B, D) 55.C. THEOREM. is admissible fiw A, then (D', B') is admissible for -At; i f D i s quasi locally closed, the converse also holds. Proof. Theorem 54.E(applied to the adjoint equation for the converse part, in which we may assume D locally closed, so that D" = D). Another proof follows from Theorems 52.F and 54.B, and 52.H. &
Remark. T h e assumption that D is quasi locally closed cannot be entirely omitted in the converse part of Theorem 55.C, since otherwise the existence of (B, D)-subspace (equivalently, the admissibility of (kB,D)), would always imply the admissibility of (B,D) itself, via the direct and converse parts of the theorem; but this is not the case, even for X = F and constant A (see remark to Theorem 52.F and Example 65.B).
56. Notes to Chapter 5 Historical notes The assumption that a certain pair of function spaces is admissible for Eq. (50.2)-an assumption that may be considered as a special instance of total stability of Eq. (50.I)-was investigated by Perron [2], who pointed out the deep significance of this condition. He limited himself, however, to the pair (C,L"), in our terminology. For this pair he proved the boundedness theorem corresponding to our Theorem 51.A. His methods are classical. Later, Bellman [I], Krein [I], and KuEer [I] extended Perron's research to the pairs (Lp, L"), using functional-analytic tools, though still for finitedimensional X. It should be noted, however, that their work is, in a sense, a step backwards from Perron's in that they only consider equations for which X,,= X. Within this restricted context they also prove our Theorem 51.A.
Complemented (B,Lm)-subspuces A recent paper by Coppel [l] has drawn our attention to a type of result that we believe worth while sketching here, although its precise status in the general theory is as yet uncertain and it does not quite seem to be in the desirable final form. For A E M(8) and 5-spaces, these results will be amply superseded by those in Section 64. We assume that the range of t is J = R, and let U be the solution of (30.3)
56. NOTESTO
CHAPTER
5
163
with U(0) =I. For every projection P in X we define the function G, : R+ x R, + 8 given by GP(t, 4
(56.1)
= ~ ~ ~ ~ ( X [ O, t P)U-'(u), 1 ~ 4 ~
and observe that (56.2)
GP(t, *) = X[o.t]U(t)U-l
+ U ( W P ( 0 ,.).
Let Y C X be a complemented subspace, P a projection 56.A. THEOREM. along Y , and assume given a function space B E b St,(or at least B E b.F 01 b 9 V with 1cB ES~). Then Y is a (B, Lm)-subspace for A if and only if there exists a number k > 0 such that
(56.3)
Gp*(t,.)x* E B'(X*),
I Gp*(t,*)x* Is, < k f m all t E R, , x* E Z(X*).
Proof. On account of Theorem 52.C we may assume B E b.FK. The condition is necessary: If Y is a (B, L")-subspace, 52.5 states that, for every f E $B(X), the function x given by x(t) = JomGp(t, u)f ( u ) du is an L"-solution of (50.2) with I x I ,< uKkl f Is , where u = max{l,I) PI]}. For any t E R, and x* E Z ( X * ) we therefore have
for a l l f ~$B(X). Then (56.3) follows by 22.U, with k
= OK;.
J-y
The condition is suficient: For every f E $B(X), x(t) = Gp(t,u)f ( u ) du gives a solution x of (50.2) with x,(O) E Y (cf. proof of 52.5). By (56.3), and 22.U, we have I(x(t), x*>l = J;I < f ( u ) ,Gp*(t,u ) x * ) du I < kI f Is for dl t E R + , x * € . Z ( X * ) ; thusxEL"(X), 1 x 1 < k l f I B . & Remark. By (56.2), Gp*(t,.)x* E B'(X*) for all t if and only if G,*(O,*)x* = U*-lP*x* E B'(X*). Use of the Banach-Steinhaus Theorem allows (56.3) to be weakened by letting k depend on x* (but not on t). '
56.B. THEOREM. Assume that B is as in Theorem projection P in X and a number k' > 0 such that (56.4)
Gp(t, .) E B'(X), I Gp(t, .)IB,
< k'
56.A. I f there exist a
for all
t
E
R+ ,
then (B, L") is admissible; conversely, if X is finite-dimensional, this condition for admissibility is also necessary. Proof. The condition is suficient: For givenf E B(X) we have Gp(t, .)fE L1(X) on account of (56.4), so that x(t) = J'," Gp(t,u ) f ( u ) du exists and, again by
164
Ch. 5. ADMISSIBILITY AND
<
RELATED CONCEPTS
.
(56.4),x E La(X), I x I k'l f le, But comparison of (56.2)and (31.3)shows that x, thus defined, is a solution of (50.2). Of course, (56.4)implies (56.3) with k = k', so that, in addition, the null-space of P is a complemented (B,L")-subspace. If X is jinite-dimensional, the condition is necessary: Assume that the norm La)-subspace. is euclidean. By Theorem 52.F, X, is a (complemented) (B, By Theorem 56.A, (56.3) holds for some projection P along X , and some k > 0. But (56.3)implies (56.4)with k' = k DimX. &
-
For B = L" and X finite-dimensional, this is Coppel's result, which generalizes a theorem of Bellman [l] for the case X , = X. co It is an open question whether the implication (56.3)+ (56.4)also holds when X is not finite-dimensional. It certainly does, for instance, if A E M(X) and B E b y , on account of Theorems 56.A, 64.A, 64.B, as the reader may easily verify.
CHAPTER 6
Admissibility and dichotomies 60. Introduction
This chapter constitutes, in a sense, the core of the book. I n Chapter 4 we described certain very neat kinds of behavior, viz., ordinary and exponential dichotomies, of the solutions of the homogeneous equation (30.1), i.e., (60.1)
X+Ax=O
( t E R, , A E L(x)). In Chapter 5 we considered the nonhomogeneous equation (30.2), i.e., (60.2)
X
+ AX = f
and discussed, for pairs (B, D ) of Banach function spaces, the relations between B-functions f and D-solutions x . This discussion gave rise to the concept of admissibility of ( B , D ) for A , and, for 9 - p a i r s and 9%-pairs, to the notion of (B,D)-manifold. T h e purpose of Chapter 6 is to establish a connection between these two aspects of the theory for equations with t ranging over J = R, (the case J = R will be dealt with in Chapter 8). An important restriction we must accept is that the pairs will consist of translation-invariant spaces, and will in fact, be F - p a i r s or F’-pairs or Y%+-pairs. Also, in most cases, only (B, D)subspaces, and regular admissibility, will be considered-no loss for finitedimensional X (but see the discussion of Section 66). On the other hand, we can deal with behavior of the solutions of (60.1) more general than dichotomies. T h e main structure of the chapter consists of “direct” and “converse” theorems. T h e typical form of a direct theorem is: “If there exists a (B, D)-subspace (or (B, D ) is regularly admissible) for A, and ..., then the solutions of (60.1) behave in sych-and-such a w a y (usually a dichotomy or 165
166
Ch. 6. ADMISSIBILITY AND
DICHOTOMIES
an incomplete imitation thereon”; the term “converse theorem” is now self-explanatory. Section 61 contains the proof of an inequality (Theorem 61.A) that is pivotal for all direct theorems, and some important corollaries. I n Section 62 we explore direct theorems short of those involving dichotomies. T h e mere existence of a (B, D)-subspace, for instance, does not in general imply a dichotomy; but the failure is, in a sense, only pointwise”, leaving aside the angular-separation condition (Diii) in the definition of a dichotomy: the growth conditions (Di), (Dii) do not hold “pointwise”, i.e., for the values 11 x(t)ll, where x is a solution of (60.1), but the analogous conditions do hold for the comparison of the “means” d-l 11 x(u)ll du or of the D-norms of “slices” ~ [ ~ , for a fixed d > 0 and different values of t . Similar results are obtained for behavior imitating an exponential dichotomy; the crucial condition here is that (B, D ) be not weaker than (L1, L : ) . T h e behavior we just sketched is uniform, in the sense, well known from stability theory, that the parameters for the comparison of slices and means at values t 2 t o , say, do not depend on t o . I n the same Section 62 we show that we can, under certain assumptions on ( B , D ) in which comparison with L1 and L“ plays a part, obtain pointwise stability results of the form of the growth conditions (Di), (Dii) or (Ei), (Eii), but at the cost of the uniformity. Making stronger assumptions on ( B , D ) we can finally obtain the central direct results in which the conclusion is the existence of an ordinary or exponential ‘dichotomy; this is done in Section 63 (Theorems 63.A, 63.G, and 63. J). There are converse theorems, characterizing the ( B , D ) for which a (B,D)-subspace exists, or that are regularly admissible, when there is an ordinary dichotomy (Theorems 63.C, 63.E) and giving a large class of such pairs when there is an exponential dichotomy (Theorem 63.K). I n the sections described u p to this point no assumption was made about A. All becomes much simpler and neater when we assume A E M(X), a situation studied in Section 64. Now any (B, D)-subspace induces a dichotomy (Theorem 64.A), an exponential dichotomy if (B, D ) is not weaker than (L1, ) : L (Theorem 64.B); and the class of pairs (B, D ) for which a (B, D)-subspace exists, or that are regularly admissible, when there is an exponential dichotomy is precisely characterized (Theorem 64.C). T h e task of showing the strength of all these results calls for several examples and accompanying comments. These are supplied by Section 65. We have endeavored to reduce the various types of examples to a minimum, and to obtain the numerous d8erent individual examples (6
J:,d
~+~p,
61. THEFUNDAMENTAL
INEQUALITIES
I67
by variations on the same theme. At the same time we have tried to give the strongest possible kind of example in each case: lowest dimension of X , greatest regularity of A, etc. We hope that this section will convince the reader that the results of the chapter are best possible from many points of view, and, in particular, that no reasonable converse theorems are possible for the “predichotomy” behavior of the solutions of (60.1) discussed in Section 62. In section 66 we turn to the consideration of the associate equations in some space X’ coupled with X . If Y is a (B,D)-subspace for A, Theorem 54.B shows that (some technical conditions being satisfied) Y o is a (D’, B’)-subspace for -A’. T h e direct theorems in Sections 62, 63, 64 can therefore readily be applied to show that the existence of a (B, D)-subspace for A has certain consequences for the behavior of the solutions of the associate homogeneous equation. 00
This exhausts the results for the associate equations that are relevant when X is finite-dimensional. Surprisingly enough, the existence of a mere (B, D)-manifold for A , or the mere admissibility of (B,D) for A , without any closedness assumptions, does provide some information on the behavior of the solutions of the associate equation. This information is obtained via an elaborate analogue of Theorem 61.A (Theorem 66.D), and is also developed in Section 66. T o conclude we list the standing assumptions that will hold throughout this chapter: the range oft is R * A E L(x); (B, D) is a 9 - p a i r or 9 + - p a i r +! or Y V + - p a i r . Further assumptions for the study of associate equations in coupled spaces will be made in Section 66.
61. The fundamental inequalities Assume that Y is a (B, D)-subspace. There exists a 61.A. THEOREM. number Kf: > 0 such that for every h > 1, A > 0, every pair of solutions y, z of (60.1) with y(0)E Y and 1) z(0)ll < hd( Y , z(O)), and all T 2 0, we have (61 .l)
Proof. Since a(B, 0) = a(1cB; A ) = cu(T-lcB; A ) (by 21.E, 23.H, and M A , XI), 52.K allows us to assume, as we now do, that B E b y and is locally closed. Under this assumption we shall prove (61.1) with K”- K‘ - K Y B , D 2SYDCyB,,(cf. Theorem 52.1).
,
+
AND Ch. 6. ADMISSIBILITY
168 to
DICHOTOMIES
If y + x = 0, the assumption implies y = x = 0, and there is nothing prove; we therefore assume w = y z # 0. We set
+
Thus x is a solution of (60.2) for f = X [ 5 , 5 + A ] s g n w (by 31.B,(b)); we have 11 f 11 = X [ ~ . T + A ] ,whence, by 23.K f E koB(X), If = B(B;A ) - ~ so that xm(0)E Y ; and 24/a(B;A ) . Further, x, = y j ,r + A 1) w ( u ) ( ~du, x(0) = -z(O) (1 w(u)\\-l du, so that (( x(0)ll X d( Y , x(0)). We may therefore apply Theorem 52.1 and obtain
IB
Jl+A
<
I x ID
(61.2)
<
< 2h dKG/.lor(B;A ) .
By Schwarz’s Inequality, (61.3)
I@ , + A J J ID <
I
II W ( U ) 11-1
11
11 du’
T+A
I @,+LY ID
1
7+A
du
5
I1 4 ~ I1 du)
7
and similarly (61 .4)
I X[o,T]’
ID
<
I
ID
/
?+A 7
These inequalities, taken together with (61.2), yield the conclusion.
9r
We now obtain some important inequalities, derived from (61.1), for the “means” and “slices” of the solutions y , z. For every pair (B, D) and A > 0 we write F(d)
(61.5)
= r(B,D; A ) =
4
a(B;d),B(D;A ) ’
a nonincreasing function of A . For real r we use the notations [ r ] =
largest integer
< r , and r+ = max(0, r}.
61.B. THEOREM. Assume that Y is a (B, D)-subspace. For every A 2 A , > 0 the following inequalities hold: (i) for every solution y of (60.1) with y(0) E Y and every to 2 0,
t 2 to
+
A0
,
61. THEFUNDAMENTAL
INEQUALITIES
169
Proof. Set d, = d/[d/dO]2 do . For the proof of part (i), let y , t o , t be as assumed. For every 7 2 0 we have, using 23.M and Theorem 61.A with z = 0, h arbitrarily close to 1, and do instead of d,
+
Taking 7 = to md,, m = 0, ..., [d/dO]-1, and adding, we find (61.6), since d,/d, = (A/d,,)/[d/dO]. Applying Theorem 61.A (with do instead of A ) and 23.M in reverse order, we prove (61.7) as follows:
For the proof of part (ii), let A, z, to , t be as assumed. For any 7 2 do the same argument as above gives
since [(T - ( t - to) - A , ) + , (T - ( t - to))+] is either (0) or an interval of length d, contained in [0, T - do]. T h e inequality between the
<
170
AND Ch. 6. ADMISSIBILITY
DICHOTOMIES
<
extreme members of (61.10) remains true for any real T , since T A, implies (T - ( t - to))+= 0 so that the first member is 0. Taking, then, T = t - mA, , m = 0, ..., [A/d0]- 1 and adding, we find (61.8). Finally, (61.9) follows by
I Xcft0-A)
ID
+.to]’
X[O.l-Aol’
ID
We prove a corollary of Theorem 61.A which describes (pointwise, but nonuniformly) the angular separation of solutions y, z of (60.1). 61.C. Assume that Y is a (B,D)-subspace. For every X > 1, A > 0, every pair of nontrivial solutions y , z of (60.1) with y(0) E Y , 11 z(O)(I< h d( Y , z(O)),and all T 2 0, we have (61.11)
y[y(T), z(T)] 2
( A K ~ T ( A ) )exp(-2f+” -~
11 A(u)11 du).
7
Proof. Set y1 = 11 JJ(T)II-~~, Theorem 61.A we have
z1 =
-11
z(~)11-%. Using 23.M
and
From (31.7) we then have
Since Y ~ ( T )z,( ~ )= ] 11 yl(T) -k zl(T)ll, the corollary is proved.
&
62. Predichotomy behavior of the solutions of the homogeneous equation Means and slices o f solutions As explained in the Introduction, we now show, in Theorem 62.A, how near we can get t o a dichotomy of the solutions of (60.1) by merely
62. PREDICHOTOMY BEHAVIOR
171
OF SOLUTIONS
assuming the existence of a (B, D)-subspace (or, in particular, the regular admissibility of some (B, D)): namely, how the growth conditions for a dichotomy, which compare values of 11 x(to)ll, 11 x(t)ll for solutions x of (60. I), are replaced by thesame comparisons for the means J'f:" 11 x(u)lI du, 11 x(u)ll du or for the D-norms of the slices ~ r f , , f o + A l Xcr,r+dlx. x, For increased symmetry, as well as for slightly greater generality, the means and slices of some of the solutions are computed on intervals of the form [(to - A ) , , to], [(t - A ) , , t ] . Theorem 62.B shows in what way the behavior of the solutions of (60.1) imitates an exponential dichotomy when (B, D ) is not weaker than (L1, L:). This is an important instance of the cardinal role played by the spaces L1,L",: L as terms of comparison.
Jy
Assume that Y is a (B, D)-subspace (in particular, 62.A. THEOREM. that (B, D ) is regularly admissible and Y = XoD).Then (i) for every d > 0 there exists M = M ( d ) > 0 such that every solution y of (60.1) withy(0)E Y satisfies, for all t 2 to 2 0,
(ii) for every h > 1, d > 0 there exists M' = M'(d, A) > 0 such that every solution z of (60.1) with 11 z(0)ll h d( Y , z(0)) satisfies, for all t 3 to 3 0,
<
t
(62.3) ( t-A ) .+
(62.4)
II 44 II du 2 M'-l
I'"
(t0-A) +
1l.wII du,
I. x [ ( t - d ) + . t J Z ID 2 M'-l I x [ ( t ~ - A ) + . t ~ lID~ ;
+
we may take, in particular, M = 1 K$F(d),M' = 1 where KF is the constant in Theorems 61.A, 61.B.
+ hKiF(d),
+
Proof. We use Theorem 61.B with do = A . For t >, to A , (61.6) implies (62.1) at once; for to < t < to d, use of (61.6) gives II y(u)ll du II y(u)ll du (1 KFW)) II y(u)lI du. T h e proof of (62.2) is entirely similar, but uses (61.7). Again for t 3 to + d, (61.8) implies (62.3); for to t to + A ,
Jl'"
< J'r,,,
<
+
+
sy
< <
AND Ch. 6. ADMISSIBILITY
172
DICHOTOMIES
by (61.8) if t 3 d, and otherwise because t < d implies [(t - 2d)+, (t - A),] = {O}. Thus (62.3) holds; and (62.4) follows similarly from (61.9). & In order to prove the next theorem we require 20.C. Assume that (B, D ) is not weaker than (L1, LT)and 62.B. THEOREM. that Y is a (B, D)-subspace (inparticular, that (B,D ) is regularly admissible and Y = XoD).Then: (i) there exists p > 0, and for ewery d > 0 there exists M = M(d) > 0, such that ewery solution of (60.1) with y(0)E Y satisfes, f o r all t 2 to 2 0, (62.5)
IX [ t . t + A l Y
(62.6)
"e"'t-to'
ID
I X[lo.to+d]J'
ID ;
(ii) for ewery h > I there exists p' = p'(h) > 0, and for ewery such h and 4 > 0 there exists M' = M'(d, A) > 0, such that ewery solution z of (60.1) with 11 z(O)11 A d( Y , z(0)) satisfies, f o r all t 2 to 2 0,
<
I X[(t-d)+.t12
(62.8)
ID
2 M'-leu'(c-t I X [ ( t o - A ) + . l o l Z
ID
.
Proof. By the assumption on ( B , D ) it follows from 23.S, with Remark 2, that either a(B; 00) = 00 or B(D; 00) = 00; therefore limA+,, r(d) = 0.
<
Proof of (i): We choose A, so large that 3KCr(d0) 1. For the given A, set p = [dO/d] 1. Now pd > d o , and (61.6) yields J W 1 + P " 11 y(u)ll du $ I1 y(u)ll du. On account of Theorem 62.A,(i) we may now apply 20.C to y ( t ) = (J'"" 11 y(u)ll du)-', with T = d o , H = < (1 KC~(A,))-~(1 K$QA))-~, r] = (unless y = 0, in which case (62.7) is trivial); we conclude that for any t 2 to 2 0,
<
+
I:pA 11
+
S T
+
< +
y(u) 11 du
< 2e-;(t-to)
where p = Ail log#. Applying now (61.6) with do= d and with to + md,m = 1, ...,p - ,I, instead of t , and adding, we obtain
to
to
62. PREDICHOTOMY BEHAVIOR
OF SOLUTIONS
173
(this includes the case p = 1). Combining, we have PttA
+l+nA
where M = M ( d ) = 2(1 entirely similar.
+ [d,/d]K;r(d)).
T h e proof of (62.6) is
Proof of (ii). We choose 0 ; = d;(A) so large that 3XK!I'(dJ With p' = [d;/d] 1, (61.8) yields
+
So+A;
II z(u) II du
\ tO+Ab-Jl'A)+
< Q fa
< 1.
II z(u) i/ du, ( to-P'A)+
so that on account of Theorem 62.A, (ii) we may apply 20.C to ~ ( t=) t T h e proof then J,l--p,d)+ 11 z(u)ll du, with 7 = dI,, H = $, r ] = $. continues as in part (i), and (62.7) holds with p' = p'(A) = (d;(A))-' log#, M' = M ' ( d , A) = 2( 1 A[d~(X)/d]K;r(d)).T h e proof of (62.8) is entirely similar. &
+
Remark 1. With the formulas given in the proof, M , M' are nonincreasing functions of d (both in Theorem 62.A and in Theorem 62.B). We may obviously assume that 0 ; is nondecreasing with A; then p' is nonincreasing with A and M' (which is linear nondecreasing with A in Theorem 62.A) is a non-decreasing function of X in Theorem 62.B. Remark 2. T h e reader will easily supply the immediate consequences of parts (ii) of Theorems 62.A, 62.B for the solutions starting from a complement of Y , if any exists. Remark 3. For a slight weakening of the assumptions on Y in Theorems 62.A, 62.B, see 62.F. We record a useful corollary of Theorems 61.A, 62.A, 62.B:
62.C. Assume that Y is a (B,D)-subspace (in particular, that (B, D) is C Y C X,, C X,,, = regularly admissible and Y = X,,). Then XOM, X,,,, = X,, . Zf D is quasi full, X,, = X,, . If (B, D ) is not weaker than (L1,L :), then all the inclusions become equalities.
174
Ch. 6. ADMISSIBILITY AND
DICHOTOMIES
Proof. For any solution z of (60.1) with z(0) 4 Y there exists X > 1 such that 11 z(O)l\ X d( Y , z(0)). It then follows from Theorem 62.A, (ii) (Eq. (62.3)-also from Theorem 61.B) that XoM,C Y; to prove the first statement it remains to show that X,, C X,,,, since the other inclusions are trivial. Let z be an M-solution of (60.1); then either z(0) E Y C X,, C X,,, , or z(0) 6 Y; in the latter case, Theorem 61.A implies, for an appropriate X > 1 and every T 2 0,
<
so that z E fD(X), z(0) E X,,, . The stated result for quasi full D is now immediate. If (B,D) is not weaker than (Ll,LT)we apply Theorem 62.B: by the same argument as above, (62.7) implies X,,C Y; and (62.5) yields Y C XoM,;therefore equality holds all around. &
Pointwise nonuniform properties of solutions
*Thetwo theorems we now prove show that by assuming slightly more about (B, D ) we can improve the conclusion of Theorems 62.A, 62.B (or rather, one-half of either at a time) to a comparison of pointwise values of the solutions of (60.1), but at the cost of the uniformity of the comparison. Assume that D is stronger than L" [and that (B, D) 62.D. THEOREM. is not weaker than (Ll,LT)],and that Y is a (B, D)-subspace (in particular, that ( B ,D ) is regularly admissible and Y = X,,,). Then there exists a positive-valued function L on R, such that every solution y of (60.1) with y(0)E Y satisfies,for all t 2 to > 0,
where p is as in Theorem 62.B]; more precisely, there exists for every
d > 0 a number Lo = L,(d) > 0 such that (62.9) [(62.10)] holds with L(t,) = L,(d) e x p ( J y 11 A(u)ll du). [Furthermore, Y = X,, = X,,, is a ( B , kD)-subspace, and (kB,kD)is regularly admissible.] Proof. Since
D is stronger than L", there is no loss in assuming
62. PREDICHOTOMY BEHAVIOR
175
OF SOLUTIONS
+
<
L". For given d > 0 we have, for all t to d (using Theorem 61.A, Theorem 62.A,(i) [Theorem 62.B,(i)], and (31.7)):
D
IIr(t)11 < I @tY I G I @tY ID
for to
< t < to + A , (31.7) yields directly
T h e conclusion follows, with
[Since decreasing exponentials are in T, 62.C implies Y C XoFC = Y, so that equality holds all around. By 52.B,(c), Y 1s a (B, kD)-marLifold; by Theorem 52.F, (kB, kD)is admissible.] &
X,,,, C X,,,
62.E. THEOREM. Assume that B is weaker than 8,Ll for some T 2 0 [and that (B,D ) is not weaker than (L1, L:)], and that Y is a (B, D)subspace (in particular, that (B,D ) is regularly admissible and Y = XoD). For every X > 1 there exists a positive-valued function L'( *, A) on R , such that every solution z of (60.1) with (1 z(O)11 X d(Y, z(0)) satisfis, for all t 2 t o 3 0,
<
(62.11) [(62.12)
II N t ) I1 2 &'(to I1 z(t) II 2 ( c o o
1
I
w1II 4 l o ) 1I w'e"'t-to)
1I z(t0) II >
where p' = p'(X) is as in Theorem 62.B]; more precisely, there exists for every X > 1, d > 0 a number Lb = Lb(d, A) > 0 such that (62.11) [(62.12)] holds with L ' ( t o , A) = L;(d, A) e x p ( J r 11 A(u) 11 du).
AND Ch. 6. ADMISSIBILITY
176
DICHOTOMIES
Proof. By the assumptions, Y is a (@Ll,D)-subspace, hence an (L1,D)-subspace (52. K); let K!l be the corresponding constant. Applying Theorem 61.A with 7 = t and letting d tend to 0, we obtain, since a(L1;A ) = 1 identically,
IX[O.t]Z ID d UK”Y1 I! z ( t ) I1 ’
(62.13)
Combining (62.13) with Theorem 62.A,(ii) PTheorem 62.B,(ii)] and (31.7) for a given value of A , and for t >, to + A , we have
<
for to t ,< to + d we apply (31.7) as in the previous proof, and obtain the conclusion with
Remark 1. If, under the assumptions of Theorem 62.D, we apply the argument of the proof to the solutions z of (60.1) with 11 z(0)ll < Ad( Y, z(O)),we obtain an inequality of the form
where L’(t, A) = Lh(d, A) exp(J:+d 1) A(u)ll du). Except for this special form of L’(*,A), the inequalities are of course trivial consequences of (31.7). A similar argument under the assumptions of Theorem 62.E yields for the solutions y of (60.1) with y(0) E Y an inequality
62. PREDICHOTOMY BEHAVIOR
OF SOLUTIONS
177
where L(t) = L,(d) exp(Jl+d 11 A(u)ll du); and except for this special form of L the inequalities are again trivial by (31.7).
Remark 2. T h e part of Theorem 62.D not in square brackets is, except for the special form of L, an immediate consequence of Theorem 33.B and (3 1.7), and is included here for the sake of symmetry (but see also Theorem 82.T, with Remark). Remark 3. T h e reader is left to supply the consequences of Theorem 62.E for the solutions starting from a complement of Y , if any exists. In Theorems 62.A, 62.B, 62.D, 62.E, as stated, the subspace Y depends on B. As a postscript to these theorems we now show that a slightly more general formulation allows us to avoid this dependence:
62.F. Theorems 62.A, 62.B, 62.D, 62.E, and also 62.C remain valid,except for the specijic values of M , M ’ in Theorem 62.A, q the assumption: “ Y is a ( B ,D)-subspace (inparticular, ( B ,D) is regularly admissible and Y = Y0J’ is replaced by the weaker assumption (cf. Theorem 52.L): “there exists a ( B ,D)manqold (in particular, ( B ,D) is admissible) and Y is a (T, D)-subspace”. Proof. For Theorem 62.A, Theorem 62.D (part not in square brackets), and 62.C (except the last statement) we simply use (T, D) instead of (B, D) (so that the assumption on (B, D) is redundant). For Theorem 62.E (part not in square brackets), we observe that Y is a (T, 1cD)-subspace and that there exists a (B, 1cD)-manifold, so that, by Theorem 54.G, Y is a (B, 1cD)-subspace; and we use this pair instead of (B, D). Finally, for Theorem 62.B, the parts in square brackets of Theorems 62.D, 62.E, and the last statement of 62.C, it follows from Theorem 54.1 that Y is actually a (B, D)-subspace, so that there was no real gain of generality. &
Miscellaneous corollaries We can now prove the promised theorems on sets of admissible .F-pairs, etc., that were mentioned at the end of Section 52. The following two results are to be contrasted with 51.M, 52.S.
62.G. Let B E b y or b y + or b.FV+ be given. There exists a full space Do E b y K such that a Y - p a i r or Y-’-pair 01 Y V + - p a i r (B,D ) with quasi full D is regularly admissible if and only if D is weaker than D o , provided
Ch. 6. ADMISSIBILITY AND
178
DICHOTOMIES
some such regularly admissible pair exists, namely, provided (B, M)is regularly admissible. Proof. There is no loss in replacing “quasi full” by “full” in the statement. Under the proviso, let 9 be the nonempty class of all full D E b y K such that (B, D) is regularly admissible. If D, ,D, E 9 we have D, v D, stronger than M;hence, using 62.C, XoDlC XO,DIV,,) C XoM= XoDl Since the correspondence D f+ D(X) preserves meets and joins, we may apply Theorem 5 1.N and conclude that there exists a space Do = hu,D that is either in b.9- or = (0) (by 23.D), and is stronger than all D € 9 , and such that (B,D,) is regularly admissible. But Do is full (by 22.H), and since B # {0},Domust contain some nonvanishing continuous function, hence Do€b y K . &
.
62.H. Assume that XoMo= XoM(in particular, that there exists an (F,G)-subspace for some (F,G ) not weaker than (Ll, L:)). Let B E b y 07 b y + or bYW+ be given. There exists D,E b y K such that a 9--pair 07 .T*-pair or Y P - p a i r (B, D) is regularly admissible ;f and only i f D is weaker than Do,provided some such regularly admissible pair exists, namely, provided (B, M)is regularly admissible. Proof. That XoMo = XoMin the stated particular case follows from 62.C. We let 9 be the nonempty class of all D E b y K such that (B, D) is regularly admissible. For every D € 9 ,62.C implies XoM,C X,, C XoM, so that equality holds. The rest of the proof is the same as for 62.G. & The following analogue of Theorem 51.G requires no reflexivity assumptions, and therefore extends the validity of the conclusion of that theorem in some cases to nonreflexive X. We include it on account of its interesting proof. Assume that (B,D) is not weaker than (Ll, ):L and 62.1. THEOREM. that there exists a (B, D)-subspace (in particular, that (B, D) is regularly admissible). Then (IcB, fD) is regularly admissible. Proof.
Let Y be a (B, D)-subspace. By 62.C, Y
= X,, = X,,,
,
so that this manifold is closed. By Theorem 52.C, Y is an (lcB,D)-sub-
, space (and 1cB E b y + ) ; set S, = S , , K, = Kyle,,, , C , = CulcB,D =K, 2SyC,. Let f E lcB(X), h > 1 be given, and set u = I f llcB For every nonnegative integer n there exists, by Theorem 52.1, a solution Y n of Yn AYn = ~ [ n . n + l ] f with Ynm(0) E Y , II~n(0)ll d(Y,Yn(O)), and I Yn I D G =!I, ~ [ n . n + l ] f IlCB XKho, IIYn(o)II ~ C F . For every n,yn coincides, in the interval [0, n ] , with a solution of (60.1), K;
+
+
.
<
<
<
63. DICHOTOMIES: GENERAL
179
CASE
to which we may apply Theorem 62.B (formula (62.7)): with d = 1, M' = M'(1, A) we find, for n = 1,2, ...,
p' = p'(A),
< M'a(D; l)e+'(n-l)
I yn ID
Using (31.7) and considering the case n (62.16)
< AKkM'a(D; l)e-"'"+l)a.
=0
IIyn(o)ll < h K C P ' %
separately we finally obtain
? = I
0,1, ...,
+ s,'
where K = max{Cy, KkM'a(D; 1) exp(p' 11 A(.)/[ du)}. Set xn = E;-'ym,n = 1,2, Then x, is a solution of jin Ax, = ~ ~ ~ By , ~ the l fassumption, . there exists another solution z, of the same equation with znm(0)E Y, I an X Y o , 11 zn(0)ll AC9. Then n-1 xn - 2, = x, - z,,, = Em=Oymm - z, is a solution of (60.1) with xn(0) - zn(0)E Y. Therefore we find, using (62.16),
... . ID <
+
<
By (62.16), (~~(0)) is a Cauchy sequence; let x be the solution of (60.2) with x(0) = limn-rmxn(0). By (31.5), xn -+ x uniformly on every interval [O, TI, T > 0. Therefore (62.17) yields
I ~ [ o . ~IDl x< limn+m SUP I x[o.~]xnIo
< A(K1 + K S y ( l
+ bz I ~ [ o . r l ( x- xn) I P(D;
- e+')-l)
If IlcB
7)
+ 0;
we conclude that x E fD(X). Therefore (IcB, fD) is admissible; and we showed above that XO,, is closed. &
63. Admissibility, (B, D)-subspaces, and dichotomies: the general case Ordinary dichotomies We have now reached the point where we can show in what cases a (B,D)-subspace induces a dichotomy for A and, conversely, for what
Ch. 6. ADMISSIBILITY AND
180
DICHOTOMIES
pairs (B, D ) a subspace that induces a dichotomy is a (B, D)-subspace. We begin with the main direct theorem for ordinary dichotomies. Assume that (B.D ) is stronger than (O,L1, L") for 63.A. THEOREM. some T 2 0, and that Y is a (B, D)-subspace (in particular, that (B, D) is regularly admissible and Y = XoD).Then Y induces a dichotomy for A. Proof. By the assumption, Y is a (8,L1,Lm)-subspace, hence an (Ll, L")-subspace (52.K); let K;, be the corresponding constant in Theorem 61.A, and recall that a(L1;d) = 1 for all d > 0. We shall prove the validity of condition (e) or (f) of Theorem 41.A. Let A > 1 be given, and consider solutions x , y , z of (60.1) with x =y z, y(0)E Y , 11 z(O)(I h d( Y , z(0)). If t 2 to > 0 we take A , 0 < d to and apply Theorem 61.A with T = to - d to find
+
<
<
I I Y ( ~ ) I1 < I @toyI < XW'
s'"
to-A
II x(u) II du;
letting d tend to 0, and using the continuity of x at to = 0, we obtain (D'i) with D = 2hK;,; if to 2 t 2 0 we apply Theorem 61.A with T = to to find
letting d tend to 0, we obtain (D'ii) with the same value of D. Another proof uses the definition of dichotomy, and verifies (Di), (Dii), with N = 2K;, ,N' = 2hK;, , by means of Theorem 61.A; and (Diii), with yo = (4AK;,)-', by means of 61.C. & The assumption on Y can be weakened a little (cf. 62.F, Theorem 52.L): 63.B. Assume that (B,D)is stronger than (O,L', L") for some 7 2 0, and that there exists a (B,D)-manifold (in particular, that ( B , D ) i s adD)-subspace, then Y induces a dichotomy for A. missible). If Y is a (T, Proof. By the assumption, there exists an (Ll, L")-manifold (52.K), and Y is a (TIL")-subspace; hence Y is an (Ll, L")-subspace (Theor e m 54.G), and Theorem 63.A yields the conclusion. &
T h e principal converse theorem for ordinary dichotomies is the following: I f the subspace Y of X induces a dichotomy for A, 63.C. THEOREM. then Y is an (Ll,L")-subspace and (L1, L") is admissible.
63. DICHOTOMIES: GENERAL
CASE
181
Proof. By definition (or Theorem 41.D), Y C X o . Let A > 1 be given; by 1 1 .E there exists a continuous ( Y , A)-splitting q of X , to which we refer condition (g) or (h) of Theorem 41.A, with the corresponding constant D = D(q). Let U denote, as usual, the solution of (30.3) with U(0) = I. Let f E koL1(X) be given. Since U - t f ~L(X) and q is continuous, the functions g = U-tf - q( U-lf ), h = q( U - l f ) are strongly measurable; since 11 q(x)ll < All x 11, g , h E L(X) and s(g), s(h) < s(f) < a;also, the values of g are in Y . By (D"i), (D"ii) we have
1 t
(63.2)
x(t) =
1
m
U(t)g(u)du -
0
t
U(t)h(u)du;
direct verification shows that x is a solution of (60.2). Now x, = U J:(')g(u) du, whence xm(0) = Js") g(u) du E Y . Further, (63.1), (63.2) 0 imply 11 x(t)ll ,< D 11 f(u)lI du = Dl f Il for all t E R, , so that Y C X , is indeed an (L1, L")-subspace, with K , D. T h e admissibility of (L', L") follows from 52.G, since L' is lean. &
Jr
<
We summarize our conclusions in Theorems 63.A, 63.C:
A subspace Y of X induces a dichotomy for A if 63.D. THEOREM. and only if Y is an (L', L")-subspace. If Xois closed, Xoinduces a dichotomy f o r A i f and only i f (L', L") is admissible. Among the immediate corollaries of this theorem we have the still outstanding proofs of some statements in Chapter 4.
Proof of Theorem 41.E. Theorem 63.D and 52.B,(b). & co
ploof of Theorem 43.E. Except for the last statement, the conclusion follows from Theorems 63.D and 53.D. If x 4 (X;)O, there exists x* E X ; such that (x, x*) # 0. Set W* = Y o Fx*, the subspace of X* spanned by YO and x*; then W* satisfies the assumptions of Theorem 43.E, so that Y n (W*)O induces a dichotomy; but x 4 (W*)O. &
+
Our next task is the study of the relations between dichotomies, Xoo , and the pair (L1, L:). We first give an alternative, simpler, but no longer purely "geometrical" proof of Theorem 41 .F.
182
AND Ch. 6. ADMISSIBILITY
DICHOTOMIES
Proof 11 of Theorem 41.F. Let N be the number corresponding to Y in the definition of a dichotomy. Let Yo be any (finite-dimensional) complement of X, with respect to Y. As in part 2 of Proof I it follows that there exists /? > 0 such that every solution yo of (60.1) with yo(0)E: Y o satisfies, for all t o ,t 2 0,
II YoO) II 2 B II YO(0) II 2 BN-l II YO(l0) II
*
Now, by Theorem 63.D, Y is an (Ll, Lm)-subspace; let K , be the corresponding constant. Let f E kJ.,l(X) and p > 1 be given and let x be a solution of (60.2) with xm(0) E Y, I x I pK,) f Il . There is a unique representation x, = y + y o , where y , y o are solutions of (60.1) with y(0)E X,, ,yo(0)E Y o . Now x = x - yo is a solution of (60.2) with zm(0)= xm(0) - yo(0) = y(0) E X,, . Further,
<
Hence X,, is also an (Ll, La)-subspace; by Theorem 63.D it induces a dichotomy. &
63.E. THEOREM. If the subspace Y of X induces a dichotomy for A and X,, has jinite codimension with respect to Y , then X,, is an (Ll, Lg)subspace and (Ll, L): is admissible. Proof. By Theorems 41.F and 63.D, X,, is an (L1,Lm)-subspace; by 52.B,(c), it is also an (L1,L:)-subspace. The admissibility of [Ll, L;) follows from 52.G. &
63.F. If X,, is closed, it induces a dichotomy for A if and only if (Ll, L;) is admissible. Proof.
Theorems 63.A and 63.E. &
Exponential dichotomies
We proceed to establish the principal direct theorem for exponential dichotomies.
63.G. THEOREM. Assume that (B,D ) is not weaker than (L1, Lz) and that Y is a (B,D)-subspace (inparticular, that (B, D ) is regularly admissible and Y = XoD).Assume further that there exists a subspace Z of X that
63. DICHOTOMIES:GENERAL
183
CASE
induces a dichotomy for A. Then Y = Z induces an exponential dichotomy for A.
Proof. By Theorem 41.D, Xoo= XOMo C 2 C XoM= X , ; but by = Y = XoM. Therefore Y = 2. 62.C, XOMo Thus Y induces a dichotomy for A; choose A > 1, and use the notations N o , NI, ,yo for the corresponding parameters. We shall establish condition (b) of Theorem 42.A for Y and this A. Let y, z be solutions of (60.1) withy(0) E Y, 11 z(O)11 A d( Y, z(0)).By Theorem 62.B and conditions (Di), (Dii) we have, for any t to 1,
<
+
< e@’NL2M’e-@‘+-to)11 s(t)11 , where M = M(1), p‘ = $(A), (Di), (Dii) imply IIy(t) 11
< e@Noe-@(t-to) IIy(to)11 ,
M’ = M’(1, A). For to
11 z(to)1)
< 1 < to + 1,
< e@‘Nie-@’(t-to)11 z ( t ))I .
Thus conditions (Ei), (Eii) of Theorem 42.A,(b) are satisfied with v = I*., v’ = p’ (since p’ depends on h we could not prove directly the
conditions for an exponential dichotomy as given in the definition),
N = epNtM, N’ = e@’Ni2M‘(we use the obvious fact that N o , N i , M , M’ 2 1). Condition (Diii) holds by assumption, with the given y o . 4, Remark. We could refine the assumption on Yin the same way as in 63.B; but in the present case Theorem 54.1 shows that Y would be a (B, D)-subspace anyway, so that the gain in generality would be only apparent.
co
A natural question to ask is whether it is not sufficient for an exponential dichotomy that Y be a mere (B, D)-manifold; a partial affirmative answer is the following result:
00
63.H. Assume that B is not stronger than L1 and that there exists a (B, D)-manifold (in particular, that (B,D) is admissible). Assume further that there exists a subspace Z of X that induces a dichotomy for A. Then Z induces an exponential dichotomy for A. Roof. A (B,D)-manifold is a (B, M)-manifold. By Theorem 63.D, Z is an (Ll, L“)-subspace, a fortiori a (T, M)-subspace. By Theorem 54.G,
184
Ch. 6. ADMISSIBILITY AND DICHOTOMIES
Z is a ( B ,M)-subspace. Since ( B ,M) is not weaker than (L1,L:), Theorem 63.G yields the conclusion. & 00
For the remaining case among pairs ( B ,D) not weaker than (Ll,L:) we accordingly formulate:
00
Query. If B is stronger than L1,but D is not weaker than L:, and there exists a ( B ,D)-manifold (or (B, D) is admissible), does a subspace Z that induces a dichotomy for A also induce an exponential dichotomy for A I
00
The following partial answer shows that at least condition (Eii) does hold for 2 and any h > 1 (besides the assumed (Diii)).
00
63.1. Assume that B is stronger than L1, that D is not weaker than L:, and that there exists a (BID)-manifold Y (in particular, that ( B , D ) is admissible and Y = X,,). Assume further that there exists a subspace Z of X that induces a dichotomy for A. Then Y C X,, C X , = 2 = X, = clY; Y is a ( B ,D A L:)-rnanifold; and there exists Y' > 0 , and for every h > 1 there exists N' = "(A) > 0, such that ewery solution z of (60.1) with 11 z(O)11 h d(Z, z(0)) satisfies (Eii).
<
Proof. See Section 66 (p. 221). J, Remark. In order to give an affirmative answer to the Query it is sufficient to do so, as 63.1 shows, for the case in which X , = X , induces D) (Theothe dichotomy and condition (Eii) holds, and for the pairs (TI rem 52.L); indeed, it is sufficient to assume either that D is locally closed, for we can replace D by D, and then by IcD, , which is also not weaker :L for 63.1 shows that we can replace than L: (by 23.S); or that D , D by D A L:.
00
<
An obvious corollary of Theorems 63.A and 63.G is:
La)for some 63.J. Assume that (B,D ) is both stronger than (O,L1, >, 0 and not weaker than (Ll,L,").I f Y is a (B, D)-subspace (inparticular, if (B,D ) is regularly admissible and Y = X,,), then Y induces an 7
exponential dichotomy f o r A. We refer the reader t o the concept of a function space thick with respect to another (Section 23, p. 71, and Section 24, p. 82), which we require to formulate the main converse theorem for exponential dichotomies:
63.K. THEOREM. I f the subspace Y of X induces an exponential dichotomy for A, and i f D is thick with respect to B,then Y = X,, is the D)-subspace and ( B ,D ) is regularly admissible. unique (B,
185
63. DICHOTOMIES: GENERAL CASE
Proof, T h e proof parallels closely the proof of Theorem 63.C, to which we refer for details. By Theorem 42.F, Y = X,, . We show that (B, D ) is admissible; the fact that Y is the unique (B, D)-subspace then follows, e.g., from 62.C. Let X > 1 be given, and q a continuous (Y, A)-splitting of X, to which condition (g) or (h) of Theorem 42.A applies with the constants v , Y’, D = D(q) > 0. Let f E B(X) be given, and define g = U-lf q( U - l f ) , h = q( U - y ) , so that g , h E L(X) (iff is continuous, so are g, h); by (E”i), (E”ii) we have (63.3)
11 U(t)g(u)11 < De-Y(t-u’IIf(u) 11 11 U(t)h(u)I( < De-’‘(u-t) Ilf(u) I1
for all
t
for all
u
2 u 2 0, 2 t 2 0.
We define x by (63.2); provided the integral exists, x is a solution
<
of (60.2), by direct verification; but (63.2) and (63.3) yield 11 x(t)ll D(J‘e-v(t-U)lIf(u)ll du J:e-v’(u-f)[lf(u)ll du), so that, by the assumption
+
on (B,D) and 23.V,( I), 24.N, the integral exists and x
E D(X).
Thus
(B, D) is admissible. If we had used 23.W,(l), the analogy with the proof of Theorem 63.C would have been a much closer one. 9, 63.L. If the subspace Y of X induces an exponential dichotomy f o r A, then the following pairs (and all weaker ones) are regularly admissible: (Bv L1,B A L“) ( afortiori (B, B)) and ((B v L1)A M,, , B A L:) for every locally closed B E b r ; i n particular, the pairs (M, L“), (Ll,T),(M,, , L:); if (B, D ) is (regularly) admissible, then so is (kB, kD). Proof. Except for the last statement, Theorem 63.K and 23.W,(3), (4), (5), (6). Assume that (B, D) is admissible; by Theorem 42.F, Y = X,, = X,,, (which is closed), and Y is a (B, D)-subspace (Theorem 52.F); by 52.B,(c), Y is a (B, kD)-subspace; by Theorem 52.F, (kB,kD)is admissible. 9, Some useful necessary and sufficient conditions form an analogue to Theorem 63.D and 63.F: (1) A subspace Y of X induces a n exponential 63.M. THEOREM. dichotomy for A if and only if Y is both a n (L1,L”)-subspace and an (L2,L2)-subspace; if and only if Y is an (M, L“)-subspace; if and only if Y is an (L1,T)-subspace; if and only if Y is an (M,, , Lr)-subspace. (2) If X,,, X,, , X,, is closed, it induces a n exponential dichotomy for A if and only if (M, L“), (Ll, T),(M,,, L:), respectiwely, is admissible. Proof. For the “if” parts: Theorems 63.D and 63.G, and 63. J. For the “only if” parts: 63.L (and Theorem 42.F). if,
186
Ch. 6 . ADMISSIBILITY AND
DICHOTOMIES
Sets o f pairs
In general, results describing classes of pairs (B,D) that are admissible, or such that there exists a (B,D)-subspace, etc., are not very satisfactory (see the appropriate subsections of Sections 51, 52, 62). When there is a dichotomy for A, however, we can give a rather precise answer t o the D) (of course F - p a i r s or F+-pairs or following questions: what pairs (B, YV+-pairs only are meant, as throughout this chapter) have a (B, D)-subspace, are admissible, are regularly admissible ? T h e question concerning mere admissibility is distinct from the one about regular admissibility only for infinite-dimensional X, and is not completely answered here. Assume that there exists a subspace that induces an ordinary dichotomy, but not an exponential dichotomy, for A (then no other subspace induces an exponential dichotomy either). We shall say that A is in case I or case II, according as X,, does or does not induce a dichotomy; and in case I or case 2, according as A,' is closed (and therefore induces a dichotomy, by Theorem 41.E) or not. There are thus four combinations: cases 11, 12,111,112. Obviously case I1 alone is possible if X is finite-dimensional (Theorem 44.A); Examples 65.A (with A = 0), 41.H, 41.1, 65.R show that all four combinations are possible if X is separable Hilbert space. We observe that in case I1 X,, has infinite codimension with respect to every dichotomy-inducing subspace (Theorem 41 .F); in particular with respect to X , in case 111.
63.N. THEOREM. Assume that there exists a subspace that induces an ordinary dichotomy, but not a n exponential dichotomy, for A. Then:
if(& D ) is:
D)-subspaces the (B, X,, is: are:
weaker than X,, (L', L") not weaker than (Ll, L") xoo but weaker than (L1, X , or C X , not weaker than (L1, L;) 00
is ( B ,D ) admissible ?
all subspaces inducing a dichotomy for A
I
case I: X , case ZI: none
none
1 no, except perhaps if B is stronger than L1, X,, (in particular, A is in case II).
yes Yes no
1
C
is (B, D) regularly admissible ?
I
case I :yes case 2: no Y" no
no
X,
= X , = dX,,
63. DICHOTOMIES: GENERAL
CASE
187
Proof. 1. Assume that Y is a (B, D)-subspace. We claim that Y induces a dichotomy for A. Indeed, Y is a (T, D)-subspace (Theorem 52.L), a fortiori a (T, M)-subspace. But the class of (L1, L")-subspaces coincides with the class of the dichotomy-including subspaces (Theorem 63.D), and this is not empty by assumption; by 54.H, every (T, M)-subspacein particular, Y-is an (Ll, L")-subspace, and induces a dichotomy for A. 2. If (B, D ) is weaker than (Ll, L"), every dichotomy-inducing subspace is an (Ll, L")-subspace, a fortiori a (B, D)-subspace; the converse follows from part 1 of the proof. Since D is weaker than L", X,, = X , (Theorem 41.D). By Theorem 63.C, (Ll, L") is admissible; a fortiori so is (B, D). If (B, D ) is not weaker than (Ll, L"), but weaker than (L1, L:), we must have B stronger than L1 and D weaker than L,: but not weaker than La. By Theorem 41.D,X,, = X,, , a subspace. I n case I, Theorem 63.E implies that (L1, L): is (regularly) admissible, and so is the weaker pair (B,D); and X,, = X,, is a (B, D)-subspace. If, conversely, Y is a (B, D)-subspace, part I of the proof and Theorem 41.D imply X,, C Y C X,, , so that there is equality all around; and Y = X,, induces a dichotomy for A, so that we are in case I. If (B, D) is not weaker than (L1, L,"), a (B, D)-subspace would induce an exponential dichotomy (Theorem 63.G), hence none exists, and (B, D ) is not regularly admissible. 00
63.H and 63.1 (the proof of the latter is pending) show that, perhaps D ) would imply with the stated exception, the mere admissibility of (B, the existence of an exponential dichotomy, and is therefore excluded. &
6 3 . 0 . Under the assumption of Theorem 63.N:
if A is in: case I1 case I 2 case I11 case 112
weaker than (L1, L,")
I
( B ,D) is regularly admissible if and only if it is:
there exists a (B,D)-subspace if and only if (B,D ) is:
weaker than (Ll, L")
I I
weaker than (Ll,L r ) weaker than (Ll, L): but not weaker than (L1, L") weaker than (Ll, L")
63.P. Assume that X is jinite-dimensional, and that there exists a subspace that induces a dichotomy, but not an exponential dichotomy, for A .
188
Ch. 6. ADMISSIBILITY AND
DICHOTOMIES
Then (B, D) is admissible if and only q, and there exists a (B, D)-subspace if and only if, (B, D) is weaker than (Ll, L,“); if, in addition, (B, D) is weaker than (L1, L“), the (B, D)-subspaces are exactly the subspaces Y with X,, C Y C X,; otherwise, the unique (B, D)-subspace is X,, . Proof. Theorems 63.N and #.A.
&
Oddly enough, the results we possess are not so clear-cut in the case of an exponential dichotomy. It is true that in that case the following facts are known: X, = X,, = X,,,(for any D E byK) is the unique subspace inducing an exponential dichotomy (or indeed, an ordinary one); and the unique (B, D)-subspace, if any exists (62.C, or part 1 of the proof of Theorem 63.N); that consequently, if there exists any (B, D)-manifold, the (B,D)-manifolds are exactly the linear manifolds contained and dense in X , (52.B,(d)); that there exists a (B, D)-manifold, and hence a (B, D)-subspace, if and only if (kB, D) is (regularly) admissible (Theorem 52.F); and that (B, D ) is admissible if and only if it is regularly admissible. We further know (Theorem 63.K) that this is the case when D is thick with respect to B;in some cases, no other (B, D ) is admissible (Theorem 64.C), but it is possible to construct even scalar equations for which additional pairs are admissible (Examples 65. N, 65.0). In view of the strength of these examples, and recalling 5l.Q, we formulate: D) that is (regularly) admissible for some Query. Is every (B, A, E L( 2) also admissible for an A E L(2) possessing an exponential dichotomy ? (Cf. Examples 65.N, 65.0, especially their presentation.)
64. Admissibility, (B, D)-subspaces, and dichotomies: the equation with A E M(2) The main theorems As noted in the Introduction, many of the complications of the previous sections cannot occur if A E M(R).This assumption has, indeed, a “smoothing” effect on the solutions of (60. l), (60.2), already recognized in Section 33, which enables us to replace the “slicewise” behavior described in Section 62 by the corresponding “pointwise” behavior without further conditions on (B,D), and thus to infer the existence of dichotomies. We proceed to give the direct theorems for ordinary and exponential dichotomies.
64. DICHOTOMIES: EQUATION
WITH
A
E
M(X)
189
64.A. THEOREM. Assume that A E M(X), and that Y is a (B, D)subspace (in particular, that (B, D ) is regularly admissible and Y = XoD). Then Y induces a dichotomy for A. Proof. Set a = I A We shall prove condition (e) or (f) of Theorem 41.A. Let X > 1 be given, and consider solutions x, y , z of (60.1) with x = y x, y ( 0 ) E Y , (1 z(O)((d X d ( Y , ~(0)). We apply Theorem 61.A with A = 1. I f t 2 to >, 0, we take T = to and find, using (31.7),
+
if to 2 t
2 0, we take T
=
to + 1 and similarly find
Thus (D'i), (D'ii) hold, with D = D(X) = $e3aK!r(l). Another proof uses the definition of dichotomy, and verifies (Di), (Dii) by means of Theorem 62.A and (Diii) by means of 61.C.
D ) is not weaker 64.B. THEOREM.Assume that A E M(X), that (B, than (Ll, L;), and that Y is a (B, D)-subspace (in particular, that (B, D ) is regularly admissible and Y = XoD).Then Y induces a n exponential dichotomy for A. Proof. Theorems 63.G and 64.A.
9,
We turn to the question of converse theorems. For ordinary dichotomies the assumption A E M(X) adds nothing to Theorems 63.C, 63.E; but for exponential dichotomies the sufficient condition for the admissibility of (B, D ) given by Theorem 63.K is also necessary: Assume that A E M(X). If there exists a subspace 64.C. THEOREM. inducing an exponential dichotomy for A , then ( B , D ) is (regularly) admissible if and only if D is thick with respect to B; and there exists a (B, D)-subspace (necessarily unique and = X,) i f and only if D is thick with respect to kB.
190
Ch. 6. ADMISSIBILITY AND DICHOTOMIES
Proof. The "if" part is proved by Theorem 63.K. We set a = I A IM and assume that (B, D) is admissible. By Theorem 42.F, X,, = X,, = Xo is the subspace inducing the exponential dichotomy (so that the admissibility is regular); therefore any bounded solution of (60.2) for any f E B(X) is also (an M-solution and therefore) a D-solution. Let x be a fixed nontrivial solution of (60.1); then x satisfies (Ei) with appropriate v, N > 0 if x(0) E Xo, and (Eii) with appropriate v', N' > 0 otherwise (we must envisage both possibilities, since Y might be = X or = (0)). Consider an arbitrary g~ E B. If x(0) E X, , consider the function y defined by
which is a solution of (60.2) for f = I q~ I sgn x (by (31.B,(b)); since I(f I( = I g~ I, and since f is continuous if q~ is continuous, f E B(X). Bounding 11 x(t)llII ~ ( u ) I labove - ~ and below by (Ei) and (31.7), respectively, we have
Since E M,23.V,( 1) implies that A:/ g~ I is bounded; hence y is bounded and, by the above observation, y E D(X). Therefore A;\ q~ I E D ; since this is true for all g~ E B,23.V1(1) implies that D is thick with respect to B.
If x(0) $ X,, , we define y(t) = - x ( t ) Jy" I v(u)l 11 x(u)ll-' du and use (Eii); otherwise the argument proceeds as above, (64.1) being replaced by e-aAil q~ I ( t ) 11 y(t)ll WA;l v I (t) (which shows that the integral defining y exists). The conclusion concerning (B, D)-subspaces follows from Theorem 52.F; the uniqueness from 62.C or Theorems 42.F and 64.A. &
<
<
Sets of pairs
Under the assumption that A E M(X), we can supplement the conclusions of Theorem 63.N and of 63.0, 63.P and give a complete description showing that the set of regularly admissible (B, D), and the set of those (BID)for which there is a (B,D)-subspace, are among only a finite number of possible sets.
64. DICHOTOMIES: EQUATION 64.D. THEOREM. Assume that A
WITH
A
E M(8)
E M(X3). Then:
there exists a (B, D)- (B, D) is regularly subspace if and only if: admissible if and only
if there is no dichotomy for A there is a dichotomy, but not an exponential dichotomy, for A :
191
if:
(mer) case I : (B, D ) is weaker than (Ll, L:)
case II: (B, D ) is weaker than (Ll, L,") case 12: (B, D ) is weaker than (Ll1L:) but not weaker than (L1, L"O)
case II: (B, D ) is weaker than (Ll, Lm)
case HI: (B,D ) is weaker than (Ll, La) case 112: (never)
there is an exponential dichotomy for A
D is thick with respect to kB
D
i s thick with respect to B.
In the cases when (B, D)-subspaces exist, they are given by Theorem 63.N f o r the case of a dichotomy, and reduce t o the single subspace X , for an exponential dichotomy. Proof.
A
6 3 . 0 , Theorems 63.N164.A, 64.C.
9.
64.E. THEOREM. Assume that X is jinite-dimensional, and that E M(a). Then:
if t h e is no dichotomy for A there is a dichotomy, but not an exponential dichotomy, for A
there is an exponential dichotomy for A
there exists a (B, D)- (B, D) is admissible if subspace if and only if: and only ;f: (ww)
(new
(B, D) is weaker than (B, D) is weaker than (Ll, L,") (namely, all (Ll, Lr) Y with X , C Y C Xo, or only X,, according as ( B ,D ) is, or is not, weaker than (Ll, Lm)) D is thick with respect D is thick with respect to kB (namely, t o B . xo = X,)
192
Ch. 6. ADMISSIBILITY AND
Proof. 63.P, Theorems 64.A, 64.C.
DICHOTOMIES
&
Remark. T h e three possibilities for A in Theorem 64.E are illustrated, respectively, by Examples 65.K, 65.L (X = R , A E C ) or Example 65.C (two-dimensional X, constant A); Example 65.A (A = 0); Example 65.B (A = &I). co
The remaining possibilities in Theorem 64.D, namely cases 12, 111, I12 for an ordinary dichotomy, are illustrated in separable Hilbert space with a symmetric- or Hermitian-valued bounded continuous A, by Examples
41.H,41.1, 65.R,respectively. 65. Examples and comments Examples with constant A We begin this section with the simplest example of all, A = 0; this will illustrate how necessary the conditions ensuring exponential behavior are in the preceding results, besides tying up several loose ends concerning "best possible" conditions in Chapter 5. We then consider the next simplest case, A = &I, and a two-dimensional example. 65.A. EXAMPLE. Let X be given arbitrarily (in particular, X may be the scalar field F) and suppose that A = 0, so that all the solutions of (60.1) are constants. As pointed out in Section 41 (Remark 4), every subspace of X induces an ordinary dichotomy; in particular, so does X,, = (0). By Theorem 63.E, or direct verification, (Ll, L,") is regularly admissible. However, not only is there no exponential dichotomy, but no nontrivial solution satisfies any of the exponential growth conditions of Theorem 62.B or of the parts in square brackets of Theorems 62.D, 62.E. Therefore the assumption that (B, D ) be not weaker than (L',L,") cannot be weakened in Theorem 62.B, Theorems 62.D, 62.E (parts in square brackets), Theorems 63.G, 64.B, and subsidiary results. Since X , = X, we find that we are in case I1 of the classification at the end of Section 63. By Theorem 63.N and 63.0, (B,D ) is regularly admissible if and only if it is weaker than (Ll, Lg); since, for any D E WK , X,, = X or X,, = (0) according as D is weaker than L" or not (Theorem 41.D), there are no further admissible pairs; there exist no (B, D)-manifolds if (B, D ) is not weaker than (Ll, L,")(otherwise X,, would be a (B, D)-subspace); if (B, D) is weaker than (Ll, LT) but not
65.
EXAMPLES AND COMMENTS
193
weaker than (L1, L"), X,, = (0) is the unique (B, D)-subspace (and -manifold); if (B,D ) is weaker than (Ll, L"), every subspace, including (0) and X , is a (B, D)-subspace, hence every linear manifold is a (B, D)-manifold. Observe that X,,= (0) # X = X,, . (Cf. Remark 2 to Theorem 52.M, and Remark to Theorem 54.F.) There exist (T,M)-subspaces (namely, all subspaces), but neither (M, M)-subspaces nor (T, T)-subspaces; by setting (B, D) equal to (T, M), (M, M), (T, T), (M, T) in turn, we illustrate the cases of 54.H (cf. Remark to that proposition) in which, of the four sets of subspaces, exactly the following are empty: none; (B, D)- and (B,M)-subspaces; (B, D)- and (T, D)-subspaces; (B, D)-, (B, M)-, and (T, D)-subspaces. Assume finally that X is finite-dimensional, with euclidean norm. We see that X is not an (Ll, L;)-subspace for A = 0, but X I = (0) is an (L1, L")-subspace for -At = 0. Since (L1)'= L", (L?)' = L', this provides the counterexample required in the Remark to Theorem 55.A. & 65.B. EXAMPLE.With X again arbitrary (and possibly = F ) , assume A = I [--I] (we could take any positive real multiple instead). T h e solutions of (60.1) are given by x ( t ) = e-'x(O) [ x ( t ) = e f x ( 0 ) ] , so that X [{O}] induces an exponential dichotomy with v = N = 1 [v' = N' = 11. By Theorem 64.C, (B,D) is (regularly) admissible if and only if D is thick with respect to B, and there exists a (B, D)-subspace (necessarily unique and = X [= {O)]) if and only if D is thick with respect to kB. This simple example-and especially the real scalar case X = R-is thus in this respect the prototype of all cases of exponential dichotomy under the condition A E M(2). I n particular, (Lz, L;) is admissible (by 63.L), but (L", L,") is not (direct verification with constant f, or 51.Q). Since L" = lcL:, L;P = kL", this provides the examples required in connection with Theorem 51.G and the Remark to Theorem 52.F. Also, X [{O}] is an (M, M)-subspace and a (T, T)-subspace, but not an (M, T)-subspace (Theorem 64.C and 23.W,(3), recalling that kM = M,). Therefore, with (B,D) equal to (M, T), we illustrate the case of 54.H in which exactly the class of (B, D)-subspaces is empty. & 65.C. EXAMPLE (Example 41.G continued). Example 41.G in twodimensional real or complex euclidean X has a constant A , and no subspace induces a dichotomy. Theorem 64.A implies that no (B,D)subspace exists. This illustrates the case of 54.Hin which all four classes of subspaces are empty. Since conditions (Di), (Dii) (but not (Diii))
194
AND Ch. 6. ADMISSIBILITY
DICHOTOMIES
for a dichotomy are satisfied for Y = X , , this example shows that (Diii) is an essential assumption in the converse result Theorem 63.C. &
Counterexamples for the direct theorems In this subsection we describe several examples with the purpose of showing that the conditions and conclusions in the direct theorems of Section 62, 63, 64 are best possible. All these examples are in X = R (but see the final Remark of this subsection) and have a continuous A 4 M.In order to make the import of the examples clear, weintroduce them by a number of remarks. For technical reasons, the order of the examples does not correspond to the order of the remarks, A good starting point is an analysis of Theorems 62.D, 62.E that shows that neither can the conclusion be strengthened nor the assumptions be weakened in general.
Remark 1. In the first place we enquire whether, in Theorem 62.D, L(to)might not always be taken independent of t o , i.e., whether condition (Di) [(Ei)] for an [exponential] dichotomy might not hold. Now if, in addition to the assumptions of Theorem 62.D, B is weaker than 8,L1for some T 0 (the assumption of Theorem 62.E), then Theorem 63.A [63. J] shows that more is true, namely that Y actually induces an [exponential] dichotomy; but without this additional assumption the answer is no: for every (B,D) with B not weaker than any O,L1 we shall construct an equation (Example 65.F) for which (B,D) is admissible, X , , = X = R is the unique (B, D)-subspace, and condition (Di) does not hold [let alone (Ei)] regardless of whether D is stronger than L" or not [or whether (B, D) is weaker than (Ll,L,")or not]. Remark 2. We claim that the assumption "D is stronger than L"" cannot be relaxed in general in Theorem 62.D. Since solutions are continuous functions, an actual weakening of this assumption occurs only if D contains an unbounded continuous function (equivalently, by 23.Q, if D, is not stronger than L"). For every (B, D) with such a D [and regardless of whether (B, D) is weaker than (L1, L,")or not] we construct an equation (Example 65.D) for which (B, D) is admissible, X,, = X = R is the unique (B,D)-subspace, and the nontrivial solutions of (60.1) are unbounded, so that (62.9) cannot even hold for to = 0 and any L(0) [let alone (62.10)]. Similar remarks apply to Theorem 62.E:
Remark 3. We ask whether, in Theorem 62.E, ,!,'(to, A) may not be taken independent of t o , i.e., whether condition (Dii) [(Eii)] might
65. EXAMPLES AND
COMMENTS
195
not hold. If, in addition, we assume that D is stronger than L" (the assumption of Theorem 62.D), Theorem 63.A [63. J] again shows that Y actually induces an [exponential] dichotomy; but without this additional assumption the answer is again no: for every (B, D) with D containing an unbounded continuous function (i.e., with D, not stronger than L"; cf. Remark 2) we construct an equation (Example 65.E) for which (B, D) is admissible, X,, = {0}, and condition (Dii) does not hold [let alone (Eii)], regardless of whether B is weaker than some 8,L' or not [or whether (B, D) is weaker than (L1, L,") or not]. Remark 4. We claim that the assumption "B is weaker than O,L1 for some 7 2 0" cannot be relaxed in general in Theorem 62.E. Indeed, for every (B, D) with B not weaker than any O,L1 [regardless of whether (B,D) is weaker than (L1,L,") or not] we construct an equation (Example 65.G) for which (B, D) is admissible, X,, = {0}, and the nontrivial solutions of (60.1) are not bounded away from 0, so that (62.1 1) cannot even hold for to = 0 and any L'(0, A) [let alone (62.12)].
T h e examples "previewed" in the preceding remarks yield illustrations of the "best-possible" character of the main direct theorems of Sections 63 and 64: Remark 5 . In Theorem 63.A [63.J], the assumption "(B,D) is stronger than (O,L1,L") for some T 0" cannot be relaxed without further conditions on A, such as A E M(%). Indeed, if the assumption fails, then either B is not weaker than any O,L1, or D is not stronger than L", or rather for an effective weakening in the latter case, D must contain an unbounded continuous function; the examples show that for any such pair there exists an equation (Examples 65.F, 65.G; Remarks 1,4, for the former case; and Examples 65.D, 65.E; Remarks 2, 3, for the latter) for which (B, D) is (regularly) admissible but there is no dichotomy [let alone an exponential dichotomy]. Remark 6. I t follows from Remark 5 with respect to 63.5 that in Theorem 63.G the assumption that some subspace induces a dichotomy is not redundant. Remark 7. All the preceding discussion shows that the assumption that A E M ( 2 ) in Theorems 64.A, 64.B is indispensable.
65.D. EXAMPLE. For every given DEW^ that contains an unbounded continuous function we construct a continuous A E L such that (with X = R ) we have X,, = XOMo = R and (M, D) is admissiblewhence (B, D) is admissible and R is. the unique (B, D)-subspace
Ch. 6. ADMISSIBILITY AND
196
DICHOTOMIES
(by 62.C) for every B E b.9- or b Y + o r b.9-V4-yet such that every nontrivial solution of (60.1) is unbounded. Let D be given, and set P(l) = P(D; I ) = I x[o.IlI D . By 23.Q, P(0) = 0, and P is continuous (see Section 23, p. 64).The function I + P(I) increases strictly from 0 to 00 as I ranges from 0 to 00, and therefore has a continuous inverse 5 that does likewise; this inverse satisfies [(u) < u for all u > 0. We set 6 ( t ) = ef J:-' Qu) du for all t E R, 6 is continuous and continuously differentiable, with $ ( t ) = -e'J';* (l,(e-f) - ( ( u ) ) du; for all t, 0 < 6 ( t ) < ((e-') < r f 1 and -e-' < $(t) < 0, so that 6 is strictly decreasing. In particular, for all t E R,
.
<
+
B ( W ) < B(t(e-") < Wt) 8(5(e-")
(65.1)
= Ct.
Set to = 0 and define the strictly increasing sequence (t,) by t , = t' < 00, we should have 6(t') = 0, t, = t,-, 6(t,-,). If t , = 00. In particular, t , = S(0). We which is absurd; hence define a function 6 by
+
0 < t < 1,
t(t)= exp(ts/2tl), (65.2) t(t
+
= exp(t
+
s(t)
-
&),
t
2 0.
Now t + 6 ( t ) is strictly increasing, since its derivative is 1 + $(t) > 0; it maps [t,-,, t,) onto [t, , t,+,), n = 1, 2, ...; since these intervals fill R+ , ( is indeed well defined on R, . Since 5 is strictly increasing, continuous, and continuously differentiable on [0, t,), it will be clear by induction on n that 8 has these properties throughout R, if only we show that &tl) = [ ( t , - 0) and &t, 0) = ( ( t , - 0). Now &O) = 1, <(+O) = 0; we then have, indeed, 5(tl) = exp(+t,) = t(tl -.O), and E(tl+ 0)(1 $(+O)) = (1 S(+O)) exp($,)l(O) 0,whence a t 1 0) = exp(&t,) = ( ( t , - 0). It is straightforward to verify, from (65.2), that
+
+
further, ((t,) = exp(t,
+
+
- Qtl)S(t,-,)
2 exp(&t1)f(tn-,); by induction,
We choose a sequence (u,) of positive numbers so large that (65.5)
lim
n-m
+
~ , , [ - ~ ( t= ~ )00,
65. EXAMPLES AND
197
COMMENTS
and a sequence (en) of positive numbers so small that cu
< &(ln - tn-l),
(65.6)
1 onrB(c.n) < a. 1
Let u be a real-valued, continuous, continuously differentiable function on R , , such that u ( t ) = 1 for all t not in the intervals J , = [t, - E , , t,], n = 1, 2, ...; in J,, 1 u(t) u(t, - 216,) = u,. Now 0 u - 1 0, xJ,; by (65.6) this last function belongs to D; also, since r, @(t,-,) < 1, we have U,E, u,fl(r,)a(D; 1) < co, so that u,xJn E L1.We conclude that u - 1 E D A L1. We set A = ( f - l - Uu-l, so that U = (-b is the solution of (30.3) with U(0) = 1. Now 4-' is bounded by a decreasing exponential, by (65.3), hence is in T and bounded. Since 5 - b = 5-'(u - 1) 6-l I f-' I (u - 1) f - l , we conclude that f-'u E D A L1. Therefore X,, = XoLl= XOMo = R . However, U(t, - ir,) = u,(-l(t, - &en) >, u,(-l(t,) -+co as n + co, by (65.5), so that every nontrivial solution of (60.1) is unbounded. I t remains to prove that (M, D) is admissible for A : in particular, that for e v e r y f g M the solution y ( t )= f-'(t)u(t) Jo ((u)u-'(u)f(u) du of (60.2) uq = belongs to D. Set q(t) = f - l ( t ) f(u)If(u)I du, so that I y I (U - I)p q. We claim that q is bounded and belongs to D; it follows that y E D , as was t o be proved. To establish our claim, consider any t t, with, say, t, t < tnfl. Since t , - tmPl = S(t,-,) t , < 1 for all m, we have, using (65.4) and the fact that f is increasing,
<
<
1;
< 1;
1;
xy
+
+
Ji
+
<
n-1
v(t)< 1 t - l ( t n ) t ( t n - T )
fn-'
I du
I~(U)
+f
ij(U)
I du
tn-1
tWr-1
T=l
<
<
<
<
< <
<
1 e - W i I j IM + 1;' m
p-tn+tl
T=l
< e-t+gti/i(l - e-ti/")-l
I f ( u ) I du
n-1
+ It"" If(zi)
I du.
t"-l
T h e first term of the last member is a decreasing exponential, which is in T,hence in D and bounded; the second term, which is constant in [ t , , t n + l ] ,is bounded by 2 l f IM; to show that it is in D we observe that it is
AND Ch. 6. ADMISSIBILITY
198
DICHOTOMIES
and compute, using (65.1) with t , instead of t, and 23.V,(1), m
B ( ~ , + I- tn) 1
f""If(.)
I du
<
m
e-tn
r1
If(u) I du
1-1
*-1
Therefore rp is bounded in [tl , co) by a function that is bounded and in D; by 22.1, rp has the same properties, as claimed. 9, 65.E. EXAMPLE.For every given D E b y K that contains an unbounded function we construct a continuous A E L such that (with X = R) we have X,, = X,, = {0} and (M, D) is admissible-whence (B,D) is admissible for any B E b y or b y + or byV-+-yet such that condition (Dii) for a dichotomy fails to hold. Let D be given, and let f , u be as in Example 65.D. We now set A = -(((-I + hu-'), whence U = f u . By (65.3), U(t) >, f ( t ) -+ 00 as t --f 00, whence X,,, = {0}, and a fortiori X,, = (0). Using (65.5) and the fact that is increasing we have u(tn)~-l(tn - $cn) = u;,-ft(tn)t-'(tn- +en)
--f
o
as n
-+co,
so that condition (Dii) is not satisfied for any nontrivial solution of (60.1). It remains to prove that (M, D) is admissible for A: in particular, that for every f E M the solution y ( t ) = -[(t)u(t) f-l(u)u-l(u)f(u) du of (60.2) exists and belongs to D. By the same argument as in Example 65.D, it is sufficient to prove that $, defined by $(t) = [ ( t ) [-'(u) I f(u)l du, belongs to D and is bounded. To establish this, consider any t > 0 with, say, tnv1 < t t, . Using (65.4) we have
Jp
JT
<
65. EXAMPLES AND
199
COMMENTS
the proof that t,b is in D and bounded is then concluded as in Example 65.D. & 65.F. EXAMPLE.For every given B E b y or b y + or b y @ ' that is not weaker than any 0,L' we construct a continuous A E L such that (with X = R) we have X,, = R and (B, T ) is admissible-whence (B, D ) is admissible and X,, = R is the unique (B, D)-subspace for every D E W-yet such that condition (Di) for a dichotomy fails t o hold. We claim that it is sufficient to consider locally closed B E b y not weaker than L'. Indeed, for any B E b y or b y - , m(B; 0) = a(lcB; 0) = a(T-lcB; 0) (21.E, 23.H); hence, by 23.R, T-lcB, a locally closed space in b y , is also not weaker than L1;since it is weaker than B, the admissibility of (T-lcB, T ) will imply that of (B, T). As for B E b y % + , such a B is stronger than L" (by 24.M,( I)), a locally closed space in b y not weaker than L1;it is therefore sufficient to consider the example for L" instead of B. Let then B E b y , locally closed and not weaker than L', be given. Then B' is locally closed and not stronger than L", hence contains an unbounded continuous function (by 23.Q). We now construct 6, u as in Example 65.D, but for B' instead of D. so that U = 6-lu-l. By (65.3) U ( t ) We set A = j6-l (-l(t) exp(-t atl), so that U E T,whence X,, = R. Using (65.5) and the fact that 6 is increasing, we have
<
+
<
+
so that condition (Di) is not satisfied for any nontrivial solution of (60.1). Now -At = - A = -(it-' 6u-l); by Example 65.E, (M, B') is admissible for -At; by Theorem 55.C (B", M'), i.e., (B, T),is admissible for A (cf. 22.M, 23.P). t
+
65.G. EXAMPLE.For every given B E b y or b y + or b F V + that is not weaker than any O,L1 we construct a continuous A E L such that (with X = R) we have X,, = (0) and (B, T) is admissible-whence X,, = (0) and (B, D ) is admissible for every D E WK-yet such that every nontrivial solution of (60.1) is not bounded away from 0. As in Example 65.F, it is sufficient to consider locally closed B E b y not weaker than L'. For such a B we construct 6, .a as in Example 65.F (i-e., as in Example 65.D, but for B' instead of D ) and set A = -it uu, so
+
200
AND DICHOTOMIES Ch. 6. ADMISSIBILITY
that U = (u-l. By (65.6), a(t) = 1 for t,-l n = 1, 2, ...; using (65.3),
= *e-tilzt,, + co
< t < *(in
as n
+ 00
+
L-1)
6,
,
;
by 23.V,(1), U 4 M. Therefore X,,, = (0). On the other hand, U(t, - *en) = uG1((tn - +en) < 0;~6(t,) -+ 0 as n -+ m, so that every nontrivial solution of (60.1) is not bounded away from 0. Now -At = -A = (6-l - 60-l; by Example 65.D, (M, B‘) is admissible for -A+. By Theorem 55.C, (B, T)is admissible for A. & Remark. Examples corresponding to Examples 65.D, 65.E, 65.F, 65.G may be set up in any Banach space X, by simply taking A = the solutions ( & ~ u -as ~ the ) I ,case may be. Then U = (%FII; of (60.1) then behave as in the real scalar case; and the question of admissibility is reduced to the same question for the corresponding real scalar equation by means of 51.S.
Counterexamples for the converse theorems We begin by showing that no converse result holds in general for the theorems of Section 62. Example 65.C, which belongs under the title of this subsection, shows that an angular-separation condition is essential, even under the strong conditions (Di), (Dii) for constant A, to ensure the admissibility of some (B,D) or the existence of a (B, D)-subspace (for a similar point with respect to conditions (Ei), (Eii), see Example 65.M). We show that no converse holds for Theorems 62.A, 62.B, 62.D, 62.E, even if the angular-separation problem is vacuous, namely, when Y = X or Y = (0). More specifically, we construct equations in X = R (but see the final Remark in this subsection), with continuous A EL, such that (T, M)is not admissible (whence no (B,D) is admissible, by 52.L), and yet such that: either X,, = R and every solution of (60.1) satisfies part (i) of the conclusion of Theorem 62.B-a fortiori of Theorem 62.A-for every D E b y K ,and also the conclusion of Theorem 62.D-even the part in square brackets-(Example 65.H); or else X,, = (0)and every solution of (60.1) satisfies part (ii) of the conclusion of Theorem 62.B-and Theorem 62.A-for every D E b y K , or the conclusion of Theorem 62.E-even the part in square brackets-(Examples 65.1, 65.J). It is noteworthy that Examples 65.1, 65 J are different ways of “turning over” Example 65.H (in the same sense as the examples in the preceding subsection follow from one another by “turning over”);
65. EXAMPLES AND
20 1
COMMENTS
in fact, Example 65.1 would yield (62.14) instead of the conclusion of Theorem 62.E. 65.H. EXAMPLE. Let u be a real-valued, continuous, and continuously differentiable function on R+ such that u(t) = 1 except on the nonoverlapping intervals Jn = [n - n-l, n], n = 1, 2, ...; on J n , u(t) 2 1 and, if un = u(u) du, we require limn+mune+ = co. e-f for all t ; We set A = 1 Uu-l, so that U ( t ) = e-b-l(t) therefore U E T, and X,, = R. We show that (T, M ) is not admissible for A . Sincef(t) = e-f is in T, it is sufficient to show that the solution y ( t ) = e-b-'(t) J: ~ ( udu ) of (60.2) is not in M (cf. 51.R).And indeed, u(t) = 1 for n t n $, n = 1, 2, ..., so that, for such t, y ( t ) 2 e- l n u(u) du e-n-lun; therefore Jn+' I y(t)l dt 2 "J: I y(t)l dt 2 le-n-l u , - + c o a s n + c o , a n d y $ M . We return to the homogeneous equation. Let d > 0 be given, and [24-1], so that p-1 < &I.We claim that there exists set p = 2 K = ~ ( d> ) 0 such that
J-z-n-l+
<
< < +
S&
+
U ( t ) + Tu=l U ( t )
(65.7)
tE
K-'e-',
R+ .
+
Assume for the moment that t 2 p. Since p 2 2, t ahd t p-' cannot both be interior points of J l s , n > p ; hence either u(t) = 1 or u(t p-l) = 1 ; then e-t-112. ~ ( t+ ) ~ ( +t p-1) = e - t ( u - l ( t ) e-+o-'(t p-1))
+
+
+
u(t)}. Since It follows that (65.7) holds with K = max{e1/2, maxnGfGP p is nonincreasing with d, K has the same property. u d - p-l, (65.7) implies For every t 2 to 2 0 and every u, 0
< <
U(t -I- u ) (65.8) U ( t p-1
+
Nowt
+ u)
< t +p-l < t It, t
1 '<
e-'t+u'
<
+ id
+ d - p-'1
= e-(t-t,)e-(t,+u)
Ke-(t-t
0)
(U(t,
+ d -p-l
u [ t + p-1,
t
+ + U(t0 4-p-l + u)). 24)
< t + d , s o that
+ d] = [ t , t + A ] .
Therefore (65.8) yields x[t.t+d]lJ
< < <
xrt.t+A-p-11U
+ TtKe-(t-
+
~rt+p-l.t+~iU
+ + (T!-to-v-' + 2Tt-t, +
) {Tt-t&[t,.t,+d-'p-+-
Ke-lt-t
I,+'p-'Xlto.to+d-~-'1~
t 1 O
Tt-t,n-'X[to+P-'.to+dl~
Tt-t,Xrto+l,--'.t0+41~}
Tt-r,+u-,-')X[to.ta-141~.
202
Ch. 6. ADMISSIBILITY AND
DICHOTOMIES
We infer that for any D E W K
in particular, for D = L ',
L",
Since the general solution of (60.1) is x = Ux(O), these inequalities show that part (i) of the conclusion of Theorem 62.B and the part in square brackets of the conclusion of Theorem 62.D hold, with p = 1, M =Lo = 4 K . &
+
65.1. EXAMPLE.With u as in Example 65.H, we set A = - 1 &u-l, so that U ( t )= e b - l ( t ) for all t. Now Jn+l U(t)dt 2 J"+*e 1dt 2 +en -+ 00 n as n -+ m, so that U 4 M, and X,, = {O}. T o she$ that (T, M) is not admissible it is sufficient, by (51.R), to show that the solution y ( t ) = - e b - l ( t ) J: e-2uo(u)du of (60.2) for f(t) = e-l, if the integral exists at all, is not in M. Now for n - 1 t < n - 8, n = 2, 3, ..., we have 'n u(t) = 1, whence, for such t, I y(t)l >, e1 Jn-n-l e-%(u) du > en-1e-2nun; therefore J"n-1 I y(t)l dt 2 ie-n-lun -+ m as n ---t m, so that indeed
<
Y4M.
For any d > 0 we choose p as in Example 65.H and prove, by the same method used for (65.7), U(t)
+ T;-lU(t) 2
K-ld,
tE
R+
with the same K. The argument continues as in Example 65.H, but with t o , t interchanged. We obtain, for any t to > 0, x[(tU-A)+.tu]'
<
Ke-'t-t
(T-(t-to-@-l) '
f
We infer that for any D E W K
and in particular for D = L ',
2T-(t-to)
+
T-(t-t~+p-'))X[(t-A)t.t]U.
65. EXAMPLESAND COMMENTS
203
These inequalities show that part (ii) of the conclusion of Theorem 62.B holds, with p' = 1, M = 4 ~ . 65.J. EXAMPLE.With u as in Example 65.H, we set A = -( 1+Uu-l), so that U(t) = efu(t) 2 e l ; therefore U $ M, and X,, = (0). Since
-At = -A = 1 + U U - ~ , Example 65.H shows that (T,M) is not admissible for -At; by Theorem 55.C, (MI, T ) , i.e., (T, M) is not admissible for A. Let A > 0 be given, and set p = 1 [A-'1. For any t p it follows that there exists t', t t' t A, such that u(t') = 1; using (31.9), el >, U(t')e-" 2 U(t)exp(--d I A(u)(du); we conclude that there exists K' = ~ ' ( d> ) 0 such that
< < + -J:,d :+A
K'et
~ ( texp(--J' )
+
I ~ ( u 1 )nu),
~ E R +
1
If
t
> to 2 0 we therefore have ~ ( t>,) et = et--toeto 2
to+A
K'-'et-'Ou(to)
exp(-j
1 ~ ( u I)du).
t0
Therefore the part in square brackets of the conclusion of Theorem 62.E holds with p' = 1, Li = K'. & The preceding examples had A + M, and for good reason: if, for instance, Y = X and A E M(X), part (i) of the conclusion of Theorem 62.A requires that Y = X induce a dichotomy (by 33.3)), hence that X be an (Ll, L")-subspace and that (Ll, La) be admissible (Theorem 63.C); and similarly for the other cases. However, we now show that if we disregard the speciaE form of L(to),L'(to,A), it is quite possible to have an equation in X = R with A E C such that either X,, = R and every solution of (60.1) satisfies the conclusion of Theorem 62.D (even the part in square brackets), or X,, = (0)and the solutions satisfy the conclusion of Theorem 62.E (ditto), and yet no (B,D) is admissible. 65.K. EXAMPLE.Set
+ 1 + sin log(t + 1) - cos log(t + l), C and U ( t ) = exp{e-" -(e-. + 1 - cos log(t + 1)) ( t + 1)) < A ( t ) = e-"
so that A
E
exp(-ee-"t), so that U E T,and X,, = R. Now for any U(t)U-l(t,)
t
2 to 2 0,
< exp{-e-"(t + 1) + (e-" + 2 ) (to + l)} = exp{ -e-"(t - to) + 2(t0 + l)}
204
Ch. 6. ADMISSIBILITY AND
DICHOTOMIES
and (2.10) holds with p = e-n, L(t,) = e2(fo+1). However, for positive integral k, U(e2kn - 1)U-l(e(Sk-l)" - 1) = exp(-e(Zk-lh + (e-" + 2)e(Sk-I)"1 = exp{(e-"
+ l)e(2k--"ff}-+
03
as
k -+ co,
so that condition (Di) does not hold, and there is not dichotomy. By Theorem 64.A, no (B, D) is admissible. &
65.L. EXAMPLE.We take the adjoint equation to that of Example 65.K, so that again A E C . Since U is the inverse of that of Example 65.K, we have U ( t ) 2 exp(e-nt), so that U 4 M,and XOM = (0); and (62.12) holds with p' = e-n, L'(to,A) = e2(lo+l); but (Dii) does not hold, there is no dichotomy, and no (B,D) is admissible. & Remark. With A as in Example 65.K [Example 65.L], we have -At = -A, and for any D E W K(XoM(-A))* , = {O}* = R = XOT(A) = XOD(A) [(xOM(-A))* = RL = {O} = XOM(A) = xOD(A)19 and ye! there is no (T,D)-subspace; this illustrates Remark 2 to Theorem 55.A and 55.B.
It was pointed out at the beginning of this subsection that Example 65.C has constant A and Y = X, satisfies (Di) and (Dii) of the definition of a dichotomy (but not (Diii)) and that no ( B , D ) is admissible. What happens if, for some subspace Y, conditions (Ei), (Eii) of the definition of an exponential dichotomy hold? If A E M(X), 42.D shows that (Diii) holds, hence there is an exponential dichotomy, and Theorem 64.C exactly describes the pairs that are admissible; otherwise, however, no (B,D) need be admissible, even for twodimensional X (for one-dimensional X the problem is vacuous) and continuous A: 65.M. EXAMPLE (Example 42.G continued). In Example 42.G, take p)(t) = e 2 f , Conditions (Ei), (Eii) of the definition of an exponential dichotomy hold. However, consider f(t) = (0, e - f ) , so that f~ T(X). The general solution x of (60.2) is x ( t ) = (xl(t), x , ( t ) ) = (cle-t
+
+ c2e3t - iet, 4c,et - Se-t),
+
where c1 = ~ ~ (-0 4x2(0) ) $, c2 = 4x2(0) Q are arbitrary. For every solution, 11 x(t)l( + 00 as t + 00, so that x 4 M. Since (T, M) is not admissible, no (B, D) is admissible. We end this subsection by showing that, in Theorem 64.C, the assumption that A E M(X) is indispensable for the necessity of the
65. EXAMPLES AND
205
COMMENTS
thickness condition on (B,D) (the sufficiency is proved in general in Theorem 63.K). We shall in fact construct, for each (B, D) in a very wide class, equations in X = R with continuous A E L for which (B, D) is admissible and, at our choice, R or {0}induces an exponential dichotomy. T h e class consists of all those (B, D) for which either D is weaker than L“ A D, , where D, E b y K contains an unbounded continuous function, or B is stronger than L1 v B, , where B, E b y or b y + or by%-’ is not weaker than any O,L1. This class obviously contains pairs with D not thick with respect to B, e.g., (M, L1), (L“, T) (cf. 23.W,(3), (4)); in particular, we point out the pair (L’ v L“, L1 A L“), strongest of all pairs formed with (generalized) Orlicz spaces (see Schaffer [l], Section 5, especially Theorem 5.2). T h e examples show, incidentally, that none of these pairs, including all pairs formed with generalized Orlicz spaces, is “absolutely inadmissible” in the sense of Section 51 (p. 135). I t is clear that it is sufficient to construct the examples for the pairs of the forms (M, L“ A D,) and (L’ v B, , T); and the discussion in Example 65.F shows that in the latter case it is sufficient t o assume that B, E b y , and is locally closed and not weaker than L’. We drop the subscript 1 in the description of the examples. 65.N. EXAMPLE.Let D E W . containing an unbounded continuous function be given, and construct the functions 6 and 5 as in Example 65.D. ) e - ‘ t ( t ) . From (65.2), q(t 6 ( t ) )= We consider the function ~ ( t = exp(-$t,)[(t); since t 6 ( t ) and 5 are increasing with t , we see that is increasing for t t,; since ~ ( t=) exp(-t t2/2t,) decreases as t goes from 0 to t , , we conclude that for any t” 2 t’ >, 0 we have ~ ( t ” ) v - l ( t3 ’ )v(t,)v-l(O)= exp( -&). Therefore
+
(65.9)
[(t”)
exp(t” - t’ - &tl)E(t’),
+
+ t”
> t’ > 0.
[A = -g5-l], so that U = 5-l [ U = 53. I t is We set A = clear from (65.9) that R [{O}] induces an exponential dichotomy for A. By 63.K, (M, L“) is admissible. T h e argument of Example 65.D [Example 65.E] shows that (M, D) is also admissible. It follows from 51 .R (or Theorem 51.N) that (M, L” A D )is admissible. & 65.0. EXAMPLE. Let B E b y be locally closed and not weaker than
L’, and construct 5 as in Example 65.F, i.e., as in Example 65.N, but
for B’ instead of D. We again set A = 8f-l [A = and find that R [{O}] induces an exponential dichotomy for A. By 63.K, (L1, T) is admissible. By Example 65.N, (M, B’) is admissible for -A+ = -A;
206
Ch. 6. ADMISSIBILITY AND
by Theorem 55.C, (B", M'),i.e., (L' v B,T)is also admissible.
DICHOTOMIES
(B,T), is admissible for A ; therefore
Remark. As in the preceding subsection, all the examples in thisexcept of course Example 65.M-may be converted into corresponding examples in any X by multiplying the real scalar A by I and applying 51.S. Examples in infinite-dimensional space We collect some examples, all in a separable (real or complex) Hilbert space X, and all with symmetric- or Hermitian-valued bounded continuous A, that depict some of the things that may go wrong if the dimension is not finite; in part, they are again used to illustrate remarks in Chapter 5. We begin by taking another look at Examples 41.H, 41.1; it was already noted that they exemplify cases 12, I11 for an ordinary dichotomy, respectively (cf. Remark to Theorems 64.D, 64.E). 65.P. EXAMPLE (Examples 33.H, 41.H, 43.F continued). In Example 41.H, X , = (0) induces a dichotomy, but X, is not closed. By Theorem 63.E, (L1, LF) is regularly admissible; but (Ll, La) is of course not regularly admissible. This illustrates Remark 2 to Theorems 51.F, 51.G, since fLc = 1cL; = L". & 65.Q. EXAMPLE (Examples 41.1, 43.F continued). In Example 41.1, X , = X induces a dichotomy, so that (Ll, L") is regularly admissible, and X i s an (Ll, Lco)-subspace,for A (Theorem 63.C). The adjoint equation --A+) = is precisely that of Example 41 .H (Example 65.P), so that Xo(L~)( &(-A+) is not closed. This illustrates the remarks introducing 53.C, Theorems 53.D, 53.E, etc. This example also shows that the assumption in Theorem 63.E that X , has finite codimension with respect to Y is not redundant, since we are in case 111, and therefore no (B,LF) can be admissible (i.e., regularly admissible, since X , = {0}) and X , cannot be a (B,L;)-subspace (Theorem 64.A or Theorem 64.D). & We can combine both examples into an instance of case I12 for an ordinary dichotomy: 65.R. EXAMPLE.Let X be the outer direct (Hilbert) sum X ( l )@ X @ ) of two copies of the (real or complex) Hilbert space P, so that X is itself
65. EXAMPLES AND
207
COMMENTS
a separable Hilbert space. Let (60.1) be the system
+ (tanh(t - n) - 1)~:)
(65.10(1))
k:)
=0
n
=
1, 2,
...,
(65.10(2))
3;) - (tanh(t - n) - l)xA2)= 0
n
=
1, 2,
...,
thus (60.1) splits into the equation of Example 41.H on X ( l ) ,and its adjoint equation, the equation of Example,41.I, on Z2). We claim that the subspace Y = {0(1)}@ S2) of X, which has the orthogonal complement 2 = X(')@ {0(2)},induces a dichotomy. Indeed, the norm of a solution of (65.10( 1)) is nondecreasing, and that of a solution of (65.10(2)) is nonincreasing; and solutions Ctvting at t = 0 from Y or 2 remain in the same subspace. Therefore condition 41.C,(d) holds, with N = N' = 1, wo = i w . Denoting by XF', X S ) the corresponding manifolds in X(i) for Eq. (65. Iqi)), i = 1,2, we obviously have X , = XA1)@ Xi2) = Xh') @ X 2 ) which , is not closed, since XA1) is not (Example 41.H); and X , = X$ @ X$) = {0}, which does not induce a dichotomy, since {0(2)} does not induce one for the solutions of (65.10(2)) (Example 41.1). & Lastly, we turn to the question whether the assumption that Y is a
(B,D)-subspace or that (B,D) is regularly admissible is indispensable in the direct theorems of the preceding sections (apart from the case in which ordinary dichotomy is presupposed, such as 63.H, 63.1, with the Query between them). Example 65.P shows that (L',L") may be admissible even if X,, is not closed. A more dramatic instance, in which no X,, is closed, no subspace Y even satisfies the conclusion of Theorem 62.A, let alone induces a dichotomy, ordinary or exponential, and yet all (B,D) with D thick with respect to B are admissible, will now be described. 65.S. EXAMPLE (Example 33.G continued). In Example 33.G, X,, is dense but properly contained in X for all D E byK ; we apply this fact to D = T,M: since any subspace Y satisfying the conclusion of Theorem 62.A must verify X,, C Y C X,,, , we should have X = clXoTC Y C X,, # X; hence such a subspace does not exist. For any f = (fn) E M(X),one solution x = (x,) of (60.2) is given by its components
Since I T~ I
< I r2 I implies
208
Ch. 6. ADMISSIBILITY AND
DICHOTOMIES
we have
[the last integral exists by 23.V,(l), since Ifn 1 EM).Since the length of an arc in X is not less than the length of the chord,
If D is thick with respect to B and f E B(X) the last member of this inequality is in D (by 23,V,(l)), so that y ED(X). Therefore (B,D) is admissible. These pairs may well be not weaker than (Ll, L,"); therefore this example illustrates the Remark to Theorem 52.E. &
Estimation o f dichotomy parameters T h e proofs of the direct theorems of Sections 63 and 64 allow u s to compute bounds for the parameters of the dichotomy, ordinary or exponential, ultimately in terms of KCand, in the case of Theorems 64.A, but not otherwise depending on A. We do not 64.B, also of I A write out this dependence explicity in the general case, but rather consider an interesting special instance, which, indeed, we shall require later (see Theorem 1 0 2 . A ) ; namely, the case when A E L"(2) and (L", L") is regularly admissible (so that, by Theorem 64.B, X,, induces an exponential dichotomy). I t turns out that under these assumptions the estimates for the bounds of the dichotomy parameters that could be obtained from the proof of Theorem 64.B are much too crude, and a method which telescopes Theorems 61.A, 64.A, 64.B yields much sharper values. We now carry out this computation.
IM,
65.T. CALCULATION. We assume that A E L"(x) and set I A I = a. We also assume that (L",L") is regularly admissible, and set K = Kw,w , S = S,; obviously, Cm," 1. Let y be a nontrivial bounded solution of (60.1). For fixed 7 > 0 we define x by x ( t ) = y ( t ) X ~ ~ , ~ Iy(u)Il-l ( U ) ~ ~du for all t E R, , so that x is (by 31.B,(b)) a solution of (60.2) for f = x[~.,]sgny. N o w f ~L"(X), I = 1; also, x(0) = 0; and x, = y 1) y(u)ll-l du is bounded, whence x is bounded. By Theorem 51.D (Remark) we have
<
J'i
J'i
If
9 ~1 x 1 d K(1 +WIfl = K(1 + S ) . (65.11) l l y ( ~ ) l l ~ l l ~ ( =~l)l xl (l ~~) \l I~ 0
65.
EXAMPLES AND
209
COMMENTS
T h e inequality between the extreme terms then holds for all + ,S)-l; if cp(t) = J 1 11 y(u)Il-l du, (65.1 1) implies whence, by integration and combkation with (65.1 l), we find
7
> 0.
w-l >,
Set v = K-l( 1
(65.12) IIy(t)j1-l
s:
> ve"(t-t')
for all t 2 t' 2 0.
IIy(u) 11-l du
Let to 2 0 be given and set t'
=
Y,
+ u-l log(1 + a+)
to
> t o . If
t 2 t' we apply (65.12) and (31.7) and find
st'
11 y ( t ) 11-1
veu(t-t')
I1 Y ( U ) 1 -1 du
to
>
1) y(to)11-l
1 t'
e-5(u-t~) du
t0
(1
+ av-l)-"+5-'u~eY't-t
1 O
II Y(10) 1I-l.
that is,
11 y ( t )11
(65.13)
if to
< (1 +
av-1)1+5-"e-
u(t-tll)
II At,) II ;
< t < t', (31.7) yields 11 y ( t ) 11 < ea't-to) 11 y(to)11 <.e(a+v)(t'-to)
e-"t-t
O)
IlY(t0)lL
i.e., again (65.13). Therefore y satisfies (Ei) with (65.14)
v = K-l(l
+ S)-l
N
=
(1
+
~v-l)l+~-'~.
T h e reader should convince himself that the value of N in (65.14) is the best obtainable from (65.12) (for any t') and (31.7). We also remark that (65.12) (with t' = to) and (31.7) yield IIy(t)ll au-l(l - e-afo)-lI1y(to)ll, so that if to is large enough N may be replaced by any number > UY-1; if, in this inequality, we set t = to and let the common value tend to co, we conclude v < a. This implies that, in (65.14), N 4uv-l. Let X > 1 be given, and let x be a nontrivial solution of (60.1) with II z(0)ll Xd(Xo, z(O)), y any bounded solution of (60.l), and set w =y z # 0. For fixed 7' > 7 2 0 define x by
<
<
< +
] w . Now f E L"(X), Therefore x is a solution of (60.2) for f = x [ ~ , , 'sgn I f I = 1; also, x(0) = -z(O) J:' 11 w(u)Il-l du, whence 11 x(0)ll <
AND Ch. 6. ADMISSIBILITY
2 10
DICHOTOMIES
Jr
Ad(Xo , x(0)); and x, = y 11 w(u)Il-l du is bounded, whence x is bounded. By Theorem 51.D we have
II 4.) II
r'
II 4 10 Ilk1 du
= It
47)It < I x I d hK(1
7
Since this holds for all T'
I = AK(1
+ 2s).
> T we have, setting v i = K-l( 1 + 2S)-l,
II z ( t )II
(65.15)
+ 2s) I f
pI1 t
w(u) 11-'
du
d ";-1.
Consider first the case y = 0. Then, if tp = J"1 11 z(u)Il-ldu, (65.15) implies +p-l -A-'v;, whence, by integration and combination with (65.15) and (31.7), we find
<
i.e.,
11 z(t)ll > (Aav;-')-' e"-"P-to' 11 z(t& so that
(65.16)
v' = k ' v ' o ,
N'
= Aavb-',
where
z satisfies (Eii) with
v; =
K-l(l
+ 2.S-l.
Assume now that y # 0 and that T >, 0 is given; we apply (65.15) to 11 J J ( T ) ~(1~z(T)~)-~z, - ~ J J , instead of y, z,and find, with t = T ,
Therefore y, z satisfy (Diii) with (65.17)
r;, = (Aavo'-1 )-1 .
We conclude that X o satisfies condition 42.A,(b) with v, v', N, N',yo given by (65.14), (65.16), (65.17). We do not quite obtain the conditions of the definition of an exponential dichotomy, since the value of v' in (65.16) depends on A. It follows from Theorem 42.A (Remark 2), however, that we may take for v f any fixed number, 0 < v f < ui ,at the cost of increasing N' = "(A) for each A. & Remark. It is not difficult to see that, conversely, we have the estimates
66. SOLUTIONS OF
THE ASSOCIATE EQUATION
21 1
The latter is obvious; the proof of the former is so similar to the proof of Theorem 63.C that we leave it to the reader (see also Schaffer[8], p. 78).
66. Behavior of the solutions of the
associate homogeneous equation Implications of the existence
of
a (B,D)-subspace
Throughout this section, we deal with coupled spaces X,X‘and Eqs. (60.1), (60.2) and (32.1), (32.2), i.e., (66.1)
9’- A’x‘ = 0
(66.2)
9’- A’s’ = ,f’,
for a given A E L(x)that has an associate. Our purpose is to study the behavior of the solutions of (66.1) when we have information concerning (60.2). In this section, (B, D) will always denote a F-pair, and we shall investigate the situation when there exists a (B,D)-manifold for A. In the last subsection we shall depart somewhat from these assumptions. The simplest case (and the only one of interest for finite-dimensiona1 X)is that in which we have a (B, D)-subspace Y for A that has the quasi-strict coupling property. We then know (Theorem 54.B)that Yo is a (D’, B‘)-subspace for -A‘, In order to apply the direct theorems of Sections 62, 63, 64, to (66.1), (66.2) we merely have to find out what additional properties of (B, D) will produce the additional properties required of (D’, B’) for this application; it is actually convenient to replace (B,D) by (B,D,), which may be done since D, contains the same continuous functions as D , with the same norms. These conditions are given in 23.T and, since they are necessary and sufficient, our discussion in Section 65 shows that the theorem we formulate will be best possible, as far as the properties of (B, D) are concerned. We state the results corresponding to the direct theorems of Section 62. The condensed form is justified by the lengthy and easily supplied conclusions. 66.A. THEOREM. Assume that Y is a (B,D)-subspace for A (in particular, that (B, D) is regularly admissible and Y = XoD)and that Y has the quasi-strict coupling property. Under the following additional assumptions, the conclusions of the following theorems hold, with (60.1) replaced by (66.l), y , z by y’, z‘, Y by Yo,D by B‘:
212
Ch. 6. ADMISSIBILITY AND DICHOTOMIES
Additionul assumption
none (B, D) not weaker than (Ll, ):L B weaker than L’ [and (B,D) not weaker than (Ll,L : ) ] D stronger than L“ [and (B,D) not weaker than (Ll,L,“)]
Theorem whose conclusion holh as modijed. Theorem 62.A Theorem 62.B Theorem 62.D Theorem 62.E
Proof. Y is a (B,D,)-subspace for A. By Theorem 54.B, Y o is a (D: , B’)-subspace for -A’. T h e conclusion follows from 23.T and Theorems 62.A, 62.B, 62.D, 62.E, applied to (66.1), (66.2), Yo, and (D;,B‘). 9. Remark 1. Observe the “cross-over” in the assumptions of Theorems 62.D, 62.E and the corresponding parts of the Theorems 66.A. Remark 2. Theorem 66.A remains valid if the assumption “Y is a D) is regularly admissible for A (B, D)-subspace for A (in particular, (B, and Y = XoD)”is replaced by the weaker assumption (cf. Theorem 52.L): “there exists a (B,D)-manifold for A (in particular, (B, D) is admissible for A) and Y is a (T,D)-subspace for A”. This follows from the proof of 62.F. There is of course an analogous theorem under the stronger assumptions of the direct theorems of Section 63 and 64, which imply the existence of a dichotomy (ordinary or exponential) for A. Such an analogue, of the form: “if Y is a (B,D)-subspacefor A with the quasistrict coupling property, and ..., then Yo induces a dichotomy (ordinary or exponential) for -A‘ ”, is then however, a simple corollary of those direct theorems via Theorem 43.A, which connects the dichotomies for A and for -A’; we leave its obvious formulation to the reader. I t is interesting, in this connection, to give a neat “instant” proof of Theorem 43.A; it has the disadvantages of involving the non-homogeneous equations and failing, for exponential dichotomies, to supply the relations, mentioned in Proof I of the theorem, between the values of v, v‘.
Proof 11of Theorem 43.A. Since Y induces an ordinary [exponential] dichotomy for A, it is an (Ll,La)-subspace [and an (L2, L2)-subspace]
66. SOLUTIONS OF THE
213
ASSOCIATE EQUATION
for A. T h e n Y o has the same properties for -A’ (since (L1)’ = L“, (L”)’ = L’, (L2)’ = L2),and therefore induces an ordinary [exponential] dichotomy for -A‘ (Theorem 54.B,and Theorem 63.D [Theorem 63.M] used twice). & 00
Implications of t h e existence of a mere ( B , D ) - m a n i f o l d The remainder of this section is of interest only if X is infinite-dimensional. In Sections 61, 62, 63, 64 the assumption that Y is closed was quite essential in the direct theorems; Example 65.S shows that there is very little definite that can be said about (60.1) if this is not the case. It is thereD)-manifold fore somewhat surprising that the existence of a mere (B, (hence, in particular, the mere admissibility of (B, D)) for A , without any closedness assumption, let alone the quasi-strict coupling condition, already implies a behavior of the solutions of the associate equation (66.1) that is at first sight quite similar to that described in the direct theorems of Sections 62,63,64 (cf. the preceding subsection). The difference, in essence, is that the behavior is not “uniform” with respect to the choice of solution. We now turn to the investigation of these results. In view of the “nonuniformity” just mentioned, it becomes interesting to state “individual” theorems, concerning isolated solutions, and “collective” theorems, concerning classes of solutions as inclusive as possible. The latter kind requires a considerably more elaborate technique, but for the sake of conciseness we shall not derive the simpler results directly (the way in which this could be done will become apparent from our proofs). The central problem is obtaining a replacement for Theorem 61.A; this is done in Theorem 66.D. We first prove a rudimentary form of it in 66.C, after recording the following almost trivial topological lemma: 66.B. Let S be any topological space, and let y be a nonnegative real-valued function on S x R , such that ~ ( p.), is nondecreasing with 1imt+“ ~ ( pt ), = co for every p E S , and v(-, t ) is continuous on S for every t E R, For any number k > 0, set B,(p) = sup{t : ~ ( pt ), k}. Then .8, is upper semicontinuous on S .
<
.
66.C. Assume that Y isa (B, D)-manifoldfor A . Then, i f K y = K y B , D ( A ) , we have:
7
(i) for every A > 0 , every solution y ’ of (66.1) with y‘(0) E Y o , and all 2 0,
(66.3)
Ch. 6. ADMISSIBILITY AND
214
(ii) for every A
%yo)I x&t,>
DICHOTOMIES
> 0, every pair of solutionsy’,
for all T > T~(z‘(O)),where function on X’ \ XiB,
.
T ~ ( x ’ )=
z’ of (66.1) withy’(0) E YO,
Ty(sgn x’) is an upper semicontinuous
Proof. Proof of (i). We may assume y’ # 0. We set
I
1
= r’(l)
11 y’(’)
x[T.,+‘](”)
0
11-l
du*
Thus x‘ is a solution of (66.2) for f ’ = sgny’; we have [ I f ’ 11 = , whence, by 23.K, f ’ E k,D’(X’), ID, = /?(D’;A ) 2A/a(D;A ) . Further, xk = y’ ST+’ 11 y ’ ( ~ ) I ldu - ~ so that xk(0) E Yo, and x’(0) = 0. By Theorem 53.A, x‘ B‘(X’) and
<
If
,Y[.,.+~]
<
I X‘ lB,< 2dKy/a(D’;A ) .
(66.5)
Using Schwam’s inequality as in (61.3) in the proof of Theorem 61.A and combining with (66.5) we obtain (66.3).
Proof of (ii). For the solutions x’ of (66.1), the function I ,y[o.Tlx’ In, is 0: indeed, using a continuous function of x‘(0) E X ’ for every fixed r (3 1.71,
11
x[o.l]x;
16‘
- I X [ o . T ] “ ; 16’
I < I x[o.l](x~ - 16’ G NB’;4 IXIO.l](~; - 4) I < B(B‘;7) II xX0) - xX0) II e x P ( r ll4u)lldu). 0
Further, for fixed x’(O), I X [ ~ , ~ ]IB, X ’ is nondecreasing as a function of T. If x‘(0) 4 XiB,, i.e., x’ 4 B’(X‘), the fact that B’is locally closed implies I,, = co (by 22.H). We may now apply 66.B to the that lim-,m I X[~,~]X‘ space S = {x’ E X ’ : 11 x’ 11 = l} \ XiB, and the function p(x‘(O), T) = I ,y[o,llx’ IB,; if C , = CyB,D(A),66.B implies the existence of an upper semicontinuous function T,(x’) = T , ( s ~x’) = BzCy(sgnx’) for x’ E X’\XiB, such that if z’ is a solution of (66.1) with z’(0)$ XiB, we have for any 7
> Ty(z’(O)),
(66.6)
I
x[O,l]z’
IB‘
> 2cYll
zf(o)~~‘
We may now assume y’, z’ as in the statement and take
r
> ~,(2’(0)).
66. SOLUTIONS OF THE Since z’(0) 4 XiB,, we have w’
ASSOCIATE EQUATION
+ z’
= y‘
215
# 0, and we may set
s n w’; again f’E k,,D’(X’), Thus x‘ is a solution of (66.2) for f’ = x[7,7+dl I 2d/a(D‘;d). Further, xk = y’ [I W’(U)II-~ du, so that xk(0) E Yo; and 11 x’(O)l[ = I( z’(O)[I )I w’(u)[I-l du. By Theorem 53.A and (66.6), x’ E B’(X’)and
f’lD, <
Jy”
Now I X[,,~IZ’ In, J-;”I( w’(u)[l-l du implies
< I x’ In,
+8
; together with (66.7) this
Using Schwarz’s inequality as in (61.3), (61.4) and combining with (66.8) we obtain (66.4). &
Assume that Y is a (B, D)-manifold for A (in partic66.D. THEOREM. ular, that (B, D) is admissible for A and Y = XoD)and that 2‘ is afinitedimensional subspace of X‘ such that Y on 2’ = (0). Then there exkt numbers c > 0,c’ 3 0 such that for any d > 0 , any pair of solutions y’, z’ of (66.1) with y’(0) E Yo, z’(0) E Z’, and any T 3 0 we have
where F(d) = P(D’, B’;A ) (61.5).
= P(B, 1cD; A )
is computed according to
Proof. 1. Set V’ = 2’ n XiB,, a finite-dimensional subspace, and let W’ be a (finite-dimensional) complement of V‘ in 2’. In particular,
W ’ n Xi,. = (0). By Theorem 53.D, M = Y n(YO V’)O= Y nV’O is a (B, D)-manifold and M o = Y o V’. If W‘ # {0},the compact nonempty set {w‘ E W’ : 11 w’ [I = I } is contained in X \ XiB,; the upper semicontinuous function ~ ~ ( 2 0 ’= ) TM(sgnw’) of 66.C has therefore a finite maximum, T , , say, on W’ \ (0);we choose a fixed T~ > T , If W‘ = {0}, we set T,, = 0. 2. Assume for the moment that V’ # {0}, dim V’ = p. Then V‘O has finite codimension p. Let V be a maximal linear manifold contained in Y such that V n V‘O = {0}, obviously a finite-dimensional subspace with
+
+
.
Ch. 6. ADMISSIBILITY AND
216
<
+ +
+
DICHOTOMIES
.
dim V p. Then Y V’O = V V‘O Since V is finite-dimensional, the sum is saturated: V V’O = ( Y V’O )” = ( Y on V’)O = {O}O = X. Therefore V is a complement of V’O in X, whence dim V = p. The continuous function I(o, o’)l is strictly positive on the compact set {v E V : 11 v I( = l } x { Y’ E V’ : 11 w‘ 11 = l}, and therefore has a positive minimum, say u > 0, there. Let w‘ be any solution of (66.1) with w’(0) E V’. There exists then a solution v of (60.1) with v(0) E V C Y such that 11 v(0)Il = 1, I(w(O), w‘(O))l> uII v’(O)11 (trivial if v’ = 0). Let y’ be any solution of (66.1) with y’(0)E Yo. By 32.C, Schwarz’s inequality, and the fact that er E D(X), we have for any d > 0, 7 3 0,
+
Set S, = S,,(A), S,, = Sv,B,(-A’). We recall that a(D;d) = P(D’; A ) < 2d/a(D’; A ) ((23.10), 23.K). We then have, using I1 v(0)ll = 1,
.
where k = 2u-?SySv~ Formula (66.10) is trivially true for V’
= (0)with
k
= 0.
+
3. Let y’, z’ be as in the statement; we may write z‘ = v’ w‘ in exactly one way, where w’, w‘ are solutions of (66.1) with s‘(0)E V‘, w’(0) E W’. Until further notice we assume T T ~ . We apply 66.C to the (B, D)-manifold M and the solutions y’ v’, w’. Using (66.3) or (66.4) according as w’ = 0 or w’ # 0, we find (since
+
7
> 7M(W’(O)))
66. SOLUTIONS OF THE
217
ASSOCIATE EQUATION
Using (66.10), (66.11), and 23.M, (66.13)
Combining (66.1 I), (66.12), (66.13) and (1 we obtain (for any 7 7 0 )
11, 11 xCO,,p'11
< 11 ZI' 11,
4. We proceed to remove the restriction 7 2 7 0 . Since a(D'; d ) 6 2a(D'; $A), /3(D'; A) < 219(D'; &A),we find I"($A) < 4F'(A). From (66.14) with $4 instead of A we have
< <
We shall now be concerned, until further notice, with 7 7 0 . We = e x p ( j 7 I( A(u)ll du). If d 2 ~ we~ have , 7o &l h o , and (31.7) yields (for 7 + A 70 ad, the other case being trivial)
set E
Hence
also
< +
<
+
Ch. 6. ADMISSIBILITY AND
218
DICHOTOMIES
Combining these computations with (66.1 5 ) and using the fact that
11 X [ ~ , ~ I 11Z < ’ 11 X [ ~ . , , ~ Z11, ’ we conclude that (66.9) holds for T < r0 with (66.16)
c = 8E(E
+ 1)K,,
C’
=
16E(E
+ l)kKM;
by (66.14), this is a fortiori true for 7 3 r0 . & 66.E. Under the assumptions of Theorem 66.D, each of the following conditions is suficient for c’ = 0 i n (66.9): (a) 2’ c XiB,; (b) Z‘n XiB, = (0); (c) D is thin with respect to B (in particular, (B,D)is not weaker than
(L1, La. Proof. We refer to the proof of 66.D. Under condition (a), W’ = (0); hence r0 = 0, z‘ = w’, and from (66.10), (66.11) we have (66.9) with c = 2KM &A, c’ = 0. Under condition (b), V’ = {0), hence k = 0, and c’ = 0 (by (66.16); a direct proof using 66.C and part 4 of the proof of Theorem 66.D is simpler in this case). Under condition (c), Theorem 53.E implies YO = XiBt,so that condition (b) is satisfied, since YO n 2’= (0). &
+
We state the conclusions reached thus far in the “individual” form. 66.F. Assume that there exists a (B,D)-manifold for A (in particular, that ( B ,D) is admissible for A). Let y‘, z’ be any solutions of (66.1) with ~’(0) E XiB,,~’(0) $ (XOD)O\ (0);andy’(0) ~’(0) $ (XOD)Ounless ~’(0) = 0. Then there exists a number c1 > 0 (depending on y’, 2 ’ )such that for all A > 0, T 2 0 we hawe
+
Remark. To gauge the weakness of the assumptions on y’, z’ we C Y oC Xi,,, for every ( B ,D)-manifold Y (Theoobserve that (XOD)O rem 53.B). Proof. We leave aside for the moment the “exceptional case” in which
~’(0) # 0 is contained in the subspace spanned by (X,,Jo and y’(0) (which
66. SOLUTIONS OF THE
ASSOCIATE EQUATION
219
requires y’(0) 4 (XOD)O,z’(0) E XiBt). Since ~’(0) E XiB, and XoD is a
(B,D)-manifold for A (52.B,(b), Theorem 52.F), Theorem 53.D implies + Fy’(0) = Y ofor some (B,D)-manifold Y. Set Z’ = Fz’(0); that (XOD)O the exceptional case being excluded, Y o nZ’ = (0). Since Z’is at most one-dimensional, we must be in case (a) or case (b) of 66.E, and the conclusion follows. We return to the exceptional case. There exists a scalar p # 0, 1 such p ~ has ‘ ~’(0) E (XOD)O. Set Y = XoD, a (B, D)-manifold, that X‘ = y’ and 2; = Fy‘(O),2; = Fz’(0). Then Y o n2; = (0) and, in the language of the proof of Theorem 66.D, Vl = Zi , i = 1,2. Applying (66.10) to V; = 2; instead of V’, y’ instead of v’, and ( p - I)-%’ instead of y‘, we ( p - l)-lx‘ = p(p - I)-l(y’ z’)), find (since y’
+
+
+
and applying (66.10) to Vi = Zi instead of V‘, z’ instead of a’, and -(p - l)-lx’ instead of y’, we find
so that (66.17) holds. ,?,
We can now use Theorem 66.D and 66.F in the place of Theorem 61.A and develop all the analogues to the direct theorems in Section 62, with a DJ, (DL , B’)as in Theorem 66.A. possible juggling of the pairs (B,D), (B, For the sake of the record, we state the “collective” form of these theorems as an adaptation of the statement of Theorem 66.A, leaving the statement of the “individual” form to the reader.
Assume that Y is a (B, D)-manifold for A (in partic66.G. THEOREM. D) is admissible for A and Y = XoD),and that Z’ is afiniteular, that (B, dimensional subspace of X’ such that Y on 2’ = (0). Then the conclusion of Theorem 66.A holds, if the condition ‘ 11 z’(0)ll h d( Y o ,~‘(0))”is replaced by “~‘(0) E Z’ ”, and dependence on h by dependence on Z’.
<
We shall be a little more explicit in the statement of the analogues of the direct theorems of Sections 63 and 64, which yield a behavior of the solutions of (66.1) that, except for its “nonuniformity” with respect to the solutions, imitates a dichotomy, ordinary or exponential. The proofs are again the same as in Sections 63 and 64, using Theorem 66.D and 66.F instead of Theorem 61.A, and Theorem 66.G instead of Theorems 62.A, 62.B, plus the relations between (B,D), (B,DJ, (Di, B’). We first state the collective^' forms:
220
Ch. 6. ADMISSIBILITY AND
DICHOTOMIES
66.H. THEOREM. Assume that Y is a (B,D)-manifold for A (in particD ) is admissible for A and Y = XoD).If ( B ,D ) is stronger ular, that (B, than (Ll, L“), or i f A E M(8), there exists N > 0, and for every finitedimensional subspace Z’ of X‘ with Yo n Z’ = (0)there exist N’ = N‘(Z’) > 0, yo = yo(Z’) > 0 such that any solutions y’, z’ of (66.1) with y‘(0) E Y o , z’(0) E Z’ satisfy (Di), (Dii), (Diii) (instead o f y , 2).
Assume that Y is a (B, D)-manifold for A (in partic66.1. THEOREM. ular, that (B,D) is admissible for A and Y = XoD),and that (B,D ) is not weaker than (Ll, Lr). Assume further that there exists a linear manifold Yl in X which, together with any finite-dimensional subspace Z’ of Xi with Y,On 2‘ = (O), satisfies the conclusion of Theorem 66.H (in particular, that (B,D ) is stronger than (Ll, L“), or that A E M(X)). Then Y o = Y: = (XoD)O= XiBs, and there exist v, N > 0, and for every Jinite-dimensional subspace Z’ of X‘ with Y o n Z’ = (0) there exist v’ = v‘(Z’) > 0, N ’ =‘N’(Z’) > 0, yo = yo(Z’) > 0 such that any solutions y’, z’ of (66.1) with y’(0) E Yo, z’(0) E Z’ satisfy (Ei), (Eii), (Diii) (instead of y, z). We finally give the “individual” forms of these results: 66. J. Assume that there exists a (B,D)-manifold for A (in particular, D ) is admissible for A). Let y‘, z’ be solutions of (66.1) such that that (B, y‘(0) E XiB,, z‘(0) 4 (X,,,)O \ (0).If (B,D ) is stronger than (L1, L“), or ;f A EM@), there exist N , N’, ‘yo > 0 such that y’, z‘ satisfy (Di), (Dii); and also (Diii), prowided y’(0) z‘(0) .$ (XOD)O.
+
66.K. Assume that there exists a (B,D)-manifold for A (in particular, D ) is admissible for A ) , and that (B,D ) is not weaker than (Ll, Lr) that (B, (then (XoD)O= Xi? by Theorem 53.E). Let y’, z‘ be solutions of (66.1) such that y’(0) E Xoe, , z’(0) .$ XiB,\ (0). If y’,z‘ satisfy the conclusion of 66. J (in particular, i f (B, D ) is stronger than (Ll, L“), or i f A E M(8)), there exist V , v’, N , N’, yo > 0 such that y’, z’ satisfy (Ei), (Eii), (Diii). We point out that the exception to the validity of (Diii) in 66.J is vacuous under the assumption of 66.K if y’, z’ # 0, and is therefore not mentioned in the statement of the latter.
a
A question ubout dichotomies We are now in a position to give a proof of 63.1, a proposition that was a partial answer to the Query following 63.H. We therefore revert to the study of Eq. (60.1) and (60.2) for any A E L(8) (without reference to a coupled space X’) and a Y-pair or Y‘-pair or YW-pair (B,D), thus
67. NOTESTO CHAPTER 6
22 1
upsetting the agreement at the beginning of this section. The reason why we deal with this proof at this point is that we can use the results of the preceding subsection with X' = X * and with T instead of B.
Proof of 63.1. The inclusions Y C X,, C X,, C Z follow from Theorem 41.D. By Theorem 52.L, Y is a (T, D)-manifold. We now set X' = X*. By Theorem 43.A, Zo induces a dichotomy for -A*. By Theorem 66.1, with Yl = ZandTinstead of B, we have Yo = Zo,whence clY = YO0 = Z; since X,, is closed, this implies X , = Z; since X , is thus the unique subspace inducing a dichotomy, Theorem 41.E implies X , = X , . Now (Theorem 63.E), hence a (B, L:)Z = X , is an (Ll, L:)-subspace subspace (52.B,(a)). It follows from 52.B,(e) that Y is a (B, DhL:)-manifold. It remains to sketch a proof of (Eii). Again by Theorem 66.1, the solutions y* of the adjoint equation (32.3) with y*(O) E Yo = Zo satisfy (Ei) with numbers that, for distinctness, we call v l , Nl. Together with condition 41.A,(g), part (D'i), for the dichotomy induced by Zo for -A*, this easily implies the validity of condition 42.A,(g), part (E'i), for the same subspace, with v1 instead of v, and a suitable D, that depends only on A. But ZO is a saturated subspace of X * ; hence it has the strict coupling property, by 12.A. We may therefore apply the argument of Proof I of Theorem 43.A (to -A* instead of A , and Zo instead of Y), and conclude that condition 42.A,(e), part (E'ii), holds for the subspace Zoo= Z with respect to A itself, with V' = v l . It follows at once that condition (Eii) holds as stated, with v' = v1 ,j
.
67, Notes to Chapter 6 The relationship between the admissibility of a pair for Eq. (60.2) and a dichotomy-like behavior of the solutions of Eq. (60.1) was established for the first time by Perron [2], using classical methods and under restrictive assumptions (the pair is (C, L"); cf. also Notes to Chapters 4 and 5 for some details). Example 65.K is essentially the same as an example used by Perron to illustrate the significance of the assumption that (C, L") be admissible, or, from another point of view, of the "uniformity" of the exponential growth conditions. Krein [l] and KuEer [l] extended Perron's results, by using methods of functional analysis, to the case where (Lp, L") is admissible, but at the cost of assuming X , = X . These results were obtained under the assumption that A EC(X). Dropping this assumption, KuEer proves that if X , = X and (Lp, L") is admissible, where p > 1, then there exist N , v > 0 such that every solution x of (60.1) satisfies
II x(t)ll d NIIX(0)ll exp(--Vt'-P-');
222
Ch. 6. ADMISSIBILITY AND
DICHOTOMIES
I( x(t)ll Q L(to)llx(to)ll exp( -v(t - t0)l+-l) for all >, to 2 0; we have here the first example of a “pointwise, nonuniform” theorem, which is of course much weaker than Theorem 62.D. In a recent paper, M. Reghig (RegiH [l]) has investigated other cases of it obviously follows that t
nonuniform behavior, especially in their relation to the admissibility of certain $-pairs that are not .T+-pairs. The idea behind the contents of Section 66 (except the rather trivial first subsection) originated with P. Hartman, and some of the proofs are essentially due to him.
CHAPTER 7
Dependence on A 70. Introduction
We shall now examine in what way the properties studied in Chapters 4,5, and 6 depend on the operator function A in Eqs. (30.1) and (30.2), i.e., (70.1)
*+fAx=O
(70.2)
*+Ax=f,
and how they vary when A varies. In a sense, therefore, this chapter contains a sort of perturbation theory, although some parts of it are more precisely concerned with “roughness” properties of the equations, i.e., insensitivity of certain patterns of behavior to small modifications of A. In this chapter we shall only deal with the case where the range of t is J = R, ,except for some material in the first subsection of Section 71, which applies equally to any interval J. There will be an occasional reminder of this agreement. The case where J = R will be dealt with in Section 84. In Section 71 we discuss to what extent the admissibility of a pair (B,D) or the existence of a (B, D)-subspace Y is unaffected by certain small variations of A, as well as how such variations affect the value of &,(A) in Theorem 51.A, or the subspace Y and the corresponding values of K,(A), S,(A). In Section 72 we derive similar insensitivity results for the existence of an ordinary or exponential dichotomy, and the variation with A of the subspaces inducing, and the parameters describing, the dichotomy. In particular, sets of A’s possessing an exponential dichotomy turn out to be open in many of the 9-spaces. This analysis is complemented “in the large” in Sections 73 and 74, where the connectivity structure of the sets of operator functions for which there is a dichotomy, ordinary or exponential, is described. 223
Ch. 7. DEPENDENCE ON A
224
The former section deals with the problem in general Banach spaces; the fragmentary results obtained are used in the latter section, dealing with the case where X is a Hilbert space, and including the finitedimensional case. To illustrate, we quote a sample result for finitedimensional X : let s2, be the class of A E L"(x) possessing an exponential dichotomy; then a, is open in L"(8) and, unless X is a real plane (where matters are complicated by the turning of solutions around 0), falls into exactly 1 + Dim X components, one for each value of Dim X,(A), and each containing a system of the f o r m j + y = 0, f - z = 0 in orthogonal subspaces of the appropriate dimensions. The corresponding class for ordinary dichotomy is connected. In this chapter we require the material in Section 13 on the set of subspaces of a Banach space, as well as the corresponding part of Section 14.
a,
71. Admissibility classes and (B, D)-subspaces Admissibility classes For the moment we let the range of t be any interval J containing 0. Let (B, D ) be an N-pair. We denote by Ad(B, D ) the class of all A E L ( x ) for which (B, D ) is admissible, and by Ad,(B, D ) , Ad,(B, D ) the subclass of those A E Ad(B, D ) for which XOD(A)is closed (i.e., (B,D) is regularly admissible for A ) , or is closed and complemented, respectively. Observe that Ad(B, ,D ) n Ad(B, , D) = Ad(B1 v Bz ,D), and similarly for Ad, , Ad,, . As usual, if (B, D ) is an 9 - p a i r or an $W'-pair, we abbreviate Ad(B(X), D ( X ) ) , etc., to Ad(B, D ) , etc.
N(x)
is such 71.A. Assume that (B, D ) is an M-pair and that F E that F - D 3 B. Assume that A, , A E L ( x ) , with A, E Ad(B, D ) , A - A , E F,KB,D(AO)I A - A , IF < 1. For any f E L(X), any Dsolution x , of (71.1)
*++fi=f
(if it exists) and any suflciently small p > 1, there exists a D-solution x of (70.2) such that (71.2)
I
- xO ID
IA -
IF(l
- fKB.D(AO) I A -
IF)-'
I'0
ID'
In particular, if A , E Ad(G, D ) for G E bM(X), then A E Ad(G, D ) and KG,D(A) KG.D(AO)(l - KB.D(AO) I A IF)-1'
71. ADMISSIBILITY CLASSES AND (B, D)-SUBSPACES
225
More in particular, the assumptions imply A E Ad(B, D) and (71.3)
2 K,:D(AO)
KB!D(A)
-
IA
-
IF.
Proof. We may assume B # {0}, the conclusion being otherwise trivially true. Set K = KB,JAO), 11 = I A - A, and let p > 1 be such that pK7 < 1. Let f, x, be as in the statement, and fixed for the time being. With yo = x, ED, we define y n recursively for n = 1, 2, ... as a D-solution of j A,y = (A, - A)ynMl with I y, ID pKKA - Aolyn-1 IB * Then IY n I D PKVIY n - 1 ID G (PKT)~I xo I D n n = 1 , 2, .... If we set x, = y i , n = 0, 1, ..., then (xn) is a D-Cauchy 0 sequence and ((A, - A)x,) a B-Cauchy sequence; if x is the D-limit of the former, (A, - A). is the B-limit of the latter; both are a fortiori L-limits. Since in Aox, = f ( A , - A)xn-, , n = 1,2, ..., by Theorem 31.D we must have (replacing x by an equivalent continuous function, if necessary) 2 A,x = f (A, - A)x, so that x is indeed a solution of (70.2); and
IF,
+ < 1
<
< 9
* a *
+
+
+
+
so that (71.2) holds. T h e particular cases follow at once by taking x, t o be a D-solution of (71.1) with I x, ID &,D(AO)I f IG;and then setting G = B. &
<
+
) any A , E L ( ~ ,the set A, F= For any space F E J V ( ~and - A, E F} and each of its subsets shall be taken to be provided, unless otherwise stated, with the translated metric and topology of F. Observe that A, F = A, F if and only if
{ AE L ( x ) : A
+
A, - A,EF.
71 .B. THEOREM. Assume that A, an N-pair. If F - D =- B, then:
E
+
+
L ( x ) , F E JV(x),and (B, D ) is
+
(a) Ad(B, D ) n (A, F)is open in A, F and &,,(A) is continuous on this set; (b) ifG E b N ( X ) , then Ad(B v G, D ) n (A, F ) is open and closed in Ad(B, D ) n (A, + F ) , and KG, D(A) is continuous on the former
+
set;
(c) with G as in (b) and for any j x e d h > 1, there exists a continuous F)) x G --+ D such that x = ( ( A , f ) function [: (Ad(B v G, D) n (A, is u D-solution of (70.2) with I x ID hKG,D(A)) f JG .
+
<
Ch. 7. DEPENDENCE ON A
226
Proof. We may again assume B # {0}, and also Ad(B,D) n (A, + F)# 0; there is no loss of generality in taking A, as any arbitrary element of this intersection. We write KB,D(A)= K ( A ) for short. Consider any A E A, F with r ) = I A - A, IF < iK-l(Ao). Then 71.A implies A E Ad(B, D) (so that Ad(B, D ) n (A, F)is open) and K-'(A) >, K-l(A,) - r). Since
+
+
<
we may apply 71.A to A, , A interchanged; hence 1 K-l(A) - K-l (A011 so that K ( A )is continuous, and part (a) is proved. With A, , A as before and G as stated, 71.A applied to A,, A in either order in turn shows that both are in Ad(B v GI D) n (A, F) as soon as either one of them is; therefore this set is open and closed in Ad(B, D) n (A, F);and again by 71.A applied to A, , A in either order,
r)
+
+
&,(A) is continuous, and (b) is proved. Consider finally the mapping A which assigns to each pair (A,f) E (Ad(B v G,D) n (A, F)) x G the set A(A,f)C D of all D-solutions of (70.2). We claim that each A(A,f)is a nonempty closed convex set in D, and that A is a lower semicontinuous carrier in the sense of Section 10. As for the first claim, A(A,f) is nonempty since A E Ad(G, D ) , ~GE; it is a translate of the linear manifold X, of all D-solutions of (70.1), hence convex and closed (by 33.A). T o prove the semicontinuity, let (A, ,fo) be given in the domain of A, and let x, E A(A, ,fo)be chosen; x, is a D-solution of i + A,x = fo . Consider then ( A , f )in the domain of A, with r ) = I A - A, IF < K-l(A,), and let p > 1 be such that pK(A,)q < 1. Let y be a D-solution of j A,y = f - fo with Iy ID p&.D(AO)lf - fo IG; then x1 = xo y is a Dsolution of (71.1). By 71.A there exists a D-solution x of (70.2), ie., x E A(A,f), such that so that
+
<
+
+
71. ADMISSIBILITY CLASSES
AND
(B, D)-SUBSPACES
227
therefore A is semicontinuous, as claimed. By 10.C we obtain the function 5 as required in (c), since hf{l x ID : x E A(A,f)} KG.D(A)lf IG ' kh
<
71.C. THEOREM. Let the assumptions of Theorem 71.B hold, and suppose F), Ad,(B, D) n (A, + F) that D L"(X). Then Ad,(B, D) n (A, are open in A, + F;SD(A) is continuous on Ad,(B, D) n (A, F);and F) into the mapping A + X,,(A) is continuous from Ad,(B, D) n (A, E ( X , 6') and from Ad,(B, D) n (A, F) into 6,(X, 6 ) .
+
<
+
+ +
Remark. The conclusion holds also without the assumption D < L"(X), but the proof is more complicated and the general result is not of such great interest. The additional assumption implies IIf(t)ll < If I < If ID for all continuous f E D and all t E J. Proof. 1. As in Theorem 71.B, we assume B # {0}, Ad,(B, D) n (A, + F)# 0, and also that A, is an arbitrary element of this nonempty intersection. We write K ( A ) = KB,D(A),S ( A ) = S,(A). Consider A E A, F with
+
it follows from 71.A that A E Ad(B, D), K-'(A) 2 K-'(A,) - q . Let x be a nonzero D-solution of (70.1); then (A - A,)x E B, and for any p > 1 there exists a D-solution y of j + A,y = (A - A,)x pK(A,)I(A - A,)x IB pK(A,)qI x ID . It follows that with Iy I D x, = x y is a D-solution of i,+ A,x, = 0, and we obtain
+
<
<
Since p was arbitrarily near 1, we have
by Theorem 33.C, XOD(A)is closed, hence A €Ad,(B,D), and Ad,(B, D) n (A, + F)is open in A, F.Furthermore,
+
(71.4)
Ch. 7. DEPENDENCE ON A
228
whence, using (71.3) and writing the unabbreviated notation for the sake of reference, (71.5)
+ SD(A))-'> K&(AO)(l + SD(AO))-'- I A - Ao IF.
K,:D(A)(l
+
Using the precise assumption on q, we obtain q < K-l(A)(1 S(A))-l, so that the preceding argument may be carried out with A,, A interchanged. Combining (713 ) with the corresponding inequality after the interchange, we have
I K-'(A) (1 + S(A))-1- K-1 (A,) (1
+
W 0 ) F 1
I d '7;
since K(A) is continuous, it follows that S(A) is also continuous on Ad,(& D) n (A0 F).
+
2. With A,, A as until now, and with the previous notation, (71.6)
11 xO(o)
- x(o) 11
= 11 Y(O) 11
< I Y ID < pK(AO)'7
I
ID
G pK(AO)S(AhII 40)Ii. By (71.4), K(Ao)S(A)q< 2K(A0)S(A0)7< 1, so that by choosing > 1 so small that pK(A,)S(A)q < 1 we obtain x, # 0 and, using 11.A, 1) sgn x(0) - sgn xo(0)ll = y[x(O), x,(O)] < 2pK(A,)S(A)q. Since x(0) was an arbitrary element of XoD(A), we find
p
{d(x, z(xOD(AO))'
z(xOD(A))}
2K(AO)S(A)'7*
Combining, we conclude that (71*7)
8"CxOD(AO),
XOdA))
+ F)
and A + Xo,(A) : Ad,(B, D) n (Ao
+ +
--+
E(X, 8') is continuous.
3. Now Ad,(B, D.) n (Ao F) is the inverse image, under this mapping, of E,(X), which is open in E(X, 8') by 13.1; hence the former set is open in Ad,(B, D) n (A, F),which in turn is open in A, F.
+
71. ADMISSIBILITY CLASSES
AND
(B,D)-SUBSPACES
229
Again by 13.1, the topologies of E,(X, 6') and S,(X, 6) coincide; hence A + X,,(A) : Ad,(B, D ) n ( A , F) -+ SJX, 6) is continuous. &
+
+
Addendum. On account of 13.E, the class Ad(B,D) n ( A , F) n ( A : X,,(A) = (0)) is open and closed in Ad,(B, D ) n ( A , + F), hence also in Ad,(B, D ) n ( A , F); for A in this class, the function f ( A , f ) of Theorem 71.B is uniquely defined and linear in f , and 11 [ ( A , .)I[ = K ( A ) ,where the norm is taken in [ B ; D]. T h e computation at the end of the proof of that theorem yields, for A,, A in this class and 7 = I A - A, IF sufficiently small, and for fo = f , x, = f ( A ,,f ), = f ( A , f1' I t ( A , f ) 9f)lD d1- f K ( A O ) 7 ) - ' K 2 ( A O ) l f l L l 7,
+
so that
II
a 4 .)
-
( ( A , .>/I < I1 [ ( A , .) Il'(1 - I1 [ ( A , .) I1 IA - A, 9
7
and the mapping A
-+
9
1J-l
IA
- A,
IF,
t ( A , .) is continuous from this class into [B; D].
For 9 - p a i r s and 9%-pairs we have the following corollary: 71 .D. Let A, E L(x)begiven, and let (B, D ) be an 9 - p a i r or 9 g - p a i r with D L". Then:
<
(a)
M B , D) n ( A ,
+ B(X)),
Ad@, D) n ( A ,
Ad@, D) n ( A ,
+ B(X))
+ B(X)),
+
are open sets in A, B(X), K,,,(A) is continuous on the first set, S,(A) on the second, and A -+ X,,(A) is continuous from the second set into E(X, 8') and from the third into S , ( X ,6 ) .
+
(b) I f G E b S or E b 9 % , respectively, Ad(B v G, D ) n ( A , B(X)) is open and closed in Ad(B, D ) n ( A , B(X)); K G , D ( Ais) continuous on the former set; and for every h > 1 there exists a continuous function t : (Ad(B vG,D) n ( A , B(X))) x G ( X )+D(X) such that x = ( ( A f) , is a D-solution of (70.2) with I x ID hKG,,,(A)If IG .
+
+
<
Proof. Theorems 71.B and 71.C, with F
=
B(X), using
B(X) * D(X) 2 B(X) (by 22.V), in the case of an S - p a i r . For an %%-pair, we observe that we may replace the pair (B(X), D(X)) by the pair (B(X), D ( X ) A C ( X ) ) (since every continuous function in D(X), hence every D-solution x, is in C ( X ) , and I x I D ( X ) A , C ( X )= sup{l x , I x I} = I x ID), and then take F = B(X), since B ( X ) (D(X) A C ( X ) )=> B(X) (by 24. J). 4,
ID
Ch. 7. DEPENDENCE ON A
230
Remark. T h e analogue of 71.D with D Ad(kB, D ) n (A, + B(X)),etc., since
< L," holds
for the sets
B(8) . D(X) * kB(X), or B(X) * (D(X)A C,(X)) =- kB(X) (by 22.V, 24. J). If J = R, , we have further results for %-pairs and 9%-pairs: 71.E. THEOREM. Assume that J = R , , and let (B, D) be an 9 - p a i r or 9 V - p a i r with D L". Then Ad(B, D ) kB(X) = Ad(B, D), and similarly for Ad,, Ad, .
<
+
Proof. T h e inclusions 3 are trivial. Let Ad stand for any one of Ad(B,D), Ad,(B,D), Ad,(B,D). We first show that if A E A and ~ B E k,B(X), then A B e Ad. Set s = s(B). For given f E B(X) there exists a D-solution x of (70.2); let y be the solution of j + (A + B)y = f with y ( s ) = x(s); then y ( t ) = x ( t ) for t 3 s, so that y E D ( X ) (by 22.1) and A B E Ad(B, D). Let U , U , be the solutions of the operator equations 0 A U = 0, 0, ( A B)U, = 0 with U(0) = U,(O) = I. T h e set of values at t = s of the D-solutions of (70.1) is U(s)XoD(A); the corresponding set for A? (A B)x = 0 is U,(s)X,,(A B). These sets coincide (again by 22.I), hence
+
+
+
+ + + +
+
is closed [and complemented] if and only if X,,(A) is closed [and complemented]. Hence A B E Ad in each case. Let now B E kB(R). Since by 71.D A E Ad n (A B(X)), an open set in A B ( 8 ) , there exists B, E k,B(W) with I B - B, I,, so small that A B - B, E Ad. Then A B = ( A B - B,) + B, E Ad by the first part of the proof. &
+
+
+
+
+
+
71.F. Assume that J = R , . Let A, E L(x)be given, and let (B, D ) be an 9 - p a i r or 9%-pair with B lean and D L". Then each of the sets Ad@, D ) n ( A , B(x)), Ad,(& D ) n ( A , B ( x ) ) , Ad,@, D) n (A, B ( 2 ) ) is either 0 or A, B(8).
+
<
+
Proof. Theorem 7 1.E.
+
+
9.
Remark. T h e conclusions of Theorem 71.E and 71.F do not remain true if we assume J = R instead of J = R , , as will be seen in Example 84.A. All cases of noncompact J can be reduced to either of
71. ADMISSIBILITYCLASSES
AND
231
(B, D)-SUBSPACES
these, as was already previously noted in the proof of Theorem 51.H. If J is compact, the same proof shows that Ad(B, D ) = Ad,(B, D ) = Ad,(B, D ) = L ( x ) for all 9 - p a i r s or 9 V - p a i r s (B, D), with X o D ( A= ) X for every A E L(x),so that the contents of Theorem 71.E and 71.F become trivial in this case. For pairs of translation-invariant spaces we have the following special result, unrelated to the preceding. 71.G. (a) Assume that J = R, and that (B, D ) is a F-pair or Y-'-pair or Y%+-pair. Let A E Ad(B, D ) be given. Then T;-A E Ad(B, D ) f o r all T E R, , and K( T;A) = KB,D( T;A) is a nonincreasingfunction of T. (b) Assume that J = R and that (B, D ) is a F-pair or Y V - p a i r . Let A E Ad(B, D ) be given. Then T,A E Ad(B, D ) for all T E R, and K( T I A )= K ( A ) .
Proof. We prove (a), the proof of (b) being similar and simpler. Let T E R, be given. For any f E B(X), we have T,+fE B(X); for any p > 1 there exists a D-solution x of i A x = T:f with Then T;x is obviously a DIx pK(A)I T,ff = pK(A)I f solution of j (T;A)y = T;Tff = f , and I Tyx I D Ix pK(A)If IB . Hence T;A E Ad(B, D ) and, since p > 1 was arbitrary, K(T;A) K ( A ) . If 7'' 2 T' 0, we apply this argument to T T A , T7,A = T>-,T,;A to show that K ( T 7 A ) K(T7A). &
ID <
+
<
IB
+
IB.
<
ID <
<
(B, D) - subspaces In this subsection, as in the remainder of the chapter, it is assumed that J = R, . If we replace admissibility of (B, D ) by existence of a (B, D)-subspace, we obtain some results analogous to those of the preceding subsection, though more restricted in scope. We usually assume B # {0}, the problem otherwise being trivial.
<
71 .H. Let (B, D ) be an 9 - p a i r or 9%-pair with D L". Assume that A,, A EL(@, A - A , E k,B(x), with s = s(A - A,), and let U, , U be the solutions of the operator equations U, A,U, = 0, 0 AU = 0, with U,(O) = U(0) = I. If Yo is a (B, D)-subspace for A,, then Y = U-l(s)Uo(s)Y o(= U-l(t)U,(t)Yofor all t 2 s ) is a (B, D)-subspacefor A ; if Yo is complemented, so is Y . Now set KO= KrJAo), So = S y J A o ) , K = K,(A), S = S,(A). If, in addition, IA - A , IB < *K;l(l So)-',
+
+
+
232
Ch. 7. DEPENDENCE ON A
then
further, S'(Y0, Y ) < 2KOSoI A and if xo is any solution of
-
(71.10)
+Ago = 0
3i',
A,
IB(
1 - Ko(1
+ S0)l A - A, I&';
with xo(0) E Y o , and p > 1 is giwen, there exists a solution x of (70.1) with ~ ( 0E) Y such that I x - XO ID PKI A - A0 IB I ~0 ID -
<
Proof. 1. Since the solutions of (71.10) and (70.1) coincide beyond s, we have U(t)U-l(s>= Uo(t)Ufl(s),t >, s, so that Y is a well-defined subspace and is complemented if and only if Yois. Y is the set of values at t = 0 of those solutions x of (70.1) that satisfy xmAO(O) E Yo;since xmAo E D(X), we conclude that Y C XoD(A). LetfE koB(X)and p > 1 be given. There exists a solution xo of (71.1) satisfying xOmAo(0) E Yo and Ix, ID pKol f IB . Let x be the solution of (70.2) such that xaA, = xomA0;it is clear that ~ ( t=) xo(t), t >, S, regardless of the value of s o ; since xmAo(0)= xOmAJO) E Y o , we have xmA(0)E Y. Finally, we know that y = x - xo is a solution o f j Ay = (A, - A)xo . Applying (31.5) to this solution in [0, s], and using the fact that it vanishes beyond s, we find:
<
+
< a@; [o, s1) I X[O.S]ID
<
+
IA
Is I xo ID exp
-
(j:11 A(u)11 d") .
<
Therefore I x ID I xo I D Ix - xo I D 0 I x, ID G pK00 I f (B , where = 1 a(B;[O,41X [ O . ~ I I D I A - A , l B exp(f, I1 A(u) II du); and Y is indeed a (B,D)-subspace for A.
0
+
2. T h e point in the remainder of the proof is that for small IB we can avoid the dependence on s. Assume then that r ) = I A - A, Is < &K;'(l So)-', and choose p > 1 so small that pKo(l + So)v < 1. Let f E koB(X)be given, and set T = max{s, so}. Let y o be a solution of (71.1) with yOmAJO) E Y o , Iyo ID < pKolf IB,
I A - A,
+
71. ADMISSIBILITY CLASSES
AND
(B, D)-SUBSPACES
233
and construct inductively the sequence ( y n ) of solutions of j n + Aoyn = (A0 - 4 y n - 1 n = 1,2, * * * I with Ynm,4,(0) E Yo, 9
co n
If we set x n = x "0 y i 3 we have z n l a A , , = y $ m A u > and (Zn), (z:nm4,) are both D-Cauchy sequences. If z , z' are their D-limits (hence their L"-limits, hence continuous functions), two applications of Theorem 31.D show that z is a solution of (70.2) and z' = zmA0; since Yois closed, zwA,(O) E Y , ,'whence zooA(0) E Y ; and
so that indeed K-' K;' - 7, as stated in (71.8). Let x be any nonzero solution of (70.1) with x(0) E Y , so that xmAo(0) E Y o . Since ( A - Ao)x E koB(X),there exists a solution y of ID ' Then with Y m A , ( O ) > I Y ID pKOTl 9 f AOy = ( A x0 = x y is a solution of (71.10) with xO(0) = xOmA,(0)= xmAO(O) ymAo(0) E Y o . We carry out the same computation as in part 1 of the proof of Theorem 71 .C and find I x ID So(1 - KO(1 So)~)-'llx(O)ll, so that S So(l- Ko(l SO)~)-l, or
<
+
+
<
+
<
+
This yields (71.9) and, together with the first inequality of (71 A), the second. We also record (71.11)
I
- %J
ID ==
Iy
ID
< fKOy I
ID'
Continuing the computation as in part 2 of the proof of Theorem 71.C, we find sup{d(y, Z( Yo)):y E Z( Y ) } 2K,Sr). Since K;l( 1 > 27, we have, from (71.8), K(l S)r) < 1, and the preceding argument may be applied to A,, A and Y o ,Y interchanged. In this way we
+
<
+
Ch. 7. DEPENDENCE ON A
234
obtain the bound for # ( Y o , Y) as in Theorem 71.C; and the formula corresponding to (71.1 1) in the interchange is precisely the last part of the statement. QI 71.1. THEOREM. Let ( B , D ) be an 9-pair or 9%-pair with B lean andD locally closed and ,< L". Assume that A, ,A E L(a), A - A, E B(X), and that there exists a [complemented] (B, D)-subspace Yofor A, . Then there exists .a [complemented] (B, D)-subspace Y for A. Prooj. 1. T o begin with, we unburden ourselves of the case of an $%-'pair (B, D). Indeed, by Theorem 52.C, a subspace Y is a (B, D)subspace for A E L(x)if and only if it is an (lcB, D)-subspace, i.e., a (klcB, D)-subspace-and indeed it is interesting, in view of some remarks made below, that KYkloB = Kncs is equal to K,, , as shown in the proof of Theorem 52.C. But (klcB, D ) is an $-pair with klcB lean, and, B being lean itself, B(X) \< klcB(a). I t is thus sufficient to prove the theorem for the 9 - p a i r (klcB,D). In the remainder of the proof, ( B , D ) denotes an 9-pair.
2. With K O ,So as in 71.H, we assume for the time being that So)-', where c, 0 < e < Q, will be fixed later. Since B is lean, there exists an increasing sequence (7,) in R, with 7, = 0, limn-," 7, = 00, such that
I A - A, Is < QcK;;'(l
+
I tI7n( A - /lo)1, < 2-'"+"cK;'(l
+
+ S,)-'.
Set A, = A, X[,.,~I(A- Ao),n = 0, 1, ..., and define U, as the solution of the operator equation 0, A,U, = 0 with U,(O) = 1. Set Y , = U;;'(T,)U~(T,)Y~; it is clear that Y , = U;E'(T,)U,-~(T,)Y,-~, n = 1, 2, ... Further,setq, = IA, - -&-I Is = I x[r,-,,T,,1(A - A,,)lB < 2-,eK,3 1 So)-', so that qn < & ('; 1 S0)-l. We claim that Y , is a (B, D)-subspace for A,, and the constants K , = KyJAn), S, = Sya(A,) satisfy
+
xy
+
+
n
(71.12)
K;1
2 K;'
-
17, > Ki'(1 1
- E(l
+ So)-l) >, Kil(1 -
€)
n
(71.13) Kil( I
+ 8,J-l 2 Ktl( 1 + So)-' - C 7i > Kil(l + So)-'( 1 - c). 1
This follows at once by induction, using 71.H: indeed, the claim holds for n = 0 by assumption; and the induction hypothesis (71.13) yields 2&(1
+ Sn)qn+l< 2K,(1 + S,)(l
- ')-'7In+1
< 2-"c(l
- €)-I
< 2-" < 1.
71. ADMISSIBILITY CLASSES
AND
(B, D)-SUBSPACES
235
If we apply (71.9) to Y , - l , Y , and use (71.12), induction yields (1
-+S")-l >, (1 + .S0)-J - Kn(1 -
n
€)-I
1 > (1 + So)-l1(vj
€(
1 - €)-I),
1
whence S, < (So(l- c) + e)(l - 2 ~ ) - l ; since either Yo = {0}, and then Y , = {0}, or So >, 1 (since D L"), this implies
<
s,
(71.14)
< S0(1- 24-1.
71.H further yields, with (71.13), 6'(Yn
9
Ynti)
+
d 2GSn~n+1(1-
< 2-%(l
Sn)vn+d-l
- (1
+ 2-("+")r)-'
< 2-,,(l
- 24-1,
so that (Y,) is a Cauchy sequence in 3 ( X , 8'); by 13.D it has a limit, which we call Y. We claim that this is the subspace required in the statement. We observe that
< c S'( Y , , Yn+l)< 2 41 - 24-1. W
6'( Yo , Y )
(71.15)
n
This, together with 13.1, implies that if Yo is complemented and Q is sufficiently small (this is the only further restriction we place on E in this proof, and it depends on Yoonly), Y is also complemented. D)-subspace for A. 3. We now substantiate our claim that Y is a (B, Let y be a solution of (70.1) with y ( 0 ) Y. ~ By 13.D there exists a sequence (y,) of solutions of j , Any, = 0 with y,(O) E Y , , lim,+my,(0) = y(0). Since A, - A vanishes on [0, T,], it follows from (31.7) that y, + y uniformly on every compact interval, a fortiori in L(X). But by (71*14), Im ID G S,ll~~(o)ll< Sn(1 - 2~)-'11~n(o)ll is bounded as n + 00. By 21.E, y E lcD(X) = D(X) and
+
I .V
ID
d So(1
- 2~)-'!2
I/ ~ n ( 0 II) = sn(1 - 2~)-lII ~ ( 011-)
Hence Y C XOD(A) and, incidentally,
s = S , ( A ) 6 So(l- 2€)4.
(71.16) Let ~ and
koB(X)be given, and let m be an integer so large that T,,, 2 s(f)
I Z
236
Ch. 7. DEPENDENCE ON A
Let z be a solution of 8 + A,z = f with zmA,(0) E Y,, I z ID < Ko(l - e)-'lfIB; this is possible, by (71.12). Since z, zmA, coincide beyond scf) we have, by (31.5) (with to = scf)) and (71.17), using the fact that A,(t) = A(t) for 0 t s(f) T ~ ' ,*
< <
<
(again by (71.12)). This implies, by induction,
using (71.18), we have
Therefore (yn) is a D-Cauchy sequence; by the argument used above, or by Theorem 31.D, the D-limit y, say, is a solution of (70.1) and y(0) = limn+wyn(0)E Y (again by 13.D). Now we consider the solution x of (70.2) that has x,, = y. Let us compare v = z - y m = z - zmAm and w = x - y = x - xmA: both functions vanish outside [0, scf)]; both are, in this interval, solutions of (70.2), since A,(t) = A(t) there. Therefore w = w, i.e., x = z (y - y,). Using (71.19),
+
1 % ID < 1'10 = KO(1
m
+ 1 1.Y"-yn-1
ID
m+l
+
1 - 2E)-l
€)(
If
< Kdl-e)-l(l
+f(2--E)(1-26)-1)
Ifl~
72. DICHOTOMY CLASSES
237
Thus Y is indeed a (B, D)-subspace for A , and we have (71 20)
4.
K
=
K,(A)
< K"(1 + r)(l
We finally drop the assumption that
- 2€)-1.
IA
-
A,
IB
is small. Since
B is lean there exists T E R , so large that I @,(A- Ao)lB< &Kil(l + So)-'; we then set A , = A, @,(A- A,), so that A - A , = X[,.,~(A - A,) E k,B(X). We apply parts 2 and 3 of the
+
proof to find a (B,D)-subspace for A,; the result for A then follows from the first part of 71.H. Q
Remark. T h e formulas (71.15), (71.16), and (71.20) show that the element-to-subset mappings that map A into the class of all (B,D)subspaces Y for A , or into the class of triples ( Y , k, s), k > K,(A), s > S,(A), are lower semicontinuous carriers in A, B(X) for each A, E L(X), if the Y's are taken in the topology of E(X, 8 ' ) ; and the same is true for the complemented (B, D)-subspaces, taken in the topology of E,(X, S ) , on account of 13.1.
+
72. Dichotomy classes Exponential dichotomies Reversing what would appear to be the natural order, we deal with exponential dichotomies first: they constitute the richer and more interesting case, and we shall go into more detail concerning them. We are again, as we shall be throughout the rest of the chapter, in J = R, . We denote by Q or, in full, Q ( X ) , the set of all A E L(X) possessing an exponential dichotomy, and by Q, = Q,(X) the subset of those for which X,,(A) (the subspace inducing the dichotomy) is complemented. We observe that the two classes coincide for finitedimensional X , and indeed for a Hilbert space, and that the question of their distinctness is open in general (Remark 4 and Query in Section 42, pp. 114-1 15). Theorem 63.M yields the relations (72.1)
R
=
Ad,(M,Lm),
52,
=
Ad,(M,L"),
which enable us t o apply the results of the preceding section. In the sequel we shall abbreviate the notations K , , " , S , to K , , S.
+
72.A. Assume A, E L(X) given. Then Q n ( A , M(X)), Q, n ( A , M(X)) are open sets in A , M(X). The function S ( A ) and the functions K,(A) for all spaces F E b F OY b 9 % , F stronger than M
+
+
Ch. 7. DEPENDENCE ON A
238
(in particular, for all spaces F E b y or by-' or b y % + ) , are continuous on Q n ( A , M(8)). The mapping A -+ X,(A) is continuous from Q n ( A , M(8)) into 3 ( X , 6') and from Q, n ( A , M(8)) into 6).
qx,
+ +
+
Proof. 71.D and (72.1), if we observe that F v M is both stronger and weaker than M,hence Ad,(F v M,La) = Ad,(M, L") = 52. ,j,
It is possible to sharpen the conclusion of 72.A concerning the mapping A --t Xo(A).To this purpose, let us denote, for any A E L(8), by X i ( A ) the set of values at t = T of all bounded solutions of (70.1). It is clear that
In particular, X;(A) is a subspace if and only if X,(A) is a subspace. 72.B. Assume A, E L ( 8 ) given. The mapping A + &(A) : 52 n (A,
is continuous unqormly in Proof.
T
E R,
+ M(X))-+
S ( X , 8')
.
We write K instead of K M . Let A be any element of
+ M(8)).By 71.G, with (B,D) = (M, L"), and (72.1), (72.2), we have T;A E Q for all R , , and K(T;A) < K ( A ) . Consider in particular A,, an arbitrary but fixed element of 52 n (A, + M(x)).
Q n (A,
T
E
By condition (Ei) for an exponential dichotomy, there exists N > 0 such that Ilxo(t)ll < N(lxo(t,)ll,t >, to 2 0, for any solution xo of (71.10) with xo(0) E X,(A,); but then, by (72.2), S( CA,) < N . With A, as just specified, let A E Q n (A, M(8)) satisfy I A - A, IM < +K-'(A,)( 1 N)-l. Using the preceding argument, I T;A - T;A, lM < I A - A, lM < +K-l(T;Ao)(1 S( T;A0))-l. Since T;A E Q n (T;A, M(8)), formula (71.7) is applicable, whence, using (72.2),
+
+
+
+
+
+
72.C. Q Mo(8)= Q, 9, Mo(8) = Q, . If A , E L(x)is given, is either li) then each of the sets Q n (A, Mo(Q),Q, n (A, W(x3)) or A, Mo(8). In particular, Q n Mo(8) = Q, n = 0.
+
+
+ w(m
72. DICHOTOMY CLASSES
239
Proof. Theorem 71.E and (72.1). T h e last statement follows because 0 4 52 (or by 42.E and 23.F). & Almost all the preceding results are particular cases of the following theorem, which, on the other hand, is their corollary:
72.D. THEOREM. Let B Eb 9 or b 9 g be stronger than M (in particular, B E b . F or b y + or by%+). Then:
+
+
(a) if A, E L(X) is given, Q n ( A , B(X)), Q, n ( A , B(2)) are open sets in A, B(X), and S ( A ) ,K B ( A )are continuous on the former set; (b) A + X,’(A) : Q n ( A , B(X))-+E ( X , 8‘) is continuous uniformly in T E R,; and A -+ X,(A) : Q, n ( A , B(X))+S,(X, 8) is continuous. If B is lean, then: (c) Q B ( 3 ) = Q, Q, B(X) = Q,; for A , E L ( ~ ) each of Q n ( A , B(X)), Q, n ( A , Bj2)) is either 0 or A, B(X); in particular, Q n B(X) = Q, n B(X) = 0.
+
+
+ +
Proof. topology and that, of KBwe
+
+ +
+
72.A, 72.B, 72.C, using the facts that the uniformity and of A B(X) are finer than those induced by A M(2); if B is lean, it is stronger than k M = M a . For the continuity set F = B in 72.A. 9,
+
+
Our next task is to study the dependence on A of the parameters involved in the exponential dichotomy. We recall the definition of the “numbers” ;(A), ;’(A) in Remark 2 in Section 42 (p. 113), and derive a new characterization of them in terms of perturbations of A.
72.E. If A E Q, for A
+ GI if
u E R , then X,(A) induces an exponential dichotomy and only if -;(A) < u < ;’(A).
Proof. The condition is suficient: Assume -;(A) every solution y of (72.3)
y
+ ( A + ul)y
< u < ;’(A). For
=0
and the solution x of (70.1) with x(0) = y(0) we have y ( t ) = e c U f x ( t ) , t E R , . By the assumption there exist v, v‘ > 0 such that -u < v < ;(A), u < Y’ < ;’(A); and on account of the definition of ;(A),;’(A)there exist N, N‘ = ”(A), yo = yo(X) that, together with these V , v‘, satisfy conditions (Ei), (Eii), (Diii) of the definition of an exponential dichotomy for A and Y = X,(A); but then the same parameters, together with v 0,
+
240
Ch. 7. DEPENDENCE ON A
v’ - u
A
> 0 instead
of v, v’, respectively, satisfy the same conditions for
+ uI and the same Y.
The condition is necessary: Assume that X,(A) induces an exponential dichotomy for A uI, with parameters v1 , v; , N , , N ; = Ni(h), yol = y,,(h). For any solution x of (70.1) and the solution y of (72.3) with y(0) = x(0) we have, if x(0) = y(0) E X,(A),
+
II 4 )II
= F t I1A t ) II
I1Y ( t 0 ) II
d Nl e+”l(t-to)
- Nl e-(vl-o)(t-t”)
<
11 x(t,) 11
for all t 2 to
0;
therefore -u < v1 - a ;(A);and by a similar argument for x(0) = y(0) $ X o ( A )we obtain u < vi u ;’(A). &
+ <
+
a}.
For any A E 52, set H ( A ) = {u E R : A ul E Since I E M(R), H ( A ) is the inverse image, under the continuous mapping u + A 01 : R -+ A M(X), of the open set 52 n (A M(2))(by 72.A). Therefore H ( A ) is open. I t contains 0; let then H,(A) denote the component of 0 in H ( A ) ;it is an open interval, bounded or unbounded.
+
+
+
C X,(A + d’l)for 72.F. If u’, u” E R, u’ < a’’, then X,(A + u’I) 01)is constant on each component any A E L(X). If A E 52, then X,(A of H ( 4 . Proof. Adding (a” - a’)I to A + 0’1 multiplies all solutions of the homogeneous equation by e-(uH-u’)t, so that the first part of the statement is trivial. Now by 72.A the mapping u -+X o ( A 01): H ( A )-+ S ( X ,Sf) is continuous. T h e first part of the proof together with 13.E now show S’(X,(A u’I),X,(A ~ “ 1< ) ) 1 if and only if X,(A u’l)= X,(A + u‘’l), which completes the proof. &
+
+
+
+
+
72.G. Ho(A) = (-;(A), ;’(A)). Proof. 72.E and 72.F. &
72.H. THEOREM. For any A, E L(B), the functions ;(A),;’(A) are lower semicontinuous on 52 n ( A , M(X));afortiori on 52 n ( A , B(2)) for any B E b y or b S % stronger than M.
+
+
Proof. Let A, be any arbitrary but fixed element of 52 n (A, +M(X)), and choose any v, Y’, 0 < v < ;(A,), 0 < Y‘ < C‘(A,). By 72.G, the set {A, UI : -v u v’} is a nonempty compact connected set containing A, and contained in the set 52 n (A, M(X)), which is open in A, M(X). Hence there exists E > 0 so small that if B E A, M(X), 1 B - ( A , uI)lM < E for some u E [-v, v’], then B E 52; in particular, if A E A, M(X), I A - A, lM < E, then A U I E Qfor alla E [-Y, Y‘].
+ +
< <
+ +
+
+
+
24 1
72. DICHOTOMY CLASSES
Therefore the connected set [-v, v'] is a subset of H ( A ) ; since it contains 0, 72.G implies [-v, v'] C H,(A) = (-;(A), ;(A)).Since v, v' were arbitrarily near ;(A,), ;'(A,), this proves the asserted semicontinuity. & Remark. If X = R, it is easy to show that A if and only if inft2t020 A(u) du > - 00 and
Ji,
;(A) = lim inf ( t - t , ) - l f t-fu+m
EQ
A(i4) du
with X,(A)
t--to--s
R
> 0;
t0
and with X,,(A) = (0) if and only if s u p l ~ I o$,~ Ao ( u ) du - ;'(A) = lim sup ( t -
=
1'
A(u) du
< co and
< 0.
1,
It follows at once that ;(A), ;'(A) are in fact continuous on Q n ( A , f M(X)) for every A , E L. This gives rise to the following question: Query. Are ;(A),;'(A) continuous on R n ( A , M(R))in the general case ? We point out, however, that this question is not so interesting as that of estimating bounds for the parameters of the exponential dichotomy in terms of K,(A), S ( A ) (B some space in b y - , say), and perhaps some norm of A , but not otherwise on A . This problem was already discussed in Section 65 (p. 208, especially the Calculation 65.T).
+
Ordinary dichotomies For ordinary dichotomies there are no satisfactory analogues of the results of the preceding subsection; we shall therefore be rather brief. Let Q, [Q,,] or, in full, Q,(X) [Q,,(X)], denote the set of all A E L(X) for which some [complemented] subspace induces a dichotomy. I n agreement with the case distinctions in Section 63 (p. 186), we denote by Q, [Q,,,.] the subset of all those A E 52, for which X,,(A) [is complemented and] induces a dichotomy, by Q, the complementary the subset of all those A E Q, for which subset of Q,; by Qol [Q,] X,,(A) is closed [and complemented], by Qoz the complementary subset of Q,. Observe that in this classification we de not exclude those A that possess an exponential dichotomy; of course Q C Q, n Qol , Q, c Q O I C n Q O l C *
72.1. A E Q, [ A E Q], if and only i f there exists a [complemented] (L1, L")-subspace for A , and Q, C Q, C Ad(L1, L"). Further, Q, = Ad,(L', L:), QOle = Ad,(L', L:), Qol= Ad,(L', L"), = Ad,(L', L").
a,,,
Ch. 7. DEPENDENCE ON A
242
If X is jinite-dimensional, Q,,, Ad(L1, L,") = Ad(L', L").
=
Proof. Theorem 63.D and 63.F.
SZ,,
= QOlc =
Q,,
= Qoc
= Q, =
&
Remark 1. I n a Hilbert space X t he indices c are superfluous, and we are left with the classes SZ, ,Q, ,SZ,,, ,Do,, Qoz . If Xis finite-dimensional,
sz,
=
sz,,
=
Q,,
.
03
Remark 2. If X is an infinite-dimensional Hilbert space, we already know by the Remark to Theorem 64.E, hence by Examples 65.A (i.e., A = 0), 41.H, 41.1, 65.R, that the intersections Q,, n Q,, , SZ,, n Go,, Q,,, n Q,, , SZ,,, n Q,, are not empty even if X is separable, and actually contain symmetric- or Hermitian-valued bounded continuous A. The inclusion Q, C Ad(L1, L") is proper, as is shown by Example 65.S with the same special properties.
0303
Remark 3. In Example 44.B the adjoint equation yields, in X = I", a constant A E (Go,\ QOIc) n In,,, (since X , = X indxes a dichotomy, and so does the uncomplemented X,,,, = 1;; see Massera and Schiffer [4], Example 4.1). In the outer direct sum X ( l )@ X ( 2 )of two copies of l", with the norm 11x(l) @ x(2)11= max{ll x(l)ll, 11 x(2)II}-a space congruent to 1" itself-we may combine that example, in the former summand, with the formal analogue in 1" of Example 41.H, in the latter summand. It is routine to verify that X , = X k ) @ X g ) = X k ) @ {0(2)},an uncomplemented subspace, induces a dichotomy; that so does the complemented subspace X ( l )@ {O(2)}; and that X , = Xtl) @ X F ) is not closed. This example therefore has (a bounded continuous) A in (Go,\ Rot,) n Qoz n Q,,.
a03
These examples show that the relations between the classes with index c and their complementary classes can be complicated in the general case (it is true that in the examples of Remark 2 the space X is neither separable nor reflexive). The following diagram shows the possible combinations of subsets of Q,:
I
1
I
tl
72. DICHOTOMY CLASSES
243
The large square represents Q, , halved vertically into Qol ,Qo,, , and horizontally into Q,, , Q,, . The inner square represents Q,,; the triangles adjoining the vertical and horizontal midlines represent Q,,, and Q,,, , respectively. The black region is the only possible case for finite-dimensional X . The region outlined in heavy lines corresponds to the cases relevant for Hilbert space. The hatched regions (including the black one) are those cases for which examples are known and described in Remarks 2 and 3 above. The remaining classes might, for all we know, be empty. We single out the most obvious open questions: 300
Que~y. Is it true for all X that Q,
= Q,
? Or that Go,, = Qol ?
72. J. THEOREM.Let QOi stand for any one of Q, , Q,, , Qo! , Q, , , Q, , Q O 2 . Then QOi L'(X) = QOi . If A, E L(X) zs given, L~(X).~n particular, then each QOin ( A , ~ ' ( r l )is) either 8 or A, L W c Q O l C nQ O l , Proof. T h e first statement follows for Q, Q, from 72.1 and Theorem 71.1; for Qol , Q o I c , Q,, , Q o l C , from 72.1 and Theorem 71.E; for 120rl,Q,, , by taking complements. T h e remainder follows trivially; the last statement is true because X , = X and X,,= {0} both induce dichotomies for 0, so that 0 E Q,,, n Qolc . & Qorl , Q,
+
+
+
We omit the obvious (and uninteresting) corollary for spaces B E b2F or b F % that are stronger than L', as well as results on semicontinuity of dichotomy-inducing subspaces, continuity of X,(A), X,,(A), S(A), K,(A), etc., which can be read off from the Remark to Theorem 71.1 and from 71.D; and estimates of the dichotomy parameters (cf. Section 65, p. 208). We close this section with a result that shows that the dichotomy classes are not very large.
If F E 9 or 9%is stronger than 72.K. THEOREM, stzff function, then Q, n F(X) is not dense in F(X).
M and contains a
Proof. We assume that v E F is stiff; we may suppose without loss that I 9 IM = I , and, if F E F W ,that v(n)= 0, n = 0, 1, ...; q~ satisfies (20.3) for some p, M > 0. Let (m,) be a sequence of positive integers with m,,, > 2( I p-')mn , n = 1, 2, ..., and set m, = 1. Define
+
c = v c (-l)"Xrwl".mn+ll E F. W
0
244
Ch. 7. DEPENDENCE ON A
For n
=
1,2,
... we have < -log
M - p(mn - mnJ
+ m,,.-l.
We set A = $Z E F(X), and claim that if B E M(X), I B - A IM < &, then B 4 52,; this will prove the proposition, since F is stronger than M. Set C = B - A , 7 = 2p-ll C lM < 1, and let V , W be the solutions CW = 0 with V(0) = of the operator equations P BV = 0, I@ W(0) = I . Then V ( t ) = W ( t )exp(-J: +(u) du). Using (31.9) for W and (72.4) we have, for every solution y of j By = 0,
+
+
+
Thus y satisfies neither condition (Di) nor condition (Dii) of the definition of a dichotomy; hence B 4 52, , as claimed. & Remark. To emphasize the role of the assumption that F contains a stiff function, we point out that 52, n F(X) is dense in F(X) when F E F or 9 % is lean; in fact, using Theorem 72.J, k,F(X) C L'(X) n Q0 , and the first set is dense in F(X). An immediate corollary for exponential dichotomies follows:
72.L. in F(X).
If F E 9 OY 9%is stronger than M,then 52 n F(X) is not dense
Proof. If F contains a stiff function, the fact that 52 C 52, and Theorem 72.K yield the conclusion; otherwise, 52 n F(X) is empty, by 42.E. &
73. CONNECTION: BANACH SPACES
245
73. Connection in dichotomy classes: Banach spaces Deformation families
We now proceed to study the connectivity properties "in the large" of the dichotomy classes, and especially to make an attempt to single out their components. We restrict ourselves to considering the classes Q, n F ( 2 ) and SZ,, n F ( X ) , where F is any space in 9or P%,not necessarily complete. In this section we deal with the case in which X is a general Banach space; the results are somewhat meager, but provide the background for the more interesting situation in which X i s a Hilbert space, which will be considered in the next section. Throughout this section and the next we shall be dealing with elements A , B, C E L(x), sometimes with various subscripts. W e agree once and for all that U , V , W , with the same subscripts, shall be, respectively, the unique solutions of the operator equations U
+ AU
= 0,
U(0) = I ;
w + cw = 0,
V + BV
= 0,
V(0) = I ;
W(0)= 1.
T h e main connection theorem we shall establish depends to a large extent on the idea of connecting two operator functions in the class under scrutiny by means of a deformation of X for each t that will leave the growth properties and the angular relationships of the solutions of the corresponding equations qualitatively undisturbed. T h e formalization of this idea is contained in the following definition and lemmas. Assume F E 9or 9%given. A function u -+ G, : [0, 11 -+ C(w)i,, , also written (G,), is called an F-DEFORMATION FAMILY if the following conditions are satisfied:
c,,
E F ( X ) for every u ; (a) G, is a primitive and (b) the mappings u .+ G, : [0, 11 -+ C ( x ) , and u + G, : [0,13 + F ( X ) are continuous.
I t follows from (b) and 24.K that D -+ G;' : [0, 11 -+ C ( X ) is also continuous; hence I Go I, I G;' I are bounded on [0, 13; we denote their least common upper bound by m or m,. 73.A. If(G,) is an F-deformation family, so is (G;l); if(GL), ((2,") are F-deformation families, so is (GLG;).
Proof.
Obvious from the definitions and 21.H, 22.V, 24. J, 30.A.
&
Ch. 7. DEPENDENCE ON A
246
73.B. Assume that A and set
E
(73.1)
C,
F ( X ) and that (G,) is an F-deformation family, G,AGil
=
-
G,Gil.
Then u + C, : [0, 11 -+ F(X) is continuous. Also (73.2)
W,
=
G,UG,’(O),
and (73.3)
x++.z = G g
is a one-to-one correspondence between the solutions of (70.2) and those of (73.4)
where g,
2
=
+ c,z
go,
G,f.
Proof. Direct verification and 21.H, 22.V, 24.J, 30.A.
&
73.C. Assume that A E F ( X ) , and that (G,) is a n F-deformation family, and define C, by (73.1). For any D E b S K , Xo,(C,) = G,(O)X,,(A); if the 9 - p a i r or 9%‘-pair (B,D ) is admissible for A , it is admissible for each C,; if Y is a ( B ,D)-manifold for A , then G,(O)Y is a (B,D)-manifold for C, , with KG,(o,y(C,) < m 2 K y ( A )and, if Y is a subspace, SG,~o,y(Co) < m2Sy(A);i f the subspace Y of Xinduces a dichotomy [an exponential dichotomy]for A , then so does G,(O)Y for C, . Proof. 73.B gives all but the last statement. This follows from the first part via Theorem 63.D [Theorem 63.M], or directly as follows: if the solutions x of (70.1) and z of i: C,z = 0 are related by (73.3), we have: if z(0)E G,(O)Y, then x(0) E Y ; if 11 z(0)ll X d(G,(O)Y, z(O)),then II x(0)ll d mllz(0)lI d mh d(G,(O)Y, G,(O)x(O)) m2X d( Y , 40)); if z (hence x) is nontrivial, we have for all t’, t” E R, , 11 z(t’)ll)I z(t”)ll-’ m211x(t‘)ll 11 x(t”)ll-’; finally, if z’, z” # 0, whence x’, X” # 0, we have
+
<
<
<
from 11.A, y [ z ’ ( t ) , z”(t)]
m-l
II I/
z’(t)11-l x ’ ( t )
-
11 z”(t)([-I
x”(t)
l
2 4m-I II x’@) II II z’@)11-1 y [ x ’ ( t ) , x‘W1 >, tm-.?[x’(t), x ” ( t ) ] .
T h e last part of the statement follows. If the parameters of the [exponential] dichotomy induced by Y for A were [v, v’,] N, ”(A), yo(h), those of the [exponential] dichotomy induced by G,(O)Y for C, are
247
73. CONNECTION: BANACHSPACES
[v, v',] m 2 N , m2N'(m2X), gm-2y,(m2X). Note that v, v' are unaltered, a fact which is not apparent from the first, indirect proof. &
Connection by arcs in dichotomy classes
a,.
We denote the sets n F(x), Q, n F(x) for short by Q, , Q O c F , respectively, for any F E 9 or 9%; if the space X requires mention, we Q,,,(X). These sets are assumed to be equipped with may write QcF(X), the metric and topology of F(X). 73.D. Assume that A E F ( ~ )and that (G,) is an F-deformation famiZy, and define C, by (73.1). If A E QOcF [ A E with, say, the complemented subspace Y inducing a dichotomy [an exponential dichotomy] for A , the mapping a + C, defines an arc in QocF [Q,,], and the mapping a -+ G,(O)Y ( a complemented subspace inducing a dichotomy [an exponential dichotomy] for C,) defines an arc in E,(X, 6 ) , hence in E,(X, 6').
a,,]
Proof. 73.B and 73.C. T h e fact that u -+G,(O)Y : [0, I] -+E(X, 6) is continuous follows as in 13.K from the continuity of u -P G,(O) : [0, I ] + which itself follows from condition (b) for an F-deformation family. &
x,
T h e next two lemmas contain the major steps required in our main connection theorem (Theorem 73.G). We assume F E 9 or 9%given.
73.E. Assume that A , B E QOcF [E Q,.,]; assume that the subspaces Y , , Y , induce dichotomies [exponential dichotomies] for A , B, respectively, and have a common complement 2; and that either U ( t ) Y , = V(t)Y,for all t E R , , or U ( t ) Z = V ( t ) Zfor all t E R, . Then there exists C E QocF [E Q,,], with the subspace Y , = Y Binducing a dichotomy [an exponential dichotomy] for C, such that A , C are connected by an arc in QOcF [in Q], and that both W ( t ) Y , = V ( t ) Y B W(t)Z , = V(t)Zfor all t E R,. Proof. We consider first the slightly more complicated alternative in which U ( t ) Z = V ( t ) Z for all t E R, . Let P , Q be the projections along Y , , Y , , respectively, onto 2. Then the projections along U ( t ) Y , , V ( t ) Y , onto U ( t ) Z = V ( t ) Zare, respectively, U(t)PU-l(t), V(t)QV-l(t). Since they have a common range, (73.5)
UPU-'VQV-'
=
VQV-'
VQV-lUPU-'
=
UPU-'.
The function S = UPU-' - VQV-l therefore satisfies S2 = 0. T h e assumption implies, by 41.B,(c) that UPU-l, VQV-l E C ( x ) , so that S E C ( 8 ) . Also S is a primitive (by 30.A), S(0) = P - Q , and
248
Ch. 7. DEPENDENCE ON A
+
= -AUPU-l+ UPU-IA RVQV-l - VQV-lR, so that E F(X) (by 22.V, 24.5). For each u E [0, 11 we define G, = Z US,a primitive in C(X),,, , since G;' = Z - US, and claim that (G,) is an F-deformation family. Indeed, = USE F(X), and the mappings u -+ G, , u ---t are linear, hence continuous. We define C, by (73.1); since Go = Z we have C, = A. We claim that C = C, satisfies the conclusion of the lemma, with Y , = G,(O)Y,. Indeed, by 73.D, A = Co and C are connected by an arc in QOcF [in Q,,]; and Y , induces a dichotomy [an exponential dichotomy] for C. Also, G,(O) = Z P - Q, Gyl(0) = Z - P Q, so that
+
c,
e,
+
+
Y c = Gl(0)YA = ( I - Q)YA C Y B = G,(O)G;'(O)YB =
-p,
G1(0)y A
9
so that equality holds throughout,and Y , = Y E further, ; G;'(O)QG,(O)= P (for t = 0, (73.5) becomes PQ = Q, QP = P). Therefore (73.2) (73.5) yield
WQW-l
+ S)CIG;'(O)QG,(O)U-'(I- S ) = ( I + S)UPU-'(I - S ) = VQV-1.
=(I
Hence the null-spaces W ( t ) Y , = W ( t ) Y , , v(t)YB, and the ranges W ( t ) Z , V(t)Z of these equal projections respectively coincide for each tER+. If we now consider the alternative assumption in which U ( t ) Y , = v ( t ) Y B , we may repeat the preceding proof, replacing Z by Y A= Y E and both Y A, Y Eby 2. T h e proof goes through as before, but is simpler, since P = Q in this case. &
73.F. Assume that A E L ( X )and that P is a projection in X with null-space Y . For every function q~ E L set (73.6)
A,
=
A
+~
( -1 2UPUp') E L(X).
(a) If Y induces a dichotomy [an exponential dichotomy] for A, for some v, then UPU-l E C ( x ) ,and Y also induces a dichotomy [an exponential dichotomy] f o r A ,for any $ > p. (b) If Y induces a dichotomy for A , then Y induces an exponential dichotomy f o r A,, provided q is stifl. (c) Zf UPU-' E C ( x ) and v > 11 A 11 [and v - 11 A 11 is $ti#] then Y induces a dichotomy [an exponential dichotomy] for A,.
249
73. CONNECTION: BANACHSPACES Proof. Direct computation shows that
Thus, for any
t o ,t E
R, ,
f/,,,(t)(I- P)u;'(~,) = ~ ( t ) (r P)u-'(~,,)exp
(-
Itt
p(u) du) ,
n
(73.7)
t
U,,,(t)pU;'(t,) = U(t)PCI-'(t,) exp
(jp(u) du) . tn
Part (a) follows at once from 41.B,(c),(d) [42.B,(c),(d)]; UPU-l E C ( x ) from (73.7) with t = t o . If y is stiff, with M , p as given by 20.D, we obtain from (73.7)
and part (b) follows from 41.B,(c) and 42.B,(d). If UPU-' E C ( x ) , then U(I - P ) W 1E C ( x ) , and we obtain from (73.7), using (31.9), for all to , t E R,
If y 2 Il,A I/ [if y - 11 A 11 is stiff, with M , p as given by 20.D], then exp( - J,,,(p)(u) - 11 A(u)ll)du) 1 [< M-le-fi(l-lO)]for t 3 to >, 0; and similarly for the other inequality for to 3 t >, 0. Part (c) then follows from 41.B,(d) [42.B,(d)]. &
<
250
Ch. 7. DEPENDENCE ON A
The assumptions of the following main theorem of this section are the same as those of 73.E, but we repeat them for the sake of easier reference.
Assume that A, B E QOcF [E SZ,,]; assume that the 73.G. THEOREM. subspaces Y , , Y Einduce dichotomies [exponential dichotomies] for A , B, respectively, and have a common complement 2; and that either U ( t ) Y , = V ( t ) Y ,for all t E R, , or U(t)Z = V ( t ) Zfor all t E R, . Then A , B are connected by an arc in QOcF [in a,,].
Proof. On account of 73.E, we may assume that Y , = Y E= Y , and that both U ( t ) Y = V ( t ) Y and U(t)Z = V ( t ) Z hold for all t E R, . Let P be the projection along Y onto 2. T h e projection along U ( t ) Y = V ( t ) Y onto U ( t ) Z = V(t)Z is then S ( t ) , where S = UPU-' = VPV-l. By the assumption and 41.B,(c), S E C ( 8 ) . For each u E [0, I] define the following elements of L ( 8 ) :
+ .(I1
+ II B ll)(I - w, H, = B + .(I1 A II + II B ll)(l - 2S), (1 - u)A + uB, 11, = c, + (11 A II + I1 R II)V - 2 s )
A,
=A
C,
=
A II
=
(1
-
u)A,
+ UB, .
Actually, by 22.V, 24. J, A,, B, , C, , D, E F(X), and the mappings [0, I] into F(X) are linear, hence continuous. Now S is a primitive, and a solution of S = -AS S A , and also of S = -B S SB, both times with S(0) = P;therefore S is a solution of u --t A, , etc., from
+
+
s = -CJ + S C , ,
(73.8)
S(0) = P
for every u. But W,PW;' is also a solution of (73.8), whence
w,PIv;'
(73.9)
=
s E C(R).
We now apply 73.F: by part (a) of that lemma, Y induces a dichotomy [an exponential dichotomy] for A , , B, , u E [0, I ] . By (73.9), C , satisfies the assumption of part (c) of 73.F in place of A ; and 11 A 11 11 B 11 - I1 C, 11 3 011 A II (1 - .)\I B 11 3 0 [and is a stiff function, since (1 A II,II B 11 are stiff, by 42.E]; therefore Y induces a dichotomy [an exponential dichotomy] for D, , u = [0, 11. Thus A = A, and A, = D o , A , = Do and D, = B, , D, = B , and B, = B are connected by arcs in GOOF [in Q,,] . &
+
+
Remark 1. We observe that along the arc connecting A , B in Q,,, [in Q], it is possible to choose the subspace Y inducing a dichotomy [an exponential dichotomy] so as to trace out an arc in E,(X, 6) (hence
74. CONNECTION: HILBERT SPACE
25 1
in E,(X, 6’)): this is the case for the arc used in 73.E-as indeed for any arc of Cu’s determined by an F-deformation family-on account of 73.D and the fact that Y , = G,(O)Y,; and it is trivially the case for the straight segments used in the proof of Theorem 73.G, since Y is fixed throughout the proof.
Remark 2. As concerns the case of Q, , the cumbersome assumption “ Y , , Y Einduce exponential dichotomies” is superfluous if we assume that X , ( A ) , X,(B) have a common complement and if we write “ U ( t ) X , ( A )= V(t)X,(B)” or, more suggestively, “X;(A) = X;(B)” instead of “ U ( t ) Y , = V(t)YB)’. Remark 3. T h e assumption “ U ( t ) Y , = V ( t ) Y , for all t E R+” may be weakened to “for each t E R,. , either U ( t ) Y , C V ( t ) Y , or U ( t ) Y , 3 V(t)Y,”; indeed, this gives Y , C Y E or Y , C Y , , but since 2 is a common complement we must have Y , = Y E ;and we then apply 13.K. Similarly, “ U ( t ) Z = V(t)Zfor all t E R+” may be weakened to “for each t E R, , either U(t)Z C V(t)Z or U ( t ) Z3 V ( t ) Z ” . We conclude this section with a theorem on the relationship between and Q, .
Q,-,,F
73.H. THEOREM. If F contains a stiff function, then QOcF (contains Q, and) is contained in cl$,, ; and every arcwise-connected component of QocF contains at least one arcwise-connected component of Q , .
Proof. QrFCQ,,, is trivial. If A EQ,,~~,let Y be a complemented subspace inducing a dichotomy for A , and P a projection along Y . For any stiff function g, E F, ug, is stiff for all real u > 0; therefore, if A,, is defined by (73.6), the fact that UPU-’ E C ( x ) (by 41.B,(c)) implies that A,, E F ( x ) , and 73.F,(b) that A u , ~ Q c Ffor all u > 0. Since u + A,, is linear, hence continuous, from [0, 13, say, into F(X), it follows that A = A , is in the F-closure of Q, , and connected to, say, A, E QcF by an arc in Q O c F . &
Remark. T h e assumption that F contains a stiff function is essential; for, if it is not satisfied, QcF is empty (by 42.E), whereas 0 E Q O c F . 74, Connection in dichotomy classes: Hilbert space
A bit
o f motivation
I n order to understand the work in this section, we glance briefly at one special case. As mentioned in the Introduction to the chapter, we
252
Ch. 7. DEPENDENCE ON A
intend to show, among other things, that for a finite-dimensional (euclidean) X , not the real plane, the components of Q, (and of other QW’s)are exactly those subsets of Q, (or QF) for which X,(A) has a certain dimension; the difficult part consists in showing that these subsets are connected-actually arcwise connected. Let us assume for definiteness that the scalar field is R and that Dim X = 3 (we recall that Dim and Codim denote orthogonal dimension and codimension), and that we are investigating that subset of Q, for which Dim X o ( A )= 1. One element of this subset is the constant operator function B = I - 2P, where P = I - P, is the orthogonal projection along the one-dimensional subspace Y, say. Let A be any member of the subset. If we could connect A in the subset itself with some C for which X o ( C )= Y = Xo(B)and W(t)Y = V(t)Y = Y for all t , we could in turn connect C to B on account of Theorem 73.G (including the Remark). T h e most obvious way of connecting A with such a C is to find an L“-deformation family (G,) that will “twist” the bounded solution x of (70.1) (unique up to multiplication by scalars) gradually into Y \ {O}-on account of connectivity, into one open half-line, say Y,, of Y-for every t. It seems most natural to wish to choose for G, a proportional interpolation, with CT E [0, I] as the parameter, of the smallest rotation that will take x ( t ) into Yl. This is not strictly possible, because x ( t ) and Y , may become antipodal, or nearly so (which is just as bad, because it spoils the boundedness of Go); it therefore becomes necessary to carry out the rotation in two steps, each through an angle uniformly bounded away from 7~ (for technical reasons, both steps are merged in one in the actual proof). The developments of this section relating to such proofs of connectedness are mainly elaborations of this idea, with variants necessitated by the fact that Dim X o ( A ) may take any value, that the manipulation with rotations is not possible in dimension 2, and that the complex scalar field imposes a different geometric structure on the “rotations” (unitary operators). We remark in passing that the deep topological reason behind the geometrical constructions is the known fact that the Grassmann manifolds (i.e., the algebraic manifolds of m-dimensional subspaces of an ndimensional real or complex vector space) are simply connected, except for the real case with n = 2, m = 1. I t will be seen that, contrary to all our work up to this point (cf. Section 15), the methods in this section strictly require the inner product and the Hilbert norm, even for finite-dimensional X ;and that although the results in this last case could be stated without reference to a special
74. CONNECTION: HILBERT SPACE
253
norm, this is highly inconvenient for part of them, as will be pointed out at the proper time. We also call attention to the unusual necessity of distinguishing between the real and complex scalar fields. Throughout this section we are dealing with Hilbert spaces, so that notation and terminology introduced in Section 14 are in force.
Two geometrical lemmas T h e following two lemmas play a fundamental part in the study of the connectivity of dichotomy classes. 74.A. Let Z be a real Hilbert space with Dim Z > 2. Let e E aZ(Z) be a given point and p a primitive on R, with values in aZ(Z). For any fixed w , $ x < w < Q x , there exists a primitive q with values in aZ(Z) such that:
(4 II 4 ll < cll P; II,
where
c = c(w) =
7r
2( -cos
(b) (p(t),q(t)) 3 cos 2w t E R,; (c)
( e , q ( t ) ) 3 cos 2w
sin w
2w
- cos w )
> n 4514 > 1;
> -1, i.e., +(p(t), q(t)) d 2w < x , for all
> -1, i.e., +(e, q ( t ) ) < 2w < x , for all
t
E
R,
.
74.B. Let Z he a complex Hilbert space. Let e E aZ(Z) be a given point and p a primitive on R, with values in aZ(Z). For any fixed w , * x < w < in, there exists a primitive q with values in aZ(2) such that: (a)
II 4 II
<
CII
$
I19
where 2n
c = c(w) =
x
-cos
2w
(b) ( p ( t ) ,q ( t ) ) is real and - 2w < &r) for all t E R,; (c)
( 3 + cos w
cos w
+ cos 2w ) > 1;
3 -cos 2w > 0 (i.e., +(p(t), q(t)) <
I(e, q(t))l = (q(t),e sgn(q(t), e ) ) 3 -cos
2w
> 0 for
all t E R,
.
Remark. In 74.A, 74.B, if p is continuous it is possible to obtain q continuous, though at the cost of an (arbitrarily small) increase of c(w). T h e proofs of these lemmas are elementary but very complicated technically, and are omitted here; they are given, together with a sketch of the proof of the Remark, in Schaffer [8], Lemmas 7.1, 7.2. We merely observe that 74.A could not be proved by setting q = @(p), where
Ch. 7. DEPENDENCE ON A
254
@ : i Z ( Z ) + aZ(Z) is a fixed continuous mapping, since @ would omit
-e from its range, and yet map no point on its antipode. From 74.B,(a), (c), and 30.B we find, as an addendum to 74.B,
Deformation families 74.C. Assume that F E 9or 9% isgiven. Assume that p , q are primitives on R , with values in the real [complex]Hilbert space X , with 1) p 11 = 11 q 1) = I and p , 4 E F ( X ) , and that for some number r , - 1 < r 1, they satisfy ( p ( t ) ,q(t)) [real and] 3 rfor all t E R , . Then there exists an F-deformation family (G,) such that:
<
(a) G,(t) is orthogonal [unitary]for each cr, t ; (b) Go = I and GI$ = q ; (c) i f v E X and ( v ,p ) = ( v , q) = 0, then G,v = v f o r all u. Proof. We define y o by cos vo = r, 0 vo < r, and the function q by cos v = ( p , q), 0 q~ y o (i.e., = Q ( p , q)), so that q~ is continuous and cos cp a primitive. In the real case, we intend to construct Gl(t) as the rotation through the angle cp(t) that carries p ( t ) onto q(t), and G,(t) by interpolating the angle of rotation linearly; this is possible since v is bounded away from r ; clearly, any element v E X that is orthogonal to all values of p and q becomes invariant for each G,(t). T h e actual construction, which follows, requires some computation; in the complex case we refrain from any geometrical picture, and therefore it is as well to carry out the construction explicitly in both cases; they can be considered simultaneously. We begin by introducing two auxiliary functions, defined in the rectangle [0, 13 x [0,yo]of the (cr, +)-plane by
<
< <
(= u at $ = 0),
we also need the following derivatives: i)rdO! $)
= qcos
$1 I/l)
=
-
u cos u$
sin J, - sin u$ cos $ sin:' $ (= - i u ( I - u2)
at
$
= 0),
74. CONNECTION: HILBERT SPACE
255
(= - 2 1 2 u2(4 - u2) at
#
= 0).
Since all these functions are continuous in their compact domain, they are uniformly continuous and bounded; let p be a common bound of I pi I, I PI I, i = 1, 2. We define the skew-symmetric-valued [skew-Hermitian-valued] primitive H = (., q)p - (-, p ) q E C ( x )and note the identities: (74.2)
Hp
=
pcos p, - q,
H 2 p = -p sin2 rp,
( H p ,p )
H3
H 2 q = -qsin2p,,
H q = p - qcosrp,
=
(Hq, q)
=
0,
= -Hsin2rp.
Direct computation also yields (74.3)
I (cos p,). I
< II b II + II 4
!I fi II < 2(ll P II
111
+ II 4 II 1,
whence (cos p))' E F, fi E F(X) (if F E 9%these functions are obviously continuous). We now define the family (G,) by (74.4)
G,
= 1 - PI(0,
rp)H
+
PdU,
V)H2
and claim that it satisfies the conditions of the statement. Indeed, because of the skewness of H , we have GZ = I pl(u, p))H p2(u, p))H2, and use of (74.2) gives, for each u,
+
G,q
=
qG0
=
=1+(2
I
+ (2P2(U, p,) 1
cos up, sin2 p,
-
-
p:<.,
p,))H"
sin2 up, sin2 p,
(I
+ Pl(.,
+
rpW4
cos ~ r p ) ~ sin2p, H 2 sin4 rp
1
-
=
I;
thus Go(t) is orthogonal [unitary] for all u, t (condition (a) of the statement) and G, E C(x)inv. From (74.4), Go = I , GI = I - H (( 1 - cos p))/sin2q ) H 2 , so that, by (74.2), C,p = p - p cos p) + q - p( 1 - cos p)) = q (condition (b) of the statement). If v is orthogonal to all values of p , q, we have Hv = (v, q)p - ( v , p ) q = 0, and by (74.4) G,v = v (condition (c) of the statement). It remains to prove conditions (a), (b) for an F-deformation family. Since cos q is a primitive and the pi, pi. are uniformly continuous the functions pi(u, p)), i = 1, 2, are primitives; hence so is G, , and
+
(74.5)
G = ( -Pi(.,
rp)H
+ pX9.
+
- P A 5 p,)h
P*(U,
v ) H 2 ) ( c o sp,).
v)(Hfi
+ AH),
256
Ch. 7. DEPENDENCE ON A
<
eu
whence, using (74.3) and 11 H II 2, we get II (I ,< 16p(ll P I\ -k 11 4 11) E F(X), and (and euis continuous if I;, 4 are continuous), whence condition (a) for an F-deformation family holds. Since p i , p ; , i = 1, 2, are bounded and uniformly continuous on their compact domain, and 9 is continuous, the mappings u -+ pi(a, q), a -+ pi(., q), i = 1, 2 from [0, 13 into C are continuous. I t then follows from (74.4), (74.5) by 21.H, 22.V, 24.J that the mappings a -+ G, : [0, I] 4 C ( 2 ) and cr -+ Go: [0, 11 -+ F ( 2 ) are continuous (condition (b)). 9,
Exponential dichotomies: the general case Since in this section X is a Hilbert space, we may write Q, , QoF for the sets Q,, , QOcF . We assume that F E 9 or 9%is given. I n the present subsection we investigate the connectivity properties of 52, excluding the case where the scalar field is real, Dim X = 2 (the “real plane”), Dim X o ( A )= 1 ; this more complicated exceptional case will be considered in the next subsection.
<
74.D. Assume that A E QF , and that Dim X o ( A ) < CO, Dim X o ( A ) Codim X o ( A )[Codim Xo(A) < 00, Codim X o ( A ) Dim Xo(A)]. If X i s real and Dim X = 2, assume further that Dim X o ( A ) # 1. Let Y be any subspace of X with Dim Y = Dim Xo(A) [Dim Y = Codim X,(A)]. Then there exists C E Q, , connected to A by an arc in Q, , such that C ( t ) YC Y a.e. (that is, for almost all t E R+),and Xo(C ) = Y [ Y is a complement of XO(C)l* Proof. If Dim X = 1, the conclusion holds trivially with C = A, since X o ( A )is either (0)or X . We shall therefore assume Dim X >, 2 throughout the proof. Set n = min{Dim Xo(A), Codim Xo(A)} < 00. Observe that, if X is real, n 2 Dim X. We shall prove the following sequence of statements, of which the conclusion of 74.D is obviously the particular case m = n:
<
+ <
< <
(P,) For given m, 0 m n, and any subspace Y of X with Dim Y = m, there exists C E Q, , connected to A by an arc in Q, , such that C ( t ) Y C Y a.e., and Dim X o(C )= n, X o ( C )3 Y [Codim X,,(C) = n, X o ( C )n Y = {O}]. T h e proof proceeds by induction in m. (Po) is trivially true with C = A. For given m, 1 m n, assume (PmLp1)true. Let Y be a subspace with Dim Y = m; since m >, 1, there exists e E Y , 11 e 11 = 1. Set Yo = Y n {e}l, a subspace of dimension m - 1. By (P7t,-l), since
< <
74. CONNECTION: HILBERT SPACE
257
connection by an arc in QF is transitive, there is no loss of generality in assuming that actually A(t)YoCY oa.e., and X o ( A ) 3Yo[Xo(A) n 17= {O}]. Let Q = I - Py0 be the orthogonal projection along Yo; then QA = QAQ. Since m - 1 < n, there exists a subspace Y’ 3 Y o with Dim Y’ = m such that Xo(A)3 Y’ [Xo(A) n Y’ = {O}]. Let x be any solution of (70.1) with x(0) E Y’ \ Y o , so that Y o and x(0) together span Y’. It follows from the fact that Yo is a.e. invariant under A(t), hence invariant under the invertible U ( t ) for all t E R, , that x ( t ) 4 Y o , i.e., Q x ( t ) # 0, for all t E R, . We set p = sgn(Qx), a primitive with values in a.Z(Y,l). By 30.B, It$ II I/ Qillill Qx I1 = /I QAx llill QxII = II QAQx llill QxII II Q A II II A II, so that p E F(X)(if A is continuous, so is i, hence so is 6).Also e E a.Z( Y;). If X is real we have Dim Yo 3 = m 2 n 2 Dim X , so that Dim Yk = Codim Yo 3 3. We may therefore apply 74.A, in the real case, or 74.B, in the complex case, with 2 = Y,L and w arbitrary but fixed in the appropriate range, and construct the corresponding function q, a primitive with values in aZ( Y;). By 74.A,(a) or 74.B,(a), as the case may be, plus the Remark to those lemmas if necessary, we find Q E F(X); in the complex case, (74.1) yields, in addition, ( e sgn(q, e)). E F(X). We now consider the pairs of functions (pi,qi), i = 1 , 2 , given by p , = p , q1 = p , = q, and q2 = e in the real, 4, = e sgn(q, e) in the complex case. From the above computations and conditions (b) and (c) of 74.A or 74.B, we conclude that either pair (pi,qi) satisfies the assumptions of 74.C, with r = cos 2w in the real, and r = -cos 2w in the complex case. Let then (Gi,u),i = 1, 2, be the corresponding F-deformation families. We define the F-deformation family (G,) by G, = G2,0Gl,u(by 73.A; this is what was meant at the beginning of the section by “carrying out the rotations simultaneously”), and define C, by (73.1). We claim that C = C, satisfies (Pm). By 74.C,(b) we have Go = I , whence Co = A . By 73.D, C i s connected to A by an arc in 9,. Since p i ,qi have their values in Y , I , 74.C,(c) implies that the restriction of G, t o Yo is the identity, i.e., G,(I - Q ) = I - Q ; hence also G;l(Z - Q ) = Z - Q, Gu(Z- Q ) = 0. Since Yo is a.e. invariant under A , G,, G;1, G,, it is a.e. invariant under C , , by (73.1), and in particular C(t)YoC Y o a.e. Now using 74.C,(b), we obtain for the special solution x of (70.1) considered above:
<
+
< + < + <
<
258
Ch. 7. DEPENDENCE ON A
Since Q x ( t ) # 0 for all t and q2 = e or q2 = e sgn(q, e ) , we see that Yoand G,(t)x(t)span Y for each t . By 73.B, z = G,x is a solution of i: + Cz = 0; therefore -CG,x = (G,x)’;but since the values of the latter function are in Y a.e., we have C(t)G,(t)x(t)E Y a.e.; together with the end of the preceding paragraph, this implies C ( t ) Y C Y a.e. By 73.C, X,(C) = G,(O)X,(A). Hence Dim X,(C) = n [Codim X,(C) = n ] . Also G,(O)Y’ is spanned by G,(0)Yo= Y o and G,(O)x(O), hence G,(O)Y’ = Y . Therefore X,(C) = G,(O)X,(A)3 G,(O)Y‘= Y [X,(C) n Y = G,(O)(X,(A)n Y ’ ) = (011. &
Remark. Along the arc connecting A and C , the subspace X,, traces out an arc in E ( X ) , on account of the construction and 73.D. We now obtain the main results of this subsection. For any two cardinals, d, c, d c = Dim X, we define the class
+
:Q,=
{ A E Q, : Dim X,(A)
=
d , Codim X,(A)
= c}
=
Q, n { A : &(A)
E :S((X)}.
74.E. ;52, is not empty i f and only if 52, is not empty, namely if and only if F contains a stiff function. If q~ E F is stqf, and ;f P is any orthogonal projection in X with Dim ( I - P ) X = Codim P X = d, Dim P X = c, then Aq,p= q ( I - 2P) E $2,.
Proof. T h e “only if” part follows from 42.E; the proof of the “if” part consists in verifying E :52,, and this follows from 73.F,(c) with 0 instead of A , hence I instead of U . & Remark. 1.
Observe that, contrary to the general rule, the case
c = d = 1, X real, is not excluded in 74.E. However, cf. 74. J and
Theorem 74.K.
Remark 2. If Y = ( I - P ) X , Z = P X , and if we identify X with the outer direct (Hilbert) sum Y @ Z,then Ap.pbecomes q(Id @ ( -Ic)), and the equation i A,,,x = 0 becomes the system
+
$+qy=O,
z-qz=o.
Remark 3 . T h e elements A,,,Pare a kind of canonical representatives of ;a,; the ambiguity in the choice of P may be apparently avoided by proceeding as in Remark 2. There is no obvious unique choice of q unless F contains the constants (in particular, if F E .Y is locally closed, by 23,s with Remark l), in which case we may choose ‘p = 1, becomes :A = I , @ ( -Ic), and the equation i A,,,x = 0 becomes the system 3 y = 0, 1 - z = 0.
+
+
74. CONNECTION: HILBERTSPACE
259
74.F. THEOREM. Assume that at least one of the two cardinals d , c is finite, and that i f X is real they are not both = 1. Then I;Q, is arcwise connected.
Proof. Consider AP,, as in 74.E, with Y = (I - P ) X , Z = P X , Dim Y = d , Dim Z = c. From the proof of 73.F we have X0(Ap,,)= Y , and U,,.,(t)Y = Y, U,.,(t)Z = Z for all t E R , . By 74.E, it is sufficient to show that any A E $2, is connected by an c, 74.D implies that there exists C, arc in $2, to AV,,. If d < 00, d connected to A by an arc in QF , such that X,(C) = Y = Xo(A9,p),and C ( t ) Y C Y a.e.; therefore Y is invariant under the invertible W ( t ) for all t E R , , so that W ( t ) Y = Y = U,,,(t)Y for all t E R, . By Theorem 73.G, C and AV,, are connected by an arc in Q,; therefore so are A and AV.,. Now along these arcs, X , describes an arc in E ( X ) , by Remark 1 to Theorem 73.G and the Remark to 74.D. But by 14.G the set zE(X) is open and closed in E ( X ) , so that X , stays in ;E(X) along those arcs; they are therefore contained in $',, and this set is arcwise connected. If c < 00, c d , 74.D implies that there exists C, connected to A by an arc in QF , such that 2 is a complement of X,(C), and of course of Y = X0(Ap,,), and C ( t ) Z C Z a.e. We deduce as above that W ( t ) Z = 2 = U,,,(t)Z for all t E R, , and another application of Theorem 73.G shows that C and ApSp are connected by an arc in Q,. T h e proof is completed as in the preceding paragraph. &
<
<
74.G. THEOREM. Assume that F is stronger than M (in particular, F E .F or .F+or F%+) and contains a stiff function. Then each :Q, is
open and closed in 52,; if either d or c is finite, and not both are is real, then :Q, is exactly one component of Q, .
=
1 if X
Proof. Since F is stronger than M, the sets Q,, $2, are the settheoretical intersections of Q, , C,QM , respectively, with F(R), and the topology of Q, is finer than that induced by Q, . For the first part of the conclusion it is therefore sufficient to consider the case F = M. Now by 72.A the mapping A + X,(A) : QM + E ( X ) is continuous, and Q :, is the inverse image of ;E(X), which is open and closed in E(X), by 14.G. Hence >QM is open and closed in SZ, . T h e second part of the conclusion now follows for every F from the first part to show that C,QF is open and closed, from 74.E to show that it is not empty, and from Theorem 74.F to show that it is connected. & Remark I . By 72.A, Q, is open in M(R), hence Q, is open in F(X); therefore the components of Q, are exactly the arcwise-connected
Ch. 7. DEPENDENCE ON A
260
components; although arcwise connectedness follows from Theorem 74.F it is therefore not necessary to point out this fact especially in Theorem 74.G.
Remark 2. T h e assumption that F is stronger than M is by no means superfluous in Theorem 74.G. If, for instance, we assume that F contains v such that limt+a :J I v(u)l du = 00, then the class U{$Q, : c or d finite} i s itself arcwise connected. T h e proof is given in Theorem 74.R (with Remark 1); it is here postponed because we have not yet dealt with the exceptional case, and because its greatest interest is for finite-dimensional X . Exponential dichotomies: the exceptional case We now consider :QF, X real, Dim X = 2. We may provide the real euclidean plane X with an ORIENTATION, i.e., appoint a certain arbitrary Cartesian coordinate system as RIGHT-HANDED. This allows us to speak of “angles” as of quantities with sign; to avoid bothering with the “choice of determinations” of these angles, we introduce the following definitions. If w E R, r ( w ) shall denote the rotation through the angle w , i.e., that element of (an orthogonal operator) that is represented in any righthanded Cartesian coordinate system by the matrix (~~~ : :). Then r ( w ’ 0”) = r(w’)r(w”),
-::
+
+
and d r ( w ) / d w = r ( w in). If x is any continuous function from R, into X \ {0}, w ( x ; 2) denotes the angle through which x has turned between 0 and t ; i.e., the unique continuous w ( x ; .) satisfying w ( x ; 0) = 0 and r ( w ( x ; .)) sgnx(0) = sgn x. If x is a primitive, it is well known that
<
so that w ( x ; .) is a primitive and I B(x; .)I 11 i11/ 1 1 x 11; if iis continuous, so is &(x; *). Assume now that A E L(w) and Dim X,(A) = 1. Let x be any nonzero solution of (70.1) wiiii x(0) E X o ( A ) ;obviously w(x; *) will not depend on the particular x chosen-they are all collinear-and we denote the common value of these w ( x ; by w A . T h u s w A is a primitive, a)
26 1
74. CONNECTION: HILBERTSPACE
<
<
with I &A I 11 f 11/11 x 11 11 A 11, and 8, is continuous if A is continuous. Also, w, is the unique continuous function satisfying wA(0)= 0 and (74.6)
r(wA(t))Xo(A) = X;(A) = U(t)Xo(A),
t E R+.
We again assume F E 9or 3 V given.
74.H. THEOREM. Let X be a real euclidean plane. If A, B
E :QF
and
q,for some orientation of X,wB - w Ais bounded, then A, B are connected by an arc in $2,.
<
Proof. There exists some real wo , say with 0 wo < 7, such that r(wo)Xo(A)= Xo(B). We define the F-deformation family (G,) by G , = r ( u ( w o + wB - o A ) )Indeed, . G,(t) is orthogonal for every a, t , and a primitive, hence G, E C(X)inv;and G, = a(&, - SA)F(u(wo oB- o A )+ &r), which is continuous if A, B are continuous, and
+
11 Gu 11
< I(.
&A
I 4-I &B 1)
<
A 11 f 11
11)s
so that G, E F(X) (condition (a) for an F-deformation family). Finally, for any a', a'' E [0,13,
+
I1 G,* - Go* II = I1 Q(0" - 0')(wn W B - W A ) ) - 1 1I < 10" - 0 ' 1 1 wO + 1, /I eu.- Gut11 < I &B - & A ]{I 0'' - 0' I 0' I/ q ( u " wf?- W A ) ) - I l l } < I 0" - 0' 1(11 A 11 + 11 1!)(1 I WO - W.4 1).
+
Hence
+
+
262
Ch. 7. DEPENDENCE ON A
so that wc0 = uwB is bounded. Further,
+ (1 -
x:(c)= r(%+ w B ( t )
0
)
in~particular, ; wc0 - w A = o(wE- w A )
~
- wA(t))Xi(A)
=~ ( ~ ~ ( ~ ) ) r ( ~ 0 ) ~ ' = ( ~x ~ i(B A )(. t ) ~ x ~ ( ~ )
By Theorem 73.G (with Remark 2), C and B are connected by an arc in Q,; but if D is any element on that arc, the proof of Theorem 73.G shows that X $ D ) = X $ C ) = X,'(B), whence, in particular, X o ( D )= Xo(B),so that the arc is in $2,; and, by (74.6), w D = w c = w E ,SO that w D - w Aremains bounded. 4p( If A , B E :Q,, we write A M B if w E - w A is bounded for some orientation of X . Obviously, M is an equivalence, and does not depend on the choice of orientation. Theorem 74.H, with some details of its proof, then implies the following corollary: 74.1.
Each equivalence class of the relation
M
in :52, is arcwise connected.
We should like to find, in analogy to 74.E, certain "simple" elements in each of these equivalence classes. T h e choice is not so obvious in this case, but we shall carry it out in such a way that the solutions of the homogeneous equation which are orthogonal at t = 0 to the solutions starting from X o ( A )remain so for all t .
74.J. Let X be an oriented real euclidean plane. If is any stiff function in F,w any real-valued primitive with w(0) = 0, d E F, and P any projection in X with Dim P X = 1, then BP,w,P = --dr(&r) q T ( w ) ( I - 2 P ) r ( - w ) E :Q,, and X o ( B P , w , p= ) ( I - P ) X , w S . W . P - w.
+
Proof. Since shows that
v, d E F, also
BP,w,P E F(X), and direct verification
t
(j d4 d") + (1 - P ) exp (- j' du) du)) = p e x p (d u ) du) + (1 - P ) exp (J cP(4d.)) F ( - 4 ) ) ;
V,.,.P(t) = r ( 4 )( pexp
0
0
t
V;,:,P(t)
Jt
0
hence
It
-
~ T * w * P ( ~ ) ( w~ ; , L P ( t " )
I1 = exP (-
j
1
d4 du) to
9
74. CONNECTION: HILBERTSPACE Therefore 20.D and 42.B,(d) imply Bq,w,PE tQ,, (I- P ) X . Finally,
+@))(r
~ ( 4 ) ~ 0 ( ~ P , W .= P )
- PIX =
w
B%W.P
= w,
by (74.6).
and Xo(Bp,u,p) =
vql,Ul,P(w - PIX =
whence
263
~ql.w*P(wo(~~.Ul.P),
&
Remark 1 . If y , z are the components of x in the right-handed Cartesian coordinate system in which P is represented by the matrix (: (there are actually two such systems; either will do), the equation 2 Bq,w,px= 0 becomes the system of two scalar equations:
0”)
+
j
+ t
Remark 2.
cos 2w.y
+ (-8 +
‘p sin
2w)z = 0,
+ (8+ p sin 2w)y - p cos 20.1.2 = 0.
Bq,,ol,P,m BC,,o,.P,if and only if
w2 - w1
is bounded.
We can now fill the gap in Theorem 74.G concerning the class
#, .
74.K. THEOREM. Assume that X is a real euclidean plane and that F is stronger than M (in particular, F E Y or 9-+ or 9-%+) and contains a stiff function. Then the components of $2, are exactly the equivalence classes of the relation M .
Proof. $2, is open and closed in SZ, (Theorem 74.G), and the equivalence classes are connected (74.1). I t will therefore be sufficient to show that they are open in iSZ, . We provide X with an orientation. Let then A E $2, be given. By 72.B and the fact that F is stronger than M there exists E > 0 so small that if B E ;QF, I B - A < E , then 6 ’ ( X i ( A ) ,XA(B))< & for all t . Let B be so chosen, and let w, be the &r. unique real number such that r ( w o ) X o ( A )= X,(B), - & < w, By (74.7), r(oo w B ( t )- w A ( t ) ) X k ( A= ) XA(B). A glance at a diagram wB(t)- wA(t))l= 6 ’ ( X i ( A ) ,X,$B)) < 9. Since shows that I sin(w, w B - w A is continuous, and its value is 0 at t = 0, this implies I wo + w B - w A I < b,and B M A . &
IF
+
<
+
To show how important is the assumption that F is stronger than M, we now give a result in the opposite direction (cf. Theorem 74.R). In order to see the contrast, we observe that, in Theorem 74.K, kF is stronger than kM = M, , and therefore contains no s t 8 functions. 74.L. Assume that X is a real euclidean plane, and that a stifl function. Then :SZ, is arcwise connected.
kF contains
Ch. 7. DEPENDENCE ON A
264
+
be the stiff function of the assumption; assume X Proof. Let oriented, and let P be any fixed orthogonal projection in X with Dim P X = 1. I n view of 74.1 and 74.J (including Remark 2), it will be sufficient to show that for every real-valued primitive w with w ( 0 ) = 0 and oj E F there is an arc in $2, connecting Bv,w,P and Bv,O,P = A v , P ;or, more specifically, that the mapping u -+ Bv,ow.P : [0, 11 ---t $2, is continuous. Since t,b E kF, there exists, for a given E > 0, some t,h0 E koF such that I - IF < E. Set s = s(+~),and observe that 11 Z - 2P (1 = 1. Then for any u', u" E [0, 11 we have
+ +,,
+
II Bv,.o'w.P- Bv.o'w.PII d I 0" - 0' I I CiJ I I $0 N11 &J"w) - V w ) II II r ( - O " w > - T ( - o ' w ) II) 2 I $ - $0 I
+
d I 0'"-
0'
+
l(1 6 I + 2 I $0 I I 0.J I)
+2!$ -
$0
I,
so that
If I u'
- u" I is sufficiently small, the second member of this inequality becomes, say, < 36. Since E > 0 was arbitrary, the mapping u .+ Bv,oo,p: [0, 11 -+ :QF is indeed continuous. &
We now return to the assumptions of Theorem 74.K. We recall that Dim Xo(A), Codim Xo(A) are invariants for the classes iQF(actually they constitute, by definition, a complete system of invariants, subject to Dim X o ( A ) Codim Xo(A)= Dim X), and they do not depend on the space F. We should like to find invariants with the same property for our components of $Ii.e., ,,for the equivalence classes of M . This will be done in Theorem 74.M. In the real sequence space I", consider the linear manifold b of all those (5,) E I" such that (C: ti)E 1"; and let (6,) -+ (4,) b denote the canonical algebraic epimorphism from 1" onto the quotient space P l b .
+
+
74.M. THEOREM. Under the assumptions of Theorem 74.K, the mapping A -,(wA(n)- wA(n - 1)) : $2, ---t 1" canonically induces a bijective mapping of the partition :QF/winto equivalence classes (which are the components) onto the quotient space 1"lb; therefore the mapping A -+
( w A ( ~ ) - w A ( ~-
1))
+ b : ;L'F
+ l"/b
is surjective and is a complete system of invariants for the relation w , i.e., f o r connection in :Q,.
74. CONNECTION: HILBERTSPACE
(Jr-,
265
F is stronger than M, A E :sZ,
implies & A E M, so that If A , B E &,' , then A rn B if and only if w B - w A is bounded; and this is the case if and only if ( w R ( n )- w A ( n ) )E I": the "only if" part is trivial; the "if" part follows 1 from I w , ( 4 - W A ( t ) l 1 %([21) - WA"tl)l J,,, I&B(.) - &A(u)l du G
Proof. Since
( w A ( n )- wA(n -
I))
=
O A ( u )du) E I".
<
li(wB(n)
- wA(n))il%
+I
- &A
&fl
+
1,.
so that A m B if and only if ((wB(n) -
-
1 ))
- w A ( n -. 1))) E b.
-
T h e preceding argument shows that the canonically induced mapping $ , / m -+ Iv/6 is well defined and injective. It remains to show that it is
surjective. Let then (4,) E I" be given. Let v E F be stiff; if F E F V , there is no loss in supposing that ~ ( n = ) 0, n = 0, I , 2, ...; we may assume that it satisfies (20.2) with d = m , a positive integer; i.e., Itni
K
=
IER+
Set
Kj
=
Jjni I j- 1 )
p)(u) du
rp(u) du
inf
>, K , j
> 0.
I
=
I , 2, ... . Define
iii
(74.7)
<
This function is measurable with I 4 I FK-iml\(5,)llm, and continuous if F E 9 V ,so that 4 E F. We set w ( t ) = Jo +(u) du, a primitive with w ( 0 ) = 0, & = t / ~ E F.Therefore w = w B,, . w . P (74.J), and it only remains to prove that ( w ( n ) - w ( n - I ) ) 6= 6. But, for any positive integer k ,
+
and therefore, if ( k
-
I)m
< km,
+
266 so that ( w ( n ) - C: to be proved. &
Ch. 7. DEPENDENCE ON A
ti)E I",
whence ( w ( n ) - w(n - I ) -
t7JE b, as was
Remark I . T h e fact that l"/b does not depend on F shows that, if F is stronger than M,each component of :Q, produces in the smaller set :Q,, with its finer topology, exactly one component, provided SZ, is not altogether empty (i.e., provided F contains a stiff function, cf. 74.E). Remark 2. We could have replaced 1" by a smaller set, e.g., by the set of bounded sequences of integers, but this seems of no real interest: there is no hope of exhibiting a complete set of representatives of P l b ; only such a set would allow us to have (by some "canonical" choice of the interpolating w and the stiff y ) a canonical element B,,,w,Pin each component of $2,. Remark 3. I" has the cardinality of the continuum; the constant sequences are pairwise inequivalent modulo b; hence P j b has the cardinality of the continuum; this is also the cardinality of the set of components of $2,. Ordinary dichotomies We again consider any Hilbert space X and assume that F E 9or 9% is given. T h e proof of the following lemma is entirely similar to that of 74.D and will be omitted. 74.N. Assume that A EQ,, , and that the subspace Y , induces a dichotomyfor A. Assume further that Dim Y , < co,Dim Y , < Codim Y , [Codim Y , < co, Codim Y , < Dim Y,]. If X is real and Dim X = 2, assume in addition that Dim Y , # 1. Let Y be any subspace of X with Dim Y = Dim Y , [Dim Y = Codim Y,]. Then there exists C E Q,, connected to A by an arc in Q,, , such that C(t)YC Y a.e., and Y induces [is a complement of a subspace inducing] a dichotomy for C. 74.0. THEOREM. Assume that A E QoF , and that a subspace of either finite dimension or finite codimension induces a dichotomy for A . Then A , 0 are connected by an arc in QoF .
Proof. Let Y , be the subspace of finite dimension or codimension inducing the dichotomy for A, and assume for the moment that we do not have X real, Dim X = 2, Dim Y, = 1. If Dim Y , Codim Y, , whence Dim Y , < 00 [Codim Y , Dim Y, , whence Codim Y, < a]
<
<
74. CONNECTION: HILBERT SPACE
267
there exists, by 74.N, a subspace Y with Dim Y = Dim Y A[Dim Y = Codim Y A ]and C, connected to A by an arc in SZ,, , such that C(t)Y C Y a.e., whence W(t)Y= Y for all t E R , , and such that Y induces [is a complement of a subspace inducing] a dichotomy for C. Since every subspace of X both induces, and is a complement of a subspace inducing, a dichotomy for 0, and is invariant under I (the U corresponding to 0), Theorem 73.G shows that C is connected to 0 by an arc in SZ,,; hence A is also connected to 0 by an arc in SZoF . Assume now that X is a real euclidean plane, which we suppose oriented, and that Dim Y A= 1. We may assume that Y A= X,(A), since otherwise X,(A) = X induces a dichotomy and we are in the preceding case. We set B = -hAI'(&r) E F(X); then V = I'(uA). Obviously every subspace induces a dichotomy for B . In particular, so does X,(A), and V(t)X,(A)= r ( w A ( t ) ) X o ( A=) U(t)X,(A). By Theorem 73.G, A and B are connected by an arc in SZ,, . But X also induces a dichotomy for B ; by the first part of the proof, B is connected to 0 by an arc in SZ,,; hence so is A. ,$
Finite-dimensional space Theorems 74.G, 74.K, 74.M, 7 4 . 0 , and 74.E, 74. J may be summarized in the following statements when X is finite-dimensional.
74.P. 'rHEoREM. Let x be a jinite-dimensional euclidean space, Dim X = n, and assume that F E 9 or 9%' is stronger than M (in particular, F E F or F+ or .FU-). I f F contains no stiff function, Q, = 0. Otherwise,SZ, is not empty and open in F(X),and, unless X is real and n = 2, its components are exactly the sets 7'jdSZ,, d = 0, ..., n ; the function Dim X,( A ) constitutes a complete system of inaariants for connection; and the elements I1jdA,= v(Id@ (- In-d)),f o r any fixed stiff v E F and any representation X = Y 0 2 , Dim Y = d, form a complete set of representatives,for the components. I n the exceptional case of the real plane, the components of SZ, are 8, (which contains - T I ) , iSZ, (which contains T I ) , and the equivalence classes of = in $2, (each containing,for some orientation, B , , o , , pfor some appropiate w and any orthogonalprojection P, Dim PX = 1); and a complete system of invariants for connection is constituted by Dim X,(A) plus, if this is = I , ( w A ( n )- u A ( n - 1)) b for some orientation of X .
+
74.Q. THEOREM. Let X be a finite-dimensional euclidean space and Then SZOF is not empty and is arcwise connected. assume that F E 9or 9%.
268
Ch. 7. DEPENDENCE ON A
T o show the significance of the assumption in Theorem 74.P that F is stronger than M, we call attention to 74.L and offer a further contrasting result.
74.R. THEOREM. Let X be a jinite-dimensional euclidean space, and assume that F E 9or 9%contains T such that liml+m'J: I ~ ( u I )du = co. Then 52, is not empty and is arcwise connected.
<
Proof. There is no loss in assuming rp 3 0. For every F, 0 < E 1, there exists ~ ( 6 E) R+ such that J ; ' v ( u ) du 3 c-l for all t 2 T ( E ) . We 1 set u(t) = max{l, rnin{J(,-,, ~ ( udu, ) ':J ~ ( udu}} ) 3 1, so that u is continuous, and t 3 T ( E ) 1 implies u(t) 3 6 - l . We then define = u-"p T, so that E F. For every E , 0 < c 1, we set t+hE = max{+ - c ~ O}; , then +Xt) = 0 for t 3 T ( C ) 1, so that + E E k,F; but 0 - $c cT, SO that I $ - + E IF €1 T IF 9 and $ E kF. We claim that is stiff. For any to E R, , the continuous function ':J y(u) du attains its maximum in [to , to T( l)] at some point t, , say, and the value of that maximum is, say, m >, 1. For every t E [ t o ,to + T ( 1) 13, either t or t - 1 is in [to , to T ( I)], and hence u(t) m. We therefore have
+
+
<
<+
<
+'
+
<
+
<
+
+
+
/
t,,+dl)+l
4 ( u ) du
<
>, m-l
'u
and our claim is proved. By 20.D there exist p, M > 0 such that +(u) du 3 p(t - to) log M for all t >, to 3 0. Let E, 0 < E I , be given. e p ( t ) - + ( t ) >, 0 for t 3 T ( E ) 1; therefore J', (~cp(u)- t,b(u)) du 3 - J"f)+l $(u) du = -A(€), say, for o any t >, to 3 0. We conclude that
ji0
+
<
+
Since that each
11 Av,pll
E
,=
+
kF is stiff, it follows from Theorem 74.F, 74.L, and 74.E is arcwise connected and contains some Ay,.p; and
n;d.Q,
+. T o
prove our proposition, it is therefore sufficient 4 E $2, for every c, 0 < c 1, where n =
+2
to show that A v , p
<
Dim X . Let then P be a fixed orthogonal projection, and let be given.
E,
0
<E
< 1,
75. NOTESTO CHAPTER 7 Set B
=
269
A,,,, + 2 4 E F. For any t >, to 3 0, (74.8) yields, with (31.9)
By 42.B,(d), X induces an exponential dichotomy for B, i.e., B E Q : .,
$,
+
Remark 1. The proof that A,,,, 2 4 E D&QF does not depend on the finiteness of the dimension of X . On account of Theorem 74.F we thus have the following statement for any Hilbert space X , to be compared to Theorems 74.G and 7 4 . 0 : If F is as in Theorem 74.R, any A , B E Q, with one of Dim Xo(A),Codim X o ( A ) ,and one of Dim Xo(B),Codim Xo(B) finite are connected by an arc in QF , i.e., {&QF : c or d finite} is arcwise connected.
u
Remark 2. I t is clear that in Theorems 74.P, 74.Q, 74.R, except for the case of the real plane in the first of these, the euclidean norm may be replaced by any Banach (Minkowski) norm. I n the exceptional case, however, the components of $2, are not easily described by means of a general Banach norm.
75. Notes to Chapter 7 As already observed in the Introduction, the contents of this chapter are, at least in part, a sort of perturbation theory; in that context it is natural that partial results were found by the authors mentioned in the Notes to Chapters 4, 5, and 6. Specifically, Satz 9 of Perron [ 2 ] , when restricted to linear perturbations, amounts essentially to the assertion that Ad(C, L") n C ( x ) , Q n C ( 2 ) are open in C ( x ) ,for finite-dimensional X . Bellman [l] similarly studies the openness in C ( 2 ) of those parts of Ad(C, La), Ad(L1, L"), and Ad(L2, L") where X,(A) = X . Krein [ 11 shows that :QLm is open in L" (where n = Dim X ) and ;(A) is semicontinuous there; KuEer [l] generalizes this to any Lp, p > 1, and also studies QoLl (which of course actually coincides with L1(x), by
270
Ch. 7. DEPENDENCE ON A
Theorem 72.J). As we see, all these authors, with the exception of Perron, restrict their attention to the case where &(A) = X. With reference to the concept of a deformation family, it should be noted that the relationship between A and each C, defined by (73.1) is one of “tsimilarity” (Conti [I]) or “kinematic similarity’’ (Markus [I]), except for the . conditions on
e,,
CHAPTER 8
Equations on R 80. Introduction In Chapters 4-7 the main objects of research were the equations (30.1) and (30.2); i.e.,
+ AX = 0,
(80.1)
f
(80.2)
f+Ax=f,
where the range of the independent variable t was J = R, , although Sections 51 and 71 contain some results valid for an arbitrary interval J. It is the purpose of this chapter to reexamine the problems studied in the preceding ones, but for J = R. T h e method for this investigation will consist in the main in cutting the domain at 0, applying the previous theory to the equations restricted to R- , R, , and finally splicing them together at 0; of course the results for arbitrary J in Chapters 5 and 7 will also be useful. T h e application of the contents of the previous chapters to the interval R- requires numerous merely verbal changes, generically subsumed under the phrase “change of variable t -+ - t ” ; we shall refer to these changes by the label “symmetrization” (already used in this sense in Section 23, p. 74) and refrain from giving the details, obvious in each case. T h e results obtained in this chapter show that, where the analogue of a theorem for equations on R , exists, it is in general a straightforward extension via the cut-and-splice method, and is less interesting than its original. We therefore do not insist on exhaustive statements, but rather on an exploration of the scope of such extensions and the proof of the most conspicuous results. This shift in emphasis explains the fact that the structure of this chapter does not reflect the structure of the preceding ones: e.g., results corresponding to parts of Chapter 5 come before the definition of an analogue of a dichotomy; and all 27 1
272
ON R Ch. 8. EQUATIONS
questions concerning associate equations are collected in a single section. We obviously rely heavily on the relations between function spaces on R and, on R- , R, , described in the subsections on “Cutting and splicing’’ in Sections 22 (p. 54) and 23 (p. 74). In particular, we shall consistently use the notations introduced there, which we recall here. All statements, formulas, etc., are understood to apply to J = R as the domain of t, unless the contrary is either explicitly stated or is implied by the following agreement: a subscript - or refers to restrictions on R- or R, respectively; the exact nature of this restriction for several objects was explained in Section 22 (pp. 54-56), 23 (pp. 74-75); and (80.1)+ is Eq. (80.1) with t restricted to R, . A superscript - or is merely a convenient index. Concerning these indices, we recall the agreement that, whenever the signs 3,-f appear in either position, all upper signs (i.e., those in both subscripts and superscripts) are to be taken together throughout the formula or sentence, and likewise for the lower signs; and the formula or sentence applies to both cases or to either, as the context may require. The main tool in dealing with Eq. (80.2) on R, in Chapters 5 and 6 was the concept of a (B,D)-manifold, where ( B , D ) is an 9 - p a i r or 9%-pair. T h e corresponding concept for Eq. (80.2) on R is obtained by splitting it into (80.2)*, and consists in a pair of linear manifolds, each related to one of these equations, and connected between themselves; we shall call this object a (B, D)-dihedron, and it indeed turns out to be a dihedron as defined in Chapter I , and a gaping one at that (Section 1 1, pp. 10-1 1 ;Section 14, p. 28).To(B, D)-subspaces for J = R, correspond closed (B, D)-dihedra for J = R. Section 81 discusses this concept and relates it to the admissibility of 9 - p a i r s and 9%-pairs, thus constituting an analogue to Section 52; the results are the only fundamental ones in this chapter that are not directly deducible from the previous work. In Section 82 we describe the behavior of the solutions of (80.1) corresponding to dichotomies and exponential dichotomies. I t turns out that the correct analogues, called double [exponential] dichotomies, consist precisely of a splicing together at 0 of [exponential] dichotomies for (80.1)+; and that a more intrinsic definition does not appear possible, except in the important special case in which the subspaces inducing the two dichotomies are disjoint. Theorems relating such double dichotomies with the admissibility, etc., of .F-pairs or FV-pairs, corresponding to those in Sections 63, 64, are then proved; their validity is a confirmation of the correctness of the definition of double dichotomies. Section 82 closes with a brief account
+
+
8 1. ( B , D)-DIHEDRA AND
ADMISSIBILITY
273
of the results on “predichotomy” behavior of the solutions of (80.1), that correspond to the theorems in Section 62. Section 83 dealswith the relations between Eqs. (80.1) and (80.2), and I t will appear that the the associate equations in coupled spaces X,X’. only meaningful analogues of previous results correspond to the case, for J = R , , of a subspace with the quasi-strict coupling propertyindeed, a complemented one-so that we are spared all the complications of Sections 53 and 64 relative to more general situations. Section 84, corresponding in subject matter to Chapter 7, contains some results on the dependence on A of the properties of (80.1) and
(80.2). 81. (B, D)-dihedra and admissibility The fundamental theorems
We consider Eqs. (80.1) and (80.2) for a given A E L(x).If D E b S K , ) of the more accurate but cumbersome we write XiOD= X k O D ( Ainstead XhOD*(A+), for the set of values at t = 0 of the D*-solutions of (80.1)* . Similar shortened notations are introduced in this chapter, and will be clear without detailed explanation. On account of 22.Q,( l), (81.1)
x-nD
”
x+OD
T h e concept we now define belongs more properly to the subject matter of Section 33; we include it here because it seems pointless to introduce there all our conventions on signs for this isolated topic. A pair ( Y - , Y + ) of linear manifolds in X is a D-GAPING DIHEDRON ( F O R A , or FOR (80.1)) if Y* C XkOD and there exists a number K’ > 0 such that for every x E X there are solutions y * of (80. I ) with y*(O)E Y+, y+(O) - y - ( 0 ) = x (so that the term “dihedron” is justified), and l y : 1, K ‘ / I x 11. We denote the infimum of the admissible values of K ‘ by K~ or, in full, K ~ ( A Y -; , Y,).
<
8 I .A. A D-gaping dihedron is a gaping dihedron. A closed dihedron ( Y - , Y,) is a D-gaping dihedron ;f (and only ;f) Y +C X*oD. Proof. Let ( Y - , Y,) be a D-gaping dihedron, and let l7+ be the monomorphism of 33.A for A* , D * . If X E X and p > 1 are given, and y * are as in the preceding definition for K ‘ = P K ~ ,we have y * ( O ) E y*cx*OD,y+(o)-y-(O) IIy*(o)ll~1117,111y11D6PKC,11171.1111xlll so that ( Y - , Y.:) is gaping, with K ( Y - , Y,) K~ max-,+ 11 11. = x 9
<
ON R Ch. 8. EQUATIONS
214
Let (Y-, Y+) be a closed dihedron with Y +C XioD.By 1 1. J, it is gaping; set K = K(Y- , Y+). Let x E X and p > 1 be given: there exist solutions y * of (80.1) with yk(0) E Y , , y+(O) - y-(0) = x, 11 y*(O)l( p ~ l xl I(. By Theorem 33.B applied to (80.1)i , I y: ID.< S+yIIyi(O)(I< pKSiyII x 11. Therefore the dihedron is D-gaping, with K~ < K max-,, S h y . t
<
We introduce a notation similar to that given in Section 52. Assume that f E L(X) has compact support, and set s+ = si(f) (cf. Section 20, pp. 39-40). If x is any solution of (80.2), we denote by x i - , or, in full, xi,“ , that solution of (80.1)+ that satisfies xi,(s+) = x(s+); obviously, x+,(t) = x ( t ) for t 2 s, , xVm(t) = x ( t ) for t < s- . We observe that, for D E b%K, x E D(X) if and only if xim(0)E XiOD (by 22.I,22.Q,( 1)). T h e problem of the admissibility of .N-pairs in general was dealt with in Section 51, and we have nothing to add for the special case J = R. In this section, therefore, (B, D ) always denotes a given %-pair OY %%-pair; at the cost of a trivial case, we add the assumption that B # (0). A pair ( Y - , Y+) of linear manifolds in X is a (B, D)-DIHEDRON (FOR A, or FOR (80.2)) if Y* C XiOD aqd there exists a number k > 0 with the following property : for every f E koB(X) there exists a solution x of (80.2) such that xkm(0) E Y* (whence x E D(X)) and I x ID < kl f I n . T h e term “dihedron” will be justified by Theorem 81.B. T h e infimum of all possible values of k we denote, as usual, by K Y ,or, in full, KyB.D(A). Since B # {0}, K Y> 0. T h e word-for-word analogue of 52.A holds, with “(B,D)-manifold” replaced by “(B, D)-dihedron” and “x,(O) E Y” by “x*,(O) E Y i ” , and defines C y= C,,B.D(A). The fundamental relations between (B, D)-dihedra for A and (Bi,D+)-manifolds for A , are given by the following theorem.
A pair ( Y - , Y+) of linear manifolds in X is a 81.B. THEOREM. (B, D)-dihedron for A if and only i f it is a D-gaping dihedron for A and Y , is a (B+,Di)-manifold for A+ . In particular, a (B, D)-dihedron is a gaping dihedron. Proof. The condition is necessary: we need only prove that ( Y - , Y+) is a D-gaping dihedron, the remaining condition following at once from the assumption and 22.Q,( I), 24.F,( 1). By the assumption we also have y* xiOD * Since B # {0},there exists q~ E koB,y 2 0, such that J’zT p)(u) du = li where s* = si(cp). Set E = exp(J’,”+11 A(u)lI du). Let x be an arbitrary solution i f (80.l), fixed for the time being, and 1 set zo(t) = x ( t ) J o y ( u )du. By 31.B,(b), zo is a solution of (80.2) for kOB(X),l f l B < E l 1B11 x(o)ll. f = vx* By(31*7),I l f 11 <Edl x(o)ll;
8 I . (B, D)-DIHEDRA AND
275
ADMISSIBILITY
Let p > 1 be given. Since (Y- , Y+) is a (B, D)-dihedron, there exists a solution z of (80.2) with z i z ( 0 )E Y i and I z ID p K Y l f IB pEK,I cp IB11 x(0)ll. Then z - zuis a solution of (80.l), and z i a = zUfm ( z - zo) = xi F(u)du + ( z - zo); therefore Z + ~ ( O ) - ~ ~ ~= ( 0 ) x(0) J:'- y ( u ) du = x(0). Further, for any t E R* , zo-lm(t)- z,(t) = x(t) cp(u) du. Therefore, using (31.7) again,
<
Jg-
<
+
JF
Since x(0) was an arbitrary element of X , (Y- , Y,) is indeed a D-gaping dihedron (just let y* be the solutions of (80.1) with yz = zim), and K~ E(K,I F ,1 -t I x [ ~ - , ,ID); ~ + it ~ is a gaping dihedron by 81 .A.
<
The condition is sujicient: Let f E k,B(X) and p > 1 be given. By the second part of the condition there exist solutions xf of (80.2) such that x:,(O) Y +, I x: I D pK+y,If*IB PKiY1.f IB> I/ x'(0)II d PC*Ylf 1 6 . By the first part of the condition, there exist solutions yi of (80.1) with y*(O)E Y , y+(O) - y-(O) = x+(O) - x-(0), and
<
<
Now x* - y + are solutions of (80.2) that coincide at t = 0, hence are identical; we denote this solution by z. T h e n zhm(0)= xza(0) I I D f Ir- ID f y'(0) y t and I I D I z- I D f I z+ ID I ID 1 y l t I D < p ( K - ~ +K+,+p.,(C-,+ c + Y ) ) l f l B , s o t h a t ( Y - ,Y + ) i s a ( B ,D)-dihedron with K , K K + , K ~ ( C _ , C,,). &
<
<
,+
<
+ xz
+
+
A corollary for closed ( B ,D)-dihedra follows:
8 1 .C. A pair ( Y- , Y k) of subspaces in X is a closed (B, D)-dihedron if and only if it is a dihedron and Y* is a (B,,D+)-subspace. Proof. Theorem 81.B and 81.A; we observe that the second part of the condition implies Y , C X,,, . & We next establish a connection between (B, D)-dihedra and the admissibility of ( B ,D), corresponding to part of Theorem 52.F (for the other part, see 81.1); we require the general boundedness theorem, Theorem 51 .A.
276
Ch. 8. EQUATIONS ON R
81.D. THEOREM. If (B, D) is admissible for A , then (X-,, a ( B ,D)-dihedron, hence a D-gaping dihedron.
, X,,,)
is
Proof. Set K = KB,D(A)(Theorem 51.A). L e t f E k,B(X) and p > 1 be given; there exists a D-solution x of (80.2) such that I x lo pKlf le; then ~ ~ ~E X*,,. ( 0 )Therefore (X,, , X,,,) is a (B, D)-dihedron with K , K . T h e last statement follows from Theorem 81 .B. &
<
<
81.E. THEOREM. (B,D) is admissible for A i f and only i f (B, , D+)is admissible for A* and (X-,, , X,,,) is a dihedron.
Proof. The condition is necessary: T h e fact that (B, , D*) is admissible for A i follows from the assumption by 22.Q,( I ) and 24.F,( I), and the rest from Theorem 8 1.D. The condition is suficient: Let f E B(X) be given. By the first part of the condition, there exists solutions x* of (80.2) with x: E D & ( X ) ; by the second part, we may set xf(0) - x-(0) = y+(O) - y-(0), where y* are solutions of (80.1) with yf(0) E X*,, , hence y * E D*(X). But then z = x+ - y + = x- - y- is a solution of (80.2) with z* = : x y: E D+(X), whence z E D ( X ) (by 22.1, 22.Q,( I)). Therefore (B, D) is admissible. 9( We state a peculiar corollary of this theorem, and provide the proof of a result mentioned in Chapter 3.
8 I .F. If D E bSKand (XODI X,,,) is a dihedron, it is a D-gaping dihedron.
Proof. Let B be the subspace of L1(or of M)of those functions that vanish a.e. outside [0, 13; then B E b s , B # (0). Now B- = {0}, so that (B-, D - ) is trivially admissible for A _ ; for every f + E B,(X) we let x+ be the solution of (80.2), with x + ( l ) = 0, so that x, = 0, whence x, E D + ( X ) ; thus (B+,D+) is admissible for A,. The proposition now follows from Theorems 8 1 .E, 8 I .D. &
co
Proof of 33.1. In our present terminology, we are to prove that if L+(x) and Dl I DZ bF+K then (X+OD,(AI)! X+OD2(AZ)) is either a gaping dihedron or no dihedron at all. Now define A EL(T)by A , = A , , and A-(--t) = -A,(t), t E R,;further, there exists, by 22.Q,(3) D E b S Ksuch that D , = D , and D- is D , symmetrized. Then X-,,(A) = X+oDl(Al),X+,,(A) = X+,,2(Az). The conclusion follows from 81 .F, 81.A. 9( 7
?
We return t o the questions concerning admissibility. It would seem, from a glance at Chapters 5 and 6 on one hand, and Theorem 81 . D on
8 1. (B, D)-DIHEDRA AND
277
ADMISSIBILITY
the other, that we were unfortunate in our general definition of regular admissibility (Section 51), at least for the case J = R, and that the assumption that X,, be closed should profitably be replaced by the requirement that both XioDbe closed. Luckily, both conditions are equivalent in the presence of admissibility: 81.G. If there exists a (B, D)-dihedron (in particular, i j (B, D) is admissible), then X+,, are closed if and only if X,, is closed.
Proof. If ( Y - , Y,) is a (B, D)-dihedron, hence a gaping dihedron X+,,) , is also a gaping dihedron (by 11.1); (Theorem 81.B), ( X n D this is the case in particular if (B, D) is admissible (Theorem 81 .D and 8l.A). T h e conclusion follows from (81.1) and 1 I.J. & Some further results
T h e preceding subsection contains the tools that are required to reduce problems about Eqs. (80. I ) , (80.2) to the corresponding problems for (80.1), , (80.2)+. According t o our plan, we do not give results that are merely technical, but only state a few representative analogues of previous theorems in Chapter 5. We call attention to the allowed omission of parentheses in such expressions as lcF+ , kF* , etc. (Section 22, p. 55, Section 23, p. 75, Section 24, p. 78). We begin with the analogue of the important Theorem 52.C. 81.H. THEOREM.A pair ( Y - , Y,) of linear manifolds in X is a (B, D)-dihedron if and only if it is an (IcB, D)-dihedron. Proof. Theorem 81 .B (necessary condition), Theorem 52.C applied to (80.2)*, and Theorem 81.B (sufficient condition). &
Remark. T h e suggested proof of Schaffer [6], Theorem 4.7 is incorrect, since it makes use of Theorem 4.1, which is the “necessity” part of our Theorem 8 1 .B with “D-gaping” replaced by “gaping”, as if this weaker condition were also sufficient. We now complete the analogy with Theorem 52.F by means of a partial converse to Theorem 81 .D. 81.1.
If there exists a (B, D)-dihedron, then (kB, D ) is admissible.
Proof. (XPnD, X+,,)
is a (B, D)-dihedron; by Theorem 81.B,
Xi,,is a (Bi , D+)-manifold for A* . By Theorem 52.F, (kB,, D*) is admissible for A , . Theorem 81.E implies the conclusion.
&
278
Ch. 8. EQUATIONS ON R
We omit the analogues of 52.G, 52.H, which follow from Theorem 81.D and 81.1 as their protoypes do from Theorem 52.F, and give the pending proofs for J = R of two theorems on admissibility.
Proof of Theorem 51 .Hfor J = R. Theorem 81 .E (necessary condition), Theorem 51.H for J = R* applied to (80.2), , and Theorem 81.E (sufficient condition). Alternatively, Theorems 81 .D, 81 .H, and 81 .I (i.e., the same proof as for J = R+). ,$, 00
a3aJ
Proof of Theorem 5 I .I for J = R. Theorem 8 I .E (nceessary condition), Theorem 5 I .I for J = R* applied to (80.2)*, and Theorem 8 1 .E (sufficient condition); the fact that (POD, XToD)is a dihedron implies that (XZOlcD, X:o,cD)is one. & Among the results that especially refer to Y-pairs and Y q - p a i r s we select the analogues of Theorem 52.L and of 52.M. 81. J. Assume that D E bYK is given. A pair ( Y - , Y + ) of linear D)-dihedron for some .F-pair or .7%-pair (B, D ) manifolds in X is a (B, if and only if it is a (T,D)-dihedron.
Proof. T h e “if” part is trivial. For the “only if” part we observe that if B Eb.F% then IcB E b 9 (24.1, 24.L,(2)); on account of Theorem 81.H we may therefore assume without loss that B E b y . T h e conclusion then follows by Theorem 81.B (necessary condition), Theorem 52.L applied to (80.2)+, and Theorem 81.B (sufficient condition), since (B* , D*) is a .%-pair. ,$,
81 .K. Assume that D E b9YK is given. Some .F-pair or T V - p a i r is D ) is [regularly] admissible. [regularly] admissible if and only if (T, Proof. T h e “if” part is trivial. For the “only if” part we may suppose, as in the preceding proof (but using Theorem 81.D and 81.1 in addition to Theorem 8 1.H) that B E b y . T h e conclusion follows by Theorem 81.E (necessary condition), 52.M applied to (80.2)+, and Theorem 81 .E (sufficient condition). & We finally’state the analogue of Theorem 62.1, which also extends the validity of Theorem 51.G to cases with nonreflexive X .
D) is not weaker than 81 .L. Assume that the .F-pair or FV-pair (B, (L1, L;) and that there exists a closed (B,D)-dihedron (in particular, that (B,D ) is regularly admissible). Then (lcB, fD) is regularly admissible.
279
82. DOUBLE DICHOTOMIES
Proof. On account of Theorem 81.H we may assume that (B,D)is a F-pair. The conclusion then follows from 81.C, Theorem 62.1 applied to (80.2)+,and Theorem 8 1 .E (since of course X-,,, XofD3 YY, = 4. &
+
+
82. Double dichotomies. Connections with admissibility and (B, D)-dihedra Double dichotomies We continue the study of Eqs. (80. I ) and (80.2) for a given A E L(X). We are faced with the task of defining forms of behavior of the solutions of (80.1) that shall be suitable analogues of the dichotomies and exponential dichotomies defined for equations on R , in Chapter 4. As we shall see, an appropriate definition runs as follows:
A closed dihedron ( Y - , Y,) INDUCES A DOUBLE DICHOTOMY (OF THE (80.1), or FOR A ) if Y* induces a dichotomy for A,; it INDUCES A DOUBLE EXPONENTIAL DICHOTOMY (OF THE SOLUTIONS OF (8o.l), or F O R A ) if Y , induces an exponential dichotomy for A , , and A is then said to POSSESS A DOUBLE EXPONENTIAL DICHOTOMY. It would be desirable to give an equivalent formulation of these definitions that would not make it so strongly dependent on the separate consideration of (80.1)- and (80.I ) + ; in particular, a formulation in terms of solutions of (80.1) that would resemble conditions (Di), (Dii), (Diii), etc., or (Ei), (Eii), etc., as the case might be. Unfortunately, this appears to be impossible beyond the trivial replacement of the phrase “Y* induces a dichotomy for A,” by its definition. We believe that the following discussion of necessary conditions and sufficient conditions will explain this state of affairs. In the case of a disjoint dihedron, however, we are able to find necessary and sufficient conditions of the desired form. We first give some necessary conditions for double dichotomies and double exponential dichotomies. SOLUTIONS OF
82.A. Assume that the closed dihedron ( Y - , Y+) induces a double dichotomy for A . Then there exists N > 0, and for every h > 1 there exist N‘ = “(A) > 0, yo = yo(h) > 0, such that any solutions y , Z* of (80.1) withy(0) E Y - n Y , , z*(O) E YT , II ~*(O)ll Ad( Y- n Y , , z*(O)), satisfy:
<
(i) (ii-)
< <
Ily(t)lI < N//y(to)lI for all t t, 0 and all t 11 z-(t)l( N’ll z-(t,)ll for all t to;
<
2 to 2 0;
Ch. 8. EQUATIONS ON R
280
<
<
(ii+) 11 z+(t)ll "(1 z+(to)ll for all t to; (iii-) y [ y ( t ) z+(t), z-(t)] >, yo for all t 0, z- # 0; @if) y [ y ( t ) z-(t), z+(t)l 2 yo for all t >, 0, zf # 0.
+
<
+
if y
+ z+,
if
+ z-,
y
<
Proof. By 11 .L, (1 z.+(O)(( AK d( Y* , z*(O)), where K = K ( Y - , Y + ) is the gape of the dihedron. T h e conclusion follows from the definition, with
N
= max
-,+
Nf ,
"(A)
-.+ N+N;(AK),
= rnax
where N * , etc., are the parameters of the dichotomy induced by Y + for A * . &
82.B. Assume that the closed dihedron ( Y - , Y+)induces a double exponential dichotomy for A. Then there exist v, v', N > 0, und for every h > 1 there exist N' = "(A) > 0, yo = y,(h) > 0, such that any solutions y , zf of(80.1) with y ( 0 ) ~ Y - Y+,z*(O)~Y,,IIx+(O)ll n < X d ( Y - n Y+,z'(O)), satisfy : t
(i) 11 y(t)ll 3 to 3 0 ;
< Ne-yIL-follly(to)ll
for all
< <
(ii-) (1 z-(t)(\ Nle-v'lr-lol I1 z-(qll (ii+) 11 z+(t)ll N'e-Y'Il-iOI II z+(t0)ll (iii*) as in 82.A.
Proof. Same as for 82.A; we find v and N , "(A),
yo(A) as in the proof of
for all
t
< to < 0 t
and all
2 to;
f o r all t d to; =
min_,+v* , v' &
=
min-.+{v, , vi},
82.A.
Remark. Conditions (iiif) yield, by specialization, the following necessary conditions for y , zfas in 82.A, 82.B: (iiiJ y [ y ( t ) ,z-(t)] 2 yo (iii:) y [ y ( t ) ,z+(t)] >, yo (iiiy) y [ z - ( t ) , z+(t)]2 yo
<
for all t 0, if y , z- # 0 ; for all t 3 0, if y , z+ # 0 ; for all t , if z-, z+ # 0.
For one-dimensional X the necessary conditions for a double dichotomy, ordinary or exponential, given in 82.A, 82.B are trivially sufficient. We show in Example 82.G that for two-dimensional X even the conditions of 82.B are not sufficient for an ordinary double dichotomy. It might be argued that we should have included additional necessary conditions on solutions w with w ( 0 ) .$ Y _ n Y,; but in the example
82.
DOUBLE DICHOTOMIES
28 I
any such conditions would be vacuous, since Y , = X. We must point out, however, that the example has A 4 M(R); it seems very unlikely that this really matters (cf. Schaffer [6], Remark 2 to Example 6.1), but still we record the question:
Query. Are the necessary conditions of 82.A [82.B] sufficient for a double [exponential] dichotomy if A E M(X) ? We now turn to the question of sufficient conditions for double dichotomies and double exponential dichotomies. .
82.C. Assume that ( Y - , Y+) is a closed dihedron. Consider the following statements: (a) [(b)] There exists N > 0, and for every [some] A > 1 there exist N' = " ( A ) > 0, yo = y,(A) > 0, such that any solutions y , zk of (80.1) with y(0) E Y- n Y,, z*(O) E Y+,11 z*(O)ll < A d ( Y - n Y + ,z*(O)), satisfy: (i), (ii*) as in 82.A; (+) y [ y ( t ) zY4, z*(t>I3 Yo
+
for all t , if y
+ z+, z i # 0.
(c) [(d)] There exists N > 0, and for every [some]A > 1 and any [some] pair q* of ( Y - n Y , , A)-splittings of Y+ there exist N' = " ( q - , qf) > 0, yo = yo(q-, q+) > 0, such that any solutionsy, z* of (80.1) w i t h y ( 0 )E Y - n Y+, z*(O) E Y , , q*(z*(O)) = z*(O), satisfy (i), (ii*), (iv'). (e) ( Y - , Y,) induces a double dichotomy. Then the following implications hold: (c) ++ (a) t)(b) + (d) + (e).
Remark. Before proceeding to the proof proper, we observe that (iv-), (iv+), taken together, are equivalent to (iii-), (iii+) of 82.A, 82.B, together with (v-1 d " ( t ) , z+(t>l3 yo (v') y [ y ( t ) ,d t ) l 3 Yo
<
for all t 0, if y, z+ # 0, for all t 3 0, if y , z- # 0,
perhaps with a different value of y o . More precisely, for t 3 0, (iv+) and (iiif) coincide; (iv-) implies (v+) with the same yo; and by l l . B (the Three-Angles Lemma), (iii+) and (v+) together imply (iv-) with t y ; instead of yo . For t 0, the symmetrical situation obtains.
<
Proof. T h e implications (a)
'
(b) (d) are trivial. (c) f (b) implies (a). Assume that (b) holds for some A,, > 1, and assume A > 1 given. Set S = S ( A ) = &(Ao - I)(+(A, I) A)-l > 0. Lety, z* be
+ +
Ch. 8. EQUATIONS ON R
282
solutions of (80.1) as specified in (a) for this A. Condition (i) is the same for (a) and (b) and therefore requires no proof. There exist solutions x* of (80.1) with x*(O) E Y - n Y+ and 11 z*(O) - x*(O)ll $(Ao 1) d( Y - n Y , , z*(O)); the solutions u* = z* - xf thus satisfy u'(0) E Y , , /I u'(0) !I ;(Ao I ) d( Y- n Y , , z*(O)) A, d( Y- n Y+, ui(0)). Also,
<
I1 X*(O) ;I
+
<
+
< 1I u*(Oj II + II Z * ( O ) It ,< (+(A, + 1) t
<
A) d(Y- rl Y ,
+
1
Z*(O)).
<
Define the solutions w* = u* Sx*. Then v*(O) E YT , 11 w*(O)ll (+(A, 1) +(A, - I)) d(Y- n Y+, z*(o)) = A, d ( Y - n Y+, v*(o)). Thus u*, w* satisfy the assumptions of (b) (for A,) instead of z*. In proving (ii+), (iv+) we may assume z+ # 0, whence u+, w+ # 0. For the moment we also assume x+ # 0. Then condition (iv+) of (b) as applied to Sx+, 0, w+ (instead of y , z-, z+) gives y[Sx+(t), v+(t)]2 yo(ho); by 1 I.A, Syo(Ao)llxf 11 211 w+ - Sx+ 1 1 = 211 u+ 11, whence
+ +
<
(82.1)
I1 Z + II -<, II u+ I!
+ !I
X+
II < (1
+ 28-ly;yAJ)
II u+ !I.
Applying now condition (iv+) of (b) to -x+, 0, uf we have y [ - x + ( t ) , u + ( t ) ] 3 yo(h,) ; using I I .A again, (82.2)
ll ut I1 < 2y,'(4)) II u+
+ x+ 11 = 2yi1(Ao)II z+ Il.
We may now drop the assumption that x+ # 0, since (82.1), (82.2) are trivially satisfied if x+ = 0, u+ = z+, because yo(A,) 2. Condition (ii+) of (a) now follows immediately from (82.l), (82.2), and condition (ii+) of (b) applied to u+; and for "(A) we find
<
2y,'(A,)(
1
+ 2W5'(A"))"(A0). +
T o prove (ivf), assume that y z- # 0; for a given to we apply (iv-1) of (b) to Ilr(t0) %-(to) II-Yr x-) - II z+(to)II-lx+, II Y(t0) z-(to) Il-lu-9 I/ &(to) 11-lu+ (instead of y , z-, z') at t = to;we use l I . A and (82.1) to find
+
+
+
Thus (iv+) is established for (a), with the last member of the preceding inequality instead of ro(A). Conditions (ii-), (iv-) follow by symmetrization.
82. DOUBLEDICHOTOMIES
283
(d) implies (e). We show that, (d) being assumed, Y+satisfies for A, the condition (d) of Theorem 41.A; the corresponding result for Yfollows by symmetrization, and the conclusion will hold. We use conditions (iiii), (+) instead of (ivi). By 1 1.M, qf induces a (Y, , AK)-splitting q of X , where K = K ( Y- , Y+); q will be the splitting in 41.A,(d). Let then u, e, be solutions of (80.1) with u(0) E Y+, q(v(0)) = v(0); this last assumption means v(0) E Y- , qf(v(0)) = v(0). Define the solutions y , w of (80.1) by w(0) = q-(u(O)), y = u - w . Then y, z- = fw, z+ = v satisfy the assumptions of (d). By (i), (ii+), (v+) applied to y , z- = - w we have, for any t 2 to 2 0, with the help of I 1 .A,
<
(even if y = 0 or w = 0, since yo(q-, q+) 2); thus (Di) holds with 2yo1(q-, q+)(N N'(q-, q f ) ) instead of N , . Condition (Dii) for v (or rather, v + ) is an immediate consequence of (ii+) for zf = v , with N'(q-, q f ) instead of N;(q); and condition (Diii) for u, v # 0 is an immediate consequence of (iii+) with yo(q-, q f ) instead of yo+(q).
+
(c) implies (a). Since (c) implies (d), and (d) implies (e), 82.A shows that all the conditions of (a) are satisfied, except possibily (v*); it therefore remains to prove this condition. Let A > 1 be given, and let q: be ( Y - n Y , , A)-splittings of Y , . Assume that (v+) of (a) were false for this A: there exist then sequences (yn), (2;) of nontrivial solutions of (80.1) such that yn(0)E Y - n Y+, z;(O) E Y , , II zi(0)II A d( Y- n Y , , z;(O)), and such that
<
inf y[y,(t), z;(t)] +0
l€R+
as
n + co.
There is no loss in assuming that d( Y- n Y , , zG(0)) = n, say. We now define a new ( Y - n Y, , A)-splitting q+ of Y- as follows: if x E Y - and x - z;(O) E Y- n Y , for some n, we set q+(x) = z;(O); for a given x, this can happen for at most one n,for it implies d( Y - n Y,.,x) = n ; otherwise we set q+(x) = q;(O). I t follows at once that q+ is indeed a ( Y - n Y , , A)-splitting of Y-; and we apply (c) to this qf and q- = 40. Since q+(z;(O)) = z;(O), condition (vf) of (c) implies y b n ( t ) , z;(t)] yo(q-, q+) for all t 2 0 and all n, contradicting the assumption on (yn), (z;). Thus (v+) of (a) holds, and the proof of (v-) follows by symmetrization. &
284
ON R Ch. 8. EQUATIONS
82.D. Assume that ( Y - , Y , ) is a closed dihedron. Consider the following statements: (a) [(b)] There exist v, v’, N > 0, and for every [some] h > 1 there exist N ’ = “(A) > 0, yo = y,(h) > 0, such that any solutions y , zf of (80.1) with y(0) E Y - n Y+, z*(O)E Y , , I/ zi(0)ll h d( Y- n Y , , z*(O)), satisfy:
<
(i), (ii*) as in 82.B; (iv*) as in 82.C. (c) [(d)] There exist Y , v’, N > 0, and for every [some] h > 1 and any [some] pair q* of ( Y - n Y+, h)-splittings of Y , there exist N’ = N‘(q-, q+) > 0, yo = yo(q-, qf) > 0, such that any solutions y , z* of (80.1) with y(0)E Y- n Y+, z*(O) E Y,, q(z*(O)) = z*(O) satisfy (i), (ii*), (iv*) as in (a). (e) ( Y - , Y + )induces a double exponential dichotomy.
Then the following implications hold: (c) tt (a) tt (b) + (d) + (e).
Proof. Same as for 82.C, using 82.B instead of 82.A and Theorem 42.A instead of Theorem 41.A. & Remark. T h e proof of the implication (c) -+ (a) was achieved by using “brute force”, e.g., by using splittings that are not continuous and do not satisfy q(ax) = a q ( x ) ; if in condition (c) the parameters N ’ , yo are assumed to depend on A, but not otherwise on q*, an immediate proof without these defects is obtained using I1.G. For one-dimensional X the sufficient conditions for double dichotomies given in 82.C, 82.D are trivially necessary. We show in Examples 82.H, 82.1, 82. J that for two-dimensional X the implications (d) 4(b), (e) + (d) are false in general, even if the former condition is “exponential” and the latter “ordinary”, and even if A E L“(x) (indeed, A may even be assumed continuous). If the dihedron ( Y - , Y+)is disjoint, however, the situation becomes much simpler: the condition (i) in 82.A, 82.B, 82.C, 82.D becomes trivial; conditions (iii*), as well as conditions (iv*), reduce to condition (iiiy) of the Remark to 82.A, 82.B; and the assumptions on z* become simply z*(O) E Y ? . Thus the set of conditions (iii), “ordinary” or “exponential”, (iiiy), is necessary and sufficient for ( Y - , Y+)to induce a double dichotomy, ordinary or exponential. This fact is recorded as part of the following two theorems.
82. DOUBLEDICHOTOMIES
285
82.E. THEOREM.Assume that ( Y - , Y , ) is a disjoint closed dihedron. The following statements are equivalent:
( Y - , Y + )induces a double dichotomy. (b) There exist N‘, yo > 0 such that any solutions z* of (80. I ) with z*(O) E Y , satisfy: (a)
/I z-(t)ll < N’ll z-(to)ll (DDi+) /I z+(t)il< N’II z+(to)ll
(DDi-)
(DDii)
y[z-(t),
z+(t)l 3 yo
for all
t 3 to;
for all
t
for all
t , if z-, z+ # 0.
< to;
(c) There exists D > 0 such that, if U is the solution of (30.3) with U ( 0 ) = I and P & is the projection along Y , onto Y , , we have
11 U(t)P_U-’(t,)ll (DD’+) I/ U(t)P+U-’(t,)ll (DD‘-)
for all t >, to; for all t
Proof. Instead of using the preceding necessary or sufficient conditions, we apply 41.B to the Eqs. (80.1)+ and the subspaces Y* with their complements .Z* = Y , and the corresponding projections P* . (a) implies (b). We use condition (a) of 41.B and obtain (DDi*), (DDii) with N ’ = max-,+ N,N;( Y T ) ,yo = min-,+ yo*( Y,). (a) implies (c). Weuse condition (c) of 41.B and obtain (DD’*) with D = D-(P-)D,(P+). ( b ) implies (a). (b) implies condition 41.B,(b) for (80.1)1, with N* = N;(Y,) = N’, Yot(Y,) = yo (c) implies (a). (c) implies condition 41.B,(d) for (80.1), , with D*(P*) = D. 9t
82.F. THEOREM.Assume that ( Y - , Y,) is a disjoint closed dihedron. The following statements are equivalent: ( Y - , Y,) induces a double exponential dichotomy. (b) There exist v’, N’, yo > 0 such that any solutions z* of (80.1) with z*(O) E Y , satisfy: (a)
(EEi-) I( z-(t)lj < N’e-v’l~-‘~i’ II z-(to)ll (EEif) 11 z+(t)ll< N’e-V‘I‘-rfi’II z+(t0)ll
for all t for all t
2 to;
< to;
(DDii) as in Theorem 82.E. This condition is redundant
if A E M(X).
C h . 8. EQUATIONS ON
286
R
(c) There exist v’, D > 0 such that, if U is the solution of (30.3) with U(0) = I and P5 is the projection along Y* onto Y , , we have (EE’-) (EE‘+)
(1 U(t)P-U-l(to)\1 < De-v’lf-tul for all t 3 to; 11 U(t)P+U-’(t,)ll < De-”lf---ln’ for all t < t o .
Proof. S a m e as for T h e o r e m 82.E, using 42.B a n d 42.D instead of 41.B. 4, Remark. If X is a Hilbert space, w e may replace (DDii) in T h e o rems 82.E, 82.F by t h e condition that there exist w o , 0 < wo in, such that:
<
(DDii,)
+(x-(t), z+(t))3 wo for all t , if z-, z+ # 0.
Examples In all the examples given here, X is the real o r complex euclidean plane with Cartesian coordinates x1 , x2 . T h e first example illustrates the discussion of the necessary conditions for double dichotomies; the others concern the sufficient conditions.
82.G. EXAMPLE.Let (80.1) be the system f,
x2
-
2x,
=0
+ 2x2 = 0
f,
+ 2x,
$2
+ 2x2
for t GO;
-
2tx2 = 0
=o
for t 2 0.
A is discontinuous at t = 0 in this example; a minor modification would remove this discontinuity without altering the conclusions. T h e general solution of the system is x = (x, , x2), where
If we wish the conditions of at least 82.A to hold, the behavior of the solutions at fa forces us to choose Y- = {x : x2 = 0) (the 3,-axis), Y , = X . We claim that then the conditions (i), (ii*), (iii*) of even 82.B are satisfied. Since Y- C Y , , (ii+) is trivial and (iii+) is vacuous, and (iii-) reduces to (iii;). Let y , z- be as in the assumption of 82.B for some X > I, which here meansy,(O) = 0, whencey, = 0, and l~r(O)1~ < (A2 - l)lz;(O)12. There is no loss in assuming z;(O) = 1, zT(0) = u, where I I u la X2. Now IIy(t)ll elt’ = l/y(O)lle-ltt for all t , so that (i) holds with v = N = 1. For t < 0, we have
+
<
82. DOUBLE DICHOTOMIES
287
A direct computation shows that, for t 2 0, ( 1 f I cr l y ( l
+
9 4 )
<1+ Iu +
t 2 12
< 2(1 +
10
1')(1
+ it4),
whence
so that (iii;) holds with yo = A-l. However, Y , = X does not induce a dichotomy for A + , let alone an exponential dichotomy (although Y- does for A-). Indeed, consider the sequence (dn)) of solutions of (80.1) defined by dn)(0) = ( - n 2 , 1); then
+ 1) 11
I1 x i n ) ( n
I1 *+(n) II
82.H.
-
(2(2n2
+ 2n + 1))1/2e-2n-2+ , x e-2n
as #-+a. &
EXAMPLE.Let (80.1) be the system with constant A , kl = 0,
2,
+ 2x,
= 0.
The general solution is x ( t ) = ( x l ( t ) , x 2 ( t ) ) = (x,(O), x2(0)e-2t). It is obvious that, if Y- is the x,-axis and Y , = X , the dihedron (Y-, Y+) induces a double dichotomy (e.g., by 41.C,(d)). Also, condition (d) of 82.C is satisfied if q+ is the orthogonal projection on the x,-axis, and q- is 0. However, for given h > I , let y , z- be the solutions given by y ( t ) = ( I , 0), ecZt),so that y(0) E Y- = Y- n Y,, 11 z-(O)11 = A = z-(t) = ((A2 tan Q(y(t), z-(t)) = e+(A2 - 1)-1'2 + 0, h d(Y- n Y,, ~ ( 0 ) ) However, . so that y[y(t),z - ( t ) ]-+ 0, as t -+ 03, and condition (b) of 82.C does not hold. tpI
Ch. 8. EQUATIONS ON R
288 82.1.
EXAMPLE.Let (80.1) be the system
x,
+ (tanh t ) x,
= 0,
f,
+ (2 + tanh t ) x,
= 0,
obtained from the preceding example by multiplying each solution by (cosh t ) - I ; the angular relationships are thus unaltered; it is easily verified that condition (d) of 82.D now holds, with q* as before, so that (Y-, Y,) induces a double exponential dichotomy; but of course condition (b) of 82.C still fails, as does in consequence condition (b) of 82.D. & Remark. An example of the falsehood of the implication ( d ) - + ( b ) in 82.D cannot have constant, periodic, or almost periodic A , since in these cases the dihedron inducing the double exponential dichotomy must be disjoint (Theorems 113.C, 102.A), so that the implication is trivially true. This is in contrast to the case of an ordinary double dichotomy illustrated by Example 82.H. 82.J.
EXAMPLE.Let (80.1) be the system
x, - 2x, x,
+ 2x,
=0 =0
for
t
< 0;
x, x,
+ 2x, - x, + 2x,
=0 =0
for t 2 0;
this is similar to Example 82.G, but has A E L"(R); the discontinuity at t = 0 is again removable without altering the conclusion. T h e general solution x = (xl, x,) is given by
We again take Y- to be the x,-axis and Y , = X . Y- obviously induces an exponential dichotomy for A- (e.g., by 41 .C,(d)). We show that Y , = X induces an exponential dichotomy for A+: let x be any nontrivial solution of (80.1); if x,(O) = 0 then x2 = 0 and )I x ( t ) ( )et = )I x(0)ll ect for all t 3 0; otherwise, set xl(0)/xz(O) = p ui (p, u real; u = 0 in the real case), so that 1) x(t)llet = 11 x,(O)II ect(1 uz ( p t ) , ) l I 2 for t 2 0. In both cases, 1) ~ ( t1) )et is decreasing for t >, 0, so that X induces an exponential dichotomy with V+ = N , = I . However, consider any solution z- of (80.1) with z;(O) # 0 (i.e., z-(0) E Y , \ (Y- n Y,)); set z;-(O)/z;(O) = p oi. If y is the solution y(t) = (e+, 0), we have tan +(y(t),z-(t))= ( 1 u2)1/2@ t ) - l + 0, so that y[y(t),z-(t)]+ 0, as t + co. Since z- was an arbitrary solution with 2-(0)E Y , \ (Y- n Y+),condition (d) of 82.D, and even of 82.C, fails for any splitting q+ (q- is trivial) because (v+) does not hold. &
+ + + + +
+
+
289
82. DOUBLEDICHOTOMIES Connections with admissibility a n d ( B , D ) - d i h e d r a
We are now in a position to test our definition of double dichotomies and double exponential dichotomies against the validity of direct and converse theorems analogous to those in Sections 63, 64. We give only the most important results, and try to reduce the proof in each case to the analogous result for (80.1)*, (80.2)1 . ( B ,D ) always denotes, until the end of this section, a given 9 - p a i r or .F%?-pair. 82.K. THEOREM. Assume that either A E M(R)or (B,D ) is stronger than (L' , L"); assume further that ( Y - , Y,) is a closed ( B ,D)-dihedron (in particular, ( B ,D) is regularly admissible and Y* = X+oD).Then ( Y - , Y + )induces a double dichotomy f o r A. Prooj. On account of Theorem 81.H and 24.1, 24.L,(2) we may restrict ourselves to .F-pairs. T h e result then follows by 81.C and either Theorem 64.A or Theorem 63.A, plus 23.Y,(l), and Remark 2 to 22.Q; the particular case of regular admissibility is included, by Theorem 81.D and 81.G. 9, A closed dihedron induces a double dichotomy for A 82.L. THEOREM.
if and only if it is a (closed) (Ll,L")-dihedron. If X , is closed, ( X - , , X,.,) induces a double dichotomy for A ;f and only if (Ll,L") is (regularly) admissible.
Proof. T h e first part by 81.C and Theorem 63.D; the second by Theorem 81.E and 63.D, using 81.G. 9,
82.M. THEOREM.If X,, is closed, (X-,,, X,,,) induces a double dichotomy if and only if (Ll, L;P)is (regularly) admissible. Proof. Theorem 81.E and 63.F, using 81.G.
9,
However, Theorem 63.E does not carry over without further assump tions (Example 82.0). What does hold is the following:
82.N. Assume that the closed dihedron ( Y - , Y + ) induces a double dichotomy for A and that X-,, X,,, = X ; assume further that the codimension of X,,, with respect to Y + is finite or, equivalently, that the codimension of X,, with respect to Y _ n Y , is finite. Then (X-,,, , X,,,) induces a double dichotomy for A and is an (Ll, L,")-dihedron, and (Ll,Lr ) is (regularly) admissible.
+
Ch. 8. EQUATIONS ON R
290
Proof. By Theorem 41.D, X*,, is a subspace contained in Y * . Since X-,, X,,, = X, we have, using the modularity of the lattice of linear manifolds,
+
whence (82.3)
+
Y - n Y+ = ( Y - n X+m) ( Y , n X-,,J.
Using the assumption again, (82.2) shows that the codimension of X+,, in Y +is finite if and only if the same is true of the codimension of X-,, n X+,, = X,, in Y k n Xroo;and (82.3) implies that this is the case (for upper and lower sign) if and only if X,, has finite codimension in Y - n Y, (indeed, the latter codimension is the sum of those of X*oo in Y*). This establishes the stated equivalence. T h e conclusion then follows from Theorems 41.F and 63.E, and 81.C and Theorem 81.E. 9, Whereas for the equation on J = R, the existence of a dichotomy :,.)implied, at least for finite-dimensional X, the existence of an (L: ., L subspace and the admissibility of (L: , Lg+), the analogous implication does not hold in general for J = R, the reason being that (X-,, , X,,,) may fail to be a dihedron:
82.0. EXAMPLE.For any X (even for X = R) and A = 0, every closed dihedron induces a double dichotomy; but X+,, = {0}, and indeed (Ll,Lg) is not admissible, as was already observed in Example 51.P. & So much for ordinary double dichotomies; we next turn to double exponential dichotomies. Assume that ( B ,D) is not weaker than (Ll,L,"),and 82.P. THEOREM. that ( Y - , Y + ) is a closed ( B ,D)-dihedron (in particular, ( B ,D ) is regularly admissible and Y + = X*oD).Assume further that there exists a closed dihedron (2- , 2,) that induces a double dichotomy for A. Then Y* = Zk, and ( Y - , Y + )induces a double exponential dichotomy for A . Prooj. As in Theorem 82.K, we may assume that ( B ,D ) is a F-pair. T h e conclusion follows by 8 I .C and Theorem 63.G. Assume that A E M(R),and that ( B , D ) is not 82.Q. THEOREM. weaker than (L1,Lorn). I f ( Y- , Y + )is a closed (B,D)-dihedron (inparticular,
82. DOUBLE DICHOTOMIES
29 1
if ( B ,D ) is regularly admissible and Y* = X*oD),then ( Y - , Y + )induces a double exponential dichotomy for A. Proof.
As for the preceding theorem, using Theorem 64.B. J,
82.R. THEOREM. Assume that the closed dihedron ( Y - , Y+)induces a double exponential dichotomy for A. If D is thick with respect to B , (B, D ) is regularly admissible; if D is thick with respect to kB, then X+on= Y + and ( Y - , Y,) is the unique (B, D)-dihedron. If A E M(X), these suflcient conditions are also necessary. Proof. By 23.Y,(4), 24.N, Theorems 63.K and 81.E, and 81.1, for the sufficiency of the conditions; by 23.Y,(4), 24.N, and Theorems 64.C, 81.D, 81.E for the necessity of the conditions when A E M(X). & T h e useful necessary and sufficient conditions for exponential dichotomies given in Theorem 63.M have their exact analogues in our present situation.
82.S. (1) A closed dihedron ( Y - , Y,) induces a double exponential dichotomy for A if and only if ( Y - , Y+)is both an (Ll,L“)-dihedron and an (L2,L2)-dihedron; if and only if ( Y - , Y,) is an ( M , L“)-dihedron; or an (Ll,T)-dihedron; or an (Mo, L,”)-dihedron. (2) If XI) XoT xOO is closed, ( X - O ! x + - O ) , (X-OT X+OT), (x-OO t X + O O ) induces a double exponential dichotomy for A if and only if ( M , L”), (Ll, T),(Mo , L,“), respectively, is admissible for A. 3
9
9
Proof. For part ( I ) : by 22.Q (Remark 2), 23.Y (Remark I ) , 81.C, and Theorem 63.M. For part (2): by 22.Q (Remark 2), 23.Y (Remark I), Theorem 81 .F, 81 .G, and again Theorem 63.M. 9, Predichotomy behavior of the solutions of the homogeneous equation Theorems 82.K, 82.P, 82.Q show that, under certain assumptions on (B,D) and other conditions, a closed (B, D)-dihedron induces a double dichotomy or a double exponential dichotomy. Under weaker assumptions, the results in Section 62 allow us to recognize a kind of “predichotomy” behavior of the solutions of (80.1). The statements of the corresponding results are lengthy and repetitious, and the proofs are not deep. We are therefore content with stating the theorems for the “exponential” case and sketching the proofs.
Ch. 8. EQUATIONS ON R
292
Assume that ( B ,D ) is not weaker than (L1, Lr) and 82.T. THEOREM. that (Y-, Y+) is a closed (B,D)-dihedron (in particular, that ( B , D ) is regularly admissible and Y* = X*oD).Then: (i) there exists p > 0 , and for eoery d > 0 there exists M = M ( d ) > 0, such that every solution y of (80.1) with y(0) E Y- n Y+ satisfies, for all t to 0 and all t 2 to 3 0,
< <
(ii) for every A > I there exist p' = p'(A) > 0,p" = p"(A) > 0 , and for every such A and A > 0 there exist M' = M'(d, A) > 0, M ' = M"(d,A) > 0, such that every solution z* of (80.1) with z*(O) E Y,, II zi(0)ll < h d( Y- n Y+, z*(O)) satisfies, for all to , t , Tft - to) 3 0,
and every solution w of (80.1) with 11 w(O)11 < A min-,+ d ( Y F ,w(0)) satisjies, to 0 and all t 3 to 3 0, for all t
< < t-A
tu-A
s"'"
All the conclusions remain oalid if we replace 11 y(u)ll du, / - A 11 y(u)ll du, etc*, b IX[lu-A.tn+AIYID I X[t-A.t+AlY ID Proof. By 81.C and 23.Y,(1) we can apply Theorem 62.B to (8O.l)*. The theorem follows, if we use I I .L to show that (1 z*(O)ll AK d ( Y i , zi(0)), where K = K(Y-,Y+),and (31.7) to show that, for any solution x of (80.1) and any t , 0 t d, J 11 x(u)II du < E II x(u)ll du I, where E = exp(SA11 A(#)]/ du), and sikiarly for the D-norms of slices. We find p = min-,+ p*, p'(A) = min-,+{p*, p&)}, p"(A) = min-,+ pi@), and M ( d ) = &E max-,+ M,t(d), M'(d, A) = e/'"A)AEmax-,+ M + ( A ) M ; ( ~ , A K ) , M"(d,A) = e14.'A)AEmax-,+ M ; ( d , A). & 3
9
<
< <
IJiA
82.U. THEOREM. Under the assumptions of Theorem 82.T, if D is stronger than L", then: (i) for every A > 0 there exists Lo = Lo(d) > 0 such that every soluiion y of (80.1) with y(0)E Y- n Y+ satisjies, for all t to 0 and all t 2 to 2 0, 11 y(t)ll < L(to)e-/llt-lulII y(to)ll,
< <
tn+A
where L(to) = Lo exp(J,,,-A11 A(#)/[ du) and p is as in Theorem 82.T;
293
83. ASSOCIATE EQUATIONS
(ii) for every A > I and every A > 0 there exists Li = Lh(d, A) > 0 such that every solution z* of (80. I ) with z*(O) E Y.F,Ij z*(O)ll ,< h d( Y- n Y,, z*(O)) satisfies, for all t o , t , F ( t - to) 2 0,
11 z+(t)/l< L’(to , A) e-/~’l/-‘Jllz+(to)ll, where L ’ ( t o ,A ) rem 82.T.
= LA exp(J:’$
/I A(u)lI du) and
p‘ = p’(A)
is as in Theo-
Proof. By 81.C and 23.Y,(I) we can apply Theorem 62.D and formula (62.14). (Remark 1 to Theorems 62.D, 62.E) to (80.l)*; using I1.L as in the preceding proof, the conclusion follows, with Lo(A) = max-,+ Lo+(d), LA(A, A ) = E max-,+ Lo+(A)LA,(A, AK). &
82.V. THEOREM. Under the assumptions of Theorem 82.T, if B is weaker than L1,then for every A > I and every A > 0 there exist LA = &(A, A) > 0, L,“ = L,”(A,A) > 0 , such that every solution z* of (80.1) with z*(O) E Y,, I( z*(O)lI ,< A d ( Y - n Y,, z*(O)) satisfies, for all t o ,t , &(t - to) 3 0 ,
/I zf(t)ll 2 (L’(to,A))-le/~”f-f~~lll z*(to)ll; and every solution w of (80.1) with 1) w(0)II for all t to 0 and all t 2 to 2 0 ,
< <
d(Y*, w(0)) satisfies,
I/ w(t)ll 3 (L”(to,A))-le/l”lt-ffl’ II w(t3111 1 +A
whereL’(to , A ) = 1,;exp(J,::-, I/ A(u)ll du),L”(to, A) =L6 exp(Jlt~~~llA(u)lldu), and p’ 5 p‘(A), p” = p“(A) are as in Theorem 82.T. Proof. By 81.C, 23.Y,(I), Theorem 62.E and formula (62.15), and 1I.L. We find &(A, A) = I; max-,, LAL(A, A K ) Lo,(A), L;(A, A) = max-,, LA*@, A). & Remark. From the statement of the “exponential” results in Theorems 82.T, 82.U, 82.V we obtain the statements of the corresponding “ordinary” results by dropping the assumption that (B, D) is not weaker than (L1, LT) and setting p = p’ = p” = 0 in the conclusions.
83. Associate equations
We consider a pair of coupled Banach spaces X,X’ and an A E L(x) that has an associate A’. I n addition to (80. I ) , (80.2) we now also consider
294
Ch. 8.
EQUATIONS ON R
the associate equations (32.1), (32.2) in X’, i.e., A’x’
== 0,
(83.1)
2’
(83.2)
2’ - A’x’ = f‘.
-
I t is our purpose to examine in what way the corresponding theory, developed in Sections 43, 53, 54, 66, has an analogue for equations on J = R. We recall that the most general analysis of the problem on J = R , was complicated by the fact that we had to deal with (B, D)manifolds that were not closed, let alone possessed the quasi-strict coupling property. I t comes as something of a surprise, pleasant or unpleasant according to the point of view, that these complications do not occur in our present situation. T o be more precise: to begin the analogy with Section 53, we should take a (B, D)-dihedron ( Y - , Y,) for A and examine the relations between the associate equations and (Y! , Y!). Now the least we ought to expect is that to every f’ E L(X’) with compact support there should correspond a solution x’ of (83.2) with xim(0) E Y!; but clearly this requires that (YO , Y:) be a dihedron: the proof is the algebraic part of the “necessity” proof of Theorem 81 .B. But this condition is extremely restrictive: by 12.B, the dihedron ( Y - , Y+) must be (X, X)-disjoint; thus Y+ are subspaces with the quasi-strict coupling property, and so are Y!; and all are saturated and complemented. Thus we are forced (and this is the unpleasant aspect) to restrict our attention to the case of (X, X’)-disjoint dihedra; but we are thus spared all the complications involved in the absence of these properties (and this is pleasant, especially in view of the major part of Section 66, for instance). I t may be argued that for finite-dimensional X these simplifications are irrelevant; but one that is certainly not is the fact that an (X, X’)-disjoint dihedron is disjoint. We have seen how important this may be in discussing the definition of double dichotomies in the preceding section. These remarks, which justify the consideration of (X, XI)-disjoint dihedra only, are, strictly speaking, merely heuristic. We complement them with the following result about an important special case.
83.A. Assume that (B,D), ( F , G ) are F-pairs, one of which is not weaker than (L1, Lo“). If there exists a (B,D)-dihedron ( Y - , Y,) for A and an (F,G)-dihedron (2’ ,2;) for -A’, then ( Y - , Y+) is (X, X ) disjoint, and 2; = Y! .
D) that is not weaker than (L1, L,”). Proof. Assume first that it is (B, By Theorems 81.B and 53.E, Y! = XLow; and by 53.H, Yf = X$, 3 X$, 3 2:. But then Yo Y: 3 2; 2; = X‘; by 12.B,
+
+
83. ASSOCIATE EQUATIONS
295
( Y - , Y+)is ( X , X)-disjoint, (Y! , Y t ) is an ( X , X)-disjoint dihedron, and since Z’+ C Y: we must have equality. If it is (F,G) that is not weaker than (L’, L,“), we interchange X , X‘ in the preceding argument, and find that ( Y - , Y + ) is equal to the ( X , X’)-disjoint dihedron (2’” , 2io);and that (2: , 2;) is ( X ’ , X)-disjoint, hence 2; saturated, so that
Y l = Z;““ = 2 ; .
,$
Remark. In view of Theorem 53.E, 83.A remains true if we replace the condition “not weaker than (Ll, L;)” by “not weaker than (Ll,Lm)”, provided A E M(X).
We now give a brief selection of the results obtained when we consider
( X , X‘)-disjoint dihedra. ( B ,D ) will denote a given YK-pair or Y - p a i r (so that the associate spaces may be defined).
83.B. THEOREM.Zf ( Y - , Y,) is an ( X , X’)-disjoint (B, D)-dihedron for A , then (Yo , Yy) is an (X‘, X)-disjoint (D’, B’)-dihedron for -A’. Proof.
81.C and Theorem 54.B, using 12.B. &
83.C. Let ( Y - , Y+)be an ( X , XI)-disjoint dihedron in X . If D is quasi locally closed, ( Y - , Y + )is a ( B ,D)-dihedron for A if and only if (Y! , Yy.) is a (an ( X ’ , X)-disjoirtt) (D’,B’)-dihedron for -A’. Proof. 81.C, 12.B, and 54.C.
&
D)-dihedron for A , then 83.D. Zf there exists a disjoint closed (B, (D’, B’) is admissible for -A* in X * . Proof. Let ( Y - , Y+)be the disjoint closed (B, D)-dihedron. By 12.D, Y! + Yy = X * ; by 81.C and Theorem 53.B, X5,. + XT,,,= X * . By 81.C and Theorems 54.E and 81.E the conclusion follows. Remurk. We cannot extend the analogy between 83.D and Theorem 54.E by replacing, in the assumption of 83.D, the existence of a disjoint closed ( B ,D)-dihedron by the regular admissibility of ( B ,D), unless we assume X , , = (0);indeed, even the analogue of Theorem 55.C for finite-dimensional X does not hold in general: the admissibility of ( B ,D ) for A need not imply the admissibility of (D’, B‘) for -At, as shown by the following example. This illustrates the insistence on disjoint dihedra that we pointed out in our introductory remarks.
83.E. EXAMPLE.Assume that X = R , and define A by A ( t ) = tanh t , so that A E C . T h e general solution of (80. I ) is x ( t ) = x(O)/cosh t ; since
Ch. 8. EQUATIONS ON R
296
<
< e-ILI cosh t I , the dihedron (R, R ) induces a double exponential dichotomy for A ; hence many F - p a i r s are admissible (Theorem 82.R). Now - A + ( t ) = -tanh t , and the general solution of the adjoint homogeneous equation is y ( t ) = y ( 0 ) cosh t , so that X*,,(-A+) = {O}; therefore these subspaces do not form a dihedron, and no F - p a i r at all is admissible for -At (Theorem 8 1 .E); this last fact may also be verified by direct computation. As usual, this example may be transformed into an example in any X by multiplying the scalar-valued A by I. & T o complete our selection, we give the important relation between double dichotomies for A and - A f , induced by ( X , X’)-disjoint and ( X , X)-disjoint dihedra.
83.F. THEOREM. A n ( X , X’)-disjoint dihedron ( Y - , Y,) in X induces a double dichotomy [a double exponential dichotomy] for A if and only if (P, Y!) induces a double dichotomy [a double exponential dichotomy]for -A‘.
Proof. 12.B and Theorem 43.A. It follows from Proof I of the latter theorem that, in the case of double exponential dichotomies, the value of the parameter v f in conditions (EEi*), (EE’*) of Theorem 82.F may be taken to be the same for A and for -A’. &
84. Dependence on A Admissibility classes and closed (B,D)-dihedra In this section we examine briefly, for Eqs. (80.1), (80.2) on J = R, the topics developed in Chapter 7 for equations on J = R, , namely, the question of the dependence of the properties of the equations and their solutions on the function A. We note, indeed, that in Section 71 admissibility classes such as Ad(B, D), Ad,(B, D), Ad,(B, D) were defined, for an arbitrary M-pair (B, D), where the range of t was any interval J , and that 71 .A and Theorems 71.B and 71.C (with Addendum) hold with the same generality. T h e same is true, for *-pairs and 3%-pairs, of the corollary 71.D of those propositions. That much, then, is already established for J = R, and need not be repeated here. Also, 71.G,(b) is a specific result for 7 - p a i r s and .F%-pairs on R. Let (B, D) be an 9 - p a i r or 3%-pair with B # {O}. T o the list of admissibility classes we add Ad,(B, D), the subclass of Ad(B, D) formed by those A in the latter class for which the dihedron (XOD , X,,,)
297
84. DEPENDENCE ON A
is disjoint; by (81.1) it is alternatively characterized by the condition X,,(A) = (0). T h e Addendum t o Theorem 71.C describes the properties of this class when D L"; in particular, Ad,(B, D ) n ( A , B(X)) is an open and closed subset of Ad,(B, D ) n ( A , B(X)) and a fortiori of Ad,.(B, D) n ( A , B(2)). As far as admissibility classes are concerned, all we intend to add is an illustration of the failure of the statement of Theqrem 71.E to hold if J = R instead of R , . This failure answers to the requirement, necessary for the admissibility of an 9 - p a i r or F % l p a i r (B, D) for A , that (XPoD, X+oD)be a dihedron (Theorem 81.D). T h e following example is very strong in this respect: indeed, X is two-dimensional, and the example shows that adding to A some B belonging to even k,F(T) for any F E 9 or 9%may destroy the admissibility of (B, D). Since such an addition does not alter the dimensions of XfOD, it is easy to see, using Theorem 81.E and 71.D, that the conclusion of Theorem 7 I .E does hold for ] = R provided X is one-dimensional.
<
+
+
+
84.A. EXAMPLE.Let X be a real or complex euclidean plane, with Cartesian coordinates x1 , x 2 . Let (80.1) be the system
x, -1-
=
0,
x*
-
x2 = 0,
with constant, symmetric- or Hermitian-valued A . Every solution is given by x ( t ) = ( x l ( t ) ,x,(t)) = (x,(O)e-l, x,(O)e!). If Y - is the x,-axis and Y , the x,-axis, it is obvious that the disjoint dihedron ( Y - , Y,) induces a double exponential dichotomy, so that every .F-pair and F%-pair (B, D) with D thick with respect to B is admissible (Theorem 82.R). Let now any space F E 9 or 9%, F # {0}, be given. There exists v E k,F, q 3 0, such that J: ~ ( udu) = Q x , where s+ = slt(cp). Consider the equation 51. ( A B)x = 0 given by the system
+
(84.1)
+
+ x1 + e - * t t ( t ) x 2
=
0,
k,
-
e"T(t)x,
-
x, =
0,
so that B is continuous if q is continuous, and
thus B E k,F(X). We claim that, if ( B , D ) is any 9 - p a i r or 9%-pair with B # (0) and such that D does not contain any function that increases exponentially, either as t -+ -a or as t + +CO (in particular, any
298
Ch. 8. EQUATIONS ON R
F - p a i r or .T%-pair), then (B, D ) is not admissible for A the solutions x = (xl, xz) of (84.1) are given by
+ B. Indeed,
(f du) r2(0)sin (1' du) 1 et x1(o) sin (fp(u) du) + x2(0) cos (1' q(u) 1. I
x , ( t ) = e-1
\.,(O) cos
y(u)
-
y(u)
0
x2(t) =
du)
The expressions in braces are constant for t therefore the assumption on D implies
x - , ~ ( -1A B ) J ( x , , x2) : x1 cos I
+
x + , ~ ( A B ) ci(x1, x2) : x1 sin
< s-
(j8-p(u) du) - r2sin n
and for t >, s,;
(r-
p(u) du) =
o1
0
(Jy
p(u, du)
+ x2 cos (j"+ p(u) du) = O (1 . 0
But, by the assumption on y , J"' y ( u ) du - J'y ( u ) du = in, so that 0 0 the one-dimensional subspaces on the right-hand side coincide; and then X-,,(A B ) X+,,(A B ) # X . By Theorem 81.D, (B, D) cannot then be admissible for A B. .&
+
+ +
+
If we continue to examine the results in Section 71 in order to see in how far they or their analogues hold for equations on R , we must consider closed (B, D)-dihedra, in place of (B, D)-subspaces. Due regard to the basic facts behind Example 84.A prevents us from seeking a full analogue t o Theorem 71.1: the steps corresponding to part 4 of the proof of that theorem cannot now be carried out. However, the remaining parts of the proof, which have to do with the case when I A - A , is small, do allow us to state the following theorem about closed (B, D)-dihedra.
IB
84.B. THEOREM. Let (B, D ) be an 9 - p a i r or 9 % - p a i r with B # (0) lean, and D locally closed and L". Assume that for A, E L(X) there exists a [disjoint] closed (B, D)-dihedron (Yo-, Yo+).Then there exists E > 0 such that i f A E A , B(8), I A - A , le < E , there exists a [disjoint] closed ( B ,D)-dihedron ( Y - , Y+)for A.
<
+
Proof. By 13. J there exists > 0 such that if Y - , Y+are subspaces in X with 6'(Y0*,Y i ) e l , then ( Y - , Y+) is a [disjoint] closed dihedron. By 81.C, Yo* is a (B* ,D+)-subspace for A,*. Now B* is lean, and D * is locally closed and LT (22.Q,(2), (3), and Remark 2; and 24.F,(3)). By the proof of Theorem 71.1 there exists E > 0 such B ( 3 ) and I A+ - A,+ E, then there exists a that, if A E A, (B* , D&subspace Y* for A+ such that 8'(Yo,, Yi) (we use
<
<
+
IB <
<
84. DEPENDENCE ON A
299
<
(71.15)). This will be the case, afortiori, if I A - A, IB e. But then ( Y - , Y,) is a [disjoint] closed dihedron; by 81.C, it is a [disjoint] closed (B, D)-dihedron for A. & Double-dichotomy classes
We denote by D or, in full Q ( X ) , the set of all A E L ( Z ) for which some closed dihedron, obviously unique and equal to ( X - , , X,,), induces a double exponential dichotomy. We could define the subset D, of all those A E D for which X , = X-, n X , , is complemented (whence X - , , X,, are complemented), but of greater interest is the smaller subset D,, , or D,(X), of those A E D for which the dihedron is disjoint, i.e., X , = {O}. I t follows from 82.S that D = Ad,(M, L"), Dd = Ad,(M, L"). More helpful is the fact, following from the very definition of a double exponential dichotomy, that A E D if and only if A&E Q* (the exponential dichotomy class corresponding to R , , described for J = R , as D in Section 72), and (&(A), X+,(A))E H ( X ) , the class of closed dihedra in X ; and that A €Dtl if and only if, in addition, ( X - , , X,,) E H,(X), the class of disjoint dihedra. Of the results corresponding to 72.A and Theorem 72.D,(a), (b), we record only the most important. In view of Example 84.A, there is no analogue of 72.B or Theorem 72.D,(c). 84.C. THEOREM. Let B E b 9 or b 9 % be stronger than M (in particular, B E b y or b F % ) . Then, ;f A , E L(X) is given, D n ( A , B(X)), D(,n ( A , B (2 ) ) are open sets in A , B(Z), and the mappings A (&,(A), -Y,,(A)): D n ( A , B(X)) 8 ( X , 6') x S ( X , &')--with range in H(X)-and A -+ X,(A) : SZ n ( A , B(X)) -+E(X, 6') are continuous.
-
+
+
+
+
--f
+
+
+
Proof. The mapping A --+ A* : A , B(X) + A,& B+(X) is continuous (22.Q,( I ) , 24.F,( I)). T h e theorem then follows from Theorem 72.D and 13.J. &
We now turn to the ordinary double dichotomies. We denote by Do the class of all A E L(rf) for which some closed dihedron induces a double dichotomy, and by D,,,, the subclass of those for which at least one such dihedron is disjoint. We do not attempt a classification corresponding to that in Section 72, but merely note that A E D, if and only if there exists a closed (Ll, L")-dihedron for A (Theorem 82.L), and A E sZ,,I if, in addition, one closed (Ll, L")-dihedron for A is disjoint. Again in view of Example 84.A, there is no analogue to Theorem 72. J, only the following weak result:
Ch. 8. EQUATIONS ON R
300
84.D. Let A, E L(x) be given. Then SZ, n ( A , $- L'(x)),SZod n ( A , + ~'(2)) are open sets in A, +
~'(x).
P ~ o o j . T h e o r e m 84.B. 9,
Connection in double-dichotomy classes We have no intention of reproducing, for equations on R , the general investigation of the connection of the dichotomy classes that was carried out in Section 73 and 74 for equations on R,. We only intend to use the results of that investigation in order to obtain the analogues of the most important theorems on the connection structure of the dichotomy classes Q, , Q, for a Hilbert space X . Our first few remarks, however, are valid for any Banach space X. We use, as far as possible, the conventions of Sections 73 and 74: e.g., U , V , W (with or without subscripts) are the solutions of 0 AU = 0, V + BV = 0, CW = 0, with U(0) = V(0)= W(0)= I . T h e concept of an F-deformation family for given F E 9 or ,F% is 'exactly as in Section 73, and 73.A, 73.B hold; so does 73.C, with subspaces replaced by closed dihedra, and dichotomies by double dichotomies; the proof follows immediately from 73.C itself. and that is that There is one difficulty concerning spaces F E 9W, given functions in F* need not be the restrictions to R* of one and the same function in F,because of lack of continuity at t = 0. This difficulty could be avoided if all functions involved vanished at t = 0. The following construction will permit us to restrict ourselves to this case without loss. It may of course be disregarded by the reader interested in F E 9 only.
+
+
84.E. CONSTRUCTION. Assume that F E ~ % We ' . suppose for the moment that there exists vo E F with ~ ~ (#0 0. ) There exists, then, T > 0 such that 0 x [ - , , ~ ]< pI vo I for some p > 0. It follows from this that any real-valued continuous function tp on R that vanishes outside [ - T , T ] is in F and satisfies I v 1, pI F, IFI v I. Define the function 0 by
<
<
<
<
Thus 0 is a primitive and 0 e(t)/t 1, d E C , I I = f , and &O) = 0. For u E [O, I] we set &(t) = ( I - u)t uO(t), a primitive coinciding with t outside [ - T , T I ; then e,(t) = t , e,(t) = O(t). For any u', u" E [0, 11 we find
(84.2)
+
84. DEPENDENCE ON A
30 I
Let A E F(8)be given. We set A,(t) = A(d,(t)) and claim that A, E F(X) and that the mapping u -+ A, : [0, I ] + F(X) is continuous. Indeed, A, - A is continuous and vanishes outside [-T, T], hence is in F(R); therefore A, E F(8). Let c > 0 be given; since A is continuous, it is uniformly continuous on [-T, TI; therefore there exists 6 > 0 such c. If u’, that t’, t“ E [-T, T], 1 1” - t’ I < 6 implies 1) A(t”) - A(t’)ll u” E [0, 11, I u” - u’ I 25d/58/16~, (84.2) yields IA,. - A,.I C, since the difference vanishes outside [ - T , T ] . Therefore I A,. - A,, pcl ‘po and the claimed continuity is proved. We set G,(t) = U(d,(t))U-l(t)and claim that (G,) is an F-deformation family. Indeed, G, is an invertible-valued primitive and coincides with the constant I outside [-T, T], hence G, E C(X)inv;and u + G, : [0, I] + C ( d )is continuous, since G,(t) is continuous in both variables U, t and is constantly equal to I for I t I 3, T and all u. Further, c, = G,A (1 - u &)A,G, E F(8); and 21.H, 22.V, 24. J, and the preceding proof of the continuity of u A,, show that u c, : [0, I ] -+ F(8) is continuous. This establishes our claim. We note in passing that G,(O) = I for all u. If we now define C, by (73.1), we find C, = ( I - u + & ) A , , W,(t) = U(O,(t)).Thus C, = A , C, = dA, , and C,(O) = 0. If, contrary to our provisional assumption, ~(0) = 0 for all rp E F, we simply set d ( t ) = t. The remaining construction goes through, but is trivial, since G, = I, C, = A ; and still C,(O) = A(0) = 0, since 11 A 11 E F. &,
<
<
IF,
< IF <
+
--+
--+
From now on we assume that X is a Hilbert space. For given F E 9or .F%f we define, as usual, Q, = Q n F(a)and Q,, = Qo n F(8). In order not to burden the reader with easily supplied arguments, we shall carry out the detailed analysis for Q, only, and later summarize the results for Q,, We ask ‘what is to take the place of the classes :Q, of Section 74. T o find an answer, let A E Q, be given, and recall that X , = X-, n X+, . Set d = Dim X , , cf = Codim X,, Obviously, ( X i , , X,l n XTo)is a disjoint closed dihedron, so that cf = Dim(Xk n Xro);but ( X k n X-,, X,l n X+,) is a disjoint closed dihedron in X,l (cf. proofs of 14.C, 14.D), so that cc+ = Dim Xd., whence d ct+ = Dim X. We therefore define, for any three cardinals d, c-, c+, d cc+ = Dim X, the class
.
.
+ +
+
C-*C$F
= { A E Q,
+ +
: Dim X,(A) = d, Codim X,,
+
= c*}.
+
It is useful to introduce the notations c = cc+, d* = d c’. We ‘ ,‘ $ , QF n { A : X,(A) E iE(X), X*,(A) E $c”((x)}. Obthen have viously, if A is in this class, then A* E $Q*, (this is the class described, for 1= R+, in Section 74); the cases where each of X-,(A), X+,(A) has =’
Ch. 8. EQUATIONS ON R
302
either finite dimension or finite codimension are exactly those in which at least two of d, c-, c+ are finite or, equivalently, at least one of c, d-, d+ is. We have the following specialized analogue of 73.D (cf. also the proof of Theorem 74.F).
-
84.F. Assume that A E c-*c$F , and that (G,,) is an F-deformation C, defines an arc in family, and define C,, by (73.I). Then the mapping u ‘-,CfdQF. Zf F E .F%, A(0) = 0 , and C;,(O) = 0 for all u, then C,,(O) = 0 for all u. Proof. T h e same proof as in 73.D shows, taking the modified form of u -+ C,, defines an arc in Q,; and since, by 73.C, as modified, X,(C,,) = G,(O)X,(A), X*,(C,) = G,,(O)X*,(A), it is clear that C,, E ‘-s‘$~. T h e last statement follows directly from (73.1). &
73.C into account, that
We next turn to the question of “canonical” representatives of the classes were “canonical” in Section 74 in the sense in which the (especially 74.E).It seems natural to choose a “canonical” A in c-’‘$2F in , as described in such a way that A* be a “canonical” representative of $2+, 74.E.We may, in addition, require that the closed dihedron (X-,(.4),X+,(A)) be perpendicular (see Section 14, p. 29). This idea is carried out as follows. C-~C&?F,
84.G. ‘-’e@F is not empty if and only if QF is not empty, namely $ and satisfies only ;f F contains a stiff function. Zf p E F is st# and, when F E 9W, p(0) = 0 ;andifP* are orthogonalprojections with P-P+ = 0 , Dim P * X = c*, Dim(Z - P-)(I - P + ) A = d , then
Proof. T h e “only if” part of the first statement follows from 42.E, since F obviously contains a stiff function if and only if F-, F , each contains one. T h e “if” part follows from the second statement, since projections with the required properties certainly exist, on account of d cc+ = Dim X , and since, if F ES%, the existence of a stiff function in F implies the existence of another that vanishes at t = 0. For the proof of the second statement, set A = A,,,P-,P+. Since )I A 11 = v, we have A E F ( & (if F E F % we use the fact that p(0) = 0). Now A* = +p+(I - 2P*) ~ Q n f - ~ ,by 74.E,and X*,(A) = ( I - P*)X, whence Codim X*,(A) = c*. Now P - P + = 0 implies that (X-,(A), X+,(A)) is a perpendicular closed dihedron, by 14.E. Therefore A E QF . Finally, X,(A) =,Y-,(A) n X,,(A) = ( I - P-)(Z- P + ) X ,so that Dim X,(A) = d . &
+ +
303
84. DEPENDENCE ON A
Remark 1. If, for F E Stv, we attempt to drop the restriction tp(0) = 0, we find that continuity of A = A,,,P-,p+at t = 0 requires 0
=
A( +O)
-
A( -0)
= 2tp(O)(Z
-
P- - P’)
= 2tp(O)(Z - P-)(I -
P);
therefore we must have y(0) = 0 unless d = 0, i.e., unless the dichotomyinducing dihedron is assumed to be disjoint.
Remark 2. If Y = ( I - P-)(I - P + ) X = ( I - P- - F ) X ,Z* = P*X, and if we identify X with the outer direct (Hilbert) sum Y @ 2- @ Z+, then A , , . P - , P tbecomes p{Zdsgn(.) @I,- @ (-I,+)}, and the equation x . 4 p - , p - . p k x= 0 becomes the system
+
y -yy
=o
t-
=
+ yz-
0
fit - yz+ = 0
for t
< 0;
y i-
fyy
+ tpz-
=o =0
for t
> 0.
a+ - yz+ = 0
We omit the special form this system takes when F E 9 contains the constants and we choose y = I . We now show that the classes c-,c$F and the “canonical” representatives we have defined are well chosen, by proving a connectivity theorem corresponding to Theorem 74.F.
84.H. THEOREM. Assume that at least two of the cardinals d, c-, c+ are jinite, and that if X is real they are not 0 , I , I in any order. Then is arcwise connected. Proof. I . We consider, for the given cardinals, some fixed Ap,p-,p+ defined as in 84.G; this exists unless QF is empty, and, on account of 84.G, it is sufficient to show that any A E C-($2F is connected to Ap.p-,Ptby an arc in this class. As a preliminary step for the case F E 93, we show that there is no loss in assuming A(0) = 0. Indeed, we carry out Construction 84.E; then 84.F implies that u .+ C, is an arc in , and we have C, = A and C,(O) = 0, so that we may replace A by C, in the rest of the proof. We therefore agree to assume that A(0) = 0 when F ~ 9 % we observe ; that, in this case, we also have A , , , p - , p + ( 0=) 0; and we intend this vanishing at t = 0 to be preserved along the whole arc joining these two.
c--ciQF
2. T h e next step consists in showing that there is no loss in assuming the closed dihedron (X-,(A), X+,(A)) to be perpendicular. We know that (omitting the argument A ) (X+,, X,l n X-,) is a disjoint closed dihedron; let Q be the projection along X+, onto X+ n X-, , and P the orthogonal projection along S+, . Then PQ = P , Q P = Q, so that ( P - Q)’ = 0.
Ch. 8. EQUATIONS ON R
304 Also, (84.3)
X:, = P X = P ( X + ,
+ ( X t n X-,)) = P ( X t n X-,).
+
Set G, = I u(P - Q); since G i l = I - u(P - Q), (G,) is an F-deformation family when considered as a family of constants. We define C,, by (73.1). By 84.F, u + C, is an arc in c-ic$2F , and C, = A; if F E S W , also C,(O) = 0. By 73.C as modified (see remarks at the beginning of this subsection), X,(C,) = G,X+,, = X+,; and (I - Q)(X,l n X-,) = {0}, so that, using (84.3), X-,(C,) = C I X , = Gl(X, (X,l n X-,)) = X, P(X,l n X-,) = X , X$,3 X$, = (X+o(Cl))L.By 1 4 4 (X-,(C,), X,(C,)) is perpendicular. We therefore assume from now on that (X4(A), X+,(A)) is perpendicular. Observe that also (X-o(A~,p-,p+)l X+O(A+,P-,P+)) = ((I - P-)X, ( I - P ) X ) is perpendicular; we intend the perpendicularity of the dichotomy-inducing di hedron to be preserved along the arc.
+
+
+
+
3. The rest of the proof will be merely sketched. Since cf, d+ = d care not both 1 if X is real, the proof of Theorem 74.F shows that A+ is connected by an arc in $2+, to p+(I- 2Q+), where Q+ is some orthogonal projection with DimQ+X = c+, CodimQ+X = d+ (we could take Q+ = Pt,but there is no real gain in this choice). This arc consists of two : subarcs in $f+, The first subarc is described in 74.D, and is the result of applying an F+-deformation family (C,') (the construction actually gave a succession of such families, but we can replace them by a single one), where each G,+ is orthogonal- or unitary-valued, and where 1) c,' 11 RII A+ 11, k being a constant (use 74.A or 74.B and the proof of 74.C for this). In particular, G,+(O)is orthogonal or unitary, and hence preserves the perpendicularity of dihedra; and if F E S V , 11 c,'(O)ll kll A(0)ll = 0, so that c,'(O) = 0. We define an F-deformation family (C,) by setting Cu+= G,' and G,- = C,'(O) (constant). The fact that we have an F-deformation family is obvious. We define C, by (73.1) and obtain an arc in '-*'dfS2,, along which C,+ describes the first subarc in $2+, mentioned above. The requirements of perpendicularity and, if F E 3W, vanishing at 0, mentioned in parts 2 and I of the proof, hold for all C,, by 73.C (modified) and 84.F, so that we set B = C, and try to continue from B. Observe that B- = G,+(O)A-G,+-'(O). The second subarc referred to above is the one joining B+ to v+(I- 2Q+) according to Theorem 73.G, with Y = X+,(B) = ( I - Q+) X, 2 = Xt,(B) = Q+X; we construct an arc u + C, in C-*e:RFbeginning at B, by letting Cu+describe that subarc and setting C,- = B- for all u; this is an arc in F(X) (if F E ~ Wthe, proof of Theorem 73.G and the facts that
<
<
305
84. DEPENDENCE ON A
B(0) = 0, ~ ( 0= ) 0 are used). Now, by the proof of Theorem 73.G, X+,(C,) = X+,(B), so that the fixed perpendicular closed dihedron ( X O ( B )X+o(B)) , = (&(c,), X+,(C,,)) induces a double exponential dichotomy for each C,; therefore the arc u 4 C, is in C-*cdfQF. We have thus succeeded in joining the original A of this part of the proof by an arc in c-*C@F to C such that C, = v+(I- 2Q+); and the perpendicularity and vanishing-at-0 requirements were satisfied along the arc, hence they hold for C. Also, C- = B- = G:(O)A-G;-’(O). 4. We now apply part 3 of this proof, symmetrized, to C instead of A , and choose Q- = P-. We thus join C by an arc in c-,ciQF (with preservation of perpendicularity and, if F E 9%of ? vanishing , at 0) to D , say, with D- = -T-(Z - 2P-); but D , = G;(O)C+G;-’(O) = v+(I- 2R+), where R+ = G;(O)Q+G,-’(O) is another orthogonal projection with Dim R+X =c+, Codim R+X = d+. Since D E C-*C&?F and (X-,(D), X+,(D)) = ((I - P-)X, ( I - R+)X)is a perpendicular closed dihedron, we must have P-R+ = 0, Dim(I - P-)(I - R+)X = Dim X,(D) = d ; therefore D = A Q , P- , R +. to A v . p - , p + .Since 5. It remains to connect AQ,p-,R+ P-X
= ( ( I - P-)X)’ C ( I - R+)X,
P-X
C (I
-
P+)X
on account of perpendicularity, there exists an orthogonal or unitary operator S on X such that S leaves P - X pointwise invariant and maps ( I - R+)X onto ( I - P + ) X ; i.e., SP-S-’ = P-, SR+S-’ = P+. Since either the orthogonal dimension d+ or the orthogonal codimension c+ of the latter two spaces is finite, we may, in the real case, choose S in such a way that it reduces to a proper rotation on a finite-dimensional subspace and to the identity on the orthogonal complement thereof; we conclude that in any case S = eH, where H is skew-symmetric or skew-Hermitian. We set S, = euH,u E [0, I]; each S, is orthogonal or unitary, and So = I , S, = S. With (S,) as an F-deformation family of constants, 84.F yields an arc in joining A q F . P - , R to+ A U , S P - S - ~ , S R t= S -Al , , , p - , p t as , required. & ‘-ar;QF
Remark. A careful analysis of the “first subarc” in parts 3 and 4 of the proof, involving a review of the proofs of 74.A, 74.B, 74.C, 74.D, would allow us to obtain R+ = P+ directly, thus avoiding part 5 of the proof.
Corresponding to Theorem 74.G we have: Assume that F is stronger than M (in particular, 84.1. THEOREM. F E 3 or F i f ) and contains a stiff function. Then each ‘-*‘iQF is open and closed in QF; if at least two of d , c-, C + are finite, and if they are not 0, I , I in any order
if X
is real, then
r--C$’F
is exactly one component of
QF
.
Ch. 8. EQUATIONS ON R
306
Proof. As in the proof of Theorem 74.G, the first part need only be proved for F = M. Now by Theorem 84.C the mapping A
+ (X-,(A),
X+,(A), Xo(A)): QM
-
E(X) X E(X) X E(X)
is continuous, and C-,ci52Mis the inverse image of :'c"(X) x $ E ( X ) x :E(X), which is open and closed in E ( X ) x E ( X ) x E ( X ) by 14.G. Hence c-vc$M is open and closed in QM . The second part of the conclusion now follows for every F from the first to show that c-*c~52F is open and closed, from 84.G to show that it is not empty, and from Theorem 84.H to show that it is connected. &
Remark 1. The components of 52, are arcwise connected, since 52, is open in F(X) (cf. Remark 1 to Theorem 74.G). Remark 2. As for Theorem 74.G, the assumption that F be stronger F than M is not redundant. In particular, if there exists ~ , E with t+l I p(u)l du = CO, then the union of all classes C-J$F with 'im,t,+m at least two of d, c-, c+ finite is arcwise connected. The proof follows, as in Remark 2 to Theorem 74.G, from the proof of Theorem 74.R.
st-,
For finite-dimensional X we conclude: Let X be a finite-dimensional euclidean space, with 84.J. THEOREM. Dim X = n , and assume that F is stronger than M (in particular, F €9or Y W ) . If F contains no stiff function, 52, = 0. Otherwise, 52, is not empty and open in F(8), and its components are, unless X is real and n = 2, exactly c+ n; the functions Codim X-,(A), the sets ,-,~~$2, , c-, c+ 2 0 , cCodim X+,(A), or the functions Dim X-,(A), Dim X+,(A) form a complete - ~ + @I,- @ set of invariants for connection; and the elements P ( I ~ - ~ - sgn(-) ( -Ic+)} for anyfixed stiff E F and any representation of X as Y @ 2- @ Z+, Dim Z* = c* form a complete set of representatives of the components. Even in the exceptional case of the real plane, '$52, , 2*$2, , o$52F are components, containing pI sgn( pl, 7 1 1 ,respectively.
+ <
a),
Remark. The study of the connectivity structure of the sets ',$?,, o$lF , $' 2, for a real euclidean plane (sets that are open and closed in 52, if F is stronger than M, by Theorem 84.1) may be carried out by an adaptation of the results of the corresponding subsection of Section 74 (i.e., 74.H-74.L, and Theorem 74.M), applied to A+ if c+ = 1, and to A- after symmetrization if c- = 1. This adaptation is straightforward and elementary, but notationally complicated; it is left to the reader. As promised, we state the results for the ordinary double dichotomy classes Q,, , corresponding to Theorems 74.0 and 74.Q.
84. DEPENDENCE ON A
307
Assume that A EQ,, , and that a closed dihedron 84.K. THEOREM. ( Y - , Y + )with at least two of Dim( Y- n Y+),Codim Y - , Codim Y+ finite induces a double dichotomy for A . Then A , 0 are connected by an arc in QoR
.
84.L. THEOREM. Let X be a finite-dimensional euclidean space. Thm is not empty and is arcwise connected.
QoF
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P A R T I11
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CHAPTER 9
Ljapunov’s method 90. Introduction
Summary of the chapter As was pointed out in the Introduction to Chapter 4, the concept of a dichotomy, ordinary or exponential, of the solutions of Eq. (30.1), i.e., (90.1)
ji.
+ A X = 0,
is closely related to the notion of uniform stability, ordinary or asymptotic, respectively. This suggests the possibility of adapting the so-called Ljapunov’s (second, or direct) method to the characterization of dichotomies. T h e present chapter provides a sketch of a treatment of this question. After some introductory material on pointwise properties of the solutions of (90.1)’ which is given in the next subsection, the definition of Ljapunov functions and their total derivatives, and the statement of the relevant assumptions concerning them, are the subject of Section 91. Sections 92 and 93 contain characterizations of exponential and ordinary dichotomies, respectively, by means of the existence of Ljapunov functions with suitable properties. T h e order is suggested by the fact that the results and their proofs are simpler, neater, and more comprehensive in the exponential case. Both sections include “direct” and “converse” theorems, the typical form of the former being: “ I f there exist Ljapunov functions satisfying such-and-such assumptions, then there exists a subspace inducing a dichotomy [an exponential dichotomy] for A”; the term “converse theorem” is then self-explanatory. Combining these theorems with the results in Chapter 6 (Sections 63, 64), we may obtain theorems of the form: “If there exist Ljapunov functions satisfying such-and-such assumptions, 31 1
312
Ch. 9. LJAPUNOV’S
METHOD
thenfor such-and-such 9 - p a i r s , etc., ( B ,D), there exists a ( B ,D)-subspace (or ( B ,D ) is admissible) for A, i.e., for (30.2): (90.2)
* +Ax
=f”,
as well as corresponding converse theorems, a matter we leave to the reader (but cf. proof of Theorem 93.A). We mentioned above that this chapter is in the nature of a sketch. Thus, we have explored neither the various regularity assumptions on the Ljapunov functions, nor the detail of the complications of the infinite-dimensional case, nor (as we have attempted to do in the preceding chapters) the precise import of the various assumptions in each theorem. For some of these matters the reader will be referred at the appropriate time to Massera and Schaffer [3]. We stress the fact that, in contrast to most of our preceding results, the assumptions of finite dimensionality and complementedness that occur in the theorems are not, in the present context, mere conveniences that may be dropped at the cost of some technical complication, but are quite essential to the results. Ljapunov’s method requires the space X to be considered as a real space, and this is accordingly assumed from Section 91 on. T h e results in Sections 92 and 93 of course concern equations on J = R, as the range of the independent variable; in connection with this restriction, see the Notes to this chapter.
Pointwise properties o f the solutions. Exceptional sets For the time being, the range of t is a given interval J with 0 E J and X is a given Banach space. It will be necessary to examine the elements of L(X), which are, strictly speaking, equivalence classes modulo null sets of strongly measurable functions, and to single out from each class a particular element that can then be studied pointwise. I f f € L(X), the limit fo(t) = limj,, (t’ - t ) - l ( ’ f ( u ) du exists a.e. in J; the null set E,(f) where the limit does not exist, and the valuef,(t) of the limit where it does, obviously depend only on the equivalence classf; and if the definition of fo is extended to all J by setting fo(t) = 0 for t E E,cf), thenf, belongs to the equivalence classf(Hil1e and Phillips [l], p. 87). We call f,, thus defined, the NATURAL REPRESENTATIVE OF f. If the equivalence class f contains a continuous function, then this is the natural representative off, and E,cf) = 0. With this terminology, iff is a primitive, the function taking as values the usual (strong) derivative:
90. INTRODUCTION
313
lim,*,, (t’ - t)-l(f(t’) - f ( t ) ) wherever the limit exists, and 0 elsewhere, is precisely the natural representative off. We now assume that A E L(x)is given and that, as usual, U denotes that solution of (30.3), i.e., (90.3)
U$AU=O
that satisfies U(0) = 1.
90.A. For every to E J we have (90.4)
lim sup(t 1+10+0
-
t0)-l 11 ( U ( t )- U(t,))U-l(t,) =
1)
l i m sup(2 - t0)-l t+tO+O
I/
t
A(u)du t0
1,
whether these limits be finite or infinite, and similarly f o r t + to - 0 (one alternative to be disregarded if to is an endpoint of]). lirn,,,? ( t - to)-’( U ( t ) - U(t,)) exists i f a n d only iflim,,,O ( t - t0)-l J;, A ( u )du extsts, and then the former limit is equal to the latter multiplied to the right by - U(to).Therefore, if 0,A stand for the natural representatives of their respective classes, Eq. (90.3) holds pointwise, i.e.,
U ( t ) + A ( t ) U ( t )= 0,
t E J.
Proof. Define the primitive V with values in 2 by (90.5)
V ( t ) = ( U ( t ) - U(t,))U-l(t,)
+ jt A(u)du. t0
I t is clearly sufficient to prove that, if either limit superior in (90.4) is finite, then lim
(90.6)
t+cO+O
( t - t0)-l
11 V ( t )11
=0
and similarly for t -+ to - 0; this last part follows by an obvious analogy and is omitted. Since U is a solution of (90.3), we have, from (90.5), V(t)=
-
it
A(u)(U(u)- U(t,))U-’(t,) du.
t0
Ch. 9. LJAPUNOV’S METHOD
314
If the first limit superior in (90.4) is u of Iipt:;p(t
-t
o y
< co,then (90.6) holds on account
II W )II t
< lim sup j, II
II II ( t - t0)-1(u(u) - u(to))u-1(t0) II nu
t+t0+0
+
Again from (90.5), V P a solution of the operator equation P A V = F, where F(t) = A(t) J1, A(u) du, and V(to)= 0. Since this equation is a particular case of (90.2) with X replaced by R, we may apply (31.5) to J‘ = [ t o , t ] for any t > to and find (t - t
0 Y
I1 W )II
If the second limit superior in (90.4) is u follows:
< co, we obtain (90.6) as
t
lit?-y$P(t
- 43-l
II W )I1 d
lim
t+to+O
t
(jtoI! 4 4 II du) exp (jtol! 4 4 II du) = 0.d
For applications to Ljapunov’s method we shall need the following three sets: E,
={t E
1: b ( t ’ - t)-l( U(t’)- U ( t ) )
E:
={t E
J : litEc+uop(t’- t)-’ 11 U(t’)- U ( t )11
E-,
={t E
J
does not exist} = m}
: lim sup(t - t‘)-lII U(t’)- U ( t )(1 = co}. t‘4-0
All these sets are null sets, since Eo = Eo(0)is one (see above) and since E; u E: C Eo . These sets may be characterized in terms of A: 90.B.
Eo = Eo(A),and EZ
=
1
t E J : litT&yp(t’ -
I
c tE C
A(#)du
t
t’
: li2n_ts+uop(t’- t ) - l
A(u) 11 du
1
{ t E J : ess:+zzup
and similarly for E;
11 I 11 t‘
t)-l
.
11 A(t’) )I = a},
1
= 031
= m
I
90. INTRODUCTION Proof. Immediate from 90.A.
315
&
90.C. If x is a solution of (90.1), then E o ( i )C E,, , and if istands for the natural representatiwe, then i ( t ) A(t)x(t)= 0 f o r all t E J\ E,; also, lim ~ u p j + (t’ ~ +-~t)-’ll x(t’) - x(t)ll < 00 for all t E J\ E: , and s i m i l a r l y f o r t ’ 4 t - O a n d t E J\E;.
+
Proof. (31.2) and 90.A.
&
T h e following lemma is valid in a real finite-dimensional space X. T h e terms “measurable”, “null set”, “almost all” refer to Lebesgue measure in J , X, X x J , as the case may require; this Lebesgue measure obviously depends for its normalization on the selection of some euclidean metric in X, but the concepts mentioned above do not depend on the normalization. We use the terminology, notations, and results of Halmos [l], especially Chapter 7. 90.D. Let X be a real finite-dimensional space. A measurable set H C X x J is a null set i f and only i f for almost all x E X the set H [ x ] = { t E J : U(t)xE H}-the projection on J of the intersection with H of the graph of the solution of (90.1) that starts from x a t t = 0 4 s a null set. Proof. We assume X to be provided with some euclidean metric, and let pJ , px , p denote Lebesgue measure in J , X, X x J, respectively; p is the completion of the product measure px x p J . Let @ be the homeomorphism of X x J onto itself defined by @(x, t ) = (U-l(t)x, t ) ; its inverse is given by @-l(x, t) = (U(t)x,t ) . We claim that if B C X x J is a Borel set, then B is null if and only if @B (also a Borel set) is null. Indeed, B is px x pJ-measurable. Assume that p ( B ) = (px x pJ)(B)= 0. Then p x ( B f )= 0 for almost all t E J; but the automorphism U-l(t) maps null sets into null sets, so that P ~ ( ( @ B= ) ~px(U-l(t)B#) ) = 0 for almost all t E J , whence p(@B)= ( p x x p J ) ( @ B= ) 0. T h e converse follows by applying the same argument to @-l; and our claim is proved. We turn to our p-measurable set H , and find H [ x ] = {t E J : @-l(x, t ) E H } = {t E J : (x, t ) E @ H } = ( @ H ) z, x E X . If p ( H ) = 0, there exists a Borel set B 3 H with p ( B ) = 0, whence p(@B) = 0. But H [ x ] C (@B)zfor all x E X, whence pJ(HIXI) p,((@B),) = 0 for almost all x E X. Assume conversely that pJ(HIXI)= 0 for almost all x E X. There exists a Borel set B C H with p ( B ) = p ( H ) . Then (@B)% C H[x], so that pJ((@B)J= 0 for almost all x E X; therefore p ( 0 B ) = 0, whence p ( H ) = p ( B ) = 0. 4,
<
Ch. 9. LJAPUNOV’S METHOD
316
91. Ljapunov functions Ljapunou functions
We continue to assume that the range of t is an interval J with 0 E J; now and henceforth in this chapter, however, X is a real Banach space; this restriction is essential for Ljapunov’s method, and when equations given in a complex space are considered, they must be replaced by the corresponding equations in the underlying real space. We are concerned with functions V : X x J + R with V(0,t) = 0 for all t E J and satisfying the following regularity conditions:
V is continuous;
(C) (Continuity):
(L) (Lipschitz condition): For any number r > 0 and any compact subinterval J’ C J there exists a number L = L(r, J’) > 0 such that I V(x”,t) - V(x‘,t)l L(I x” - x’ 11 for all x’, x” E r Z ( X ) and t E J’. There is no loss in assuming that L has the least possible value; it is then nondecreasing with r and J’.
<
Such functions, especially when considered together with their total derivatives to be defined in the next subsection, are called (GENERALIZED)LJAPUNOV FUNCTIONS. V is (ABSOLUTELY) HOMOGENEOUS OF DEGREE K ( K 2 1) if V(px, t) = I p lKV(x,t ) for all x E X, t E J , p E R. We sometimes require that V satisfy some of the following stronger regularity conditions:
(C‘) (Absolute continuity): For any numbers e, r > 0 and any compact subinterval J‘ C J there exists a number 6 = a(€, r, J’) > 0 such that for any finite set of disjoint intervals (t,, ti + 6,) C J’, i = 1, ..., n, with 6, 6 and any corresponding xt E r Z ( X ) we have I V(x, 9 ti 8,) - V(x, td)l e;
c:
+
xy
<
9
<
(L’) (Uniform Lipschitz condition): The function L in condition (L) may be chosen independent of J’. (L’) trivially implies (L), and (C’) and (L) together imply (C). Other, still stronger, regularity conditions are pertinent in the context of Ljapunov’s method, but we refrain from going into further detail (cf. Massera and Schaffer [3], pp. 545-546). 91.A. If the Ljapunov function V satisjies (C‘) and if x : J-. X is a primitive, then the function V, : J + R defined by Vz(t)= V(x(t),t ) is a primitive.
91.
LJAPUNOV FUNCTIONS
317
Proof. Let J‘ be a given compact subinterval; we have to prove that V , is absolutely continuous on J‘. Let c > 0 be given and set Y = suple,. 11 x(t)ll, 6‘ = S ( i c , r, J’) (condition (C‘)), L = L(Y,J’) (condition (L)). Since x is absolutely continuous on J‘, there exists 6, 0 i 6 S‘, such that for every finite set of disjoint intervals (ti , ti Si) C J’, i = 1, ..., n,with 1 :6, 6 we have En1 11 x(ti + Si) x(tJ €/2L.Since 6 S’, we have
+
<
<
1
<
<
I
A continuous nondecreasing function c : R, -+ R, with c(r) = 0 if and only if Y = 0 is termed a COMPARISON FUNCTION. T h e following assumptions on a Ljapunov function V are in common use: V is UNIFORMLY SMALL (AT 0) if I V(x, t)1 a(II x 11) for some comparison function a and all x E X,t E J ; in Ljapunov’s terminology, “V has an infinitely small upper bound”. We observe that (L’) implies that V is uniformly small, with any a such that a(r) 2 rL(r). V is POSITIVE DEFINITE if V ( x ,t ) >/ b(ll x 11) for some comparison function b and all x E X,t E J ; V is NEGATIVE DEFINITE if - V is positive definite. A positive definite Ljapunov function W that is independent of t (and therefore satisfies (C’) and (L’) and is uniformly small) is termed a GAUGE FUNCTION. Some gauge functions always exist: W(x)= 11 x 1 ‘ defines one for each K >/ I , and this function is in fact homogeneous of degree K .
<
Total derivatives We now consider Ljapunov functions in connection with a given equation (90.1). Let A E L(x)be given, and let V be a Ljapunov function. T h e UPPER RIGHT DERIVATIVE OF V ALONG THE SOLUTIONS OF (90.1)or, if A is understood, the UPPER RIGHT TOTAL DERIVATIVE OF V is the extended-real-valued function V‘+ on X x J defined by
Ch. 9. LJAPUNOV’S METHOD
318
where A stands for the natural representative. T h e LOWER RIGHT,UPPER LEFT,LOWER LEFT, TOTAL DERIVATIVES V l , V‘-, VL are defined correspondingly (by lim infh,+, , lim S U ~ ~ + -lim ~ , infh,-,), and V’ stands for any of these total derivatives, of unspecified type. An endpoint of J at which the particular total derivative under study is undefined can and shall be disregarded in the sequel. If V has a pseudodiflerential dV[(x,t ) ; (Ax, At)] (see Alexiewicz and Orlicz [l]), we have V ( x , t ) = d V [ ( x ,t ) ; (-A(t)x, l)] for each total derivative. 91.B.
v I + ( x , t ) = f o o i f a n d only i f D:V(x, t ) = l i y y p kl(V(x, t
+-h)
-
V(x, t ) ) = f 00,
respectively; and correspondingly for the other total derivatives and partial derivate numbers. Proof. By (L) there exists L real h with t h E J ,
+
I h-l{ V ( X- h ~ ( t ) xt ,
>0
such that for sufficiently small
+ h ) - V ( X , t j) - h-i{ v ( ~t +, h) - qx1 t)) 1
a finite number independent of h. $, Remark.
On account of 91.B, the sets
depend only on V , and not on A ; the same is true of the sets G+(&moo), G-( fco), G-(kco) corresponding to the other total derivatives; in particular, all these sets are empty if, for each x, V(x, -) satisfies a local Lipschitz condition with respect to t. We now justify the term “derivative along the solutions of (90.1)” by relating the total derivatives of V to the derivate numbers of the function V,-defined in the preceding subsection by V,(t) = V ( x ( t ) ,t)-for a solution x of (90.1). T h e derivate numbers are indicated by the prefixes D+, D, , D-, D- . 91.C. Let x be any solution of (90.1). D+V,(t) = V’+(x(t),t ) for all t g J \ E o . If t E J \ E L , then D+V,(t)= k c o i f a n d o n l y i f ( x ( t ) , t ) E G+(f a), respectively. And correspondingly for the other derivate numbers and total derivatives.
9 1. LJAPUNOV FUNCTIONS
319
Proof. For each t E J , condition (L) implies the existence of L such that, for sufficiently small real h with t h E J,
+
>0
-+
If t E J \ E, and h 0, the second member tends to 0 by 90.C, and hence the difference quotients in the first member have equal limits superior, viz., D + V z ( t ) and V ’ + ( x ( t ) t, ) . If t E :‘\ E : , the second member remains bounded, by 90.C; hence those limits superior are either both finite, or both infinite and equal. 9, T h e clue to the application of Ljapunov’s method under the more general nonclassical assumptions in which we have placed ourselves (CarathCodory’s conditions and, less important, general Banach spaces) lies in the following definitions and in Theorem 91.D (Main Lemma). A total derivative V‘ of the Ljapunov function V is ALMOST MINORIZED [ALMOST MAJORIZED]BY the continuous function cp : X + R if the following conditions are satisfied (since these conditions involve V itself and A , this terminology is somewhat improper): (a) Either V satisfies (C’); or the corresponding Em is countable and the projection on J of every relatively compact subset of the corresponding G( -03) [the corresponding G( + a ) ] is also countable; (B) If H = {(x, t ) : V’(x,t ) < V(X, t ) } [ H = {(x, t ) : V ( X , t ) > V(X, t ) } ] , then either the projection on ] of every relatively compact subset of H is null, or X is finite-dimensional and H itself is null. I t is important to observe that these conditions are so framed that no knowledge of the solutions of (90.1) is required for their verification (cf. 90.B). T h e most important instances of these definitions deserve a special terminology. A total derivative I/’ is ALMOST POSITIVE DEFINITE if it is almost minorized by c(ll x II), and ALMOST NEGATIVE DEFINITE if it is almost majorized by -c(ll x II), where c is some comparison function; and V’ is ALMOST POSITIVE [ALMOST NEGATIVE] SEMIDEFINITE if it iS almost minorized [almost majorized] by 0.
x
(Main Lemma). Assume that the Ljupunov function 91 .D. THEOREM V has some total derivative that is almost minorized [almost majorized] by the continuous function q. Then for every solution x of (90.1) and
320 all t o , t, E J , to
Ch. 9. LJAPUNOV’S METHOD
< t , , we have
Proof. 1. I t is sufficient to carry out the proof for almost minorized V’+;indeed: (a) if V i is almost minorized by v, so is afortiori V’+,and the proof applies; (b) if V: or V’+ is almost majorized by 9,then (- V)’+= - V i or (- V)’f = - V’+ is almost minorized by --v, and we apply the former proof to -V, -rp; (c) the proof for the left total derivatives follows from that for the right ones by a change of variables t+ -t, say, and interchange of “upper” and “lower” as well as of “almost minorized” and “almost majorized” . 2. Assume that the first alternative in condition (p) holds, and let x, to , t, be given as stated. If V satisfies (C’)-the first alternative of (ar)-91.A implies that V , is a primitive, since x is one; by 91.C (first coincides a.e. with V’+(x(t),t); part), (the natural representative) by condition (p), T,(t) 2 v ( x ( t ) , t ) for almost all t E [ t o , t,] (since { ( x ( t ) , t) : t E [to , t,]) is compact), and the conclusion follows by Lebesgue integration between to and t , . Under the second alternative of (a),the set of t E [to , t,] for which ( x ( t ) , t ) E G + ( - o ~ )is countable; by 91.C (second part) and the assumption on E : , D+V,(t) > - 03 for all t E [ t o ,t,] except for a countable set; and by (p) and the first part of 91.C, D+Vz(t)>, p)(x(t),t) for almost all t E [ t o , t,]. T h e conclusion then follows from a result in the theory of the Perron integral (Saks [ 11, p. 204). 3. Assume now that the second alternative in condition (/I holds. ) Then V’+(x(t),t) >, y ( x ( t ) , t) for all t E J\ H[x(O)], where H [ x ] is the set defined in 90.D (here H[x(O)] = {t : ( x ( t ) , t) E H}). By 90.D, H [ x ( O ) ] is a null set for almost all x(0) E X; the argument in part 2 of the proof thus establishes the conclusion for “almost all” solutions of (90.1), i.e., all except those starting from a certain null set in X at t = 0; since the complementary set is dense, the conclusion follows by continuity (condition (C) and (31.2)) for the exceptional solutions also. +j,
v,(t)
92. Exponential dichotomies In this section, as in the next, the range of t is J = R,; X is still a given real Banach space. We begin with the “direct” theorem.
32 1
92. EXPONENTIAL DICHOTOMIES
92.A. THEOREM.Assume that A E M(X), and that a Ljapunov function V exists that is uniformly small and has one of its total derivatives almost negative definite. Then X,,is closed, all unbounded solutions tend to co in norm as t + GO, and there exist numbers v, v‘, N , Nk , y 5 > 0 such that any solutions y , z of (90.I ) with y ( 0 ) E X,, , z(0) 4 X,,(or z = 0 ) satisfy (Ei), (Eii,), (Eiii,) (cf. Theorem 42.A,(am))-the last two conditions with T = T ( z ) = min{t E R, : I/ z(t’)ll 3 1; z(0)llf o r all t’ 3 t}. If X,, has finite codimension, then X,, induces an exponential dichotomy f o r A. Proof. 1. T h e essential idea of the proof consists in classifying the solutions x of (90.1) into those for which V,(t) >, 0 for all t and those for which V x ( t l )< 0 for some 2,; these will turn out to be the bounded solutions and those tending to co in norm, respectively. Since V , is strictly decreasing for x # 0, by Theorem 91.D, V,(t) 3 0 for all t implies V,(t) > 0 for all t unless x = 0. We divide the proof into several steps. We let a, c denote the comparison functions involved in the assumptions of uniform smallness of V and almost negative definiteness of some total derivative; we also set E = exp(1 A IM). 2. We first show that, if x is a solution of (90.l), Vz(tl)< 0 for some t , if and only if limt+511 x(t)ll = co. T o prove the “only if” part, if 6 > 0 is so small that a(6) < - Vx(tl),then, since V , is decreasing, a(l1 x(t)ll) 3 I V,(t)l 3 - V z ( t l ) > a(6) for all t 3 t , , whence 11 x(t)ll >, 6 for such t. Again by Theorem 91 .D, V,(t) V,(tl) - ( t - t,)c(6) -+ -CO as t + co, so that a(ll x(t)ll) 3 j V,(t)l -+ co and hence 11 x(t)ll+ co as t + 00. For the “if” part: if lim,+m11 x(t)ll = CO, then infleR+11 x(t)ll = .$ > 0; Theorem 91.D yields V,(t) V,(O) - t c ( 0 -+ - CO, so that V,(t) < 0 for large enough t. 3. For any nontrivial bounded solution y , V,(t) > 0 for all t, by the preceding argument. We show that, conversely, V,(t) > 0 for all t implies not only that y is bounded, but also that there exist v, N > 0 such that all such y satisfy (Ei); it follows from Theorem 33.C that X,, is closed. Specifically, we claim that if y is a solution of (90.1) with V v ( t ) > 0 for all t, then y satisfies (Ei), where N is any positive number so large that N > 2E, a(2N-l) < c(E-’), and v = c( l ) ~ ~ ( i log V ) 2. We first show that
<
<
We consider the solution y 1 = 2 N - 1 ~ ~ y ( t , , ) ~ ~by - 1 part y ; 2 of the proof, V J t ) > 0 for all t , and we are to show that 11 yl(t)ll $iVl yl(to)ll = 1 for all t 3 t o .If t,, t ,< t,, I , ( 3 I .7) indeed yields llyl(t)ll Ell yl(t,,)ll
<
+
<
<
<
322
Ch. 9. LJAPUNOV’S METHOD
&Nllyl(to)]l;assume that 1) yl(tl)ll > 1 for some t , >, to + 1 ; then, by (31.7), IIyl(t)ll > E-’ for t , - 1 < t < t , . Theorem 91.D yields the contradiction -42N-’)
=
<
- 4 l Y 1 ~ ~ 0 ~ l lVt,,(fl) ~
<
- V?,,(tO)
-
J
11
4llYl(t) II) dt
11,
< Next, set
-
J;;-lc(E-l) dt
= -C(P).
u(N)c-l(l), and consider, for given t o , the solution 11 y2(t1)11 1 for some t , , to t , to T ; if this were not the case, Theorem 91.D would imply the contradiction T
=
< <
<
y, = N \ ~ y ( t o ) l ~ -We l y . claim that
+
+ <
<
Therefore, by (92.1), IIr(t0 4 11 m Y ( t l ) l l = Bllr(t0)ll IIyz(t1)ll &lly(to)ll. We can therefore apply 20.C to 11 y 11-l with the given T, 7 = 2, H = 2N-’, and obtain (Ei) with the stated Y, N. 4. Let z be an unbounded solution; by parts 2 and 3, limt+m11 z(t)l(= 00; therefore T = T(z),as given in the statement, is well defined. We claim that z satisfies (Eii,) with this T, and NL = 2Em+l, where m is any integer such that m 2 2 4 I), and v’ = &( I)u-l(Nk) log 2. We first prove that
1) z ( t ) 1) >, 2N:’ 1) z(to)1)
(92.2)
for all
t
>, to >, T.
5
<
Suppose (92.2) fails for some t to >, T; we can find t , , t , , to t , < 2, t such that II z(t,)ll = &V ‘ l, z(t2)ll = Em+lll z(t,)ll and I( z(t,)ll > 11 z(u)ll > 11 z(t& for t , < u < t , . By (31.7) we must have t , - t , > m 2 4 I)c-l( I ) . Consider the solution z1 = 11 ~ ( t , ) ~ ~ so - ~ zthat , 1) z,(O)\( 1. Theorem 9 1 .D yields the contradiction
<
<
-241)
< -4z&,) II) - 41ZI(0) II) ,< hJh) < c(ll 44 II) du d - ( t z - t I k ( 1 ) . -
-
I’q(0)
s”
-
V,,(t,)
tl
Next, we set T = 2a(Nk)c-’(l) and consider, for given to 3 T, the solution z2 = &VI:l ~ ( t ~ ) l / -By ~ z(92.2), . 11 z,(t)ll 1 for all t 2 t o . Therefore Theorem 91.D implies, for any p > I ,
-4%(to
+ P 7 ) It) - 4 N ; )
< -411 Z,(tO -L- P.) It) @(It zz(t0) I l l < I’z,(to + p 7 ) - V,,(to) < -
-pTC(l)
==
-2pU(NL).
92. EXPONENTIAL DICHOTOMIES
323
+ p~)ll)> a(Nk),whence 11 zz(to+ p ~ ) l l> N;; since p > 1 was arbitrary, II z(to + .)I1 2 K 3 l z(to)llII i- 2 211 z(t0)11. We can therefore apply 20.C to /I z (1 (but only for t 2 to 2 T) with the Therefore a(ll zz(to
=
given
Nk
.
T,
~2(t,
7 = 2, H = 2N;-,-’, and obtain (Eii,)
with the stated T, v’,
5. It now follows from 42.D that there exists ya > 0 such that y, z as given in the statement satisfy (Eiii,) with T = T(z). 6 . Assume that X, has finite codimension. If X o = X , X o obviously induces an exponential dichotomy for A ; we assume, therefore, that Z # (0) is a (finite-dimensional) complement of X,. By the usual compactness argument applied to {x E 2 : I( x 11 = I}, it follows that there exists To 2 0 such that T ( z ) To for all solutions z of (90.1) with z(0) E Z. Let z be such a solution; if 0 to t To , (31.7) yields 11 z(t)ll 3 (1 z(t,)lI e v ’ ( t - l o ) exp(-v’T0 - 1 ;, 11 A(u)ll du). Combining this with (Eiim),we obtain (Eii) with
<
< < <
By 42.D there exists yo > 0 such that solutions y, z with y(0) E X , , z(0) E 2 satisfy (Eiii). By 42.B,(a) or (b), X o induces an exponential dichotomy for A. 9,
Remark 1. If A 4 M(X), the remaining assumptions of Theorem 92.A do not ensure that even conditions (Di) or (Dii) hold, even if X is onedimensional, A is continuous, V satisfies (C’) and ( L f ) ,and indeed V f ( x ,t) = - 1 x I: see Massera and Schaffer [3], Example 3.1. This is in contrast with the classical Ljapunov Theorem for asymptotic stability, where V is positive definite, and (Ei) holds for all solutions regardless of whether A E M(X)or not (see Massera [2], Theorems 13, 7(f); the same proof works for discontinuous A). 00
Remark2. If X,, has infinite codimension, it need not induce an exponential dichotomy even if all other assumptions in Theorem 92.A hold: Massera and Schaffer [ 3 ] , Example 3.2 illustrates this for a separable Hilbert space X , A E C ( ~ (the ) adjoint equation to the equation of Example 33.G) and an extremely regular V . However, part 6 of the proof remains applicable to any finite-dimensional subspace 2 of X with X,n 2 = {0}, and yields a “collective” behavior of the solutions that is precisely like the one described in Theorem 66.1 for the solutions of an associate equation, except that in addition Y’ is independent of 2.
Ch. 9. LJAPUNOV'S METHOD
324
We next consider the converse theorem.
92.B. THEOREM. Assume that A possesses an exponential dichotomy and that X , is complemented. Then there exist nonnegative Ljapunov functions V , , V , satisfying (C') and ( L ' ) and hence uniformly small, such that every total derivative of V , , V , is almost negative semidefinite, almost positive semidefinite, respectively, and every total derivative of the Ljapunov function V = Vo - V , is almost negative definite. If A E M(R), Vo , V1 may be so chosen that V , V , is positive definite. In any case, V,, , V , may be chosen homogeneous of the same prescribed degree K 2 1.
+
Proof. 1. We apply condition 42.B,(c) for an exponential dichotomy to X , and some projection P along X , . It is convenient to rewrite conditions (E:i), (E:ii) with t o , t replaced by t , u, respectively:
(92.3)
I( U(u)(Z - P)U-'(t) 11 < De-'(u-l' for all (1 U(u)PU-'(t)I\ < De-Y'(t-u) for all
u
t
2 t 2 0, >, u 2 0.
If we multiply (90.3) from left and right by U-' and integrate between t ' , t" (t" >, t' 2 0), we obtain U-l(t"> - U-l(t') = U-l(s)A(s)ds; from this and (92.3) we immediately obtain
(1 U(u)(l - P)(U-l(t")- U-l(t'))1)
< De-
,.l* " ~ l " )t ,
J )I A(s)11 ds
for
(92.4)
(1 U(u)P(U-'(t")- U-l(t'))I(
< De-
u'(t'-u)
u
2 t" 2 t' 2 0,
J:: (1 A($)(1 ds for
t" 2 t' 2 u 2 0.
2. Let W be a fixed gauge function; let L*(r) be the corresponding Lipschitz constant (condition (L) or (L')) and b* the comparison function involved in the assumption that W is positive definite. We define V , , V , as follows, for each x E X, t E R,: Vo(x,t )
=
SrnW(U(u)(Z
-
P)U-'(t)x) du
+
I"
1
(92.5) V,(x, t )
=
W(PU-'(t)x)
W(U(u)PU-'(t)x) du,
0
= V , - V , . Since (92.3) yields W(U(u)(Z - P)U-l(t)x) < Dll xllL*(DII x (I)e-v(U-f)for all u 2 t, the integral defining Vo is finite.
and set V
The first term in the expression defining V , may be omitted without altering the conclusions, except the positive-definiteness of V , Vl when A E M ( a ) (part 4 of the proof). Obviously V , , V , are nonnegative
+
92. EXPONENTIAL DICHOTOMIES
325
and satisfy Vn(0, t ) = Vl(O, t ) = V(0,t ) = 0 for all t E R,; if W is homogeneous of degree K (e.g., W(x)= 11 x I ‘), then so are V , , V , , V . For given Y > 0 and any x’, x” E r.Z(X), t” 3 t’ 3 0, we have, by (92.5), (92.3), (92.4), (92.6)
I V ~ ( X t””,) - P’~(X’,t ’ ) I
< f: W(U(u)(l
-
P)U-l(t’)x’)du
jml: + “(u)(l
tL*(DY)
/I(U)(Z -
P)U-’(t”)(x”- x’)
1”
-
P)( U-’(t”)-
t”
< DrL*(nr)(t”- t ‘ ) + v-’DL*(Dr)(lI
X” -
< L ~ ( Y ) ( V Y ( t~‘ ‘) ’+ 1) -
where L,(r) (92.7)
I
=
X” - X’
u-yt‘))x’ II du x’
/I + 11 X’ (1 j f ,I1 A(s)11 ds)
I::
11 4-Y
/ / A($)11 ds),
v-’DL*(Dr); similarly,
V~(X”,t ” ) - V~(X’,t ’ ) I
< L,(Y)(v’r(t”
-
t’)
+ 11 x” - N’ 11 +
If,11 A(s)I( ds) , 1”
Y
+
where L,(r) = ( 1 v’-’)DL*(Dr). It follows from (92.6) and (92.7) that V , , V , , hence also V , satisfy (C) and (L’); they are therefore Ljapunov functions and are uniformly small. We next prove that V , , V , , hence also V , satisfy (C’). Let E, Y > 0 and the compact interval J‘ C R, be given, and choose 6 > 0 so small that rL,(r)(vG ~j:’”ll A(s)jl ds) E for any finite set of disjoint intervals (ti, ti Si)C J’, i = 1, ..., n, with 1 :ai 6. Consider any such set of intervals, and any xi E Y Z ( X ) ,i = 1, ..., n. By (92.6),
+ cy +
I
Vn(X1
1
tt
<
+ 8,) - J’n(.v,
>
t,) I
< yLn(y) ( v
<
8, I
+ f jtZi6’ II A(s)II ds) S 1
E.
‘t
T h e proof for V , , based on (92.7), is entirely similar. 3. If x is any given solution of (90.l), set y = U(Z - P)x(O) = U(I - P)U-’x, z = UPx(0) = UPU-lx; we find x = y + z , and y ( u ) = U(u)(I - P)U-’(t)x(t),z(u) = U(u)PU-’(t)x(t)for any u , t E R, . Thus
vn+(f)=
S
IfIJt)
W 4 0 ) )4- W(z(u))du,
(92.8) =
m
W ( ~ ( t 4du, ))
c1
* n
Ch. 9. LJAPUNOV'S METHOD
326
so that V , , V,, are not only primitives, but continuously differentiable functions, and
Po&) = - W ( y ( t ) )< 0,
(92.9)
Vl&)
= W(z(t))2 0.
Now V&x(t),t) = vOz(t) for all t E R, \ E,, , by 91.C, and similarly V' and Qz, where V i , V ; , V' are any total derivatives for V ; and of V , , V , , V , respectively. Since these Ljapunov functions satisfy (C') and Eo is a null set, (92.9), (92.10) show that Vk is almost negative semidefinite, V ; is almost positive semidefinite, and V ' is almost negative definite (with the comparison function c(r) = b * ( i r ) ) .
vlz,
4. If A E M(X),set E = exp(1 A by (92.8) and (31.7),
1
t+1
v0.m
2
1
W ( Y ( 4 )fiu 2
1;'
IM).
Let x be any solution of (90.1);
b*(llY(U) 11) fiu
z b*(E-'
I1Y ( t )II) for t >, 0 for O < t < l
du 3
J
6*(11 z(u) 11) du 3 b*(E-' /I z ( t )11)
for t
2 1.
1-1
Therefore
+
V,(x, t ) 2 b(ll x /I), where the comparison function b is Thus Vo(x,t) given by b(r) = b*(QE-'r), and V , V , is positive definite. &
+
Remark 1 . If A satisfies further regularity conditions, an appropriate choice of Wimplies that V,, , V , , Vsatisfy stronger regularity conditions; for certain such conditions, the existence of a sufficiently regular W depends on the nature of X. See Massera and Schaffer [3], Theorem 3.2. mco
Remark 2. The assumption that X,,is complemented cannot be avoided in the proof by the use of splittings, unless the existence of some (X, , .)splitting satisfying a Lipschitz condition at the very least were assumed; no natural conditions, short of complementedness itself, for the existence of such a splitting seem to be known.
93. ORDINARY DICHOTOMIES
327
93, Ordinary dichotomies In contrast to the case of exponential dichotomies and to the classical uniform-stability theorem in Ljapunov's method, we require a pair of Ljapunov functions to characterize ordinary dichotomies, instead of a single one; but no restrictive assumption is made on A. 93.A. THEOREM. Assume that there exist two nonnegative Ljapunov functions V,, , V , that are homogeneous of the same degree, such that Vo V , is uniformly small and positive definite, and such that some total derivative of V , is almost negative semidejinite and some total derivative of V , is almost positive semidefinite. Then X,, is a subspace, and there exists N > 0 such that every solution y of (90.1) with y(0) E X,, satisjies (Di). Zf V , satisfies (L') and X,,has finite codimension, then X, is also a subspace, and every subspace Y , X,,C Y C X,, induces a dichotomy for A.
+
Proof. 1. If K >, 1 is the common degree of homogeneity, the assumptions imply the existence of numbers a, b > 0 such that hK11 x IjK
< Vo(x,t ) + V , ( x , t ) < a" 11 x I/*
for all x
E
X,
tE
R,.
T h e assumptions on the total derivatives of V , , Vl imply, by Theorem 91.D, that VoJis nonincreasing and V,, is nondecreasing for any solution x of (90.1).
2. A fundamental role is played by the set Y oC X of all values y(0) of the solutions y of (90.1) for which V,, = 0; on account of the homogeneity, Yo is a symmetric cone. If y is such a solution and t to 0, we have b"JJ y(t)l\" V,,(t) Vo,(t,) a*\\y(to)JJK, whencey satisfies (Di) with N = ab-'; in particular, YoC X , . If y o is a solution of (90.1) with yo(0)E X,,, then 0 Vly, ,< aKllyo IIK and the fact that Vluo is nondecreasing imply that V?,,,= 0, i.e., yo(0)E Y o .Thus X,, C Y o ,and indeed every solution starting from X,, satisfies (Di). On account of Theorem 33.C, X,,is closed.
<
<
> >
<
<
3. Assume from now on that V , satisfies (L') and set L = L(1). Let 2 be any finite-dimensional subspace with Yo n 2 = (0). Since V , is homogeneous and V,, is nondecreasing with limt+mVl,(t) > 0 for every solution z of (90.1) with z(0)E 2 \ {0}, the usual compactness argument yields the existence of T = T ( 2 ) 3 0 such that V l z ( T )> 0 for all such z. Since V , is homogeneous and Vozis nonincreasing, use of compactness yields the existence of a number p = p(Z), 0 < p < 1,
Ch. 9. LJAPUNOV'S METHOD
328
such that V,,(t) >, p"(1 - p")-lV,,(t) for all t >, T. Since V,, is nondecreasing, a" I1 4
+
t ) II" >/ VI1Z(t) b I'lZ(t0) >, p"6" 11 z(to)11"
3 P Y V O Z ( t 0 ) VlZ(t0)) for all t > to 2 T .
< < <
Use of (31.7) for 0 to t T and combination with the preceding inequality shows that z satisfies (Dii) with
(I
T(Z)
N'
= " ( Z ) = ab-'p-'(Z)
exp
11 A(u) 11 du) .
0
Let y, z be nontrivial solutions of (90.1) with y(0) E Y o , z(0) E 2. For any given to >, T, set y1 = 11 y(to)ll-ly, z1= 11 ~ ( t , ) I \ - ~ Then z. P"6"
< P"(Voz,(to) +
< L II %(to) - Yl(t0) I1
Using (31.10) for 0
<
~IZ,(~ON
VlZl(t0)
= VlLJtO) - V1rl(to)
= J W Y @ o ) ,4 t O ) l .
< to < T, we conclude that y , z satisfy (Diii) with
yo = y0(z)= P ~ " ( z ) Lexp(-2 -~
~ : ~ ) l l ~ ( u ) ldu). l
4. We finally assume from now on that the subspace X,, has codimension m < co ; hence X, is also a subspace. In order to prove that X,, or X, or an intermediate subspace induces a dichotomy, we have a choice, similar to that in Theorem 41 .F, between a long and messy proof (actually, several long and messy proofs) utilizing only the geometry of the solutions of (90.l), with copious applications of the Three-Angles Lemma to establish angular separation; and a short proof that uses the connection between dichotomies and the admissibility of (Ll, L") for A, and therefore introduces the extraneous properties of the nonhomogeneous equation. An instance of a proof of the former type is given in Massera and Schaffer [3], Theorems 4.2 and 4.3, which show that X , induces a dichotomy; the complete conclusion follows by Theorems 41.F and 41 .E. Here we give, for the sake of variety and simplicity, a proof of the second type. Since X, is a subspace, it is sufficient to prove that (L',L") is admissible for A, on account of Theorems 63.D, 41.F, 41.E. 5. We claim that there exists a finite-dimensional subspace 2, of X such that Yon 2, = (0) and Yo 2, = X.T h e class of all subspaces 2 with Yon 2 = (0) is not empty, since it contains (0);since X,, C Y o , any subspace 2 in the class has dim 2 m ; there exists therefore one, say 2, , of maximum dimension. If we had xo E X \ ( Y o Z,), the subspace 2, Rx, spanned by 2, and x, would satisfy
+
<
+
+
yon (zo RX,) = p j
+
93. ORDINARY DICHOTOMIES
329
(since Y o is a symmetric cone) and would have a greater dimension than 2, , which is absurd; therefore Y o 2, = X, and our claim is established. Parts 2 and 3 of the proof show that “ Y owith the complement 2, , induces a dichotomy for A” with the given N , N’ = N’(Zo),yo = yo(Zo), except that of course Yoneed not be a linear manifold at all! (If it were, our worries would be over.) However, it is tempting to apply the proof of Theorem 63.C to show that (Ll,L”) is admissible. This will indeed be done-and it will not matter that Y o is a cone, or that 2, is a linear manifold, but only that Yo 2, = X and that (Di), (Dii), (Diii) hold. T h e technical difficulty is that, although for each X E X we have x =y z, y E Y o, z E 2, , there is in general no way of choosing y, z (a “splitting”) so that x-+y and x - + z are continuous; we shall overcome this difficulty by means of a trick.
+
+
+
6. Let wi , i = 1, ..., n be a finite family of solutions of (90.1). There exist solutions yi , zi of (90.1) with yi(0)E Y o , zi(0)E 2, , such that yi zi = w i . For any i and any u E R , we have IIyi(u)ll, (1 zi(u)ll 2y;’ll wi(u)II, by (Diii) and 11 .A if y i , zi # 0, trivially otherwise (since yo 2). Using (Di), (Dii), we have, for every i,
+ <
(93.1)
<
I/y i ( t )/I < 11 11 w,(u) 11 1) z i ( t )1) < D 1) wi(u) 11
for all
u 20
t
for all u 2 t
>0,
where D = 2y;’ max{N,N’}. Let Ji C R , , i = 1, ..., n, be nonoverlapping compact intervals. We define x by x ( t ) = C: (yi(t) xf,(u)du - zi(t)J,“xf,(u)du). T h u s x is a solution of (90.2) with f = C: x f , w i . Further (93.1) implies
Ji
<
and hence x E L“(X), I x I Dlf Il . T o complete the proof that (L’,L”) is admissible for A it remains to show, on account of 51.B, that the linear manifold of “solution-step functions” f = xJiwi (the linearity is obvious) is dense in L1(X) or, equivalently, in koL1(X) (since L’is lean). But let g E koL1(X) and E > 0 be given, and set s = s(g). There exists a “constant-step function” h =1 :xf,h<, /i c [o,$3 nonoverlapping compact intervals, hi E X, i = 1, ..., n, such that I U--lg- h Il c exp(11 A(u)ll du) (Hille and
c:
<
Ji
Ch. 9. LJAPUNOV'S METHOD
330
Phillips [I], p. 86). We then set wi = U h , , solutions o f (90.1), and f = Uh = C"1 x J i w i ,and we use (31.9) to find
Ig a l
- .f 11 =
I u(u-lg - h ) l1 < I u-lg - h l1exp
(1'II
~ ( u II )du)
<
6.
9,
Remark. Massera and Schaffer [3], Theorems 4.1,4.2, and Examples4. I , 4.2, 4.3 (in separable Hilbert space) show quite precisely how much may be salvaged if the codimension of X , is not finite; the existence of a subspace inducing a dichotomy for A is not included. We now turn to the corresponding converse theorem. T h e converse theorems for exponential and ordinary dichotomies resemble each other in statement and proof much more closely than do the direct theorems. 93.B. THEOREM. Assume that the complemented subspace Y induces a dichotomy for A. Then there exist nonnegative Ljapunov functions V,, , V , satisfying (C') and ( L ' ) and hence uniformly small, and homogeneous of the same prescribed degree K 2 1, such that V , V , is positive definite and every total derivative of V,, , V , is almost negative semidefinite, almost positive semidejinite, respectively.
+
Proof. 1. We apply condition 41.B,(c) for a dichotomy to Y and some projection P along Y. We rewrite (Dii), (D:ii) with t,, , t replaced by t , u, respectively: (93.2)
I( U(u)(l - P)U-'(t)II 11 U(u)PU-'(t)1)
< r)
for all u >, t >, 0
for all t 3 u 2 0.
As in the proof of Theorem 92.B we obtain
11 u ( ~ )-( IP)(U-'(~'')-
1 11 11 I/ < D 1 A(s) 11 <
~ r - ~ t ' 11) ) D
(93.3)
t"
~ ( s )
ds for u 2 t" 3 t' 2 o
I'
t"
(1 U(u)P(II-l(t") - U-'(t'))
I1
ds for t" >, t ' >, u >, 0 ;
1'
a similar argument yields (93.4)
I\ (U(t")- U(u))(l- P ) U - l ( t ' )Il,II (U(u) - U(t'))PU"t'') I1
< I) I,, 11 ~ ( s )11 ds 1'
for
t" >, u >, t' 2 0.
2. Let W be any fixed gauge function that is homogeneous of K >, I (e.g., W ( x ) = )I x 1 '); the Lipschitz constant and the
degree
93. ORDINARY DICHOTOMIES
33 1
comparison function for the positive-definiteness of W may be taken as > 0. We define Vo, V , as
L * ( Y )= L*rK-l,b*(r) = ( b * ~ )with ~ , L*, b* follows, for each _x E XI t E R,:
vo/o(x, t ) = sup W(U(u)(Z- P)U-'(t)x) u>t
(93.5) V1(x,t )
sup
=
W(U(u)PU-'(t)x).
o
By (93.2) these functions are nonnegative and homogeneous of degree K , and satisfy V,,(O, t) = V,(O, t) = 0 for all t E R, .. For given Y > 0 and any x', x" E r Z ( X ) , t" 2 t' 2 0, we have, by (93.9, (93.2), (93.3), (93.4),
and similarly (with the same bound) for V , . I t follows as in the proof of Theorem 92.B that V , , V , satisfy (C') and (L') and are therefore uniformly small Ljapunov functions. Since U ( t ) ( l - P)U-'(t)x + U(t)PU-'(t)x = x for all x, t, we must have max(l1 U(t)(Z - P)U-'(t)xJI,IIU(t)PU-'(t)x 11) 2 &I x 11. Therefore (93.5) implies Vo(x,t )
+ V1(x,t ) 2 wcU(t)(r
-
P)U-'(2)x)
+ W(U(t)PU-'(t)x) 2 (P*II x IIYI
so that V ,
+ V , is positive definite.
3. If x is any solution of (90.l), we set y = U(Z - P)x(O), z = UPx(0); arguing as in the proof of Theorem 92.B,
332
Ch. 9. LJAPUNOV’S METHOD
Thus, for every solution x, V,, is nonincreasing, and each derivate number DV,,(t) is nonpositive for every t E R, . By 91.C, every total derivative VA(x, t ) of V , is nonpositive for all x E X and all t E R, \ E,,; since E, is a null set and V,,satisfies (C’), we have proved that VA is almost negative semidefinite. T h e proof that every total derivative of V1 is almost positive semidefinite is entirely similar.
Remark. Under supplementary assumptions on A and on the existence of certain kinds of gauge function W , V , and V , satisfy stronger regularity conditions. For one of these see Massera and Schaffer [3], Theorem 4.4. 94. Notes to Chapter 9 The paper of Maizel’ [I] may be considered as the first attempt to characterize exponential dichotomies by means of an extension of Ljapunov’s method. Results of Krasovskii [l] extending Ljapunov’s method to the case of conditional asymptotic stability in the nonlinear case, may also be adapted to this end, by using Theorem 45.A to translate his “uniformly noncritical behavior” into the language of exponential dichotomies (cf. Notes to Chapter 4). We have already pointed out in the Introduction and in Section 91 that Ljapunov’s method is essentially “real”. It may be asked whether the method permits a characterization of double dichotomies, ordinary or exponential, when the range of the independent variable is J = R . The following extremely simple examples show that this is not the case in any natural way.
94.A. EXAMPLE (Example 83.E continued). In Example 83.E, X = R , A(t) = tanh t , so that A E C, and the dihedron (R,R) induces a double exponential dichotomy for A. Let V be a Ljapunov function that is uniformly small and has some almost semidefinite total derivative; for any solution x of (90.1), V , is monotone by Theorem 91.D; but limltl+mI x ( t ) l = 0, hence limltl+m Vz(t)= 0, and this implies V , = 0. Thus the only such Ljapunov function is V = 0. $, 94.B. EXAMPLE (Example 83.E continued). For the adjoint equation to the equation of the preceding example, A(t)= -tanh t , so that A E C, and no dihedron at all induces even an ordinary dichotomy. Yet the function V defined by V(x,t ) = -x2 tanh t is a Ljapunov function satisfying (C’) and (L’); it is uniformly small and homogeneous of degree 2; these properties are immediately verified. Its unique total derivative V’ is negative definite: V‘(x, t ) = -x2(1
+ tdnh* t ) .
9,
CHAPTER 10
Equations with almost periodic A 100. Introduction Summary o f the chapter
This chapter deals with a topic more special than those discussed up to now; indeed, we intend to study Eqs. (30.1) and (30.2),i.e., (100.1)
X+Ax=O
(100.2)
X+Ax=f
when A is a (uniformly) almost periodic function of t ; often, it will also be assumed that f is almost periodic, and we shall of course be particularly interested in almost periodic solutions of (lOO.l), (100.2), but they will not be our only object of study. T h e problems considered in this chapter are only to a small extent a generalization of those for periodic equations. The latter will be considered in the next chapter. We do not consider any but “uniformly”, or Bohr, almost periodic functions; certain extensions to Stepanoff almost periodic functions (related to M as the uniformly almost periodic ones are to C)are possible, and perhaps desirable in order to include all integrable periodic functions; the topic does not seem to warrant more than an invitation to the interested reader to carry out this straightforward generalization himself. Later in this introduction we establish the terminology and notation we intend to use concerning almost periodic functions, and state some of the auxiliary results we require. A somewhat vexing question concerns the choice of the appropriate range of the independent variable t . T h e natural setting for almost periodic functions is the whole real line R ; yet the “differential-equation-theoretic” aspect of our subject suggests R , 333
334
Ch. 10. EQUATIONS WITH
ALMOST PERIODIC
A
as more adequate whenever functions occur that are not assumed to be almost periodic. We compromise the question by using R as the basic range of t , and letting this choice dictate the proofs to be used, but including those results for the equations on R+ that seem of sufficient interest (cf. Section 104). Results for R- are omitted, and can be obtained by symmetrization. Thus, the range of t is R throughout the chapter, unless otherwise noted, and the notation introduced in the subsections on “Cutting and splicing” in Chapter 2 (pp. 54,74), and used in the whole Chapter 8 in order to relate this range to R- , R , is in force (see Introduction of Chapter 8), and is supplemented below for our present purposes. This introduction ends with general remarks on Eqs. (100.1) and (100.2) with almost periodic A. The bulk of the chapter is motivated to a large extent by the fact that, if X is finite-dimensional, the admissibility, for almost periodic A, of a pair consisting of spaces of almost periodic functions is equivalent to the existence of a double exponential dichotomy for A, induced by a disjoint dihedron (Theorem 103.A). Various approaches to this result are made in a more general setting. Section 101 deals, in particular, with the assumption that (100.1) has no nontrivial almost periodic solution, or at least none of some special type; this assumption is of course verified in the finite-dimensional situation described above. We give a weak sufficient condition for this assumption, and derive some consequences concerning admissibility. Cases of admissibility of pairs of spaces of almost periodic functions in which this assumption fails, if they should indeed exist, lack interest; for this reason, as well as for that of triviality, we omit the straightforward specialization to the almost periodic case of those results in Sections 5 I and 71 that concern general .N-pairs. Section 102 is devoted to double exponential dichotomies and their relation to the admissibility of certain pairs. T h e important Theorem 102.A relates such double exponential dichotomies for A to exponential dichotomies for the restriction A+ to R , . Section 103 deals with the cases of reflexive X, for which there are important simplifications, and finite-dimensional X. Section 104 (Notes) includes a discussion of the results of Favard for almost periodic equations.
Spaces o f almost periodic functions We assume known the essential results of the theory of (uniformly) almost periodic functions with values in a Banach space; this theory
100. INTRODUCTION
335
was developed by Bochner [2] and closely parallels the case of realvalued functions. For a recent account we refer to Corduneanu [3], pp. 131-144. We omit the adverb "uniformly" in the sequel. We consider the modules (additive subgroups) of real numbers, henceforth simply called MODULES and denoted in general by German letters. They constitute a complete lattice under inclusion; the lattice meet and join operations are intersection and addition, respectively; the least element is {0}, and the greatest is R itself. Except for (0) and the modules with a single generator, viz., those of the form {nw : n integral} for some real w # 0, all modules are dense in R; we call them DENSE modules. Every almost periodic function f has a Fourier series C a,exp(iAnt); we denote by m(f) the MODULE OF^, i.e., the (at most count"a1e) module generated by the spectrum {An} off. We require a relation, due to Favard, between modules and translation numbers: 100.A. Iff,g are almost periodic functions with values in any Banach spaces X, Y, then B(f)C %R(g) if and only if for every real r > 0 there exists a real 6 > 0 such that every &translation number of g is an r-translation number off. Proof. See Favard [l], pp. 4243. This proof applies, unchanged, to functions with values in Banach spaces. & If X is a Banach space and a is a module, we denote by A,(X) the space of all almost periodic functionsf : R --t X with %(j) C a, equipped with the supremum norm. Again there will be no confusion between this notation and the corresponding ones for 9-spaces and g v - s p a c e s . A,(X) is isometrically embedded as a subspace in C(X), hence in L"(X). Ab(X) if and only if Thus A,(X) E b.N(X), and obviously A,(X) a C b. We agree to drop the subscript if the module is R itself: thus A(X) is the space of all almost periodic functions on R with values in X. As usual, we also omit the argument X whenever X = R. If a is, in turn, {0},{nu : n integral}, {rw : r rational} for some real w # 0, A,(X) is, respectively, the space of all constant functions, of all continuous periodic functions with period 2r/I w 1, of all limit-periodic functions with period 2.rr/l w I (cf. Bohr [I]). T h e frequent restriction, in this chapter, to dense modules emphasizes the difference between the periodic and the properly almost periodic case, already hinted at above.
<
100.B. Zf a is a dense module, lc(A,(X))
<
= L"(X).
L"(X) and IcC(X) = L"(X) (by 24.I), it is Proof. Since A,(X) enough to show that C ( X ) lc(A,(X)). Let then f~ C(C(X)) and a
<
336
Ch. 10. EQUATIONS WITH
ALMOST PERIODIC
A
compact interval J' C R be given; set I = p(J'). Since a is dense, there exists w E a, 0 < w < 2x11. There obviously exists a continuous functiong : R --t X , periodic with period 2x/w > I, that coincides with f on J' and satisfies l g I IxJ.fI < I f I < 1. Now m(g) C {nu : n integral} C a , so that g E Z(Aa(X)).Since J' was arbitrary, it follows that f E cl, Z(Aa(X)). 9,
<
Remark. The assumption that a is dense is essential: the space of constants is itself locally closed; and local closure preserves any period common to all functions of a space. 100.C. If a, b, c are modules such that a, b C c , and Banach spaces over the same scalar field, then Aa([Y;4) * Ab([X; YI)
if X , Y , 2 are
M [ X ; 21).
Proof. From the general theory, if U :R + [ Y ;21and V : R + [X; Y] are almost periodic, so is U V , and m( U V ) C m(U ) m(V ) . &
+
Remark. Important special cases of 100.C are obtained by choosing each of X , Y, 2 to be either a Banach space X or its scalar field F; cf. Remark to 21.H. We require some information about the relationship between almost periodic functions and their restrictions to R, . We of course omit the corresponding results for R- , which follow by symmetrization. One tool is the following trivial lemma: 100.D.
Iff. A ( X ) , then I f I = lim SUP^.++^ 11 f(t)ll.
We denote by A,+(X) the class cf+ :f E A,(X)}, with the supremum norm; thus A,+(X)is isometrically embedded in C + ( X ) ,hence in L y ( X ) . Observe that A , ( X ) 4 9'3'(X), so that we cannot use the definitions and results in Section 24 (p. 78). Actually, however, the situation here is much simpler, since the mapping f +f + : A , ( X ) + A,+(X) is not merely an epimorphism: 100.E. For every module a, A , ( X ) is ,congruent to A,+(X) under the mapping f +f + ; therefore A,+(X) E bX+(X).
Proof. The mapping is linear and surjective; it is isometric by 100.D. t 100 F. Assume that f , g E A ( X ) . Thenf + is a primitive and cf+)' = g+, ;f and only ;ff is a primitive and f = g .
100. INTRODUCTION
337
Proof. T h e “if” part is trivial. For every fixed T > 0 the function h defined by h(t) = f(t) - f(t - T) - Jtt--r g ( u ) du is almost periodic, since f and g are. Iff+ is a primitive and (f+)‘= g, , then h(t) = 0 for all t 3 T ; hut then 1OO.D implies h = 0. Hence f ( - ~ ) = f ( 0 ) J i T g ( u )du, and this holds for all T > 0, so that f- is a primitive, with cf-)’ = g_ . T h e “only if” part of the statement follows. 9,
+
Almost periodic equations and solutions. Preliminary facts
We state a lemma on the restrictions of almost periodic solutions to R, , and on the modules of such solutions. 100.G. . Let A E A(X), f E A(X) be given. If x is a solution of (100.2), then x+ E A,.(X)if and only if x E A(X).I n that case 9R(f) C m(A)+ ’.)3t(x).
Proof.
The “if” part is trivial. T o prove the “only if” part, let y+ = x, . Then apply 100.F toy, f - A y instead off, g , to show that y is a solution of (100.2). But x+ = y+ , and the uniqueness of solutions implies x = y E A(X). T h e relation between the modules follows from the trivial remark that, since x, f E A(X),
y
E A(X) be such that
W ( f ) = ‘rn(X).
9,
A matter of notation: we shall in general replace the subscript
A,(X) by the subscript *a-hence A(X) by *-as, e.g., in Xo,,(A) = XoAflcx,(A On ) . account of 100.G and 100.D, we are able to do without the corresponding notations, such as X+o+,(A)= X+oa c x , ( A + )for , the such as equations restricted to R + . We shall also use abbr:$ations (A,, Ab)for the pair (A,(X), Ab(X)),etc. ’
It is known (and we shall come to this point again, see Theorems 113.A, 113.B) that if A is periodic and X o is closed, then all nontrivial bounded solutions of (100. I ) are bounded away from 0, and indeed, if X is finitedimensional, are almost periodic. If A is almost periodic and %t(A)is dense, this is not necessarily so, even for one-dimensional real X,as regards the first point, and for two-dimensional real, or one-dimensional complex, X,as regards the second; the following examples are due to Bohr [2], to whom we refer for details. 100.H. EXAMPLE.Let a be any given dense module. There exists in a, such that C“ nu, < m. then a sequence (w,) of positive numbers 1 a3 1 Set rp(t) = Cy nu, sin u,t, #(t)= J y(u) du = C n( I - cos ~ ~ ~Then 1 ) . .o 1 E A,, # 0 is a primitive and Z/J = y , and #(7r/wn)>, 2n, so that I/I is unbounded.
>
Ch. 10. EQUATIONS WITH
338
ALMOST PERIODIC
With X = R, set A = v, so that U = exp(-+) bounded away from 0, and not almost periodic. ,$,
A
is bounded, not
100.1. EXAMPLE.With q,9 as in Example 100.H, let X be the complex field, and set A = itp, so that U = exp( -$); every nontrivial solution of (100.1) is bounded and bounded away from 0, but none is almost periodic. Such an example is impossible for X = R. ,$, 00
cow
In general, X,, need not be closed: Massera and Schaffer [4], Example 2.1 illustrates this for a constant A in separable Hilbert space; this example is easily adapted to show that, for any dense module a, X,,. need not be closed. The denseness of a is essential here (see Theorem 112.B). Even if closed, X,, need not be complemented: Massera and Schaffer [4], Example 2.2, has constant A in X = P, p # 2 (this space is reflexive, indeed uniformly convex and separable, if p # 1, a); all bounded solutions are constant, and X, is an uncomplemented subspace; hence X,,, = X , has this property for any module a whatever.
101. The condition X,,,
= (0)
In the introduction we hinted at the significance, for the theory of Eqs. (100.1) and (100.2), of assuming that Xo*a= (0) for a given module a. We first prove a lemma of a somewhat more general scope, which is applicable to the situation just mentioned, as well as to another problem, to be dealt with in the next section. 101.A. Assume that A E A(B), and that D is a subspace of L"(X) such that i f f ~ D then A ~ E D and TJED for all T E R (e.g., D = L"(X), or D = A,(X) with %l(A)C a). Assume further that X,, = (0) and that (D, D ) is (regularly) admissible for A . I f f E D is almost periodic, and x is the unique D-solution of (100.2), then x is also almost periodic, and !IR(x) !IR(A)= %."(f) !lll(A).If a is a module such that %](A)C a, then (D A A,(X), D A A,(X)) is regularly admissible.
+
+
Proof. Set K
=
K',.,(A). For the given f and 6 = K-'( 1
+ K If
E
> 0, set
I)-'€.
Let T be any &translation number common to f and A . We have i Ax = f , (T,x)' (T,A)(T,x)= TTf, so that y = x - T,x is the unique D-solution of j Ay = g , where
+
g
+
=
+
[f- T J ) - ( A - [TTA))(T,x)= f
- TTf- A ( T 9 )
+ T,(Ax) E D .
101. THECONDITION
X,,, = (0}
339
Therefore Ix-T,.l
= I -y l < K l g l < ~ q +~I T F I I ) < K q l + K I f l ) = c .
Thus 7 is an €-translation number of x . Therefore x is almost periodic, and 100.A, 100.G yield 'm(x)C m ( f ) m(A)C m ( x ) m(A).Addition of !l.Jt(A)yields the required equality. T h e last statement is an immediate corollary. ,j,
+
+
We now show that, if a is a dense module, the condition X,,, = {0} is actually a consequence of a very mild admissibility assumption on A or on A , .
I01 .B. THEOREM. Assume that a is a dense module. If D is a subspace of L"(X) containing A,(X) and (A,, D ) is regularly admissible for A ; or if D+ is a subspace of Ly(X) containing A,+(X) and (A,+(X), D+) is regularly admissible for A+;then X,,,= (0). Proof. We carry out the proof in the former alternative; it is the same in the latter, except for an added appeal to l00.G. By Theorem 51.D (Remark) there exists K' > 0 such that if j~ A,(X) and the solution x of (100.2) with x ( 0 ) = 0 is in D, then I x I K'I f I. Let y be any nontrivial A,-solution of (100.1). Now y is continuous and, by 100.D, lim sup,,, 11 y(t)ll = Iy 1; since a is dense, there exists w E a such that 2K'w < 1 and IIy(rr/2w)ll 41y I. Consider the function x defined by x ( t ) = y ( t ) sin w t ; by 100.C, x E A,(X) C D ; and x ( 0 ) = 0. But direct computation shows that x is a solution of (100.2) for f ( t ) = w y ( t ) cos w t , and again j~ A,(X). But then
<
a contradiction. Hence the assumption that a nontrivial A, solution of (100.1) exists is absurd. & An immediate corollary is: 101.C. Assume that a is a dense module, b a module, b C a. If (A, , L") or (A,, Ab)is regularly admissible for A , or ( A a + , LT) or (A,+, Ab+) is regularly admissible for A + , then X,,,= (0).
Remark 1. T h e role of D in Theorem 101.B is peculiar: indeed, the only D-solution that appears in the proof is actually in A,(X), and the D-norm equals the Lm-norm. We do not choose D = A,(X), since the pair (A,, A,) might not be admissible; nor D = L"(X), since
340
Ch. 10. EQUATIONS WITH
ALMOST PERIODIC
A
(AP,L") might not be regularly admissible. With regard to the last point, we have: 00
Que~y. May the assumption on D or D+ in Theorem 101.B be replaced by the mere admissibility of (A, , L") or (A,+,Ly),respectively, perhaps with the added assumption that X,,, is closed I
Remark 2. The assumption that A is almost periodic was neither made nor used in Theorem 101.B and 101.C.
00
The question remains, whether the assumption that a be a dense module may be dropped in Theorem 101.B and 101.C. We give a partially affirmative answer for finite-dimensional X ; for separable Hilbert space, on the other hand, the answer is negative, even for constant A and a = {nu}, as Example 1 12.D shows. 101.D. Assume that a is a given module, and that either a is dense or X isjinite-dimensional. If (A, ,A,) is regularly admissible for A , or (A,+ ,A,+) is regularly admissible for A + , then Xo*, = (0).
Prooj. For dense a the statement follows from 101.C with b = a. Assume that a = {nw}, w > 0, and that (A,, A,) is admissible. Suppose there exists a nontrivial A,-solution yo (i.e., periodic with period 27r/w) of (lOO.l), and construct the sequence (y,) in A, by letting Y , + ~be an A,-solution of (100.2) with f = -y, , n = 0, 1, .... Define the sequence of continuous functions (xn) by xn(t) = C: ( t i / j ! y,+r(t), ) n = 0, 1, ... . A direct computation shows that x, is a solution of (100.1) for every n. But for integral m, x,(2nm/w) = mn{(2n/w)nyo(0) o(l)} as m -+00, so that the set {x,} is linearly independent over the scalars; hence {x,(O)} is an infinite linearly independent set in X, and X cannot be finite-dimensional. If a = {0}, the same proof applies. &
+
Remark. If a = {0}, the assumption of 101.D implies that A is constant; if a = {nu} and X is finite-dimensional, another proof can be given using the techniques of Section I12 (cf. Theorem 112.B); they can be used to show that actually A must be periodic with period 2?r/w; if this is assumed, the conclusion is indeed well known.
We now combine 101.A and 101.D, but we omit the periodic case, which is dealt with in greater generality in the next chapter (Section 112). 101.E. Assume that a is a dense module and that A E A,(X). If (A, , A,) is regularly admissible for A, then every f E A,(X) satisfies, together with the unique A,-solution x of (100.2), W(x) m(A)= W(f ) I111(A).
+
+
102. EXPONENTIAL DICHOTOMIES
34 1
Proof. By 101.D, Xo*, = (03. T h e conclusion follows from 101.A, with D = A,(X). & 101.F. Assume that a is a dense module, and that A, E A(X). Then Ad,(A, , A,,) n ( A , A,,(X)) is open in A , A,,(@; and i f , for each A in that set, [ ( A )E (A,(X))- is the monomorphism that mapseach f E A,,(X) into the unique A,-solution of (100.2), then the mapping
+
A
+
+
( ( A ): Ad,(& , A") n (A"
+
&(If))
+
(A,(X))"
is continuous. Proof. T h e first part follows from Theorem 71.C, using 1OO.C; [ ( A ) is well defined, since Xo+,= (0) by 101.D, and obviously linear, injective, and bounded (by Theorem 51 .A); the conclusion follows from the Addendum to Theorem 71.C. & 101.G. Ad,(A, A) n A(X) is open in A(X); and i f , f o r every A in that set, [ ( A ) E (A(X))- is the monomorphism that maps each f E A ( X ) into the unique almost periodic solution of (100.2), then the mapping A -+ [ ( A ) : Ad,(A, A) n A(X) --+ (A(X))- is continuous.
+
are added Remark. 101.E, 101.F, 101.G remain true if subscripts everywhere, so as to obtain results for the equations on R,; to see this, it is sufficient to use 100.E and 1OO.G.
102, Exponential dichotomies We prove an important theorem to the effect that an almost periodic A possesses a double exponential dichotomy if and only if A+ possesses an exponential dichotomy, and indeed the closed dihedron inducing the former is disjoint.
102.A. THEOREM. Assume that A
E A(X).
The following statements
are equivalent: (a) A possesses a double exponential dichotomy, and the closed dihedron ( X - o , X+o)inducing it is disjoint; (b) A possesses a double exponential dichotomy; (c) A+ possesses an exponential dichotomy. Proof. 1. T h e implications (a) --+ (b) + (c) are trivial; it remains t o prove (c) --+ (a), so that we assume that (c) holds.
Ch. 10. EQUATIONS WITH ALMOST PERIODIC A
342
On account of Theorem 72.D applied to Ly, there exists E > 0 such that if B+ E Ly(X), I B+ - A+ I < E, then B+ possesses an exponential dichotomy, (Ly, Ly) is regularly admissible for B+, and S+(B+)< 2S+(A+), Kt(B+) 2K+(A+), say, where K+ = K+, . Since A is almost periodic, we may choose an increasing sequence (~(n))of +translation numbers of A such that ~ ( n 2 ) n, n = 1, 2, ... . Now 1(TT(,,A)+- A+ I = I T,,,,A - A I r ; therefore (T,(,,A)+ possesses an exponential dichotomy, and S+((TT(,,A)+)< 2S+(A+), K+((T7(,,A)+)2K+(A+). By Calculation 65.T,the parameters v, N corresponding to A+ and to each (TT(,,A)+may, and shall, be taken to have a common value each, depending only on I A I, S+(A+),K+(A+) (for I (T,(n,A)+ I = I A I, by 100.D). 2. We consider in what follows the equations
<
<
<
(102.1)
6
+ (T,(,)A)X= 0
(102.2)
X
+ (TT(n)A)*x
= Tr(n1f
for each n. For any f E L(X), x is a solution of (102.2) if and only if T-,(,,x is a solution of (100.2); and similarly for (102.1) and (100.1). Let y be a bounded solution of (100.1); then (TTc,,y)+is a bounded solution of (102. I)+ . By (Ei) for ( TT(,,A)+,
II ~ ( 0II )= !I T r ( n ) A ~ ( nII) )< Ne-yT(nl II TT(n)y(O)II = Ne-YT(n) II Y ( - T ( ~ ) II)
< NecYnI y I. Since this holds for all n, we must have y(0) = 0, i.e., X,(A) = (0). 3. Our next aim is showing that (L", L") is admissible for A. L e t f E L"(X) be given. For each n, (T,(,,j)+ E Ly(X); therefore (102.2) has a solution, which we write T,(,,y, , with (TT(,)yn)+ E Ly(X), I(T~(,)Y,)+ I < ~K+((T,,,,,A)+)I(T~,,,~+ I G 4 ~ + ( ~ + )I. iobserve f that each yn is a solution of (100.2); therefore ( T,(,,yn)+ and ( T,(,,Y,+~)+ are bounded solutions of (102.2)+; and ( T,(JY,+~- y,,))+ is a bounded solution of (102.1)+ . Again by (Ei), llyn I I ( 0 )
-
~ n ( 0 II) == II T7(n)yn+,(~(n)) - T 7 ( n ) ~ d ~ (IIn ) )
.< N e - v T ( n ) II TT(n)yrt+l(O) - y'T(o)Yn(O) II
< Ne-'n(l T~(n+l)Yn+l(7(n < 8Ne-mK,(A.,) I f l .
+
-
dn>)11 + 11 T~(nlYn(o) 11)
Therefore ( ~ ~ ( is0 a) Cauchy ) sequence; let y be the solution of (100.2) with y(0) = limn+" ~ ~ ( 0 ) .
103. REFLEXIVE AND
FINITE-DIMENSIONAL SPACES
343
Let t E R be given; there exists m such that ~ ( m ) --t. n >, m we have t 7 ( n ) 2 0, so that
For any
+
IlYn(t)/I
=
II TTcn)yn(t+ .(.)I
II < I ( L w ) y n ) +I d 4K+(A+)If I*
By (31.7) applied to the solutions y n - y of (lOO.l), we have y ( t ) = limn+" y , ( t ) , so that y(t)ll 4K+(A+)If I. Since t was arbitrary, y is bounded, and (L", L") is indeed admissible for A, withK,(A) <44K+(A+). 4. Since Xo(A)= (0) (part 2 of the proof), (L",L") is regularly admissible for A ; by Theorem 82.Q, the closed dihedron (X-,(A), X+o(A)),which is disjoint by (81 .I) induces a double exponential dichotomy for A, so that (a) is verified. &
<
An incidental corollary for equations on R+,already mentioned in previous chapters, follows: 102.B. If A + E A+(X) possesses an exponential dichotomy, then X+,(A+) is complemented. Proof. There exists A E A(X) such that A+ = A+. The assumption is then statement (c) of Theorem 102.A; this is equivalent to statement (a) of the same theorem, so that X+,(A+) = X+,(A) has the complement
X-,(A).
9t-
A more important consequence of Theorem 102.A concerns the admissibility of pairs of spaces of almost periodic functions.
Let a be agiven module, and assume that A E A,(X). 102.C. THEOREM. If A possesses a double exponential dichotomy, or if A+ possesses an exponential dichotomy, then (A,, A,) is regularly admissible for A, and (Aa+,A"+)is regularly admissible f o r A+ .
Proof. T h e alternative assumptions are equivalent, by Theorem 102.A, and imply, by the same theorem, that X,(A) = X,,,(A) = (0). By Theorem 82.R, or by the proof of Theorem 102.A, (L",L") is regularly admissible for A. T h e regular admissibility of (A,, A,) for A follows from IOl.A, with D = L"(X); the corresponding result for A , is then obtained using 1OO.E and 1OO.G. & 103. Reflexive and finite-dimensional spaces If X is reflexive, the various conditions we have been discussing in the preceding section fall into a simple pattern, described by the following
344
Ch. 10. EQUATIONS WITH
ALMOST PERIODIC
A
theorem. We leave the periodic case to be discussed in the next chapter (Remark 2 to Example 113.N). 103.A. THEOREM.Assume that X is rejexive and that a is a dense module. Let A E A(X) be given, and set b = a Y.JI(A).The following statements are equivalent, and imply X, = X,,,= XO*a= (0):
+
A possesses a double exponential dichotomy, and the closed dihedron
X+,)inducing it is disjoint; A possesses a double exponential dichotomy; A+possesses an exponential dichotomy; (L",L") is regularly admissible for A ; (Ly, Ly) is regularly admissible for A+; ( A b , Ab)is (regularly) admissible for A , and X, is closed; (Ab+ , Ab+)is (regularly) admissible for A+ , and X+,is closed; (A, , L") is regularly admissible for A ; ( A a + , Ly) is regularly admissible f o r A+ . Proof. (a), (b), (c) are equivalent by Theorem 102.A; (b) is equivalent to (d) by Theorems 82.Q and 82.R; (c) is equivalent to (e) by Theorem 64.C. T h e first five conditions are thus equivalent. They imply (f) and (8) by Theorem 102.C (the term "regularly" in these conditions may be retained or omitted, by 33.D), and these respectively, and trivially, imply (h), (i). That much is true without the assumption of reflexivity of X or of denseness of a; we now proceed to use these assumptions. Since 1OO.B states that lc(A,(X)) = L"(X), we may apply Theorem 51.F and thus show that (h) implies (d); but 100.B implies at once that also lc(A,+(X)) = Ly(X); an application of Theorem 51.F to A+ then shows that (i) implies (e). & If X is finite-dimensional, Theorem 103.A is simplified by the redundancy of the word "regularly" in statements (d)-(i), and of the conditions on X , , X+,in (f), (g), respectively; under the same assumption, Theorem 101.B and its corollaries 101.C, 101.D (for dense a) are contained in Theorem 103.A, since (h), and also (i), implies that X,= Xo*a= (0). For a special case in finite-dimensional X we mention a necessary and sufficient condition for the equivalent statements: 103.B. If X is jnite-dimensional and A E A(X) is represented, for some basis of X,by a triangular matrix of functions (in A(F)), then A possesses a double exponential dichotomy (and all the statements of
104. NOTESTO CHAPTER 10
345
Theorem 103.A hold) i f and only if each diagonal element has a nonzero mean value.
Proof. See Massera [3].
&
Closely connected with this criterion is the following remark: Theorem 103.A shows that, for finite-dimensional X , t h e admissibility of (A,, , A,,) (a a dense module) for almost periodic A implies (via t h e admissibility of (A,, L")), Xo*, = (0). I t is well known that in t h e periodic case ( A periodic with period 2 x / u , and a = { n u ) ) the converse also holds (cf. Theorem 112.B); for dense a and 91(A) this is false, however, even for X = R:
103.C. EXAMPLE (Example 100.H continued). I n Example IOO.H, X,,, = Xn* = (0). If (A,,, A,) were admissible, Theorem 103.A would imply X , = (0);b u t Xn= R . (And indeed, t h e mean value of tp is 0 by construction). & 104. Notes to Chapter 10
Equations on
R,
In the Introduction we explained our choice of R as the range of the independent variable as being the natural domain of definition of the almost periodic functions. The general point of view taken in this book raises the question, however, of the possibility of developing the theory for R , as the range of t , without using functions on R in an auxiliary capacity. It makes sense to ask this question for equations with almost periodic A , since almost periodic functions on R+-more precisely, elements of A+(X) for a Banach space X-can be characterized without having recourse to an extension to R : specifically, a function f E C + ( X )belongs to A+(X) if and only if there exists, for every > 0, a number I ( € ) > 0 such that every interval 1'C R , of length ~(1') = l(r) contains some T such that I f - T;f I c. It is then possible to prove directly, and without major changes in the proofs, all the results in this chapter that refer to equations on R,, with the following qualifications: we have not been able to prove 102.B without using Theorem 102.A; and Theorem 102.C, insofar as it concerns equations on R,, requires a different proof, since it is not true in general that X,, = {0}, and the analogue of 101.A cannot therefore be used. The required proof can be carried out using asymptotically almost periodic functions: see Schaffer [7], Theorem 3.4 and Corollary 3.4. Asymptotically almost periodic functions were introduced by Frkchet; see Frkchet [ I]; they were first applied to almost periodic differential equations by Reuter [ 11.
<
346
Ch. 10. EQUATIONS WITH
The theory
ALMOST PERIODIC
A
of Favard
All the results in this chapter that concern almost periodic solutions were “global”, in that Eq. (100.2) was considered for a whole class of “test functions” f . There are some important “individual” results, concerning a single f; it is then not sufficient in general, however, to examine a single A. For completeness’ sake we include a brief survey of these results; the basic one is due to Favard [ I ] (Third Theorem; the proof is identical with that of the apparently weaker Second Theorem); Favard [2], pp. 85-95 reproduces this theorem with its proof. For any A E A(a) the set Tr(A) = cl,( T,A : T E R } is a compact subset of A(X). Favard’s result is then:
Assume that X is finite-dimensional, and that A , f are 104.A. THEOREM. almost periodic. Assume further that, for every B E Tr(A), every nontrivial By = 0 is bounded away from 0. If (1 00.2) has a bounded bounded solution of y solution, thenit hasanalmostperiodicsolutionxwith‘%I(x)+“(A) =’m(f ) +Q(A).
+
co
This result can be generalized, without modifying Favard’s proof, to a space
X that is (isomorphic to) a uniformly convex space, but in general at the cost of weakening the conclusion to the existence of a solution x that is WEAKLY ALMOST PERIODIC in the sense of Amerio [I], i.e., such that (x, a * ) is almost
periodic for each a* E X ” . This generalization is carried out in detail in Amerio [2], Theorem 11, which actually refers to a slightly more general situation, and differs somewhat in the assumptions; but since the proof is a straightforward adaptation of Favard’s, we feel justified in omitting it. In the case in which X o ( B ) = (0) for all B E T r ( A ) (Favard’s First Theorem) the generalization may actually be carried out if X is merely reflexive. We summarize: 03
104.B. Assume that X is reflexive and that A , f are almost periodic. Assume further that either X o ( B ) = (0) for each B E Tr(A), or X is uniformly convex and, for every B E Tr(A), every nontrivial bounded solution of y By = 0 is bounded away from 0. If (100.2) has a bounded solution, then it has a weakly almost periodic solution.
+
co
Under what conditions is there actually an almost periodic solution? One answer, that appears to be a little narrow in its assumption, can be gleaned from Amerio [2], Theorem I11 (with Amerio [3]), which uses the fact that a weakly almost periodic function is almost periodic if and only if the closure of its range is compact:
co
104.C. Assume that X is unqormly convex, and that A ,f are almost periodic. Assume further that for every B E Tr(A) there exists a, > 0 such that mery
104. NOTESTO CHAPTER 10
347
+
solution of By = 0 satisfies 11 x(t)ll 3 uBll x(0)ll for all t E R. If (100.2) has a bounded solution, then it has an almost periodic solution.
Amerio [4] shows that, even for the equation with A = 0, which satisfies the assumptions of 104.C, the statement is false if X is the space of continuous real-valued functions on [0, I] with the C-norm, a separable, nonreflexive space; whether reflexivity without uniform convexity is sufficient, even in this special case, seems to be an open question. Remark. A final observation about the case in which A is periodic (it need not then be continuous, but we disregard this refinement here): the set of translatesof A is then closed, so that the assumptions made “for every B E Tr(A)” in Theorem 104.A, 104.B, 104.C need be made only for A itself. If &(A) is closed, the “bounded away” condition of Theorem 104.A and 104.B holds and need not be assumed (Theorem 113.A); and in finite-dimensional X (but not even in separable infinite-dimensional Hilbert space) every bounded solution of (100.1) is (weakly) almost periodic (Theorem I13.B, Example 113.G), so that Theorem 104.A becomes:
104.D. Assume that X is finite-dimensional and that A is continuous and periodic, and f is almost periodic. Every bounded solution of (100.2) is then almost periodic; and if any exist there is one, say x, with W(x) 91(A) = m ( f ) %R(A).
+
+
CHAPTER 11
Equations with periodic
A
110. Introduction Summary of the chapter
Continuing with our study of special topics, we now consider Eqs. (30.1) and (30.2), i.e., (110.1)
k+Ax=O
(110.2)
i+Ax=f
when A is periodic; we are of course particularly interested in the case in whichfis also periodic, as well as in the existence of periodic solutions. It was already mentioned in the preceding chapter that the periodic case cannot be subsumed under the almost periodic: in the first place, almost-periodicity presupposes continuity, whereas periodicity does not; this is, however, a minor difficulty, which could be overcome either by extending the almost periodic theory to include Stepanoff almostperiodic functions-an effort hardly justified by this special circumstance -or by observing that, where the question of continuity is the only obstacle, the methods of the almost periodic case apply, often with considerable simplifications, to the periodic one; secondly, we have seen in Chapter 10 that the assumption that a certain module be dense is essential for some results (e.g., Theorem 103.A), which therefore have no analogue for periodic equations; finally, the periodic case is by far the richer in structure. As in the preceding chapter, the range o f t is R, unless otherwise noted, and the same notations and conventions used there and in Chapter 8 to relate this range to R- , R , are in force. For the sake of convenience, we assume the period of A to be 1 ; this entails no loss of generality. Indeed, if A is periodic with period T > 0, say, we can always transform Eqs. (1 10.1) and ( 1 10.2) into equations of 348
110. INTRODUCTION
349
the same form with A having period 1 , by replacing x ( t ) , A(t),f(t) by We observe that this change of variables leaves invariant:
x ( ~ t )T, A ( T ~~ f)(,~ t ) ,respectively.
(a)
the boundedness and continuity properties, if any, of x, A , f;
(b) the periodicity, if any, except for a division by T of every period; (c) the expressions JII A(t)ll dt, Jllf(t)ll dt, JII i(t)ll dt, where the integral is taken over one period; and the bounds of 11 x 11. Thus, unless specially qualified, “periodic” means “periodic with period 1 ”. This introduction concerns the necessary information on periodic functions, as well as the basic facts about the solution U of (30.3). For periodic equations in finite-dimensional X we have available the well-known canonical reduction to equations with constant A , associated with the name of Floquet [l], and the theory of the latter kind of equations is exhaustively known. However, such a “Floquet representation” is in general not possible in infinite-dimensional X ; this topic is discussed in Section 1 1 1. This is one reason why we seek results by other methods; another is that it thus becomes possible to avoid, even in the finite-dimensional case, the usual investigation by reduction to canonical matrix form of the constant A , a method that is indelibly finite-dimensional and often unnecessarily elaborate. Section 112 contains results of a “global” nature on periodic solutions (110.1) and ( I 10.2) with periodic f. Section 113 concerns the solutions of ( 1 10.1), and especially dichotomies and exponential dichotomies. It is shown in particular that the existence of double dichotomies, ordinary or exponential, may be inferred from the study of X*OD, for D E b Y K without , involving admissibilities, etc.; there are also results corresponding t o those in Sections 102 and 103. Section 114 deals with results of an “individual” kind, especially necessary and sufficient conditions that Eq. (1 10.2) with agiven periodicfhave a periodic solution.
Spaces o f periodic functions For a given Banach space X , the functions f E L(X) that are periodic (with period I ) obviously constitute a subspace of M(X); we denote this subspace, provided with the norm of M(X), by P(X), with the usual conventions about dropping the argument X . We remark that, on account of the periodicity, J‘” Ilf(u)lI du = for every f~ P(X) I and all t E R . In addition to P(X), we consider the continuous functions on R with
If IM
Ch. 11. EQUATIONS WITH PERIODIC A
350
values in X that are periodic; they constitute a subspace of C(X); we provide it with the norm of this space and denote it by PC(X). Obviously, PC(X) = P(X) A C(X) and, in the notation of the preceding chapter, PC(W = A{*nn)(X). The operation of restriction to the interval [0, 13 is a congruence of P(X) onto ro,llL1(X) and a congruence of PC(X) onto the hyperplane of to,llC(X)consisting of those functions f for which f( 1) = f(0). This suggests the consideration of periodic extensions of other spaces in ro,119(X) or to,llg6%'(X);but all such spaces of periodic functions are stronger than P(X) and, provided they contain a continuous function that vanishes nowhere, weaker than PC(X); their properties will therefore be substantially accounted for in the sequel by those of P(X) and PC(X), and we shall not take them specially into account. Properties of U
We assume that A E P(x). As usual, U denotes the solution of the operator equation (30.3), i.e., O+AU=Q,
( I 10.3)
with U(0) = I. Following a familiar argument, the periodicity of A implies that T-,U and UU(1) are solutions of (1 10.3), and have the common value U(1) at t = 0. We set U(1) = U , once and for all throughout this chapter, and conclude that T-,U = UU,;in general, U ( t ) = U(t - n)U;
(1 10.4)
for all t and all integers n. We record one simple consequence: 1 10.A. For given c
> 0 there exists 6 > 0 such that II W U - Y ~ o )- Ill < p
all t o , t
~ O Y
E
R,I t
- to
I
< 6.
Proof. Since U is continuous, such 6 certainly exists under the t o , t E [--I, I]; we may assume 6 < 1. For any given
restriction
<
t o , t E R, I t - to I 6, there exists an integer n I to - n [ < 1. But then I(t - n) - (to- n)I 11 U(t)U-1(t0)- I l l = (1 U(t - n)U-'(to - n) - 111
such that
< 6, and by < z. &
I t - n I,
As a matter of notation, we shall always write E = exp(1 A that, e.g., (31.9) implies 11 U,II < E.
(1 10.4)
Iw),So
11 1. FLOQUET REPRESENTATION
351
11 1. Floquet representation We continue our study of equations with A E P(x), an assumption we need not repeat. We say that A HAS A FLOQUET REPRESENTATION OF ORDER m, where m is a positive integer, if there exists B E 13 such that P ( t ) = U(t)etBis periodic with period m. In that case, P is an invertiblevalued primitive with P(0) = Z and U ( t ) = P(t)e-IB. The significance of a Floquet representation lies in the fact that the change of variables x = P y transforms (1 lO.l), ( I 10.2) into the equations j~ By = 0, j By = g , where B is constant and g = P - v . Obviously, if A has a Floquet representation of order m, it has one of every order that is a multiple of m.
+
+
11 1.A. A has a Floquet representation of order m i f and only i f Uy has a logarithm in 13.
Proof. If U y has a logarithm, we take this to be -mB; using (1 10.4), P ( t m) = U ( t + m)e't+m'B = U ( t ) U ~ e m B e= t e U(t)efB= P ( t ) , If, conversely, A has a Floquet representation of order m, U y = U ( m ) = P(m)e-mB= e-mB. .j,
+
The following result is usually known as Floquet's Theorem. A proof, based on 11 1.A, can be found in Coddington and Levinson [l], pp. 78, 81, 106-107, where there is a good discussion of the real case. llI.B. THEOREM. If X is finite-dimensional, A has a Floquet representation of order at most 2 i f X is real, and of order 1 if X is complex. We next look for general conditions that ensure the existence of a Floquet representation of order 1; they are of interest for infinitedimensional X, and also for real finite-dimensional X. 111.C. Zf U , + oZ is invertible for all u E R, , then A has a Floquet representation of order 1.
+
Proof. (1 - a)Z oU, is invertible for all o E [0, 13, so that Z is connected to U , by an arc of invertible elements in the (commutative) closed subalgebra of a generated by U , . A theorem of Nagumo (Rickart [ 13, Theorem (1.4.12)) implies that U , has a logarithm (actually belonging to this subalgebra). The conclusion follows from 111.A. 4, For Banach spaces in general, we mention the following result; a slightly stronger form is given by Schaffer [lo].
352
Ch. 11.
EQUATIONS WITH PERIODIC A
There exists a constant po = po(X)3 log 4 such I 1 I .D. THEOREM. that if I A 1, < po then A has a Floquet representation of order 1. po 2 $Io , where 1, is the infimum of the lengths of all arcs on the boundary of Z ( X ) that connect opposite points. Proof. See Schaffer [lo]. The proof for Massera and Schiiffer [2], Theorem 2.1.
IA
IM < log 4
is given by
Remark I. It is an easy consequenceof Theorem 111.D that the assumption ‘‘I A IM < TO)’ may be weakened slightly to “there exist p EP and an invertible C E 8 such that I P A C - pl 1, < po” (cf. Massera and Schaffer [2], Corollary 2.1). Remark 2. If the conjecture of Schaffer [lo] that 1, 3 3 for all X is verified, we have po 3 -$ > log 4 for all X. Remark 3. In contrast to the usual situation in this book, the statement of Theorem 1 I 1.D depends strongly on the particular norm of X.
We turn to a more thorough scrutiny of the case in which X is a Hilbert space.
11 l.E. THEOREM. If X is a real or complex Hilbert space and 1.
I A lM < 7,then A has a Floquet representation of order
Proof. Assume that the conclusion were false. T h e set of u E R, for which U , uI is invertible is open and contains 0. Since by l l l . C it does not cover R, , it has a boundary element, say oo > 0. I t is then standard (see, e.g., Rickart [l], pp. 278-279) to infer the existence of a sequence (x,) in X, 11 x, 11 = 1, such that y, = (o;*U, I ) x , -+0 as n-+oo. We now use the fact that X is a Hilbert space. Consider, for each n, the solution Ux, of ( I 10.1). We have 11 ox, 11 11 A 11 11 U x , 11, and therefore, using 30.B,
+
+
<
T h e last member is the length of an arc on aZ(X)that connects sgn x , = sgn(-x, y,) = -sgn(x, - y,). Therefore
+
x , with sgn Ulx,, IAl,>
Q ( x n , - ~ n
+ ~ n )=
r-
Q(.rtr,~,-~y,)>,~-arcsinll~nII-~
as n - . i o ,
so that
=
I A IM 2 n, in contradiction
with the assumption.
&
I 1 I , FLOQUET REPRESENTATION
353
Remark 1. As for Theorem I I l.D (Remark I ) we may replace A J M < T” in Theorem I I I .E by “there exists g~E P and an invertible C E ;3 such that I C-’AC - TI IM < n .
“I
9,
Remark 2. For real X, the bound T in Theorem 11 I .E is best possible, as Example 1 1 I .F will show for continuous A and two-dimensional X. If X = R, however, A always has a Floquet representation of order I , with B = A(t)dt.
Ji
co
Remark 3. For infinite-dimensional real or complex Hilbert spaces, the bound n in Theorem I 1 1.E is almost best possible in a stronger sense: Example 1 I I.G. in a separable Hilbert space, will show that for any larger bound no Floquet representation at all need exist. This leaves only the following case open (cf. Schaffer [9]):
co
Query. If X is an infinite-dimensional Hilbert space and I A 1, = n, does A have a Floquet representation of any order? Of order I if A’ is complex ? Of order 2 if X is real ?
I I1.F. EXAMPLE.Let X be a real euclidean plane, with Cartesian co-ordinates x, , x 2 . Let A ( t ) be represented by the matrix
in (-1
sin 2ni - cos 2ni
-sin cos2nt 2nt) -
’
so that A E PC(x). It is immediately verified that for every u E X we have A(t)u = n(u,a(t))b(t), where a ( t ) = (cos nt, sin nt), b(t) = (sin nt, -cos nt). Since 11 a 11 = 11 b (1 = 1, we have 11 A 11 = n,a constant, and I A 1, = n. Now U ( t ) is represented by the matrix
(Z,P,S:i
nt cos at - sin nt nt sin nt cos vt
+
1. 9
and therefore U , by (-A 1;). This operator clearly has no (real) square root, let alone a logarithm, in X.
co
1 1 I .G. EXAMPLE.Let X be the real or complex separable Hilbert space of all square-summable sequences x = (xn) of real or complex numbers, where n runs over all integers. The set {e?,,}, = (emn)= (a,,,), is a complete orthonormal set. We let V be the orthogonal or unitary SHIFT OPERATOR, defined by Ve,n = Krabbe [ I ] has shown that V = ely, where W is the skew-symmetric or skew-Hermitian operator defined by (We,,, , e,) = (- I)nr-n(m- n)-l for m # n; and 11 WIJ = r (the spectrum of W is the line segment with endpoints -Ti, +Ti).
Ch. 1 I .
354
EQUATIONS WITH
PERIODIC A
Let a real number E > 0 be given and fixed for the time being. Let R be the symmetric or Hermitian operator defined by Re, = 0 if m < 0, Re,,, = ce, if m 3 0. Then eRew,= e, if m < 0, eRew,= e'e, if m 2 0. We now set A(t) = - W - 6t( I - t)e""Re-"" for 0 t 1 (so that A( 1) = A(0) = - W), and define A elsewhere so as to make it periodic. Actually, therefore, A E PC(x). Direct verification shows that V ( t ) = e'wef1(3--2"R for 0 t 1, so that V , = eweR = VeR. It was shown by Schaffer [I I], Theorem 3, that V: has no roots of any order that does not divide h, hence no logarithm, for any positive integer h. By I I ].A, A has no Floquet representation of any order. Now e"" is orthogonal or unitary for every t ; therefore 1) A(t) WII = 6 4 I - t)ll R 1) = 6t( I - t ) for ~ 0 t I , and consequently I A + W I = i c , I A W ( , = E , whence I A 1, r z. Since c > 0 was arbitrarily small, A differs by as little as we please, both in P(8) and in PC(& from the constant -W, and I A 1, exceeds r by as little as we please. &
< <
< <
+
< < < +
+
112. Periodic equations and periodic solutions As before, we assume that A E P ( ~throughout ) this section. Our present purpose is the study of the existence and properties of periodic solutions of ( I lo.]), ( I 10.2), the latter with periodic f . We first have to consider arbitrary solutions of ( I 10.2) with periodic f . Assume then that f EP(X).If we use (31.3) and take ( 1 10.4) and the periodicity into account, we find that any solution x of ( I 10.2) satisfies, for every integer n, (112.1)
x(n
+ I)
@f
=
=
U,x(n)
+ @f,
where 1
( I 12.2)
IT,
[
v-'(t),f(t)dt.
I n order to study more closely the transformation @, we introduce a useful class of functions. For any y E X we define the function f , : R ---F X by fv(t) = 6t( 1 - t ) U ( t ) U y l y for 0 t 1, and by periodicity elsewhere; thus fu E PC(X).
< <
1 12.A. Define rP by ( I 12.2) and Y by Y y = f , for every y E X . Then: @ is an epimorphism from P(X) onto X and (its (a) @Y= I E restriction to PC(X) i s ) an epimorphism from PC(X) onto X; Y is a monomorphism from X into P(X) and a monomorphism from X intoPC(X).
x;
I 12. PERIODIC EQUATIONS
AND PERIODIC SOLUTIONS
355
(b) Y X = {f, :y E X } is a complemented subspace of P(X) and of PC(X), and the respective null-space of @ is a complement; in either case, Y@is the projection along this complement onto Y X .
Proof. @, Y are linear. For any y E X ,
so that @Y = I. This shows that Y is injective and that @ is surjective from PC(X), a fortiori from P(X), onto X. It is a trivial algebraic consequence of @Y = I that, in P(X) or in PC(X), Y@ is the algebraic projection along the null-space of @ onto YX. To complete the proof, it is sufficient to show that @, Y a r e bounded. For any f EP(X),(31.9) and ( 1 12.2) yield 11 @fll E l f l M , so that @ E [P(X);X I . Since PC(X) P(X),we have a fortiori @ E [PC(X);XI. For any y E X, (31.9) implies I Y y IM Ell y 11 6t( I - t ) dt = Elly 11, and I Y y I QElly 11; thus Y E [X; P(X)], Y E [X; PC(X)]. In fact, II @ IlrPrx,:xl > /I @ IlrPccx,:xl II Ilrx;Pcx,l E9 II Ilrx;Pc(x,l 9t
<
<
9
<
< <
Ji
< w-
We now come to the consideration of periodic solutions of ( 1 10.2) with f E P(X). I t follows from ( I 12.1) and the periodicity of A that a solution x of ( I 10.2) is periodic if and only if (112.3)
(I
-
U,)X(O) = Oj.
112.B. THEOREM. XOp= Xnp,is the (closed) null-space of ( I - Ul). Equation ( 1 10. I ) has no nontrivial periodic solution (i.e., Xnp= (0)) if and ody i f I - U , is injective; Eq. ( I 10.2) has a periodic solution for every f~ P(X), or for every f E PC(X) (i.e., (P, PC), or (PC, PC), is (regularly) admissible) i f and only if I - U , is surjective; this periodic solution exists and is unique i f and only if I - U , is invertible. All three statements are equivalent i f X is jnite-dimensional. Proof. T h e first statement follows from ( I 12.3) on setting f = 0. T h e rest follows from ( I 12.3) if we recall from 1 12.A that the range of Q, is all X; and we use the fact that a bijective operator is invertible. On a finite-dimensional space, every injective or surjective operator is invertible. & Remark 1. Theorem I12.B shows that the admissibility of (PC, PC) implies that of (PIPC). An analysis of the proof shows that even the admissibility of (YX, PC(X)) implies that of (P, PC).
356
Ch. 1 1 . EQUATIONS WITH PERIODIC A
Remark 2. In Section 100 we mentioned that Massera and Schaffer [4], Example 2.2 illustrates the fact that X,,, need not be complemented, even if A is constant and X is separable and uniformly convex, hence reflexive. Remark 3. The well-known equivalence, for finite-dimensional X , of the three statements given in Theorem 112.B is not valid in general if X is not finite-dimensional; indeed, not even for constant A in separable Hilbert space, as the following examples illustrate. 112.C. EXAMPLE.Set X = P, a real or complex separable Hilbert space. Let S be the (unilateral) shift operator on 12, defined by Se, = em+,, M = 1 , 2, ...; its adjoint St is described by Stel,, = en,+, , m = 2, 3, ..., and S'e, = 0. Now StS = I # SSt, so that S is injective, S' is surjective, but neither is invertible. Now 11 S 1) = I ; therefore I - 4shas a logarithm, say T. Set A = -T. Then U(t) = e l T , so that I - U , = &S. By Theorem 112.B, ( I 10.1) has no nontrivial periodic solutions, but (PC,PC) is not admissible. & 112.D. EXAMPLE.With X , S, T as in the preceding example, set A = - Tt. Now I - U , = $St.By Theorem I 12.B, (P,PC)is admissible, but (1 10.1) does have nontrivial periodic solutions. ,$ Even if (P, PC) or, equivalently, (PC, PC) should not be admissible, it remains possible to say something about the periodic solutions of( 110.2) for those periodic f for which they may exist.
1 12.E.
THEOREM.
The following statements are equivalent:
(a) ( I - UJX is closed (this is true in particular if X is finitedimensional). (b) The set {f E P(X): there exists a periodic solution of ( I 10.2)) is closed in P(X). ~ there exists a periodic solution of ( I 10.2)) is (c) The set { f PC(X): closed in PC(X). (d) [(e)] There exists a number k > 0 such that if f E P(X) [if f~ PC(X)] and there exists a periodic solution of ( 1 10.2), there exists one periodic solution x with I x I KI f IM [I x I kl f 11.
<
<
Proof. We prove the equivalence of (a), (b), (d); the proof of t h e equivalence of (a), (c), (e) is entirely similar. (a) implies (b). By ( I 12.3), f belongs t o the set described in (b) if and only if @f E (I - U,)X; therefore the set in question is precisely
I 12. PERIODIC EQUATIONS
357
AND PERIODIC SOLUTIONS
the inverse image under the continuous linear mapping @ : P ( X ) + X of the subspace (I - U,)X; hence it is itself a subspace. (b) implies (d). Under (b), @-I((Z - U,)X), provided with the norm of M(X), is a Banach space; now (@-l((Z - U,)X), PC(X)) is admissible; therefore (d) holds by Theorem 51 .A.
co
(d) implies (a). Consider the quotient space Y = X/X,,, a Banach space. On account of the fact that X,, is the null-space of Z - U , , this operator induces a monomorphism SZ : Y -+X, with SZY = (I - U,)X, 11 SZ (1 = 11 I - U, 1 . For any y E Y, set f
=
YSZy E Y(Z - U,)X c @ - 1 ( ( Z
-
U,)X);
by ( I I2.3), the periodic solutions of ( I 10.2) are exactly the solutions x with x(0) E Y . Among them, by (d), is one, say x, , with I x, I k l f J M Then IlY
d 11 xO(o)ll
6 I xO I
It follows that SZY
'
= (I -
<
<
.
'IySZy IM kEII Q-Y 11 d - ul 11 Ily I I Y U,)X is complete, so that (a) holds. &
*
Remark. Since X,, is closed, by Theorem I12.B, conditions (b), (c) imply the regular admissibility of (@-l((Z - U,)X), PC(X)); we leave it to the reader to formulate the more precise statements, replacing (d), (e), that are obtained by applying Theorem 51.D; we only state the corresponding consequence for complemented X,, .
I12.F. Assume that X,,,has the complement 2. If any one of the equivalent statements of Theorem 112.E. holds, there exists a number k' > 0 such that i f f E P ( X ) [ i f f E PC(X)] and there exists a periodic solution of ( 1 10.2), the unique periodic solution x with x(0) E 2 satisjies I x I ,<
k ' l f IM [I I k ' l f 11. Proof. 51.E applied to B = @-l((Z = PC(X). &
-
U J X ) with the norm of
M(X) [of C(X)], and D
Since X,,, = X,,, is always closed, admissibility and regular admissibility of (P,P C ) and of (PC, P C ) coincide. In particular, Ad(P, PC) n P ( 2 ) = 4d,(P, PC) n P(X),
etc. T h e results concerning these admissibility classes can be read off from Theorems 71.B, 71.C, since obviously P ( 8 ) P C ( X ) 3 P(X), PC(x) * PC(X) PC(X).We omit the specific form of these statements for periodic equations.
Ch. 1 I . EQUATIONS WITH
358
PERIODIC
A
113. The solutions of the homogeneous equation D- solutions It is to be expected that the periodicity of A imposes strong restrictions on the bounded solutions and other D-solutions, for D E WK , of (1 10.I). This is indeed the case, as we shall presently see. We continue to assume, without further mention, that A E P(x).
113.A. THEOREM. ( I ) If D E WKand X,, is closed, then either = {0},according as D is weaker than L" or not. In the former case, there exists N > 0 such that eoery D-solution (i.e., eoery bounded solution) x of ( 1 10. I ) satisfies 11 x(t)ll < NII x(to)lI for all t o , t, and is thus bounded away from 0, unless it is the trivial solution. (2) If D+ E WK+ and X+,,+is closed, then either X+,,,+ = X+,, or X+,,+ = X+,,, or X+,,+ = X+,,, according as D+ is weaker than Ly ,not weaker than Ly but weaker than L,",, or not weaker than L;+ . In any case, there exists N + > 0 such that eoery solution x of ( 1 10.1) with x+ E D+(X) satisJies (1 x(t)(l < N+lJx(t,)JJfor all t >, to . In the third case, and also in the second if X+,,,+ = X+, is finite-dimensional, there exist Y+, N + > 0 such that this solution x satisfies 11 x ( t ) 11 < N+e-'+(t--to) [[ x(to) [I for all t > t o .
X,, = X, or X,,
Proof. We shall systematically set n = [to]; then (31.7) implies Ell 4to)ll. Proof of (1). Since A E M(X), every M-solution of (1 lO.l), a fortiori every D-solution, is bounded (Section 33, pp. 92-93). Replacing D by D A L" if necessary, there is no loss in assuming D L", whence X,, C Xo . Assume that X,, # (0); let x be a nontrivial D-solution of (1 10.I ) and let to , t be given. Now T-,x is also a D-solution of ( I 10.I ) ; by Theorem 33.C we have, setting S = S, ,
II x ( 4 II
<
<
11 x ( t ) l l
ID =
I T - ~ IxD < As I
11
=
z'-?Ix(o)
''11
x ( n ) 11
Thus 11 x 11 E D is bounded away from 0 by (SE)-'I x must be weaker than L"; and in this case obviously conclusion holds with N = SE.
< SE 11
I > 0,
11.
so that D
X,, = X,. T h e
Proof of (2). As in the proof of (I), there is no loss in assuming that D+ < Ly , so that X+,,+C X+,; equality holds if D+ is weaker than Ly . Let in any case x be a solution of (1 10. I ) with x+ E D+(X), and let t >, to be given. Then T-,x is also a solution of (1 10.1); if n 0, (T-,x)+ = T;x+ E D+(X); if n < 0, TI,(T-,x)+ = ( T ,T-,,x)+ = x+ E D+(X); but
1 13. SOLUTIONS OF THE
HOMOGENEOUS EQUATION
359
@-,( T-,x)+ = T_f,T:,( T-,x)+ E D+(X) differs from (T-,x)+ only on [o, -n], and both functions are continuous there; by 22.1, (T-,x)+ E D+(X) in this case too. By Theorem 33.C we have, setting S+ = S+D+,
Thus 11 x(t)ll < N+II x(t,)ll with N+ = S+E. Assume that D+ is not weaker than Ly . For any solution x of (1 10.1) with x+ E D+(X) we have lim 1) x(t))l S+Einf,,, 11 x(t)ll = 0. Therefore X+OD+ C X+oo,with equality if D+ is weaker than L;+ . Consider this case, with finite-dimensional X+,,. By the usual compactness argument there exists T > 0 such that 1) x(t)ll *E-lIl x(0)ll for every solution x of (1 10.1) with x+ E D+(X) and all t >, T . If x is any such nontrivial solution, and to is given, T-,x is another such solution and we have
<
<
Using (1 13.1) and applying 20.C to 11 x 11-l we conclude that 11 x(t)ll < N+ e-”+(t-l,) (1 x(t,)ll for all t 2 t o , where v+ = 7-1log 2, N+ = 2S+E. Assume, finally, that D+ is not weaker than Lg+.By 23.S we may choose 7 > 0 so large that /3(7)= /3(D+; T ) >, 4S,2E2. Let x be a nontrivial solution of (1 10.1) with x+ E D+(X),and let to be given; T-,x is another such solution. By 23.M and (1 13.I),
We conclude again that 11 x ( t ) 1)
< N+e--v’t(t-to) 11 x ( t ) 11 for all t >, t o , with
v+, N + given by the same formulas. I t follows that X+OD+ C X+,,, and
the reverse inequality holds becauseD+ is weaker than T+(by 23. J).
<
Remark. In Theorem l13.A,(2), the inequality 1) x ( t ) 11 Nfe-” +(l--t”) 11 x(to) 11 for solutions with x+ E D+(X) certainly need not hold if D+ is weaker than Ly , as the case A = 0, D+ = LT illustrates.
Ch. 11. EQUATIONS WITH
360 00
PERIODIC
A
Neither need it hold if D+ is not weaker than Ly but weaker than L& and X,, is not finite-dimensional, as Example 113.F, with constant symmetric or Hermitian A in separable Hilbert space, shows. If x is a bounded solution of ( 1 lO.l), we have 11 x ( t ) - x(to)lJ< I( U(t)U-l(t0)- ZI(I x I, and llO.A implies that this solution is uniformly continuous. If X is finite-dimensional, it is known that every bounded solution is in fact almost periodic: the Floquet representation reduces the proof to the corresponding one for constant A , and the latter is usually carried out by representing A by a matrix in canonical form. We give a direct proof that includes some information for the infinitedimensional case as well.
If X o is closed, every solution of (1 10.1) the closure 113.B. THEOREM. of whose range is compact is almost periodic. I n particular, if X is jinitedimensional, every bounded solution of (1 10.1) is almost periodic. Proof. T h e set {x(n) : n integral} is contained in a compact set, and is therefore totally bounded. For given E > 0, let n, , ..., nk be distinct integers, which we assume ordered by increasing size, such that for every integer n we have 11 x(n) - x(nj)ll S-% for some j , 1 j k, where S = S , (Theorem 33.C). Set 1 = I ( € ) = n, - n, 1. For nk - m to 1. given to E R, set m = [n, - t o ] ; then to n, - m Let j be such that 11 x(m) - x(nj)ll S - k Now T-,,,x - T-,,x is a bounded solution of (1 10.1). By Theorem 33.C,
<
< <
< < < +
+
<
I * - T ( , , , - m P I = I T-,"X - T,,* I < s /I T3-n,(0) - L , ( O ) I1 = s1) x ( m ) - x(n,) 11 < f. Thus [ t o , to 00
+ I]
contains the €-translation number nj
-
m of x.
9,
The assumption that the closure of the range is compact (an obvious necessary condition) is not redundant in general, as Example 113.G, with constant skew-symmetric or skew-Hermitian A in separable Hilbert space, illustrates.
Double dichotomies We anticipated in the Introduction the remarkable fact that if A is periodic it is possible to infer the existence of double dichotomies, ordinary or exponential, from the properties of X*oDfor some D E W K . The behavior of the solutions, as described by Theorem 113.A, makes this plausible; we now proceed to prove it. We begin with the simpler case of the double exponential dichotomy.
I 13. SOLUTIONS OF T H E
HOMOGENEOUS EQUATION
36 1
113.C. THEOREM.Assume that D E b y K is not weaker than L“; if (XPnn , X,,,) is a closed dihedron (which must be disjoint) and either X is jinite-dimensional or D is not weaker than L;, then (X-,,D,X,,,) induces a double exponential dichotomy f o r A . Conversely, i f a closed dihedron ( Y - , Y,) induces a double exponential dichotomy for A , it is disjoint, and coincides with (X-,,D, X+,,,)for every D E b,FK .
Proof. Under the assumptions of the “direct” statement, the closed is disjoint, by Theorem 1 l3.A,(1). We apply dihedron (XPon,X,,,) Theorem I 13.A,(2) to D ’ = D, and again, under symmetrization, to D- = D.. , and find that the dihedron satisfies the conditions (EEi*) of Theorem 82.F,(b), with v’ = min-.+ v*, N’ = max-.+ N*. Since A E M(X), Theorem 82.F shows that the dihedron induces a double exponential dichotomy. T o prove the converse, 42.F implies Y + = X + , , for any D E byK; since this includes T, which is not weaker than Lt, Theorem 1 l3..4,( I ) again implies that the dihedron is disjoint. 9, co
Remark. The alternative assumption that X be finite-dimensional or D not weaker than Lr is not redundant, as we show in Example I13.F with a constant A in a separable Hilbert space, with ( X o ,0X+m)= ({0},X ) , a disjoint closed dihedron, but with no double exponential dichotomy. We come to the case of ordinary double dichotomies, which is complicated by the possible lack of disjointness of the dihedron. I n view of Theorems 113.C and I13.A,(2) it would seem to be sufficient to take D = L“ for the “direct” theorem, but there is no special gain for the proof in this restriction.
If D E b y , and (X-(,,, X+,,) is a closed dihedron, 1 13.D. THEOREM. this closed dihedron induces a double dichotomy f o r A . Proof. 1 . We prove that X+,, induces a dichotomy for A + by establishing statement (d) of Theorem 41 .A. T h e corresponding statement for X-,D and A - follows by symmetrization, and both taken together constitute the conclusion. Let N , Nf, N - be the constants of Theorem 1 13.A, part ( I ) , part (2) with Df = D, , and part (2) symmetrized with D- = D- , respectively (if X,, = XPonn X,,, = {0}, the choice of N >, 1 is arbitrary). Let h > 1 be given, and let q-+ be an (X,,, A)-splitting of XPoD.If K = K ( X - , , , X,,,) is the gape of the dihedron, q+ induces, by I I.M, an (X+oD, hK)-splitting q of X . This will be the splitting in statement 41 .A,(d).
Ch. 11. EQUATIONS WITH PERIODIC A
362
Let y, z be solutions of (1 10.1) with y+ , z+ as required in that statement, i.e., y(0) E X+,, , q(z(0))= z(0); the latter condition means z(O) E X-,,, qf(z(0)) = z(0) and implies 11 z(0)lI hd(XO,, ~(0)).Now the growth conditions (Di), (Dii) are satisfied by y+ , z+ on account of Theorem l13.A,(2) (including symmetrization), with N+ = N+, N ; = N-. It therefore only remains to prove the angular-apartness condition (Diii). 2. We claim that
<
where x' = ( A
(113.3)
+ 1)"-
+ 1 > 1.
We split the proof of (113.2) into two cases. In both we consider an arbitrary solution u of (110.1) with u(0) E X , , and observe that T,u(O) E X,, , hence z(0) - T,u(O) E X-,,. We assume that n is given. Assume that hN-11 z(n)ll < A'II z(0)ll. Then Theorem l13.A,(2) (symmetrized) implies
11 z(n)11 < h'(m-)-l 11 z(0) 11 d A'(N-)-' d(XOD , z(0)) Q
.) Q x' II 4
(I 4 0 ) - TnU(O) II - T,u(n) II = A' I1 z(n) - 4 0 ) 11;
since u(0) E X,, was arbitrary, (1 13.2) holds in this case. Assume that AN-[]z(n)ll > A'll z(O)11. Again by Theorem l13.A,(2) (symmetrized), and by Theorem 113.A,(l),
"-
II 4 .)
II .(n) - T n u ( 4 II 2 N II 40)- T n U ( 0 ) II 2 N II 4-4 II - N It 4 0 ) II 2 II 4 0 ) II - N It 40)I1 2 I1 44 \I - I1 44 - 4 0 ) I1 - N II 4 0 ) 11,
- 4 0 ) I1 = "-
whence, using (113.3), ("-
+ 1) (I z(n) - 4 0 ) II 2 II 44II - NII 4 0 ) II 2 (1 - h'-'A"-) = A'-'("-
II z(n) II
+ 1) (I s(n)11;
since u(0) E X,, was arbitrary, (1 13.2) holds in this case too. 3. By 1 1.L, (1 13.2) implies
11 d n ) 11 < h' d(XOD z(n)) < h t K d(X+,, z(n)), I
9
n
==
0, 1, ... .
I 13. SOLUTIONS OF THE Now sgn y ( n ) = sgn T-,,y(O)E X,,, (1 13.4)
Y[Y(~), 4n)l
F 11 sari
363
HOMOGENEOUS EQUATION
. Therefore, if y , z # 0,
4.) - SPY(")
II 3 (I ~ ( n1)1-l d(XAoD , ~ ( n ) )
>, ( A ' K ) - ' . Finally, if t
E
R , , (3 I . 10) and ( I 13.4) yield Y [ Y ( l ) ,4 t ) l
2 '2E-4[y([rl), z([t])l 3 w A ' K ) - l .
+
Therefore (Diii) holds, with y o = yo(q) = g ( E 2 ~ ( ( h I)NN-
+ l))-l.
&
As noted above, the strongest case of Theorem 113.D is: Zf ( X - , , X+,) is a closed dihedron, then it induces a double dichotomy for A . A trivial kind of weak converse is: Zf some closed dihedron ( Y - , Y + ) induces a double dichotomy for A , then (I-, , X+,J is a dihedron (since Y + C X*,). W e thus have:
113.E. If X is finite-dimensional, there exists a closed dihedron that X+, = X . induces a double dichotomy for A , if and only if X-,
+
00
00
00
Remark. If X is not finite-dimensional, two questions arise in this connection: firstly, is it true that the existence of a closed dihedron inducing a double dichotomy implies that the dihedron (X-, , X+,) is closed or, equivalently, that X , is closed ? The answer is negative: a counterexample in a separable Hilbert space is given in Example 113.H; we have not found one with constant A. The second question remains open, although we conjecture a negative answer: Query. If (X-,, X+,) is a dihedron that is not closed (but perhaps gaping), does there exist a closed dihedron that induces a double dichotomy ? We remark in concluding that: r f X isfinite-dimensional and (-3-, ,X+,) is a dihedron, then actually X = X , + X-, + X+, . This follows rather easily for constant A from the explicit form of the solutions and, via the Floquet Theorem, for all periodic A. It is no longer true if X is not finitedimensional: Examples 113.1 (in separable Hilbert space) and 113.J (in a nonreflexive, nonseparable space, but with constant A) illustrate this; we lack a counterexample with constant A in a (separable) Hilbert space or even only a reflexive space.
co
Examples I13.F. EXAMPLE.Let X be the real or complex separable Hilbert space of square-integrable real- or complex-valued functions x = ( x ~ ) ,
364
Ch. 1 1 . EQUATIONS WITH
PERIODIC
A
u E [0, 11, with
the L2-norm. Define the symmetric or Hermitian operator A by (Ax), = uxo,u E [0, I]. Then U(t)= e-tA, so that ( U ( ~ ) X=) e-%, ~ , and limt+w(I U(t)x (I2 = limt+a J: e-210(x, l2 do = 0 by Lebesgue's Domi-' nated-Convergence Theorem. Therefore X,, = X . On the other hand, limt--m 11 U(t)x11 = co for all x # 0, so that X-, = (0). N + e-ut't-to) II U(t0)xII We claim that we cannot have 11 U(t)x11 for all x E X+M)= X and all t >, t o , let alone an exponential dichotomy induced by the closed dihedron (X-, , X,,,) = ({0}, X ) . Indeed, this would imply 11 U(t)ll N + e--'+t -+ 0 as t + co. But if xLn) E X is defined by xp) = n1/2~~~,~-1](u), n = I , 2, ..., we have 1) xtn)ll = 1, and, = n J:-' cZto du > / r z t l n+ 1 as n 00, for fixed t 2 0, (1 U(t)x(n)((2 so that 11 U(t)ll 2 I for all t >, 0 (actually, equality holds). ,$,
<
<
--f
In the remaining examples, X is the real or complex separable Hilbert space of square-summable sequences x = (xn) of real or complex numbers, where n runs over all the integers; the notation is as in Example I I I .G; in particular, V = ew is the shift operator, Ve, = en,+l. 113.G. EXAMPLE.We set A = -W, a skew-symmetric or skewHermitian constant, so that U ( t ) = etw is orthogonal or unitary. Every solution x of ( I 10.1) is bounded: indeed, /I x (1 is constant. Moreover, since X is reflexive, the weak closure of the range of x is weakly compact. However, we now show that no nontrivial solution is weakly almost periodic, let alone almost periodic. Let x be a nontrivial solution with, say, xtn(0) # 0. Now x(n) = Vnx(0), so that x,(n) = x,-~(O) -+ 0 as n + CO. We show, using an argument common in the theory of almost periodic functions, that this suffices to prevent xm = (x, e), from being almost periodic. For assume that it were almost periodic, and let r, 0 < 4r < I x,,,,(O)l, be given. Since an almost periodic function is uniformly continuous, we may choose 6 > 0 so small that t , t' E R,I t' - t I 6 implies I xm(t') - x?,,(t)l E . Let p be an integer so large that 1 xn,(n)l = 1 x,&O)( r for all n 3 p . Let ( T ~ be ) a sequence of r-translation numbers of x,, with T ~ -+ T~~ 3 p . Since the fractional part of T~ lies in the compact set [0, I], there exist K, K', K' > K, and an integer Y 2 p such that I Y - ( T ~ , T ~ ) I 6. But then we derive the contradiction
<
<
<
<
113.H. EXAMPLE.We consider the adjoint equation to the equation in Example l l l .G; i.e., with the notations there introduced, A(t) =
113. SOLUTIONS OF THE
+ 6t( 1
t)etWRe-tWfor 0
HOMOGENEOUS EQUATION
< t < 1, and otherwise periodic, so that 0 < t < 1, and U , = = VePR.
--
W
A
E PC(8), U ( t ) = e'we-t"13--2t'R for
-
365
Since ,Iw is unitary and e-t"3--2t)Re, = em or = e-t2(3--2t)f em according as m < 0 or m 3 0, ( I 10.4) implies that 11 Ux 11 is nonincreasing for each x; therefore X,, = X, and the closed dihedron ({0}, X) induces a double dichotomy; further, X , = X-, n X,, = K O . Now if n is a positive integer, U(-n) em = (eRV-')"ern= ern-" if m < 0, and U(-n)e,,, = min{efR,efnt)e,-n if m 0. Since (1 Ue,,, I( is nonincreasing, 11 U( -#)en, 11 = max{ 1 , efm} < 03. Therefore 1imt++== llU(t)en,ll= limn+or, X - - X , contains all e,,, and is therefore dense in X. However, if O -m x = m-le, EX, we have I( U(-n)x I/ 2 I( U(-n)x), 1 = n-'efn + 03 as n -+ co, so that x 4 X-, . Thus X-, = X , is not closed. & 9
>
C,
113.1. EXAMPLE (Example 1 1 1 .G continued). In example 1 I1.G we have I/ Ux 11 nondecreasing for each x E X (proof as in the preceding example) so that X-, = X, X,,, = {0}, and (X, (0)) induces a double dichotomy. For each m, U(n)e, = (VeR)"em= min{efn,ef(nfm) }ern+, for all sufficiently large positive integers n, so that limt-,a 11 U(t)x(1 = 03 for all x # 0, and X,, = (0). Finally, U(-n)em = (e-RV-l)nem = em+ if m < 0, and max{ePfn,e-fm}em--n if m 2 0, for all n 2 0, so that limt+-m(1 U(t)x 11 > 0 for all x # 0, and X-, = (0). We obtain
+ x o o + x,,,
= (0)f
x. d
1 13.J. EXAMPLE (Example 44.B continued). If we consider the adjoint equation (in X * =).I in Example 44.B, with the constant -A*, as defined on all R, all solutions of ( I 10.1) are obviously nonincreasing, and are unbounded (except the trivial one) as t + -a.Therefore ({0}, X*) = ( P oX:,) , induces a double dichotomy. However, X z , C X z , = {0}, and X:, = 1: (Example 44.B), so that X: Xz, XTw = I," # I" = X*. &
+
+
Exponential and double exponential dichotomies I n this subsection we examine another question, namely the between double exponential dichotomies for A and exponential mies for A,, and their consequences, in analogy to Sections for almost periodic equations. T h e fundamental result here is analogue of Theorem 102.A.
relations dichoto-
102, 103 a n exact
The following statements are equivalent: 1 13. K. THEOREM. (a) A possesses a double exponential dichotomy, and the closed dihedron ( X - , , X,o) inducing it is disjoint;
366
Ch. I I . EQUATIONS WITH
PERIODIC
A
(b) A possesses a double exponential dichotomy; (c) A+ possesses an exponential dichotomy. Proof. T h e implications (a) -+ (b) -+ (c) are trivial. T h e proof of ) n, (c) +(a) is entirely similar to that of Theorem 102.A (with ~ ( n = n = 1, 2, ...), except that part 1 of the proof is replaced by the remark that T,A = A, n = 1, 2, ... . (The implication (b) -+ (a) is contained in Theorem 113.C.) &
I 13.L. If A+possesses an exponential dichotomy, then X+,(A) = X+,(A+) is complemented. Proof. By the implication (c) -+ (a) of Theorem 113.K, X+,(A) has the complement .%'-,(A). &
I 13.M. THEOREM. If either A possesses a double exponential dichotomy PC)and (PC, PC)are or A+ possesses an exponential dichotomy, then (P, regularly admissible for A , and X,, = X , = (0). Proof. T h e alternative assumptions are equivalent, by Theorem 113.K, and imply, by the same theorem, that X , = (0). By 82.S, (M, L") is admissible for A. For any f E P(X) M(X), a fortiori for any f E PC(X),let x be the unique bounded solution of ( I 10.2). On account of the periodicity, T,x is also a solution of ( 1 10.2), therefore T,x = x, and x is periodic. &
<
In contrast to the almost periodic case (Theorem 103:A), the converse of Theorem 113.M is false, even for constant A in one-dimensional complex, or two-dimensional real, X: 113.N. EXAMPLE. Let X be (identified with) the complex scalar field, and set A = i. Then U ( t ) = ecit, and 1 - U , = 1 - e c i is invertible. By Theorem I12.B, (P,PC) and (PC, PC)are admissible. Indeed, more is true: a change of variable shows that for every f E L(X) that is periodic with a period other than a multiple of 27r, ( 1 10.2) has a unique solution with the same period. However, all solutions of (1 10.1) have constant norm, so that A does not possess a double exponential dichotomy. Taking X to be the underlying real euclidean plane, the same equation yields a real two-dimensional counterexample for the same purpose. &
Remark 1. If X = R , the converse of Theorem 113.M does hold. PC) or (PC, PC) Indeed, set 1: A(t) dt = u. T h e admissibility of (P,
113. SOLUTIONS OF THE
367
HOMOGENEOUS EQUATION
for A implies, by Theorem 112.B, that I u # 0. If u > 0 we have, for any t 2 to,
-
U,= 1 - r u# 0, i.e.,
thus ({0}, X) induces a double exponential dichotomy. If u < 0, it follows by symmetrization that (X,{O})induces a double exponential dichotomy.
Remark 2. We can now answer the question as to what becomes of Theorem 103.A when the modules a, b are not dense (in that case, of course, we can drop the assumption of continuity for A , n . If we use Theorem 113.K, it follows as in Theorem 103.A that the conditions (a), (b), (c), (d), (e) are still equivalent and imply (f), (g), while these in turn imply (h), (i), respectively, all this being independent of the assumption that X is reflexive. It will follow from Theorem 114.C that, if this assumption is added, (f) and (h) are indeed equivalent, as are (g) and (i), and in fact all four conditions are equivalent if X is finitedimensional. However, Example 113.N shows that none of these four conditions need imply any of the first five, even if A is constant and X is one-dimensional complex or two-dimensional real. There is an important characterization of those A E P(x)that possess a double exponential dichotomy, in terms of the spectrum of U , . In order to make this characterization available for the case of real X also, we have to use the formalism of the COMPLEXIFICATION X iX of X (see, e.g., Rickart [l], pp. 5-9); this is the set of (formal) elements x + iy,x, y E X, with the obvious algebraic rules for complex scalars, and
+
11 x
+ iy 11 = mfx (11 x cos 0 - y sin 6 11 + II x sin 6 + y cos 6 I l ) / f i ;
+
+
the algebra of operators (X iX)" consists of the elements A 23, A, BE with the obvious rules for operating on the elements. It is customary to call "spectrum of A" for A E 13 the spectrum of A i0 E (X iX)".It is obvious that A E P(3) possesses a double exponential dichotomy if and only if A + i0 E P((X + iX)")does; and if U,corresponds to A, U , i0 is the corresponding operator for A i0. The following theorem states the promised characterization. It could
x,
+
+
+
+
368
Ch. 11. EQUATIONS WITH
PERIODIC
A
be adapted to yield another proof of Theorem 113.K, but we prefer to consider it separately, in view of the quite different functionalanalytic context to which it belongs.
A possesses a double exponential dichotomy (equiv1 13.0. THEOREM. alently, A+ possesses an exponential dichotomy) i f and only i f the spectrum of U , does not meet the unit circumference. Proof. In view of the preceding remarks we may, and do, restrict the proof to the case of complex X. T h e spectrum of an operator U will be denoted by A( U). The condition is necessary: Let u be any fixed real number. T h e solutions of 9 (A i d l y = 0 differ from those of (1 10.1) by a factor e-iuf, so that the closed dihedron that induces a double exponential dichotomy for A does the same for A i d . By Theorems 113.M and 112.B, applied to A i d , it follows that I - eciUU, is invertible, hence efu4 A( U,).Since u was an arbitrary real number, we conclude that A( U,) does not meet the unit circumference. The condition is sujicient: Since A( U,) is compact and does not meet the unit circumference, we have A( U,) = A- u A+, where A-, A+ are compact sets, not both empty, respectively exterior and interior to the unit circle. By the Spectral-Decomposition Theorem (see, e.g., Riesz and Sz.-Nagy [ I ] , p. 417), there exists a disjoint closed dihedron ( Y - , Y+) with Y,t invariant under U , and such that, if U* is the restriction of U , to Y* (so that( U*)" is the restriction of Uy to Y* for every integer n), we have A( U*) = A*. Since A( U+),A(( U-)-l) are both interior to the unit circle, there exists Y' > 0 such that the spectral radii of both U+ and (U-)-1 are < e-u'. Now for all sufficiently large positive integers m we have II( U+)" 11, [I( U-)-m 11 < e-v'm;it follows that there exists N i > 0 such that
+ +
+
+
(1 13.5)
11 (U+).l11, 1) ( U-)-'jl 11
< Ni e-"'nr,
m
= 0,1,
... .
If A- or A+ were empty, the corresponding part of the argument is vacuous, and should be omitted. Consider a solution z- of ( I 10.1) with z-(0) E Y+ , and take t >, to . Set n = [ t ] , no = [to], so that n 2 no and I(t - to) - ( n - no)l < 1. By ( I 10.4), z-(n) = UFz-(O) = (U+)"z-(O),z-(no) = ( U+)%-(O). Using (113.5) and (31.7) we have
11 z - ( t ) 11
< E (1 z-(n) 11 Q Ell
(17+)"-~0
< -. "p ' i t - t II z-(to) II, )
0
11 11 z-(n,) 11
< E2NA e-"'+"
0
II z-(tn) I/
1 14. INDIVIDUAL PERIODIC
369
EQUATIONS
where N ’ = E2Niev’.Thuscondition (EEi-) of Theorem 82.F is satisfied. t o , the same For a solution z+ of (1 10.1) with z+(O) E Y - , and t argument yields z+(n)= ( U-)-cnU-”)z+(n,,), and therefore condition (EEi+) of the same theorem. Since A E M ( X ) , Theorem 82.F implies that A possesses a double exponential dichotomy. &
<
If we apply Theorem 113.0 to the case of a constant A we obtain the generalization of a familiar result in finite-dimensional X :
If A E X , the constant A possesses a double expo113.P. THEOREM. nential dichotomy (equivalently, A + possesses an exponential dichotomy) if and only if the spectrum of A does not meet the imaginary axis. Proof. We may again assume that X is complex. Now U , = e-*. By the Spectral-Mapping Theorem (see, e.g., Riesz and Sz.-Nagy [l], p. 428), the mapping A+e-I of the complex plane maps A ( A ) onto A ( e d ) , and the imaginary axis onto the unit circumference. Since the imaginary axis is the complete inverse image of the unit circumference, A ( A ) meets‘ the former if and only if A( U , ) = A ( e d A )meets the latter. T h e conclusion follows by Theorem 113.0. &
1 14. Individual periodic equations In this section we consider Eq. (I 10.2) with A E P(x) and a giwen ~ E P ( X(these ) assumptions are in force throughout the section), and we are mainly interested in conditions ensuring the existence of a periodic solution; the conditions concern the assumed existence of a solution of a certain kind (in particular, a bounded solution). I t is interesting that the property assumed of this solution, such as boundedness, need in general refer only t o its restriction to R+ (or, by symmetrization, to R-). T h e results of this section may be compared with Favard’s theorems for the almost periodic case, discussed in Section 104. T w o main approaches are used: one is based on Tychonoff’s FixedPoint Theorem, the other on the Hahn-Banach Theorem. T h e most general result obtained by the first method is:
Assu,me that (1 10.2) has a solution x such that the 114.A. THEOREM. closed convex hull of the set { x ( n ) : n = 0, 1, ...} is weakly compact. Then ( I 10.2) has a periodic solution. Proof. By ( I 12.1) the set {x(n)} is invariant under the nonhomogeneous linear continuous mapping x -+ U,x + @ f ; hence its closed convex hull K is also invariant. Since the mapping is linear and con-
370
WITH Ch. 11. EQUATIONS
PERIODIC
A
tinuous, it is also continuous in the weak topology, in which K is compact. By Tychonoffs Fixed-Point Theorem (Tychonoff [l]), there exists a fixed point yo of the mapping in K,i.e., (I- Ul)yo= @f. By (112.3) the solution y of (110.2) with y(0) = yo is periodic. & We present two applications: in the first, a solution with relatively compact range (at least on ++) is assumed to exist; in the second, the assumed solution is merely bounded (on R or on R+),but some assumption of reflexivity must be added. 114.B. THEOREM. If the equation (1 10.2) has a solution x such that the closure of the range of x+ is compact, then it has a periodic solution.
Proof. By the assumption, the closure C of {x(n) : n = 0, 1, ...} is compact. The closed convex hull K of { ~ ( n )is} the closed convex hull of C, hence K is itself compact, aforfiori weakly compact. The conclusion follows by Theorem 114.A. & 114.C. THEOREM. Assume that the closure of Xo [of X,,] is reflexiwe (in particular, that X is refledwe). If Eq. (1 10.2) has a bounded solution [a solution x with bounded x+], then it has a periodic solution.
Proof. If x is bounded, so is T-,x, n = 0,1, ...; and yn = T-,x - x is a bounded solution of (1 10.1). Since yJ0) = x(n) - x(O), (m(O)} is a bounded set in Xo , and its closed convex hull KOis a bounded set in clX, . If clXo is reflexive, KOis compact in the weak topology of clXo , which coincides, by the Hahn-Banach Theorem, with the topology induced on clXo by the weak topology of X. Hence KOis weakly compact in X,and so is K = x(0) K O ,the closed convex hull of {x(n)}. The conclusion follows by Theorem 114.A. The proof under the partly weaker and partly stronger assumption that x+ is bounded and clX,, is reflexive is entirely similar. &
+
Remark 1. For finite-dimensional X, the proof of Theorem 114.C via Theorem 114.A reduces to an application of Brouwer’s Fixed-Point Theorem. An alternative proof may be based on the Floquet representation. Still another is a consequence of Theorem 114.F. Remark2. It may be asked whether the compactness assumptions in Theorems 114.A, 114.B, 114.C (here appearing as reflexivity) might be dropped, i.e., whether the existence of a bounded solution, or even one bounded on R, only, is enough to ensure the existence of a periodic solution. An example given by Massera and Schiiffer [2] (Example 3.3) in the separable (but of course nonreflexive) space Z1-a space congruent to this is actually used-emphatically suggests a negative answer.
I 14. INDIVIDUAL PERIODIC oooo co
EQUATIONS
37 1
Added in proof. This negative answer can indeed be established by means of that example, in view of the fact, recently discovered by D. Arlt (reported in Douady [I]), that the set of invertible operators on 1," is connected in (Lon)-. Remark 3. It is obvious from the proofs that in Theorems I14.B and I 14.C the assumption of relative compactness of the range and of boundedness, respectively, need not be made about x, or x itself, but only about the set of its values at integral values of t . However, it is easily shown that this is no actual weakening of the assumption (cf. Massera and Schaffer [2], Lemma 3.1). We now turn t o the fundamental result of the other method of approach. T h e device used in the proof is d u e to H. F. Bohnenblust, and was applied (in finite-dimensional X ) by Massera [I].
I 14.D. THEOREM. If ( I 10.2) has a solution x such that
n integral (in particular, such that x.+ is bounded), then @f lies in the closure of the range of I - U , . Proof.
From ( 1 12.1) we obtain 11-1
(1 14.1)
+ C lJl@f,
~ ( n= ) Ur~(0)
n
= 0,
I , ... .
t=O
Take any linear functional x * E ((I- U1)X)O,where the polar set is taken in X * (i.e., x * is in the null-space of (I - U,)*).T h e n ( U , . , x*) = (*, x*), so that (1 14.1) yields
II .(n) I1 !I x * II 3 I
(+)l
x*>
I
=
I W),x r >
+
<@h x r ) I.
Dividing by n and taking the limit inferior as n + 00 we find = 0. Since x* E ((I- U,)X)O was arbitrary,
(@f, x*)
@!€((I
-
U,)X)W = cl((f
-
Up). &
114.E. Under the assumption of Theorem I 14.D, f lies in the P-closure of the set of functions in P ( X ) for which ( I 10.2) has a periodic solution. I f , in addition, f E P C ( X ) , then f lies in the PC-closure of the corresponding set in P C ( X ) .
372
Ch. 11. EQUATIONS WITH
PERIODIC
A
Proof. T h e set in question is @-l((I - U,)X). By 112.A, Y is an isomorphism of X onto the subspace Y X of P(X), and this subspace has the complement @ - l ( { O } ) ; further, @-I( V ) = Y V @-]({O}) for any set V C X . By Theorem I14.D,
+
f € @-l(cl((l - U , ) X ) ) = Y cl((l - U , ) X ) + @-I({O}) cl,(Y(I - U , ) X ) + @-'({O}) = clp(Y(I - U,)X @-'({O})) =
+
=
clp(@-'((I
-
(I])X)).
T h e proof for f E PC(X) is entirely similar.
,j,
a3
Remark. It is not true in general that if @f belongs to the closure of ( I - U,)X but not to ( I - U,)X itself, (1 10.2) has a bounded solution: indeed, if X is reflexive, this is precluded by Theorem 114.C. The following question remains open:
co
Quay. If @ f ~cl((I - Cr,)X) \ ( I - U,)X, does ( I 10.2) always have a solution with lim infn+- 11 x(n)ll/n = 0 ?
We come to the main application of Theorem I14.D:
114.F. THEOREM.Assume that the range of I - U , is closed (or that any of the equivalent conditions of Theorem I12.C holds). If Eq. ( I 10.2) has a solution x with lim inf,,, 11 x(n)ll/n = 0 ( n integral) or, equivalently, lim inf,,, 11 x(t)ll/t = 0-in particular, a solution with bounded x+-, then it has a periodic solution. Proof. T h e equivalence of the assumptions on the limits inferior follows from the inequality (1 x(n)ll E(ll x(t)ll lM), valid, by t
<
<
+
+ If
CHAPTER 12
Higher-order equations 120. Introduction
Summary o f the chapter In all preceding chapters the linear differential equations that were considered were (30. I), (30.2), i.e., (120.1)
X+Ax=O
(120.2)
a+Ax=f
These are first-order equations (although the values of x and f lie in a Banach space X) in that only the first derivative of the solution is involved. In this chapter our objects of study are the higher-order equations
c A,,w(A) rri
(120.3)
w(r!itIl
+
=0
0
(1 20.4)
where m is some positive integer, and the values of w , h are in some Banach space W, while the values of each Ak are in It is usually claimed that (120.3), (120.4) may be reduced to (l20.1), (120.2) in an appropriate larger space X (namely the outer direct sum of m 1 copies of W ) , and that therefore the theory of the higherorder equations can be subsumed under that of the first-order equations. T h e first part of this statement is substantially correct, and we intend to describe the claimed reduction in Section 121; the last part, however, does not follow in general. This is best illustrated in our context by the concept of admissibility of a pair of function spaces for Eq. (120.4). If B, D E b.N( W )are given, 373
m.
+
374
Ch. 12. HIGHER-ORDER EQUATIONS
it is reasonable to call the pair (B, D) admissible for (1 20.4) if for every h E B there exists a solution w E D of (120.4). If we consider the corresponding “reduced” equation (1 20.2) in the direct-sum space X = WOO... 0 W,,, , say, this condition means that for every f with the mth “component” in B and all others 0 there exists a solution x of (120.2) with its 0th “component” in D. Now the f ’ s involved do form a space in b N ( X ) , but if B were an 9 - s p a c e or a F-space this would no longer be the case for the corresponding space off’s; a more serious obstacle in the way of relating this admissibility of (B, D) with some admissibility for (120.2) and then applying the results of previous chapters is the lack of a priori restrictions on any “component” of the solution x except the 0th. Fortunately, the difficulties just described can in the main be overcome, by means of some technical devices, in the most important case of .F-pairs and similar pairs, provided certain mild assumptions are made on the A k . I t thus becomes possible to establish results substantially similar to those in Chapter 6, relating the admissibility of a 9-pair, or a similar condition, to the behavior of the solutions of the homogeneous equation (120.3) and their derivatives. We remark that, with a somewhat greater effort at technical refinement, the same methods can be applied to systems of equations with each equation of a different order, yielding essentially the same results. We do not go into this matter here. Another, and more fundamental, discrepancy between Eqs. ( 120.9, (120.4) on the one hand and ( 1 20. l), ( I 20.2) on the other is the difficulty in setting up the concept of an adjoint equation in the former case. More precisely, if we reduce (120.3), say, to the form (120.1) in the usual way and then consider the corresponding adjoint equation i* - A*x* = 0, this equation does not have the form appropriate to the reduction of a presumptive “adjoint” equation to (120.3). On the other hand, the usual definition of such an adjoint equatioipresupposes a sufficient smoothness of the A, and is not symmetrical even if the possible asymmetry of W , W* is disregarded. The bilimr functional involved in the appropriate form of Green’s Formula is iself unsymmetrical and depends on t (cf. Hartman [l], Section 12). Hartman [ I ] has studied these questions and related ntters extensively; he bases his work on a very general abstract Ethod that is adaptable to other cases besides the subject of the 86ent chapter, not necessarily involving differential equations. Our p u r p in including this chapter is merely to give the reader a glimpse of tenature of the results and of the special problems they involve; it isherefore much more restricted, as concerns both the method and t scope of the
120. INTRODUCTION
375
results. We shall be content with a bare sketch of the central theorems of the type of those of Chapter 6, as referred to above. We omit all discussion, for the higher-order equations, of the questions dealt with in Chapters 7-11. We use several proofs in Hartman's paper, but do not attempt to describe his general method. We refer to that paper for further information, especially on the adjoint equations and on certain second-order equations with different assumptions on A,, A,. See also Hartman [2], Chapter XIII. Section 121 contains a discussion of Eqs. (120.3), (120.4) and their reduction to the forms (120.1), (120.2), and an analysis of the solutions. There is an important theorem (Theorem 121.B) giving bounds for the derivatives of a solution in terms of bounds ior the solution itself and for h. Section 122 deals with the concepts of admissibility and of (B,D)-subspaces for F-pairs. Section 123 contains the main theorems.
n th primitive functions In order to speak properly of a solution of (120.3) or (120.4) we must generalize the concept of a primitive function. We define recursively an nTH P R I M I T I V E ( F U N C T I O N ) on an interval J and with values in a Banach space X:f is a first primitive iff is a primitive; f is an ( n 1)st primitive if f is a primitive and f is an nth primitive, n = 1, 2, ... . I f f is an nth primitive, we denote by f ( k ) ,k = 0, ...,n , its ~ T DHE R I V A T I V E ,defined recursively by f ( O ) = f, f ( k + l )= ( f ( k ) ) ;' each, except the last, is an ( n - k)th primitive. We prove a lemma giving estimates for the derivatives f ( O ) , ...,f ( m ) of an ( m + 1)st primitive in terms off and f ( m + l ) .
+
120.A. For every positive integer m there exists a real number c, > 0 with the following property: iff : J + X is an ( m 1)st primitive, and a compact interval [ T , T A ] C J is given, then
+
+
111
(120.5)
Ah I!f("(t) I/
< c , ,i=O1l l f ( ~
+jd/m)(1
+ Am f" l ~ f ( m + l ~ (11udu,) T
~ E [ T , T + A ] , k = O ,..., m.
Proof, It is sufficient to verify the corresponding inequality (120.6)
Ch. 12. HIGHER-ORDER EQUATIONS
376
is a real-valued ( m + 1)st primitive on [0, 13. Indeed, if + d] and k are given, there exists x * E X * such that 11 x * 1) = 1, < f ( k ) ( t l )x*) , = Ilf(k)(tl)ll.We set ~ ( s )= Re(f(7 + sd), x*), so that gP)(s) = diRe(f(i)(T + sd), x*), i = 0, ..., m + 1, and (120.6) at s =
where
y~
t , E [T,T
(tl - ~ ) / d implies (120.5) at t = t , . We proceed to prove (120.6). We consider the interpolation polynomials r,, of degree m that satisfy r,,(i/m) = 6,, , i , j = 0, ..., m ; actually, r m j ( s )= l&+j (ms - i)/(j- z]. There exists a common bound c,, > 0 for all I rg; I,j, k = 0, ..., m, on the interval [0, 13. Now y~ = # + C L o ( p ( j / m ) n m jwhere , # is an ( m 1)st primitive that vanishes at s = j / m , j = 0 , ..., m. By Rolle's Theorem there exists sk E [o, 13 such that #(k)(sk)= 0, k = 0, ..., m. Therefore
+
Using induction on m
- k we find
Since rT1) = 0 for all j , we have follows. &
#(m+l) = ~ ( ~ + land ) ,
(120.6)
Remark. A detailed analysis of the polynomials rmjand their derivatives shows that c,,, may be taken t o be m m ( g )if m is even, and mm(i(m?,J if m is odd.
121. The ( m
+ 1)st-order equation
Although we might, of course, take a more general point of view, we agree that throughout this chapter the range of t shall be J = R , . We shall always consider a positive integer m, a Banach space W, and the functions Ak E L( k = 0, ..., m. If h E L(W ) , a SOLUTION of Eq. (120.4) is an (m + 1)st primitive w : R , + Wthat, together with its derivatives, satisfies (120.4), considered as an equation in L(W). For h = 0, we obtain a SOLUTION of (120.3).
w),
121. THE( m
+ I)ST-ORDER
377
EQUATION
I n order to “reduce” Eqs. ( 1 20.3), (120.4) to the forms ( 120. I), (1 20.2), m we introduce the Banach space X = 0 W, the outer direct sum o f o m m 1 times W , provided with the supremum norm; i.e., if x = @ xj E X , 0 xi E W (superscripts will denote the components of elements in X ) , then 11 x l I x = maxi 11 xj I l w ; there is no risk in omitting the subscripts of the norms from now on. We remark that this choice of norms is so convenient that we permit it to infringe the previous agreement that the norm in a finite-dimensional space X is assumed to be euclidean (Section 15, p. 32). For given Al:€L ( r ) , k = 0, ..., m, and h E L(W) we define the Eq. (120.2) in X , given by the system
+
‘
21 - xj+I = 0 ,
(121.1)
J =
0 , ..., m
-
1
911
A?‘
+ 1A p h = h, 0
as the RED.UCED FORM of (120.4). It is easy to see that A and that, with the given norm on X ,
Iljll
=
E
L(x),f E L(X)
II h II.
Equation (120.1) with the same A is the REDUCED FORM of (120.3). T h e precise relation between (120.4) and its reduced form (120.2) is given by the following lemma, the trivial proof of which is omitted.
+
121.A. A n ( m mI)st primitive w : R, 4 W is a solution of (120.4) and only x = @ w ( j ) is a solution of the reduced form (120.2). 0 Conversely, a primitive x : R, -+ X is a solution of (120.2) if and only if xo is an ( m I)st primitive and a solution of (120.4), and xj = xo(j), j = 0, ..., m. Consequently, for any to E R, and any xo E X there exists a unique solution w of (120.4) with w ( J ) ( t o= ) xj0 ’ j = 0, ..., m.
+
T h e fundamental tool allowing us to work with solutions of (120.4) and of its reduced form is the following estimate for the derivatives of a solution of (120.4). T h e number c,,, is the constant given by 120.A.
p
121.R. THEOREM. For given to 3 0 there exists a positive integer such that
= p(tJ
Ch. 12. HIGHER-ORDER EQUATIONS
378
for any such p , and any h E L(W ) ,eoery solution w of (120.4) satisfies ma
(121.4) 11 w Y t ) l (
< 2e!~'cn,p"C (1 4 s +j / m p ) 11 + 3e:"
II h ( r ) I/ dr,
["I
j=U
Y
O<s
K = O ,..., m,
s
[
(121.5) 11 d r ) ( t )I1 < 2 e i p (c,pnhilm
llw(u)ll du
+
t+1+1/nrp
Ilh(zr)ll did) ,
k = O ,..., m.
O
Proof. The existence of p = p(to) follows from the continuity of the left-hand member of (121.3) as a function of s. For given s, 0 s to there exists, on account of (l21.3), an integer i, 0 i < p , such that
< <
<
(121.6)
II A ( r ) (I dr
[:A
< 4,
=s
T
+ i/p,
A
=
l/p.
In
We consider the solution x = @ w ( j ) of the reduced form (120.2) of 0 (120.4), Using 120.A we find, for almost all t, 7 < t ,< 7 + d,
II w(n'+l)(t) II < I1 20) II < II 4 )II I1 JEWII
+ II
m
< I1 4I1 (cn,P"'x II
4 7
+ j / m p ) /I
1=0
I/
+ f'"
117~(m+1)(y)ll
dr) 4- Ilh(t)ll.
T
Integrating over the interval and using (121.6) we obtain
We use 120.A again, and find
< t < s + 1, (31.5), (121.2), (121.3) imply 11 x(t)ll < 11 h(r)ll dr), and this, together with (121.7), yields (121.4). Let now t,"O < t < to - limp, be given. Consider any s, t < s < t + limp < t o , and carry out the preceding argument as far as (121.7). Now 0 < s t < - t < (s + 1 lip) ( s - limp) < I and For any
e*P(ll
t,
+
x(~)ll
s
JH+'
-
7
-
-
121. THE(m
+ <
d s $- 1 ( 121.7) imply 7
+ I)ST-ORDEREQUATION
< t + 1 + l/mp.
379
Therefore (31.5), (121.2), (121.3),
Integrating the first and last members with respect to s over the interval [t, t limp], we obtain (121.5). Q
+
A first important consequence of Theorem 121.B is a closedness theorem analogous to Theorem 31.D. Let (Ak,,), k = 0, ..., m, (h,) be sequences in 121.C. THEOREM. L( @’), L( W), respectively, and let w, be some solution of
n = 1 , 2, ... .Zf the limits = A, , k = 0, ..., m,1imnjm Lhn = h, limn+mLw, = w exist, then the function w is (except for equivalence ) on every modulo a null set) a solution of (120.4), and wkk’ -+ w ( ~uniformly compact subinterval of R+ as n 400, for k = 0, ..., m.
Proof. With Eq. (121.8) we associate its reduced form
and with (120.4) its reduced form (120.2). Obviously, limnjm LA,n = A, m L f n = f. Now x, = @ w::) is a solution of (121.9). If we can k=O
prove that limn+mLx, = x exists the conclusion will follow at once from Theorem 31.D applied to the reduced forms, and from 121.A. We proceed to prove the L-convergence of (x,). (The actual proof anticipates a part of the proof of Theorem 3 I .D, as we shall see.) We choose an arbitrary tn > 1 and fix it for the time being. Since (A.,) converges in L ( x ) , there exists a positive integer p such that J8+l 11 A.,(r)\l dr < $p for all s, 0 < s < t, and n = I , 2, ...; then the
380
Ch. 12. HIGHER-ORDER EQUATIONS
limit A satisfies (121.3). From Theorem 121.B (specifically, (121.5)) applied to the solution w, of (121.8) we find
n = l , 2 ,....
0
Since (w,), (h,) converge in L( W) and towas arbitrarily large, we conclude that (x,) is uniformly bounded on every compact subinterval of R, . Now x, is a solution of i, Ax, = ( A - A,,)x, f, ,which is the reduced form of
+
+
m
,Arn+" + C A,W;'
= gn
k-0
+
where g, = (A, - A,,,)eoik) h, . T h e uniform boundedness of (xn) on every compact subinterval of R, and the convergence of (A,.,), (h,) in L(P), L(W), respectively, imply that (g,) converges in L(W) (its limit is, in fact, h). We again choose an arbitrary to > 1, and a corresponding p such that (121.3) is satisfied. Applying Theorem 121.B as before, but to w,, - wn as a solution of (120.4) with h replaced by g,. - g, , we find
II x n * ( t ) - x n ( t ) II ,< 2 e + (cmpm+'m ~
+
s'"" )I
g,*(u)
0
- g,,(u) 1) du) ,
t,+l 0
II wnc(u) - W n ( t l ) II du
0s t
< t" - 1
I
71, 71' =
1 2, 9
... .
Therefore(x,) converges uniformly on [0, to - I]. Since towas arbitrarily large, (x,) converges uniformly on every compact subinterval of R, , a fortiori in L(X). 9. Other consequences of Theorem 121.B pertain to the case in which A, E M( k = 0, ..., m ; this is equivalent, on account of (121.2), to A E M(a). T h e following lemma is a trivial consequence of Theorem 121.B.
r),
121.D. Assume that A, E M(P), k = 0, ..., m, and t h a t p is a positive integer such that p 21 A IM . Assume that h E L( W ) , and define the function H by H ( t ) = ':J 11 h(u)ll du. Then every solution w of (120.4) satisfies
122. ADMISSIBILITY AND (B, D)-MANIFOLDS
38 I
We can use this lemma to obtain an interesting result concerning solutions of (120.4) that belong to some space D( W), where D E b y K , Such solutions are of course again called D-SOLUTIONS.
m),
121.E. THEOREM. Assume that A, E M( k = 0, ..., m, and that m D E b y K. A solution w of (120.3) is a D-solution if and only if x = @ w ( j ) 0 is a D-solution of the reduced form (120.1). If h E L(W ) has compact support, or if h E B(W ) where B E b y or b y + or by%‘+ and D is thick with respect to B, then a solution w of (120.4) is a D-solution i f and only m if x = @ w ( j ) is a D-solution of the reduced form (120.2). 0
Proof. T h e “if” parts are obvious. Let p , H be as in 121.D. In the homogeneous equation, h = 0, hence H = 0, so that w ED(W) implies X E D ( X )by (l21.10), and in fact
If h has compact support, H has compact support and is continuous, so that H ED ; the last fact also holds if h E B(W ) and D is thick with respect to B. In both cases, (121.10) implies that if w E D ( W ) then EED(X). &
122. Admissibility and (B, D)-manifolds In the Introduction we mentioned that we should be interested in the “admissibility” of a Y-pair (B, D) with respect to (120.4), i.e., the existence of a D-solution of (120.4) for every h E B(W); and also in the extension to higher-order equations of the concept of a (B, D)-manifold. I t is tempting to regard the results of the preceding section, and especially Theorem 121.E, as a means of reducing these problems for Eq. (120.4) to corresponding problems for the reduced form (120.2), for which the machinery of Chapter 5 is available. This is indeed possible, but involves a certain amount of technical juggling of function spaces that would delay our reaching the important results. We therefore prefer to follow the course of paralleling, for the higher-order equation, the essential features of the first-order theory of Chapter 5, and using the reduced form as a convenient auxiliary device. We assume A, E L( k = 0, ..., m, given throughout the section. Although our attention will later be focused on Y-pairs and the like, the first few definitions and results are meaninful in a wider context.
m),
382
Ch. 12. HIGHER-ORDER EQUATIONS
For any space D E b M ( W ) , a solution w of (120.3) or (120.4) is a D-SOLUTION if w E D , a terminology anticipated in the preceding section. m We define the set X,,,, = {@ w ( j ) ( O ): w a D-solution of (l20.3)}, a linear manifold in X. In full, i?should be denoted by X,,,,(A, , ..., A,,&). An N-pair -more precisely, an Jv(W)-pair -(B, D ) is ADMISSIBLE (FORA , , ..., A,,L, or, more loosely, FOR (120.4)) if for every h E B there exists a D-solution w of (120.4). T h e pair is REGULARLY ADMISSIBLE if, in addition, X,,,) is closed. Concerning the concept of admissibility we merely establish the analogue of Theorem 5 I .A. 122.A. THEOREM. If the pair (B, D) is admissible for A , , ..., A , , , there exists a number K > 0 such that for every h E B and every number p > I there is a D-solution w of (120.4) with I w ID pKI h .
<
In
Proof. T h e proof is quite similar to that of Theorem 51.A (the essential change is the choice of the closedness theorem), but in view of the importance of the result we give it in full. Let V be the linear manifold of all D-solutions of (120.4) for all h E B,i.e., of all ( m l)st primitives w E D that satisfy w ( ~ + ~C" ) A , w ( ~E) B. T h e mapping 17 : V + B defined by I7w = ~ ( ~ + l ) A , w ' ~ )is linear and, by assumption, surjective. Since B,D are stronger than L(W), Theorem 121.C (with fixed A , , ..., A?,J implies that the graph of I7 is closed in D x B.Since B, D are Banach spaces, the Open-Mapping Theorem implies the existence of a number k > 0 such that for every h E B there exists w E n - l ( { h } ) with I w ID kl h T h e conclusion then holds with K equal to the infimum of all possible values of k. 9,
+
+ +
<
In.
We now specialize our study to the case we are actually interested in: we assume from now on that A , E M(W),k = 0, ..., m, and that (B, D) is a Y - p a i r , a .T+-pair, or a .T%'+-pair. Since D E b y K , Theorem 121.E implies (with the usual dropping of the arguments W , X) that (122.1)
Xn(DdAo
1
..., Ant)
= &,(A).
For this reason we simply write X,, . For such pairs as were just mentioned, we define the concept of a (B, D)-manifold as follows. (FORA , , ..., A , , or A linear manifold Y in X is a (B,D)-MANIFOLD FOR (120.4)) if Y C X n D , and if there exists a number K , > 0 such that for every h E k,B( W) and every number p > 1 there exists a solution m w of (120.4) such that, if x = 0w ' j ) is the corresponding solution 0 of (120.2), we have x z ( 0 ) E Y (whence x, E D(X), whence x E D(X),
122. ADMISSIBILITY AND (B, D)-MANIFOLDS
383
<
whence w E D (W))and I w 1, pKYlh In . A closed (B, D)-manifold is a (B,D)-SUBSPACE. We next make a quick survey of the essential theorems about (B, D)manifolds and (B, D)-subspaces, using as much as possible the analogy with the corresponding results in Section 52. 122.B. If Y is a (B, D)-manifold for A,, ..., A,, there exists a number C , 2 0 such that, i f h E k,B( W), p > 1, and w is a solution .of (120.4) m such that x = @ w(j) has xm(0)E Y and I w ID pKyI h I,, , then II x(0)ll PCYl h ;1 ’ Proof. If p is a positive integer, p 2 21 A IM, Theorem 121.B (formula 121.5)) implies, in particular,
<
<
I1 ~(0) I1 < 2eiP(cmpm+’mor(D;1
+ l/mP) Iw l o +
1
+ l/mP) Ih In),
so that the conclusion holds with
+
C Y = 2ef”{cn,pn1+1mKya(D; 1 I&)
+ a(B; 1 + l/ntp)).
122.C. Assume that Y is a (B, D)-manifold. Then: (a) Y is a (B, , D,)-manifold for any F-pair or F+-pair or Y W - p a i r (B, ,D,) (such that (kB, ,D1) is) weaker than (B, D ) ; (b) any linear manifold Z such that Y C 2 C X,, is also a (B, D)manifold; in particular, so is X,, itself; (c) if Y C X,,, , Y is a (B, kD)-manifold. Proof.
Same as for 52.B.
9,
T h e next few results allow us to restrict our attention substantially to F - p a i r s only.
A linear manifold Y in X is a 122.D. THEOREM. for A, , ..., A,, i f and only i f it is an (lcB, D)-manifold.
(B,D)-manifold
Proof. T h e proof follows the lines of that of Theorem 52.C, but moves back and forth between (120.4) and the reduced form. T h e “if” part is trivial by 122.C,(a). Assume that Y is a (B, D)-manifold; (lcB, D) is a F - p a i r or a F--pair. Let h E k,lcB(W) and p > 1 be given and set s = s(h). As in the proof of Theorem 52.C there existsg E B( W) such that II g I1 < Ilf II (whenceg E k,B(W) with s(g) < s) and I g le < I h llcs and
384
Ch. 12. HIGHER-ORDER EQUATIONS
+
By the assumption, there exists a solution v of dm+l) 1; =g m such that y = @ v ( j ) satisfies ym(0)E Y and I v 1, < pKYl gmlr < pKyl h llcB . Let k be the solution of (120.4) such that x = @ w ( j ) 0 satisfies x(s) = y(s), so that x, = y, , xm(0)E Y. I t then follows from (121.2) and (31.5) as in Theorem 52.C that I x - y ID < ( p - 1)1 h Ilea; thus
I
ID
ID
+ I w-v
ID
dI
ID
+ I "-.Y
10
d (pKY
+ (P-l))
I
llcB
.&
122.E. If Y is a (B, D)-manifold for A,, ..., A,, , then Y is a (T-lcB, D)-manifold (where the latter pair is a F - p a i r weaker than (B, D ) and T-lcB is locally closed). Proof. T h e proof is related to that of 52.J as the proof of Theorem 122.D is related to that of Theorem 52.C. T h e details are left to the reader. &
122.F. Assume that D E b y K is given. The linear manifold Y in X is a (B,D)-manifold for A,, ..., A, for some F - p a i r or Y j - p a i r or F P - p a i r (B, D ) if and only i f i t is a (T, D)-manifold. Proof. Same as for Theorem 52.K, using 122.C, 122.E instead of 52.B, 52.5. &
Our next result concerns the connection between admissibility and (B, D)-manifolds. 122.G. THEOREM. I f (B, D ) is admissible for A, , ..., A,,, , then X,, is a ( B ,D)-manifold. Proof. Same as the proof of Theorem 52.F (first part), using Theorem 122.A instead of Theorem 51.A. & Remark. We do not give the (valid) partial converse, analogous to the second part of Theorem 52.F.
Next, we obtain the fundamental theorem on (B, D)-subspaces, analogous to Theorem 52.1. I n this theorem, S , = S,,(A) for some subspace Y C X,,(A) (Theorem 33.B). 122.H. THEOREM. Assume that Y is a (B, D)-subspacefor A,, ..., A,. For every h E k,B( W )and every A > 1 [and every (Y, A)-splitting q of XI m there exists a solution w of (120.4) such that x = @ w ( j )satisfies xm(0)E Y 0 and 11 x(0)ll Ad( Y , x(0)) [q(x(O)) = x(O)]; every solution w with these
<
122. ADMISSIBILITY AND (B, D)-MANIFOLDS
385
Proof. T h e proof is almost identical with that of Theorem 52.1, but since that proof is itself referred to the proof of a previous theorem, we give our present proof explicitly. T o prove the existence of w in the first part of the statement, we let q be a ( Y ,A)-splitting of X . By the assumption, there exists a solution w‘ m of (120.4) such that x‘ = 0w’(i) has xL(0) E Y , and I w’ I D hKYlh IB. m 0 Let w be the solution of (120.4) such that x = @ w ( j ) satisfies x(0) = q(x‘(0)). Then q ( x ( 0 ) ) = x(O), 11 x(0)ll h d( Y , l(O)), and x - x‘ = x, - x6 is a solution with x,(O) - xL(0) = x(0) - x’(0) E Y . Therefore
<
<
Y. T o prove the second part, we let the solution w (and x) be as stated, and let p > 1 be given. By the definition and by 122.B there exists a solution w‘ of (120.4) that, with the corresponding x’, satisfies x‘(0) E Y , I w’ ID pKYlh I,, , 11 x’(0)ll < pCyl h l B . Now x - x‘ = x, - x, is a solution of (120. I ) with initial value in Y , so that 11 x(0)ll < h d( Y , x(0)) = h d( Y , x’(0)) < All x’(0)lI < phCyI h In . By Theorem 33.B, I w - w‘ ID < I x - 3’ ID S y I I ~ ( 0 ) x’(0)ll p( I X ) S y C y I h l e a Thus Xm(O) E
<
I
<
Since p
>I
<
.
+
was arbitrary, the conclusion follows.
&
Finall.y, we record the almost obvious fact that admissibility of a .F-pair or related pair for A implies admissibility of the same pair for A,, ..., A,,, , and that a similar implication holds with regard to (B, D)manifolds. 122.1. THEOREM. If (B, D ) is [regularly] admissible f o r A , it is [regularly] admissible f o r A,, ..., A,,, . If Y is u (B, D)-manifold for A , Y is a (B, D)-manifold for A,, ..., A,,, .
Proof. Assume that (B, D ) is admissible for A. If h E B(W), the corresponding f in the reduced equation (12b.2) satisfies f E B(X) by ( l 2 l . l ) , (121.2). By the assumption, the reduced equation has a D-solution, say x. By 121 .A, xo is a D-solution of (120.4), so that (B, D) is admissible for A, , ..., A,),. T h e conclusion about regular admissibility follows from (122.1). If Y is a (B, D)-manifold for A , and h E k,B( and p > 1, then f E k,B(X) and the reduced equation (120.2) has a solution x with ~ ~ (E 0Y ), I x ID pKYlf Is = pKYlh I, , where
w>
<
Ch. 1 1. HIGHER-ORDER EQUATIONS
386
m
K , = KYB,D(A).By 121.A, xo is a solution of (120.4), with @ xO‘j)= x, 0 and I xo ID I x ID < pK,1 h Is. Since by the assumption Y C X,D, Y is also a ( B ,D)-manifold for A,, ..., A,,, . &
<
123. The main theorems T h e results we intend to prove in this section correspond closely to those given in Chapter 6 (especially Section 64) for first-order equations. T h e same terminology about “direct” and “converse” theorems will be used: a “direct” theorem states the way in which the regular admissibility of a certain pair (B, D) or the existence of a ( B ,D)-subspace for A,, ..., A,, implies a certain type of behavior of the solutions of the homogeneous equation (120.3), or rather of the reduced form (120.1) -in our study, more precisely the existence of a dichotomy or an exponential dichotomy for A. “Converse theorem” has then the obvious meaning.
We only deal with the case in which A, E M(@’), k = 0, ..., m, an assumption we make throughout the section. (B, D ) always denotes a given 9 - p a i r or F+-pair orYV-+-pair. As we shall see, the main reason why our direct theorems do not have the neat aspect of those of Section 64, in spite of the assumption that A, E M( k = 0, ..., m, is the need to impose additional conditions on the coefficients A , , ..., A,; observe that A, is not affected by these additional conditions, and that they are satisfied anyway in the important case that A, E La( k = 1, ..., m. T h e converse theorems, by contrast, are immediate corollaries of the converse theorems of Chapter 6 as applied to the reduced forms of the homogeneous and nonhomogeneous equations.
m),
m),
123.A. THEOREM. Assume that Y is a ( B ,D)-subspace for A,, ...,A,,, (in particular, that ( B ,D ) is regularly admissible for A, , ..., A,rl and Y = XoD).Assume further that , Y [ ~ , ~ +E, ~T-lcB A , ( a space >, B ) for all t E R , , k = 1, ..., m, with (123.1)
“’tp
I X [ t . t + l ] A k ( T I C B < 00,
=
1,
...,
(in particular, that A, E L“(@), k = 1, ..., m, or that B is weaker than 0,Ll for some T E R+). Then Y induces a dichotomy for A. Proof. 1. We observe first that the parenthetical conditions that follow (123.1) do imply (123.1). If A,: E L “ ( m , then II X ~ ~ . ~I1 + , ~ A ~
<
123. THEMAIN
387
THEOREMS
I A , Ix[I,I+lIE T-lcB, since T-lcB E b y K(23.1,23.K, 24.1, 24.L,( I)), and I x [ ~ , ~ . ,IT-lcB . ~ ~ A<, ~ I A , IP(T-lcB; 1 ) and this bound does not depend
< <
on t . If B is weaker thin O,L1, then T-lcB is weaker than L', say L1 PT-lcB; and X [ l . f + l I A k E L'(W, I XrI.f+1IAkIT-lcB \< P Jf6+lII Ak(U)lldu pI A , l M , which again does not depend on t . Next, we may and do assume without loss that B E b y K and B is locally closed, since we may otherwise replace B on account of 122.E by the weaker space T-lcB, which has these properties. Once this assumption is made, B coincides with T-lcB, so that X[f,f+lIAk E B for all t~ R + , k = I , ..., m,and (123.1) is replaced by
Throughout the proof, p denotes a positive integer such that P 2 2 1 A l M *
2. In order to enter the proof proper, we choose a real-valued ( m 1)st primitive q~ 3 0 defined on R+ , vanishing outside [i,I], and such that 2,:J ~ ( tdt) = I . Let c > 0 be a common bound for I ~ ( I,~ k = 0, ..., m. We intend to show that statement (e) or (f) of Theorem 41.A is satisfied by Y with respect to A. Let h > I be given and let y , z be . solutions of (120.1) such that y(0) E Y , 11 z(0)lI Ad( Y , ~ ( 0 ) ) Let A 2 1 and 7 E R+ be chosen and define w by
+
<
+
are ( m I)st primitives and, by 121.A, yo(k)= Y , Now yo, z0, zk,k = 0, ..., m. Therefore
z O ( ~ )=
+ wr.(t),
K
= 0,
..., m
+ 1,
where
k
so that
= 0,
..., m
+ 1,
1
Ch. 12. HIGHER-ORDER EQUATIONS
388
implies that zrNow (123.3) , and (123.5) yields
+
w is a solution of (120.4) for h = v , , + ~
m
We set x = @ w ( j ) . Taking into account that each v k vanishes, by (123.5), outside'(7 + &I, 7 + A ) , k = 0, ..., m, it follows from (123.3) and the assumption on q that (123.7)
=
@,+AX
in particular, x, (123.8)
m+d,
= dy
X[0,7+&AlX
=
-~X[u.r+iAlz;
and x(0) = -dz(O), so that
Xm(0) E
y,
I/ x(0) I1 d M Y , x(0)).
3. Thus far, we have let d 2 1 be arbitrary, because we shall require the preceding argument in the next proof. But now we choose d = 1. We find that (123.6) implies, with (123.2) and (31.7), that h E koB(W )with I h IB 011 y(7) z(7)11, where (T = 2m+1eh(&B; 1) ak) does not depend on h, y, x, 7 . We may now apply Theorem 122.H to w , x in view of (123.8): we find I w ID < AKll h IB hoKkll Y ( T ) z(7)ll; by (123.7) we conclude that
+
<
xy
<
+
+
(1 23.9)
4. Let now t o , t E R, be given. If t 2 to we apply Theorem 121.B to yo (on account of (121.2), limp l/p 6 in formula (121.5)). We use (31.7) and (123.9) for 7 = t o , and find
<
<
+
II ~ ( t!I )< e*" II ~ ( t 1 ) II < 2ePCmPm+lm
< 2epcmpnr+1ma(D;3) I @ t 0 + i ~ O ID < D I l ~ ( t o )+ z(to) 11, where D = D(X) = 2he%r,,pm+1ma(D;+)OK;. If t < to we apply Theorem 121.B to xo, and use (31.7), and (123.9) for 7 = to + I , and find
< e-*PDI1 A t o + 1) + 4 1 , + 1) I1 d D II rcto)+ 4 t o )!I, with the same D is proved. &
=
D(h). Therefore condition (e) or (f) of Theorem 41.A
123. THEMAIN
389
THEOREMS
123.B. THEOREM. Let the assumptions of Theorem 123.A hold. Suppose, in addition, that (B, D) is not weaker than (L1,L,"); and that, ifD is weaker than Lz, then (123.10)
lirn d-'s;p
'1b.X
I ~[,,,+~p4,IT-l,-B
k = 1, ..., m
= 0,
(this last condition holds if A, E L"( @) or A , E T-lcB( @), k Then Y induces an exponential dichotomy for A.
=
1, ..., m).
Proof. 1. We observe that the parenthetical conditions following ( 123.10) do imply ( 1 23.10) when D is weaker than L," and consequently B is not stronger than L'. This is obvious, even without this assumption on B, when A, E T-lcB(m). If A, e L m ( r ) ,we use 23.K for the locally closed space T-lcB and find 0-' supl I X [ , , ~ +IT-lcB ~~A, d-'/3( T-lcB; d)l A, I 21 A, I/&( T-lcB; 0 ) = 21 A, 1/a(B; 0) -+ 0 as d + 00 (Remark 2 to 23.S, also applicable to B E bFV'). As in the proof of Theorem 123.A, we may and do assume that B E bF' and B is locally closed, since otherwise B may be replaced, on account of 122.E, by the weaker space T-lcB. Under this assumption, ( I 23. lo), when applicable, becomes
<
<
(123.1 1)
lim +=d-I syp I ~[,,,+,p4~, lB = 0.
A
k
=
1, ...,m.
T h e positive integer p and the real-valued function q~ are as in the preceding proof. By Theorem 123.A, Y induces a dichotomy for A ; let N o , N i = Ni(h) be the parameters in conditions (Di), (Dii) of the definition of a dichotomy. In order to prove the theorem, we show that Y satisfies condition (b) of Theorem 42.A for any given A > 1 ; we need only prove (Ei), (Eii), since (Diii) is satisfied on account of the ordinary dichotomy induced by Y .
2. We assume first that D is not weaker than Lr, so that (by 23.S) lim,+mB(D; 1) = 00. T o prove (Ei), we choose 1 >, 2 so large that B(D; 1) > 3 0 e f ~ ' ~ ~ p ~ + ~ m N , u K ; ,
where CT is as in part 3 of the preceding proof. Let y be any solution of (120.1) with y(0) E Y . We carry out parts 2 and 3 of the proof of Theorem 123.A for this y , for 0 instead of x, for d = 1, and for any T E R, . Since here h > 1 is arbitrarily close to 1 , ( 1 23.9) is replaced by
( I 23.12)
I @,do ID < OK; IIAd 11.
Ch. 12. HIGHER-ORDER EQUATIONS
390
Now we use Theorem 121.B and (123.12), (31.7), as well as 23.M, and find
II Y('
/
+ 1) II d NO(1 -
T+l
I/Y W II du Tfl
< 2 e * 1 t m p m + 1 m ~ , ( 1-
11-1
/ '+' ';1 du
11 y'(s) 11 ds
T+l
+ >
>
since 1 2 implies ( ( I - 1 ) / ( 1 + &))P(D;1 i) &?(D;I). It follows from (Di) and 20.C applied to ]I y 11-l (when y # 0) that (Ei) holds with N = 2 N 0 , v = I-' log 2. To prove (Eii), we consider A > 1 given, and choose I' = l'(A) >/ 2 so large that P(D;1') 2 30Ae*~c,pm+1mN~(h)oK~. Let z be any solution of (120.1) with 11 z(O)11 A d( Y , ~(0)).We carry out parts 2 and 3 of the proof of Theorem 123.A for this h and z, for 0 instead of y, for d = 1, and for T 1' instead of T . Proceeding as above and using Theorem 121.B, (123.9), (31.7), and 23.M, we find
<
+
I1 4 ') II < %(I'
- 11-l
J
5+L'-l
;I z(u) II du T
I t follows from (Dii) and 20.C that (Eii) holds with N'
=
2Ni(A),
v' = l'-l log 2. Observe that this value of v' depends on A; for this
reason we have proved condition (b) of Theorem 42.A, rather than condition (a). 3. We assume now that D is weaker than Lr;then B is not stronger than L1, so that (since B is locally closed) lim,,,d-'P(B;d) limA+m2 / 4 B ; d ) = 0 (23.K, Remark 2 to 23.S); further, (123.1 1) holds. T o prove (Ei), we choose d 2 I so large that
<
123. THEMAIN
39 1
THEOREMS
Let y be any solution of (120.1) with y(0)E Y. We carry out part 2 of the proof of Theorem 123.A for thisy, for 0 instead of z, for the specified d, and for any T E R+ . Now (Di) and (123.6) imply h E k,B(W) with
I
IS
+
11 Y ( T ) 11 (&D;
2ms'cN0
m
'YP
I X[t.t+dIAk
6=1
IS)
< 2m+1CNo€AIIY(T) 1).
Continuing as in part 3 of that proof, but with this A and with h > 1 arbitrarily close to I , we find I w ID ,< K;I h IS whence, using (123.7),
I @~+dy'
(1 23.13)
ID
I
d
ID
d 2n'+1cN0K&11 y ( T )11'
By Theorem 121.B and (123.13),
(1 y
+
( ~ A ) (1
< 2efpc,pnr+*m+;$/
I1Y " 4 I1du
< 2ef"r,pnb+1ma(D;2) I&@,, ID < 2my2etPc p'"+'ma(D; $)cN,K;EI I ~ ( T ) in
11
=
4 I]JJ(T) 11.
By (Di), and 20.C applied to (1 y (I-l (when y # 0), (Ei) holds with N = 2N0 ,v = d-1 log 2. T o prove (Eii), we consider h > 1 given, and choose A' = d'(h) 2 3 so large that
+c m
d'-'(fi(B; A')
I X[f.l+d']Ak
IS)
d
k=l
= (2m+3he~~c,pm+1ma(D; $)cN;(h)K;)-'.
<
Let z be any solution of (120.1) with 11 z(0)II h d(Y,z(0)).We carry out part 2 of the proof of Theorem 123.A for this A and z,for 0 instead of y, A' instead of A , and any T E R , . Now (Dii) and (123.6) imply h E k,B( W) with
Continuing as in part 3 of that proof, but with A' instead of 1, we find, using (1 23.7), (123.14)
I X [ O . . r + i)lZ0
ID
A'-1
392
Ch. 12. HIGHER-ORDER EQUATIONS
By Theorem 121.B and (123.14),
11 Z ( T ) 11
< 2et%,,,pm+lm I( z0(u)(1 du < 2etPc,pm+'ma(D; 3) I ~[~,,+ip" ID < 2m+zAet"cpm+lma(D;#)clVdK;e' I( + A') 11 < .$ )I + d') (1. m
Z(T
By (Dii), 20.C implies that (Eii) holds with N'= 2Ni, v'=
Z(T
A'-l log2.
&
As was mentioned at the beginning of this section, there is a great contrast between the preceding rather formidable direct theorems and the largely trivial converse theorems. I f the subspace Y of X induces a dichotomy for A, 123.C. THEOREM. then Y is an (Ll,L")-subspace for A,, ...,A, and (Ll, L") is admissible for A,, ..., A, . If X,, has finite codimension with respect to Y , then X,, is an (Ll,Lg)-subspace for A, , ..., A, and (Ll,L;P) is admissible for A,, ..., A, . Proof. Theorems 63.C, 63.E, and 122.1. & Combining Theorems 123.A and 123.C, we have: 123.D. THEOREM. A subspace Y of X induces a dichotomy for A i f and only i f Y is an (Ll,La)-subspace for A , , ..., A, , If X , is closed, X , induces a dichotomy for A if and only i f (Ll,L") is (regularly) admissible for A, , ..., A,; i f X,, is closed, X,, induces a dichotomy for A i f and only if (Ll,Lg) is (regularly) admissible for A,, ..., A,, . For exponential dichotomies the converse theorem is as follows. 123.E. THEOREM. If there exists a subspace of X inducing an exponential dichotomy for A , then ( B ,D ) is (regularly) admissible for A,, ..., A,,, i f D is thick with respect to B ; and there exists a ( B ,D)-subspace for A, , ..., A , (necessarily unique and = XoD)i f D is thick with respect to kB. Proof. Theorems 64.C and 122.1. & Remark. We do not discuss the assumptions on A,, ..., A, that would ensure the necessity of these conditions. Combination of Theorems 123.B and 123.E yields the following result.
A subspace Y of X induces an exponential dichot123.F. THEOREM. omy for A i f and only i f it is an (M, L")-subspace for A,, ..., A,; and i f a n d only i f Y = X , and (M, L") is (regularly) admissible for A, , ..., A,,, .
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2861. A. F. RUSTON I. Conjugate Banach spaces. R o c . Cambridge Philos. SOC.53 (1957), 576-580 [MR 19 (1958), 7551. JA. B. RUTICKI'~, see M. A. KRASNOSEL'SKI'~. S. SAKS I . Theory of the integral. Monografie Matematyczne, T. 7 (2nd ed.). Hafner, New York, 1937. C. SANSONE and R. CONTI 1. Equo2ioni diffmenziali non lineari. Cons. Naz. delle Ricerche, Monografie Mat. No. 3. Cremonese, Rome, 1956 [ M R 19 (1958), 5471. J. J. SCHAFFER(see also J.L. MASSERA) 1. Function spaces with translations. Math. Ann. 137 (1959), 209-262 [ M R 21 (ISM), #287]. 2. Addendum: Function spaces with translations. Math. Ann. 138 (1959), 141-144 [MR 21 (1960), #6529]. 3. Linear differential equations and functional analysis. V. Math. Ann. 140 (1960), 308-321 [ M R 22 (1961), #8182]. 4. A result in the theory of function spaces with translations. Univ. Replib. Far. Ingen. Agrimens. Monteoideo PI&. Inst. Mat. Estadist. 3 (1962), 147-151 [MR 27 (1964), #l8l8]. 5. Linear differential equations and functional analysis. VI. Math. Ann. 145 (1962), 354-400 [MR25 (1963), #4193]. 6. Linear differential equations and functional analysis. VII. Equations on (-a, 00). Math. Ann. 149 (1963), 1-32 [ M R 26 (1963), #6499]. 7. Linear differential equations and functional analysis. IX. Almost periodic equations. Math. Ann. 150 (1963), 111-118 [ M R 27 (1964), #5961]. 8. Linear differential equations and functional analysis. VIII. Math. Ann. 151 (1963), 57-100 [MR 2 7 (1964), #5960]. 9. On Floquet's Theorem in Hilbert spaces. Bull. Amer. Math. SOC.70 (1964), 243-245 [MR 28 (1964), #5243]. 10. On Floquet's Theorem in Banach spaces. Uniw. Reptib. Far. Ingen. Agrimens. Montevideo Publ.Inst. Mat. Estadist. 4 (1964), 13-21. 1 I . More about invertible operators without roots. R o c . Amer. Math. SOC.16 (1965), 213-219 [ M R 30 (1965). #3374]. J. T. SCHWARTZ, see N. DLJNFORD. K. SUNDARESAN I. Orthogonality in normed linear spaces. J . Indian Math. SOC.22 (1958), 99-107 [MR 21 (1960), #5881]. B. SZ-NAGY,see F. RIBZ. A. E. TAYLOR I . Introduction to functional analysis. Wiley, New York. 1958 [MR 20 (1959), #541 I]. A. TYCHONOFF 1. Ein Fixpunktsatz. Math. Ann. 111 (1945), 767-776. D. G. WERTHEIM, see G . G. LORENTZ. A. C. ZAANEN, see W. A. J. LUXEMBURG.
+
Index Author and subject Subject entries are made, in general, under a noun; see, in particular, “function”, “space”. Author entries do not include complete references; see pages 393-398.
A ACHIESER, N. I., see AHIEZER, N. I. Admissibility, see Pair, admissible regular, see Pair, regularly admissible a.e., 36 AHIEZER, N. I., 30 ALEXIEWICZ, A., 318 AMERIO,L., 346, 347 Angle, 27 ANTOSIEWICZ, H. A., xii ARLT,D., 371 Associate (to have), 90 Automorphism, 5 acceptable, I9
Case I, 11, 1, 2 , 11, 12, 111, 112, 186 J. A., 31, 85 CLARKSON, Closure local, 44 radial, 34 CODDINGTON, E. A., 98, 351 Complement, 5 Complexification, 367 Congruence, congruent, 5 CONTI,R., 98, 270 Coordinate system, right-handed, 260 COPPEL,W. A., 162, 164 CORDUNEANU, C., ix, 335 Coupling property, [quasi-]strict, I5
B Behavior, uniformly noncritical, see Solution(s), uniformly noncritical behavior of BELLMAN, R., viii, 162, 164, 269 BERKSON, E. R., 18, 20, 31 BOCHNER, S., 85, 335 BOHNENBLUST, H. F., 371 BOHR,H., 335, 337 BOURBAKI, N., 13, 14, 31, 83
C Carrier, 6 lower semicontinuous, 6
D DAY, M. M., 31, 143 Deformation family, F-,245 DELPASQUA, D., 31 Derivative, 85 kth, 375 along solutions (of equation), upper right, 317 total, almost majorized [minorized] (by function), 3 19 almost negative [positive] definite, 319 almost negative [positive] semidefinite, 319 399
INDEX lower left [right], 318 upper left [right], 3 17-3 I8 Dichotomy, I02 double, 279 double exponential, 279 exponential, I10 allowable parameters, see Parameters, allowable ordinary, 102 D I E U D O N NJ.,~ ,83 Dihedron, 4 (B,D)-, 274 closed, 5 inducing double [exponential] dichotomy, see Dichotomy, double [exponential] disjoint, 4 ( X , X’)-disjoint [(X’, X)-disjoint], 16 gaping, 10-1 I D-gaping, 273 perpendicdar, 29 Distance angular, 7 HausdorfT, 20 DIXMIER, J., 13, 15 DOMSLAK, Ju. I., x DOUADY, A., 371 DUNFORD, N., 31, 85
E ELLIS,H. W., 83 Endomorphism, 5 Envelope, 76 Epimorphism, 5 Equation adjoint, 90 associate, 90 reduced form of, see Form, reduced
M., 345 FR~CHET, Function comparison, 3 I7 gauge, 317 Ljapunov, see Ljapunov function noncollapsing to left [right], 73 positive, 36 primitive, see Primitive stiff, 39 weakly almost periodic, 347
Gape, I I GLASMANN, I. M., see GLAZMAN, I. M. GLAZMAN, I. M., 30 GOHBERC, I. C., 31 Green’s Formula, 91 Gronwall’s Lemma, 88, 98
H HALILOV, 2. I., x HALMOS, P. R., xii, 315 HALPERIN, I., 56, 83 HARTMAN, P., ix, x, xiii, 222, 374, 375 HILLE,E., 31, 80, 83, 90, 98, 126, 312, 329 Holder’s Inequality, 38, 51 Homomorphism, 5
I Interval, 36 (of functions), 36 Isomorphism, 5
K
F Family, F-deformation, see Deformation family, FFAVARD, J., 334, 335, 346, 369 FLOQIJET, G., 349 Floquet representation, of order rn, 351 Floquet’s Theorem, 351 Form, reduced (of equation), 377
S., 38 KAKIITANI, KOTHE,G., 83 KRABBE, G. L., 353 KRASNOSEL’SKI~, M. A., 31, 83 N. N., 122, 332 KRASOVSKI~, K R E ~ NM. , G., viii, 31, 162, 221, 269 K I I ~ E RD., L., viii, 162, 221, 269 KURATOWSKI, K., 20, 21
AUTHORAND L LEVINSON, N., 98, 351 Limit, S-, 4 LINDENSTRAUSS, J., 24 LJAPUNOV, A. M., 31 I Ljapunov function (generalized), 316 homogeneous of degree K , (absolutely),
SUBJECT
shift, 353 translation, 40 truncation, 40 Orientation (of real plane), 260 ORLICZ,W., 318
P
316 negative [positive] definite, 31 7 uniformly small (at 0), 3 I7 LORENTZ, G. G., 83 LUXEMBURG, W. A. J., 51, 83
M Main Lemma, 319-320 MAIZEL',A. D., viii, 122, 332 MALKIN, I. G., 122 Manifold (B,D)-, 139, 382 linear, 4 Mapping along solutions (of equation), 87 MARKUS, A. S., 31 MARKUS, L., 210 MASSERA, J. L., viii, ix, x, 31, 83, 96, 122,
123, 242, 3 12, 3 16, 323, 326, 328, 330, 332, 338, 345, 352, 356, 370, 371 MICHAEL, E., 6 MIL'MAN,D. P., 31 Module, 335 dense, 335 (of function), 335 Monomorphism, 5 MORSE,A. P., 85
N NAGUMO, M., 351
0
O'NEIL, R., 46 ONUCHIC, N., ix, x Opening, 31 Operator, 5 associate, 39
401
Pair(s), 125 admissible, 126, 382 equivalent, 125 3-,S K - , Sg-,125
."-, 125 regularly admissible, 127, 382 stronger [weaker] than (another), 125 Y-, Y + - , . T V - ,.F%'*-, 125 Parameters, allowable, 113 PERRON,O., viii, ix, 122, 162, 221, 269,
270 PERSIDSKI~, K. P., 122 PHILLIPS, R. S., 31,80, 83,90,98, 126,312,
330 Primitive, 85 nth, 375 Projection, 5 algebraic, 4 along M onto N, 4 associated with (dihedron), 4 Property, [quasi-]strict coupling, see Coupling property, [quasi-]strict
R RECHIS, M., 222 REG& M., see RECHIS,M. Representation, Floquet, see Floquet representation Representative, natural, 312 REUTER, G. E. H., 345 RICKART, C. E., 351, 352, 367 RIESZ,F., 31, 368, 369 RUSTON,A. F., 13, 14 RUTICKII,JA. B., 83
S SAKS,S., 320 SANSONE, G., 98
402
INDEX
SCHXFFER, J. J. ix, x, xi, 17, 31, 34, 38, 43, 44,47, 54, 55, 56, 58, 59, 61, 62, 63, 65, 66, 77, 19, 81, 83, 94, 96, 122, 123, 205, 211, 242, 253, 277, 281, 312, 316, 323, 326, 328, 330, 332, 338, 345, 351, 352, 353, 354, 356, 370, 371 SCHWARTZ, J. T., 31 Selection, continuous, 6 Sequence, increasing, 36 Set convex, balanced, 34 radially closed, 35 polar, 13 Solution(s), 84, 376 D-,92, 381, 382 invertible-valued, 87 uniformly noncritical behavior of, 122 Space(s) associate (of another), 52 coupled, 13 lean, 48, 56, 77, 79-80 norm-equivalent, 35 [quasi] full, 48, 56 [quasi] locally closed, 44 stronger [weaker] than (another), 35 thick (with respect to another), 71, 82 thin (with respect to another), 52, 78
A)-, 9 Splitting, (Y, Subspace, 5 (B,D)-, 139, 383 complemented, 5 inducing dichotomy [exponential dichotomy], see Dichotomy, exponential saturated, 14 SUNDARJSAN, K., 31 Supremum, essential, 36 Symmetrization, 74 SZ-NAGY,B., 31, 368, 369
T TAYLOR, A. E., 31, 38 Three-Angles Lemma, 8 A., 370 TYCHONOFF,
W WERTHEIM, D. G., 83
Z ZAANEN, A. C., 51, 83
Notation Symbols standing for generic numbers, functions, sets, spaces, etc., are omitted when consistent with intelligibility and do not affect alphabetical order when present.
(C),(C?, 3 16 C,jC, Co , 76; Co,81 C(X)inv, 80 cl, cl, , 4 Codim, 26 d(., .), 5 D+,D + ,D-, D - , 318 (Di), (Dii), (Diii), 102; (D'i), (D'ii), (D"i), (D"ii), 103; (D:i), (Dlii), 104; (DiiiH), 105 (DDi-), (DDi+), (DDii), (DD'-), (DD'+), 285; (DDiiH), 286 Dim, 26
403
NOTATION
e> 5 e, 76, 78 £„(), 312; 314 (Ei), (Eii), (Eiioo), (Eiiioo), 110; (E'i), (E'ii), (E'i), (E^ii), 111; (E "i), (E "ii), 112; (Eiii^H), H3 (EEi"), (EEi+), 285; (EE'"), ( E E ) , 286 ess inf, ess sup, ess lim, ess lim inf, ess lim sup, 36-37 f, 48, 56 F, 4 (F), 46; (F*), 55 ^, 46; & 49; & ± & 54; ^ ( • ) , 55; J^(-), 56 (FC), 76; (FC*), 79 ^f, 76; 78; « ^ ( ) , 79; ^ ± ( ) , 80 G+( ± 00), G ( ± 00), G~(± 00), G_( ± 00), 318 /, 4 inf, 36 k, k , 48, 56, 77, 79 K, KB,D(-)> > * V > ^ n u > ( ) > 139, 274, 382 m
c
c
/+
K %
t
K ± >
+
0
1 2 6
T, 6 1 - 6 2 ^ , 57;
58; F
59; JT , , 74; 75 $~<€^, 81; ^ ( ) , ^ f - ( - ) , 82 T r ( ) , 346 316 V V\ V' , V' , V'__, 317-318 * o , * o o > 93; AT , X ( ) , 93, 382; -^o(D)» -^o(D)(*» •••» ')» 382; -X±OD , *±ODO, 273; * 0 .aO, 337; A T J ( ) , 238 X ,93 (Z), 57; (Z^), 75 K)
+
^(), «^(), ^ 0), +
99
+
+
oD
o D
D
(a),
319
«(•; 70, 42; (0), 319
<*(•; /), 59; <*(•; 0), «(•; 00), 64
j8(;/), 61; « • ; < » , a>), 64 Y[*,y], 7; y [ y , ^ ] , y [ F , Z ] , 8 y(-; 70, 77 T(J), T ( B , D ; J ) , 168 T(a>), 260 A * ) , 41 8(-, ), 19; 8'(«, 0,20 ^(', ), 20 H(),# (),24 0('> ), 31 e , 40; e F , 58; 0 F()> 75 0
(L),(L'), 316 L, L ( ) , L ( ) , 38; L , 5 4 L°°, L (-), yL"(-), L°°() , L°°(), L«, 48 lc, 44, 55, 56, 80 lim^, 4 <0l(), 335 M, M , 60 7
T
±
p
7
37;
0
^ ^ ( • ) , y ^ ( 0 , 41
P y , 26 P ( ) , 349; P C ( ) , 350
K ' , ' W ' , ' ) , 19 i*, 4; R-,R
+i
36
rad, 35 *•(•). *-(•), *+(') (as in *(/), 5(F), etc.), 39-40 S , S ( - ) , 5 , 5 ( ) , 94 S, 61-62 sgn, 4-5 sup, 36 T~, 60, 75 T (on i*, on R ), 40; T+, T7, 40, 74; T+F, 58; T+F(), 75 (T), 57; (T-), 58; (T*), (T^), 75 r
T
A
(on J*), 69 ft, 36
A-,
^1+
*>*(•), n-)> in
£(0,
(N), 42
D
T
«(*, ), 11; * D , *D04; *, 0, 273 A > <> (on *+)> 69; 4, ,
r
K D
+
18-19, 30; Sc(-), 1 8 - 1 9 ; £(•, 8), S (-, 8), £(., 80, *c(-, 8'), 20; «£(•), 31 m, 26 *(', •), 14 *()> 5 36; x~> X > 54 «>(*; )» <»A > 260 £ (), 0 , £(•), Q , Od'), 237, 299; Q*, 299; £ , 0 ( ) , 241, 299; tf , 0 (-), ^ o l » ^olc » ^OlI » ^01 » , ^ 0 2 » 241; Q , 256, 301; £ , fioa, 299; Q ^CF(-), ^OCF , ^OCFO, 247; 5 % , 258; J ^ , 301 (as in r ) , 69, 168 _ + ± (as in
+
c
d
0
O
oc
Ff
C
+
OC
o¥
c F
F
+
T
+
+
+
404
* *a
INDEX I
331
139; , +m + m I~ 214 63 ‘ (as in F’), 50; (as in A’), 14, 89 * (as in X * ) , 5; (as in A * ) , 14, 90; +, 26 O, 13; 1,26 -3 5 ., 85 !I * 11, II . Iix, I . I, 4; I * I, I . I , , 31; I * IF, 37, 55, 18, 19 .(.I (as in F(X)), 55, 79 (., 26; (., .>,13 m m
c,
~
