Vector-valued Laplace transforms and Cauchy problems

Monographs in Mathematics Vol. 96 Managing Editors: H. Amann Universität Zürich, Switzerland J.-P. Bourguignon IHES, B...

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Monographs in Mathematics Vol. 96

Managing Editors: H. Amann Universität Zürich, Switzerland J.-P. Bourguignon IHES, Bures-sur-Yvette, France K. Grove University of Maryland, College Park, USA P.-L. Lions Université de Paris-Dauphine, France Associate Editors: H. Araki, Kyoto University F. Brezzi, Università di Pavia K.C. Chang, Peking University N. Hitchin, University of Warwick H. Hofer, Courant Institute, New York H. Knörrer, ETH Zürich K. Masuda, University of Tokyo D. Zagier, Max-Planck-Institut Bonn

Wolfgang Arendt Charles J.K. Batty Matthias Hieber Frank Neubrander

Vector-valued Laplace Transforms and Cauchy Problems Second Edition

Wolfgang Arendt Angewandte Analysis Universität Ulm 89069 Ulm Germany [email protected]

Charles J.K. Batty St. John’s College Oxford OX1 3JP UK [email protected]

Matthias Hieber Fachbereich Mathematik TU Darmstadt Schlossgartenstr. 7 64289 Darmstadt Germany [email protected]

Frank Neubrander Department of Mathematics Louisiana State University Baton Rouge, LA 70803 USA [email protected]

2010 Mathematics Subject Classification: 35A22, 46F12, 35K25 ISBN 978-3-0348-0086-0 e-ISBN 978-3-0348-0087-7 DOI 10.1007/978-3-0348-0087-7 Library of Congress Control Number: 2011924209 © Springer Basel AG 2011 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use permission of the copyright owner must be obtained. Cover design: deblik, Berlin Printed on acid-free paper Springer Basel AG is part of Springer Science+Business Media www.birkhauser-science.com

Contents Preface to the First Edition

ix

Preface to the Second Edition

xii

I

Laplace Transforms and Well-Posedness of Cauchy Problems

1

The Laplace Integral 1.1 The Bochner Integral . . . . . . . . . . . . . . . . 1.2 The Radon-Nikodym Property . . . . . . . . . . 1.3 Convolutions . . . . . . . . . . . . . . . . . . . . 1.4 Existence of the Laplace Integral . . . . . . . . . 1.5 Analytic Behaviour . . . . . . . . . . . . . . . . . 1.6 Operational Properties . . . . . . . . . . . . . . . 1.7 Uniqueness, Approximation and Inversion . . . . 1.8 The Fourier Transform and Plancherel’s Theorem 1.9 The Riemann-Stieltjes Integral . . . . . . . . . . 1.10 Laplace-Stieltjes Integrals . . . . . . . . . . . . . 1.11 Notes . . . . . . . . . . . . . . . . . . . . . . . .

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The 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8

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63 . 65 . 68 . 73 . 77 . 80 . 84 . 89 . 100

2

Laplace Transform Riesz-Stieltjes Representation . . . . . . . . . . . A Real Representation Theorem . . . . . . . . . Real and Complex Inversion . . . . . . . . . . . . Transforms of Exponentially Bounded Functions Complex Conditions . . . . . . . . . . . . . . . . Laplace Transforms of Holomorphic Functions . . Completely Monotonic Functions . . . . . . . . . Notes . . . . . . . . . . . . . . . . . . . . . . . .

1 5 6 15 21 27 32 36 40 44 48 55 59

vi 3

CONTENTS Cauchy Problems 3.1 C0 -semigroups and Cauchy Problems . . . . . . . 3.2 Integrated Semigroups and Cauchy Problems . . 3.3 Real Characterization . . . . . . . . . . . . . . . 3.4 Dissipative Operators . . . . . . . . . . . . . . . 3.5 Hille-Yosida Operators . . . . . . . . . . . . . . . 3.6 Approximation of Semigroups . . . . . . . . . . . 3.7 Holomorphic Semigroups . . . . . . . . . . . . . . 3.8 Fractional Powers . . . . . . . . . . . . . . . . . . 3.9 Boundary Values of Holomorphic Semigroups . . 3.10 Intermediate Spaces . . . . . . . . . . . . . . . . 3.11 Resolvent Positive Operators . . . . . . . . . . . 3.12 Complex Inversion and UMD-spaces . . . . . . . 3.13 Norm-continuous Semigroups and Hilbert Spaces 3.14 The Second Order Cauchy Problem . . . . . . . . 3.15 Sine Functions and Real Characterization . . . . 3.16 Square Root Reduction for Cosine Functions . . 3.17 Notes . . . . . . . . . . . . . . . . . . . . . . . .

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107 108 121 132 137 141 145 148 162 171 184 188 197 201 202 217 222 230

II

Tauberian Theorems and Cauchy Problems

4

Asymptotics of Laplace Transforms 4.1 Abelian Theorems . . . . . . . . . . . . . . . . . . 4.2 Real Tauberian Theorems . . . . . . . . . . . . . . 4.3 Ergodic Semigroups . . . . . . . . . . . . . . . . . 4.4 The Contour Method . . . . . . . . . . . . . . . . . 4.5 Almost Periodic Functions . . . . . . . . . . . . . . 4.6 Countable Spectrum and Almost Periodicity . . . . 4.7 Asymptotically Almost Periodic Functions . . . . . 4.8 Carleman Spectrum and Fourier Transform . . . . 4.9 Complex Tauberian Theorems: the Fourier Method 4.10 Notes . . . . . . . . . . . . . . . . . . . . . . . . .

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243 244 247 261 272 288 295 306 318 325 329

Asymptotics of Solutions of Cauchy Problems 5.1 Growth Bounds and Spectral Bounds . . . . . 5.2 Semigroups on Hilbert Spaces . . . . . . . . . 5.3 Positive Semigroups . . . . . . . . . . . . . . 5.4 Splitting Theorems . . . . . . . . . . . . . . . 5.5 Countable Spectral Conditions . . . . . . . . 5.6 Solutions of Inhomogeneous Cauchy Problems 5.7 Notes . . . . . . . . . . . . . . . . . . . . . .

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337 338 351 352 360 371 378 386

5

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239

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CONTENTS

III 6

vii

Applications and Examples

397

The 6.1 6.2 6.3 6.4

Heat Equation The Laplacian with Dirichlet Boundary Conditions Inhomogeneous Boundary Conditions . . . . . . . . Asymptotic Behaviour . . . . . . . . . . . . . . . . Notes . . . . . . . . . . . . . . . . . . . . . . . . .

7

The 7.1 7.2 7.3

Wave Equation 417 Perturbation of Selfadjoint Operators . . . . . . . . . . . . . . . . . 417 The Wave Equation in L2 (Ω) . . . . . . . . . . . . . . . . . . . . . 423 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 427

8

Translation Invariant Operators on Lp (Rn ) 8.1 Translation Invariant Operators and C0 -semigroups 8.2 Fourier Multipliers . . . . . . . . . . . . . . . . . . 8.3 Lp -spectra and Integrated Semigroups . . . . . . . 8.4 Systems of Diﬀerential Operators on Lp -spaces . . 8.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . .

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401 401 408 412 415

429 430 435 441 449 458

A Vector-valued Holomorphic Functions

461

B Closed Operators

467

C Ordered Banach Spaces

477

D Banach Spaces which Contain c0

481

E Distributions and Fourier Multipliers

485

Indexes Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

493 493 525 533

Prefaces Preface to the First Edition Linear evolution equations in Banach spaces have seen important developments in the last two decades. This is due to the many diﬀerent applications in the theory of partial diﬀerential equations, probability theory, mathematical physics, and other areas, and also to the development of new techniques. One important technique is given by the Laplace transform. It played an important role in the early development of semigroup theory, as can be seen in the pioneering monograph by Hille and Phillips [HP57]. But many new results and concepts have come from Laplace transform techniques in the last 15 years. In contrast to the classical theory, one particular feature of this method is that functions with values in a Banach space have to be considered. The aim of this book is to present the theory of linear evolution equations in a systematic way by using the methods of vector-valued Laplace transforms. It is simple to describe the basic idea relating these two subjects. Let A be a closed linear operator on a Banach space X. The Cauchy problem deﬁned by A is the initial value problem u (t) = Au(t) (t ≥ 0), (CP ) u(0) = x, where x ∈ X is a given initial value. If u is an exponentially bounded, continuous function, then we may consider the Laplace transform ∞ u ˆ(λ) = e−λt u(t) dt 0

of u for large real λ. It turns out that u is a (mild) solution of (CP ) if and only if (λ − A)ˆ u(λ) = x

(λ large).

(1)

Thus, if λ is in the resolvent set of A, then u ˆ(λ) = (λ − A)−1 x.

(2)

x

PREFACES

Now it is a typical feature of concrete evolution equations that no explicit information on the solution is known and only in exceptional cases can the solution be given by a formula. On the other hand, in many cases much can be said about the resolvent of the given operator. The fact that the Laplace transform allows us to reduce the Cauchy problem (CP ) to the characteristic equation (1) explains its usefulness. The Laplace transform is the link between solutions and resolvents, between Cauchy problems and spectral properties of operators. There are two important themes in the theory of Laplace transforms. The ﬁrst concerns representation theorems; i.e., results which give criteria to decide whether a given function is a Laplace transform. Clearly, in view of (2), such results, applied to the resolvent of an operator, give information on the solvability of the Cauchy problem. The other important subject is asymptotic behaviour, where the most challenging and delicate results are Tauberian theorems which allow one to deduce asymptotic properties of a function from properties of its transform. Since in the case of solutions of (CP ) the transform is given by the resolvent, such results may allow one to deduce results of asymptotic behaviour from spectral properties of A. These two themes describe the essence of this book, which is divided into three parts. In the ﬁrst, representation theorems for Laplace transforms are given, and corresponding to this, well-posedness of the Cauchy problem is studied. The second is a systematic study of asymptotic behaviour of Laplace transforms ﬁrst of arbitrary functions, and then of solutions of (CP ). The last part contains applications and illustrative examples. Each part is preceded by a detailed introduction where we describe the interplay between the diverse subjects and explain how the sections are related. We have assumed that the reader is already familiar with the basic topics of functional analysis and the theory of bounded linear operators, Lebesgue integration and functions of a complex variable. We require some standard facts from Fourier analysis and slightly more advanced areas of functional analysis for which we give references in the text. There are also four appendices (A, B, C and E) which collect together background material on other standard topics for use in various places in the book, while Appendix D gives a proof of a technical result in the geometry of Banach spaces which is needed in Section 4.6. Finally, a few words should be said about the realization of the book. The collaboration of the authors is based on two research activities: the common work of W. Arendt, M. Hieber and F. Neubrander on integrated semigroups and the work of W. Arendt and C. Batty on asymptotic behaviour of semigroups over many years. Laplace transform methods are common to both. The actual contributions are as follows. Part I: All four authors wrote this part. Part II was written by W. Arendt and C. Batty. Part III was written by W. Arendt (Chapters 6 and 7) and M. Hieber (Chapter 8).

PREFACES

xi

C. Batty undertook the coordination needed to make the material into a consistent book. The authors are grateful to many colleagues and friends with whom they had a fruitful cooperation, frequently over many years, which allowed them to discuss the material presented in the book. We would especially like to acknowledge among them H. Amann, B. B¨aumer, Ph. B´enilan, J. van Casteren, R. Chill, O. ElMennaoui, J. Goldstein, H. Kellermann, V. Keyantuo, R. deLaubenfels, G. Lumer, R. Nagel, J. van Neerven, J. Pr¨ uss, F. R¨abiger, A. Rhandi, W. Ruess, Q.P. V˜ u, and L. Weis Special thanks go to S. Bu, R. Chill, M. Haase, R. Nagel and R. Schnaubelt, who read parts of the manuscript and gave very useful comments. The enormous technical work on the computer, in particular typing large parts of the manuscript and unifying 4 diﬀerent TEX dialects, was done with high competence in a most reliable and eﬃcient way by Mahamadi Warma. To him go our warmest thanks. The authors are grateful to Professor H. Amann, editor of “Monographs in Mathematics”, for his support. The cooperation with Birkh¨ auser Verlag, and with Dr. T. Hintermann in particular, was most enjoyable and eﬃcient. Ulm, Oxford, Darmstadt, Baton Rouge August, 2000

Wolfgang Arendt Charles Batty Matthias Hieber Frank Neubrander

xii

PREFACES

Preface to the Second Edition Ten years after the publication of the ﬁrst edition of this monograph, it is clear that vector-valued Laplace transform methods continue to play an important role in the analysis of partial diﬀerential equations and other disciplines of analysis. Among the most notable new achievements of this period are the characterization of generators of cosine functions on Hilbert space due to Crouzeix, and quantitative Tauberian theorems for Laplace transforms with applications to energy estimates for wave equations. In this second edition, the new developments have been taken into account by updating the Notes on each Chapter and the Bibliography. For example, the characterization of generators of cosine functions on Hilbert space by a purely geometric condition on the numerical range is precisely stated in Theorem 3.17.5. The main text has not been substantially changed, except in Section 4.4 where some results are now presented in quantitative forms. Their applications in the study of damped wave equations are explained in detail in the Notes of the section. A few minor mathematical gaps and typographical errors have been corrected, and we are grateful to M. Haase, J. van Neerven, R. Schumann and D. Seifert for alerting us to some of them. September 2010

The Authors

Part I

Laplace Transforms and Well-Posedness of Cauchy Problems

Part I

3

As a guide-line for Part I, as well as for the entire book, we have in mind the formula u ˆ(λ) = R(λ, A)x (3) saying that a mild solution of the Cauchy problem, u (t) = Au(t) with initial value x, is given by the resolvent of the underlying operator A evaluated at x. Thus, if we want to ﬁnd solutions, we ﬁrst have to characterize those functions which are Laplace transforms; i.e., we study representation theorems. Correspondingly, on the side of evolution equations, we investigate existence and uniqueness of solutions of the Cauchy problem. Other subjects treated here include regularity and positivity. Part I contains three chapters as follows: 1. The Laplace Integral 2. The Laplace Transform 3. Cauchy Problems We start with an introduction to the vector-valued Lebesgue integral; i.e., the Bochner integral. For our purposes it suﬃces to consider functions deﬁned on the real line. Then we introduce the Laplace integral and investigate its analytic properties, giving special attention to its diverse abscissas. This will play an important role when solutions of the Cauchy problem are considered, as the abscissas give information about the asymptotic behaviour for large time. Operational properties of the Laplace integral are also discussed. Finally, we introduce functions of (semi) bounded variation deﬁned on the half-line and the Laplace-Stieltjes transform. They will be needed when we study resolvent positive operators (Section 3.11) and Hille-Yosida operators (Section 3.5). The vector-valued Fourier transform on the line is introduced in Section 1.6 and we prove the Paley-Wiener theorem for functions with values in a Hilbert space. This is the ﬁrst of several representation theorems for Laplace transforms which we present in this book. In Chapter 2, real representation theorems are the central subject. We prove a vector-valued version of Widder’s classical theorem which describes those functions which are Laplace transforms of bounded measurable functions. The vector-valued version (Section 2.2) will lead directly to generation theorems in Chapter 3 for semigroups and integrated semigroups (Section 3.3) and for cosine functions (Section 3.15). A particularly simple representation theorem is valid for holomorphic functions (Section 2.6). The Laplace transform is an isomorphism between certain classes of holomorphic functions deﬁned on sectors in the complex plane. This will lead directly to the generation theorem for holomorphic semigroups in Section 3.7. The third representation theorem is a vector-valued version of Bernstein’s theorem describing Laplace-Stieljes transforms of monotonic functions (Section 2.7). It

4

Part I

has its counterpart for Cauchy problems in Section 3.11 where resolvent positive operators are considered. The uniqueness theorem for Laplace transforms is easy to prove (Section 1.7), but it has striking consequences. It gives directly an approximation result for sequences of Laplace transforms. In Chapter 3 we ﬁnd its counterpart for Cauchy problems in the form of the Trotter-Kato theorem (Section 3.6). For Cauchy problems, the most satisfying situation is when there corresponds exactly one (mild) solution to each initial value. This notion of well-posedness is equivalent to existence of a C0 -semigroup (Section 3.1). We also consider weaker forms of well-posedness which are characterized by the existence of integrated semigroups. In applications, they allow one to describe precise regularity for certain partial diﬀerential equations in Lp (Rn ), and Chapter 8 is devoted to this. Here in Part I, there are three situations where integrated semigroups occur in a natural way. Operators satisfying the Hille-Yosida condition generate locally Lipschitz continuous integrated semigroups. Using convolution properties established in Section 1.3, we prove a beautiful existence and uniqueness theorem due to Da Prato and Sinestrari for the inhomogeneous Cauchy problem deﬁned by such operators. The second interesting class of examples are resolvent positive operators which always generate twice integrated semigroups. This will be proved in Section 3.11. In Chapter 6 a resolvent positive operator will provide an elegant transition from elliptic to parabolic problems. Finally, in Section 3.14 we show that the second order Cauchy problem is well-posed on a space X if and only if the associated canonical system generates an integrated semigroup on the product space X × X. In Section 3.10 we show that integrated semigroups and semigroups are equivalent, up to the choice of the underlying Banach space. This choice is particularly interesting in the context of the second order Cauchy problem. In Section 3.14 we will show the remarkable result that the space of well-posedness is unique, and we ﬁnd the phase space associated to the second order problem. In the applications to the wave equation given in Chapter 7 we will see how this space is well adapted to perturbation theory, allowing us to prove well-posedness of the wave equation deﬁned by very general second order elliptic operators. Special attention is given to C0 -groups; i.e., to Cauchy problems allowing unique mild solutions on the line. In Section 3.9 we study when a holomorphic semigroup of angle π/2 has a boundary group. This problem will occur again in Section 3.16 where we investigate which cosine functions allow a square root reduction. A striking theorem due to Fattorini shows that on UMD-spaces a square root reduction is always possible; i.e., each generator A of a cosine function is of the form A = B 2 − ω where B generates a C0 -group and ω ≥ 0. This beautiful result concludes the three sections on the second order Cauchy problem, applications of which will be given in Chapters 7 and 8.

Chapter 1

The Laplace Integral The ﬁrst three sections of this chapter are of a preliminary nature. There, we collect properties of the Bochner integral of functions of a real variable with values in a Banach space X. We then concentrate on the basic properties of the Laplace integral ∞

fˆ(λ) :=

τ

e−λt f (t) dt := lim

τ →∞

0

e−λt f (t) dt

0

for locally Bochner integrable functions f : R+ → X. In Section 1.4 we describe the set of complex numbers λ for which the Laplace integral converges. It will be shown that the domain of convergence is non-empty if and only if the antiderivative of f is of exponential growth. In Section 1.5 we discuss the holomorphy of λ → fˆ(λ) and in Section 1.7 we show that f is uniquely determined by the Laplace integrals fˆ(λ) (uniqueness and inversion). In Section 1.6 we prove the operational properties of the Laplace integral which are essential in applications to diﬀerential and integral equations. In particular, we show that the Laplace integral of the convolution t k ∗ f : t → 0 k(t − s)f (s) ds of a scalar-valued function k with a vector-valued function f is given by ˆ fˆ(λ) (k ∗ f )(λ) = k(λ) ˆ if fˆ(λ) exists and k(λ) exists as an absolutely convergent integral. In Section 1.8 we consider vector-valued Fourier transforms and we show that Plancherel’s theorem and the Paley-Wiener theorem extend to functions with values in a Hilbert space. Finally, after introducing the basic properties of the Riemann-Stieltjes integral in Section 1.9, we extend in Section 1.10 the basic properties of Laplace integrals to Laplace-Stieltjes integrals ∞ τ (λ) := dF e−λt dF (t) := lim e−λt dF (t) 0

τ →∞

0

of functions F of bounded semivariation.

W. Arendt et al., Vector-valued Laplace Transforms and Cauchy Problems: Second Edition, Monographs in Mathematics 96, DOI 10.1007/978-3-0348-0087-7_1, © Springer Basel AG 2011

5

6

1. THE LAPLACE INTEGRAL

If f is Bochner integrable, then the normalized antiderivative t → F (t) := (λ) f (s) ds is of bounded variation. We will see that fˆ(λ) exists if and only if dF 0 (λ). Thus, the Laplace-Stieltjes integral is a natexists, and in this case fˆ(λ) = dF ural extension of the Laplace integral. This extension is crucial for our discussion of the Laplace transform in Chapter 2 since there are many functions r : (ω, ∞) → X which can be represented as a Laplace-Stieltjes integral, but not as a Laplace integral of a Bochner integrable function. Examples are, among others, Dirichlet series ∞ (λ), where F is the step function ∞ an χ(n,∞) , or r(λ) = n=1 an e−λn = dF n=1 (λ), where F is of bounded semivariation, but not the any function r(λ) = dF antiderivative of a Bochner integrable function. t

1.1

The Bochner Integral

This section contains some properties of the Bochner integral of vector-valued functions. We shall consider only those properties which are used in later sections, and we shall assume that the reader is familiar with the basic facts about measure and integration of scalar-valued functions. Let X be a complex Banach space, and let I be an interval (bounded or unbounded) in R, or a rectangle in R2 . A function f : I → X is simple if it is n of the form f (t) = r=1 xr χΩr (t) for some n ∈ N := {1, 2, . . .}, xr ∈ X and Lebesgue measurable sets Ωr ⊂ I with ﬁnite Lebesgue measure m(Ωr ); f is a step function when each Ωr can be chosen to be an interval, or a rectangle in R2 . Here χΩ denotes the characteristic (indicator) function of Ω. In the representation of a simple function, the sets Ωr may always be arranged to be disjoint, and then xr (t ∈ Ωr ; r = 1, 2, . . . , n) f (t) = 0 otherwise. A function f : I → X is measurable if there is a sequence of simple functions gn such that f (t) = limn→∞ gn (t) for almost all t ∈ I. Since any χΩ for Ω measurable is a pointwise almost everywhere (a.e.) limit of a sequence of step functions, it is not diﬃcult to see that the functions gn can be chosen to be step functions. When X = C, this deﬁnition agrees with the usual deﬁnition of (Lebesgue) measurable functions. It is easy to see that if f : I → X, g : I → X and h : I → C are measurable, then f + g and h · f are measurable. Moreover, if k : X → Y is continuous (where Y is any Banach space), then k ◦ f is measurable whenever f is measurable. In particular, f is measurable. If X is a closed subspace of Y , and f is measurable as a Y -valued function, then f is also measurable as an X-valued function. To verify measurability of a function we often use the characterization given by Pettis’s theorem below. We say that f : I → X is countably valued if there is a countable partition {Ωn : n ∈ N} of I into subsets Ωn such that f is constant on each Ωn ; it is easy to see that f is measurable if each Ωn is measurable (and

1.1. THE BOCHNER INTEGRAL

7

conversely, {t ∈ I : f (t) = x} is measurable whenever f is measurable and x ∈ X). Also, f : I → X is called almost separably valued if there is a null set Ω0 in I such that f (I \ Ω0 ) := {f (t) : t ∈ I \ Ω0 } is separable (equivalently, f (I \ Ω0 ) is contained in a separable closed subspace of X); f is weakly measurable if x∗ ◦ f : t → f (t), x∗ is Lebesgue measurable for each x∗ in the dual space X ∗ of X. Here and subsequently, ·, · denotes the duality between X and X ∗ . For a subset D of X, we shall denote the closure of D in X by D. For x ∈ X and ε > 0, we shall let BX (x, ε) := B(x, ε) := {y ∈ X : y − x < ε} and B(x, ε) := {y ∈ X :

y − x ≤ ε}. We shall also use this notation when X = Rn or X = C, when it will be implicit that the norm is the Euclidean norm. Theorem 1.1.1 (Pettis). A function f : I → X is measurable if and only if it is weakly measurable and almost separably valued. Proof. If f is measurable, then there exist a null set Ω0 and simple functions gn such that gn → f pointwise on I \ Ω0 . The simple functions x∗ ◦ gn converge to x∗ ◦ f on I \ Ω0 for all x∗ ∈ X ∗ . Therefore, f is weakly measurable. The values taken by the functions gn form a countable set D and f (I \ Ω0 ) ⊂ D. Thus, f is almost separably valued. To prove the converse statement one may replace X by the smallest closed subspace which contains f (I \ Ω0 ) and then choose a countable dense set {xn : n ∈ N}. By the Hahn-Banach theorem, there are unit vectors x∗n ∈ X ∗ with |xn , x∗n | = xn . For any ε > 0 and x ∈ X there exists xk such that x − xk < ε. Hence, sup |x, x∗n | n

≤

x ≤ xk + ε = |xk , x∗k | + ε

≤ |x − xk , x∗k | + |x, x∗k | + ε ≤ sup |x, x∗n | + 2ε. n

So

x = sup |x, x∗n | n

for all x ∈ X.

(1.1)

This implies that t → f (t)−x = supn |f (t)−x, x∗n | is measurable for all x ∈ X. Let Δ := {t ∈ I \ Ω0 : f (t) > 0} and Δn,ε := {t ∈ Δ : f (t) − xn < ε} for ε > 0 and n ∈ N. The sets Δn,ε are measurable and n Δn,ε = Δ. For ﬁxed ε > 0, the sets Ω1,ε := Δ1,ε and Ωn,ε := Δn,ε \ k
f (t) − gε (t) < ε

for all t ∈ I \ Ω0 .

8

1. THE LAPLACE INTEGRAL

This shows that f is the uniform limit almost everywhere of a sequence of measurable, countably valued functions. Let (In ) be an increasing sequence of bounded subintervals of I with I = I . For each n ∈ N, deﬁne a simple function hn := g2−n χHn , where Hn := In ∩ nkn n −n −n and kn is chosen such that the Lebesgue measure m(In \ Hn ) < 2 . i=1 Ω i,2 ∞ If t ∈ n=k Hn for some k ≥ 1, then

f (t) − hn (t) = f (t) − g2−n (t) < 2−n ∞ ∞ for all n ≥ k. Thus, limn→∞ hn (t) = f (t) for all t ∈ k=1 n=k Hn . For k ≥ j, ∞ ∞

m Ij \ Hn ≤ m(In \ Hn ) < 2−k+1 . n=k

n=k

∞ ∞

Hence, Ij \ k=1 n=k Hn is null, for each j. This shows that limn→∞ hn (t) = f (t) for almost all t ∈ I. Corollary 1.1.2. Let f : I → X. Then the following statements hold: a) The function f is measurable if and only if it is the uniform limit almost everywhere of a sequence of measurable, countably valued functions. b) If X is separable, then f is measurable if and only if it is weakly measurable. c) If f is continuous, then it is measurable. d) If fn : I → X are measurable functions and fn → f pointwise a.e., then f is measurable. Proof. The statement b) is an immediate consequence of Pettis’s Theorem 1.1.1. For d), observe ﬁrst that f is weakly measurable. Deﬁne Ω0 := ∪n Ωn where Ωn is a null set such that fn (I \ Ωn ) is separable. Then m(Ω0 ) = 0 and Δ := n fn (I \ Ω0 ) is separable. Since the least closed subspace containing Δ is separable and contains f (I \ Ω0 ) it follows that f is almost separably valued. Thus, f is measurable. If f is continuous, then f is weakly measurable and the countable set {f (t) : t ∈ Q} is dense in the range of f . Again by Pettis’s theorem, f is measurable. One implication of a) was established in the proof of Pettis’s theorem and the converse follows from d). Pettis’s theorem can be improved considerably in the following way. A subset W of X ∗ is called separating if for all x ∈ X \ {0} there exists x∗ ∈ W such that x, x∗ = 0. Corollary 1.1.3. Let f : I → X be an almost separably valued function. Assume that x∗ ◦ f is measurable for all x∗ in a separating subset W of X ∗ . Then f is measurable.

1.1. THE BOCHNER INTEGRAL

9

Proof. Changing f on a set of measure 0 and replacing X by a subspace, we can assume that X is separable. Let Y := {x∗ ∈ X ∗ : x∗ ◦ f is measurable}. Then Y is a subspace of X ∗ which contains W . By the Hahn-Banach theorem, Y is weak* dense in X ∗ . Let Y1 = Y ∩ B X ∗ (0, 1). We show that Y1 is weak* closed. Let x∗ be in the weak* closure of Y1 . Since X is separable, the weak* topology on B X ∗ (0, 1) is metrizable (see [Meg98, Theorem 2.6.23], for example). Thus, there exists a sequence (x∗n )n∈N in Y1 converging to x∗ . Hence, x∗n ◦ f → x∗ ◦ f as n → ∞ pointwise on I. Thus, x∗ ◦ f is measurable; i.e., x∗ ∈ Y1 . This proves the claim. It follows from the Krein-Smulyan theorem (Theorem A.6) that Y is weak* closed. Since Y is weak* dense, we have Y = X ∗ ; i.e., f is weakly measurable. Now the result follows from Theorem 1.1.1. n For a simple function g : I → X, g = i=1 xi χΩi , we deﬁne n g(t) dt := xi m(Ωi ) I

i=1

where m(Ω) is the Lebesgue measure of Ω. It is routine to verify that the deﬁnition n is independent of the representation g = i=1 xi χΩi , and the integral so deﬁned is linear. A function f : I → X is called Bochner integrable if there exist simple functions gn such that gn → f pointwise a.e., and limn→∞ I f (t) − gn (t) dt = 0. If f : I → X is Bochner integrable, then the Bochner integral of f on I is f (t) dt := lim gn (t) dt. n→∞

I

I

It is easy to see that this limit exists and is independent of the choice of the sequence (gn ). If Ω is measurable with ﬁnite measure, then χΩ can be approximated in L1 -norm by step functions, and it follows that the functions gn can always be chosen to be step functions. The integral I f (t) dt lies in the closed linear span of {f (t) : t ∈ I}. The set of all Bochner integrable functions from I to X is a linear space and the Bochner integral is a linear mapping. When X = C, the deﬁnitions of Bochner integrability and integrals agree with those of Lebesgue integration theory. When I is a rectangle, we may denote a Bochner integral by I f (s, t) d(s, t). It is one of the great virtues of the Bochner integral that the class of Bochner integrable functions is easily characterized. Theorem 1.1.4 (Bochner). A function f : I → X is Bochner integrable if and only if f is measurable and f is integrable. If f is Bochner integrable, then

f (t) dt ≤ f (t) dt. (1.2)

I

I

10

1. THE LAPLACE INTEGRAL

Proof. If f is Bochner integrable, then there exists an approximating sequence of simple functions gn . Thus f and f are measurable. The integrability of f

follows from

f (t) dt ≤ gn (t) dt + f (t) − gn (t) dt. I

Moreover,

I

I

f (t) dt = lim gn (t) dt

n→∞

I

≤

I

=

lim

gn (t) dt n→∞ I

f (t) dt. I

To prove the converse statement, let (hn ) be a sequence of simple functions approximating f pointwise on I \ Ω0 , where m(Ω0 ) = 0. Deﬁne simple functions by hn (t) if hn (t) ≤ f (t) (1 + n−1 ), gn (t) := 0 otherwise. Then gn (t) ≤ f (t) (1 + n−1 ) and limn→∞ gn (t) − f (t) = 0 for all t ∈ I \ Ω0 . Because the functions f and gn − f are integrable and gn (t) − f (t) ≤ 3 f (t) , we can apply the scalar dominated convergence theorem and obtain that limn→∞ I gn (t) − f (t) dt = 0. Example 1.1.5. a) Let X be the Lebesgue space L∞ (0, 1) of all (equivalence classes of) bounded measurable functions from (0, 1) to C. Let f : (0, 1) → L∞ (0, 1) be given by f (t) := χ(0,t) . Then f is not almost separably valued since f (t)−f (s) = 1 for t = s. Thus, f is not measurable and therefore not Bochner integrable. b) Let X be the Banach space c0 of all complex sequences x = (xn )n∈N such that ∗ 1 limn→∞ xn = 0, with x = supn |xn |. Identify ∞ X with the space of all complex sequences a = (an )n∈N such that a := n=1 |an | < ∞. Let f : [0, 1] → c0 be given by f (t) := (fn (t))n∈N where fn (t) := nχ[0, n1 ] (t). Let x∗ = (an )n∈N ∈ 1 . Then ∞ t → f (t), x∗ = n=1 nan χ[0, n1 ] (t) is measurable on [0, 1]. Since c0 is separable, it follows from Pettis’s theorem that f is measurable. Moreover, 1 ∞ |f (t), x∗ | dt ≤ |an | = x∗ < ∞. 0

n=1

1 However, f (t) = n for t ∈ ( n+1 , n1 ], so t → f (t) is not integrable on [0, 1]. Thus, f is not Bochner integrable on [0, 1].

Now we will consider the behaviour of the Bochner integral under linear operators. The following result is a straightforward consequence of the deﬁnition of the Bochner integral, and we shall use it frequently without comment, especially in the case of a linear functional (Y = C).

1.1. THE BOCHNER INTEGRAL

11

Proposition 1.1.6. Let T : X → Y be a bounded linear operator between Banach spaces X and Y , and let f : I→ X be Bochner integrable. Then T ◦f : t → T (f (t)) is Bochner integrable and T I f (t) dt = I T (f (t)) dt. We shall also need a version of Proposition 1.1.6 for a closed operator A on X (see Appendix B for notation and terminology). Proposition 1.1.7. Let A be a closed linear operator on X. Let f : I → X be Bochner integrable. Suppose that f (t) ∈ D(A) for all t ∈ I and A ◦ f : I → X is Bochner integrable. Then I f (t) dt ∈ D(A) and A

f (t) dt =

I

A(f (t)) dt. I

Proof. Consider X × X as a Banach space in the norm (x, y) = x + y . The graph G(A) of A is a closed subspace of X × X. Deﬁne g : I → G(A) ⊂ X × X by g(t) = (f (t), A(f (t))). It is easy to see that g is measurable and

g(t) dt = f (t) dt + A(f (t)) dt < ∞. I

I

I

By Theorem 1.1.4, g is Bochner integrable. Moreover, I g(t) dt ∈ G(A). Applying Proposition 1.1.6 to the two projection maps of X × X onto X shows that g(t) dt = f (t) dt , A(f (t)) dt . I

I

I

This gives the result. Now we give vector-valued versions of two classical theorems of integration theory. Theorem 1.1.8 (Dominated Convergence). Let fn : I → X (n ∈ N) be Bochner integrable functions. Assume that f (t) := limn→∞ fn (t) exists a.e. and there exists an integrable function g : I → R such that fn (t) ≤ g(t) a.e. for all n ∈ N. Then f is Bochner integrable and f (t) dt = limn→∞ I fn (t) dt. Furthermore, I

f (t) − f (t)

dt → 0 as n → ∞. n I Proof. The function f is Bochner integrable since it is measurable (by Corollary 1.1.2) and since f is integrable (because f (t) ≤ g(t) a.e.). Deﬁne hn (t) :=

f (t) − fn (t) for t ∈ I. Since |hn (t)| ≤ 2g(t) and hn (t) → 0 a.e., the scalar dominated convergence theorem implies that I f (t) − fn (t) dt → 0 as n → ∞. By (1.2),

f (t) dt − fn (t) dt → 0.

I

I

12

1. THE LAPLACE INTEGRAL

Theorem 1.1.9 (Fubini’s Theorem). Let I = I1 × I2 be a rectangle in R2 , let f : I → X be measurable, and suppose that

f (s, t) dt ds < ∞. I1

I2

Then f is Bochner integrable and the repeated integrals f (s, t) dt ds, f (s, t) ds dt I1

I2

I2

I1

exist and are equal, and they coincide with the double integral

I

f (s, t) d(s, t).

Proof. Since any measurable function is almost separably valued, we may assume that X is separable. The scalar-valued case of Fubini’s theorem implies that f is integrable on I, I2 f (s, t) dt exists for almost all s ∈ I1 , and for each x∗ ∈ X ∗ the repeated integrals f (s, t), x∗ dt ds, f (s, t), x∗ ds dt I1

I2

I2

I1

exist and are equal. It follows from Theorem 1.1.4 that f : I → X is Bochner integrable and I2 f (s, t) dt exists for almost all s ∈ I1 , and from Theorem 1.1.1 that s → I2 f (s, t) dt is measurable. Moreover,

f (s, t) dt ds ≤

f (s, t) dt ds < ∞,

I1

I2

so Theorem 1.1.4 shows that

I1

I1

I2

I2

f (s, t) dt ds exists. Since

f (s, t) ds dt =

I2

I1

it follows similarly that

f (s, t) dt I1

I2

I2

f (s, t) ds dt exists. For any x∗ ∈ X ∗ , I1

I2

f (s, t) dt ds, I1

ds, x∗

=

f (s, t), x∗ dt ds

I1 I2 = f (s, t), x∗ d(s, t) I = f (s, t) d(s, t), x∗ I = f (s, t), x∗ ds dt I2 I1 ∗ = f (s, t) ds dt, x . I2

I1

1.1. THE BOCHNER INTEGRAL

13

The Hahn-Banach theorem implies that f (s, t) dt ds = f (s, t) d(s, t) = f (s, t) ds dt. I1

I2

I

I2

I1

Let L1 (I, X) denote the space of all Bochner integrable functions f : I → X, and let

f 1 := f (t) dt. I

In the usual way, we shall identify functions which diﬀer only on sets of measure zero. Then · 1 is a norm on L1 (I, X). Theorem 1.1.10. The space L1 (I, X) is a Banach space. Proof. Let (fn ) be a sequence in L1 (I, X) with

fn 1 < ∞. Bythe monotone convergence theorem for series of positive scalar-valued functions,

fn (t) < ∞ ∞ a.e., n=1 fn (·) is integrable, and ∞

fn (t) dt =

I n=1

∞ n=1

fn (t) dt.

I

∞ Hence, n=1 fn (t) converges a.e. to a sum g(t) in the Banach space X. By Corol∞ lary 1.1.2, g is measurable. Moreover, g(t) ≤ n=1 fn (t) , so g is integrable. By Theorem 1.1.4, g is integrable. Finally,

k k ∞

fn ≤ g(t) − fn (t) dt ≤

fn (t) dt → 0

g −

I I n=1

1

n=1

n=k+1

as k → ∞. Thus, L1 (I, X) is a Banach space. By the deﬁnition of Bochner integrability, the simple functions are dense in L1 (I, X) and, by the remarks following the deﬁnition, the step functions are dense. It follows easily that the inﬁnitely diﬀerentiable functions of compact support are also dense in L1 (I, X). We shall be particularly interested in the case when I = R+ := [0, ∞). If f ∈ L1 (R+ , X), an application of the Dominated Convergence Theorem shows that ∞ τ f (t) dt = lim f (t) dt. (1.3) 0

τ →∞

0

L1loc (R+ , X)

When f ∈ (i.e., f is Bochner integrable on [0, τ ] for every τ ∈ R+ ), the limit in (1.3) may exist ∞ without f being Bochner integrable on R+ . If the limit exists, we say that 0 f (t) dt converges as an improper (or principal value) integral, and we deﬁne ∞ τ f (t) dt := lim f (t) dt. 0

τ →∞

0

14

1. THE LAPLACE INTEGRAL

∞ When f ∈ L1 (R+ , X), i.e. 0 f (t) dt < ∞, we say that the integral is absolutely convergent . For 1 < p < ∞, let Lp (I, X) denote the space of all measurable functions f : I → X such that

1/p

f p :=

f (t) p dt

< ∞.

I

Let L∞ (I, X) be the space of all measurable functions f : I → X such that

f ∞ := ess sup f (t) < ∞. t∈I

Note that the spaces Lp (I, C) (1 ≤ p ≤ ∞) are the usual Lebesgue spaces which we shall denote simply by Lp (I). With the usual identiﬁcations, each of the spaces Lp (I, X) becomes a Banach space. The proofs of completeness are similar to the scalar-valued cases. The proof of Theorem 1.1.4 shows that the simple functions are dense in Lp (I, X) for 1 < p < ∞ (so Lp (I, X) can also be deﬁned in a similar way to the Bochner integrable functions). It follows that the step functions, and the inﬁnitely diﬀerentiable functions of compact support, are also dense. By considering such functions ﬁrst and then approximating, one may show as in the scalar-valued case, that if f ∈ Lp (I, X) and f (s − t) if s − t ∈ I, ft (s) := 0 otherwise, then t → ft is continuous from R into Lp (I, X) for 1 ≤ p < ∞. We have presented the theory above in the case when I is an interval in R (or, for Fubini’s theorem, I is a rectangle in R2 ). Almost all the integrals which appear in this book will indeed be over intervals in R (or repeated integrals in R2 ). However, the entire theory works, with no changes in the proofs, when I is a measurable set in Rn (or in Rm ×Rn , in Fubini’s theorem). Since the step functions are dense in each of the spaces Lp (I × J, X) for 1 ≤ p < ∞, it is easy to see from Fubini’s theorem that there is an isometric isomorphism between Lp (I × J, X) and Lp (I, Lp (J, X)) given by f → g, where (g(s))(t) := f (s, t). This enables many properties of the spaces Lp (I, X) when I is a rectangle in Rn to be deduced from the case n = 1. Finally in this section, we introduce notation for spaces of continuous and diﬀerentiable functions. Let I be an interval in R. We denote by C(I, X) the vector space of all continuous functions f : I → X. For k ∈ N, we denote by C k (I, X) the space of all k-times diﬀerentiable functions with continuous kth derivative, and we

1.2. THE RADON-NIKODYM PROPERTY

15

∞ put C ∞ (I, X) := k=1 C k (I, X). When I is compact, C(I, K) is a closed subspace of L∞ (I, X) and therefore a Banach space with respect to the supremum norm

· ∞ . When I is not compact, we let Cc (I, X) be the space of all functions in C(I, X) with compact support, and Cc∞ (I, X) := Cc (I, X) ∩ C ∞ (I, X). Thus Cc∞ (I, X) is a dense subspace of Lp (I, X) for 1 ≤ p < ∞. When I = R+ or I = R, we shall also consider the space C0 (I, X) of all continuous functions f : I → X such that lim|t|→∞,t∈I f (t) = 0 and the space BUC(I, X) of all bounded, uniformly continuous functions f : I → X. These are both Banach spaces with respect to

· ∞ , and C0 (I, X) ⊂ BUC(I, X) ⊂ L∞ (I, X). When X = C, we shall write C(I) in place of C(I, C), etc., and we shall extend this notation to cases when I is replaced by an open subset Ω of Rn . Note that Cc∞ (Ω) coincides with the space D(Ω) of test functions on Ω (see Appendix E), and we shall use both notations according to context. Furthermore, when Ω is any locally compact space, we shall let C0 (Ω) be the Banach space of all continuous complex-valued functions on Ω which vanish at inﬁnity, with the supremum norm. When K is any compact space, we shall let C(K) be the Banach space of all continuous complex-valued functions on K, with the supremum norm.

1.2

The Radon-Nikodym Property

In this section we consider properties of functions F obtained as indeﬁnite integrals. If f : [a, b] → X is Bochner integrable, we say that F : [a, b] → X is an antiderivative or primitive of f if

t

F (t) = F (a) +

f (s) ds

(t ∈ [a, b]).

a

Given a function F : [a, b] → X and a partition π, a = t0 < t1 < . . . < tn = b, let V (π, F ) :=

n

F (ti ) − F (ti−1 ) .

i=1

Then F is said to be of bounded variation if V (F ) := V[a,b] (F ) := sup V (π, F ) < ∞, π

where the supremum is taken over all partitions π of [a, b]. We say that F isabsolutely continuous on [a, b] if for every ε > 0 there exists δ > 0 such that i F (bi ) − F (ai ) < ε for every ﬁnite collection {(ai , bi )} of disjoint intervals in [a, b] with i (bi − ai ) < δ. We say that F is Lipschitz continuous if there exists M such that F (t) − F (s) ≤ M |t − s| for all s, t ∈ [a, b]. Clearly, any Lipschitz continuous function is absolutely continuous.

16

1. THE LAPLACE INTEGRAL

Proposition 1.2.1. Let F : [a, b] → X be absolutely continuous. Then F is of bounded variation. Moreover, if G(t) := V[a,t] (F ), then G is absolutely continuous on [a, b]. Proof. Take ε > 0, and let δ be as in the deﬁnition of absolute continuity of F . Then V (F ) ≤ ε whenever {(ai , bi )} is a ﬁnite collection of disjoint subintervals [a i ,bi ] i of [a, b] with i (bi − ai ) < δ. In particular, F is of bounded variation on any subinterval of length less than δ. Since [a, b] is a ﬁnite union of such intervals, F is of bounded variation on [a, b]. Moreover,

|G(bi ) − G(ai )| =

i

V[ai ,bi ] (F ) < ε.

i

Thus, G is absolutely continuous. A point t ∈ [a, b] is said to be a Lebesgue point of f ∈ L1 ([a, b], X) if t+h limh→0 h1 t f (s) − f (t) ds = 0. It is easy to see that any point of continuity is a Lebesgue point, and the following proposition shows that almost all points are Lebesgue points. t Proposition 1.2.2. Let f : [a, b] → X be Bochner integrable and F (t) := a f (s) ds (t ∈ [a, b]). Then a) F is diﬀerentiable a.e. and F = f a.e. b) limh→0

1 h

t+h t

f (s) − f (t) ds = 0 t-a.e.

c) F is absolutely continuous. d) V[a,b] (F ) =

b a

f (s) ds.

Proof. To show a) and b) let gn be step functions such that f (t) = lim gn (t) a.e. and

lim

n→∞

For h > 0,

1 t+h

f (s) ds − f (t)

h t

≤ ≤

1 h 1 h +

n→∞

t+h

b

f (t) − gn (t) dt = 0.

a

f (s) − f (t) ds

t

t+h

f (s) − gn (s) ds

t

1 h

t+h t

gn (s) − gn (t) ds + gn (t) − f (t) .

1.2. THE RADON-NIKODYM PROPERTY

17

Since gn is a step function and s → fn (s) − gn (s) is Lebesgue integrable, it follows from Lebesgue’s theorem [Rud87, Theorem 8.17] that

1 t+h

1 t+h

lim sup f (s) ds − f (t) ≤ lim sup

f (s) − f (t) ds

h t

h t h↓0 h↓0 ≤ 2 gn (t) − f (t)

for all t ∈ [a, b] \ Ωn and some null set Ωn . Taking the limit as n → ∞ yields the right-diﬀerentiability of F and 1 lim h↓0 h

t+h

f (s) − f (t) ds = 0

t

for all t ∈ [a, b] \ n∈N Ωn . The left-hand limits are similar. For c), let ε > 0. There exists δ > 0 such that Ω f (s) ds < ε whenever collection of disjoint subintervals of [a, b] with μ(Ω) < δ. If {(ai , bi )} is a ﬁnite i (bi − ai ) < δ, then taking Ω = i (ai , bi ), we deduce that

b

i

F (bi ) − F (ai ) = f (s) ds ≤

f (s) ds < ε.

ai

Ω i

i

To prove the statement d), observe ﬁrst that, for any partition π of [a, b],

t b

i

V (π, F ) = f (s) ds ≤

f (s) ds.

ti−1

a i

b

Thus, V (F ) ≤ a f (s) ds. Conversely, given ε > 0, we may choose a step function b g such that a f (s) − g(s) ds < ε. There is a partition π of [a, b] such that g is constant on each interval (ti−1 , ti ). Then a

b

f (s) ds − V (F )

≤

b

f (s) ds − V (π, F )

b

t

i

≤

g(s) ds + ε − f (s) ds

ti−1

a i

ti

t

i

= g(s) ds − f (s) ds + ε

ti−1

ti−1

i b ≤

f (s) − g(s) ds + ε a

a

≤ 2ε. Since ε > 0 is arbitrary, this completes the proof of d).

18

1. THE LAPLACE INTEGRAL

In the scalar case, the fundamental theorem of calculus [Rud87, Theorem 8.18] states that any absolutely continuous function F : [a, b] → C is diﬀerentiable t a.e., f := F is Lebesgue integrable, and F (t) − F (a) = a f (s) ds for all t ∈ [a, b]. We will see below (Example 1.2.8) that the fundamental theorem does not hold for Lipschitz continuous functions with values in arbitrary Banach spaces. The following weaker statement holds for all Banach spaces. Proposition 1.2.3. Let F : [a, b] → X be absolutely continuous, and suppose that t f (t) := F (t) exists a.e. Then f is Bochner integrable and F (t) = F (a)+ a f (s) ds for all t ∈ [a, b]. Proof. Since f (t) = limn→∞ n(F (t + 1/n) − F (t)), it follows from Corollary 1.1.2 that f is measurable. Let G(t) := V[a,t] (F ), so G : [a, b] → R is absolutely continuous by Proposition 1.2.1. Hence G is diﬀerentiable a.e. and G ∈ L1 [a, b]. Since

F (t + h) − F (t) ≤ V[t,t+h] (F ) = G(t + h) − G(t),

f (t) ≤ G (t) a.e. Hence f ∈ L1 [a, b], so f is Bochner integrable by Theorem 1.1.4. For x∗ ∈ X ∗ , F (t), x∗ = =

t F (a), x∗ + f (s), x∗ ds a t F (a) + f (s) ds, x∗ a

by the scalar fundamental theorem of calculus. By the Hahn-Banach theorem, t F (t) = F (a) + a f (s) ds. Let I be any interval in R. A function F : I → X is said to be absolutely continuous if it is absolutely continuous on each compact interval of I. We now consider the property that every absolutely continuous function F : I → X is diﬀerentiable a.e. It is easy to see that this property is independent of the interval I, so it is a property of X alone. Proposition 1.2.4. For any Banach space X the following are equivalent: (i) Every absolutely continuous function F : R+ → X is diﬀerentiable a.e. (ii) Every Lipschitz continuous function F : R+ → X is diﬀerentiable a.e. Proof. Clearly, (i) implies (ii). Assume that statement (ii) holds and let F : R+ → X be absolutely continuous. By Proposition 1.2.1, F is locally of bounded variation and G is absolutely continuous where G(t) := V[0,t] (F ). Let h(t) := G(t) + t. Then h is strictly increasing, h(0) = 0, and h(R+ ) = R+ . Moreover,

F (t) − F (s) ≤ G(t) − G(s) ≤ h(t) − h(s)

1.2. THE RADON-NIKODYM PROPERTY

19

for all t ≥ s ≥ 0. Hence, F ◦h−1 : R+ → X is Lipschitz continuous. By assumption, F ◦ h−1 is diﬀerentiable a.e. Since |h(t) − h(s)| ≥ |t − s|, h−1 maps null sets to null sets. Moreover, h is diﬀerentiable a.e., since G is absolutely continuous. It follows from the chain rule that F is diﬀerentiable a.e. Deﬁnition 1.2.5. A Banach space X is said to have the Radon-Nikodym property if the equivalent conditions of Proposition 1.2.4 are satisﬁed. By the remarks above, the space X has the Radon-Nikodym property if and only if every Lipschitz continuous function F : [0, 1] → X is diﬀerentiable almost everywhere. It is clear that a closed subspace of a space with the Radon-Nikodym property has the Radon-Nikodym property. Next we exhibit a class of spaces having the Radon-Nikodym property. Theorem 1.2.6 (Dunford-Pettis). Let X = Y ∗ where Y is a Banach space, and suppose that X is separable. Then X has the Radon-Nikodym property. Proof. By Proposition 1.2.4, it suﬃces to show that every Lipschitz function F : R+ → X is diﬀerentiable a.e. We may assume that F (0) = 0 and the Lipschitz constant is 1, so that F (t) − F (s) ≤ |t − s|. For y ∈ Y , the function y, F (·) : R+ → C is Lipschitz with Lipschitz constant y , so there exists gy ∈ L∞ (R+ ) such that gy ∞ ≤ y and t y, F (t) = gy (s) ds (t ∈ R+ ). 0

Moreover, gy is unique up to null sets. Since Y ∗ is separable, Y is also separable (see [Meg98, Theorem 1.12.11]). n Let D be a countable dense subset of Y . Suppose that y = i=1 αi yi for some n ∈ N, αi ∈ Q + iQ and yi ∈ D. Then t n n y, F (t) = αi yi , F (t) = αi gyi (s) ds, so gy (s) =

n i=1

0 i=1

i=1

αi gyi (s) a.e. Hence,

n n

αi gyi (s) ≤ α i yi

i=1

(1.4)

i=1

for almost all s ∈ R+ . This holds for all possible n ∈ N, αi ∈ Q + iQ and yi ∈ D, but there are only countably many such possibilities. Hence there is a null subset Ω of R+ such that (1.4) holds for all s ∈ R+ \ Ω and all n ∈ N, αi ∈ Q + iQ and yi ∈ D. It follows immediately that (1.4) holds for all αi ∈ C. This shows that for s ∈ R+ \ Ω, the map y → gy (s) from D to C extends to a unique f (s) ∈ Y ∗ = X with f (s) ≤ 1. For y ∈ D, y, f (·) is measurable and bounded, and t y, F (t) = y, f (s) ds. (1.5) 0

20

1. THE LAPLACE INTEGRAL

By density of D and the dominated convergence theorem, (1.5) is valid for y ∈ Y . Since Y is weak* dense in X ∗ = Y ∗∗ and f is separably valued, it follows from Corollary 1.1.3 that f : R+ → X is measurable. Since f is bounded, f is locally Bochner integrable and it follows from (1.5) that t F (t) = f (s) ds. 0

By Proposition 1.2.2 a), F is diﬀerentiable a.e. Corollary 1.2.7. Every reﬂexive space has the Radon-Nikodym property. Proof. Since a continuous function has separable range, it suﬃces to show that every separable reﬂexive space has the Radon-Nikodym property. This follows from Theorem 1.2.6. Next we give some examples of spaces which do not have the Radon-Nikodym property. Example 1.2.8. a) Let X = C[0, 1] and deﬁne F (t − s)χ[0,t] (s). For 0 ≤ t1 ≤ t2 ≤ 1, ⎧ ⎪ ⎨t2 − t1 F (t2 )(s) − F (t1 )(s) = t2 − s ⎪ ⎩ 0

: [0, 1] → C[0, 1] by F (t)(s) := (0 ≤ s ≤ t1 ), (t1 < s ≤ t2 ), (t2 < s ≤ 1).

Thus F (t2 ) − F (t1 ) ∞ = |t2 − t1 |, so F is Lipschitz continuous. However, limh→0 h1 (F (t + h) − F (t)) does not exist in the norm topology, so C[0, 1] does not have the Radon-Nikodym property. b) It follows from a) that L∞ (0, 1) does not have the Radon-Nikodym property either. However, when F is as in a) and L∞ (0, 1) is identiﬁed with L1 (0, 1)∗ , the weak* derivative Fw∗ (t) := w*- limh→0 h1 (F (t + h) − F (t)) = χ[0,t] exists in ∞ L (0, 1). Note that Fw∗ is not measurable and therefore not Bochner integrable (Example 1.1.5 a)). We shall see in Section 1.9 that Fw∗ is Riemann integrable t and F (t) = 0 Fw∗ (s) ds in the sense of Riemann integrals. More generally, it follows from the next result that every Banach space containing a closed subspace isomorphic to c0 (see Example 1.1.5 b) and Appendix D) does not have the Radon-Nikodym property. Proposition 1.2.9. The space c0 does not have the Radon-Nikodym property. Proof. Let F (t) := (Fn (t))n∈N , where Fn (t) := n1 sin(nt) (n ∈ N). Then F : R+ → c0 is Lipschitz continuous since t |Fn (t) − Fn (s)| = cos(nr) dr ≤ |t − s| (t, s ≥ 0, n ∈ N). s

1.3. CONVOLUTIONS

21

However, F is nowhere diﬀerentiable, since Fn (t) = cos(nt) and (cos(nt))n∈N ∈ c0 . Thus, c0 does not have the Radon-Nikodym property. It follows from Theorem 1.2.6 that the space l1 = c∗0 has the Radon-Nikodym property. However, L1 (0, 1) does not, even though c0 is not a closed subspace of L1 (0, 1) (see Appendix D). Proposition 1.2.10. The space L1 (0, 1) does not have the Radon-Nikodym property. Proof. In fact, deﬁne F : [0, 1] → L1 (0, 1) by F (t) = χ[0,t] . Then F is clearly Lipschitz continuous. Let 0 < t < 1. Then F is not diﬀerentiable at t. In fact,

1

(F (t + h) − F (t)) − 1 (F (t + 2h) − F (t)) = 1

h

2h 1 for 0 < h < (1 − t)/2.

1.3

Convolutions

For k, h ∈ L1 (R), standard arguments with Fubini’s theorem (see [Rud87, Theorem 7.14]) show that the convolution (k ∗ h)(t) := k(t − s)h(s) ds R

exists a.e. and k∗h ∈ L1 (R). Moreover, convolution is commutative and associative. In this section, we consider convolutions involving vector and operator-valued functions. If k : R → C and f : R → X are measurable, we deﬁne the convolution by (k ∗ f )(t) = k(t − s)f (s) ds (1.6) R

whenever this exists (as a Bochner integral). Since (k ∗ f )(t) = we may write f ∗ k in place of k ∗ f .

R

k(s)f (t − s) ds,

Proposition 1.3.1. Let k, h ∈ L1 (R) and f ∈ L1 (R, X). Then a) (k ∗ f )(t) exists for almost all t ∈ R and k ∗ f ∈ L1 (R, X). b) h ∗ (k ∗ f ) = (h ∗ k) ∗ f a.e. Proof. These results may be deduced from the vector-valued version of Fubini’s theorem (Theorem 1.1.9) in the same way as in the scalar case. (Alternatively, they may be deduced from their scalar cases, using Theorem 1.1.4 and the Hahn-Banach theorem). Many standard facts about convolutions of scalar-valued functions extend to the vector-valued case. We summarize some of them here.

22

1. THE LAPLACE INTEGRAL

Proposition 1.3.2. a) (Young’s inequality) Let 1 ≤ p, q, r ≤ ∞ satisfy 1/p + 1/q = 1 + 1/r. If k ∈ Lp (R) and f ∈ Lq (R, X), then k ∗ f ∈ Lr (R, X) and

k ∗ f r ≤ k p f q .

b) Let 1 < p, p < ∞ satisfy 1/p + 1/p = 1. If k ∈ Lp (R) and f ∈ Lp (R, X), then k ∗ f ∈ C0 (R, X). c) If k ∈ L1 (R) and f ∈ L∞ (R, X), or if k ∈ L∞ (R) and f ∈ L1 (R, X), then k ∗ f ∈ BUC(R, X). d) If k ∈ L1 (R) and f ∈ C0 (R, X), or if k ∈ C0 (R, X) and f ∈ L1 (R), then k ∗ f ∈ C0 (R, X). Proof. Under the assumptions of a), the scalar-valued version of Young’s inequality [RS72, Section IX.4] shows that (|k| ∗ f )(t) exists a.e., and |k| ∗ f ∈ Lr (R). If (|k| ∗ f )(t) exists, then (k ∗ f )(t) exists and

(k ∗ f )(t) ≤ (|k| ∗ f )(t). Moreover, k ∗ f is weakly measurable by the scalar-valued theory, and hence measurable by Pettis’s theorem (Theorem 1.1.1). Thus, k ∗ f ∈ Lr (R, X). The proofs of the remaining parts are similar to the scalar-valued case [Rud62, Theorem 1.1.6]. Suppose that 1 ≤ p < ∞ and 1/p + 1/p = 1. For t, h ∈ R,

(k ∗ f )(t + h) − (k ∗ f )(t) ≤ (k(t + h − s) − k(t − s)) f (s) ds

R

1/p

≤ → 0

|k(s + h) − k(s)| ds p

R

f p

as h → 0. This shows that k ∗ f is uniformly continuous. When k ∈ L∞ (R) and f ∈ L1 (R, X), uniform continuity is established in a similar way with the roles of k and f reversed. Boundedness follows from a). This proves c). For b) and d), observe that k ∗ f has compact support if both k and f have compact support, and then the results follow by density arguments. When k and f are deﬁned on R+ , or on [0, τ ] where τ > 0, then k ∗ f may be deﬁned by (1.6) by taking k(t) and f (t) to be zero for other values of t. Then (k ∗ f )(t) =

0

t

k(t − s)f (s) ds

(t ≥ 0).

It is immediate that Propositions 1.3.1 and 1.3.2 remain valid in these contexts. Note that if k ∈ L1loc (R+ ) and f ∈ L1loc (R+ , X), then k ∗ f ∈ L1loc (R+ , X).

1.3. CONVOLUTIONS

23

As a tool, we shall need the notion of regularization from harmonic analysis. A molliﬁer is a sequence (ρn )n∈N in L1 (R) of the following form. The function ρ1 ∈ 1 L (R) satisﬁes R ρ1 (t) dt = 1, and ρn ∈ L1 (R) is given by ρn (t) = nρ1 (nt) (t ∈ R, n ∈ N). The next lemma shows that any molliﬁer acts as an approximate unit on various function spaces. Lemma 1.3.3. Let (ρn )n∈N be a molliﬁer. a) Let f ∈ BUC(R, X). Then limn→∞ f ∗ ρn − f ∞ = 0. b) Let f ∈ L1 (R, X). Then limn→∞ f ∗ ρn − f 1 = 0. Proof. a) Let ε > 0. There exists c > 0 such that 2 f ∞ |s|≥c |ρ1 (s)| ds ≤ ε. Then for t ∈ R, n ∈ N,

(f ∗ ρn )(t) − f (t) = (f (t − s) − f (t))ρn (s) ds

R

s

= (f (t − ) − f (t))ρ1 (s) ds

n R

s

≤ − f (t) |ρ1 (s)| ds

f t − n |s|≤c

s

+ − f (t) |ρ1 (s)| ds

f t − n |s|>c ≤

sup f (t − h) − f (t) ρ1 1 + ε

|h|≤c/n

≤ 2ε for all suﬃciently large n, since f is uniformly continuous. b) Let f ∈ L1 (R, X). Then

(f (t − s) − f (t))ρn (s) ds dt

f ∗ ρn − f 1 =

R

s

dt = f (t − − f (t))ρ (s) ds 1

n R R

s

≤ − f (t) dt |ρ1 (s)| ds.

f t − n R R As observed in Section 1.1, limn→∞ R f (t − ns ) − f (t) dt = 0 for all s ∈ R. Now the claim follows from the dominated convergence theorem. Lemma 1.3.3 is also valid for the spaces Lp (R, X) (1 < p < ∞) (see Remark 1.3.8 b)). The notion of molliﬁer can be extended to a family {ρε : 0 < ε ≤ 1} where ρε (t) = ε−1 ρ(t/ε), and Lemma 1.3.3 remains valid in that case. The theory of vector-valued convolutions and molliﬁers extends, with almost no changes, to the case of functions on Rn for n ≥ 1.

24

1. THE LAPLACE INTEGRAL

Now we move on to consider convolutions of vector-valued functions with operator-valued functions. The space of all bounded linear operators from a Banach space X into a Banach space Y is denoted by L(X, Y ), or simply by L(X) when Y = X. A function T : R+ → L(X, Y ) is strongly continuous if t → T (t)x is continuous for all x ∈ X. By the uniform boundedness principle, a strongly continuous function T is locally bounded. Note also that T is lower semi-continuous and hence measurable. We state the convolution results for strongly continuous functions T : R+ → L(X, Y ), but they are also valid for T : (0, ∞) → L(X, Y ) if T is strongly continuous on (0, ∞) and bounded on (0, 1). There are similar results for compact intervals [0, τ ]. Proposition 1.3.4. Let f ∈ L1loc (R+ , X) and let T : R+ → L(X, Y ) be strongly continuous. Then the convolution t (T ∗ f )(t) := T (t − s)f (s) ds 0

exists (as a Bochner integral) and deﬁnes a continuous function T ∗ f : R+ → Y . Proof. Fix t ≥ 0. First, we show that s → T (t − s)f (s) is measurable on [0, t]. When f (s) = χΩ (s)x for some measurable Ω ⊂ R+ and x ∈ X, then T (t − s)f (s) = χΩ (s) · T (t − s)x. This is measurable, being the product of a measurable scalar-valued function and a continuous vector-valued function. By linearity, T (t − ·)f (·) is measurable when f is a simple function. For measurable f , there is a sequence of simple functions fn → f a.e. and then T (t − s)fn (s) → T (t − s)f (s) s-a.e., so T (t − ·)f (·) is measurable. Since

T (t − s)f (s) ≤ T (t − s) f (s)

it follows from Theorem 1.1.4 that (T ∗ f )(t) exists. Continuity of T ∗ f follows from the dominated convergence theorem (Theorem 1.1.8). Now we state the analogue of Proposition 1.3.2 for operator-valued functions on R+ . Proposition 1.3.5. Let f ∈ L1loc (R+ , X) and T : R+ → L(X, Y ) be strongly continuous. a) (Young’s inequality) Let 1 ≤ p, q, r ≤ ∞ satisfy 1/p + 1/q = 1 + 1/r. If ∞ p

T (t)

dt < ∞ and f ∈ Lq (R+ , X), then T ∗ f ∈ Lr (R+ , Y ) and 0

T ∗ f r ≤ f q

∞ 0

1/p

T (t) dt p

.

1.3. CONVOLUTIONS

25

∞ b) Let 1 < p, p < ∞ satisfy 1/p + 1/p = 1. If 0 T (t) p dt < ∞ and f ∈ Lp (R+ , X), then T ∗ f ∈ C0 (R+ , Y ). ∞ c) If 0 T (t) dt < ∞ and f ∈ BUC(R+ , X), or if T is bounded and f ∈ L1 (R+ , X), then T ∗ f ∈ BUC(R+ , Y ). ∞ d) If 0 T (t) dt < ∞ and f ∈ C0 (R+ , X), or if limt→∞ T (t) = 0 and f ∈ L1 (R+ , X), then T ∗ f ∈ C0 (R+ , Y ). Proof. The proofs are similar to Proposition 1.3.2, with the exception of the uniform continuity of T ∗ f when T is integrable and f is bounded and uniformly continuous. Then, for 0 ≤ t ≤ t + h,

(T ∗ f )(t + h) − (T ∗ f )(t)

t+h

t

≤ T (s)f (t + h − s) ds + T (s)(f (t + h − s) − f (t − s)) ds

t

0 ∞ t+h ≤

T (s) ds f ∞ +

T (s) ds sup f (s + h) − f (s)

0

t

s≥0

→ 0 uniformly in t as h → 0. If f or T is more regular, then T ∗ f is continuously diﬀerentiable. We give two such results. Proposition 1.3.6. Let T : R+ → L(X, Y ) be strongly continuous and bounded, t x ∈ X, f ∈ L1loc (R+ , X), f (t) = x + 0 f (s) ds (t ≥ 0). Then T ∗ f ∈ C 1 (R+ , Y ) and (T ∗ f ) (t) = (T ∗ f )(t) + T (t)x. Proof. Let u(t) := (T ∗ f )(t) + T (t)x. Then by Proposition 1.3.4, u ∈ C(R+ , Y ). By Fubini’s theorem we have t t s t u(s) ds = T (r)f (s − r) dr ds + T (s)x ds 0 0 0 0 t t t = T (r)f (s − r) ds dr + T (s)x ds 0 r 0 t t−r t = T (r)f (s) ds dr + T (s)x ds 0 0 0 t t = T (r)(f (t − r) − x) dr + T (s)x ds 0

=

(T ∗ f )(t)

0

(t ≥ 0).

By Proposition 1.2.2 a), this proves the claim.

26

1. THE LAPLACE INTEGRAL

Proposition 1.3.7. Let T : [0, τ ] → L(X, Y ) be Lipschitz continuous with T (0) = 0, and let f ∈ L1 ([0, τ ], X). Then T ∗ f ∈ C 1 ([0, τ ], Y ). Proof. First, suppose that f ∈ C 1 ([0, τ ], X). By Proposition 1.3.6, T ∗ f has a derivative g ∈ C([0, τ ], Y ). For 0 ≤ r ≤ t ≤ τ ,

t

= (T ∗ f )(t) − (T ∗ f )(r)

g(s) ds

r

t

≤

T (t − s)f (s) ds +

0

r t

≤

r

L (t − s) f (s) ds +

r 0

r

(T (t − s) − T (r − s))f (s) ds L (t − r) f (s) ds

≤ L (t − r) f 1 , where L is a Lipschitz constant for T , so that T (t2 ) − T (t1 ) ≤ L|t2 − t1 | and, in particular, T (s) ≤ Ls since T (0) = 0. It follows that g(s) ≤ L f 1 for all s ∈ [0, τ ]. Now, consider f ∈ L1 ([0, τ ], X). There is a sequence (fn ) in C 1 ([0, τ ], X) such that fn − f 1 → 0. By Proposition 1.3.5, (T ∗ (fn − f ))(t) → 0. By the ﬁrst paragraph,

(T ∗ fn ) − (T ∗ fm ) ∞

≤ →

L fn − fm 1 0,

so ((T ∗ fn ) ) converges uniformly to a function g ∈ C([0, τ ], Y ). Since t t (T ∗ f )(t) = lim (T ∗ fn )(t) = lim (T ∗ fn ) (s) ds = g(s) ds, n→∞

n→∞

0

0

it follows that T ∗ f ∈ C 1 ([0, τ ], Y ). Remark 1.3.8. a) There is an analogous theory of operator-valued convolutions on R. One may deﬁne (T ∗ f )(t) :=

R

T (t − s)f (s) ds

for almost all t, if T : R → L(X, Y ) is strongly continuous, R T (t) p dt < ∞ and f ∈ Lq (R, X) where 1/p + 1/q ≥ 1, and Proposition 1.3.5 is valid in this case. b) Convolutions of scalar, vector or operator-valued functions can sometimes be considered as Bochner integrals with values in a function space. Suppose that 0 < τ ≤ ∞, k ∈ L1 (0, τ ) and f ∈ Lp ((0, τ ), X) where 1 ≤ p < ∞. Deﬁne G : (0, τ ) → Lp ((0, τ ), X) by f (t − s) (0 < s < t < τ ) G(s)(t) := 0 otherwise.

1.4. EXISTENCE OF THE LAPLACE INTEGRAL

27

Then G is continuous and G(s) p ≤ f p . We can therefore form the Bochner τ integral 0 k(s)G(s) ds in L1 ((0, τ ), X). Then

τ 0

k(s)G(s) ds = k ∗ f

a.e. in (0, τ ).

This can beproved by ﬁrst considering step functions, or by considering integrals τ of the form 0 (k ∗ f )(t), x∗ g(t) dt for arbitrary x∗ ∈ X ∗ and g ∈ Lp (0, τ ). Thus,

k ∗ f p ≤ k 1 f p , a special case of Young’s inequality. This approach can also be used for convolutions on R, when the spaces Lp (R, X) can be replaced by BUC(R, X) or C0 (R, X). For example, this leads to a very short proof of Lemma 1.3.3 for all these spaces. The same idea can be used in the operator-valued case. Suppose that 0 < τ ≤ ∞, T : (0, τ ) → L(X, Y ) is strongly continuous and f ∈ Lp ((0, τ ), X), where 1 ≤ p < ∞. Let T (s)f (t − s) (0 < s < t < τ ), H(s)(t) := 0 otherwise. p Then H(s) ∈ Lp ((0, τ ), Y ) τ τ ), Y ), H(s) p ≤ T (s) f p , and H : (0, τ ) → L ((0, τ is continuous. If 0 T (s) ds < ∞, one may form the Bochner integral 0 H(s) ds in Lp ((0, τ ), Y ) and it coincides a.e. with T ∗ f .

1.4

Existence of the Laplace Integral

Let X be a complex Banach space and L1loc (R+ , X) := {f : R+ → X: f is Bochner integrable on [0, τ ] for all τ > 0}. This section is concerned with the existence of the Laplace integral ∞ τ fˆ(λ) := e−λt f (t) dt := lim e−λt f (t) dt 0

τ →∞

0

τ

for f ∈ L1loc (R+ , X) and λ ∈ C. Note that 0 e−λt f (t) dt exists as a Bochner ∞ integral, and if 0 e−λt f (t) dt exists as a Bochner integral then it agrees with the deﬁnition above, by the dominated convergence theorem. Of special interest will be the abscissa of convergence of fˆ, given by abs(f ) := inf{Re λ : fˆ(λ) exists}. It will be shown that the set of those λ ∈ C for which the Laplace integral converges is either empty or a right half-plane whose left boundary point abs(f ) coincides with the exponential growth bound of the antiderivative t → F (t) − F∞

(t ≥ 0),

28

1. THE LAPLACE INTEGRAL

t where F (t) := 0 f (s) ds, F∞ := limt→∞ F (t) if the limit exists, and F∞ := 0 otherwise. Therefore, one particular result of this section will be that a locally Bochner integrable function t f is Laplace transformable if and only if its normalized antiderivative F : t → 0 f (s) ds is exponentially bounded. Proposition 1.4.1. Let f ∈ L1loc (R+ , X). Then the Laplace integral fˆ(λ) converges if Re λ > abs(f ) and diverges if Re λ < abs(f ). Proof. Clearly, fˆ(λ) does not exist if Re λ < abs(f ). For λ0 ∈ C deﬁne G0 (t) := t −λ s e 0 f (s) ds (t ≥ 0). Then, for all λ ∈ C and t ≥ 0, integration by parts gives 0

t

e−λs f (s) ds =

0

t

e−(λ−λ0 )s e−λ0 s f (s) ds

0

= e

−(λ−λ0 )t

t G0 (t) + (λ − λ0 ) e−(λ−λ0 )s G0 (s) ds. (1.7) 0

If fˆ(λ0 ) exists, then G0 is bounded. Moreover, it follows from (1.7) that fˆ(λ) exists if Re λ > Re λ0 and ∞ fˆ(λ) = (λ − λ0 ) e−(λ−λ0 )s G0 (s) ds (Re λ > Re λ0 ). (1.8) 0

This shows that fˆ(λ) exists if Re λ > abs(f ). If fˆ(λ) converges for all λ ∈ C, then abs(f ) := −∞. If the domain of convergence is empty, then abs(f ) := ∞. A function f is called Laplace transformable if abs(f ) < ∞. It follows from Proposition 1.4.1 that the interior of the domain of convergence of fˆ(λ) is the open right half-plane {Re λ > abs(f )}. As the following example shows, the domain of convergence may or may not include points on the boundary abs(f ) + iR. Example 1.4.2. Let f (t) := (1 + t)−1 . Then abs(f ) = 0 since the Laplace integral fˆ(λ) converges for λ > 0 but not for λ = 0. If λ = ir (r = 0), then integration by parts implies that fˆ(ir) converges and 1 1 fˆ(ir) = − ir ir

∞ 0

e−irt dt. (1 + t)2

Thus, the domain of convergence of fˆ(λ) is {Re λ ≥ 0, λ = 0}. For f (t) := 1, the domain of convergence of fˆ(λ) is the open half-plane {Re λ > 0}; for f (t) := (1 + t2 )−1 it is the closed half-plane {Re λ ≥ 0}. From the proof of Proposition 1.4.1 and the uniform boundedness principle one also obtains the remarkable result that the abscissa of convergence of fˆ(λ) is

1.4. EXISTENCE OF THE LAPLACE INTEGRAL given by

t −λs ∗ ∗ ∗ abs(f ) = inf λ ∈ R : sup e f (s), x ds < ∞ for all x ∈ X . t>0

29

(1.9)

0

To see this we denote the right-hand side of (1.9) by absw (f ). Clearly, absw (f ) ≤ abs(f ). Assume that absw (f ) < abs(f ). Then there exists λ0 such that absw (f ) < λ0 < abs(f ) and supt>0 |G0 (t), x∗ | < ∞ (x∗ ∈ X ∗ ), where G0 is as in the proof of Proposition 1.4.1. It follows from the uniform boundedness principle that G0 is bounded. Thus, by (1.7), fˆ(λ) exists for all λ > λ0 . Since this contradicts λ0 < abs(f ) one obtains that absw (f ) = abs(f ). Next abs(f ) will be described by the exponential growth of f and its antiderivatives. For f : R+ → X the exponential growth bound is given by ω(f ) := inf ω ∈ R : sup e−ωt f (t) < ∞ . t≥0

It is obvious that abs(f ) ≤ abs( f ) ≤ ω(f ).

(1.10)

It will be shown in Example 1.4.4 below that there are cases in which one has strict inequalities in (1.10). In fact, it is possible that abs(f ) = −∞ and ω(f ) = ∞. It will be shown next that abs(f ) is determined by the exponential growth of t the antiderivative t → F (t) − F∞ (t ≥ 0), where F (t) := 0 f (s) ds, F∞ := limt→∞ F (t) if the limit exists, and F∞ := 0 otherwise. Theorem 1.4.3. Let f ∈ L1loc (R+ , X). Then abs(f ) = ω(F − F∞ ).

t Proof. Suppose that abs(f ) < ∞. For λ0 > abs(f ) deﬁne G0 (t) := 0 e−λ0 s f (s) ds (t ≥ 0). Then G0 is continuous and convergent as t → ∞, so G0 is bounded. To prove that abs(f ) ≥ ω(F − F∞ ) one considers the three cases abs(f ) > 0, abs(f ) = 0, and abs(f ) < 0. First, let abs(f ) > 0. Then F∞ = 0 and, for λ0 > abs(f ), integration by parts gives t t t F (t) = f (s) ds = eλ0 s e−λ0 s f (s) ds = eλ0 t G0 (t) − λ0 eλ0 s G0 (s) ds. 0

0

0

It follows that F (t) ≤ Ce + C(e − 1) ≤ 2Ce for t ≥ 0, where C := sups≥0 G0 (s) . This shows that abs(f ) ≥ ω(F − F∞ ) if abs(f ) > 0. Second, let abs(f ) = 0. If F∞ = 0, then the same procedure as above yields abs(f ) ≥ ω(F − F∞ ). If abs(f ) = 0 and limt→∞ F (t) = F∞ exists, then it follows from the continuity of F that supt≥0 F (t) − F∞ < ∞. Thus, abs(f ) = 0 ≥ ω(F − F∞ ). This shows that abs(f ) ≥ ω(F − F∞ ) if abs(f ) = 0. Third, let abs(f ) < 0. Choose abs(f ) < λ0 < 0. For r ≥ t ≥ 0 one has r F (r) − F (t) = eλ0 s e−λ0 s f (s) ds t r λ0 r = e G0 (r) − eλ0 t G0 (t) − λ0 eλ0 s G0 (s) ds. λ0 t

λ0 t

λ0 t

t

30

1. THE LAPLACE INTEGRAL

Since G0 is bounded and limr→∞ F (r) = F∞ , it follows that

∞

λt

−|λ0 |s 0

≤ 2Ceλ0 t

F∞ − F (t) = e G (t) + λ e G (s) ds 0 0 0

t

for t ≥ 0. This proves that abs(f ) ≥ ω(F − F∞ ). To show the reverse inequality, suppose that ω(F − F∞ ) < ∞ and let ω > ω(F − F∞ ). Since F is continuous, there exists M ≥ 0 such that F (t) − F∞ ≤ M eωt for all t ≥ 0. Let λ > ω > ω(F − F∞ ). Using the fact that F − F∞ is an antiderivative of f , integration by parts yields t t e−λs f (s) ds = e−λt (F (t) − F∞ ) + F∞ + λ e−λs (F (s) − F∞ ) ds. 0

0

Hence, fˆ(λ) exists for λ > ω(F − F∞ ) and is given by fˆ(λ) = F∞ + λ(F − F∞ )(λ). ∞ −λt This shows that abs(f ) ≤ ω(F − F∞ ). Since λF (λ) = λ e F dt = F∞ if ∞ ∞ 0 Re λ > 0, fˆ(λ) = λFˆ (λ) if Re λ > max{abs(f ), 0}. (1.11) If ω ≥ 0, then the triangle inequality implies that ω(F ) ≤ ω if and only if ω(F − F∞ ) ≤ ω. Thus, a locally Bochner integrable function f is Laplace transt formable if and only if its antiderivative F (t) = 0 f (s) ds is exponentially bounded; and, abs(f ) ≤ ω ⇐⇒ ω(F ) ≤ ω (if ω ≥ 0). (1.12) The following is a ﬁrst example of a function which is Laplace transformable but not exponentially bounded; i.e., abs(f ) < ω(f ) = ∞. t

t

Example 1.4.4. For t ≥ 0 let f (t) := et ee cos(ee ). Then ω(f ) = ∞. Since

t

F (t) =

f (s) ds = 0

t

s es

es

ee

e e cos (e ) ds = 0

t t

cos(u) du = sin ee − sin(e),

e

it follows that F∞ = 0. Thus, by Theorem 1.4.3, abs(f ) = ω(F ) = 0. Finally, the results of this section will be formulated for strongly continuous operator-valued functions T : R+ → L(X, Y ). We deﬁne the exponential growth bound of T by −ωt ω(T ) := ω( T ) = inf ω ∈ R : sup e T (t) < ∞ . t≥0

By the uniform boundedness principle, ω(T ) = sup{ω(ux ) : x ∈ X},

1.4. EXISTENCE OF THE LAPLACE INTEGRAL

31

where ux (t) := T (t)x. t If T : R+ → L(X, Y ) is strongly continuous, and λ ∈ C, then 0 e−λs T (s) ds t −λs denotes the bounded operator x → 0 e T (s)x ds, and we deﬁne t −λs abs(T ) := inf Re λ : e T (s) ds converges strongly as t → ∞ 0

=

sup{abs(ux ) : x ∈ X}.

Here and in what follows, to say S : R+ → L(X, Y ) converges strongly as t → ∞ refers to the strong operator topology; i.e., it means that limt→∞ S(t)x exists in Y for all x ∈ X. Proposition 1.4.5. Let T : R+ → L(X, Y ) be strongly continuous, let S(t) = t T (s) ds, and S∞ be the strong limit of S(t) as t → ∞ if it exists, and S∞ := 0 0 otherwise. Then t a) limt→∞ 0 e−λs T (s) ds exists in operator norm whenever Re λ > abs(T ). t b) abs(T ) = inf λ ∈ R : supt>0 0 e−λs T (s)x, y ∗ ds < ∞ for all x ∈ X and y ∗ ∈ Y ∗ . c) abs(T ) = ω(S − S∞ ). t Proof. If 0 e−λ0 s T (s) ds converges strongly, then it is uniformly bounded in operator norm. Thus, a) follows from (1.7), while b) is immediate from (1.9). To prove c), it is possible to repeat the proof of Theorem 1.4.3. Alternatively, one may deduce c) from Theorem 1.4.3 as follows. Let ux (t) := T (t)x, vx (t) := S(t)x − S∞ x, and S(t)x − lim S(s)x if the limit exists, s→∞ vx (t) := S(t)x otherwise. By Theorem 1.4.3, abs(ux ) = ω(vx ). Moreover, ω(vx ) and ω(vx ) coincide if either of them is strictly positive, since vx − vx is a constant function. First suppose that abs(T ) < 0. Then ω(vx ) = abs(ux ) ≤ abs(T ) < 0, so limt→∞ vx (t) exists for all x ∈ X. Thus, S(t) converges strongly, so vx = vx and abs(T )

= sup{abs(ux ) : x ∈ X} = sup{ω(vx ) : x ∈ X} = sup{ω(vx ) : x ∈ X} = ω(S − S∞ ).

Next, suppose that ω(S − S∞ ) < 0. Then again, S(t) converges strongly and, as above, abs(T ) = ω(S − S∞ ).

32

1. THE LAPLACE INTEGRAL Next, suppose that abs(T ) > 0. Then abs(T )

=

sup{abs(ux ) : x ∈ X, abs(ux ) > 0}

= =

sup{ω(vx ) : x ∈ X, ω(vx ) > 0} sup{ω(vx ) : x ∈ X, ω(vx ) > 0}

=

ω(S − S∞ ).

Finally, suppose that ω(S − S∞ ) > 0. Then the same argument as in the previous paragraph can be applied to show that abs(T ) = ω(S − S∞ ). Remark 1.4.6. If S∞ is the norm-limit of S(t) as t → ∞ if this exists and S∞ ∈ L(X, Y ) is arbitrary otherwise, and S∞ is as in Proposition 1.4.5, then it is trivial that ω(S − S∞ ) = ω(S − S∞ ). If T : R+ → L(X, Y ) is strongly continuous and abs(T ) < ∞, we deﬁne the Laplace integral of T by ∞ t −λs ˆ T (λ) := e T (s) ds := lim e−λs T (s) ds (Re λ > abs(T )), t→∞

0

0

where the right-hand integral is interpreted as above and the limit exists in operator norm (Proposition 1.4.5).

1.5

Analytic Behaviour

Let f ∈ L1loc (R+ , X). In this section it will be shown that λ → fˆ(λ) is holomorphic for Re λ > abs(f ). In general, fˆ need not have a singularity on the boundary abs(f ) + iR of its domain of convergence and may be extended holomorphically to a strictly larger half-plane. However, it will be shown that abs(f ) is a singularity of fˆ if X is an ordered Banach space with a normal positive cone and f (t) ≥ 0 a.e. (see Appendix C). We put N0 := N ∪ {0}. Theorem 1.5.1. Let f ∈ L1loc (R+ , X) with abs(f ) < ∞. Then λ → fˆ(λ) is holomorphic for Re λ > abs(f ) and, for all n ∈ N0 and Re λ > abs(f ), ∞ fˆ(n) (λ) = e−λt (−t)n f (t) dt (1.13) 0

(as an improper Bochner integral). Proof. Deﬁne qk : C → X (k ∈ N0 ) by qk (λ) :=

0

k

N λn k (−t)n f (t) dt. N →∞ n! 0 n=0

e−λt f (t) dt = lim

1.5. ANALYTIC BEHAVIOUR

33

The limits exist uniformly for λ in bounded subsets of C. By the Weierstrass convergence theorem (a simple special case of Vitali’s Theorem A.5), the functions k (j) qk are entire and qk (λ) = 0 e−λt (−t)j f (t) dt for all j ∈ N0 . t Let Re λ > λ0 > abs(f ) and deﬁne G0 (t) := 0 e−λ0 s f (s) ds. By Proposition 1.4.1, G0 is bounded and integration by parts gives fˆ(λ) − qk (λ)

∞

=

e−(λ−λ0 )s e−λ0 s f (s) ds

k −(λ−λ0 )k

= −e

∞

G0 (k) + (λ − λ0 )

e−(λ−λ0 )s G0 (s) ds.

k

It follows that qk converges to fˆ uniformly on compact subsets of {λ : Re λ > abs(f )}. Again by the Weierstrass convergence theorem, fˆ is holomorphic and (j) qk (λ) → fˆ(j) (λ) as k → ∞, for Re λ > abs(f ). If abs(f ) < ∞, then the abscissa of holomorphy of fˆ is denoted by hol(fˆ) := inf{ω ∈ R : fˆ extends holomorphically for Re λ > ω}. By Theorem 1.5.1, hol(fˆ) ≤ abs(f ).

(1.14)

In general, equality does not hold in (1.14) (see Example 1.5.2 below). However we shall see that equality does hold for positive functions on ordered spaces (Theorem 1.5.3) and for exponentially bounded functions which extend holomorphically into a sector {| arg(λ)| < α} for some 0 < α < π2 (see Section 2.6). Furthermore, in Chapter 4 (Theorem 4.4.19) we shall see that abs(f ) ≤ hol0 (fˆ) whenever f is exponentially bounded, where hol0 (fˆ) := inf{ω ∈ R : fˆ has a bounded holomorphic extension for Re λ > ω} is the abscissa of boundedness of fˆ. The following example shows that it may happen that hol(fˆ) < abs(f ) < ω(f ). Example 1.5.2. Let f (t) = et sin et (t ≥ 0). Obviously, ω(f ) = 1. Since F (t) = t f (s) ds = cos 1 − cos et , one obtains from Theorem 1.4.3 that abs(f ) = 0. It 0 follows from

34

1. THE LAPLACE INTEGRAL

t 0

t

d (− cos es ) ds ds 0 t = −e−λt cos et + cos 1 − λe−λs cos es ds 0 t d = −e−λt cos et + cos 1 − λ e−(λ+1)s (sin es ) ds ds 0

e−λs f (s) ds =

e−λs

−e−λt cos et + cos 1 − λe−(λ+1)t sin et t +λ sin 1 − λ(λ + 1) e−(λ+2)s es sin es ds

=

0

that fˆ(λ) = cos 1 + λ sin 1 − λ(λ + 1)fˆ(λ + 2) if Re λ > 0. Thus, fˆ has a holomorphic extension to Re λ > −2, then to Re λ > −4, etc. This shows that hol(fˆ) = −∞. Another of this function is that abs(f ) < abs(|f |). ∞ remarkable property ∞ Clearly, 0 e−λt |f (t)|dt = 1 u−λ | sin u| du converges for λ > 1. Since √ n−1 (4j+3)π/4 √ n−1 nπ 1 2 1 2 2π | sin u| du ≥ du ≥ , u 2 j=1 (4j+1)π/4 u 2 j=1 4j + 3 1 it follows that abs(|f |) = 1. Theorem 1.5.3. Let X be an ordered Banach space with a normal positive cone. Assume that abs(f ) < ∞ and f (t) ≥ 0 a.e. Then hol(fˆ) = abs(f ). If, in addition, abs(f ) > −∞, then fˆ has a singularity at abs(f ). Proof. If abs(f ) = −∞, then the statement follows from (1.14). If abs(f ) > −∞, then one may assume that abs(f ) = 0. Otherwise replace f by t → e−ωt f (t), where ω := abs(f ). By (1.14), λ → fˆ(λ) is holomorphic for Re λ > 0. Assume that fˆ is holomorphic at λ = 0. Then there exists a full circle with centre at 1 and radius 1 + 2λ0 for some λ0 > 0 such that fˆ is holomorphic on it (the extension of fˆ to points in the complex plane with Re λ ≤ 0 is denoted by the same symbol). Hence, fˆ(−λ0 ) = ∞

∞ k=0

(−1)k (1 + λ0 )k

1 ˆ(k) f (1). k!

Next it will be shown that 0 e f (t), x∗ dt converges for all x∗ ∈ X ∗ , which is ∗ a contradiction to (1.9). Let g(t) := f (t), x∗ . Since X+ is generating (Proposition ∗ C.2), one can assume that x ≥ 0 and thus g(t) ≥ 0. Then, by Theorem 1.5.1 and the monotone convergence theorem, ∞ ∞ k 1 gˆ(−λ0 ) = (1 + λ0 ) e−t tk g(t) dt k! 0 k=0 ∞ ∞ −t (1+λ0 )t = e e g(t) dt = eλ0 t g(t) dt. 0

λ0 t

0

1.5. ANALYTIC BEHAVIOUR

35

The following example shows that there are positive functions with hol(fˆ) < ω(f ). 3 2 Example 1.5.4. Let Ω := n∈N [n, n + e−n ) and f (t) := et χΩ (t). Then f (t) ≥ 0 ∞ ∞ n+e−n3 −λs s2 and ω(f ) = ∞. However, 0 e−λs f (s) ds = n=0 n e e ds converges ˆ for all λ ∈ R. Hence, hol(f ) = abs(f ) = −∞. Finally in this section, we consider operator-valued functions. Let T : R+ → L(X, Y ) be strongly continuous, where X and Y are arbitrary Banach spaces, and assume that abs(T ) < ∞. As in Section 1.4, let ux (t) := T (t)x (x ∈ X, t ≥ 0), and t ˆ T (λ) := lim e−λs T (s) ds (Re λ > abs(T )), t→∞

0

where the integral is interpreted as in Section 1.4 and the limit exists in operator norm (Proposition 1.4.5). Since Tˆ(λ)x = u x (λ), it follows from Theorem 1.5.1, (1.8) and Proposition A.3 that Tˆ : {Re λ > abs(T )} → L(X, Y ) is holomorphic. We deﬁne hol(Tˆ ) := inf ω ∈ R : Tˆ extends to a holomorphic function from {Re λ > ω} into L(X, Y ) .

Proposition 1.5.5. Let T : R+ → L(X, Y ) be strongly continuous with abs(T ) < ∞. Then hol(Tˆ) = sup {hol( ux ) : x ∈ X} . Proof. Since Tˆ(λ)x = u x (λ) for Re λ > abs(T ), it is clear that hol(Tˆ) ≥ hol( ux ) for all x ∈ X. We have to show that if ω < abs(T ) and each u x extends holomorphically to {Re λ > ω} and Tˆ (λ)x = u x (λ) (Re λ > ω), then Tˆ is a holomorphic function of {Re λ > ω} into L(X, Y ). Take λ0 with Re λ0 > abs(T ). For |λ − λ0 | < abs(T ) − ω, Tˆ (λ)x =

∞ (n) u x (λ0 ) (λ − λ0 )n . n! n=0

Moreover, for 0 < r < abs(T ) − ω,

m (n)

u

x (λ0 )

n sup (λ − λ0 ) : m ∈ N, |λ − λ0 | < r < ∞

n! n=0 for each x ∈ X. By the uniform boundedness principle, Tˆ (λ) ∈ L(X, Y ) and Tˆ is bounded on {|λ − λ0 | < r}. It follows from Proposition A.3 that Tˆ : {Re λ > ω} → L(X, Y ) is holomorphic.

36

1. THE LAPLACE INTEGRAL

1.6

Operational Properties

The importance of Laplace integrals in applications to diﬀerential equations lies in the fact that they transform the analytic operations of diﬀerentiation, integration and convolution into algebraic operations of multiplication. In this section, we establish these and other basic properties of Laplace transforms. Proposition 1.6.1. Let f ∈ L1loc (R+ , X), μ ∈ C and s ∈ R+ . Let g(t) fs (t) hs (t)

:= e−μt f (t) (t ≥ 0), := f (s + t) (t ≥ 0), f (t − s) (t ≥ s), := 0 (0 ≤ t < s).

Let λ ∈ C. Then a) gˆ(λ) exists if and only if fˆ(λ + μ) exists, and then gˆ(λ) = fˆ(λ + μ). b) fs (λ) exists if and only if fˆ(λ) exists, and then s λs −λt ˆ fs (λ) = e f (λ) − e f (t) dt . 0

s (λ) = e−λs fˆ(λ). c) hs (λ) exists if and only if fˆ(λ) exists, and then h Proof. These results follow immediately from the formulae: τ τ −λt e g(t) dt = e−(λ+μ)t f (t) dt, 0 0 s+τ τ s −λt λs −λt −λt e fs (t) dt = e e f (t) dt − e f (t) dt ,

0 τ

0

e−λt hs (t) dt

= e−λs

0 τ −s

0

e−λt f (t) dt

(τ > s).

0

Proposition 1.6.2. Let f ∈ L1loc (R+ , X) and T ∈ L(X, Y ), and let (T ◦ f )(t) = T (f (t)). Then T ◦ f ∈ L1loc (R+ , Y ). If fˆ(λ) exists, then T ◦ f (λ) exists and equals T (fˆ(λ)). Proof. By Proposition 1.1.6, T ◦ f ∈ L1loc (R+ , Y ) and τ τ e−λt (T ◦ f )(t) dt = T e−λt f (t) dt. 0

0

The second statement follows on letting τ → ∞.

1.6. OPERATIONAL PROPERTIES

37

Proposition 1.6.3. Let f ∈ L1loc (R+ , X) and A be a closed operator on X. Suppose that f (t) ∈ D(A) a.e. and A ◦ f ∈ L1loc (R+ , X). Let λ ∈ C. If fˆ(λ) and A ◦ f (λ) ˆ ˆ both exist, then f (λ) ∈ D(A) and A ◦ f (λ) = A(f (λ)). Proof. By Proposition 1.1.7, τ e−λt (A ◦ f )(t) dt = A 0

τ

e−λt f (t) dt.

0

Since A is closed, the second statement follows on letting τ → ∞. Now we consider convolutions. Proposition 1.6.4. Let k ∈ L1loc (R+ ), f ∈ L1loc (R+ , X), λ ∈ C, and suppose that ˆ fˆ(λ). Re λ > max(abs(|k|), abs(f )). Then (k ∗ f )(λ) exists and (k ∗ f )(λ) = k(λ) Proof. Replacing k(t) by e−λt k(t) and f (t) by e−λt f (t), and using Proposition 1.6.1, we may assume that λ = 0. First, we give the simple proof in the case when f ∈ L1 (R+ , X). Then Fubini’s theorem gives that k ∗ f ∈ L1 (R+ , X) and ∞ (k ∗ f )(0) = (k ∗ f )(t) dt 0 ∞ t = k(t − s)f (s) ds dt 0 0 ∞ ∞ = k(t − s) dt f (s) ds 0

s

ˆ fˆ(0). = k(0) Now, assume only that fˆ(0) exists. Replacing f (t) by f (t) − e−t fˆ(0) (and ˆ using the previous case),

τ

we may assume that f (0) = 0. Let ε > 0. There exists K such that 0 f (s) ds < ε whenever τ > K. Then τ (k ∗ f )(t) dt = (1 ∗ (k ∗ f ))(τ ) = (k ∗ (1 ∗ f ))(τ ). 0

Hence,

τ

(k ∗ f )(t) dt

0

t

K

≤ k(τ − t) f (s) ds dt

0

0

τ t

+ k(τ − t) f (s) ds dt

≤ M

K τ

τ −K

0

|k(s)| ds + ε

τ −K 0

|k(s)| ds,

38

1. THE LAPLACE INTEGRAL

t

where M := supt≥0 0 f (s) ds < ∞. Letting τ → ∞,

τ

≤ ε k 1 . lim sup (k ∗ f )(t) dt

τ →∞

0

Since ε > 0 is arbitrary, it follows that (k ∗ f )(0) = 0, as required. As a corollary, we recover a simple result which was already observed in (1.11). Corollary 1.6.5. Let f ∈ L1loc (R+ , X) and let F (t) = fˆ(λ) exists, then Fˆ (λ) exists and Fˆ (λ) = fˆ(λ)/λ.

t 0

f (s) ds. If Re λ > 0 and

Proof. This is immediate from Proposition 1.3.1 with k(t) = 1. Corollary 1.6.6. Let f : R+ → X be absolutely continuous and diﬀerentiable a.e. If Re λ > 0 and f (λ) exists, then fˆ(λ) exists and f (λ) = λfˆ(λ) − f (0). t Proof. By Proposition 1.2.3, f ∈ L1loc (R+ , X) and f (t) − f (0) = 0 f (s) ds. The result follows from Corollary 1.6.5. √ Now we want to consider the substitution of λ for λ; we will ﬁnd a function √ ˆ h such that h(λ) = fˆ( λ). For this we ﬁrst calculate the Laplace integral of a special function. Lemma 1.6.7. Let s > 0 and 2

φs (t)

=

e−s /4t √ , πt

ψs (t)

=

se−s /4t √ 2 πt3/2

s (λ) φ

=

√ 1 √ e−s λ , λ

s (λ) ψ

= e−s

2

(t > 0).

Then

√ λ

(Re λ > 0).

Proof. First, we show that, for α > 0,

∞

−((α/u)−u)2

e 0

du = 0

∞

√ α −(v−α/v)2 π e dv = . 2 v 2

The ﬁrst equality follows from the change of variable v := α/u. Taking the average

1.6. OPERATIONAL PROPERTIES

39

and making the change of variable w := u − α/u, this gives ∞ 2 2 1 ∞ α e−((α/u)−u) du = 1 + 2 e−(u−α/u) du 2 u 0 0 1 ∞ −w2 = e dw 2 −∞ √ π = . 2 Now, for λ > 0, ∞ √ √ √ 2 1 s (λ) = e−s λ √ e−( λt−s/(2 t)) dt φ πt 0 ∞√ √ 2 λ −(s√λ/(2u)−u)2 = e−s λ √ e du πλ 0 2u2 √ √ 2 π = e−s λ √ 2 πλ √

e−s λ √ , = λ √ s (λ) = e−s λ ψ

∞

√ √ 2 s e−( λt−s/(2 t)) dt 3/2 2 πt 0 ∞ √ √ 2 2 = e−s λ √ e−(s λ/(2u)−u) du π 0

= e−s

√ λ

√

.

For Re λ > 0, the results follow by uniqueness of holomorphic extensions. Proposition 1.6.8. Let f ∈ L1loc (R+ , X) with ω(f ) < ∞, and let ∞ −s2 /4t e √ g(t) = f (s) ds, πt 0 ∞ −s2 /4t se √ h(t) = f (s) ds. 2 πt3/2 0 √ √ √ ˆ Then gˆ(λ) = fˆ( λ)/ λ and h(λ) = fˆ( λ), whenever Re λ > (max{ω(f ), 0})2 . √ Proof. If Re λ > (max{ω(f ), 0})2 , then Re λ > max{ω(f ), 0}, and Fubini’s theorem and Lemma 1.6.7 give ∞ ∞ −s2 /4t e √ gˆ(λ) = e−λt f (s) dt ds πt 0 0 ∞ −s√λ e √ f (s) ds = λ 0 √ √ = fˆ( λ)/ λ.

40

1. THE LAPLACE INTEGRAL

√ ˆ Similarly, h(λ) = fˆ( λ).

1.7

Uniqueness, Approximation and Inversion

In this section we shall show that any f ∈ L1loc (R+ , X) with abs(f ) < ∞ is uniquely determined by its Laplace transform fˆ and we shall give the Post-Widder inversion formula (Theorem 1.7.7). Other inversion theorems appear in Section 2.3. The following elementary statement will be used in the proofs of many results in this and the following sections. Lemma 1.7.1. Let a, b > 0 and deﬁne λn := a + nb, e−λn (t) := e−λn t (n ∈ N0 , t ≥ 0). Then {e−λn : n ∈ N0 } is total in L1 (R+ ). Proof. By the Stone-Weierstrass theorem, the linear span of the set P := {t → a−1 tbn/a : n ∈ N0 } is dense in C[0, 1] and thus in L1 (0, 1). Now the statement follows from the fact that Φ : L1 (0, 1) → L1 (R+ ) deﬁned by (Φg)(t) := ae−at g(e−at ) is an isometric isomorphism which maps P onto the exponential functions {e−λn : n ∈ N0 }. Proposition 1.7.2. Let f ∈ L1loc (R+ , X) with abs(f ) < ∞, let a > abs(f ), b > 0 and λn := a + nb. If fˆ(λn ) = 0 for all n ∈ N, then f (t) = 0 a.e. t Proof. One can assume that a > max{abs(f ), 0}. Deﬁne F (t) := 0 f (s) ds (t ≥ 0). Then 0 = fˆ(λn ) = λn Fˆ (λn ) for all n ∈ N0 (see Corollary 1.6.5). It follows from Theorem 1.4.3 that a > ω(F ). Thus G(t) := e−at F (t) is continuous and bounded on R+ , and G(nb) = Fˆ (a+nb) = 0 (n ∈ N). For x∗ ∈ X ∗ deﬁne gx∗ (t) := G(t), x∗ (t ≥ 0). Then gx∗ ∈ L∞ (R+ ) = L1 (R+ )∗ and ∞ ˆ e−nb , gx∗ = e−nbt gx∗ (t) dt = G(nb), x∗ = 0. 0

Since {e−nb : n ∈ N} is a total subset of L1 (R+ ) by Lemma 1.7.1, it follows that gx∗ (t) = 0 for all t ≥ 0 and x∗ ∈ X ∗ . This implies that F (t) = eat G(t) = 0 for all t ≥ 0 and thus f (t) = 0 a.e. Because of its importance we reformulate this result in the following form which is used frequently in the book. Theorem 1.7.3 (Uniqueness Theorem). Let f, g ∈ L1loc (R+ , X) with abs(f ) < ∞ and abs(g) < ∞, and let λ0 > max(abs(f ), abs(g)). Suppose that fˆ(λ) = gˆ(λ) whenever λ > λ0 . Then f (t) = g(t) a.e. Remark 1.7.4. A sequence (λn )n∈N of complex numbers is called a uniqueness sequence for the Laplace transform if f = 0 a.e. whenever f ∈ L1loc (R+ , X), abs(f ) < Re λn for all n, and fˆ(λn ) = 0 for all n. It was shown in Proposition 1.7.2 that

1.7. UNIQUENESS, APPROXIMATION AND INVERSION

41

equidistant sequences λn = a + nb (b > 0) are examples of uniqueness sequences. In particular, this shows that a function of the form λ → q(λ) sin(λ) cannot have a representation as a Laplace transform. A ﬁnite sequence (λ1 , · · · , λn ) is not a uniqueness sequence since the function λ → (λ+μ)−2n (λ−λ1 ) · · · (λ−λn ) (Re λ > −μ) is the Laplace transform of the convolution product of the functions t → (1 − (λi + μ)t)e−μt (1 ≤ i ≤ n). A characterization of uniqueness sequences will be given in the Notes to this section. One can deduce the following fundamental result on approximation from the uniqueness theorem by a quotient argument (see also the proof of Vitali’s theorem in Appendix A). Theorem 1.7.5 (Approximation). Let fn ∈ C(R+ , X) with fn (t) ≤ M eωt for some M > 0, ω ∈ R and all n ∈ N. Let λ0 ≥ ω. The following are equivalent: (i) The Laplace transforms fˆn converge pointwise on (λ0 , ∞) and the sequence (fn )n∈N is equicontinuous on R+ . (ii) The functions fn converge uniformly on compact subsets of R+ . Moreover, if (ii) holds, then fˆ(λ) = limn→∞ fˆn (λ) for all λ > λ0 , where f (t) := limn→∞ fn (t). Proof. The space c(X) := {(xn )n∈N : xn ∈ X and limn→∞ xn exists} is a closed subspace of l∞ (X) := {(xn )n∈N : xn ∈ X and supn∈N xn < ∞}. Deﬁne w : R+ → l∞ (X) by w(t) = (fn (t))n∈N . Assume that (i) holds. The equicontinuity of fn implies the continuity of w, and the convergence of fˆn (λ) implies that w(λ) ˆ = (fˆn (λ))n∈N ∈ c(X) for all λ > λ0 . Consider the quotient mapping q : l∞ (X) → l∞ (X)/c(X). Then (q ◦ w)(λ) = q(w(λ)) ˆ = 0 for all λ > λ0 . Since q ◦ w : R+ → ∞ l (X)/c(X) is continuous, it follows from the uniqueness theorem that q◦w(t) = 0 for all t ≥ 0; i.e., w(t) ∈ c(X) for all t ≥ 0. Hence (fn )n∈N converges pointwise. Since (fn )n∈N is equicontinuous, this implies uniform convergence on each compact subset of R+ . Conversely, assume that (ii) holds. Clearly, uniform convergence implies equiˆ continuity. Let f (t) = limn→∞ fn (t). Then limn→∞ f n (λ) = f (λ) for all λ > λ0 by the dominated convergence theorem. We point out that one cannot omit the condition that the sequence (fn )n∈N is equicontinuous. To give an example, let X = C and fn (t) := eint (t ≥ 0, n ∈ N). 1 Then f n (λ) = λ−in converges to 0 as n → ∞ for all λ > 0. But fn (t) does not converge as n → ∞ if t ∈ R+ \ 2πZ. Another quotient argument enables us to deduce the converse of Proposition 1.6.3 from the uniqueness theorem. Proposition 1.7.6. Let A be a closed linear operator on X, let f, g ∈ L1loc (R+ , X) such that abs(f ) < ∞ and abs(g) < ∞, and let ω > max{abs(f ), abs(g)}. Then the following assertions are equivalent:

42

1. THE LAPLACE INTEGRAL

(i) f (t) ∈ D(A) and Af (t) = g(t) a.e. on R+ . (ii) fˆ(λ) ∈ D(A) and Afˆ(λ) = gˆ(λ) for all λ > ω. Proof. The implication (i) ⇒ (ii) has already been proved in Proposition 1.6.3. (ii) ⇒ (i): Let G(A) be the graph of A, which is a closed subspace of X × X, and let q : X × X → (X × X)/G(A) be the quotient map. Deﬁne h : R+ → ˆ (X × X)/G(A) by h(t) = q(f (t), g(t)). Then h(λ) = q(fˆ(λ), gˆ(λ)) = 0 for all λ > ω, by (ii). By the uniqueness theorem, h(t) = 0 a.e. This proves that (i) is true. Recall from Proposition 1.2.2 that t is a Lebesgue point of f ∈ L1loc (R+ , X) t+h if limh→0 h1 t f (s) − f (t) ds = 0 and that almost all points t are Lebesgue points of f . We now prove the Post-Widder inversion formula. Theorem 1.7.7 (Post-Widder). Let f ∈ L1loc (R+ , X). Assume that abs(f ) < ∞ and that t > 0 is a Lebesgue point of f . Then f (t) = lim (−1)k k→∞

1 k!

k+1 k k fˆ(k) . t t

Proof. To explain the structure of the proof, we ﬁrst consider the special case when f is a bounded continuous function. By Theorem 1.5.1, 1 (−1) k! k

k+1 ∞ k k (k) ˆ f = ρk (s)f (s) ds, t t 0

! " 1 k k+1 −ks/t k where ρk (s) := k! e s (s > 0). The functions ρk are “approximate t ∞ Dirac δ-functions”; i.e., ρk ≥ 0, 0 ρk(s) ds = 1, and for all ε > 0 and all open intervals I ⊂ R+ containing t we have s∈I ρk (s) ds < ε for all suﬃciently large k / (see below). Since f is assumed to be bounded and continuous, it follows from

∞

∞

ρk (s)f (s) ds − f (t) = ρk (s)(f (s) − f (t)) ds

0 0 ≤ 2 f ∞ ρk (s) ds + sup f (s) − f (t)

s∈I /

∞

s∈I

that 0 ρk (s)f (s) ds → f (t) as k → ∞. This proves the statement for bounded and continuous functions f . Now let f ∈ L1loc (R+ , X) and max(abs(f ), 0) < ω < ∞. By (1.12), ω > ω(F ), s where F (s) := 0 f (r) dr. In the following let t > 0 be a ﬁxed Lebesgue point of f , and let k ∈ N such that k > ωt. Let s G(s) := (f (r) − f (t)) dr = F (s) − F (t) − f (t)(s − t) (s ≥ 0). t

1.7. UNIQUENESS, APPROXIMATION AND INVERSION

43

ωs Since ω > ω(F ), there exists M > 0 such that G(s) ≤ M e for all s ≥ 0. 1 k+1 ∞ −λs k Since k! λ e s ds = 1 for λ > 0 (by induction and integration by parts), 0 it follows from Theorem 1.5.1 and integration by parts that

Jk

k+1 1 k k (k) ˆ (−1) f − f (t) k! t t k+1 ∞ 1 k e−ks/t sk f (s) ds − f (t) k! t 0 k+1 ∞ 1 k e−ks/t sk (f (s) − f (t)) ds k! t 0 k+1 ∞ s 1 k k e−ks/t sk−1 − 1 G(s) ds k! t t 0 k+2 ∞ k e−ku uk−1 (u − 1)G(ut) du. k!t 0 k

:= = = = =

Let ε > 0. Since t is a Lebesgue point of f , there exists 0 < δ < 1 such that

tu

1

ε 1

G(ut) = (f (r) − f (t)) dr

≤ 3 |u − 1| t t t if |u − 1| ≤ δ. Deﬁne J1,k :=

k k+2 k!t

1+δ 1−δ

e−ku uk−1 (u − 1)G(ut) du.

Then

J1,k

≤ ≤ =

ε k k+2 3 k!

1+δ

1−δ k+2 ∞

e−ku uk−1 (u − 1)2 du

! " εk e−ku uk+1 − 2uk + uk−1 du 3 k! 0 ε k k+2 (k + 1)! 2k! (k − 1)! ε − k+1 + = k+2 k 3 k! k k k 3

for all k ∈ N with k > ωt. Let J2,k

k k+2 := k!t

0

1−δ

e−ku uk−1 (u − 1)G(ut) du.

If k > 1/δ, the function u → e−ku uk−1 is increasing on (0, 1 − δ). Thus,

J2,k ≤

k k+2 −k(1−δ) e (1 − δ)k−1 k!t

0

1−δ

(1 − u) G(ut) du =: bk .

44

1. THE LAPLACE INTEGRAL

k+2 Since bk+1 /bk = 1 + k1 eδ−1 (1 − δ) → eδ (1 − δ) < 1 as k → ∞, one obtains that J2,k < ε/3 for all suﬃciently large k. Finally let 1 1 k+2 ∞ −ku k−1 J3,k := k e u (u − 1)G(ut) du. t k! 1+δ

The function u → e−mu um is decreasing on (1 + δ, ∞) for all m ∈ N. Choose k0 > tω and let k > k0 . Then

k+2 ∞

k

−(k−k0 )u k−k0 −k0 u k0 −1

J3,k = e u e u (u − 1)G(ut) du

k!t 1+δ ∞ k k+2 −(k−k0 )(1+δ) ≤ e (1 + δ)k−k0 e−k0 u uk0 −1 (u − 1)M eωut du k!t 1+δ =: ck . ! "k+2 −1−δ Since ck+1 /ck = 1 + k1 e (1 + δ) → e−δ (1 + δ) < 1, one obtains that

J3,k < ε/3 for all suﬃciently large k. It follows from Jk = J1,k + J2,k + J3,k that Jk → 0 as k → ∞.

1.8

The Fourier Transform and Plancherel’s Theorem

In this section, we give a brief summary of the properties of vector-valued Fourier transforms, and we extend Plancherel’s theorem and the Paley-Wiener theorem (characterizing the Laplace transforms of L2 -functions) to functions with values in a Hilbert space. For f ∈ L1 (R, X), the Fourier transform of f is the function Ff : R → X deﬁned by ∞

(Ff )(s) :=

e−ist f (t) dt.

−∞

We also deﬁne

∞

(Ff )(s) :=

eist f (t) dt = (Ff )(−s) = (F fˇ)(s),

−∞

where fˇ(t) := f (−t). Many properties of the Fourier transform on L1 (R, X) can be proved in exactly the same way as for the scalar-valued case (some can also be proved by applying linear functionals and using the scalar-valued results and the Hahn-Banach theorem), and we quote some of them here. Proofs of the scalar-valued cases may be found in standard textbooks such as [Rud87], [RS72], [Yos80], [Rud91]. Theorem 1.8.1. Let f ∈ L1 (R, X) and g ∈ L1 (R). a) F(g ∗ f )(s) = (F g)(s)(Ff )(s).

1.8. THE FOURIER TRANSFORM AND PLANCHEREL’S THEOREM b)

∞ −∞

g(t)(F f )(t) dt =

∞ −∞

45

(F g)(t)f (t) dt.

c) (Riemann-Lebesgue Lemma) Ff ∈ C0 (R, X). d) (Inversion Theorem) If Ff ∈ L1 (R, X), then f =

1 2π F(F f )

a.e.

For scalar-valued functions, Plancherel’s theorem [Rud87, Theorem 19.2] √ shows that Ff ∈ L2 (R) and F f 2 = 2π f 2 whenever f ∈ L1 (R) ∩ L2 (R), and hence the restriction of F to L1 (R) ∩ L2 (R) has a unique extension to a bounded linear operator on L2 (R) (also denoted by F ), √12π F is a unitary operator on

L2 (R), and Theorem 1.8.1 b) holds for f, g ∈ L2 (R). Hence, F −1 = (2π)−1 F, where (Ff )(t) = (F f )(−t). Plancherel’s theorem is not true for vector-valued functions, except when the space X is a Hilbert space (see the Notes). If X is a Hilbert space with inner product (·|·)X , then L2 (R, X) is also a Hilbert space with inner product ∞ (f |g)L2 (R,X) := (f (t)|g(t))X dt. −∞

As observed in Section 1.1, the simple functions are dense in L2 (R+ , X). In this Hilbert space context, this can also be shown by computing the orthogonal complement of the simple functions. Theorem 1.8.2 (Plancherel’s Theorem). Let X be a Hilbert space. Then F f ∈ √ L2 (R, X) and Ff L2 (R,X) = 2π f L2 (R,X) for all f ∈ L1 (R, X) ∩ L2 (R, X). The restriction of F to L1 (R, X) ∩ L2 (R, X) extends to a bounded linear operator F on L2 (R, X) and √12π F is a unitary operator on the Hilbert space L2 (R, X). Moreover, ∞

−∞

∞

((F f )(t)|g(t))X dt =

−∞

(f (t)|(Fg)(−t))X dt

(1.15)

for all f, g ∈ L2 (R, X). Proof. Let f ∈ L1 (R, X) ∩ L2 (R, X). To prove that √

Ff L2 (R,X) = 2π f L2 (R,X) , it suﬃces to assume that X is separable, since f is almost separably valued. Let {en : n ∈ N} be an orthonormal basis of X, and let fn (t) := (f (t)|en )X . Then fn ∈ L1 (R) ∩ L2 (R) and ((F f )(s)|en )X = (F fn )(s). Now, using the scalar-valued Plancherel theorem, ∞ ∞ ∞

(F f )(s) 2 ds = |(Ffn )(s)|2 ds −∞

=

−∞ n=1 ∞ ∞

2π

=

−∞ n=1 ∞

2π −∞

|fn (t)|2 dt

f (t) 2 dt.

46

1. THE LAPLACE INTEGRAL

This proves the ﬁrst part of the result. Since L1 (R+ , X) ∩ L2 (R+ , X) is dense in L2 (R+ , X), F extends uniquely to a bounded linear operator on L2 (R, X) such that √12π F is an isometry. One may prove (1.15) in a similar way, using Parseval’s formula and the corresponding scalar-valued result (Theorem 1.8.1 b)). This implies that the adjoint operator of F in the sense of Hilbert spaces is F, where (Ff )(t) = (Ff )(−t). Hence, FF = 2πI so F is surjective, and it follows that F is surjective. We remark that Plancherel’s theorem extends to functions of several variables with values in a Hilbert space X: the normalized Fourier transform (2π)−n/2 F is a unitary operator on L2 (Rn , X). This can be deduced from the scalar-valued case as in the proof of Theorem 1.8.2. Alternatively, it may be deduced from Theorem 1.8.2 by induction, using the identiﬁcation L2 (Rn+1 , X) = L2 (R, L2 (Rn , X)). When f ∈ L1 (R+ , X), we consider f as being a member of L1 (R, X) with f (t) = 0 for t < 0, so ∞ (F f )(s) = e−ist f (t) dt = fˆ(is), 0

where, as usual, fˆ is the Laplace transform of f . Similarly, if t → e−at f (t) belongs to L1 (R+ , X) (for example, if f ∈ L2 (R+ , X) and a > 0), then its Fourier transform is s → fˆ(a + is). Thus, Plancherel’s theorem can be used to study Laplace transforms of functions in L2 (R+ , X) when X is a Hilbert space. Let C+ := {λ ∈ C : Re λ > 0} and H 2 (C+ , X) be the space of all holomorphic functions g : C+ → X such that ∞

g 2H 2 (C+ ,X) := sup

g(α + is) 2 ds < ∞. α>0

−∞

For scalar-valued functions, the Paley-Wiener theorem [Rud87, Theorem 9.13] shows that g ∈ H 2 (C+ ) := H 2 (C+ , C) if and only if g = fˆ|C+ for some f ∈ L2 (R+ ). Then ∞ ∞ 2 sup |g(α + is)| ds = lim |g(α + is)|2 ds. α>0

−∞

α↓0

−∞

Moreover, g has Ff as a boundary function in the sense that g(α + is) → (F f )(s) s-a.e. and in L2 -norm, as α ↓ 0. In addition, g is the Poisson integral of F f : α ∞ (Ff )(r) g(α + is) = dr 2 π −∞ α + (s − r)2 [Dur70, Chapter 11], [Koo80, Chapter VI]. Again, these results are not true for vector-valued functions in general, but they are true in the case of Hilbert spaces.

1.8. THE FOURIER TRANSFORM AND PLANCHEREL’S THEOREM

47

Theorem 1.8.3 (Paley-Wiener Theorem). Let X be a Hilbert space. Then the map f → fˆ|C+ is an isometric isomorphism of L2 (R+ , X) onto H 2 (C+ , X). Moreover, for f ∈ L2 (R+ , X), α ∞ (F f )(r) ˆ f (α + is) = dr. (1.16) π −∞ α2 + (s − r)2 ∞ As α ↓ 0, fˆ(α+is)−(F f )(s) → 0 (s)-a.e. and −∞ fˆ(α+is)−(F f )(s) 2 ds → 0. Proof. Let f ∈ L2 (R+ , X). For α > 0, ∞ ! " fˆ(α + is) = e−ist e−αt f (t) dt. 0

By Plancherel’s Theorem 1.8.2, ∞

2

ˆ

f (α + is) ds = 2π −∞

∞

0

e−2αt f (t) 2 dt ≤ 2π f 22 .

Thus fˆ ∈ H 2 (C+ , X). Moreover, Plancherel’s theorem and the dominated convergence theorem give ∞ ∞

fˆ(α + is) − (F f )(s) 2 ds = 2π |e−αt − 1|2 f (t) 2 dt → 0 −∞

0

as α ↓ 0. For α > 0 and x ∈ X, ∞ α ∞ ((F f )(r)|x)X (Ff )(r) x (fˆ(α + is)|x)X = dr = dr , 2 2 2 2 π −∞ α + (s − r) −∞ α + (s − r) X which establishes (1.16). The proof that fˆ(α + is) − (F f )(s) → 0 a.e. is similar to the scalar-valued case. Conversely, let g ∈ H 2 (C+ , X). Then gˆ is separably valued, so we may assume that X is separable. Let {en : n ∈ N} be an orthonormal basis of X, and let gn (λ) = (g(λ)|en )X . Then gn ∈ H 2 (C+ ), so the scalar-valued case implies that there exists fn ∈ L2 (R+ ) such that gn = fn |C+ . Moreover, ∞ ∞ ∞ ∞ 2 |fn (t)| dt = lim e−2αt |fn (t)|2 dt 0

α↓0

n=1

n=1

0

∞ ∞

1 lim |gn (α + is)|2 ds 2π α↓0 n=1 −∞ ∞ 1 = lim

g(α + is) 2 ds < ∞. 2π α↓0 −∞ ∞ ∞ Hence n=1 |fn (t)|2 converges, and therefore n=1 fn (t)en converges to a sum f (t) in X, for almost all t. Now, f ∈ L2 (R+ , X) and, for λ ∈ C+ , =

fˆ(λ) =

∞ n=1

fn (λ)en = g(λ).

48

1.9

1. THE LAPLACE INTEGRAL

The Riemann-Stieltjes Integral

b This section is an introduction to the Riemann-Stieltjes integral a g(t) dF (t) of a vector-valued function F and a scalar-valued function g on [a, b]. Such integrals play an important role in the approach to Laplace transform theory taken in Chapter 2. More precisely, let f : R+ → X be a bounded measurable function. t Then t → F (t) := 0 f (s) ds is Lipschitz continuous and we will show in Sections 1.10 and 2.1 that ∞ ∞ fˆ(λ) = e−λt f (t) dt = e−λt dF (t) = TF (e−λ ) 0

∞

0

for all λ > 0, where TF : g → 0 g(t) dF (t) is a bounded linear operator from L1 (R+ ) to X and e−λ (t) := e−λt . Thus, the Laplace integrals fˆ(λ) are evaluations of a bounded linear operator TF at the exponential function e−λ . Since the map ΦS : F → TF turns out to be an isometric isomorphism between the Lipschitz continuous functions F : R+ → X and L(L1 (R+ ), X) (see Section 2.1), many functional analytic arguments can be applied to Laplace transform theory. A function F : [a, b] → X is of bounded semivariation if there exists M ≥ 0 such that i (F (ti ) − F (si )) ≤ M for every choice of a ﬁnite number of nonoverlapping intervals (si , ti ) in [a, b]. Recall from Section 1.2 that F is of bounded variation if there exists M ≥ 0 such that i F (ti ) − F (ti−1 ) ≤ M for every ﬁnite partition a = t0 < t1 < . . . < tn = b of [a, b]. Further, F is of weak bounded variation if x∗ ◦ F : t → F (t), x∗ is of bounded variation for every x∗ ∈ X ∗ . The set of functions F : [a, b] → X of bounded semivariation is denoted by BSV([a, b], X). A function F : R+ → X is in BSVloc (R+ , X) if it is of bounded semivariation on every compact subinterval of R+ . As remarked in Section 1.2, any Lipschitz function is of bounded variation, and it is easy to see that any function of bounded variation is of bounded semivariation. We show in the following proposition that the function F : [0, 1] → L∞ [0, 1] deﬁned by F (t) = χ[0,t] is of bounded semivariation. Since F (t) − F (s) = 1 for all t = s, F is not of bounded variation, not separably valued, and not measurable (see Example 1.1.5). Hence, functions of bounded semivariation may not be measurable. Proposition 1.9.1. Let X be an ordered Banach space with normal cone. Let F : [a, b] → X be increasing. Then F is of bounded semivariation. Proof. Let (si , ti ) (i = 1, 2, . . . , n) be disjoint intervals in [a, b]. Then 0≤

n

(F (ti ) − F (si )) ≤ F (tn ) − F (s1 ) ≤ F (b) − F (a).

i=1

Hence

n

(F (ti ) − F (si )) ≤ c F (b) − F (a) ,

i=1

1.9. THE RIEMANN-STIELTJES INTEGRAL

49

where c is a constant associated with the normal cone (see Appendix C). In the context of Proposition 1.9.1, it is easy to see directly that F is of weak bounded variation. In fact, there is the following general result. Proposition 1.9.2. A function F : [a, b] → X is of bounded semivariation if and only if it is of weak bounded variation. Proof. Assume that F is of weak bounded variation. Let SΩ := i (F (ti ) − F (si )), where Ω is the union of ﬁnitely many disjoint intervals (si , ti ) in [a, b]. For each x∗ ∈ X ∗ , there exists Mx∗ := V[a,b] (x∗ ◦ F ) such that |SΩ , x∗ | ≤ Mx∗ for all such Ω. It follows from the uniform boundedness principle that F is of bounded semivariation. Now, let F be of bounded semivariation. Then there exists a constant M ≥ 0 such that i (F (ti ) − F (si )) ≤ M for any choice of a ﬁnite number of disjoint intervals (si , ti ) in [a, b]. To obtain the weak bounded variation, one writes x∗ = x∗1 + ix∗2 , where x∗j is a real-linear functional, and distinguishes between the subintervals on which the numbers F (ti ) − F (ti−1 ), x∗j are either positive or negative. Let F, g be two functions deﬁned on an interval [a, b], one with values in X and the other with values in C. If π denotes a ﬁnite partition a = t0 < t1 < . . . < tn = b of [a, b] with partitioning points ti and with some intermediate points si ∈ [ti−1 , ti ] (i = 1, . . . , n), we denote by |π| = maxi (ti − ti−1 ) the norm of π, and by n S(g, F, π) := g(si ) (F (ti ) − F (ti−1 )) i=1

the Riemann-Stieltjes sum associated with g, F and π. We say that g is RiemannStieltjes integrable with respect to F if

b

g(t) dF (t) := lim S(g, F, π) a

|π|→0

exists in the norm topology of X. Here π runs through all partitions of [a, b] with intermediate points, and the limit must be independent of the choice of intermediate points. It is immediate from the deﬁnition that the set of all functions g which are Riemann-Stieltjes integrable with respect to a ﬁxed function F is a linear space, and the Riemann-Stieltjes integral is linear in g (and also in F ). If F is of bounded b variation, g is bounded and a g(t) dF (t) exists, then

b

g(t) dF (t) ≤ sup g(t) V[a,b] (F ).

a

t∈[a,b]

(1.17)

50

1. THE LAPLACE INTEGRAL

If F is of bounded semivariation, then it follows from the proof of Proposition 1.9.2 that

b

g(t) dF (t) ≤ 4M sup g(t) , (1.18)

a

t∈[a,b] where

M := sup (F (ti ) − F (si )) : (si , ti ) disjoint subintervals of [a, b] .

i (1.19) The Riemann-Stieltjes integral respects closed operators; there are easy analogues of Propositions 1.1.6 and 1.1.7, both when g is scalar-valued and F is vector-valued and in the alternative case. When F (t) = t, we write S(g, π) for S(g, F, π) and call it the Riemann sum associated with g and π. We say that g is Riemann integrable on [a, b] if g is Riemann-Stieltjes integrable with respect to F (t) = t, and we write

b

g(t) dt := lim S(g, π). a

|π|→0

In the scalar-valued case, g : [a, b] → C is Riemann integrable if and only if g is bounded and continuous a.e., and the Riemann and Lebesgue integrals are then equal [Rud76, Theorem 11.33]. By applying linear functionals, it follows that if g : [a, b] → X is Riemann integrable, then it is bounded and weakly measurable. If X is separable, then g is measurable by Pettis’s theorem 1.1.1 and hence Bochner integrable by Theorem 1.1.4. Whenever g is both Riemann and Bochner integrable, the two integrals coincide (so our notation should not cause confusion). However, Riemann integrable functions with values in an inseparable space may be nowhere continuous and not even measurable (see Example 1.9.7 below). Now we return to Riemann-Stieltjes sums and integrals of two functions F and g. Let π be a partition of [a, b] with partitioning points a = t0 < t1 < . . . < tn = b and intermediate points si ∈ [ti−1 , ti ]. If one chooses s0 = a and sn+1 = b, then we obtain a partition π with partitioning points a = s0 ≤ s1 ≤ . . . ≤ sn+1 = b, with intermediate points ti ∈ [si , si+1 ], and |π | ≤ 2|π|. Moreover, S(F, g, π) = g(b)F (b) − g(a)F (a) − S(g, F, π ). It follows that if g is Riemann-Stieltjes integrable with respect to F , then so is F with respect to g (and vice versa, by symmetry) and the following formula of integration by parts holds:

b a

g(t) dF (t) = g(b)F (b) − g(a)F (a) −

b

F (t) dg(t). a

(1.20)

1.9. THE RIEMANN-STIELTJES INTEGRAL

51

Example 1.9.3. Let a ≤ c ≤ d ≤ b and let I be an interval with endpoints c and d. Let F : [a, b] → X. If a ∈ I and b ∈ I, then F and χI are Riemann-Stieltjes integrable with respect to each other if and only if F is continuous at c and d, and then b

b

χI (t) dF (t) = −

a

F (t) dχI (t) = F (d) − F (c).

a

If a = c ∈ I and b ∈ I, then F and χI are Riemann-Stieltjes integrable with respect to each other if and only if F is continuous at d, and then b b χI (t) dF (t) = F (d) − F (a), F (t) dχI (t) = −F (d). a

a

If a ∈ I and b = d ∈ I, then F and χI are Riemann-Stieltjes integrable with respect to each other if and only if F is continuous at c, and then b b χI (t) dF (t) = F (b) − F (c), F (t) dχI (t) = F (c). a

a

Proposition 1.9.4. Let F : [a, b] → X and g : [a, b] → C. If one function is continuous and the other is of bounded semivariation, then F and g are RiemannStieltjes integrable with respect to each other. Proof. a) Assume that F is of bounded semivariation and that g is continuous. Let ε > 0. Then there exists δ > 0 such that |g(s1 ) − g(s2 )| < ε whenever |s1 − s2 | < δ. Let πj , (j = 1, 2), be two partitions of [a, b] with |πj | < δ/2. Let a = t0 < t1 < . . . < tn = b be the partitioning points of π1 and π2 together. Then S(g, F, πj ) =

n

g(sj,i )(F (ti ) − F (ti−1 ))

i=1

where sj,i , ti and ti−1 belong to the same subinterval of πj . In particular, |s1,i − s2,i | < δ. For x∗ ∈ X ∗ , n ∗ ∗ |S(g, F, π1 ) − S(g, F, π2 ), x | = (g(s1,i ) − g(s2,i ))F (ti ) − F (ti−1 ), x <

i=1 n

ε

|F (ti ) − F (ti−1 ), x∗ |

i=1

≤ 4εM x∗ , where M is deﬁned by (1.19). Now, Cauchy’s convergence criterion implies that b g(t) dF (t) exists. a b) Assume that F is continuous and g is of bounded semivariation. By Proposition 1.9.2, g is of bounded variation. Given ε > 0 there exists δ > 0 such that

F (s1 ) − F (s2 ) < ε whenever |s1 − s2 | < δ. Similarly to a), one shows that

S(F, g, π1 ) − S(F, g, π2 ) < εV[a,b] (g)

52

1. THE LAPLACE INTEGRAL

whenever |π1 | < δ/2 and |π2 | < δ/2, where V[a,b] (g) is the total variation of g. b Hence a F (t) dg(t) exists. Corollary 1.9.5. Let g : [a, b] → C be piecewise continuous, and F : [a, b] → X be continuous and of bounded semivariation. Then F and g are Riemann-Stieltjes integrable with respect to each other. Proof. Since g is piecewise continuous, g = g1 + g2 where g1 is continuous and g2 is a step function. Now the result follows from Proposition 1.9.4 and Example 1.9.3. Corollary 1.9.6. Let X be an ordered Banach space with normal cone. Let f : [a, b] → X be increasing. Then f is Riemann integrable. Proof. This is immediate from Propositions 1.9.1 and 1.9.4. Example 1.9.7. Let f : [0, 1] → L∞ [0, 1] be deﬁned by f (t) := χ[0,t] , so f is increasing, nowhere continuous and not measurable (Example 1.1.5 a)). By Corollary 1.9.6, f is Riemann integrable and 1 n 1 f (t) dt = lim χ[0,i/n] . n→∞ n 0 This shows that then

1 0

i=1

f (t) dt is the function s → 1−s. Similarly, if F (t) :=

t 0

f (r) dr,

F (t)(s) = (t − s)χ[0,t] (s) (see Example 1.2.8 b)). Proposition 1.9.8. Let F : [a, b] → X be of bounded semivariation and g : [a, b] → C be of bounded variation. Then gF is of bounded semivariation. Proof. There exists M > 0 such that |g(t)| ≤ M and F (t) ≤ M for all t ∈ [a, b]. The function gF is of bounded semivariation since it is of weak bounded variation. This follows from the assumptions and the estimates: |g(ti )F (ti ) − g(ti−1 )F (ti−1 ), x∗ | i

≤

i

+

|(g(ti ) − g(ti−1 ))F (ti ), x∗ |

|g(ti−1 )(F (ti ) − F (ti−1 )), x∗ |

i

≤ M x∗

i

∗

|g(ti ) − g(ti−1 )| + M

|F (ti ) − F (ti−1 ), x∗ | ,

i

∗

for all x ∈ X . In the remainder of this section, we give some results which reduce RiemannStieltjes integrals to Riemann or Bochner integrals when g or F has a derivative in an appropriate sense.

1.9. THE RIEMANN-STIELTJES INTEGRAL

53

Proposition 1.9.9. Let F : [a, b] → X be of bounded semivariation and g ∈ C 1 [a, b]. Then F g is Riemann integrable and

b

F (t) dg(t) = a

b

F (t)g (t) dt.

a

Proof. a) We show that hF is Riemann integrable for each h ∈ C[a, b]. If h is a step function, then hF is of bounded semivariation and hence Riemann integrable, by Proposition 1.9.4. Since each continuous function h on [a, b] is a uniform limit of step functions and F is bounded, the claim follows since a uniform limit of Riemann integrable functions is Riemann integrable. b) There exists M > 0 such that |g (t)| ≤ M and F (t) ≤ M for all t ∈ [a, b]. For ε > 0 there exists δ > 0 such that |g (s) − g (s )| < ε whenever |s − s | < δ. Let π be a partition of [a, b] with |π| < δ and with partitioning points ti and intermediate points si . By the mean value theorem, there exist si ∈ (ti−1 , ti ) such that g(ti ) − g(ti−1 ) = g (si )(ti − ti−1 ). Let π be the partition with partitioning points ti and intermediate points si , so |π | = |π|. Then S(F, g, π )

=

F (si )g (si )(ti − ti−1 )

i

= S(F g , π ). Letting |π| → 0, it follows that

b a

F (t) dg(t) =

b a

F (t)g (t) dt.

Proposition 1.9.10. Let F : [a, b] → X be of bounded semivariation and g, h ∈ t C[a, b]. Then G(t) := a h(s) dF (s) is of bounded semivariation on [a, b] and

b

g(t) dG(t) =

b

g(t)h(t) dF (t).

a

a

Proof. Let M be such that |h(t)| ≤ M for all t ∈ [a, b]. Let π be a partition of [a, b] with partitioning points ti and intermediate points si . By (1.17), for x∗ ∈ X ∗ , i

|G(ti ) − G(ti−1 ), x∗ |

=

ti h(s) dF (s), x∗ ti−1 i

≤ M V[a,b] (x∗ ◦ F ).

It follows from Proposition 1.9.2 that G is of bounded semivariation. By Proposib b tion 1.9.4, a g(t) dG(t) and a g(t)h(t) dF (t) both exist. For ε > 0 there exists δ > 0 such that |g(s ) − g(s)| < ε whenever |s − s| < δ.

54

1. THE LAPLACE INTEGRAL

If |π| < δ, then

# $ b ∗ g(t)h(t) dF (t), x S(g, G, π) − a ti ∗ = (g(si ) − g(t))h(t) dF (t), x ti−1 i

≤ εM V[a,b] (x∗ ◦ F ). It follows that

# a

b

$

b

g(t) dG(t) −

∗

g(t)h(t) dF (t), x

= 0.

a

The result follows from the Hahn-Banach theorem. The following result gives analogues of a special case of Proposition 1.9.10 and of Proposition 1.9.9, with Riemann integrals replaced by Bochner integrals. Proposition 1.9.11. Let g : [a, b] → C and F : [a, b] → X. If F is an antiderivative b of a Bochner integrable function f and if g is continuous, then a g(s) dF (s) exists b and equals the Bochner integral a g(s)f (s) ds. If F is continuous and g is absob b lutely continuous, then a F (s) dg(s) equals the Bochner integral a F (s)g (s) ds. Proof. Assume that g is continuous t and that there exists a Bochner integrable function f such that F (t) = F (a) + a f (s) ds for all t ∈ [a, b]. Then F is of bounded b variation (Proposition 1.2.2) and the Riemann-Stieltjes integral a g(s) dF (s) exb ists by Proposition 1.9.4. The Bochner integral a g(s)f (s) ds exists by Theorem 1.1.4, since g is bounded and measurable. For ε > 0 there exists δ > 0 such that |g(s ) − g(s)| < ε whenever |s − s| < δ. For any partition π with |π| < δ,

b

ti

g(s)f (s) ds = (g(si ) − g(s))f (s) ds

S(g, F, π) −

a t i−1 i b ≤ ε

f (s) ds. a

Letting |π| → 0 and ε → 0, the result follows. The proof of the second statement is analogous and is omitted. Combining Propositions 1.9.4, 1.9.8 and 1.9.9 with the integration by parts formula (1.20) one obtains the following statement which will be used frequently in later sections. Let F : [0, t] → X be of bounded semivariation. Then t t e−λs dF (s) = e−λt F (t) − F (0) + λ e−λs F (s) ds (1.21) 0

0

1.10. LAPLACE-STIELTJES INTEGRALS

55

t for all λ ∈ C. One should notice that the integral 0 e−λs F (s) ds is a Riemann integral if F is of bounded semivariation, by Propositions 1.9.4 and 1.9.9. If F is also continuous, then the integral can be taken in the Bochner sense.

1.10 Laplace-Stieltjes Integrals This section contains the essential properties of the Laplace-Stieltjes integral ∞ τ −λt dF (λ) := e dF (t) := lim e−λt dF (t), τ →∞

0

0

where F ∈ BSVloc (R+ , X); i.e., F : R+ → X is of bounded semivariation on each compact subinterval of R+ . First we observe that the Laplace-Stieltjes integral is a generalization of the Laplace integral. t (λ) Proposition 1.10.1. Let f ∈ L1loc (R+ , X) and F (t) := 0 f (s) ds. For λ ∈ C, dF (λ) = fˆ(λ). exists if and only if fˆ(λ) exists, and then dF Proof. By Proposition 1.9.11, τ −λt e dF (t) = 0

τ

e−λt f (t) dt

0

(τ ≥ 0),

and the result follows by letting τ → ∞. The results obtained in Sections 1.4, 1.5 and 1.6 for Laplace integrals carry over to Laplace-Stieltjes integrals with only minor modiﬁcations of the proofs, and we give these below, starting with elementary properties. Throughout this section, integrals over R+ are to be understood as improper Riemann-Stieltjes (or Riemann) integrals. Thus ∞ τ g(t) dF (t) := lim g(t) dF (t), τ →∞ 0 0 ∞ τ F (t) dt := lim F (t) dt. τ →∞

0

∞

0

It is easy to see that 0 F (t) dt exists if F ∈ BSVloc (R+ , X) and F (t) ≤ h(t) (t ≥ 0) for some h ∈ L1 (R+ ). Recall from Section 1.4 that the exponential growth bound of a function F ∈ BSVloc (R+ , X) is deﬁned by ω(F ) := inf ω ∈ R : sup e−ωt F (t) < ∞ . t≥0

It follows from (1.21) that abs(dF ) ≤ ω(F ) and ∞ (λ) = −F (0) + λ dF e−λt F (t) dt

(Re λ > ω(F )).

0

Note that (1.22) is a generalization of both (1.11) and Corollary 1.6.5.

(1.22)

56

1. THE LAPLACE INTEGRAL

Proposition 1.10.2. Let f ∈ BSVloc (R+ , X) and F (t) := locally Lipschitz continuous, and

t 0

f (s) ds. Then F is

(λ) = −f (0) + λ2 Fˆ (λ) df (λ) = −f (0) + λdF whenever Re λ > ω(f ). Proof. Since f is locally bounded, (1.17) implies that F is locally Lipschitz continuous and ω(F ) ≤ ω(f ). In particular, F ∈ L1loc (R+ , X) ∩ BSVloc (R+ , X). By (1.21) and Proposition 1.9.10, τ τ e−λs df (s) = e−λτ f (τ ) − f (0) + λ e−λs dF (s). 0

0

Letting τ → ∞ gives

(λ) df (λ) = −f (0) + λdF

(λ) = λFˆ (λ). whenever Re λ > ω(f ). By (1.22), dF Now we give a generalization of Proposition 1.6.1 a). t Proposition 1.10.3. Let F ∈ BSVloc (R+ , X), μ ∈ C and let G(t) := 0 e−μs dF (s) (λ + μ) exists, and then (t ≥ 0). For λ ∈ C, dG(λ) exists if and only if dF dG(λ) = dF (λ + μ). Proof. By Proposition 1.9.10, τ e−λt dG(t) = 0

τ

e−(λ+μ)t dF (t),

0

and the result follows on letting τ → ∞. For F ∈ BSVloc (R+ , X), let

(λ) exists . abs(dF ) := inf Re λ : dF

(λ) converges if Re λ > Proposition 1.10.4. Let F ∈ BSVloc (R+ , X). Then dF abs(dF ) and diverges if Re λ < abs(dF ). (λ) does not exist if Re λ < abs(dF ). For λ0 ∈ C deﬁne G0 (t) := Proof. Clearly, dF t −λ s 0 e dF (s) (λ ∈ C, t ≥ 0). Then, by Proposition 1.9.10, 0 t t −λs e dF (s) = e−(λ−λ0 )s dG0 (s) (λ ∈ C , t ≥ 0). 0

0

Integration by parts (1.20), and Proposition 1.9.9, yield t t e−λs dF (s) = e−(λ−λ0 )t G0 (t) + (λ − λ0 ) e−(λ−λ0 )s G0 (s) ds. 0

0

(1.23)

1.10. LAPLACE-STIELTJES INTEGRALS

57

(λ0 ) exists, then G0 is bounded. Therefore, dF (λ) exists if Re λ > Re λ0 and If dF ∞ (λ) = (λ − λ0 ) dF e−(λ−λ0 )s G0 (s) ds (Re λ > Re λ0 ). (1.24) 0

(λ) exists if Re λ > abs(dF ) and, as for the Laplace integral This shows that dF (see (1.9)), t abs(dF ) = inf λ ∈ R : sup e−λs dF (s), x∗ < ∞ for all x∗ ∈ X ∗ . (1.25) t>0

0

Theorem 1.10.5. Let F ∈ BSVloc (R+ , X) and let F∞ := limt→∞ F (t) if the limit exists, F∞ := 0 otherwise. Then abs(dF ) = ω(F − F∞ ). t Proof. For λ0 > abs(dF ) deﬁne G0 (t) := 0 e−λ0 s dF (s) (t ≥ 0). Then G0 is bounded. To prove that abs(dF ) ≥ ω(F − F∞ ) one considers the two cases abs(dF ) ≥ 0 and abs(dF ) < 0. First, let abs(dF ) ≥ 0 and λ0 > abs(dF ). It follows from Proposition 1.9.10, integration by parts (1.20), and Proposition 1.9.9 that t t F (t) = F (0) + eλ0 s dG0 (s) = F (0) + eλ0 t G0 (t) − λ0 eλ0 s G0 (s) ds 0

0

−λ0 t

for all t ≥ 0, so supt≥0 e (F (t) − F∞ ) < ∞. Thus ω(F − F∞ ) ≤ abs(dF ) if abs(dF ) ≥ 0. Second, let abs(dF ) < 0. Choose abs(dF ) < λ0 < 0. For r ≥ t ≥ 0 one has r r λ0 s λ0 r λ0 t F (r) − F (t) = e dG0 (s) = e G0 (r) − e G0 (t) − λ0 eλ0 s G0 (s) ds. t

t

Thus,

lim F (r) = F∞ = F (t) − e

λ0 t

r→∞

∞

G0 (t) − λ0

eλ0 s G0 (s) ds

t

exists and supt≥0 e−λ0 t (F (t) − F∞ ) < ∞. Therefore, ω(F − F∞ ) ≤ abs(dF ) if abs(dF ) < 0. To show the reverse inequality let ω > ω(F − F∞ ). Then there exists M ≥ 0 such that F (t) − F∞ ≤ M eωt for all t ≥ 0. Let λ > ω > ω(F − F∞ ). Integration by parts (see (1.21)) yields t t e−λs dF (s) = e−λt (F (t) − F∞ ) + F∞ − F (0) + λ e−λs (F (s) − F∞ ) ds. 0

0

(λ) exists for λ > ω(F − F∞ ) and is given by Hence, dF ∞ (λ) = F∞ − F (0) + λ dF e−λs (F (s) − F∞ ) ds. 0

This shows that abs(dF ) ≤ ω(F − F∞ ). Note that (1.26) is a generalization of (1.22).

(1.26)

58

1. THE LAPLACE INTEGRAL

Theorem 1.10.6. Let F ∈ BSVloc (R+ , X) and assume that abs(dF ) < ∞. Then (λ) is holomorphic for Re λ > abs(dF ), and λ → dF ∞ (n) (λ) = dF e−λt (−t)n dF (t) (Re λ > abs(dF ), n ∈ N0 ) 0

(as an improper Riemann-Stieltjes integral). k Proof. Let qk (λ) := 0 e−λt dF (t). It follows from (1.18) that N λn k (−t)n dF (t). N→∞ n! 0 n=0

qk (λ) = lim

By the Weierstrass convergence theorem (a special case of Vitali’s Theorem A.5), qk is entire and k (j) qk (λ) = e−λt (−t)j dF (t) 0

for all j ∈ N0 . Let Re λ > λ0 > abs(dF ), and deﬁne t G0 (t) := e−λ0 s dF (s). 0

Then G0 is bounded and it follows from Proposition 1.9.10, (1.20) and Proposition 1.9.9 that ∞ dF (λ) − qk (λ) = e−(λ−λ0 )s dG0 (s) k ∞ = −e−(λ−λ0 )k G0 (k) + (λ − λ0 ) e−(λ−λ0 )s G0 (s) ds. k

uniformly on compact subsets of {λ : Re λ > abs(dF )}. Hence qk converges to dF is holomorphic and q(j) (λ) → Again by the Weiestrass convergence theorem, dF k (j) dF (λ) as k → ∞, if Re λ > abs(dF ). Finally in this section, we consider operator-valued Laplace-Stieltjes integrals. Let S : R+ → L(X, Y ) be a function. By the uniform boundedness principle, S ∈ BSVloc (R+ , L(X, Y )) if and only if vx := S(·)x ∈ BSV(R+ , Y ) for all x ∈ X. When S ∈ BSVloc (R+ , L(X, Y )), we let −ωt S(t) < ∞ , ω(S) := inf ω ∈ R : sup e abs(dS)

t≥0 t

e−λs dS(s) converges strongly as t → ∞

:=

inf Re λ :

=

sup{abs(dvx ) : x ∈ X}.

0

The following analogue of Proposition 1.4.5 follows from Theorems 1.10.5 and 1.10.6.

1.11. NOTES

59

Proposition 1.10.7. Let S ∈ BSVloc (R+ , L(X, Y )) and let S∞ be the strong limit of S(t) as t → ∞ if it exists, and S∞ := 0 otherwise. Then t a) limt→∞ 0 e−λs dS(s) exists in operator norm whenever Re λ > abs(dS), b) abs(dS) = ω(S − S∞ ). t Proof. If 0 e−λ0 s dS(s) converges strongly as t → ∞, then it is uniformly bounded in operator norm. Thus, a) follows from (1.23). Hence, t −λs abs(dS) = inf Re λ : e dS(s) converges in norm as t → ∞ 0

= ω(S − S∞ ), by Theorem 1.10.6, where S∞ is the norm limit of S(t) as t → ∞, if this exists, S∞ := 0 otherwise. It is trivial that ω(S − S∞ ) = ω(S − S∞ ), so b) is proved.

1.11 Notes Section 1.1 The Bochner integral is an extension of the Lebesgue integral to functions with values in Banach spaces. Introduced around 1930 by Bochner, it has become a widely used integral in inﬁnite dimensional applications. Much of Section 1.1 follows the treatment of the Bochner integral in Chapter III of [HP57], where many references to the original literature can be found. Comprehensive treatments of the Bochner integral and vectorvalued measures, as well as references to the literature are contained in the monograph [DU77] by Diestel and Uhl. Corollary 1.1.3 is taken from [Are01]. One reason for introducing the Riemann integral here is that increasing functions with values in an ordered Banach space with normal cone are always Riemann integrable (Corollary 1.9.6). However, if the space is not separable, Riemann integrable functions are not necessarily Bochner integrable and their antiderivative may be nowhere diﬀerentiable. One way to circumvent these diﬃculties is to consider generalizations of the Riemann integral. This allows a version of the fundamental theorem of calculus where all continuous functions f : [0, 1] → X (where X is a Banach space) with f (0) = 0 are diﬀerentiable in the mean and coincide with the generalized Riemann integral of their derivatives (see [BLN99]). Section 1.2 The Radon-Nikodym property was identiﬁed in the 1970s as an important property in the theory of vector measures and also in the geometry of Banach spaces. Our treatment is based on [DU77]. Section 1.3 Most of the results are vector-valued versions of standard material. Proposition 1.3.7 is contained in [KH89]. Section 1.4 The Laplace transform has a long history, dating back to Euler’s paper ‘De constructione

60

1. THE LAPLACE INTEGRAL

aequationum’ from 1737, Lagrange’s ‘M´emoire sur l’utilit´e de la m´ethode de prendre le milieux entre les r´esultats de plusieurs observations’ from 1773, and Laplace’s ‘M´emoire sur les approximations des formules qui sont ﬁnctions de tr`es grands nombres’ from 1785. Since then it has been widely used in mathematics and engineering (in particular in ordinary diﬀerential, diﬀerence and functional equations, electrical engineering and applications to signal processing problems). Modern Laplace transform theory began to emerge at the end of the 19th century when Heaviside popularized a user-friendly and powerful operational calculus within the engineering community in connection with his research in electromagnetism [Hea93]. Since his methods were to a large degree based on purely formal operations with few mathematical justiﬁcations, many mathematicians at the beginning of the 20th century began to strive for a solid mathematical foundation of Heaviside’s operational calculus by virtue of the Laplace transform. These eﬀorts culminated in Widder’s books ‘The Laplace Transform’ [Wid41] and ‘An Introduction to Transform Theory’ [Wid71] as well as Doetsch’s ‘Theorie und Anwendung der LaplaceTransformation’ [Doe37] and his monumental, three volumed ‘Handbuch der LaplaceTransformation’ [Doe50]. These monographs have been among the best introductions to the subject and have become classic texts. A ﬁrst comprehensive look at Laplace transform theory for functions with values in a Banach space X is contained in Hille’s monograph ‘Functional Analysis and Semi-Groups’ from 1948 [Hil48]. Many historical notes on Laplace transform theory can be found in the books of Doetsch and in survey articles by Deakin [Dea81], [Dea82], and Martis in Biddau [Bid33]. One weakness of Laplace transform theory—compared to Heaviside’s operational calculus—are the restrictions concerning the growth of the functions at inﬁnity. To remove these restrictions, Vignaux introduced in 1939 an asymptotic version of the Laplace transform [Vig39], [VC44]. For an extension of the asymptotic Laplace transform to vector-valued functions, and references to the literature, see [LN99] and [LN01]. The characterization of the abscissa of convergence by the exponential growth of the antiderivative are vector-valued versions of classical results due to Landau (1906) and Pincherle (1913) that can be found in [Doe50, Volume I,Theorems 2.2.7 and 2.2.8], or [HP57, Section 1.6.2]. Section 1.5 Theorem 1.5.1 is due to Pincherle and Landau (1905); Theorem 1.5.3 is due to Landau (1906). The proofs given here, as well as Example 1.5.2, follow [Doe50, Volume I, Sections 3.2–3.4] where further references to the classical literature can be found. Section 1.6 The results of this section are straightforward vector-valued versions of standard results in the classical theory of Laplace transforms (see [Doe50, Volume I, Sections 2.14, 2.15], for example). Section 1.7 Theorem 1.7.5 has been proved by Ti-Jun Xiao and Jin Liang [XL00], but a special case was given by Kurtz [Kur69] and the general result was mentioned by Chernoﬀ [Che74, p.106]. In fact, the proof of [Che68, Proposition] carries over to the case considered in Theorem 1.7.5. The short proof given here appeared in [Bob97b] and [Are01]. The Inversion Theorem 1.7.7 is due to Post (1930) and Widder (1934) (see [Wid41, Section 7.6], and [Doe50, Volume I, Section 8.2]). Numerically more eﬃcient Post-Widder type inversion formulas for the Laplace transform can be found in [Jar08, Theorem 4.1]

1.11. NOTES

61

and [JNO08]. They are derived from rational approximation methods for operator semigroups developed by Hersh and Kato [HK79] and Brenner and Thom´ee [BT79] (see also [Kov07]). The Uniqueness Theorem 1.7.3 was mentioned ﬁrst by Pastor [Pas19] in 1919 and is a special case of the following result of Shen [She47] (see also [Doe50, Volume I, Section 2.9], [BN94], [Mih09]). Theorem 1.11.1. Let f ∈ L1loc (R+ , X) with abs(f ) < ∞. Let (λn ) be an inﬁnite sequence with no accumulation point and Re λn ≥ a > 0 for all n ∈ N and some a > abs(f ). If λn − 1 = ∞, 1 − λn + 1 n=1 ∞

then (λn )n∈N is a uniqueness sequence; i.e., fˆ(λn ) = 0 (n ∈ N) implies that f = 0. Conversely, let (λn ) be a sequence with λn = 0 and | arg(λn )| ≤ θ < π/2 (n ∈ N). If (λn ) has no accumulation point and the sum above is ﬁnite, then there exists 0 = f ∈ L1loc (R+ , X) with fˆ(λn ) = 0 for all n ∈ N. Consider the horizontal sequences λn := a + nγ b for a, b > 0. If 0 < γ ≤ 1 or γ < 0, then (λn ) is a uniqueness sequence. If γ > 1 or a = 0 and γ < −1, then {λn } is the set of zeros of a non-trivial Laplace transform. For example, if f (t) = √1t sin( 1t ), √ π −√2λ then fˆ(λ) = e sin( 2λ) which has zeros for λn = 2n2 π 2 (n ∈ N). The vertical λ sequences λn = 1 + inγ are uniqueness sequences if 0 < γ ≤ 12 . If γ > 12 , then {λn } is the set of zeros of a non-trivial Laplace transform. Uniqueness sequences are important in the discussion of Cauchy problems which are well-posed in the regularized sense (see [B¨ au01] or [LN99] for deﬁnitions and references). Section 1.8 For 1 ≤ p ≤ 2, a Banach space X is said to have Fourier type p if the Fourier transform on L1 (R, X) ∩Lp (R, X) extends to a bounded linear operator of Lp (R, X) into Lp (R, X), where 1/p + 1/p = 1. The Hausdorﬀ-Young inequalities show that C has Fourier type for every p ∈ [1, 2]. Every Banach space has Fourier type 1, and a space with Fourier type p also has Fourier type q whenever 1 ≤ q ≤ p. Theorem 1.8.2 shows that Hilbert spaces have Fourier type 2, and conversely Kwapie´ n [Kwa72] showed that a space with Fourier type 2 is isomorphic to a Hilbert space. A space of the form Lp (Ω, μ), where 1 ≤ p < ∞ and (Ω, μ) is any measure space, has Fourier type min(p, p ). The spaces with non-trivial Fourier type (i.e., Fourier type p for some p > 1) have been characterized by Bourgain [Bou82], [Bou88] (see also [Pis86]). Every superreﬂexive space (a Banach space with an equivalent uniformly convex norm) has non-trivial Fourier type, but there exist reﬂexive spaces which do not have non-trivial Fourier type and there exist non-reﬂexive spaces which do have non-trivial Fourier type. A Banach space X is said to have the analytic Radon-Nikodym property (ARNP) if each function g ∈ H 2 (R, X) has a boundary function, i.e. limα↓0 g(α + is) exists s-a.e. This property was ﬁrst considered by Bukhvalov [Buk81], [BD82] using functions on the unit disc rather than C+ , and H p -spaces for p = 2, but this formulation is equivalent. Thus, Theorem 1.8.3 shows in particular that Hilbert spaces have the (ARNP). Every reﬂexive space has the (ARNP), and more generally, any space with the Radon-Nikodym property, and also any space of the form L1 (Ω, μ), has the (ARNP). On the other hand,

62

1. THE LAPLACE INTEGRAL

c0 does not have the (ARNP), and there exist spaces with non-trivial Fourier type which do not have the (ARNP) (see [HN99]). Section 1.9 This section contains some of the basic properties of the Riemann-Stieltjes integral; see [HP57] and [Wid41] for further details and references to the original literature. Section 1.10 In the classical Laplace transform literature, many authors preferred Laplace-Stieltjes ∞ ∞ integrals 0 e−λt dF (t) since they include Laplace integrals 0 e−λt f (t) dt (when F is ∞ −λti diﬀerentiable a.e.) and Dirichlet series (when F is a step function). The i=1 ai e importance of the Laplace-Stieltjes integral for our purposes is that many classical results for Laplace-Stieltjes integrals of complex-valued functions F can be extended to functions with values in arbitrary Banach spaces X, whereas vector-valued extensions of the corresponding Laplace transform results often require additional assumptions on X (see, for example, [Zai60]). All results of this section are vector-valued versions of classical results for Laplace-Stieltjes transforms in [Wid41] or [Wid71].

Chapter 2

The Laplace Transform In this chapter the emphasis of the discussion shifts from Laplace integrals fˆ(λ) and (λ) to the Laplace transform L : f → fˆ and to the Laplace-Stieltjes transform dF . The Laplace transform is considered ﬁrst as an operator acting on LS : F → dF L∞ (R+ , X) and the Laplace-Stieltjes transform as an operator on Lip0 (R+ , X) := F : R+ → X : F (0) = 0, F Lip0 (R+ ,X) :=

F (t) − F (s)

sup <∞ . |t − s| t,s≥0 These domains of L and LS are relatively easy to deal with and have immediate and important applications to abstract diﬀerential and integral equations. The following observation is the key to one of the basic structures of Laplace t transform theory. If f ∈ L∞ (R+ , X), then t → F (t) := 0 f (s) ds belongs to Lip0 (R+ , X) and ∞ ∞ L(f )(λ) = e−λt f (t) dt = e−λt dF (t) = TF (e−λ ), ∞

0

0

where TF : g → 0 g(s)dF (s) is a bounded linear operator from L1 (R+ ) into X, and where e−λ denotes the exponential function t → e−λt . The operator TF is fundamental to Laplace transform theory. In Section 2.1 it is shown that ΦS : F → TF is an isometric isomorphism between Lip0 (R+ , X) and L(L1 (R+ ), X) (Riesz-Stieltjes representation theorem). This representation is crucial for the following reason. The main purpose of Laplace transform theory is to translate properties of the generating function F into properties of the resulting ∞ function λ → ∞ r(λ) = 0 e−λt dF (t) and vice versa. Since F (t) = TF χ[0,t] = 0 χ[0,t] (s) dF (s) ∞ and r(λ) = TF e−λ = 0 e−λs dF (s), the generating function F as well as the resulting function r are evaluations of the same bounded linear operator acting on diﬀerent total subsets of L1 (R+ ).

W. Arendt et al., Vector-valued Laplace Transforms and Cauchy Problems: Second Edition, Monographs in Mathematics 96, DOI 10.1007/978-3-0348-0087-7_2, © Springer Basel AG 2011

63

64

2. THE LAPLACE TRANSFORM

In Section 2.2, the range of the Laplace-Stieltjes transform acting on Lip0 (R+ , X) is characterized. It is shown that a function r : R+ → X has a Laplace-Stieltjes representation r = LS (F ) for some F ∈ Lip0 (R+ , X) if and only if r is a C ∞ -function whose Taylor coeﬃcients satisfy the estimate λn+1 (n)

r (λ) < ∞. n∈N0 λ>0 n!

r W := sup sup

(2.1)

This can be rephrased by saying that the Laplace-Stieltjes transform is an isometric isomorphism between the Banach spaces Lip0 (R+ , X) and ∞ CW ((0, ∞), X) := {r ∈ C ∞ ((0, ∞), X) : r W < ∞}.

If the Banach space X has the Radon-Nikodym property (see Section 1.2), then (and only then) “Widder’s growth conditions” (2.1) are necessary and suﬃcient for r to have a Laplace representation r = L(f ) for some f ∈ L∞ (R+ , X); i.e., Banach spaces with the Radon-Nikodym property are precisely those Banach spaces in which the Laplace transform is an isometric isomorphism between L∞ (R+ , X) and ∞ CW ((0, ∞), X). For X = C, this is a classical result usually known as “Widder’s Theorem”. If r = LS (F ) for some F ∈ Lip0 (R+ , X), then the inverse Laplace-Stieltjes transform has many diﬀerent representations. A few of them, such as F (t)

∞ r(λ) dλ = lim (−1)j+1 etnj r(nj) n→∞ λ Γ j=1 k+1 k k d r(λ) k 1 = lim (−1) , k→∞ k! t dtk λ λ=k/t

=

1 2πi

eλt

will be proved in Section 2.3. In Section 2.4, the results of the previous sections are extended to functions with exponential growth at inﬁnity; i.e., we investigate the Laplace transform acting on functions f with ess supt≥0 e−ωt f (t) < ∞. In applications it is usually impossible to verify whether or not a given function r satisﬁes Widder’s growth conditions (2.1). Thus, in Sections 2.5 and 2.6 some complex growth conditions are discussed which are necessary (and in a certain sense suﬃcient) for a holomorphic function r : {Re λ > ω} → X to have a Laplace representation. In Section 2.5, the growth condition considered is supRe λ>ω λ1+b r(λ) < ∞ for some b > 0. In Section 2.6, we discuss functions r which are holomorphic in a sector Σ := {| arg(λ)| < π2 + ε} and satisfy supλ∈Σ λr(λ) < ∞. We will see that any such r is the Laplace transform of a function which is holomorphic in the sector {| arg(λ)| < ε}. The ﬁnal class of functions which we will consider are the completely monotonic ones; i.e., C ∞ -functions r with values in an ordered Banach space such that (−1)n r (n) (λ) ≥ 0 for all n ∈ N0 and λ > ω. In the scalar case,

2.1. RIESZ-STIELTJES REPRESENTATION

65

Bernstein’s theorem states that a function r is completely monotonic if and only if it is the Laplace-Stieltjes transform of an increasing function. In Section 2.7 we investigate for which ordered Banach spaces Bernstein’s theorem holds.

2.1

Riesz-Stieltjes Representation

In the following sections the emphasis will be on the properties of the Laplace . As is transform L : f → fˆ and the Laplace-Stieltjes transform LS : F → dF the case with all linear operators, the choice of the domain is crucial. For the Laplace-Stieltjes transform LS the most convenient choice of the domain space is Lip0 (R+ , X) := F : R+ → X : F (0) = 0, F Lip0 (R+ ,X) :=

F (t) − F (s)

sup <∞ . |t − s| t,s≥0 If F (t) =

t 0

f (s) ds for f ∈ L∞ (R+ , X), then F ∈ Lip0 (R+ , X) and

∞

−λt

e 0

∞

dF (t) =

e−λt f (t) dt

(λ > 0),

0

by Proposition 1.10.1. Thus, any result for LS acting on Lip0 (R+ , X) translates into one for L acting on L∞ (R+ , X). However, since there are Banach spaces in which not every Lipschitz continuous function is the antiderivative of an L∞ function (see Section 1.2), the Laplace-Stieltjes transform is a true generalization of the Laplace transform. It is the generalization needed to deal eﬀectively with Laplace transforms of vector-valued functions. In this section we investigate the Riesz-Stieltjes operator ΦS which assigns to F ∈ Lip0 (R+ , X) a bounded linear operator TF : L1 (R+ ) → X such that ∞ τ TF f := f (s) dF (s) := lim f (s) dF (s), 0

τ →∞

0

1

when f ∈ L (R+ ) is continuous. It will be shown that ΦS is an isometric isomorphism between Lip0 (R+ , X) and L(L1 (R+ ), X), the space of all bounded linear operators from the Banach space L1 (R+ ) into X (Riesz-Stieltjes representation). This observation is fundamental for the whole chapter. To see why the RieszStieltjes representation is such an important tool, observe that (λ) = TF e−λ (λ > 0). F (t) = TF χ[0,t] (t ≥ 0) , and dF Thus, if one knows F , then the operator TF is speciﬁed on the set of characteristic functions χ[0,t] (t > 0), which is total in L1 (R+ ). Therefore, TF and, in partic (λ) (λ > 0) are completely determined. ular, the Laplace integrals TF e−λ = dF

66

2. THE LAPLACE TRANSFORM

(λ) determine TF on the set of exponential Conversely, the Laplace integrals dF functions e−λ (λ > 0), which is also total in L1 (R+ ) (Lemma 1.7.1). Hence, the (λ) determine the properties of TF and, in particular, the Laplace integrals dF properties of F (t) = TF χ[0,t] (t ≥ 0). Theorem 2.1.1 (Riesz-Stieltjes Representation). There exists a unique isometric isomorphism ΦS : F → TF from Lip0 (R+ , X) onto L(L1 (R+ ), X) such that TF χ[0,t] = F (t)

(2.2)

for all t ≥ 0 and F ∈ Lip0 (R+ , X). Moreover, TF g = lim

t→∞

t

∞

g(s) dF (s) := 0

g(s) dF (s)

(2.3)

0

for all continuous functions g ∈ L1 (R+ ). Note that it is part of the claim that the improper integral in (2.3) converges. We shall call the isomorphism ΦS the Riesz-Stieltjes operator. Proof. Let D := span{χ[0,t) : t > 0}, the space of step functions, which is dense in L1 (R+ ). For each f ∈ D there exists a unique representation f=

n

αi χ[ti−1 ,ti ) ,

i=1

where 0 = t0 < t1 < . . . < tn , αi ∈ C (i = 1, . . . , n). Let F ∈ Lip0 (R+ , X). Deﬁne TF : D → X by n n TF (f ) = TF αi χ[ti−1 ,ti ) := αi (F (ti ) − F (ti−1 )). i=1

i=1

Then,

TF (f ) ≤ F Lip0 (R+ ,X)

n

|αi |(ti − ti−1 ) = F Lip0 (R+ ,X) f 1 .

i=1

Hence, TF has a unique extension TF ∈ L(L1 (R+ ), X). Moreover,

TF ≤ F Lip0 (R+ ,X) . Conversely, if T ∈ L(L1 (R+ ), X), let F (t) := T χ[0,t) for t ≥ 0. Then for t > s ≥ 0,

F (t) − F (s) = T χ[s,t) ≤ T χ[s,t) 1 = T (t − s).

2.1. RIESZ-STIELTJES REPRESENTATION

67

Thus, F ∈ Lip0 (R+ , X) and F Lip0 (R+ ,X) ≤ T . It follows from the deﬁnitions that T = TF and if T = TG then F = G. This shows that F → TF is an isometric isomorphism. Finally, let g ∈ L1 (R+ ) be a continuous function and let F ∈ Lip0 (R+ , X). Take t > 0, and let π be a partition of [0, t] with partitioning points 0 = t0 < t1 < . . . < tn = t and intermediate points si ∈ [ti−1 , ti ]. Let fπ :=

n

g(si )χ[ti−1 ,ti ) .

i=1

Thus, S(g, F, π) = TF (fπ ). As |π| → 0, fπ − gχ[0,t) 1 → 0, so

t

0

g(s) dF (s) = TF (gχ[0,t) ).

As t → ∞, gχ[0,t) − g 1 → 0, so

∞ 0

g(s) dF (s) = TF (g).

We conclude this section by discussing convergence of functions and their Laplace-Stieltjes transforms. In fact, the Laplace-Stieltjes transform allows us to give a purely operator-theoretic proof of the following approximation theorem. Note, however, that the essential implication (i) ⇒ (iv) can also be obtained with the help of Theorem 1.7.5 (which may easily be strengthened by merely considering convergence on a sequence of equidistant points). Theorem 2.1.2. Let M > 0, Fn ∈ Lip0 (R+ , X) with Fn Lip0 (R+ ,X) ≤ M for all n ∈ N, and rn = LS (Fn ). The following are equivalent: (i) There exist a, b > 0 such that limn→∞ rn (a + kb) exists for all k ∈ N0 . (ii) There exists r ∈ C ∞ ((0, ∞), X) such that rn → r uniformly on compact subsets of (0, ∞). (iii) limn→∞ Fn (t) exists for all t ≥ 0. (iv) There exists F ∈ Lip0 (R+ , X) such that Fn → F uniformly on compact subsets of R+ . Moreover, if r and F are as in (ii) and (iv), then r = LS (F ). Proof. By the Riesz-Stieltjes Representation Theorem 2.1.1, there exist Tn ∈ L(L1 (R+ ), X) such that Tn = Fn Lip0 (R+ ,X) ≤ M, Tn e−λ = rn (λ), and Tn χ[0,t] = Fn (t) (n ∈ N, t ≥ 0, λ > 0). Each of the statements imply that the uniformly bounded family of operators Tn converges on a total subset of L1 (R+ )

68

2. THE LAPLACE TRANSFORM

(see also Lemma 1.7.1). By equicontinuity (see Proposition B.15), for any uniformly bounded sequence of operators, the topology of simple convergence on a total subset equals the topology of simple convergence and the topology of uniform convergence on compact subsets. Thus there exists T ∈ L(L1 (R+ ), X) such that Tn g → T g as n → ∞ for all g ∈ L1 (R+ ) (simple convergence). For all b > 0 the sets Kb := {χ[0,t] : 0 ≤ t ≤ b} and Eb := {e−λ : 1b ≤ λ ≤ b} are compact in L1 (R+ ) (continuous images of compact sets are compact). Hence, Tn → T uniformly on Kb and Eb (uniform convergence on compact subsets). Now the statements follow from the Riesz-Stieltjes representation.

2.2

A Real Representation Theorem

acting In this section the range of the Laplace-Stieltjes transform LS : F → dF on Lip0 (R+ , X) will be characterized. Since λ → dF (λ) = λF (λ) is holomorphic and, by Proposition 1.7.2, functions like λ → (sin λ)x (x ∈ X) cannot be in the range of LS , the range must be a proper subset of C ∞ ((0, ∞), X). The following observations will lead to a complete description of the range. Let F ∈ Lip0 (R+ , X) and TF := ΦS (F ), where ΦS is the Riesz-Stieltjes operator of Section 2.1. Deﬁne ∞ (λ) = r(λ) := dF e−λt dF (t) (λ > 0). 0

Then, by Theorem 1.10.6, r ∈ C ∞ ((0, ∞), X) and ∞ (n) r (λ) = e−λt (−t)n dF (t) = TF kn,λ , 0

where kn,λ (t) := e−λt (−t)n (t ≥ 0, λ > 0, n ∈ N0 ). Since kn,λ 1 = = n!/λn+1 and TF = F Lip0 (R+ ,X) , it follows that

∞ 0

e−λt tn dt

r(n) (λ) ≤ F Lip0 (R+ ,X) n!/λn+1 for all n ∈ N0 and λ > 0. Thus, r is a C ∞ -function whose Taylor coeﬃcients satisfy

r W :=

sup λ>0,k∈N0

λk+1 (k)

r (λ) ≤ F Lip0 (R+ ,X) . k!

maps Lip (R+ , X) This shows that the Laplace-Stieltjes transform LS : F → dF 0 into the space ∞ CW ((0, ∞), X) := {r ∈ C ∞ ((0, ∞), X) : r W < ∞}.

In 1936, Widder showed that the Laplace transform maps L∞ (R+ , R) onto ∞ CW ((0, ∞), R). The following result is the vector-valued version of Widder’s classical theorem.

2.2. A REAL REPRESENTATION THEOREM

69

Theorem 2.2.1 (Real Representation Theorem). The Laplace-Stieltjes transform ∞ LS is an isometric isomorphism between Lip0 (R+ , X) and CW ((0, ∞), X). ∞ Proof. We have already shown that LS maps Lip0 (R+ , X) into CW ((0, ∞), X) and = 0 for some F ∈ Lip (R+ , X), that LS (F ) W ≤ F Lip0 (R+ ,X) . If LS (F ) = dF 0 ∞ (λ) = 0 for all λ > 0. Since the exponential then TF e−λ = 0 e−λt dF (t) = dF functions e−λ (λ > 0) are total in L1 (R+ ) (Lemma 1.7.1), it follows that TF = 0. In particular, TF χ[0,t] = F (t) = 0 for all t ≥ 0. Thus, LS is one-to-one. ∞ The hard part of the proof is to show that LS is onto. Let r ∈ CW ((0, ∞), X). 1 Deﬁne Tk ∈ L(L (R+ ), X) by k+1 ∞ 1 k k Tk f := f (t)(−1)k r(k) dt (k ∈ N0 ). k! t t 0

The operators Tk are uniformly bounded by r W since Tk f ≤ r W f 1 for all f ∈ L1 (R+ ). We will show below that Tk e−λ → r(λ) as k → ∞ for all λ > 0. Since the exponential functions e−λ (λ > 0) are total in L1 (R+ ) it then follows from Proposition B.15 that there exists T ∈ L(L1 (R+ ), X) with T ≤ r W such that Tk f → T f for all f ∈ L1 (R+ ). In particular, r(λ) = lim Tk e−λ = T e−λ . k→∞

The Riesz-Stieltjes Representation Theorem 2.1.1 then yields the existence ∞ of some F ∈ Lip0 (R+ , X) with F Lip0 (R+ ,X) = T ≤ r W such that T g = 0 g(t) dF (t) for all continuous functions g ∈ L1 (R+ ). Hence, for all λ > 0, ∞ (λ). r(λ) = T e−λ = e−λt dF (t) = dF 0

W = F Lip (R ,X) for F ∈ Lip (R+ , X). Thus, LS is onto and LS (F ) W = dF 0 + 0 It remains to be shown that Tk e−λ → r(λ) as k → ∞ for all λ > 0. Observe that k+1 ∞ k k −λt k 1 (k) Tk e−λ = e (−1) r dt k! t t 0 ∞ 1 = (−1)k e−λk/u uk−1 r(k) (u) du (k − 1)! 0 ∞ % k−1 j 1 k j d −λk/u k−1 (k−j−1) = (−1) (−1) e u r (u) (k − 1)! j=0 duj u=0 & ∞ k d +(−1)k e−λk/u uk−1 r(u) du . k 0 du ! "k−1 To discuss the derivatives of u → e−λk/u uk−1 , deﬁne G(x, u) := e−x/u ux . Then G(sx, su) = G(x, u) for all s > 0. Diﬀerentiating both sides of the last

70

2. THE LAPLACE TRANSFORM

∂G equality with respect to s and then setting s = 1 yields x ∂G ∂x (x, u)+u ∂u (x, u) = 0 1 ∂G 1 ∂G or x ∂u (x, u) = − u ∂x (x, u). This implies that k−1 k−2 ∂ ∂ −x/u u −x/u u e =− e . ∂u xk ∂x xk−1

By induction on j, it follows that k−1 j k−j−1 ∂j −x/u u j ∂ −x/u u e = (−1) e ∂uj xk ∂xj xk−j or

(0 ≤ j ≤ k),

−x/u j ∂ j −x/u k−1 e j k k−j−1 ∂ e u = (−1) x u . ∂uj ∂xj xk−j

Hence, h(u)

:=

k−1

(−1)j

j=0

=

k−1

∂ j −x/u k−1 (k−j−1) e u r (u) ∂uj

∂j x ∂xj j=0 k

e−x/u xk−j

Since

uk−j−1 r (k−j−1) (u) ≤

uk−j−1 r (k−j−1) (u).

r W (k − j − 1)! , u

one obtains that

h(u) ≤

k−1 j=0

−x/u

r W (k − j − 1)! k ∂ j e . x j k−j u ∂x x

It follows that limu→∞ h(u) = 0 = limu→0 h(u). Therefore, letting x = λk, ∞ k 1 d −λk/u k−1 Tk e−λ = e u r(u) du. (k − 1)! 0 duk Since by (2.4), k k ∂ k −x/u k−1 xk −x/u kx ∂ −x/u e u = (−1) e = e , ∂uk u ∂xk uk+1 it follows that Tk e−λ

= =

∞ λk k k 1 e−λk/u k+1 r(u) du (k − 1)! 0 u λk k k+1 ∞ −λkt k−1 1 e t r dt. k! t 0

(2.4)

2.2. A REAL REPRESENTATION THEOREM

71

Deﬁne f (t) := 1t r( 1t ) and s := λ1 . Then Tk e−λ

k+1 ∞ k e−kt/s tk f (t) dt s 0 k+1 1 k k k (k) ˆ = s(−1) f . k! s s =

s k!

Finally, one concludes from the Post-Widder Inversion Theorem 1.7.7 that 1 lim Tk e−λ = sf (s) = r = r(λ) k→∞ s for all λ > 0. For later use in Section 2.5, we observe that in the Widder conditions it is not necessary to consider all values of k. Proposition 2.2.2. Let r ∈ C ∞ ((0, ∞), X), and suppose that limλ→∞ r(λ) = 0 and m+1 there exist M > 0 and inﬁnitely many integers m such that supλ>0 λ m! r (m) (λ)

∞ ≤ M . Then r ∈ CW ((0, ∞), X) and r W ≤ M . Proof. It suﬃces to show that if r(m) (λ) ≤ M m!/λm+1 , for all λ > 0, then

r(k) (λ) ≤ M k!/λk+1 for all λ > 0 and 0 ≤ k < m. Let ∞ (−1)m r˜(λ) := (λ − μ)m−1 r(m) (μ) dμ. (m − 1)! λ Note that the integral is absolutely convergent, r˜(m) (λ) = r (m) (λ), and the substitution t = λ/μ gives

˜ r(λ) ≤ M m

∞

λ

(μ − λ)m−1 Mm dμ = μm+1 λ

0

1

(1 − t)m−1 dt =

M . λ

Hence r − r˜ is a polynomial and limλ→∞ (r − r˜)(λ) = 0, so r = r˜. It follows that

∞

(−1)m

(k) m−k−1 (m)

r (λ) = (λ − μ) r (μ) dμ

(m − k − 1)! λ ∞ M m! (μ − λ)m−k−1 ≤ dμ (m − k − 1)! λ μm+1 M k! = λk+1 for λ > 0 and 0 ≤ k < m. Now it will be shown that the Laplace transform is an isometric isomorphism ∞ between L∞ (R+ , X) and CW ((0, ∞), X) if and only if the Banach space X has the

72

2. THE LAPLACE TRANSFORM

Radon-Nikodym property. Recall from Section 1.2 that X has the Radon-Nikodym property if every F ∈ Lip0 (R+ , X) is diﬀerentiable a.e., or equivalently if every absolutely continuous function F : R+ → X is diﬀerentiable a.e. As shown in Theorem 1.2.6 and Corollary 1.2.7, every separable dual space (for example, l1 ) and every reﬂexive Banach space have the Radon-Nikodym property. However, L1 (R+ ) and c0 do not have the property (Propositions 1.2.9 and 1.2.10). Theorem 2.2.3. Let X be a Banach space. The following are equivalent: (i) X has the Radon-Nikodym property. (ii) The Laplace transform L : f → fˆ is an isometric isomorphism between ∞ L∞ (R+ , X) and CW ((0, ∞), X). ∞ (iii) The Riesz operator Φ : f → Rf , Rf g := 0 g(t)f (t) dt is an isometric isomorphism between L∞ (R+ , X) and L(L1 (R+ ), X). Proof. Deﬁne the normalized antiderivative I : L∞ (R+ , X) → Lip0 (R+ , X) by t I(f ) := F , F (t) := 0 f (s) ds (t ≥ 0). Then I is one-to-one and I(f ) Lip0 (R+ ,X) ≤

f ∞ for all f ∈ L∞ (R+ , X). If I is onto, then X has the Radon-Nikodym property (see Proposition 1.2.2). Conversely, if X has the Radon-Nikodym property and F ∈ Lip0 (R+ , X) then f (t) := F (t) exists for almost all t ≥ 0. Since f (t) = (t) limh→0 F (t+h)−F a.e., one concludes that f ∞ ≤ F Lip0 (R+ ,X) . In particular, h ∞ f ∈ L (R+ , X) and by Proposition 1.2.3, F = I(f ). Thus X has the RadonNikodym property if and only if I is an isometric isomorphism. ∞ The Riesz-Stieltjes operator ΦS : F → TF , where TF g = 0 g(t) dF (t) for 1 all continuous g ∈ L (R+ ), is an isometric isomorphism between Lip0 (R+ , X) and L(L1 (R+ ), X), and the Laplace-Stieltjes transform ∞ , dF (λ) = LS : F → dF e−λt dF (t), 0

∞ is an isometric isomorphism between Lip0 (R+ , X) and CW ((0, ∞), X). When F = ∞ 1 I(f ), TF g = 0 g(t)f (t) dt for all g ∈ L (R+ ), by Proposition 1.9.11 and continuity in L1 -norm. Now the statements follow from the fact that Φ = ΦS ◦ I and L = LS ◦ I on L∞ (R+ , X).

Example 2.2.4. a) Consider X = L1 (R+ ). Let F (t) := χ[0,t] (t ≥ 0) and r(λ) := e−λ (Re λ > 0), where e−λ (t) = e−λt . Then F ∈ Lip0 (R+ , L1 (R+ )) and ∞ (λ). r(λ) = e−λt dF (t) = dF 0

Since F is nowhere diﬀerentiable (see Proposition 1.2.10), there does not exist f ∈ L∞ (R+ , L1 (R+ )) such that ∞ r(λ) = e−λt f (t) dt. 0

2.3. REAL AND COMPLEX INVERSION

73

b) Consider C0 (R+ ) as a subspace of L∞ (R+ ). Deﬁne F : R+ → C0 (R+ ) by F (t)(s) := (t − s)χ[0,t] (s), and f : R+ → L∞ (R+ ) by f (t) := χ[0,t] . Then F ∈ t Lip0 (R+ , C0 (R+ )) and F (t) = 0 f (s) ds as a Riemann integral in L∞ (R+ ), but F is nowhere diﬀerentiable and f is not measurable (see Examples 1.2.8 and 1.9.7). Moreover, ∞ ∞ 1 e−λ = e−λt dF (t) = e−λt f (t) dt λ 0 0 as (improper) Riemann-Stieltjes and Riemann integrals, but λ → λ1 e−λ is not the Laplace transform of any function in L1 (R+ , L∞ (R+ )).

2.3

Real and Complex Inversion

We have shown in Section 2.2 that the Laplace-Stieltjes transform LS is an iso∞ metric isomorphism between Lip0 (R+ , X) and CW ((0, ∞), X). In this section we will derive several representations of the inverse Laplace-Stieltjes transform L−1 S . Theorem 2.3.1 (Post-Widder Inversion). Let F ∈ Lip0 (R+ , X), r = LS (F ), and t > 0. Then k+1 k k d r(λ) k 1 F (t) = lim (−1) . k→∞ k! t dλk λ λ=k/t Proof. Since ω(F ) ≤ 0 and F (0) = 0, it follows from (1.22) that ∞ r(λ) = e−λt F (t) dt λ 0 for all λ > 0, where the integral is an absolutely convergent Bochner integral. Now the statement follows from Theorem 1.7.7. k k (k) Applying Leibniz’s rule (f · r) = j=0 f (k−j) r (j) to f (λ) := λ1 and j r one can rewrite the Post-Widder inversion of the Laplace-Stieltjes transform as j k 1 k k F (t) = lim (−1)j r (j) (t > 0). (2.5) k→∞ j! t t j=0 Compared to the Post-Widder inversion, it is remarkable that in the following Phragm´en-Doetsch inversion formula only the values r(k) for large k ∈ N are needed and that the convergence is uniform for all t ≥ 0. Theorem 2.3.2 (Phragm´en-Doetsch Inversion). Let F ∈ Lip0 (R+ , X) and r = LS (F ). Then

∞ j+1

(−1) tkj

F (t) −

≤ c r W e r(kj)

k j!

j=1 for all t ≥ 0 and k ∈ N, where c ≈ 1.0159..., and r W = F Lip0 (R+ ,X) .

74

2. THE LAPLACE TRANSFORM

Proof. By the Riesz-Stieltjes Representation Theorem 2.1.1 and the Real Representation Theorem 2.2.1, there exists T ∈ L(L1 (R+ ), X) such that r(λ) = ∞ −λt e dF (t) = T e−λ (λ > 0), T χ[0,t] = F (t) (t ≥ 0) and T = r W = 0

F Lip0 (R+ ,X) . Thus,

∞ ∞ j+1

(−1) tkj j+1 1 tkj

F (t) −

≤ T χ[0,t] −

e r(kj) (−1) e e −kj .

j! j!

j=1 j=1 Deﬁne pk,t (s) := 1 − e−e

k(t−s)

χ[0,t] − pk,t 1

= =

1

∞

=

j+1 1 tkj e e−kj (s). j=1 (−1) j!

t

Then,

∞

|pk,t (s) − 1| ds + |pk,t (s)| ds 0 t t ∞ k(t−s) k(t−s) e−e ds + 1 − e−e ds 0

=

1 k

≤

1 k

t

ekt

e

1

∞ 1

1 1 1 − e−u du + du u k 0 u 1 e−u 1 − e−u du + du u u 0 −u

∞ for all t ≥ 0 and k ∈ N. Now the claim follows from the fact that 1 u1 e−u du + 1 1−e−u du = −2 Ei(−1) + γ ≈ 1.0159..., where Ei(z) is the exponential integral u 0 and γ is Euler’s constant (see [Leb72, Section 3.1]). The following corollary shows that the Phragm´en-Doetsch inversion is invariant under exponentially decaying perturbations for small values of t. Corollary 2.3.3. Let F ∈ Lip0 (R+ , X), r = LS (F ), and q(λ) = r(λ)+a(λ) (λ > 0), where a : (0, ∞) → X is a function such that lim supn→∞ n1 log a(n) ≤ −T for some T > 0. Then ∞ (−1)j+1 tkj F (t) = lim e q(kj) k→∞ j! j=1 for all 0 ≤ t < T . Proof. Let 0 < T0 < T and choose k0 such that a(k) ≤ e−T0 k for all k ≥ k0 . Then,

∞

(−1)j+1 tkj

F (t) − e q(kj)

j! j=1

∞

∞

(−1)j+1 tkj

(−1)j+1 tkj

+

≤ F (t) − e r(kj) e a(kj)

j! j! j=1 j=1 ∞

≤

1 −(T0 −t)k 2 2

r W + etkj e−T0 kj ≤ r W + ee − 1. k j! k j=1

2.3. REAL AND COMPLEX INVERSION

75

The Post-Widder inversion and the Phragm´en-Doetsch inversion are called real inversions of the Laplace-Stieltjes transform since they use only properties of r(λ) for large real λ. For the following complex inversion formula we use the ∞ fact that if r(λ) = 0 e−λt dF (t) (λ > 0) for some F ∈ Lip0 (R+ , X), then r admits a holomorphic extension for Re λ > 0 which we denote by the same symbol (see Theorem 1.10.6). We shall give here a proof based on the Riesz-Stieltjes representation, but we shall give another, rather simple, proof in Section 4.2. Theorem 2.3.4 (Complex Inversion). Let F ∈ Lip0 (R+ , X) and r = LS (F ). Then c+ik 1 r(λ) F (t) = lim eλt dλ, k→∞ 2πi c−ik λ where the limit is uniform for t ∈ [0, a] for any a > 0, and c > 0 is arbitrary. Proof. By the Riesz-Stieltjes Representation Theorem 2.1.1, there exists T ∈ L(L1 (R+ ), X) such that r(λ) = T e−λ (Re λ > 0) and F (t) = T χ[0,t] (t ≥ 0). Thus,

c+ik c+ik

1 1

λt r(λ) λt e−λ e dλ ≤ T χ[0,t] − e dλ .

F (t) −

2πi c−ik λ 2πi c−ik λ 1

Now the statement follows from the next lemma. Lemma 2.3.5. Let t ≥ 0 and a, c > 0. Then the functions c+ik 1 e−λ hk,t := eλt dλ 2πi c−ik λ converge towards χ[0,t] in L1 (R+ ) as n → ∞, uniformly for t ∈ [0, a]. t Proof. Let hk,t − χ[0,t] 1 = Ak + Bk , where Ak := 0 |hk,t (s) − 1| ds and Bk := ∞ |hk,t (s)| ds. We show ﬁrst that limk→∞ Ak = 0. The residue of the function t λ → eλ(t−s) /λ at the point 0 is 1. By Cauchy’s theorem, λ(t−s) 1 e hk,t (s) − 1 = − − dλ, 2πi λ Γ+ Γ− Γ0 where Γ± := {λ : λ = u ± ik; 0 ≤ u ≤ c}, Γ0 := {λ : λ = keiu ; π/2 ≤ u ≤ 3π/2}. Along Γ+ , and similarly along Γ− , it follows from 0 ≤ s ≤ t that c e(u+ik)(t−s) eλ(t−s) ec(t−s) dλ = du ≤ c . Γ+ λ u + ik k 0 Along Γ0 , for 0 ≤ s < t, 3π/2 3π/2 λ(t−s) iu e k(t−s)e = dλ e du ek(t−s) cos u du. ≤ λ Γ0 π/2 π/2

76

2. THE LAPLACE TRANSFORM

Hence, Ak

=

t

|hk,t (t − s) − 1| ds t cs 3π/2 ce 1 + eks cos u du ds πk 2π π/2 0 0

≤

→ 0 as k → ∞, uniformly for t ∈ [0, a] for all a > 0, by the monotone convergence theorem, or by explicit estimation. ± := {λ : λ = u ± ik ; c ≤ u ≤ k}, Γ 0 := In order estimate Bk , we deﬁne Γ √ to iu {λ : λ = k 2e ; −π/4 ≤ u ≤ π/4}. By Cauchy’s theorem, 1 hk,t (s) = 2πi

−

+ Γ

+

− Γ

+

0 Γ

eλ(t−s) dλ. λ

+ , and similarly along Γ − , it follows from s − t ≥ 0 that Along Γ eλ(t−s) dλ Γ+ λ

k e(u+ik)(t−s) 1 k = du ≤ e−u(s−t) du c k c u + ik =

e−c(s−t) − e−k(s−t) . k(s − t)

0, Along Γ eλ(t−s) dλ λ Γ0

π/4 √ π/4 √ iu = ek 2(t−s)e du ≤ ek 2(t−s) cos u du −π/4 −π/4 π/4 √ π √ π = 2 ek 2(t−s) cos(u) du ≤ ek 2(t−s) cos(π/4) = e−k(s−t) . 2 2 0

Hence, for all t ≥ 0,

∞

|hk,t (s)| ds ≤

t

=

1 ∞ e−c(s−t) − e−k(s−t) 1 ∞ −k(s−t) ds + e ds π t k(s − t) 4 t ∞ 1 1 zk (s) ds + , π 0 4k

1 where zk (s) := ks (e−cs − e−ks ) ≤ e−cs for k ≥ c by the mean value theorem −x applied to e over [cs, ks]. By the dominated convergence theorem, or by explicit estimation, Bk → 0 as k → ∞, uniformly for t ∈ [0, a] for all a > 0.

2.4. TRANSFORMS OF EXPONENTIALLY BOUNDED FUNCTIONS

2.4

77

Transforms of Exponentially Bounded Functions

So far in this chapter, Laplace transforms have been considered for bounded or globally Lipschitz continuous functions. We shall now adapt the results of the previous sections to functions with exponential growth at inﬁnity, by an elementary “shifting” procedure (see Proposition 1.6.1 a) and Proposition 1.10.3). More precisely, for ω ∈ R we consider the Laplace-Stieltjes transform acting on Lipω (R+ , X) := G : R+ → X : G(0) = 0,

G(t) − G(s)

G Lipω (R+ ,X) := sup <∞ t t>s≥0 eωr dr s and the Laplace transform acting on ∞ 1 −ωt Lω (R+ , X) := g ∈ Lloc (R+ , X) : g ω,∞ := ess sup e g(t) < ∞ . t≥0

It is easy to see that

G Lipω (R+ ,X)

⎧

G(t) − G(s)

⎪ ⎪ ⎨sup0≤s
G(t) − G(s)

⎪ ⎪ ⎩sup0≤s
if ω ≥ 0, if ω ≤ 0.

It is clear that the multiplication operator Mω : g → e−ω· g(·) is an isometric ∞ isomorphism between L∞ ω (R+ , X) and L (R+ , X), and we now set up the corresponding isomorphism between Lipω (R+ , X) and Lip0 (R+ , X). For G ∈ Lipω (R+ , X) and f ∈ BSVloc (R+ ), it follows from the deﬁnition of the Riemann-Stieltjes integral that

b

b

f (t) dG(t) ≤ G Lipω (R+ ,X) |f (t)|eωt dt (0 ≤ a ≤ b). (2.6)

a

a Let

(Iω G)(t) :=

t

e−ωs dG(s).

0

Then (2.6) implies that Iω G ∈ Lip0 (R+ , X) and Iω G Lip0 (R+ ,X) ≤ G Lipω (R+ ,X) . Similarly if F ∈ Lip0 (R+ , X) and (Jω F )(t) :=

0

t

eωs dF (s),

78

2. THE LAPLACE TRANSFORM

then Jω F ∈ Lipω (R+ , X) and Jω F Lipω (R+ ,X) ≤ F Lip0 (R+ ,X) . Moreover, Jω Iω G = G and Iω Jω F = F , by Proposition 1.9.10. Hence, Iω is an isometric isomorphism of Lipω (R+ , X) onto Lip0 (R+ , X). Note that if G ∈ L∞ ω (R+ , X) then ω(G) ≤ ω and abs(dG) ≤ ω by Theorem 1.10.5. Thus, the Laplace-Stieltjes transform ∞ (LS,ω G)(λ) := dG(λ) = e−λt dG(t) 0

exists for λ > ω. By Proposition 1.10.3, (LS,ω G)(λ) = (LS Iω G)(λ − ω). Let ∞ CW ((ω, ∞), X) :=

(2.7)

r ∈ C ∞ ((ω, ∞), X) :

r W :=

sup λ>ω,k∈N0

(λ − ω)k+1 (k)

r (λ) < ∞ . k!

This is a Banach space, and it is clear that the shift Sω : r → r(· − ω) is an ∞ ∞ isometric isomorphism of CW ((0, ∞), X) onto CW ((ω, ∞), X). The equation (2.7) may be written as LS,ω = Sω ◦ LS ◦ Iω . Now we can give the following reformulation of the Real Representation Theorem 2.2.1. Theorem 2.4.1. Let ω ∈ R. The Laplace-Stieltjes transform is an isometric iso∞ morphism of Lipω (R+ , X) onto CW ((ω, ∞), X). In particular, for M > 0 and ∞ r ∈ CW ((ω, ∞), X), the following are equivalent: 1 (k) (i) (λ − ω)k+1 k! r (λ) ≤ M

(λ > ω, k ∈ N0 ).

(ii) There exists G : R+ → X satisfying G(0) = 0 and G(t + h) − G(t) ≤ t+h ∞ M t eωr dr (t, h ≥ 0), such that r(λ) = 0 e−λt dG(t) for all λ > ω. Proposition 1.6.1 a) gives Lω = Sω ◦ L ◦ Mω where L and Lω are the Laplace transforms on L∞ (R+ , X) and L∞ ω (R+ , X). Hence Theorem 2.2.3 can be reformulated as follows. Theorem 2.4.2. Let M > 0, ω ∈ R. If X has the Radon-Nikodym property then ∞ for any r ∈ CW ((ω, ∞), X) the following are equivalent: 1 (k) (i) (λ − ω)k+1 k! r (λ) ≤ M

(λ > ω, k ∈ N0 ).

(ii) There existsg ∈ L1loc (R+ , X) with g(t) ≤ M eωt for almost all t ≥ 0 such ∞ that r(λ) = 0 e−λt g(t) dt for all λ > ω.

2.4. TRANSFORMS OF EXPONENTIALLY BOUNDED FUNCTIONS

79

As in Theorem 2.1.1 one shows that there exists an isometric isomorphism ΦS,ω between the spaces Lipω (R+ , X) and L(L1ω (R+ ), X), where ∞ L1ω (R+ ) := h ∈ L1loc (R+ ) : h ω,1 := eωt |h(t)| dt < ∞ . 0

The isomorphism ΦS,ω assigns to every function G ∈ Lipω (R+ , X) an operator T ∈ L(L1ω (R+ ), X) with T = G Lipω (R+ ,X) such that ∞ Th = h(t) dG(t) 0

for all continuous functions h ∈ L1ω (R+ ), T χ[0,t] = G(t) for all t ≥ 0, and T e−λ = dG(λ) if Re λ > ω. The inversion theorems in Section 2.3 all remain valid, with almost no changes in the proofs (the version of Theorem 2.3.4 for Lipω (R+ , X) can be deduced directly for some from the case ω = 0 by using the isomorphism Iω ). Thus, if r = dF F ∈ Lipω (R+ , X), then k+1 k k d r(λ) k 1 F (t) = lim (−1) . (2.8) k→∞ k! t dλk λ λ=k/t If c > max(ω, 0), then 1 k→∞ 2πi

c+ik

F (t) = lim

c−ik

eλt

r(λ) dλ, λ

(2.9)

where the limit exists uniformly on compact subsets of R+ . Finally, F (t) = lim

k→∞

∞ j=1

(−1)j+1

1 tkj e r(kj), j!

(2.10)

where the limit exists uniformly on R+ . The following is a consequence of the Phragm´en-Doetsch inversion (2.10). Proposition 2.4.3. Let ε > 0 and f ∈ L1loc (R+ , X) with abs(f ) < ∞. The following are equivalent. (i) lim supλ→∞

1 λ

log fˆ(λ) ≤ −ε.

(ii) f = 0 a.e. on [0, ε]. t t Proof. Let F (t) := 0 f (s) ds and G(t) := 0 F (s) ds. Since abs(f ) < ∞, ω(F ) < ∞ by Theorem 1.4.3 and hence G ∈ Lipω (R+ , X) for some ω ∈ R. By Corollary 1.6.5 and Proposition 1.10.1, fˆ(λ) = λF (λ) = λdG(λ) = λ2 G(λ)

80

2. THE LAPLACE TRANSFORM

for Re λ > ω. Deﬁne

1ˆ f (λ) = F (λ) = dG(λ) λ for λ > ω. If (i) holds, then lim supλ→∞ λ1 log r(λ) ≤ −ε. Let 0 < ξ < ε. Then there exist M, λ0 > 0 such that r(λ) ≤ M e−λξ for all λ > λ0 . Let t ∈ [0, ξ). Then, for λ0 < k ∈ N,

∞

∞ (t−ξ)k

(−1)j+1 tkj

1 (t−ξ)kj e

≤M e r(kj) e = M e − 1 →0

j! j!

j=1

j=1 r(λ) :=

it follows from (2.10) that G = 0 on [0, ξ) for all as k → ∞. Since r = dG, 0 < ξ < ε. Thus, G = 0 on [0, ε] and hence f = 0 a.e. on [0, ε], by Proposition 1.2.2. This proves that (i) ⇒ (ii). Suppose that (ii) holds. Then F = 0 on [0, ε]. Thus ∞ ∞ ∞ r(λ) = e−λt F (t) dt = e−λt F (t) dt = e−λε e−λt F (t + ε) dt. 0

0

ε

∞ Since t → F (t+ε) is exponentially bounded, it follows that 0 e−λt F (t+ε) dt ≤ C for some C > 0 and therefore r(λ) ≤ Ce−ελ for all suﬃciently large λ. This proves that (ii) ⇒ (i). If f ∈ L1loc (R+ , X) with abs(f ) < ∞, then it follows from Corollary 1.6.5 and the exponential boundedness of F that there exist M, λ0 > 0 such that fˆ(λ) ≤ M for all λ > λ0 . Thus, lim supλ→∞ λ1 log fˆ(λ) ≤ 0. This and the previous proposition yield the following corollary. Corollary 2.4.4. Let f ∈ L1loc (R+ , X) with abs(f ) < ∞. Then the following are equivalent: (i) lim supλ→∞

1 λ

log fˆ(λ) = 0.

(ii) For every ε > 0, the restriction of f to [0, ε] does not vanish a.e.

2.5

Complex Conditions

It was shown in the previous section that a holomorphic function q : {Re λ > ω} → X has a Laplace-Stieltjes or multiplied Laplace representation ∞ ∞ q(λ) = e−λt dF (t) = λ e−λt F (t) dt 0

0

1 (k) if there exists a constant M > 0 such that the Taylor coeﬃcients k! q (λ) are k+1 bounded by M/(λ − ω) for all λ > ω and k ∈ N0 . Since only properties of the function q along the real half-line (ω, ∞) are involved, Widder’s growth conditions

2.5. COMPLEX CONDITIONS

81

are also referred to as “real conditions”. In many instances, these real conditions are too diﬃcult to be checked because all derivatives of q have to be considered, whereas the growth of q in a complex half-plane Re λ > ω can be estimated. In these cases one can apply the following representation theorem. Theorem 2.5.1 (Complex Representation). Let ω ≥ 0, let q : {Re λ > ω} → X be a holomorphic function with supRe λ>ω λq(λ) < ∞ and let b > 0. Then there exists f ∈ C(R+ , X) with supt>0 e−ωt t−b f (t) < ∞ such that q(λ) = λb fˆ(λ) for Re λ > ω. Proof. Let α > ω and deﬁne 1 f (t) := lim R→∞ 2πi

α+iR

α−iR

q(λ) eλt b λ

1 dλ = 2π

∞

−∞

e(α+ir)t

q(α + ir) dr. (α + ir)b

Observe that the latter integral is absolutely convergent, by the assumption on q, so the limit exists uniformly for t in compact subsets of R+ . Hence, f is continuous on R+ . By applying Cauchy’s theorem over rectangles with vertices α±iR, β±iR, and using the assumption on q, it is easy to see that the deﬁnition of f is independent of α > ω. For α > ω and R > 0, let Γα,R be the path consisting of the vertical half-line π {α + ir : r < −R}, the semicircle {α + Reiθ : −π 2 ≤ θ ≤ 2 }, and the half-line {α + ir : r > R}. By Cauchy’s theorem, f (t)

= =

1 q(λ) eλt b dλ 2πi Γα,R λ −R 1 q(α + ir) e(α+ir)t dr 2π −∞ (α + ir)b π/2 iθ 1 q(α + Reiθ ) iθ + e(α+Re )t Re dθ 2π −π/2 (α + Reiθ )b ∞ 1 q(α + ir) + e(α+ir)t dr. 2π R (α + ir)b

Hence,

f (t)

≤ =

dr M π/2 e(α+R cos θ)t + dθ b+1 2π −π/2 Rb R r M eαt M eαt π/2 Rt cos θ + e dθ, πbRb πRb 0 M eαt π

∞

where M := supRe λ>ω λq(λ) . Choosing R = 1/t, we obtain that f (t) ≤ Ctb eαt for some C independent of α > ω. Hence, f (t) ≤ Ctb eωt .

82

2. THE LAPLACE TRANSFORM

Given λ with Re λ > ω, choose ω < α < Re λ. By the dominated convergence theorem and Fubini’s theorem, ∞ ∞ α+iR 1 q(z) e−λt f (t) dt = lim e−λt ezt b dz dt R→∞ 2πi z 0 0 α−iR α+iR 1 q(z) = lim dz. R→∞ 2πi α−iR (λ − z)z b By Cauchy’s residue theorem around the path consisting of the semicircle {α + Reiθ : −π/2 ≤ θ ≤ π/2} and the line-segment {α + ir : −R ≤ r ≤ R}, α+iR π/2 1 q(z) 1 q(α + Reiθ )Reiθ q(λ) dz = dθ + b b 2πi α−iR (λ − z)z 2π −π/2 (λ − α − Reiθ )(α + Reiθ )b λ →

q(λ) λb

as R → ∞, using the assumption on q. We mention that Theorem 2.5.1 does not hold for b = 0. In fact, Desch and Pr¨ uss [DP93] construct a scalar-valued holomorphic function q on C+ satisfying sup q(λ) (1 + |λ|) < ∞

Re λ>0

such that q is not the Laplace transform of a function f ∈ L∞ loc (0, ∞). On the other hand, if λq(λ) and λ2 q (λ) are bounded on the right half-plane, then q is the Laplace transform of a bounded continuous function, as we show in the following corollary. Corollary 2.5.2 (Pr¨ uss). Let q : {Re λ > 0} → X be holomorphic. If there exists M > 0 such that λq(λ) ≤ M and λ2 q (λ) ≤ M for Re λ ∞> 0, then there exists a bounded function f ∈ C((0, ∞), X) such that q(λ) = 0 e−λt f (t) dt for ∞ Re λ > 0. In particular, q ∈ CW ((0, ∞), X). Proof. It follows from Theorem 2.5.1 that there are functions fi ∈ C(R+ , X) (i = 0, 1) and C > 0 such that fi (t) ≤ Ct for t > 0, ∞ ∞ q(λ) = λ e−λt f0 (t) dt, and λq (λ) = λ e−λt f1 (t) dt 0

0

for Re λ > 0. By Theorem 1.5.1, ∞ q (λ) = e−λt f0 (t) dt − λ 0

0

∞

e−λt tf0 (t) dt =

Integration by parts (or Corollary 1.6.5) yields t ∞ λ e−λt f0 (s) ds − tf0 (t) dt = λ 0

0

∞ 0

∞ 0

e−λt

e−λt f1 (t) dt.

t 0

f1 (s) ds dt.

2.5. COMPLEX CONDITIONS

83

t Since the Laplace transform is one-to-one, it follows that tf0 (t) = 0 f0 (s) ds − t f (s) ds. Thus, f0 ∈ C 1 ((0, ∞), X) and tf0 (t) = −f1 (t). Therefore, f0 (t) ≤ C 0 1 for all t > 0 and ∞ ∞ −λt q(λ) = λ e f0 (t) dt = e−λt f0 (t) dt (Re λ > 0). 0

0

Remark 2.5.3. If f ∈ L∞ ((0, ∞), X), then r = fˆ is holomorphic on the right half-plane and

λr(λ)

≤

λ2 r (λ)

≤

|λ|

f ∞ , Re λ 2 |λ|

f ∞ Re λ

(Re λ > 0).

In particular, λr(λ) and λ2 r (λ) are bounded on each sector Σα = {reiγ : r > 0, |γ| < α} where α ∈ (0, π/2). In Corollary 2.5.2 the estimate is required uniformly on the right half-plane, which is more. On the other hand, continuity is obtained as an additional result. We close this section with a characterization of Laplace transforms of functions in L1loc (R+ , X) with f (t) ≤ M tn for some M, n ≥ 0 and almost all t ≥ 0 (if X has the Radon-Nikodym property) or the Laplace-Stieltjes transforms of t functions H : R+ → X with H(0) = 0 and H(t) − H(s) ≤ M s r n dr for some M > 0 and all 0 ≤ s ≤ t (for general X). Corollary 2.5.4. Let M > 0, n ∈ N0 , and r ∈ C ∞ ((0, ∞), X). The following are equivalent: k+n+1

(i) λ(k+n)! r (k) (λ) ≤ M

(λ > 0, k ∈ N0 ).

(ii) There exists H : R+ → X satisfying H(0) = 0 and H(t) − H(s) ≤ t ∞ M s r n dr (0 ≤ s ≤ t), such that r(λ) = 0 e−λt dH(t) for all λ > 0. Proof. By the Real Representation Theorem 2.2.1, the statement holds for n = 0. Therefore, let n ≥ 1. To show that (i) implies (ii), deﬁne ∞ 1 (u − λ)n−1 r(u) du m(λ) := (−1)n (n − 1)! λ k+1

for λ > 0. Then, m(k) (λ) = r (k−n) (λ) for all k ≥ n and λ > 0. Since λ k! m(k) (λ)

k+1 = λ k! r (k−n) (λ) ≤ M for all λ > 0 and k ≥ n, it follows from Proposition ∞ 2.2.2 that m ∈ CW ((0, ∞), X) and m W ≤ M . By Theorem 2.2.1, there exists G : R+ → X with G(0) = 0 and G(t) − G(s) ≤ M |t − s| for all t, s ≥ 0 such that ∞ m(λ) = 0 e−λt dG(t) for all λ > 0. By Theorem 1.5.1 and Proposition 1.9.10, ∞ ∞ r(λ) = m(n) (λ) = e−λt (−t)n dG(t) = e−λt dH(t), 0

0

84

2. THE LAPLACE TRANSFORM

t where H(t) := 0 (−s)n dG(s). Now the statement (ii) follows from H(t)−H(s) = t t

s (−r)n dG(r) ≤ M s r n dr for all 0 ≤ s ≤ t. To show that (ii) implies (i), let x∗ ∈ X ∗ . The function x∗ ◦ H is locally Lipschitz continuous, hence absolutely continuous and diﬀerentiable a.e. If ∞ d h(t) := dt H(t), x∗ , then |h(t)| ≤ M tn x∗ and r(λ), x∗ = 0 e−λt h(t) dt, by Proposition 1.9.11. Hence, k+n+1 k+n+1 ∞ λ λ (k) ∗ −λt k r (λ), x = e (−t) h(t) dt (k + n)! (k + n)! 0 ≤ M x∗ . Now (i) follows from the Hahn-Banach theorem.

2.6

Laplace Transforms of Holomorphic Functions

In this section those functions are characterized which are Laplace transforms of holomorphic, exponentially bounded functions deﬁned on some open sector Σα := {reiγ : r > 0, −α < γ < α} for some 0 < α ≤ π/2. The closure of Σα is denoted by Σα . We shall use the same notation for 0 < α < π. Note that Σ π2 = C+ := {Re λ > 0}. Theorem 2.6.1 (Analytic Representation). Let 0 < α ≤ → X. The following are equivalent:

π 2,

ω ∈ R and q : (ω, ∞)

(i) There is a holomorphic function f : Σα → X such that supz∈Σβ e−ωz f (z)

< ∞ for all 0 < β < α and q(λ) = fˆ(λ) for all λ > ω. (ii) The function q has a holomorphic extension q˜ : ω + Σα+ π2 → X such that supλ∈ω+Σγ+ π (λ − ω)˜ q (λ) < ∞ for all 0 < γ < α. 2

Proof. Assume that (i) holds. Let 0 < β < α. Then there exists M > 0 such that

f (z) ≤ M |eωz | for all z ∈ Σβ \ {0}. Deﬁne paths Γ± by Γ± := {se±iβ : 0 ≤ s < ∞}. By Cauchy’s theorem, ∞ −λt q(λ) = e f (t) dt = e−λz f (z)dz 0

= e±iβ

∞

Γ±

e−λse

±iβ

f (se±iβ ) ds

(2.11)

0

for all λ > ω. Let 0 < ε < π2 − β, and let λ ∈ C with − π2 − β + ε < arg(λ − ω) < π π π iβ iβ 2 − β −ε. Then − 2 + ε < arg((λ − ω)e ) < 2 − ε, so Re((λ− ω)e ) ≥ |λ− ω| sin ε. It follows that iβ

e−λse f (seiβ ) ≤ M e−s|λ−ω| sin ε .

2.6. LAPLACE TRANSFORMS OF HOLOMORPHIC FUNCTIONS

85

Consequently, the integral q+ (λ) := eiβ

∞

iβ

e−λse f (seiβ ) ds

0

is absolutely convergent and deﬁnes a holomorphic function in the region − π2 − β + ε < arg(λ − ω) < π2 − β − ε, with (λ − ω)q+ (λ) ≤ M/ sin ε. Similarly, q− (λ) := e−iβ

∞

e−λse

−iβ

f (se−iβ ) ds

0

deﬁnes a holomorphic function in the region − π2 + β + ε < arg(λ − ω) < π2 + β − ε, with (λ − ω)q− (λ) ≤ M/ sin ε. By (2.11), both q+ and q− are extensions of q, and together they deﬁne a holomorphic extension q˜ to ω + Σ π2 +β−ε , satisfying

(λ − ω)˜ q (λ) ≤ M/ sin ε in the sector. Since β < α and 0 < ε < π2 − β are arbitrary, this proves (ii). Assume that (ii) holds. Let 0 < γ < α and δ > 0. There exists M > 0 such that (λ − ω)˜ q (λ) ≤ M for all λ ∈ (ω + Σγ+ π2 ) \ {ω}. Consider an oriented path Γ (depending on γ and δ) consisting of ' ( ' ( Γ± := ω + re±i(γ+π/2) : δ ≤ r and Γ0 := ω + δeiθ : −γ − π2 ≤ θ ≤ γ + π2 . Let 0 < ε < γ and consider z ∈ Σγ−ε . For λ = ω + re±i(γ+π/2) ∈ Γ± , Re(λz)

=

ω Re z + r|z| cos(arg z ± (γ + π/2))

≤ ω Re z − r|z| sin ε. Hence,

eλz q˜(λ) ≤ eω Re z e−r|z| sin ε This shows that f (z) :=

1 2πi

M (λ ∈ Γ± ). r

(2.12)

eλz q˜(λ) dλ

(2.13)

Γ

is absolutely convergent, uniformly for z in compact subsets of Σγ , and therefore deﬁnes a holomorphic function in Σγ . By Cauchy’s theorem, this function is independent of δ > 0, and also independent of γ < α so long as arg z < γ (here we use the assumption on q˜ to estimate the integral of eλz q˜(λ) over arcs {ω + Reiθ : γ1 + π2 ≤ |θ| ≤ γ2 + π2 }). Hence (2.13) deﬁnes a holomorphic function f ∈ Σα . To estimate f (z), we choose δ = |z|−1 , and choose γ and ε such that γ < α and | arg z| < γ − ε. On Γ0 , λ = ω + |z|−1 eiθ (−γ − π/2 ≤ θ ≤ γ + π/2), so

1

2πi

Γ0

eλz q˜(λ) dλ

≤

1 2π

γ+π/2

eω Re z ecos(arg z+θ) M dθ

−γ−π/2

≤ M e1+ω Re z .

(2.14)

86

2. THE LAPLACE TRANSFORM

On Γ± , λ = ω + re±i(γ+π/2) , and the estimate (2.12) gives

∞

1

1 M

λz e q ˜ (λ) dλ ≤ eω Re z e−r|z| sin ε dr

2πi Γ±

2π |z|−1 r M eω Re z ∞ e−r sin ε = dr 2π r 1 M eω Re z ≤ . 2π sin ε

(2.15)

Now (2.14) and (2.15) establish that sup e−ωz f (z) < ∞ z∈Σγ−ε

for any 0 < ε < γ < α. Next we will show that fˆ(λ) = q(λ) whenever λ > ω. Given such λ, choose 0 < δ < λ − ω, and 0 < γ < α. Then λ is to the right of the path Γ, and Fubini’s theorem and Cauchy’s residue theorem imply that ∞ 1 fˆ(λ) = e−λt eμt q˜(μ) dμ dt 2πi Γ 0 1 q˜(μ) = dμ 2πi Γ λ − μ 1 q˜(μ) = q˜(λ) + lim dμ, R→∞ 2πi Γ λ −μ R R := {ω + Reiθ : −γ − π/2 ≤ θ ≤ γ + π/2}. Then where Γ

γ+π/2

q˜(μ) M

≤ dμ dθ

λ−μ iθ − λ| |ω + Re ΓR −γ−π/2 → 0 as R → ∞. This proves that fˆ(λ) = q(λ). When f is as in Theorem 2.6.1 (i), it is an easy consequence of Cauchy’s integral formula for derivatives that

sup z k e−ωz f (k) (z) < ∞ z∈Σβ

for all 0 < β < α. Recall from Sections 1.4 and 1.5 that, for f ∈ L1loc (R+ , X), ω(f ) := inf ω ∈ R : sup e−ωt f (t) < ∞ , abs(f ) hol(fˆ)

:= :=

'

t≥0

( inf Re λ : fˆ(λ) exists , ' ( inf ω ∈ R : fˆ has a holomorphic extension for Re λ > ω .

2.6. LAPLACE TRANSFORMS OF HOLOMORPHIC FUNCTIONS

87

Moreover, hol(fˆ) ≤ abs(f ) ≤ ω(f ). We will now show that equalities hold when f is holomorphic and exponentially bounded on a sector. Theorem 2.6.2. Let 0 < α < π/2, let f : Σα → X be holomorphic, and suppose that there exists ω ∈ R such that supz∈Σα e−ωz f (z) < ∞. Then hol(fˆ) = abs(f ) = ω(f ). Proof. By Theorem 2.6.1, there exists γ > 0 such that fˆ has a holomorphic extension (also denoted by fˆ) to ω + Σγ+π/2 and C := supλ∈Σγ+π/2 (λ − ω)fˆ(λ) < ∞. By deﬁnition of hol(fˆ), fˆ also has a holomorphic extension to hol(fˆ) + Σπ/2 = {Re λ > hol(fˆ)}. Let ω > hol(fˆ). There exists γ > 0 such that ω + Σγ +π/2 ⊆ (ω + Σγ+π/2 ) ∪ (hol(fˆ) + Σπ/2 ). Hence, fˆ is holomorphic on ω + Σγ +π/2 and continuous on the closure. Let ' ( U := λ ∈ (ω + Σγ +π/2 ) ∩ (ω + Σγ+π/2 ) : |λ − ω | < 2|λ − ω| . If λ ∈ U , then (λ − ω )fˆ(λ) ≤ 2C. Moreover, (ω + Σγ +π/2 ) \ U is compact. Hence, supλ∈ω +Σγ +π/2 (λ−ω )fˆ(λ) < ∞. It follows from Theorem 2.6.1, and the

fact that the Laplace transform is one-to-one, that supz∈Σβ e−ω z f (z) < ∞ for some β > 0, and in particular, ω(f ) ≤ ω . Since this holds whenever ω > hol(fˆ), it follows that ω(f ) ≤ hol(fˆ), completing the proof. In the remainder of this section we will consider asymptotic behaviour of f (t) as t → ∞ and as t → 0. In the case of holomorphic functions deﬁned on a sector it can be described completely in terms of the Laplace transform. This is not the case in general, and in Chapter 4 a systematic treatment of this question will be given. However, here we can use contour arguments directly on the basis of the representation formula (2.13). First we show that asymptotic behaviour along one ray and on the whole sector are equivalent. This is a consequence of Vitali’s theorem (Theorem A.5). Proposition 2.6.3. Let 0 < α ≤ π and let f : Σα → X be holomorphic such that sup f (z) < ∞ z∈Σβ

for all 0 < β < α. Let x ∈ X. a) If limt→∞ f (t) = x, then lim |z|→∞ f (z) = x for all 0 < β < α. z∈Σβ

b) If limt↓0 f (t) = x, then lim |z|→0 f (z) = x for all 0 < β < α. z∈Σβ

88

2. THE LAPLACE TRANSFORM

Proof. a) Let fk (z) = f (kz). It follows from Vitali’s theorem that limk→∞ fk (z) = x uniformly on compact subsets of Σα . Let 0 < β < α. Let ε > 0. There exists k0 ∈ N such that fk (z) − x ≤ ε whenever z ∈ Σβ , 1 ≤ |z| ≤ 2, k ≥ k0 . Let z ∈ Σβ , |z| ≥ k0 . Choose k ∈ N such that k ≤ |z| < k + 1. Then

f (z) − x = fk (z/k) − x ≤ ε. This proves a). b) This follows by applying a) to the function z → f (z −1 ). Now we consider the asymptotic behaviour of f (t) as t → ∞ and t ↓ 0. Theorem 2.6.4 (Tauberian Theorem). Consider the situation of Theorem 2.6.1, and let x ∈ X. a) One has limt↓0 f (t) = x if and only if limλ→∞ λq(λ) = x. b) Assume that ω = 0. Then limt→∞ f (t) = x if and only if limλ↓0 λq(λ) = x. Proof. We can assume that ω = 0 for both cases a) and b) by replacing f (z) by e−ωz f (z) otherwise. Replacing f (t) by f (t) − x, we can also assume that x = 0. For simplicity, we shall denote the function q˜ of Theorem 2.6.1 by q. Assume that limλ→∞ λq(λ) = x. Let 0 < γ < α. By Proposition 2.6.3, lim |λ|→∞ λq(λ) = x. Let ε > 0. There exists δ0 > 0 such that λq(λ) ≤ ε λ∈Σγ+π/2

whenever |λ| ≥ δ0 , λ ∈ Σγ+ π2 . Let 0 < t ≤ 1/δ0 . Now we choose the contour Γ as in the proof of Theorem 2.6.1, (ii) ⇒ (i), with δ = 1/t. Then

1 γ+π/2 iθ eiθ ieiθ

1

λt e

e q(λ) dλ = e q dθ

2πi

2πi −γ−π/2

t t Γ0 γ+π/2 ε ≤ ecos θ dθ 2π −γ−π/2 ≤ ε e, and

1

λt e q(λ) dλ

2πi Γ±

1

∞

t·re±i(γ+π/2) ±i(γ+π/2) ±i(γ+π/2) dr = e q(re )re

2πi 1/t r

∞

1 s ±(γ+π/2) s ±i(γ+π/2) ds se±i(γ+π/2)

= e q( e ) e

2πi t t s 1 → 0

as t ↓ 0 by the dominated convergence theorem. It follows from the representation (2.13) that lim supt↓0 f (t) ≤ ε e.

2.7. COMPLETELY MONOTONIC FUNCTIONS

89

The converse implication is easy and does not depend on holomorphy. Assume that limt↓0 f (t) = 0. Let ε > 0. There exists τ > 0 such that f (t) ≤ ε for all t ∈ [0, τ ]. Then lim sup λq(λ)

λ→∞

≤

lim sup λ λ→∞

τ

e 0

≤ ε + lim sup λ λ→∞

= ε + lim sup M λ→∞

−λt

∞

f (t) dt + λ

∞

e

−λt

f (t) dt

τ

e−λt M eωt dt

τ

λ e−(λ−ω)τ = ε, λ−ω

where ω > ω(f ) and M is suitable. This completes the proof of a). The assertion b) is proved in the same way as a).

2.7

Completely Monotonic Functions

Throughout this section, X will be an ordered Banach space with normal cone (see Appendix C). Let f : R+ → X be increasing. Then f is of bounded semivariation on each interval [0, τ ], by Proposition 1.9.1. Assume that ω(f ) < ∞. Then the Laplace-Stieltjes transform τ ∞ df (λ) = lim e−λt df (t) =: e−λt df (t) (2.16) τ →∞

0

0

converges on the half-plane {Re λ > abs(df )}, and deﬁnes a holomorphic function df on {Re λ > abs(df )}. Recall from Theorem 1.10.5 that abs(df ) < ∞ if and only if ω(f ) < ∞. Theorem 2.7.1. Let f : R+ → X be an increasing function. Assume that −∞ < abs(df ) < ∞. Then abs(df ) is a singularity of df . t Proof. Replacing f (t) by 0 e− abs(df )s df (s), we can assume that abs(df ) = 0. Assume that df has a holomorphic extension to a neighbourhood of 0. Then there exists δ > 0 such that df (−δ) =

∞ n=0

(−1)n (1 + δ)n

(df )(n) (1) . n!

∗ Let x∗ ∈ X+ . Then

df (−δ), x∗ =

∞ (1 + δ)n ∞ −t n e t df (t), x∗ . n! 0 n=0

90

2. THE LAPLACE TRANSFORM

Since all expressions are positive we may interchange the sum and the integral and obtain ∞ ∞ δt ∗ e df (t), x = e−t e(1+δ)t df (t), x∗ 0

0

=

∞ (1 + δ)n ∞ −t n e t df (t), x∗ n! 0 n=0

=

df (−δ), x∗ < ∞.

∗ Since X+ spans X ∗ (see Proposition C.2), it follows that abs(x∗ ◦ f ) ≤ −δ for ∗ all x ∈ X ∗ . It follows from (1.25) that abs(df ) ≤ −δ, which contradicts the assumption.

Corollary 2.7.2. Let f ∈ L1loc (R+ , X) such that f (t) ≥ 0 a.e. Assume that −∞ < abs(f ) < ∞. Then abs(f ) is a singularity of fˆ. Hence, hol(fˆ) = abs(f ). Proof. This is immediate from Proposition 1.10.1 and Theorem 2.7.1. Our aim is to characterize functions of the form df where f : R+ → X is increasing. Then ∞ n (n) (−1) df (λ) = e−λt tn df (t) ≥ 0 0

for all n ∈ N0 , λ > ω. Thus df is completely monotonic in the sense of the following deﬁnition. Deﬁnition 2.7.3. A function r : (ω, ∞) → X is completely monotonic if r is inﬁnitely diﬀerentiable and (−1)n r (n) (λ) ≥ 0

for all λ > ω, n ∈ N0 .

(2.17)

In the following, we shall assume that ω = 0 for simplicity (otherwise, we t can replace r(λ) by r(λ + ω) and f (t) by 0 e−ωs df (s)). Recall that by Theorem 1.10.5 abs(df ) ≤ 0 if and only if ω(f ) ≤ 0. Deﬁnition 2.7.4. We say that Bernstein’s theorem holds in X if for every completely monotonic function r : (0, ∞) → X there exists an increasing function f : R+ → X such that ω(f ) ≤ 0 and r(λ) = df (λ) for all λ > 0. Bernstein’s theorem does hold in X = R; this is just Bernstein’s classical theorem from 1928 [Ber28]. Here we will prove it, as a special case of Theorem 2.7.7, with the help of the Real Representation Theorem 2.2.1. Deﬁnition 2.7.5. The space X has the interpolation property if, given two sequences (xn )n∈N , (yn )n∈N in X such that xn ≤ xn+1 ≤ yn+1 ≤ yn

(n ∈ N)

(2.18)

2.7. COMPLETELY MONOTONIC FUNCTIONS

91

there exists z ∈ X such that xn ≤ z ≤ yn

for all n ∈ N.

(2.19)

Examples 2.7.6. a) Assume that X = Y ∗ where Y is an ordered Banach space with normal cone. Then X has the interpolation property. ∗ Proof. Let x∗n ≤ x∗n+1 ≤ yn+1 ≤ yn∗ (n ∈ N). Replacing x∗n by x∗n − x∗1 and yn∗ by ∗ ∗ yn − x1 we can assume that x∗n ≥ 0. Deﬁne z ∗ ∈ X ∗ by x, z ∗ = supn∈N x, x∗n . Then z ∗ is linear and positive, and hence continuous (see Appendix C).

b) If X is reﬂexive, then X has the interpolation property. This follows from a). c) Each von Neumann algebra (i.e., a ∗-subalgebra of L(H) which is closed in the strong operator topology, where H is a Hilbert space) has the interpolation property. This follows from a) and [Ped89, Theorem 4.6.17]. d) Every σ-order complete Banach lattice (i.e., a Banach lattice in which each countable order-bounded set has a supremum) has the interpolation property. e) If X has order continuous norm (i.e., each decreasing positive sequence converges) then X has the interpolation property. f) The space C[0, 1] does not have the interpolation property. See the Notes for further comments on the interpolation property. Now we can formulate the following characterization, which is the main result of this section. Theorem 2.7.7. Bernstein’s theorem holds in X if and only if X has the interpolation property. The proof of Theorem 2.7.7 will be carried out in several steps. On the way we will prove a characterization of completely monotonic functions which is valid without restrictions on the space. First, we study convex functions. Let J ⊂ R be an interval. A function F : J → X is called convex if F (λs + (1 − λ)t) ≤ λF (s) + (1 − λ)F (t) for all s, t ∈ J, 0 < λ < 1. Many order properties of convex functions carry over from the scalar case since for x ∈ X we have x≥0

if and only if

x, x∗ ≥ 0

∗ for all x∗ ∈ X+ .

For example, a twice diﬀerentiable function F is convex if and only if F ≥ 0. Lemma 2.7.8. Let [a, b] be a closed interval in the interior of J and let F : J → X be convex. Then F is Lipschitz continuous on [a, b]. Moreover, if F (J) ⊂ X+ and F (a) = 0, then F is increasing on [a, b].

92

2. THE LAPLACE TRANSFORM

Proof. Let c < a, d > b such that [c, d] ⊂ J. Then for a ≤ t < s ≤ b, F (a) − F (c) F (s) − F (t) F (d) − F (b) ≤ ≤ . a−c s−t d−b Since the cone is normal this implies that F is Lipschitz continuous on [a, b]. The second assertion is easy to see. We notice in particular that every convex function deﬁned on an open interval is continuous. Let −∞ < a < b ≤ ∞ and let f : [a, b) → X+ be increasing. Then f is Riemann integrable on [a, t] whenever a ≤ t < b (see Corollary 1.9.6). Let

t

F (t) :=

f (s) ds (a ≤ t < b).

(2.20)

a

Then F : [a, b) → X+ is convex. If X has the interpolation property, then the following converse result holds. Proposition 2.7.9. Assume that X has the interpolation property. Let F : [a, b) → X+ be convex such that F (a) = 0, where −∞ < a < b ≤ ∞. Then there exists an increasing function f : [a, b) → X+ such that (2.20) holds. Proof. The following two properties follow from convexity: a) Let a ≤ s < b. Then the diﬀerence quotient 1 (F (s + h) − F (s)) h is positive and increasing for h ∈ (0, b − s). b) Let a ≤ s < s + h ≤ t < t + k < b. Then 1 1 (F (s + h) − F (s)) ≤ (F (t + k) − F (t)) . h k

(2.21)

Put f (a) = 0. It follows from the interpolation property, a) and b) that for each t ∈ (a, b) there exists f (t) ∈ X such that 1 1 (F (s + h) − F (s)) ≤ f (t) ≤ (F (t + k) − F (t)) h k

(2.22)

whenever a ≤ s < s + h ≤ t < t + k < b. It follows from (2.21) and (2.22) that f : [a, b) → X+ is increasing. t Let G(t) := a f (s) ds. We show that F = G. Let a < t < b. Let a ≤ t0 < t1 < . . . < tn = t be a partition of [a, t]. Setting hi := ti − ti−1 , we obtain from

2.7. COMPLETELY MONOTONIC FUNCTIONS

93

(2.22) that n

f (ti−1 )(ti − ti−1 ) ≤

i=1

=

n 1 (F (ti−1 + hi ) − F (ti−1 )) hi h i=1 i n

(F (ti ) − F (ti−1 ))

i=1

= F (t) − F (a) = F (t). It follows from the deﬁnition of the Riemann integral that G(t) ≤ F (t). Also by (2.22), n

f (ti )(ti − ti−1 ) ≥

i=1

n F (ti ) − F (ti−1 ) i=1

ti − ti−1

(ti − ti−1 )

= F (t). Hence G(t) ≥ F (t). Next, we prove a converse version of Proposition 2.7.9. Proposition 2.7.10. Assume that for every convex function F : R+ → X+ with F (0) = 0 and ω(F ) = 0 there exists an increasing function f : R+ → X+ such t that F (t) = 0 f (s) ds (t ≥ 0). Then X has the interpolation property. Proof. Let xn ≤ xn+1 ≤ yn+1 ≤ yn (n ∈ N). We can assume that x1 ≥ 0 (replacing xn by xn − x1 and yn by yn − x1 otherwise). Deﬁne f : R+ → X by ⎧ n xn if t ∈ [ n−1 , n+1 ); n ≥ 1, ⎪ n ⎪ ⎪ ⎨y n+1 n if t ∈ [ , n n n−1 ); n ≥ 2, f (t) := ⎪ y1 if t ∈ [2, ∞), ⎪ ⎪ ⎩ 0 if t = 1. t Then f ∈ L1loc (R+ , X). Let F (t) := 0 f (s) ds. Then F : R+ → X+ is convex and F (0) = 0. By assumption, there exists an increasing function g : R+ → X such t that F (t) = 0 g(s) ds (t ≥ 0). Then F (t − h) − F (t) F (t + h) − F (t) ≤ g(t) ≤ −h h for all t > 0 and h > 0 small enough. It follows that g(t) = F (t) whenever F n is diﬀerentiable at t. Consequently, g(t) = xn if t ∈ ( n−1 , n+1 ) and g(t) = yn if n n+1 n t ∈ ( n , n−1 ). Hence, xn ≤ g(1) ≤ yn . Thus, z := g(1) interpolates between the two sequences. For completeness, we also give the usual representation of convex functions as a corollary of Proposition 2.7.9.

94

2. THE LAPLACE TRANSFORM

Corollary 2.7.11. Assume that X has the interpolation property. Let F : (a, b) → X be convex, and let c ∈ (a, b). Then there exist x ∈ X and an increasing function f : (a, b) → X such that t F (t) = F (c) + (t − c)x + f (s) ds c

for all t ∈ (a, b). Proof. We may assume that c = 0. It follows from convexity that 1 1 (F (0) − F (−t)) ≤ (F (s) − F (0)) t s whenever 0 < s < b, 0 < t < −a. Moreover, the left-hand diﬀerence quotient is decreasing in t, and the right-hand one is increasing in s. By the interpolation property, there exists x ∈ X such that 1 1 (F (0) − F (−t)) ≤ x ≤ (F (s) − F (0)) t s for all 0 < t < −a, 0 < s < b. In particular, the function G(t) := F (t) − F (0) − tx

(t ∈ (a, b))

is positive, convex and satisﬁes G(0) = 0. By Proposition 2.7.9, there exist increasing functions f1 : [0, b) → X+ and f2 : [0, −a) → X+ such that t G(t) = f1 (s) ds for t ∈ [0, b) and 0 t G(−t) = f2 (s) ds for t ∈ [0, −a). 0

We can assume that f1 (0) = f2 (0) = 0. Let f (t) := f1 (t) for t ∈ [0, b) and t f (t) := −f2 (−t) for t ∈ (a, 0). Then f is increasing and G(t) = 0 f (s) ds for all t ∈ (a, b). Now we will study completely monotonic functions. We need the following formulas (2.23) and (2.24) (the latter is merely needed for n = 1 and n = 2). In the remainder of this section we shall sometimes use loose notation such as r(λ) to λ (n) r(λ) denote the function λ → r(λ) and r(λ) to denote its derivatives λ , and λ λ of orders 1 and n. Lemma 2.7.12. Let r ∈ C ∞ ((0, ∞), X). Then (n) n (−1)n n+1 r(λ) (−1)m m (m) λ = λ r (λ) n! λ m! m=0

(2.23)

2.7. COMPLETELY MONOTONIC FUNCTIONS and

k+n

λ

r(λ) λn

(k) (n)

= λk r(k+n) (λ)

95

(2.24)

for all λ > 0, k, n ∈ N0 . In particular, if r is completely monotonic, then λ → r(λ)/λ is also completely monotonic. Proof. The ﬁrst formula (2.23) is immediate from Leibniz’s rule. It follows that if r is completely monotonic, then λ → r(λ)/λ is also completely monotonic. We show by induction over n that (2.24) holds for all k ∈ N0 . It is obvious for n = 0. Moreover, (k+1) (k+1) (k) r(λ) r(λ) r(λ) λk r (k+1) (λ) = λk λ = λk λ + (k + 1) λ λ λ (k) r(λ) = λk+1 λ for λ > 0. This shows that (2.24) holds for n = 1. Now assume that (2.24) holds for a ﬁxed n ∈ N and k ∈ N0 . Then, applying (2.24) to r yields (k) (n) r (λ) λk r (k+n+1) (λ) = λk+n (2.25) λn for λ > 0. Observe that λk+n+1 (r(λ)/λn+1 )(k) = λn · λk+1 (r(λ)/λn+1 )(k) = nλn−1 λk+1 (r(λ)/λn+1 )(k) + λn λk+1 (r(λ)/λn+1 )(k) = nλn−1 λk+1 (r(λ)/λn+1 )(k) + λn λk (r(λ)/λn )(k+1) , by applying (2.24) for n = 1 to the function r(λ)/λn instead of r. Hence, λk+n+1 (r(λ)/λn+1 )(k) ! "(k) ! "(k) = nλn+k r(λ)/λn+1 + λn+k r (λ)/λn − nr(λ)/λn+1 = λn+k (r (λ)/λn )

(k)

for λ > 0. It follows from (2.25) that (n+1) (n) = λn+k (r (λ)/λn )(k) = λk r(λ)(k+n+1) . λk+n+1 (r(λ)/λn+1 )(k) Thus, (2.24) holds when n is replaced by n + 1.

96

2. THE LAPLACE TRANSFORM

Proposition 2.7.13. Let F ∈ Lip0 (R+ , X) and let ∞ r(λ) = λdF (λ) = λ e−λt dF (t)

(λ > 0).

0

Then r is completely monotonic if and only if F is convex and F (t) ≥ 0 (t ≥ 0). ∞ −λt Proof. Assume that r is completely monotonic. Note that r(λ) dF (t). λ = 0 e Thus, by the Post-Widder formula (Theorem 2.3.1), for t > 0 we have F (t) = limk→∞ Fk (t), where Fk (t) := Gk (k/t),

Gk (λ) :=

"(k) (−1)k k+1 ! λ r(λ)/λ2 . k!

By Lemma 2.7.12, λ → r(λ)/λ2 is completely monotonic, and it follows that ! " Fk (t) ≥ 0. We show that Fk is convex; i.e., that Fk (t) = − kt−2 Gk (k/t) ≥ 0. d Let H(λ) := −λ2 kGk (kλ). Then Fk (t) = dt H(1/t) = −t−2 H (1/t). Thus it suﬃces to show that H (λ) ≤ 0 or equivalently 2λkGk (kλ) + λ2 k 2 Gk (kλ) ≥ 0 for λ > 0. Letting μ := kλ we have to show that

(μGk (μ)) = 2Gk (μ) + μGk (μ) ≥ 0

(μ > 0).

This is true since (2.24) for n = 2 gives (μGk (μ)) =

(−1)k k+2 (−1)k k (k+2) μ (r(μ)/μ2 )(k) = μ r (μ) ≥ 0 k! k!

(μ > 0).

This proves one implication. ∗ Conversely, suppose that F is convex and F (t) ≥ 0 for all t ≥ 0. Let x∗ ∈ X+ . ∗ Then x ◦ F is convex, positive and Lipschitz continuous. There is an increasing, d bounded function g : R+ → R+ such that g(t) = dt F (t), x∗ a.e., and F (t), x∗ = t g(s) ds for all t ≥ 0 (see Proposition 2.7.9). We may assume that g(0) = 0. By 0 Proposition 1.10.1 and (1.22), (λ), x∗ = λˆ r(λ), x∗ = λdF g (λ) = dg(λ)

(λ > 0).

∗ Hence, x∗ ◦r is completely monotonic for all x∗ ∈ X+ and therefore r is completely monotonic.

Next we prove a representation theorem for completely monotonic functions deﬁned on R+ (and not merely (0, ∞)). Proposition 2.7.14. Let r ∈ C ∞ (R+ , X) such that (−1)n r (n) (λ) ≥ 0 (λ ≥ 0). Then there exists a convex function F ∈ Lip0 (R+ , X) such that F (t) ≥ 0 (t ≥ 0) and (λ) r(λ) = λdF

(λ > 0).

(2.26)

2.7. COMPLETELY MONOTONIC FUNCTIONS

97

Proof. It follows from (2.23) that for k ∈ N and λ > 0, (−1)k k+1 pk (λ) := λ k!

r(λ) λ

(k) =

k (−1)m m (m) λ r (λ) ≥ 0. m! m=0

Moreover, limλ↓0 pk (λ) = r(0). It follows from (2.24) for n = 1 that pk (λ) =

(−1)k k (k+1) λ r (λ) ≤ 0 k!

(λ > 0).

Thus 0 ≤ pk (λ) ≤ r(0) for all λ > 0. Since the cone is normal, this implies that the ∞ function r(λ) λ is in CW ((0, ∞), X). By Theorem 2.2.1, there exists F ∈ Lip0 (R+ , X) r(λ) such that λ = dF (λ) (λ > 0). It follows from Proposition 2.7.13 that F is positive and convex. Theorem 2.7.15. A function r : (0, ∞) → X is completely monotonic if and only if there exists a convex function F : R+ → X+ satisfying F (0) = 0 and ω(F ) ≤ 0 such that ∞

r(λ) = λ

e−λt dF (t)

(λ > 0).

(2.27)

0

In that case, F is uniquely determined by r. ∗ Proof. a) Assume that r is of the form (2.27). Let x∗ ∈ X+ . Then there exists an increasing function f : R+ → R+ such that f (0) = 0 and

F (t), x∗ = Thus r(λ), x∗ =

0 ∞

t

f (s) ds (t ≥ 0).

e−λt df (t)

(λ > 0).

0

Hence, r(·), x∗ is completely monotonic and (−1)n r (n) (λ), x∗ = (−1)n

d dλ

n

r(λ), x∗ ≥ 0.

∗ Since x∗ ∈ X+ is arbitrary, it follows that r is completely monotonic. b) Conversely, let r be completely monotonic. By Proposition 2.7.14, there exists ∞ a convex function G ∈ Lip0 (R+ , X) such that G(t) ≥ 0 (t ≥ 0) and r(λ+1) = λ 0 e−λt dG(t) (λ > 0). Let

F (t) := 0

t

(1 − (t − s))es dG(s).

98

2. THE LAPLACE TRANSFORM

∗ Then F is positive and convex. In fact, let x∗ ∈ X+ . Then there exists an increast ∗ ing function g : R+ → R+ such that G(t), x = 0 g(s) ds and g(0) = 0. By Proposition 1.9.10, Fubini’s Theorem and (1.20),

∗

t

F (t), x =

e g(s) ds − 0 t s = e g(s) − 0

t = 0

t

s

(t − s)es g(s) ds r e g(r) dr ds

0 s

0

s

(t ≥ 0).

er dg(r) ds

0

∗ Thus x∗ ◦ F is positive and convex for all x∗ ∈ X+ , so F is positive and convex. By Proposition 1.10.1 and (1.22),

r(λ + 1), x∗ = λˆ g (λ) =

∞

e−λt dg(t)

0

for λ > 0. By Proposition 1.10.3, for λ > 1, ∞ r(λ), x∗ = e−λt et dg(t) = 0

where

f (t) := 0

t

∞

e−λt df (t),

0

e dg(s) = e g(t) − s

t

t

es g(s) ds,

0

t ∞ by (1.20). Since F (t), x∗ = 0 f (s) ds, it follows that r(λ) = λ 0 e−λt dF (t) . Moreover, applying for λ > 1. By Theorem 2.7.1, abs(dF ) is a singularity of dF Proposition 2.7.14 to r(· + δ) shows that r has a holomorphic extension to {λ ∈ C : Re λ > δ} for all δ > 0, and hence to {λ ∈ C : Re λ > 0}. It follows from (λ) = r(λ) for the uniqueness of holomorphic extensions that abs(dF ) ≤ 0 and dF λ > 0. By Theorem 1.10.5, ω(F ) ≤ 0 (actually, ω(F ) = 0 unless r ≡ 0). Finally, uniqueness of F follows from the Post-Widder formula (Theorem 2.3.1). Theorem 2.7.16. Assume that X has the interpolation property. Let r : (0, ∞) → X be completely monotonic. Then there exists an increasing function f : R+ → X+ such that f (0) = 0, ω(f ) ≤ 0 and r(λ) =

∞

e−λt df (t)

(λ > 0).

0

Proof. By Theorem 2.7.15, there exists a convexfunction F : R+ → X+ satisfying ∞ F (0) = 0 and ω(F ) ≤ 0 such that r(λ) = λ 0 e−λt dF (t) for all λ > 0. By Proposition 2.7.9, there exists an increasing function f : R+ → X+ such that

2.7. COMPLETELY MONOTONIC FUNCTIONS

99

t F (t) = 0 f (s) ds (t ≥ 0). We can assume that f (0) = 0. Let ω > 0. There exists M ≥ 0 such that F (t) ≤ M eωt . Since f is increasing we have

t f (t/2) ≤ 2

t

f (s) ds ≤ F (t).

t/2

It follows that ω(f ) ≤ 0 (actually, ω(f ) = 0 unless r ≡ 0). By Proposition 1.10.2, 0

∞

e−λt df (t) = λ

∞

e−λt dF (t) = r(λ)

(λ > 0).

0

Now we can prove Theorem 2.7.7. Proof of Theorem 2.7.7. One direction is given by Theorem 2.7.16. In order to prove the other, assume that Bernstein’s theorem holds in X. We show that X has the interpolation property. Let F : R+ → X+ be convex such that F (0) = 0 and ω(F ) = t 0. By Proposition 2.7.10, it suﬃces to show that F (t) = 0 f (s) ds (t ≥ 0) for some ∞ increasing function f : R+ → X. By Proposition 2.7.13, r(λ) := λ 0 e−λt dF (t) deﬁnes a completely monotonic function on (0, ∞). By assumption, there exists an increasing function f : R+ → X such that

∞

r(λ) =

e−λt df (t).

0

t We may assume that f (0) = 0. Let H(t) := 0 f (s) ds. Using Proposition 1.10.2 and (1.22), λ2 H(λ) = df (λ) = r(λ) = λ2 F (λ) for all λ > 0. It follows from the uniqueness theorem that H(t) = F (t) for all t ≥ 0. If r : (0, ∞) → X is completely monotonic, there may be many increasing functions f : R+ → X+ such that r = df . However, if X has order continuous norm, then we may pick out a normalized version of f . Let f : R+ → X be increasing and assume that X has order continuous norm. For t ≥ 0 we deﬁne f (t+) = lims↓t f (s), and for t > 0 we let f (t−) = lims↑t f (s). We say that f has a jump at t > 0 if f (t+) = f (t−). Lemma 2.7.17. Assume that X has order continuous norm and that f : R+ → X is increasing. Then the number of jumps of f is countable. Proof. Let τ > 0 and J := {t ∈ (0, τ ) : f (t+) = f (t−)}. Let ε > 0 and Jε := {t ∈ J : f (t+) − f (t−) ≥ ε}. We claim that Jε is ﬁnite. Otherwise there exist m tn ∈ Jε (n ∈ N), tn = tm for n = m. Let xn = f (tn +) − f (tn −). Then xn ≤ f (τ ) − f (0) for all m ∈ N. Since X has order continuous norm, the n=1 ∞ sum n=1xn converges. Hence, xn → 0 as n → ∞. This is a contradiction. Since J = n∈N J1/n , it follows that J is countable.

100

2. THE LAPLACE TRANSFORM

We continue to assume that X has order continuous norm. Let f : R+ → X be increasing. We deﬁne the normalization f ∗ : R+ → X of f by f (0+) if t = 0, ∗ f (t) = 1 (f (t+) + f (t−)) if t > 0. 2 The function f is called normalized if f = f ∗ . It follows from the deﬁnition of the Riemann-Stieltjes integral that t t g(s) df (s) = g(s) df ∗ (s) 0

0

for every t > 0 and every continuous function g : [0, t] → C. In fact, one may take a sequence of partitions (πn )n∈N with intermediate points which avoid the jumps of f ). Then S(g, f, πn ) = S(g, f ∗ , πn ) for all n ∈ N, and so 0

t

g(s) df (s) = lim S(g, f, πn ) = lim S(g, f ∗ , πn ) = n→∞

n→∞

t

g(s) df ∗ (s).

0

In conclusion, we obtain the following result. Theorem 2.7.18 (Bernstein’s theorem). Assume that X has order continuous norm. Let r : (0, ∞) → X be completely monotonic. Then there exists a unique normalized increasing function f : R+ → X such that f (0) = 0, ω(f ) ≤ 0 and ∞ r(λ) = e−λt df (t) (λ > 0). 0

Proof. Since X has the interpolation property, ∞ existence follows from Theorem 2.7.16. For uniqueness, suppose that r(λ) = 0 e−λt df (t) (λ > 0). By Proposition ∞ t 1.10.2, r(λ) = λ 0 e−λt dF (t) (λ > 0) where F (t) := 0 f (s) ds. It follows from Theorem 2.7.15 that F is uniquely determined by r. Since F (t+) := lim

1 (F (t + h) − F (t)) = f (t+) h

F (t−) := lim

1 (F (t) − F (t − h)) = f (t−) h

h↓0

if t ≥ 0, and h↓0

if t > 0, the normalized function f is also unique.

2.8

Notes

Section 2.1 Representation of operators from a space of the form L1 (Ω, μ) into a Banach space

2.8. NOTES

101

X by vector measures is a classical subject (see [DU77, Section III.1]). In view of the applications to Cauchy problems, Stieltjes integrals seem more appropriate than vector measures in our context. In the context of Laplace transform theory, the Riesz-Stieltjes Representation Theorem 2.1.1 appeared in a paper of Hennig and Neubrander [HN93] (see also [Neu94] and [BN94]). For a discussion of the representation of bounded linear operators in L(Lp (R+ ), X) as functions of bounded p -variation (1/p + 1/p = 1, p > 1), see the work of Weis [Wei93] and Vieten [Vie95]. Section 2.2 For real-valued functions, Theorem 2.2.1 was proved by Widder in 1936 [Wid36] (see also [Wid41]). In trying to extend scalar-valued Laplace transform theory to vector-valued functions, Hille [Hil48] remarks on several occasions that Widder’s theorem can be lifted to inﬁnite dimensions if the space is reﬂexive, but not in general (see [Hil48, p.213] or [Miy56]). In fact, it was shown by Zaidman [Zai60] (see also [Are87b] or Theorem 2.2.3) that Widder’s theorem extends to a Banach space X if and only if X has the RadonNikodym property (for example, if X is reﬂexive). In 1965, Berens and Butzer [BB65] gave necessary and suﬃcient complex conditions for the Laplace-Stieltjes representability of functions in reﬂexive and uniformly convex Banach spaces. However, these results were of limited applicability. In general, important classes of Banach spaces that appear in studying evolution equations do not possess the Radon-Nikodym property. As a consequence, in the 1960s and 1970s Laplace transform methods were applied mainly to special vector-valued functions, like resolvents and semigroups, which have nice additional algebraic properties. In the theory of C0 -semigroups the link between the generator A and the semigroup T is given via the Laplace transform ∞ (λ − A)−1 x = e−λt T (t)x dt (x ∈ X). 0

The crucial algebraic property which made it possible to extend classical Laplace transform results to this abstract setting is the algebraic semigroup law T (t + s) = T (t)T (s), (t, s ≥ 0). Hille and Phillips comment in the foreword to [HP57] that “.... in keeping with the spirit of the times the algebraic tools now play a major role....” and that “.... the Laplace-Stieltjes transform methods..... have not been replaced but rather supplemented by the new tools.” The major disadvantage of the “algebraic approach” to linear evolution equations becomes obvious if one compares the mathematical theories associated with them (for example, semigroup theories, cosine families, the theory of integro-diﬀerential equations, etc.). It is striking how similar the results and techniques are. Still, without a Laplace transform theory for functions with values in arbitrary Banach spaces, every type of linear evolution equation required its own theory because the algebraic properties of the operator families changed from one case to another. In the late 1970s, in search of a general analytic principle behind all these theories, the study of Laplace transforms of functions with values in arbitrary Banach spaces was revitalized by Sova (see [Sov77] up to [Sov82]). An important result for Laplace transforms in Banach spaces is Theorem 2.6.1, proved by Sova in 1979 [Sov79b], [Sov79c]. This analytic representation theorem is behind every generation result for analytic solution families of linear evolution equations. The Real Representation Theorem 2.2.1 shows that the statement of Widder’s Theorem extends to arbitrary Banach spaces if the Laplace transform is replaced by the Laplace-Stieltjes transform. It is due to [Are87b] where it was deduced from the scalar result by Widder [Wid41] by duality arguments. The proof of Theorem 2.2.1 given here is

102

2. THE LAPLACE TRANSFORM

a modiﬁcation of Widder’s original proof given in [Wid41]; see [HN93]. Further extensions of these results are given in [Bob97a], [Bob97b], [Kis00], [Bob01] and [Cho02]. The characterization of the range of the Laplace-Stieltjes transform acting on Lip0 (R+ , X) given in Theorem 2.2.1 is based on the Post-Widder inversion formula in Theorem 1.7.7. Corresponding to other inversion formulas, equivalent descriptions can be formulated. Employing the complex inversion formula (see [Sov80b], [BN94]), or the Phragm´en-Doetsch inversion (see [PC98]), one can prove that the following growth and regularity conditions are equivalent. Theorem 2.8.1. Let r : (0, ∞) → X be continuous. The following are equivalent: (i) r ∈ C ∞ ((0, ∞), X) and

k+1 λ (k) < ∞. sup r (λ) k! λ>0

k∈N0

(ii) limλ→∞ r(λ) = 0 and r has an extension to a holomorphic function r : {Re λ > 0} → X such that, for all γ > 0, supRe λ>γ r(λ) < ∞ and ∞ 1 r(γ + it) < ∞. sup dt 2π k+2 s>0 −∞ (1 − ist) k∈N0

(iii) supλ>0 λr(λ) < ∞ and ∞ (−1)j−1 jk sup e λr(jλ) < ∞. (j − 1)! λ>0 j=1 k∈N

For a discussion of the Lp -conditions ∞ k+1 p 1 (k) k k r dt ≤ M t k! t 0

for all k ≥ 0,

and their connection to the representability of r as the Laplace transform of a function of bounded p-variation (p > 1), see [Wid41, Chapter VII], [Lev69], [Sov81a], [Wei93], and ∞ [Vie95]. It is shown in [KMV03] τ −λtthat a function r ∈ C ((0, ∞), X) is the ﬁnite LaplaceStieltjes transform r(λ) = 0 e dF (t) of a Lipschitz continuous function F : [0, τ ] → X with F (t) − F (s) ≤ M |t − s| for all 0 ≤ t, s ≤ τ if and only if k+1 λ (k) ≤M sup sup r (λ) k! k∈N0 λ>k/τ and sup

sup

k∈N λ∈(0,k/τ )

−k λτ (k) τ e r (λ) < ∞.

Section 2.3 Theorem 2.3.2 goes back to Phragm´en’s proof of the Uniqueness Theorem 1.7.3 (see [Phr04]), and to Doetsch [Doe37] who recognized the usefulness of the formula as an inversion procedure (see also [Doe50, Volume I, Section 8.1]). The Phragm´en-Doetsch inversion formula shows that a Laplace transformable function f is determined by the

2.8. NOTES

103

values of fˆ(λn ), where λn = n ≥ n0 . An extension of the Phragm´ en-Doetsch inversion −1 to arbitrary M¨ untz sequences (λn ) ⊂ R+ (i.e., λn+1 − λn ≥ 1 and ∞ n=1 λn = ∞), has been obtained by B¨ aumer [B¨ au03] (see also [BLN99]). There does not seem to be any inversion formula that holds for arbitrary uniqueness sequences (see Theorem 1.11.1). Corollary 2.3.3 is taken from [BN96] and is one of the key ingredients in the theory of asymptotic Laplace transforms (see [LN99], [LN01]). Whereas the complex inversion formula in Theorem 2.3.4 (the proof given here is from [HN93]) is in general aﬀected by exponentially decaying perturbations of the Laplace transform, the following modiﬁcation, due to Lyubich [Lyu66], gives a complex inversion formula which holds locally even if the transform undergoes such perturbations. ∞ Theorem 2.8.2. Let τ > 0, ω > 0, F ∈ Lip0 (R+ , X), and q(λ) = 0 e−λt dF (t) + a(λ) (λ > 0), where a ∈ L1loc (R+ , X) is a function with lim supλ→∞ λ1 log a(λ) ≤ −τ . Then ∞ 1 q(t) H(μ) := eμt dt 2πi ω t is well deﬁned for Re μ < 0, has a holomorphic continuation to the sliced half-plane {μ : Re μ < τ } \ [0, τ ), and F (t) = lim (H(t + iε) − H(t − iε)) ε→0

for all t ∈ [0, τ ).

Haase [Haa08] has given a diﬀerent approach to Theorem 2.3.4 and Lemma 2.3.5. Section 2.4 With the exception of Proposition 2.4.3 which is due to Doetsch (see [Doe50, Volume I, Section 14.3]) and Corollary 2.4.4, the results are straightforward reformulations of the main theorems of the sections 2.1–2.3. Using a Phragm´ type inversion formula en-Doetsch −1 along sequences (λn ) ⊂ R+ with λn+1 − λn ≥ 1 and ∞ untz sequences), n=1 λn = ∞ (M¨ one can strengthen the statement of Proposition 2.4.3 as follows (see [B¨ au03]). Theorem 2.8.3. Let 0 ≤ τ and let f ∈ L1loc (R+ , X) with abs(f ) < ∞. Then the following are equivalent: (i) f (t) = 0 almost everywhere on [0, τ ] and τ ∈ supp(f ). (ii) Every M¨ untz sequence (βn ) satisﬁes lim supn→∞

1 βn

log fˆ(βn ) = −τ.

(iii) For every M¨ untz sequence (βn ) there exists a M¨ untz subsequence (βnk ) such that lim

k→∞

1 log fˆ(βnk ) = −τ. β nk

(iv) There exists a M¨ untz sequence (βn ) with lim supn→∞ (v) lim supλ→∞

1 λ

1 βn

log fˆ(βn ) = −τ.

log fˆ(λ) = −τ .

As a consequence of these equivalences one obtains the following short proof of Titchmarsh’s theorem (see [B¨ au03], [BLN99] or [MB87, Section VI.7]). Corollary 2.8.4 (Titchmarsh’s Theorem). Let k ∈ L1 [0, τ ] with 0 ∈ supp(k) and f ∈ L1 ([0, τ ], X). If k f = 0 on [0, τ ], then f = 0.

104

2. THE LAPLACE TRANSFORM

Proof. We extend k and f by zero to R+ . Then, by Proposition 2.4.3 and Corollary ˆ 2.4.4, lim supλ→∞ λ1 log |k(λ)| = 0 and lim supλ→∞ λ1 log k f (λ) ≤ −T . By taking subsequences, it follows from the theorem above that there exists a M¨ untz sequence (βn ) ˆ n )| = 0 and such that limn→∞ β1n log |k(β −τ ≥ lim

n→∞

1 log k f (βn )

βn

= =

1 ˆ n )| + lim 1 log fˆ(βn )

log |k(β n→∞ βn βn 1 lim log fˆ(βn ) . n→∞ βn lim

n→∞

Thus, f = 0 on [0, τ ]. A function k ∈ L1loc (R+ ) with abs(k) < ∞ is a regularizing function if lim sup λ→∞

1 ˆ log |k(λ)| = 0, λ

or, equivalently, if 0 ∈ supp(k) (by Corollary 2.4.4). By the Titchmarsh-Foia¸s theorem (see [BLN99]), the condition 0 ∈ supp(k) t is necessary and suﬃcient for the convolution operator K : f → k ∗ f , (k ∗ f )(t) := 0 k(t − s)f (s) ds to be an injective operator on C(R+ , X) with dense range in the Fr´echet space C∗ (R+ , X) of all continuous functions g : R+ → X such that g(0) = 0, equipped with the seminorms g n := supt∈[0,n] g(t) . Moreover, f K,n := supt∈[0,n] Kf (t) deﬁnes a family of seminorms on C(R+ , X) and K extends to an isomorphism between the Fr echet completion C [k] (R+ , X) of C(R+ , X) with respect to that family of seminorms and the Fr´echet space C∗ (R+ , X). Typical examples of regularizing functions are k(t) =

kδ (t) = Note that k1/2 (t) =

1 2πi

tb−1 Γ(b)

with δ

etλ−λ dλ

1 ˆ k(λ) = b (b > 0), λ with

or δ

ˆδ (λ) = e−λ (0 < δ < 1). k

ω+iR

1 √ t−3/2 e−1/4t 2 π

(see Lemma 1.6.7).

If k is a regularizing function, then the elements of the Fr´echet space C [k] (R+ , X) are called k-generalized functions. A k-generalized function u is said to be Laplace transformable if the continuous function f := k ∗ u ∈ C∗ (R+ , X) is Laplace transformable and the Laplace transform of u is deﬁned as u ˆ(λ) :=

fˆ(λ) . ˆ k(λ)

Let H = {λ : Re λ > ω} and m : H → C be holomorphic. A meromorphic function q : H → X is said to have an m-multiplied Laplace representation if there exists f ∈ ˆ for some regularizing C∗ (R+ , X) with abs(f ) ≤ ω such that mq = fˆ on H. If m = k function k, then the meromorphic function q has a Laplace representation q = u ˆ for u = K−1 f ∈ C [k] (R+ , X) (see [B¨ au97], [BLN99], and [LN99]). Section 2.5 Theorem 2.5.1 is a standard result of Laplace transform theory. Corollary 2.5.2 is due to

2.8. NOTES

105

Pr¨ uss [Pr¨ u93], the proof given here is from [BN94]. Corollary 2.5.4 is a special case of results in [DVW02] (see also [DHW97]). Theorem 2.5.1 can be interpreted in terms of k-generalized functions and Laplace transforms (see the Notes of Section 2.4; we use the same notation here). Let q : H → X be holomorphic with supλ∈H λq(λ) < ∞. As shown in Theorem 2.5.1, for all b > 0 there ˆ exists f ∈ C∗ (R+ , X) such that q(λ) = λb fˆ(λ) on H. Thus, q(λ) = u ˆ(λ) = f (λ) , where ˆ k(λ)

1 k(t) = Γ(b) tb−1 and u = K−1 f ∈ C [k] (R+ , X) coincides with the b-th (distributional) derivative of f . More generally, if q is a meromorphic function on some half-plane H with ˆ0 (λ)q(λ) is holomorphic on H and values in X for which λ → λk

ˆ0 (λ)q(λ) < ∞ sup λk

λ∈H

for some regularizing function k0 , then it follows from Theorem 2.5.1 that there exists ˆ0 (λ)q(λ) = k(λ)q(λ) ˆ f ∈ C∗ (R+ , X) such that λ1 k = fˆ(λ) or q(λ) = u ˆ(λ), where k := [k] 1 ∗ k0 and u ∈ C (R+ , X) is a generalized function such that f = k ∗ u. Notice that if ki are regularizing functions and k1 ∗ k2 = k3 , then C [k1 ] (R+ , X) is continuously embedded in C [k3 ] (R+ , X). Thus, a faster growing q will have a less regular u such that q = u ˆ. Section 2.6 Theorem 2.6.1 is due to Sova [Sov79b] and Theorem 2.6.2 is taken from [Neu89b]. Section 2.7. In 1893, Stieltjes proved in a letter to Hermite that a bounded continuous function f : R+ → R is positive if and only if fˆ(n) (λ) ≥ 0 for all n ∈ N0 and all λ suﬃciently large (see [BB05]). Bernstein proved his theorem in 1928 [Ber28]. The characterization of those ordered Banach spaces in which Bernstein’s theorem (Theorem 2.7.7) holds is due to Arendt [Are94a]. The interpolation property is of particular interest for spaces of the form C(K), where K is a compact space. Then it can be described in terms of K: the space C(K) has the interpolation property if and only if K is an F -space (i.e., if A, B ⊂ K are open and disjoint Fσ -sets, then A ∩ B = ∅). Note that C(K) is σ-order complete if and only ¯ is open). For example, if K is quasi-stonean (i.e., if A ⊂ K is an open Fσ -set, then A ˇ K := βN \ N is a F -space which is not quasi-stonean (where βN denotes the Stone-Cech compactiﬁcation of N). Whereas every quasi-stonean space K is totally disconnected (i.e. the connected component of each point x is {x}), there exist connected compact F spaces. One reason why these spaces have been studied is that C(K) has the Grothendieck property (see Section 4.3) if K is an F -space. We refer to the article by Seever [See68] for this and further information. The interpolation property is also equivalent to two other vector-valued versions of classical theorems; namely, Riesz’s representation theorem for positive functionals on C[0, 1] and Hausdorﬀ’s theorem on the moment problem. More precisely, the following is proved in [Are94a]. Theorem 2.8.5. Let X be an ordered Banach space with normal cone. The following are equivalent: (i) X has the interpolation property. (ii) For every positive T ∈ L(C[0, 1], X) there exists an increasing function f : [0, 1] → 1 X such that T g = 0 g(t) df (t) for all g ∈ C[0, 1].

106

2. THE LAPLACE TRANSFORM

(iii) For each completely monotonic sequence 1 (xn )n∈N in X there exists an increasing function f : [0, 1] → X such that xn = 0 tn df (t) (n ∈ N). Here, a sequence x = (xn )n∈N is called completely monotonic if (−Δ)k x ≥ 0 for all k ∈ N where Δ : X N → X N is given by Δx = (xn+1 − xn )n∈N . Bernstein’s theorem in ordered Banach spaces with order continuous norm (Theorem 2.7.18) is proved in [Are87a] with the help of the classical scalar theorem. A ﬁrst vector-valued version of Bernstein’s theorem is due to Bochner [Boc42]. But Bochner considered convergence in order, whereas for our purposes norm convergence of RiemannStieltjes sums and improper integrals is essential in order to make the results applicable to operator theory. Here we deduce Bernstein’s theorem from the Real Representation Theorem 2.2.1. One can obtain Widder’s theorem (the scalar case of Theorem 2.2.1) as an easy corollary of Bernstein’s classical result (see [Wid71, Section 6.8]). However this argument is restricted to the scalar case. On the other hand, it is possible to deduce the vectorvalued version of Theorem 2.2.1 from the scalar case by a duality argument (see [Are87b] and the Notes of Section 2.2).

Chapter 3

Cauchy Problems In this chapter we study systematically well-posedness of the Cauchy problem. Given a closed operator A on a Banach space X we will see in Section 3.1 that the abstract Cauchy problem u (t) = Au(t) (t ≥ 0), u(0) = x, is mildly well-posed (i.e., for each x ∈ X there exists a unique mild solution) if and only if the resolvent of A is a Laplace transform; and this in turn is the same as saying that A generates a C0 -semigroup. Well-posedness in a weaker sense will lead to generators of integrated semigroups (Section 3.2). The real representation theorem from Section 2.2 will give us directly the characterization of generators of C0 -semigroups in terms of a resolvent estimate; namely, the Hille-Yosida theorem. When the operators are not densely deﬁned, we obtain Hille-Yosida operators which are studied in detail in Section 3.5. Also for results on approximation of semigroups in Section 3.6 we can use corresponding results on Laplace transforms from Section 1.7. Much attention is given to holomorphic semigroups which are particularly simple to characterize by means of the results of Section 2.6. We consider not only holomorphic semigroups which are strongly continuous at 0, but more general holomorphic semigroups which will be useful in applications to the heat equation with Dirichlet boundary conditions in Chapter 6. When the holomorphic semigroup exists on the right half-plane, the boundary behaviour is of special interest. If the semigroup is locally bounded, then a boundary C0 -group is obtained on the imaginary axis. This case is particularly important for fractional powers (see also the Notes of Section 3.7) and for the second order problem (Section 3.16). When the holomorphic semigroup is polynomially bounded we obtain ktimes integrated semigroups where the k depends on the degree of the polynomial. A typical example is the Gaussian semigroup. Its boundary is governed by the Schr¨ odinger operator iΔ, which we study in Section 3.9 and in Chapter 8. The last

W. Arendt et al., Vector-valued Laplace Transforms and Cauchy Problems: Second Edition, Monographs in Mathematics 96, DOI 10.1007/978-3-0348-0087-7_3, © Springer Basel AG 2011

107

108

3. CAUCHY PROBLEMS

three sections are devoted to the second order Cauchy problem; i.e., to the theory of cosine functions. A central result will be to establish a unique phase space on which the associated system is well-posed. This is a particularly interesting special case of the intermediate spaces which are constructed in Section 3.10 for integrated semigroups. In two places we will give results for UMD-spaces which are not valid in general Banach spaces: in Section 3.12 where we establish a particularly simple complex inversion formula for semigroups, and in Section 3.16 where we prove Fattorini’s remarkable theorem on the square root reduction. There is no special section on perturbation theory, but we prove perturbation results for Hille-Yosida operators, integrated semigroups and generators of cosine functions in the corresponding sections. For holomorphic semigroups we consider not only “relatively small perturbations” but also “compact perturbations” with respect to A. This chapter contains some interesting examples of holomorphic semigroups in Sections 3.7 and 3.9, but for real applications we refer to Part III. Throughout this chapter we will make extensive use (sometimes without comment) of notation, terminology and basic properties of closed operators which may be found in Appendix B. In some examples we shall use some basic notions of distributions and Sobolev spaces which may be found in Appendix E.

3.1

C0 -semigroups and Cauchy Problems

Let A be a closed operator on a Banach space X. We consider the abstract Cauchy problem u (t) = Au(t) (t ≥ 0), (ACP0 ) u(0) = x, where x ∈ X. By a classical solution of (ACP0 ) we understand a function u ∈ C 1 (R+ , X) such that u(t) ∈ D(A) for all t ≥ 0 and (ACP0 ) holds. If a classical solution exists, then it follows that x = u(0) ∈ D(A). It will be useful to ﬁnd a weaker notion of solution where x may be arbitrary. This can be done by integrating the equation. Assume that u is a classical solution. Since A is closed, it follows from Proposition 1.1.7 that t t u(s) ds ∈ D(A) and A u(s) ds = u(t) − x (t ≥ 0). (3.1) 0

0

Deﬁnition 3.1.1. A function u ∈ C(R+ , X) is called a mild solution of (ACP0 ) if (3.1) holds. The following assertion shows that mild and classical solutions diﬀer merely by regularity. Proposition 3.1.2. A mild solution u of (ACP0 ) is a classical solution if and only if u ∈ C 1 (R+ , X).

3.1. C0 -SEMIGROUPS AND CAUCHY PROBLEMS

109

Proof. Assume that u ∈ C 1 (R+ , X). Let t ≥ 0. Then 1 1 (u(t + h) − u(t)) = A h h

t+h

u(s) ds t

for all h = 0 small enough (h > 0 if t = 0). Since A is closed, it follows that 1 u(t) = lim h→0 h u (t) = Au(t).

t+h

u(s) ds ∈ D(A) and

t

Next we want to characterize mild solutions with the help of Laplace transforms. Let u ∈ C(R+ , X). Recall from (1.12) that abs(u) < ∞ if and only if

t

ωt

u(s) ds (t ≥ 0) (3.2)

≤ Me 0

for some M, ω ≥ 0. As before, we denote by ∞ u ˆ(λ) := e−λt u(t) dt (λ > ω) 0

the Laplace transform of u. Theorem 3.1.3. Let u ∈ C(R+ , X) such that (3.2) holds. Then the following assertions are equivalent: (i) u is a mild solution of (ACP0 ). (ii) u ˆ(λ) ∈ D(A) and λˆ u(λ) − Aˆ u(λ) = x for all λ > ω. Proof. (i) ⇒ (ii): Let u be a mild solution. Let λ > ω. We know from (1.11) that ∞ t −λt u ˆ(λ) = λ e u(s) ds dt. 0

0

Since A is closed, it follows from Proposition 1.6.3 that u ˆ(λ) ∈ D(A) and ∞ t Aˆ u(λ) = λ e−λt A u(s) ds dt 0 0 ∞ = λ e−λt (u(t) − x) dt 0

= (ii) ⇒ (i): Let v(t) :=

t 0

λˆ u(λ) − x.

u(s) ds. Then by (1.11), vˆ(λ) = u ˆ(λ)/λ ∈ D(A) and

Aˆ v (λ) = Aˆ u(λ)/λ = u ˆ(λ) − x/λ = fˆ(λ)

(λ > ω),

110

3. CAUCHY PROBLEMS

where f (t) := u(t) − x (t ≥ 0). It follows from Proposition 1.7.6 that v(t) ∈ D(A) and Av(t) = f (t) = u(t) − x for all t ≥ 0; i.e., u is a mild solution of (ACP0 ). Let u be a mild solution of (ACP0 ) satisfying (3.2). Assume that ω < λ ∈ ρ(A). Then it follows from Theorem 3.1.3 that u ˆ(λ) = R(λ, A)x. Thus the Laplace transform of a mild solution is the resolvent applied to the initial value. This leads us to consider operators whose resolvent exists on a half-line and is a Laplace transform. For this, let T : R+ → L(X) be strongly continuous. Recall from Proposition 1.4.5 that abs(T ) < ∞ if and only if

t

ωt

T (s)x ds (t ≥ 0, x ∈ X) (3.3)

≤ M e x

0

for some ω ≥ 0, M ≥ 0. In that case, abs(T ) ≤ ω and t ˆ T (λ)x := lim e−λs T (s)x ds t→∞

0

deﬁnes a bounded operator Tˆ (λ) ∈ L(X) whenever Re λ > abs(T ). Moreover, Tˆ : {Re λ > abs(T )} → L(X) is holomorphic (see Section 1.5). As in Sections 1.4 and 1.5, we denote by ω(T ) the exponential growth bound of T , and by hol(Tˆ ) the abscissa of holomorphy of Tˆ . Recall that hol(Tˆ) ≤ abs(T ) ≤ ω(T ). We will consider Laplace transforms of operator-valued functions on many occasions in this and subsequent chapters. The following deﬁnition will be helpful. Deﬁnition 3.1.4. Let λ0 ∈ R and let R : (λ0 , ∞) → L(X) be a function. We say that R is a Laplace transform if there exists a strongly continuous function T : R+ → L(X) such that abs(T ) ≤ λ0 and R(λ) = Tˆ (λ)

(λ > λ0 ).

The following proposition is a simple consequence of the uniqueness theorem. Proposition 3.1.5. Let T : R+ → L(X) be strongly continuous such that abs(T ) < ∞. Let ω > abs(T ). Then the following hold: a) If B ∈ L(X) such that B Tˆ(λ) = Tˆ (λ)B for all λ > ω, then BT (t) = T (t)B for all t ≥ 0. b) In particular, if Tˆ(μ)Tˆ (λ) = Tˆ (λ)Tˆ(μ) for all λ, μ > 0, then T (t)T (s) = T (s)T (t) for all t, s ≥ 0. Proof. a) For x ∈ X and λ > ω, one has ∞ e−λt T (t)Bx dt = Tˆ (λ)Bx = B Tˆ (λ)x = 0

0

∞

e−λt BT (t)x dt.

3.1. C0 -SEMIGROUPS AND CAUCHY PROBLEMS

111

It follows from the uniqueness theorem that T (t)Bx = BT (t)x for all t ≥ 0. b) Let μ > ω. It follows from a) that Tˆ(μ)T (t) = T (t)Tˆ (μ) for all t ≥ 0. Fixing t ≥ 0 and applying a) to B := T (t) shows that T (s)T (t) = T (t)T (s) for all s ≥ 0. Now we introduce C0 -semigroups. Deﬁnition 3.1.6. A C0 -semigroup is a strongly continuous function T : R+ → L(X) such that T (t + s) T (0)

= T (t)T (s)

(t, s ≥ 0),

= I.

In the next theorem we show that C0 -semigroups are exactly those strongly continuous operator-valued functions whose Laplace transforms are resolvents. It is remarkable that C0 -semigroups are automatically exponentially bounded. Theorem 3.1.7. Let T : R+ → L(X) be a strongly continuous function. The following assertions are equivalent: (i) abs(T ) < ∞ and there exists an operator A such that (λ0 , ∞) ⊂ ρ(A) and Tˆ (λ) = R(λ, A)

(λ > λ0 )

for some λ0 > abs(T ). (ii) T is a C0 -semigroup. In that case, ω(T ) < ∞, {Re λ > hol(Tˆ)} ⊂ ρ(A) and Tˆ(λ) = R(λ, A) whenever Re λ > hol(Tˆ). Proof. a) Let T be a C0 -semigroup. We show ﬁrst that ω(T ) < ∞. Let M := sup0≤t≤1 T (t) . Then M < ∞ by the uniform boundedness principle. Let ω = log M . Let t ∈ R+ . Take n ∈ N0 and s ∈ [0, 1) such that t = n + s. Then

T (t) = T (s)T (1)n ≤ M M n = M eωn ≤ M eωt . b) Assume that abs(T ) < ∞. Let μ > λ > abs(T ). Then integration by parts yields for x ∈ X, ∞ ∞ Tˆ(λ)x − Tˆ(μ)x 1 = e(λ−μ)t Tˆ (λ)x dt − e(λ−μ)t e−λt T (t)x dt μ−λ μ − λ 0 0 ∞ ∞ (λ−μ)t −λs = e e T (s)x ds dt 0 0 ∞ t − e(λ−μ)t e−λs T (s)x ds dt ∞0 ∞0 (λ−μ)t = e e−λs T (s)x ds dt 0

t

112

3. CAUCHY PROBLEMS

∞

=

e 0

=

∞

−μt

∞

∞

t

e−μt

0

e−λ(s−t) T (s)x ds dt e−λs T (s + t)x ds dt.

0

On the other hand, Tˆ (μ)Tˆ(λ)x = 0

∞

e−μt

∞

e−λs T (s)T (t)x ds dt.

0

So it follows from the uniqueness theorem (Theorem 1.7.3) that (Tˆ (λ))λ>abs(T ) is a pseudo-resolvent (see Appendix B) if and only if T satisﬁes T (s + t) = T (s)T (t) (s, t ≥ 0). Now assume that Tˆ is a pseudo-resolvent. Then T (0) is a projection. Moreover, T (0)x = 0 if and only if T (t)x = T (t)T (0)x = 0 for all t ≥ 0. Thus by the uniqueness theorem, T (0)x = 0 if and only if Tˆ (λ)x = 0 (λ > ω(T )). By Proposition B.6, (Tˆ(λ))λ>abs(T ) is a resolvent if and only if T (0) = I. This proves that (i) ⇔ (ii). c) It follows from (i) and Proposition B.5 that {Re λ > hol(Tˆ )} ⊂ ρ(A) and ˆ T (λ) = R(λ, A) whenever Re λ > hol(Tˆ). Deﬁnition 3.1.8. Let T be a C0 -semigroup. The generator of T is deﬁned as the operator A on X such that (ω(T ), ∞) ⊂ ρ(A) and Tˆ(λ) = R(λ, A) for all λ > ω(T ). Thus, an operator A is the generator of a C0 -semigroup if and only if its resolvent is a Laplace transform in the sense of Deﬁniton 3.1.4. In the following proposition we collect the diverse relations of a C0 -semigroup and its generator. These properties will be used frequently without further reference. Proposition 3.1.9. Let T be a C0 -semigroup on X and let A be its generator. Then the following properties hold: a) limλ→∞ λR(λ, A)x = x for all x ∈ X; in particular, A is densely deﬁned. b) For all x ∈ X, the function ux (t) := T (t)x is a mild solution of (ACP0 ). c) R(λ, A)T (t) = T (t)R(λ, A) for all λ ∈ ρ(A) and t ≥ 0. d) x ∈ D(A) implies T (t)x ∈ D(A) and AT (t)x = T (t)Ax. t t e) 0 T (s)x ds ∈ D(A) and A 0 T (s)x ds = T (t)x − x for all x ∈ X and t ≥ 0. t f) Let x, y ∈ X. Then x ∈ D(A) and Ax = y if and only if 0 T (s)y ds = T (t)x − x for all t ≥ 0. g) Let x ∈ X. Then x ∈ D(A) if and only if y = limt↓0 1t (T (t)x − x) exists. In that case, Ax = y.

3.1. C0 -SEMIGROUPS AND CAUCHY PROBLEMS

113

h) T (·)x is a classical solution of (ACP0 ) if and only if x ∈ D(A). i) If λ ∈ C then (eλt T (t))t≥0 is a C0 -semigroup and A + λ is its generator. j) Let x ∈ X and λ ∈ C. Then x ∈ D(A) and Ax = λx if and only if T (t)x = eλt x for all t ≥ 0. Proof. a) follows from the following Abelian argument. There exist M ≥ 0 and ω ∈ R such that

T (t) ≤ M eωt (t ≥ 0). Let ε > 0 and x ∈ X. There exists τ > 0 such that T (t)x−x ≤ ε for all t ∈ [0, τ ]. Therefore lim sup λR(λ, A)x − x

λ→∞

∞

−λt

≤ lim sup λ e (T (t)x − x) dt

λ→∞ 0 τ ∞ −λt −λt ωt ≤ lim sup λ e ε dt + λ e (M e + 1) dt x

λ→∞

0

τ

= ε.

b) By Theorem 3.1.3, a function u ∈ C(R+ , X) is a mild solution if and only if u ˆ(λ) = R(λ, A)x = Tˆ (λ)x for all λ > ω(T ). So the claim follows from the uniqueness theorem. c) follows from Proposition 3.1.5. d) follows from c) (by Proposition B.7). e) follows from b). t f) Let x ∈ D(A). Then by d), T (s)Ax = AT (s)x. Hence by e), 0 T (s)Ax ds t t = A 0 T (s)x ds = T (t)x − x. Conversely, let x, y ∈ X such that 0 T (s)y ds = T (t)x − x for all t ≥ 0. Then ∞ t −λt R(λ, A)y = λ e T (s)y ds dt 0 0 ∞ = λ e−λt (T (t)x − x) dt 0

= λR(λ, A)x − x. Thus x ∈ D(A) and y = λx − (λ − A)x = Ax. t g) Let x ∈ D(A). Then by f), 1t (T (t)x − x) = 1t 0 T (s)Ax ds → Ax as t → 0. t Conversely, let x, y ∈ X such that y = limt↓0 1t (T (t)x − x) = limt↓0 1t A 0 T (s)x ds. Since A is closed, it follows that x ∈ D(A) and Ax = y. t h) Let x ∈ D(A). Then T (t)x = x + 0 T (s)Ax ds by e). Thus, T (·)x ∈ C 1 (R+ , X) and the claim follows from Proposition 3.1.2. Conversely, if T (·)x is a classical solution, then x = T (0)x ∈ D(A) by deﬁnition.

114

3. CAUCHY PROBLEMS

i) follows from Theorem 3.1.7. j) Replacing A by A − λ we may assume that λ = 0. Now the claim follows from f). Property g) is sometimes expressed by saying that A is the inﬁnitesimal generator of T . Since almost all the C0 -semigroups which arise naturally from diﬀerential operators cannot be written down explicitly, we do not give examples in this section. However the reader who wishes to see explicit examples may look already to Examples 3.3.10, 3.4.8, 3.7.5, 3.7.6 and 3.7.9, and to various examples in Chapter 5. We note here that if T : R+ → L(X) satisﬁes T (t + s) = T (t)T (s) (t, s ≥ 0) and limt↓0 T (t)x − x = 0 (x ∈ X), then T is a C0 -semigroup. To see this, we have to show that T is strongly continuous at t > 0. Right-continuity follows immediately from the estimate T (t + h)x − T (t)x ≤ T (t) T (h)x − x . For left-continuity, note that the assumptions imply that there exist M > 0 and δ > 0 such that T (h) ≤ M whenever 0 < h < δ (otherwise, there exist tn ↓ 0 such that T (tn ) → ∞ and, by the uniform boundedness theorem, there exists x ∈ X such that (T (tn )x) is unbounded, which is a contradiction). Hence for 0 < h < δ, we have (T (t − h)x − T (t)x ≤ T (δ − h) T (t − δ) x − T (h)x → 0 as h ↓ 0. Since T (0)T (t)x = T (t)x, letting t ↓ 0 shows that T (0) = I. The following result characterizes C0 -semigroups which are norm-continuous on R+ . It also describes the situation when the generator A of a C0 -semigroup T is bounded. Since A is closed, this is equivalent to saying that D(A) = X. Theorem 3.1.10. Let A be the generator of a C0 -semigroup T . The following assertions are equivalent: (i) The operator A is bounded; i.e., D(A) = X. (ii) limt↓0 T (t) − I = 0. In that case, T (t) = etA :=

∞ k=0

tk Ak k!

(t ≥ 0).

∞ tk Ak Proof. (i) ⇒ (ii): Assume that A is bounded. Then clearly, T (t) := k=0 k! deﬁnes a continuous mapping T : R+ → L(X) such that T (0) = I and T (t) ≤ et A . Let λ > A . Then ∞ ∞ ∞ Ak ∞ −λt k e−λt T (t) dt = e t dt = Ak λ−(k+1) = R(λ, A). k! 0 0 k=0

k=0

Thus, T is a C0 -semigroup and A is its generator by Deﬁnition 3.1.8. (ii) ⇒ (i): It follows from Proposition 4.1.3 or direct computation as in the proof of Proposition 3.1.9 a) that limλ→∞ λR(λ, A) − I = 0. Thus, there exists λ > ω(T ) such that λR(λ, A) − I < 1/2. This implies that λR(λ, A) is invertible in L(X). In particular, D(A) = λR(λ, A)X = X. Now we consider uniqueness of mild solutions of (ACP )0 .

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115

Proposition 3.1.11. Let T be a C0 -semigroup and A be its generator. Let τ > t 0, x ∈ X. Let u ∈ C([0, τ ], X) such that 0 u(s) ds ∈ D(A) and

t

u(s) ds = u(t) − x

A 0

for all t ∈ [0, τ ]. Then u(t) = T (t)x. t Proof. Let v(t) = 0 (u(s) − T (s)x) ds. Then by hypothesis and by Proposition 3.1.9 e), v(t) ∈ D(A) (0 ≤ t ≤ τ ). Moreover, v (t) = Av(t) (0 ≤ t ≤ τ ) and t v(0) = 0. We show that v ≡ 0. Let S(t)y := 0 T (s)y ds. Then S(t)y ∈ D(A) and AS(t)y = T (t)y − y for all y ∈ X, by Proposition 3.1.9 e). Let 0 < t ≤ τ, w(s) := S(t − s)v(s), 0 ≤ s ≤ t. Then w (s)

= −T (t − s)v(s) + S(t − s)v (s) = −T (t − s)v(s) + S(t − s)Av(s) = −T (t − s)v(s) + AS(t − s)v(s) = −v(s).

Since w(t) = w(0) = 0, we conclude that 0 = w(t) = 0

t

w (s) ds = −

t

v(s) ds. 0

Since t ∈ (0, τ ] is arbitrary, it follows that v(s) = 0 for s ∈ [0, τ ]. Proposition 3.1.9 and Proposition 3.1.11 show in particular that the abstract Cauchy problem (ACP0 ) is well-posed (in the sense of mild solutions) whenever the operator A generates a C0 -semigroup T . Moreover, the orbits are given by T (·)x where x is the initial value. Now we show the converse assertion. If (ACP0 ) is mildly well-posed (i.e., for each x there exists a unique mild solution), then the operator generates a C0 -semigroup. More precisely, we have the following result. Theorem 3.1.12. Let A be a closed operator. The following assertions are equivalent: (i) For all x ∈ X there exists a unique mild solution of (ACP0 ). (ii) The operator A generates a C0 -semigroup. (iii) ρ(A) = ∅ and for all x ∈ D(A) there exists a unique classical solution of (ACP0 ). When these assertions hold, the mild solution of (ACP0 ) is given by u(t) = T (t)x.

116

3. CAUCHY PROBLEMS

Proof. (i) ⇒ (ii): Let ux be the mild solution for the initial value x ∈ X. It follows from uniqueness that ux (t) is linear in x. So for each t ≥ 0 there exists a linear mapping T (t) : X → X such that T (t)x = ux (t) for all x ∈ X. We show that T (t) is continuous. Denote by Φ : X → C(R+ , X) the mapping Φ(x) = ux . Note that C(R+ , X) is a Fr´echet space for the topology of uniform convergence on intervals of the form [0, τ ] where τ > 0. The mapping Φ is linear. We show that Φ has a closed graph. In fact, let xn → x in X and uxn → u in C(R+ , X). Let t > 0. Then t t t u (s) ds converges to 0 u(s) ds as n → ∞. Since A 0 uxn (s) ds = uxn (t) − 0 xn t t xn and since A is closed, it follows that 0 u(s) ds ∈ D(A) and A 0 u(s) ds = limn→∞ uxn (t) − xn = u(t) − x. Thus u(t) = T (t)x; i.e., u = Φ(x). It follows from the closed graph theorem that Φ is continuous. This implies that T (t) ∈ L(X) for all t ≥ 0. Let u be a mild solution of (ACP0 ) with initial value x. Then it is easy to see that u(· + s) is a mild solution for the initial value u(s). So uniqueness implies that T (t + s)x = T (t)T (s)x. We have shown that T is a C0 -semigroup. Let B be the generator of T . Then R(λ, B) = Tˆ(λ) (λ > ω(T )). On the other hand, by Theorem 3.1.3, Tˆ(λ)x ∈ D(A) and (λ − A)Tˆ (λ)x = x for all x ∈ X and λ > ω(T ). Thus, D(B) ⊂ D(A) and (λ − A)R(λ, B)x = x (x ∈ X) if λ > ω(T ). If we show that (λ − A) is injective, then it follows that λ ∈ ρ(A) and R(λ, A) = R(λ, B). Thus, A = B. Assume that λ > ω(T ) and let x ∈ D(A) such that (λ − A)x = 0. Then u(t) := eλt x is a mild solution. Thus, T (t)x = eλt x. Since ω(T ) < λ, it follows that x = 0. (ii) ⇒ (iii) follows from Proposition 3.1.9 b) and Proposition 3.1.11. (iii) ⇒ (i): Let λ ∈ ρ(A). Let x ∈ X. There exists a classical solution v of (ACP0 ) with initial value R(λ, A)x. It is easy to check that u(t) := (λ − A)v(t) deﬁnes a mild solution of (ACP0 ) with initial value x. This shows existence. In order toshow uniqueness, let u be a mild solution for the initial value x = 0. Then t v(t) := 0 u(s) ds deﬁnes a classical solution for the initial value 0. Hence v(t) = 0 for all t ≥ 0 by assumption. It follows that u(t) = 0 (t ≥ 0). As a corollary of Theorem 3.1.12 we show that one can also characterize C0 semigroups and their generator by property e) of Proposition 3.1.9. This will be useful later. Corollary 3.1.13. Let A be a closed operator on X and T : R+ → L(X) be strongly t continuous such that 0 T (s)x ds ∈ D(A) and

t

A 0

T (s)x ds = T (t)x − x

for all x ∈ X, t ≥ 0. Assume that T (t)x ∈ D(A) and AT (t)x = T (t)Ax for all x ∈ D(A), t ≥ 0. Then T is a C0 -semigroup and A is its generator. Proof. Let x ∈ X. Then u(t) := T (t)x deﬁnes a mild solution of (ACP0 ). As in the

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117

proof of Proposition 3.1.11, u is the unique mild solution. Now the claim follows from Theorem 3.1.12. We show by an example that the condition that A has non-empty resolvent set cannot be omitted in assertion (iii) of Theorem 3.1.12; i.e., it can happen that the abstract Cauchy problem (ACP0 ) is well-posed in the sense of classical solutions without A being the generator of a C0 -semigroup. Example 3.1.14. Let B be a densely deﬁned closed operator on a Banach space Y such that D(B) = Y . Consider the operator A on X := Y × Y given by 0 B A= 0 0 with domain Y × D(B). Then A is closed and densely deﬁned. Moreover, for all (x, y) ∈ D(A), u(t) = (x + tBy, y) (t ≥ 0) is the unique classical solution of (ACP0 ). However, there does not exist a mild solution for an initial value (x, y) if y ∈ Y \ D(B). This is easy to see. Now we consider the inhomogeneous Cauchy problem. In contrast to the homogeneous case, we consider this on a bounded interval [0, τ ] where τ ∈ (0, ∞), but results on R+ can be deduced by letting τ vary. We shall apply Laplace transform techniques to inhomogeneous Cauchy problems on R+ in Section 5.6. Let A be a closed operator and let f ∈ L1 ([0, τ ], X) where τ > 0. We consider the inhomogeneous Cauchy problem u (t) = Au(t) + f (t) (t ∈ [0, τ ]), (ACPf ) u(0) = x, where x ∈ X. A function u ∈ C([0, τ ], X) is called a mild solution of (ACPf ) if t u(s) ds ∈ D(A) and 0 u(t) = x + A

t

t

u(s) ds + 0

0

f (s) ds (t ∈ [0, τ ]).

Assume that f ∈ C([0, τ ], X). Then we deﬁne a classical solution as a function u ∈ C 1 ([0, τ ], X) such that u(t) ∈ D(A) for all t ∈ [0, τ ] and such that (ACPf ) is valid. Note that in that case, since Au(t) = u (t) − f (t) (t ∈ [0, τ ]), one has u ∈ C([0, τ ], D(A)), where D(A) is seen as a Banach space with the graph norm. Since A is closed, the proof of Proposition 3.1.2 is also valid in the inhomogeneous case, so the following holds. Proposition 3.1.15. Let f ∈ C([0, τ ], X) and u ∈ C([0, τ ], X) be a mild solution of (ACPf ). Then u is a classical solution if and only if u ∈ C 1 ([0, τ ], X).

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3. CAUCHY PROBLEMS

In the case when A generates a C0 -semigroup there always exists a mild solution. Proposition 3.1.16. Let A be the generator of a C0 -semigroup T on X. Then for every f ∈ L1 ([0, τ ], X) the problem (ACPf ) has a unique mild solution u given by

t

u(t) = T (t)x + 0

T (t − s)f (s) ds

(t ∈ [0, τ ]).

(3.4)

Sometimes, (3.4) is called the variation of constants formula for the solution. Proof. Uniqueness: Let u1 , u2 ∈ C([0, τ ], X) be two mild solutions of (ACPf ). Then t u := u1 − u2 ∈ C([0, τ ], X), u(0) = 0 and A 0 u(s) ds = u(t) for all t ∈ [0, τ ]. It follows from Proposition 3.1.11 that u ≡ 0. Existence: We have seen that T (·)x is a mild solution of the homogeneous t Cauchy problem. It remains to show that v(t) := 0 T (t − s)f (s) ds is a mild solution of (ACPf ) with initial value x = 0. Extending f by 0 to R+ , Proposition 1.3.4 shows that v ∈ C([0, τ ], X). Using Proposition 3.1.9 e) and Fubini’s theorem we obtain t t s A v(s) ds = A T (s − r)f (r) dr ds 0 0 0 t t = A T (s − r)f (r) ds dr

0

t

=

r t−r

A 0

T (s)f (r) ds dr 0

t

(T (t − r)f (r) − f (r)) dr t = v(t) − f (r) dr. =

0

0

This proves the claim. Corollary 3.1.17. Let A be the generator of a C0 -semigroup. Let x ∈ D(A), f0 ∈ t X, f (t) = f0 + 0 f (s) ds (t ∈ [0, τ ]) for some function f ∈ L1 ([0, τ ], X). Then the function u deﬁned by (3.4) is a classical solution of (ACPf ). This follows from Proposition 3.1.16, Proposition 1.3.6 and Proposition 3.1.15. This result will later be extended to a class of operators which are not densely deﬁned (Theorem 3.5.2). Finally, given a closed operator A on X, we consider the Cauchy problem on the line u (t) = Au(t) (t ∈ R), ACP0 (R) u(0) = x,

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119

where x ∈ X. A function u ∈ C(R, X) is called a mild solution of ACP0 (R) if t u(s) ds ∈ D(A) and 0 t A u(s) ds = u(t) − x for all t ∈ R. 0

Proposition 3.1.18. Assume that A is an operator such that A generates a C0 semigroup T+ and −A generates a C0 -semigroup T− . Deﬁne T+ (t) if t ≥ 0, U (t) = (3.5) T− (−t) if t < 0. Then U : R → L(X) is strongly continuous, U (0) = I and U (t + s) = U (t)U (s) (t, s ∈ R). Proof. Note ﬁrst that T+ (t)T− (s) = T− (s)T+ (t) for s, t ≥ 0, by Proposition 3.1.5. The only assertion which is not obvious is to show that U (t − s) = U (t)U (−s) if t ≥ 0, s ≥ 0. We can assume that 0 ≤ s ≤ t (replacing A by −A for the other case). Let x ∈ X, t ≥ 0, v(s) := T+ (t − s)x for s ∈ [0, t]. Then for 0 ≤ r ≤ t, r r t −A v(s) ds = −A T+ (t − s)x ds = −A T+ (s)x ds 0

0

t−r

= T+ (t − r)x − T+ (t)x = v(r) − T+ (t)x. Thus, v is a mild solution of the problem v (s) = −Av(s) v(0) = T+ (t)x.

(0 < s ≤ t),

It follows from Proposition 3.1.11 that v(s) = T− (s)T+ (t)x. Hence, U (t − s)x = T+ (t − s)x = v(s) = T− (s)T+ (t)x = U (−s)U (t)x. Deﬁnition 3.1.19. An operator A on X is said to generate a C0 -group if A and −A generate C0 -semigroups. In that case, the function U : R → L(X) deﬁned by (3.5) is called the C0 -group generated by A. Proposition 3.1.20. A closed operator A generates a C0 -group if and only if for every x ∈ X there exists a unique mild solution u of ACP0 (R). In that case, u(t) = U (t)x (t ∈ R), where U is the C0 -group generated by A. If x ∈ D(A), then d U (·)x ∈ C 1 (R, X), U (t)x ∈ D(A) for all t ∈ R and dt U (t)x = AU (t)x (t ∈ R). Proof. Assume that ACP0 (R) is mildly well-posed; i.e., for all x ∈ X there exists a unique mild solution of ACP0 (R). Then it is clear that for each x ∈ X there exists a mild solution of u (t) = ±Au(t) (t ≥ 0), (CP )± u(0) = x.

120

3. CAUCHY PROBLEMS

The solutions of (CP )+ and (CP )− are both unique. In fact, let u ∈ C(R+ , X) be a mild solution of (CP )+ with initial value u(0) = 0. Then extending u by 0 on (−∞, 0) one obtains a mild solution of ACP0 (R). Hence u ≡ 0 by assumption. The same argument is valid for (CP )− . Now it follows from Theorem 3.1.12 that A and −A are both generators of C0 -semigroups. Conversely, if A generates a C0 -group, then it is easy to see that U (·)x is a mild solution of ACP0 (R). The remaining properties follow directly from the corresponding results for semigroups. If A generates a C0 -semigroup T , then mild solutions of ACP0 (R) can be described diﬀerently. Deﬁnition 3.1.21. A function u ∈ C(R, X) is called a complete orbit of T if u(t + s) = T (t)u(s)

for all t ≥ 0, s ∈ R.

Proposition 3.1.22. Let A be the generator of a C0 -semigroup T and let u ∈ C(R, X), x = u(0). Then u is a mild solution of ACP0 (R) if and only if u is a complete orbit. Proof. Assume that u is a complete orbit. Let x = u(0). Since T (t)x = u(t) for t ≥ 0, we have t A u(s) ds = u(t) − x (t ≥ 0). 0

For t < 0 we have T (−t)u(t) = u(0) = x and t 0 A u(r) dr = A u(r + t) dr = −A 0

−t

= −A

0

−t

−t

u(r + t) dr 0

T (r)u(t) dr = u(t) − T (−t)u(t) = u(t) − x.

Thus u is a mild solution of ACP0 (R). Conversely, assume that u is a mild solution of ACP0 (R) with x = u(0). Let s ∈ R. Then for t ≥ 0, t s+t A u(r + s) dr = A u(r) dr = u(t + s) − u(s). 0

s

Thus u(· + s) is a mild solution of (ACP0 ) for x = u(s). Hence u(t + s) = T (t)u(s) for t ≥ 0. If a C0 -semigroup T extends to a C0 -group, then T (t) is invertible for all t > 0. The following converse statement is sometimes useful (and will be needed in Proposition 4.7.2, for example). Proposition 3.1.23. Let A be the generator of a C0 -semigroup T . If there exists t0 > 0 such that T (t0 ) is invertible, then A generates a C0 -group.

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121

Proof. a) Let t > 0. We show that T (t) is invertible. Let T (t)x = 0. Choose n ∈ N such that nt0 > t. Then T (nt0 )x = T (nt0 − t)T (t)x = 0. Since T (nt0 ) = T (t0 )n is invertible, it follows that x = 0. Thus, T (t) is injective. Let y ∈ X. Let x := T (nt0 − t)T (nt0 )−1 y. Then T (t)x = y. Thus, T (t) is surjective. b) Deﬁne U (t) := T (t) for t ≥ 0 and U (t) := T (−t)−1 for t < 0. Then U : R → L(X) satisﬁes U (t + s) = U (t)U (s) for all t, s ∈ R. Let x ∈ X, t0 ∈ R. Let t1 > max{−t0 , 0}. Then lim U (t)x = T (t1 )−1 lim T (t + t1 )x = U (t0 )x.

t→t0

t→t0

Thus U is strongly continuous. c) We show that the generator of (U (−t))t≥0 is −A. Let x ∈ D(A). Then 1 1 lim (U (−t)x − x) = lim U (−1) T (1 − t)x − T (1)x t↓0 t t↓0 t = T (1)−1 (−AT (1)x) = −Ax. Conversely, if x ∈ X such that y := limt↓0 1t (U (−t)x − x) exists, it follows as above that −y = limt↓0 1t (T (t)x − x). Thus, x ∈ D(A) and −Ax = y. Now the claim follows from Proposition 3.1.9.

3.2

Integrated Semigroups and Cauchy Problems

Let T be a C0 -semigroup on a Banach space X with generator A. For k ∈ N we deﬁne S : R+ → L(X) by

t

S(t)x := 0

(t − s)k−1 T (s)x ds (k − 1)!

(t ≥ 0, x ∈ X).

By Theorem 3.1.7, there exist M, ω ≥ 0 such that T (t) ≤ M eωt for all t ≥ 0. Taking Laplace transforms and integrating by parts yields ∞ R(λ, A) = λk e−λt S(t) dt (λ > ω). (3.6) 0

Here, the Laplace integral is understood in the sense of Section 1.4, but, for each ∞ x ∈ X and λ > ω, one has R(λ, A)x = λk 0 e−λt S(t)x dt as an absolutely convergent Bochner integral. The above formula (3.6) is the basic idea behind the following deﬁnition. Consider an arbitrary strongly continuous function S : R+ → L(X). We recall from Proposition 1.4.5 that abs(S) < ∞ if and only if there exist constants M, ω ≥ 0 such that

t

ωt

S(s) ds (t ≥ 0). (3.7)

≤ Me 0

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3. CAUCHY PROBLEMS

In that case, the Laplace integral ∞ ˆ S(λ)x := e−λt S(t)x dt := lim τ →∞

0

τ

e−λt S(t)x dt

0

exists for all λ ∈ C with Re λ > ω and all x ∈ X and deﬁnes a bounded operator ˆ S(λ) on X. Hence, the following deﬁnition is meaningful. Deﬁnition 3.2.1. Let A be an operator on a Banach space X and k ∈ N0 . We call A the generator of a k-times integrated semigroup if there exist ω ≥ 0 and a strongly continuous function S : R+ → L(X) such that abs(S) ≤ ω, (ω, ∞) ⊂ ρ(A) and ∞ R(λ, A) = λk e−λt S(t) dt (λ > ω). (3.8) 0

In this case, S is called the k-times integrated semigroup generated by A. If k = 1 we also use the notion once integrated semigroup. By Theorem 3.1.7, a 0-times integrated semigroup is the same as a C0 semigroup. The discussion above shows that if A generates a 0-times integrated semigroup, then A generates a k-times integrated semigroup for every k ∈ N. The same argument shows that if A generates a k-times integrated semigroup, then A generates an n-times integrated semigroup for every n > k. As in the situation of C0 -semigroups we collect diverse relations of an integrated semigroup and its generator. Lemma 3.2.2. Let k ∈ N and let S be a k-times integrated semigroup on X with generator A. Then the following hold: a) R(μ, A)S(t) = S(t)R(μ, A)

(t ≥ 0, μ ∈ ρ(A)).

b) If x ∈ D(A), then S(t)x ∈ D(A) and AS(t)x = S(t)Ax for all t ≥ 0. c) Let x ∈ D(A) and t ≥ 0. Then t tk S(s)Ax ds = S(t)x − x. k! 0 In particular,

d dt

(S(t)x) = S(t)Ax +

tk−1 (k−1)! x.

t d) Let x ∈ X and t ≥ 0. Then 0 S(s)x ds ∈ D(A) and t tk A S(s)x ds = S(t)x − x. k! 0 In particular, S(0) = 0. t k e) Let x, y ∈ X such that 0 S(s)y ds = S(t)x− tk! x for all t ≥ 0. Then x ∈ D(A) and Ax = y.

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123

Proof. By deﬁnition and assertion (3.7), there exist constants M, ω ≥ 0 such that t (ω, ∞) ⊂ ρ(A) and || 0 S(s) ds|| ≤ M eωt for t ≥ 0. In the following let λ > ω. a) follows from Proposition 3.1.5. b) is implied by a) (by Proposition B.7). c) Let x ∈ D(A). Integrating by parts yields ∞ tk λk+1 e−λt x dt = R(λ, A)(λ − A)x k! 0 ∞ ∞ k+1 −λt k = λ e S(t)x dt − λ e−λt S(t)Ax dt 0 0 ∞ k+1 = λ e−λt S(t)x dt 0 ∞ t k+1 −λt −λ e S(s)Ax ds dt. 0

0

The uniqueness theorem implies the assertion. d) Let μ ∈ ρ(A) and x ∈ X. By a) and c) we have t t tk S(s)x ds = μR(μ, A) S(s)x ds − R(μ, A)S(t)x + R(μ, A)x. k! 0 0 t Hence 0 S(s)x ds ∈ D(A) and t t tk (μ − A) S(s)x ds = μ S(s)x ds − S(t)x + x. k! 0 0 t k e) Let x, y ∈ X such that 0 S(s)y ds = S(t)x − tk! x for all t ≥ 0. Then ∞ ∞ k+1 −λt k R(λ, A)(λx − y) = λ e S(t)x dt − λ e−λt S(t)y dt 0 0 ∞ ∞ t = λk+1 e−λt S(t)x dt − λk+1 e−λt S(s)y ds dt = x.

0

0

0

Hence x ∈ D(A) and λx − y = λx − Ax, which implies that Ax = y. Remark 3.2.3. Observe that in contrast to the situation of C0 -semigroups, generators of k-times integrated semigroups for k ≥ 1 need not be densely deﬁned. However, assertion d) of Lemma 3.2.2 implies that S(t)x ∈ D(A) for t ≥ 0 and x ∈ X. We saw in Theorem 3.1.7 that C0 -semigroups are precisely those operatorvalued functions whose Laplace transforms are resolvents R(λ, A) of operators A. By deﬁnition, k-times integrated semigroups are exactly those operator-valued functions whose Laplace transforms are λ−k R(λ, A) for operators A. In the following proposition we show that this property corresponds to the functional equation (3.9) for S.

124

3. CAUCHY PROBLEMS

Proposition 3.2.4. Let S : R+ → L(X) be a strongly continuous function satisfying t

0 S(s) ds ≤ M eωt (t ≥ 0) for some M, ω ≥ 0. Let k ∈ N. For λ > ω set ∞ k R(λ) := λ e−λt S(t) dt. 0

Then the following assertions are equivalent: (i) There exists an operator A such that (ω, ∞) ⊂ ρ(A) and R(λ) = (λ − A)−1 for λ > ω. (ii) For s, t ≥ 0, S(t)S(s)

=

1 (k − 1)!

)

t+s

(s + t − r)k−1 S(r) dr * s k−1 − (s + t − r) S(r) dr , t

(3.9)

0

and S(t)x = 0 for all t ≥ 0 implies x = 0. Proof. We ﬁrst claim that {R(λ) : λ > ω} is a pseudo-resolvent if and only if (3.9) holds. Since ∞ ∞ R(λ) R(μ) −λt = e e−μs S(t)S(s) ds dt (λ, μ > ω), λk μk 0 0 the claim follows from the uniqueness theorem (Theorem 1.7.3) provided we are able to prove that 1 1 1 (R(λ) − R(μ)) (3.10) μ − λ λk μk equals the term ∞ ∞ s+t 1 e−λt e−μs (s + t − r)k−1 S(r) dr ds dt (k − 1)! 0 t ∞0 ∞ s 1 −λt −μs − e e (s + t − r)k−1 S(r) dr ds dt. (3.11) (k − 1)! 0 0 0 Notice that (3.10) equals R(λ) R(μ) 1 1 1 1 1 R(μ) − + − =: I + II. μk μ − λ λk μk μ − λ μk λk μk As in the proof of Theorem 3.1.7 we see that term I equals ∞ ∞ 1 −μt e e−λs S(t + s) ds dt μk 0 0 ∞ ∞ t+s (t + s − r)k−1 = e−λt e−μs S(r) dr ds dt. (k − 1)! 0 0 t

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125

Hence, it remains to show that term II is equal to the second term in (3.11). This follows from the following computation: −II

=

k−1

ˆ λ−(j+1) μ(j−k) S(μ)

j=0

=

k−1

λ−(j+1)

k−1 ∞

e−λt

j=0 0 ∞ −λt

=

e−μs

0

j=0

=

∞

e

0

0

tj dt j!

∞

s

e−μs

0

∞

(s − r)k−j−1 S(r) dr ds (k − j − 1)!

e−μs

0

s

0

s 0

(s − r)k−j−1 S(r) dr ds (k − j − 1)!

(s + t − r)k−1 S(r) dr ds dt. (k − 1)!

Finally, recall that {R(λ) : λ > ω} is the resolvent of an operator A in X if and only if Ker R(λ) = {0} (see Proposition B.6). This is equivalent to the fact that S(t)x = 0 for all t ≥ 0 implies x = 0. In contrast to the situation for semigroups the functional equation (3.9) in Proposition 3.2.4 does not imply that abs(S) < ∞. This will be shown at the end of this section in Remark 3.2.15. It follows from Proposition 3.1.5 or from the above Proposition 3.2.4 that for a k-times integrated semigroup S we have S(t)S(s) = S(s)S(t)

(s, t ≥ 0).

(3.12)

The above elementary properties of integrated semigroups will be used in the following without further notice. A particular example of an integrated semigroup is the antiderivative of a semigroup which is not necessarily strongly continuous at 0. In order to make this more precise, consider a strongly continuous function T : (0, ∞) → L(X) satisfying a) T (t + s) = T (t)T (s)

(s, t > 0),

b) there exists c > 0 such that ||T (t)|| ≤ c for all t ∈ (0, 1], c) T (t)x = 0 for all t > 0 implies x = 0. Then by the proof of Theorem 3.1.7, there exist constants M, ω ≥ 0 such that ||T (t)|| ≤ M eωt for all t > 0. For t ≥ 0 set t S(t) := T (s) ds. 0

Then (S(t))t≥0 satisﬁes condition (ii) of Proposition 3.2.4 with k = 1. Hence, there exists an operator A such that (ω, ∞) ⊂ ρ(A) and ∞ ∞ R(λ, A) = λ e−λt S(t) dt = e−λt T (t) dt (λ > ω). (3.13) 0

0

126

3. CAUCHY PROBLEMS

Deﬁnition 3.2.5. Let T : (0, ∞) → L(X) be a strongly continuous function satisfying assumptions a), b) and c) above. Let A be deﬁned as in (3.13). Then T is called a semigroup and A is called its generator. We will see in the following Section 3.3 that a semigroup T on X is a C0 semigroup on X if and only if D(A) = X. Proposition 3.2.6. Let A be the generator of a k-times integrated semigroup S on X for some k ∈ N and let a ∈ C. Then A − a generates a k-times integrated semigroup Sa on X which is given by t k (t − s)j−1 −as k −at Sa (t) = e S(t) + aj e S(s) ds. j (j − 1)! 0 j=1

Proof. Taking Laplace transforms of Sa we obtain, for μ suﬃciently large ∞ ∞ −μt e Sa (t) dt = e−(μ+a)t S(t) dt 0

0

+

∞

e−(μ+a)t S(t) dt

0

= =

R(μ + a, A) 1 (μ + a)k μk

k j=0

k k aj μ−j j j=1

k j

aj μk−j

R(μ + a, A) (μ + a)k R(μ, A − a) = . k k (μ + a) μ μk

Hence, the assertion follows directly from Deﬁnition 3.2.1. Proposition 3.2.7. Let A be an operator on X and let μ ∈ ρ(A), k ∈ N. Then A generates a k-times integrated semigroup S on X if and only if there exists ω ∈ R such that (ω, ∞) ⊂ ρ(A) and R(·, A)R(μ, A)k is a Laplace transform Tˆ in the sense of Deﬁnition 3.1.4. In that case, ω(T ) < ∞ if and only if ω(S) < ∞. Proof. By Proposition 3.2.6, the operator A generates a k-times integrated semigroup if and only if A−μ does so. By Deﬁnition 3.2.1 and Proposition 1.6.1 a), this is equivalent to λ → (λ − μ)−k R(λ, A) being a Laplace transform. The resolvent equation implies that R(λ, A)R(μ, A)k =

R(λ, A) R(μ, A) R(μ, A)2 R(μ, A)k − − −... − (3.14) k k k−1 (μ − λ) (μ − λ) (μ − λ) (μ − λ)

for λ, μ ∈ ρ(A), λ = μ. The ﬁrst assertion follows easily. Moreover, each step in the passage between S and T preserves exponential boundedness. In the following we characterize those operators which generate a k-times integrated semigroup for some k ∈ N simply by the fact that the resolvent is polynomially bounded on a half-plane. The real characterization given in the next section determines precisely the order of integration.

3.2. INTEGRATED SEMIGROUPS AND CAUCHY PROBLEMS

127

Theorem 3.2.8. Let A be an operator and let k ∈ N. a) Assume that there exists ω ≥ 0, M ≥ 0, b > 0 such that λ ∈ ρ(A) and

R(λ, A) ≤ M |λ|k−1−b whenever Re λ > ω. Then A generates a k-times integrated semigroup S satisfying ω(S) ≤ ω. b) Conversely, if A generates a k-times integrated semigroup S such that ω(S) < ∞, then for ω > max{ω(S), 0} there exists M such that λ ∈ ρ(A) and

R(λ, A) ≤ M |λ|k whenever Re λ > ω. Proof. a) Apply Theorem 2.5.1 to q(λ) := λb R(λ, A)/λk . b) Let max{ω(S), 0} < ω1 < ω. There exists M1 ≥ 0 such that S(t) ≤ M1 eω1 t (t ≥ 0). Hence, by Proposition B.5, λ ∈ ρ(A) and

∞

k

−λt

R(λ, A) = λ e S(t) dt

0

≤

|λ|k M1 (Re λ − ω1 )−1 ≤ |λ|k M1 (ω − ω1 )−1

whenever Re λ > ω. We now turn our attention to the inhomogenous Cauchy problem u (t) = Au(t) + f (t) (t ∈ [0, τ ]), (ACPf ) u(0) = x,

(3.15)

where τ > 0, f ∈ L1 ([0, τ ], X), x ∈ X and A is assumed to be the generator of a k-times integrated semigroup S on X for some k ∈ N. Recall from Section 3.1 that by a mild solution of (ACPf ) we understand a function u ∈ C([0, τ ], X) such t t t that 0 u(s) ds ∈ D(A) and u(t) = A 0 u(s) ds + x + 0 f (s) ds for all t ∈ [0, τ ] and that by a classical solution of (ACPf ) we mean a function u ∈ C 1 ([0, τ ], X) ∩ C([0, τ ], D(A)) satisfying (ACPf ) for all t ∈ [0, τ ]. For x ∈ X consider the function v given by t

v(t) := S(t)x + 0

S(s)f (t − s) ds

(t ∈ [0, τ ]).

(3.16)

It follows from Proposition 1.3.4 that v ∈ C([0, τ ], X). Lemma 3.2.9. a) If there is a mild solution u of (ACPf ), then v ∈ C k ([0, τ ], X) (k) and u = v . b) If there exists a classical solution u of (ACPf ), then v ∈ C k+1 ([0, τ ], X) and u = v (k) . s s Proof. a) For 0 ≤ s ≤ t ≤ τ set w(s) := S(t − s) 0 u(r)dr. Since 0 u(r)dr ∈ D(A) for s ∈ [0, τ ], it follows from Lemma 3.2.2 c) that s (t − s)k−1 s w (s) = −S(t − s)A u(r) dr − u(r) dr + S(t − s)u(s) (k − 1)! 0 0 s (t − s)k−1 s = − u(r) dr + S(t − s) x + f (r) dr (s ∈ [0, t]). (k − 1)! 0 0

128

3. CAUCHY PROBLEMS

Since 0 = w(0) − w(t) = − 0

t

t 0

S(t − s) x +

w (s) ds we have

s

f (r) dr

(t − s)k−1 (k − 1)!

t

ds =

0

0

s

u(r) dr ds. 0

Using this and Proposition 1.3.6, it follows that t dk+1 (t − s)k−1 s u(t) = u(r) dr ds dtk+1 (k − 1)! 0 0 t dk = S(t)x + S(s)f (t − s) ds = v (k) (t). dtk 0 b) This follows immediately from Proposition 3.1.15. Lemma 3.2.10. Let v be deﬁned by (3.16). Assume that v ∈ C k ([0, τ ], X). Then u := v (k) is a mild solution of (ACPf ). Moreover, if v ∈ C k+1 ([0, τ ], X), then u := v (k) is a classical solution of (ACPf ). Proof. By Fubini’s theorem, t t t v(s) ds = S(s)x ds + 0

0

0

t−r

S(s)f (r) ds dr

0

t t−r for t ∈ [0, τ ]. By Lemma 3.2.2 d), 0 S(s)x ds ∈ D(A), 0 S(s)f (r) ds ∈ D(A) t−r t k and A 0 S(s)f (r) ds = S(t−r)f (r)− (t−r) k! f (r). By Proposition 1.1.7, 0 v(s) ds ∈ D(A) and t t tk (t − r)k A v(s) ds = v(t) − x − f (r) dr. (3.17) k! k! 0 0 Since A is closed and v ∈ C k ([0, τ ], X), it follows from (3.17) that v (j−1) (t) ∈ D(A) for t ∈ [0, τ ] and that Av (j−1) (t) = v (j) (t) −

tk−j x− (k − j)!

0

t

(t − r)k−j f (r) dr (k − j)!

(3.18)

for j = 1, . . . , k. Since v(0) = 0, this implies that v (j) (0) = 0 for j = 1, 2, . . . , k − 1. It now follows from (3.18) for j = k that u := v (k) is a mild solution of (ACPf ). If v ∈ C k+1 ([0, τ ], X), we may diﬀerentiate (3.18) once more and see that v(k) (t) ∈ D(A) and that Av (k) (t) = v (k+1) (t) − f (t) (t ∈ [0, τ ]). (3.19) Hence u := v (k) satisﬁes u (t) = Au(t) + f (t) for t ∈ [0, τ ]. Also, by (3.18) for j = k, u(0) = v (k) (0) = x. Combining the above Lemmas 3.2.9 and 3.2.10 with Lemma 3.2.2 c) and Proposition 1.3.6, we obtain the following corollary.

3.2. INTEGRATED SEMIGROUPS AND CAUCHY PROBLEMS

129

Corollary 3.2.11. Let A be the generator of a k-times integrated semigroup on X for some k ∈ N. a) Assume that x ∈ D(Ak+1 ). Then there exists a unique classical solution of (ACP0 ). b) Assume that f ∈ C k+1 ([0, τ ], X) and that there exist xj ∈ D(A) for j = 0, 1, . . . , k satisfying x0 = x, xj+1 = Axj + f (j) (0) (j = 0, 1, . . . , k). Then there exists a unique classical solution of (ACPf ). c) Assume that f ∈ C k ([0, τ ], X) and that there exist xj ∈ D(A) for j = 0, 1, . . . , k − 1 satisfying x0 = x, xj+1 = Axj + f (j) (0) (j = 0, 1, . . . , k − 1). Then there exists a unique mild solution of (ACPf ). Remark 3.2.12. We remark that in contrast to the case of a C0 -semigroup (see Corollary 3.1.17) a mere regularity condition on the function f does not suﬃce to ensure the existence of a classical solution of (ACPf ). Indeed, let A be the generator of a once integrated semigroup S on X such that A does not generate a C0 -semigroup on X. Then there exists y ∈ X such that S(·)y ∈ / C 1 ([0, τ ], X). Consider the function f deﬁned by f (t) := y for t ∈ [0, τ ]. If x = 0, then v(t) = t S(s)y ds, but v ∈ / C 2 ([0, τ ], X). 0 In the following we turn our attention to the converse of Corollary 3.2.11; i.e., we are aiming to show that A is the generator of an integrated semigroup whenever the associated Cauchy problem (ACP0 ) admits a unique classical solution for all initial data x belonging to the domain of some power of A. To this end, we restrict ourselves to the case of generators of exponentially bounded k-times integrated semigroups; i.e., we assume that the function S in Deﬁnition 3.2.1 satisﬁes in addition the property that S(t) ≤ M eωt for all t ≥ 0 and some suitable constants M, ω ≥ 0. For x ∈ X consider then the “(k + 1)-times integrated version” of (ACP0 ), which is to ﬁnd v ∈ C 1 (R+ , X) ∩ C(R+ , D(A)) satisfying ⎧ tk ⎨ v (t) = Av(t) + x (t ≥ 0), (ACPk+1 ) (3.20) k! ⎩v(0) = 0. Assume that A generates an exponentially bounded k-times integrated semigroup t S on X and deﬁne v by v(t) := 0 S(s)x ds for t ≥ 0. Then by Lemma 3.2.2 d), t tk v (t) = S(t)x = A S(s)x ds + x (t ≥ 0) (3.21) k! 0 and v(0) = 0. Hence v is a classical solution of (ACPk+1 ). It is unique by Lemma 3.2.9 and exponentially bounded since S is so. We have therefore proved the implication (i) ⇒ (ii) of the following result. Recall that a 0-times integrated semigroup is the same as a C0 -semigroup, so the following may be compared with Theorem 3.1.10.

130

3. CAUCHY PROBLEMS

Theorem 3.2.13. Let A be a closed operator on X and let k ∈ N0 . The following assertions are equivalent: (i) A generates an exponentially bounded k-times integrated semigroup on X. (ii) For all x ∈ X there exists a unique classical solution of (ACPk+1 ) which is exponentially bounded. (iii) ρ(A) = ∅ and for every x ∈ D(Ak+1 ) there exists a unique classical solution of (ACP0 ) which is exponentially bounded. Proof. The remarks before Theorem 3.2.13 imply the assertion (i) ⇒ (ii). Moreover, the implication (i) ⇒ (iii) follows from Corollary 3.2.11. For the proof of the implication (ii) ⇒ (i) we need the following “uniform exponential boundedness principle”. Lemma 3.2.14. Let X, Y be Banach spaces and let V : R+ → L(X, Y ) be a function. Assume that V (·)x is exponentially bounded for all x ∈ X. Then there exist constants M ≥ 0, ω ∈ R such that

V (t) ≤ M eωt

(t ≥ 0).

Proof. Observe that, for n ∈ N, the set Xn deﬁned by ' ( Xn := x ∈ X : V (t)x ≤ nent for all t ≥ 0 is a closed subset of X. The hypothesis implies that X = n∈N Xn . Hence, by Baire’s theorem, there exists n0 ∈ N such that Xn0 has non-empty interior. It follows that there exist z ∈ X, ε > 0, M ≥ 0 and ω ∈ R such that

V (t)x ≤ M eωt

(t ≥ 0)

provided x − z ≤ ε. For y ≤ 1 we have εe−ωt V (t)y ≤ e−ωt V (t)(εy + z) + e−ωt V (t)z ≤ 2M for t ≥ 0. Thus V (t) ≤

2M ωt e ε

for t ≥ 0.

The above Lemma 3.2.14 enables us now to prove the implication (ii) ⇒ (i) in Theorem 3.2.13. (ii) ⇒ (i): Denote by V (·)x the solution of (ACPk+1 ). The mapping V (t) : X → D(A) is linear by uniqueness. We even have that V (t) ∈ L(X, D(A)). Indeed, the space C(R+ , D(A)) is a Fr´echet space for the seminorms pm (f ) := sup f (t) D(A) . 0≤t≤m

Deﬁne a mapping Φ : X → C(R+ , D(A)) by Φ(x) = V (·)x. Then Φ is closed and the closed graph theorem implies that Φ is continuous (see the proof of Theorem

3.2. INTEGRATED SEMIGROUPS AND CAUCHY PROBLEMS

131

3.1.12). In particular, the mapping X → D(A), x → V (t)x is continuous for t ≥ 0. The hypothesis together with Lemma 3.2.14 implies that V (t) ≤ M eωt (t ≥ 0) ∞ for suitable constants M, ω ≥ 0. Therefore Q(λ)x := λk+1 0 e−λt V (t)x dt is well deﬁned for λ > ω. Since

t

k+1 k+1

= V (t)x − t

≤ M eωt + t AV (s)x ds x

x ,

(k + 1)! (k + 1)! 0 it follows from Theorem 1.4.3 that abs(AV (·)x) ≤ ω. By Proposition 1.6.3, Q(λ)x ∈ D(A) for all x ∈ X, all λ > ω and ∞ ∞ k+2 −λt k+1 (λ − A)Q(λ)x = λ e V (t)x dt − λ e−λt AV (t)x dt 0 0 ∞ ∞ d = λk+2 e−λt V (t)x dt − λk+1 e−λt V (t)x dt dt 0 0 ∞ k t + λk+1 e−λt x dt k! 0 = x for λ > ω. In order to show that λ−A is injective for λ > ω assume that (λ−A)x = 0 for some x ∈ D(A) and λ> ω. Then the solution V (t)x of (ACPk+1 ) is given t k by V (t)x = 0 (t−s) eλs ds x. Since V (t)x ≤ M eωt x for all t ≥ 0, it follows k! that x = 0. Hence, R(λ, A) = Q(λ) for λ > ω and V is a (k + 1)-times integrated k d semigroup generated by A. By hypothesis S(t)x := dt V (t)x = AV (t)x + tk! x exists for all t ≥ 0 and all x ∈ X and V (0) = 0. Integrating by parts shows that ˆ R(λ, A) = λk S(λ), so S is a k-times integrated semigroup generated by A. (iii) ⇒ (ii): For the time being, assume that 0 ∈ ρ(A). Let x ∈ X and let u be the solution of (ACP0 ) with initial value u(0) = A−k−1 x. Then v given by v(t) := u(t) − A−k−1 x − tA−k x − . . . −

tk−1 tk A−2 x − A−1 x (k − 1)! k!

is an exponentially bounded solution of (ACPk+1 ). Let v¯ be another solution of (ACPk+1 ). Then u = v − v¯ solves (ACP0 ) with initial value u(0) = 0. Hence, u ≡ 0 and we have proved (ii) provided 0 ∈ ρ(A). In the case where 0 ∈ ρ(A), we have that 0 ∈ ρ(A − μ) for some μ ∈ C. Hence, the preceding argument shows that (ii) holds for (A − μ). The implication (ii) ⇒ (i) and Proposition 3.2.6 imply that A generates a k-times integrated semigroup which, as we have seen, implies (ii). Remark 3.2.15. a) The assumption that ρ(A) = ∅ in Theorem 3.2.13 (iii) cannot be omitted even if k = 0 (see Example 3.1.11). b) The assumption of exponential boundedness in Theorem 3.2.13 (ii) and (iii) cannot be omitted as the following example shows: Let 1 ≤ p < ∞ and X be the

132

3. CAUCHY PROBLEMS

space p of all complex sequences x = (xn )n∈N such that x := ( ∞. Deﬁne the closed operator A on X by

∞ n=1

1/p

|xn |p )

<

D(A) := {x ∈ X : (an xn ) ∈ X}, where an := n + ien

2

Ax := (an xn )n∈N t for n ∈ N. For t ≥ 0 set S(t)x := ( 0 esan ds)xn . n

Then S(t) ∈ L(X) for t ≥ 0 and S(·)x is strongly continuous. For x ∈ X let t v(t) := 0 S(s)x ds. We verify that v (t) = Av(t) + tx for t ≥ 0; i.e., v is the unique solution of (ACP2 ). However, v is not exponentially bounded if xn = 0 for all s+t t n ∈ N. Observe also that S satisﬁes S(s)S(t) = s S(r)dr − 0 S(r)dr. However, S is not Laplace transformable since v is not exponentially bounded.

3.3

Real Characterization

In Section 3.1 (respectively, 3.2) we proved that the Cauchy problem u (t) = Au(t)

(t ≥ 0),

u(0) = x,

possesses a unique classical solution for all x ∈ D(A) (respectively, x ∈ D(Ak+1 )) provided the operator A generates a C0 -semigroup (respectively, k-times integrated semigroup) on X. It is therefore interesting to characterize generators of C0 -semigroups (respectively, integrated semigroups) by properties of the operators A or their resolvents. In the following we characterize generators of C0 -semigroups (respectively, exponentially bounded integrated semigroups) in terms of estimates for the resolvents and all their powers for real λ. Recall that an operator A was deﬁned to be the generator of a k-times integrated semigroup S on X for some k ∈ N0 if (ω, ∞) ⊂ ρ(A) for some ω ≥ 0 and there exists a strongly continuous function S : R+ → L(X) satisfying abs(S) ≤ ω and ∞ R(λ, A) = λk e−λt S(t) dt (λ > ω). 0

By applying the Real Representation Theorem 2.4.1 to the special case of resolvents, we obtain the following characterization. Here and in what follows, we use the notation (R(λ, A)/λk )(n) to denote the nth derivative of the function λ → R(λ, A)/λk . Note that the ﬁrst derivative of R(λ, A) is −R(λ, A)2 (see Corollary B.3). Theorem 3.3.1. Let A be a linear operator on X. Let M ≥ 0, ω ∈ R and k ∈ N0 . Then the following assertions are equivalent: (i) (ω, ∞) ⊂ ρ(A) and sup sup (λ − ω)n+1 (R(λ, A)/λk )(n) /n! ≤ M. n∈N0 λ>ω

3.3. REAL CHARACTERIZATION

133

(ii) A generates a (k + 1)-times integrated semigroup Sk+1 on X satisfying

t

Sk+1 (t) − Sk+1 (s) ≤ M

eωr dr

(0 ≤ s ≤ t).

s

Proof. The implication (i) ⇒ (ii) follows from Theorem 2.4.1 and assertion (1.22). Conversely, assume that (ii) holds. By Deﬁnition 3.2.1, there exists ω ≥ ω such ∞ that R(λ,A) = λ 0 e−λt Sk+1 (t) dt for all λ > ω . By Proposition B.5, (ω, ∞) ⊂ λk ρ(A) and (i) follows from Theorem 2.4.1 and (1.22). When k > 0 and X = {0}, conditions (i) and (ii) of Theorem 3.3.1 cannot be satisﬁed for ω < 0. Note that, given a linear operator A satisfying condition (i) above, one cannot improve the order of integration of Sk+1 , in general (see Example 3.3.10 below). However, if A is densely deﬁned, we obtain the following characterization. Theorem 3.3.2. Let A be a densely deﬁned operator on X. Let M ≥ 0, ω ∈ R and k ∈ N0 . Then the following assertions are equivalent: (i) (ω, ∞) ⊂ ρ(A) and sup sup (λ − ω)n+1 (R(λ, A)/λk )(n) /n! ≤ M. n∈N0 λ>ω

(ii) A generates a k-times integrated semigroup Sk on X satisfying

Sk (t) ≤ M eωt

(t ≥ 0).

The following lemma will be useful in the proof of Theorem 3.3.2. In addition to the space Lipω (R+ , X) deﬁned in Section 2.4, we also set 1 1 −ωt Cω (R+ , X) := f ∈ C (R+ , X) : f (0) = 0, sup ||e f (t)|| < ∞ t≥0

=

1

C (R+ , X) ∩ Lipω (R+ , X).

Lemma 3.3.3. For ω ∈ R, the space Cω1 (R+ , X) is a closed subspace of Lipω (R+ , X). In particular, if S ∈ Lipω (R+ , L(X)), then {x ∈ X : S(·)x ∈ C 1 (R+ , X)} is a closed subspace of X. Proof. Let (fn ) ⊂ Cω1 (R+ , X) such that (fn ) converges to f in Lipω (R+ , X). t Then fn (t) = 0 fn (s) ds for t ≥ 0. Since sups∈[0,t] fn (s) − fm (s) ≤ eωt fn − fm Lipω (R+ ,X) for n, m ∈ N0 , it follows that (fn ) converges uniformly on comt pact sets to a function g ∈ C(R+ , X) and that f (t) = 0 g(s) ds for t ≥ 0. Hence f ∈ Cω1 (R+ , X). The ﬁnal statement follows, since x → S(·)x is continuous from X into Lipω (R+ , X).

134

3. CAUCHY PROBLEMS

Proof of Theorem 3.3.2. Assume that (ii) holds. Then A also generates a (k + 1)times integrated semigroup Sk+1 on X which in addition satisﬁes assertion (ii) of Theorem 3.3.1. Hence assertion (i) follows from that theorem. Conversely, assume that (i) holds. By Theorem 3.3.1, A generates a (k + 1)times integrated semigroup Sk+1 on X such that Sk+1 ∈ Lipω (R+ , L(X)) with

Sk+1 Lipω (R+ ,L(X)) ≤ M . By Lemma 3.2.2 c), Sk (t)x :=

d Sk+1 (t)x dt

(3.22)

exists for all x ∈ D(A) and t → Sk (t)x is continuous. By Lemma 3.3.3, the deﬁnition of Sk (t)x given in (3.22) is also meaningful for x ∈ D(A) and t → Sk (t)x is also continuous for x ∈ D(A). By assumption D(A) = X and therefore A is the generator of the k-times integrated semigroup Sk on X which clearly satisﬁes

Sk (t) ≤ M eωt

(t ≥ 0).

Notice that the special case k = 0 in Theorem 3.3.2 is precisely the classical Hille-Yosida theorem (in the general form presented here due to Hille, Yosida, Feller, Miyadera and Phillips), which we state explicitly due to its special importance. Theorem 3.3.4 (Hille-Yosida). Let A be a densely deﬁned operator on X. Then A generates a C0 -semigroup on X if and only if there exist constants M ≥ 0, ω ∈ R such that (ω, ∞) ⊂ ρ(A) and

(λ − ω)n+1 R(λ, A)(n) /n! ≤ M

(λ > ω, n ∈ N0 ).

(3.23)

It is immediate from Theorem 3.3.2 and the relation (−1)n R(λ, A)(n) /n! = R(λ, A)n+1

(λ ∈ ρ(A), n ∈ N0 )

(3.24)

(see Corollary B.3) that the generator of a C0 -semigroup T of contractions may be characterized as follows. Corollary 3.3.5. Let A be a densely deﬁned operator on X. Then A generates a C0 -semigroup on X satisfying T (t) ≤ 1 for all t ≥ 0 if and only if (0, ∞) ⊂ ρ(A) and

λR(λ, A) ≤ 1 (λ > 0). (3.25) It is possible to express the semigroup in terms of the resolvent via “Euler’s formula” for exponentials, which is well known in the scalar case. Corollary 3.3.6. Let A be the generator of a C0 -semigroup T . Then ! "−n T (t)x = lim I − nt A x n→∞

for t > 0 and x ∈ X.

(3.26)

3.3. REAL CHARACTERIZATION

135

Proof. By (3.24) we have ! "−n (−1)n−1 n I − nt A = λn R(λ, A)n = λ R(λ, A)(n−1) (n − 1)! where λ = nt . Thus assertion (3.26) is precisely the Post-Widder inversion formula proved in Theorem 1.7.7. For a densely deﬁned operator A on X denote its adjoint by A∗ . Then R(λ, A)∗ = R(λ, A∗ ) for all λ ∈ ρ(A) = ρ(A∗ ) (see Proposition B.11). As a direct consequence of Theorem 3.3.1 and Theorem 3.3.4 we obtain the following result. Corollary 3.3.7. Let A be the generator of a C0 -semigroup on X. Then the adjoint A∗ of A generates a once integrated semigroup on X ∗ . We remark that if the underlying Banach space X is reﬂexive, then the adjoint A∗ of A even generates a C0 -semigroup on X ∗ . This follows from Theorem 3.3.4, since A∗ is densely deﬁned (see Proposition B.10). In fact, the following proposition shows that operators satisfying the Hille-Yosida condition (3.23) acting on reﬂexive spaces are necessarily densely deﬁned. Proposition 3.3.8. Let A be a linear operator on a reﬂexive Banach space X. Assume that there exist constants M, ω ≥ 0 such that (ω, ∞) ⊂ ρ(A) and λR(λ, A)

≤ M (λ > ω). Then A is densely deﬁned. Proof. Let x ∈ X and for n ∈ N with n > ω set an := R(n, A)x. By assumption, (nan )n∈N,n>ω is a bounded sequence. Let z be a weak limit point of the relatively weakly compact set {nan : n ∈ N}. Since nan − Aan = x and limn→∞ an = 0, the closedness of A implies that x = z. But z is in the weak closure of D(A) and hence in the norm closure of D(A). Corollary 3.3.9. Let X be a reﬂexive Banach space and assume that A generates a C0 -semigroup on X. Then the adjoint A∗ of A generates a C0 -semigroup on X ∗ . Example 3.3.10. Let 1 ≤ p < ∞ and X := Lp (R). Consider the operator Ap f := f on Lp (R) with domain D(Ap ) := W 1,p (R). For the deﬁnition of the Sobolev space W 1,p (R), see Appendix E. Then Ap generates the C0 -semigroup Tp on Lp (R) given by (Tp (t)f )(x) = f (x + t) (x ∈ R, t ≥ 0).

Identifying Lp (R)∗ with Lp (R) where 1/p + 1/p = 1, Corollary 3.3.9 implies that −Ap generates a C0 -semigroup on Lp (R), provided p > 1. In fact, this is evident since Tp extends to a C0 -group. If p = 1, then by Corollary 3.3.7, −A∞ generates a once integrated semigroup S on L∞ (R), where A∞ f := −f with t domain D(A∞ ) := W 1,∞ (R). This is given by (S(t)g)(x) = 0 g(x − s) ds. As a further consequence of Theorem 3.3.1 and Theorem 3.3.2 we note the following corollary.

136

3. CAUCHY PROBLEMS

Corollary 3.3.11. Let T be a semigroup on X in the sense of Deﬁnition 3.2.5 and let A be the generator of T . Then T is a C0 -semigroup on X if and only if D(A) is dense in X. Finally, given an operator A in X and a closed subspace Y of X, we deﬁne the part AY of A in Y by := {y ∈ Y ∩ D(A) : Ay ∈ Y }

D(AY ) AY y

:=

Ay.

(3.27)

If D(A) ⊂ Y , then ρ(A) ⊂ ρ(AY ) and R(λ, AY ) = R(λ, A)|Y for all λ ∈ ρ(A). An important case is Y = D(A) when A is not densely deﬁned. Then it may well happen that AY is not densely deﬁned either (a concrete example is the Poisson operator considered in Section 6.1). Nevertheless the following holds true. Lemma 3.3.12. Let A be an operator on X such that (ω, ∞) ⊂ ρ(A) and M := sup λR(λ, A) < ∞ λ>ω

for some ω ∈ R. Let Y = D(A). Then a) limλ→∞ λR(λ, A)x = x for all x ∈ Y . b) D(AY ) is dense in Y . c) If A satisﬁes the Hille-Yosida condition (3.23), then AY generates a C0 semigroup on Y . Proof. a) The assumption implies that limλ→∞ R(λ, A)x = 0 for all x ∈ X. Hence, limλ→∞ λR(λ, A)x = limλ→∞ (x + R(λ, A)Ax) = x if x ∈ D(A). Consequently, limλ→∞ λR(λ, A)x = x if x ∈ D(A) = Y . b) Since R(λ, A)x ∈ D(AY ) if x ∈ Y , this follows from a). c) This follows from the Hille-Yosida theorem. Although the part of an operator A in D(A) may not be densely deﬁned, we obtain the following result from the proof of Theorem 3.3.2. Corollary 3.3.13. Let A be an operator satisfying the equivalent conditions of Theorem 3.3.1 for some k ∈ N0 and let Y = D(A). Then the part AY of A in Y generates a k-times integrated semigroup on Y . Proof. By Lemma 3.2.2, Sk+1 (·)x ∈ C 1 (R+ , X) provided x ∈ D(A). It follows from Lemma 3.3.3 that Sk+1 (·)x ∈ Cω1 (R+ , X) for all x ∈ D(A) = Y . For t > 0 and d x ∈ Y let Sk (t)x = dt Sk+1 (t)x. Since Sk+1 (t)D(A) ⊂ D(A) for t > 0, it follows that Sk (t)x ∈ Y for all x ∈ Y . Thus, Sk : R+ → L(Y ) is strongly continuous and ∞ ∞ R(λ, A)x = λk+1 e−λt Sk+1 (t)x dt = λk e−λt Sk (t)x dt 0

for all x ∈ Y and λ suﬃciently large.

0

3.4. DISSIPATIVE OPERATORS

137

Proposition 3.3.14. Let T be a C0 -semigroup on X with generator A. Let X := D(A∗ ). Then T (t) := T (t)∗ |X deﬁnes a C0 -semigroup whose generator A is the part of A∗ in X . The C0 -semigroup T is known as the sun-dual of T . Proof. It follows from Lemma 3.3.12 that the part of A∗ in X generates a C0 semigroup T on X . For x ∈ X and x∗ ∈ X , Corollary 3.3.6 shows that + , + , x, T (t) x∗ = lim x, (I − nt A )−n x∗ n→∞ + , = lim x, ((I − nt A)−n )∗ x∗ n→∞ + , = lim (I − nt A)−n x, x∗ n→∞

= T (t)x, x∗ = x, T (t)∗ x∗ . It follows that T (t) = T (t)∗ |X for all t > 0.

3.4

Dissipative Operators

In the previous section we saw that, by the Hille-Yosida characterization, the generator of a contraction semigroup may be characterized in terms of a resolvent estimate for real λ. It is the aim of this section to give a second characterization of such semigroups, which turns out to be quite useful in particular when dealing with diﬀerential operators. In order to do so, we deﬁne for x ∈ X the subdiﬀerential dN(x) of the norm N : X → R+ , N (x) = x at x by dN(x) := {x∗ ∈ X ∗ : x∗ ≤ 1, x, x∗ = x } .

(3.28)

The Hahn-Banach theorem implies that dN(x) = ∅ for all x ∈ X. Deﬁnition 3.4.1. An operator A on X is called dissipative if for every x ∈ D(A) there exists x∗ ∈ dN(x) such that ReAx, x∗ ≤ 0.

(3.29)

A useful characterization of dissipative operators is the following. Lemma 3.4.2. An operator A on X is dissipative if and only if

(λ − A)x ≥ λ x

(x ∈ D(A), λ > 0)

or equivalently,

x − tAx ≥ x

(x ∈ D(A), t > 0).

(3.30)

138

3. CAUCHY PROBLEMS

Proof. Assume that A is dissipative. Let x ∈ D(A), t > 0. Let x∗ ∈ dN(x) satisfy (3.29). Then for t > 0,

x − tAx

≥ Rex − tAx, x∗ = x − t ReAx, x∗ ≥ x .

Conversely, let x ∈ D(A) and assume that x − tAx ≥ x (t > 0). Choose x∗t ∈ dN(x − tAx) and let x∗ be a weak* accumulation point of x∗t as t ↓ 0. Then

x∗ ≤ 1. Since x − tAx = x − tAx, x∗t , letting t ↓ 0 shows that x = x, x∗ . Thus, x∗ ∈ dN(x). Moreover,

x ≤ x − tAx

= ≤

Rex, x∗t − t ReAx, x∗t

x − t ReAx, x∗t .

Thus, ReAx, x∗t ≤ 0. Letting t ↓ 0 shows that ReAx, x∗ ≤ 0. Example 3.4.3. a) Dissipative operators acting on Hilbert spaces or Lp -spaces may be characterized as follows: (i) Let A be an operator on a Hilbert space H. Denote by (·|·) the inner product in H. Then A is dissipative if and only if Re(Ax|x) ≤ 0 for all x ∈ D(A). (ii) Let Ω ⊂ Rn be open, 1 < p < ∞, set X := Lp (Ω) and identify X ∗ with Lp (Ω) where 1/p + 1/p = 1. For f ∈ X \ {0} we deﬁne sign f ∈ L∞ (Ω) by ⎧ ⎨0 if f (x) = 0, (sign f )(x) := f (x) ⎩ if f (x) = 0. |f (x)| ( −(p−1) ' Then dN(f ) = f p sign f¯ · |f |p−1 , where f¯ denotes the complex conjugate function of f . Therefore an operator A on X is dissipative if and only if Re Af · sign f¯ · |f |p−1 dx ≤ 0 (3.31) Ω

for all f ∈ D(A). b) If A is dissipative, then cA is dissipative for all c > 0. c) If B ∈ L(X), then B − B is dissipative. Lemma 3.4.4. Let A be a densely deﬁned dissipative operator on X. Then A is closable and A is dissipative. Proof. Let (xn ) ⊂ D(A) such that xn → 0 and Axn → y for some y ∈ X as n → ∞. We show that y = 0. To this end, let z ∈ D(A). It follows from Lemma

3.4. DISSIPATIVE OPERATORS

139

3.4.2 that (I − tA)x ≥ x for all t > 0 and all x ∈ D(A). Hence xn + tz ≤ xn + tz − tA(xn + tz) for all n ∈ N and all t > 0. This implies that

tz ≤ tz − ty − t2 Az and hence z ≤ z − y − tAz for all t > 0. Letting t → 0 we obtain z ≤ z − y for all z ∈ D(A). Since D(A) is dense, it follows that y = 0 which means that A is closable. Taking limits in (3.30) shows that A is dissipative. The following theorem due to Lumer and Phillips characterizes generators A of C0 -semigroups of contractions in terms of dissipativity of A. Theorem 3.4.5 (Lumer-Phillips). Let A be a densely deﬁned operator on X. Then A generates a C0 -semigroup of contractions on X if and only if a) A is dissipative, and b) (λ − A)D(A) = X for some (or all) λ > 0. Proof. Let A be the generator of a C0 -semigroup of contractions. Then assertion b) holds by the Hille-Yosida theorem (Corollary 3.3.5). Moreover, the Hille-Yosida theorem combined with Lemma 3.4.2 implies assertion a). In order to prove the converse implication note that by Lemma 3.4.2 we have

(λ − A)x ≥ λ x

(x ∈ D(A), λ > 0).

(3.32)

Since (λ0 − A)D(A) = X for some λ0 > 0, it follows from (3.32) that λ0 − A is invertible and that R(λ0 , A) ≤ λ−1 0 . We show that this property holds for all λ > 0. In fact, let Λ := ρ(A) ∩ (0, ∞). Then Λ = ∅ and therefore A is closed. Furthermore let (λn ) ⊂ Λ such that limn→∞ λn = λ > 0. By Corollary B.3, dist(λn , σ(A)) ≥ R(λn , A) −1 ≥ λn for all n ∈ N and it follows that λ ∈ Λ. This shows that Λ is closed in (0, ∞). Since Λ is obviously open, it follows that Λ = (0, ∞) and therefore (0, ∞) ⊂ ρ(A). Inequality (3.32) implies that R(λ, A) ≤ λ−1 for all λ > 0 and the Hille-Yosida theorem ﬁnally implies the assertion. By the same proof with Theorem 3.3.1 replacing the Hille-Yosida theorem we obtain the following characterization in the case when D(A) is not necessarily dense. Corollary 3.4.6. Let A be an operator on X. The following assertions are equivalent: (i) A is dissipative and (λ − A)D(A) = X for some (or all) λ > 0. (ii) A generates a once integrated semigroup S satisfying

S(t) − S(s) ≤ |t − s|

(t, s ≥ 0).

In concrete examples, dissipativity is often relatively easy to verify whereas the range condition b) in Theorem 3.4.5 is hard to show. However, in the following example of the Dirichlet-Laplacian, the range condition is just a consequence of the Riesz-Fr´echet lemma.

140

3. CAUCHY PROBLEMS

Example 3.4.7 (The Laplacian with Dirichlet boundary conditions). Let Ω ⊂ Rn be an open set. Consider the operator A : D(A) → L2 (Ω) deﬁned by D(A)

:= {u ∈ H01 (Ω) : Δu ∈ L2 (Ω)},

Au :=

Δu.

1 Here, n H02 (Ω) is the Sobolev space deﬁned in Appendix E, and Δu is deﬁned to be E). Denoting by (·|·) j=1 Dj u in the sense of distributions (see also Appendix the inner product in L2 (Ω), we see that (Au|u) = Ω (Δu)¯ u = − Ω |∇u|2 ≤ 0 for u ∈ D(A). Hence, by Example 3.4.3 a), the operator A is dissipative. In order to prove the range condition b) of Theorem 3.4.5, let f ∈ L2 (Ω). Then the mapping Φ : v → Ω f v deﬁnes a continuous linear functional on the Hilbert space H01 (Ω). By the Riesz-Fr´echet lemma, there exists a unique u ∈ H01 (Ω) such that Φ(v) = (v|¯ u)H01 (Ω) for all v ∈ H01 (Ω). Here,

(v|¯ u)H01 (Ω) :=

uv + Ω

n j=1

Ω

Dj uDj v

denotes the inner product in H01 (Ω). Considering in particular v ∈ D(Ω), it follows that u − Δu = f in D(Ω) . This implies that u ∈ D(A) and u − Au = f . Obviously, D(A) is dense in L2 (Ω) and by the Lumer-Phillips theorem, A generates a contraction semigroup on L2 (Ω). We call A the Laplacian with Dirichlet boundary conditions on L2 (Ω), and we denote it by ΔL2 (Ω) . We remark that in the case where Ω is a bounded domain with boundary of class C 2 , it can be shown (see [Bre83, Th´eor`eme IX.25]) that D(ΔL2 (Ω) ) = H 2 (Ω) ∩ H01 (Ω). Example 3.4.8. Consider the Hilbert space X := L2 (0, 1) and the operator := {u ∈ H 1 (0, 1) : u(0) = 0},

D(A)

:= u .

Au

1 Then Re (u|Au) = 12 0 (u(x)u (x)+u (x)u(x)) dx = 12 |u(1)|2 ≥ 0. For f ∈ L2 (0, 1) and λ ∈ C, deﬁne u by x u(x) := e−λ(x−y) f (y) dy (x ∈ (0, 1)). 0

Then u ∈ D(A) and λu + u = f . Hence, the range condition (λ + A)D(A) = X is fulﬁlled for all λ ∈ C and by the Lumer-Phillips theorem, −A generates a contraction semigroup T on X. It is not diﬃcult to see that T is given by f (x − t) (t ≤ x), (T (t)f )(x) = (3.33) 0 (t > x).

3.5. HILLE-YOSIDA OPERATORS

141

∞ x In fact, the mapping x → 0 e−λt (T (t)f )(x) dt = 0 e−λ(x−y) f (y) dy belongs to ∞ ∂ D(A) and (λ + ∂x ) 0 e−λt (T (t)f )(x) dt = f (x) for x ∈ (0, 1). Since (λ + A) is ∞ invertible for all λ ∈ C, we see that (λ + A)−1 = 0 e−λt T (t) dt. The representation (3.33) implies that T also deﬁnes a C0 -semigroup of positive contractions on Lp (0, 1) for 1 ≤ p < ∞. Its generator is given by −Ap where D(Ap ) = {u ∈ W 1,p (0, 1) : u(0) = 0}

3.5

and Ap u = u .

(3.34)

Hille-Yosida Operators

In this section we consider operators which satisfy the Hille-Yosida condition (3.23) but are not necessarily densely deﬁned. Deﬁnition 3.5.1. A linear operator A on X is called a Hille-Yosida operator if there exist ω ∈ R, M ≥ 0 such that (ω, ∞) ⊂ ρ(A) and

(λ − ω)n R(λ, A)n ≤ M

(n ∈ N0 , λ > ω).

(3.35)

We note that by the Hille-Yosida theorem and the identity (−1)n R(λ, A)(n) /n! = R(λ, A)n+1 , the class of densely deﬁned Hille-Yosida operators coincides with the class of generators of C0 -semigroups on X. We also observe that by Theorem 3.3.1 an operator A on X is a Hille-Yosida operator if and only if A generates a once integrated semigroup S on X satisfying t

S(t) − S(s) ≤ M eωr dr (0 ≤ s ≤ t) (3.36) s

for some ω ∈ R, M ≥ 0. The above estimate (3.36) implies in particular that S is a locally Lipschitz continuous function on R+ . This fact will be of crucial importance in the proof of the following result on the inhomogeneous Cauchy problem for operators which are not necessarily densely deﬁned. More precisely, consider the problem u (t) = Au(t) + f (t) (t ∈ [0, τ ]), (ACPf ) (3.37) u(0) = x, where f : [0, τ ] → X is a given function and x ∈ X. When D(A) = X, then the inhomogeneous problem can be solved in the classical sense by means of the variation of constants formula provided x ∈ D(A) and f ∈ C 1 ([0, τ ], X) (see Corollary 3.2.11 b)). Note that this method cannot be used when D(A) = X and when f (t) ∈ D(A). The method which we use in the following to treat (ACPf ) is based

142

3. CAUCHY PROBLEMS

on the fact that a Hille-Yosida operator generates a once integrated semigroup which is locally Lipschitz continuous. Employing the results of Section 3.2 in the present situation, we see that existence and uniqueness results for (ACPf ) are equivalent to the fact that v given by

t

v(t) = S(t)x + 0

S(t)f (t − s)ds

is suﬃciently regular. More precisely, the following holds true. Theorem 3.5.2 (Da Prato-Sinestrari). Let A be a Hille-Yosida operator on X and let τ > 0. a) Let f ∈ L1 ([0, τ ], X) and x ∈ D(A). Then there exists a unique mild solution of (ACPf ). t b) Let f (t) = f0 + 0 g(s) ds where f0 ∈ X and g ∈ L1 ([0, τ ], X). Let x ∈ D(A) and assume that Ax+f0 ∈ D(A). Then there exists a unique classical solution of (ACPf ). Remark 3.5.3. a) Note that x ∈ D(A) is a necessary condition for a mild solution t t u to exist, because limt→0 1t 0 u(s) ds = x and 0 u(s) ds ∈ D(A) by deﬁnition of a mild solution. b) If a classical solution u of (ACPf ) exists, then x ∈ D(A) and Ax + f (0) ∈ D(A). In fact,

(3.38)

Au(0) + f (0) = u (0) = lim 1t (u(t) − u(0)) ∈ D(A). t→0

Proof of Theorem 3.5.2. Let A be a Hille-Yosida operator on X. By Theorem 3.3.1, A generates a once integrated semigroup S on X which satisﬁes estimate (3.36). In order to prove assertion a), it suﬃces by Lemma 3.2.9, Proposition 1.3.7 and Lemma 3.2.10 to show that t → S(t)x belongs to C 1 ([0, τ ], X). This follows from the assumption that x ∈ D(A), Lemma 3.2.2 c) and Lemma 3.3.3. Thus, we have proved assertion a). In order to prove assertion b), it suﬃces by Lemma 3.2.9 and Lemma 3.2.10 to verify that v ∈ C 2 ([0, τ ], X). Since x ∈ D(A) it follows from Lemma 3.2.2 c) and Proposition 1.3.6 that v (t) = x + S(t)Ax + S(t)f0 +

t 0

S(s)g(t − s) ds.

Now, by Proposition 1.3.7 the convolution term on the right hand side above belongs to C 1 ([0, τ ], X) and t → S(t)(Ax + f0 ) belongs to C 1 ([0, τ ], X) by the

3.5. HILLE-YOSIDA OPERATORS

143

argument given in the proof of assertion a), since Ax + f0 ∈ D(A) by assumption. Given a Hille-Yosida operator A, we consider now the problem whether or not the sum A + B of A and some operator B is again a Hille-Yosida operator. We start with the following renorming lemma. Lemma 3.5.4. Let A be a Hille-Yosida operator satisfying estimate (3.35) for some M > 0 and ω = 0. Then there exists an equivalent norm | · | on X such that

x ≤ |x| ≤ M x for x ∈ X and |λR(λ, A)x| ≤ |x|

(x ∈ X, λ > 0).

Proof. For μ > 0 and x ∈ X set

x μ := sup μn R(μ, A)n x . n≥0

Then

x ≤ x μ ≤ M x

and

μR(μ, A) μ ≤ 1.

(3.39)

Let λ ∈ (0, μ] and set y := R(λ, A)x. It follows that y = R(μ, A)(x + (μ − λ)y) and hence that y μ ≤ μ1 x μ + (1 − λμ ) y μ . Therefore, λR(λ, A) μ ≤ 1 and it follows from (3.39) that

λn R(λ, A)n x ≤ λn R(λ, A)n x μ ≤ x μ

(0 < λ ≤ μ).

(3.40)

Hence x λ ≤ x μ for 0 < λ ≤ μ. Deﬁning |x| := lim x μ , μ→∞

the assertion follows by taking n = 1 in (3.40) and letting μ → ∞. Theorem 3.5.5. Let A be a Hille-Yosida operator on X and let B ∈ L(D(A), X). Then A + B is a Hille-Yosida operator. Proof. Replacing A by A − ω we may assume that the estimate (3.35) is satisﬁed for A with ω = 0. Denote by | · | the norm introduced in Lemma 3.5.4. It follows from that lemma that |λR(λ, A)| ≤ 1 (λ > 0). Note that λ−(A+B) = (I −BR(λ, A))(λ−A) for λ > 0. Since |BR(λ, A)| ≤ |B|/λ, the operator I − BR(λ, A) is invertible for λ > |B| and |(λ − (A + B))−1 | ≤ |(λ − A)−1 | |(I − BR(λ))−1 | ≤

1 1 1 = λ 1 − |B|λ−1 λ − |B|

for those λ. Hence |(λ−|B|)R(λ, A+B)| ≤ 1 for λ > |B|. Returning to the original norm we have for x ∈ X,

(λ − |B|)n R(λ, A + B)n x ≤ |(λ − |B|)n R(λ, A + B)n x| ≤ |x| ≤ M x

144

3. CAUCHY PROBLEMS

for λ > |B|. Thus A + B is a Hille-Yosida operator. Taking into account the Hille-Yosida theorem and the fact that generators of C0 -semigroups are densely deﬁned we obtain the following corollary. Corollary 3.5.6. Let A be the generator of a C0 -semigroup on X and let B ∈ L(X). Then A + B generates a C0 -semigroup. Next we consider perturbation by operators deﬁned on the domain of the given operator. If A is a closed operator, we consider D(A) with the graph norm

x D(A) := x + Ax for which it is a Banach space. Two operators A, deﬁned ˜ deﬁned on a second Banach space X, ˜ are called similar if there exists on X, and A, ˜ such that an isomorphism U : X → X ˜ = U D(A) D(A)

and

˜ x = Ax U −1 AU

for all x ∈ D(A).

In that case, A and A˜ have similar properties. For example, if A generates a C0 ˜ given by semigroup T , then A˜ generates the C0 -semigroup T˜ on X T˜ (t)y = U T (t)U −1 y

˜ t ≥ 0). (y ∈ X,

Similarly, if A generates an integrated semigroup S, then A˜ generates the inte˜ given by grated semigroup S˜ on X ˜ S(t)y = U S(t)U −1 y

˜ t ≥ 0). (y ∈ X,

Theorem 3.5.7. Let A be an operator such that (ω, ∞) ⊂ ρ(A) and M := supλ>ω λR(λ, A) < ∞ for some ω ∈ R, and let B ∈ L(D(A)). Then there exists a bounded operator C ∈ L(X) such that A + B and A + C are similar. In particular, if A is a Hille-Yosida operator, then A + B is also a Hille-Yosida operator. For the proof we need the following well-known result. Lemma 3.5.8. Let U, V ∈ L(X). If I − U V is invertible, then so is I − V U . Proof. One has (I − V U )−1 = I + V (I − U V )−1 U . Proof of Theorem 3.5.7. Choose λ0 > ω and let S := (λ0 − A)BR(λ0 , A) ∈ L(X). Choose λ > λ0 such that SR(λ, A) < 1. Then I − (λ0 − A)BR(λ, A)R(λ0 , A) = I − SR(λ, A) is invertible. It follows from Lemma 3.5.8 that I − BR(λ, A) is also invertible. Let C := (λ − A)BR(λ, A) ∈ L(X). We show that A + B and A + C are similar. Let U := I − BR(λ, A). Then U is an isomorphism on X such that U D(A) = D(A). Moreover, U (A + C)U −1

= U (A − λ + C)U −1 + λ = U [A − λ − (A − λ)BR(λ, A)]U −1 + λ = U (A − λ)[I − BR(λ, A)]U −1 + λ = U (A − λ) + λ = A − λ + B + λ = A + B.

3.6. APPROXIMATION OF SEMIGROUPS

145

This proves the claim. Now the second assertion follows from Theorem 3.5.5. Finally, we collect several examples of Hille-Yosida operators. Example 3.5.9. a) Let A be the generator of a C0 -semigroup on X. Then the adjoint A∗ of A is a Hille-Yosida operator on X ∗ . b) As a concrete example, consider X = L∞ (R) and deﬁne A by Au := −u with D(A) := W 1,∞ (R). By Example 3.3.10, (0, ∞) ⊂ ρ(A) and R(λ, A) ≤ 1/λ for λ > 0. c) Let A be the generator of a semigroup T in the sense of Deﬁnition 3.2.5. Since t S(t) := 0 T (s) ds fulﬁlls assumption (ii) of Theorem 3.3.1 for k = 0 it follows from that theorem that A is a Hille-Yosida operator. d) Let X := C[0, 1] and deﬁne an operator A on X by Au D(A)

:= −u ' ( := u ∈ C 1 [0, 1] : u(0) = 0 .

Then D(A) = {u ∈ C[0, 1] : u(0) = 0} = X, (0, ∞) ⊂ ρ(A) and ||R(λ, A)|| ≤ 1/λ for λ > 0. In fact, for f ∈ X and λ > 0 set x u(x) := e−λy f (x − y) dy (x ∈ [0, 1]). 0

Then u ∈ D(A), (λ − A)u = f and sup |u(x)| ≤ f

x∈[0,1]

∞

e−λy dy =

0

1

f , λ

which implies the assertions above.

3.6

Approximation of Semigroups

In this section we study convergence of semigroups. It is interesting that we obtain the main result (Theorem 3.6.1) directly as a consequence of the approximation theorem for Laplace transforms given in Section 1.7, which in turn was proved by a simple functional analytic argument. At the end of the section we give a second proof of the Hille-Yosida theorem, as a simple corollary of the approximation theorem. Let Tn be a C0 -semigroup on X with generator An (n ∈ N). We suppose that

Tn (t) ≤ M

(t ≥ 0, n ∈ N).

If for each x ∈ X lim Tn (t)x =: T (t)x

n→∞

(3.41)

146

3. CAUCHY PROBLEMS

converges uniformly on [0, τ ] for each τ > 0, then it is easy to see that T is a C0 -semigroup. Denote its generator by A. Then it follows from the dominated convergence theorem that limn→∞ R(λ, An )x = R(λ, A)x for all x ∈ X, λ > 0. We now show the converse assertion. Theorem 3.6.1 (Trotter-Kato). Let Tn be a C0 -semigroup on X with generator An (n ∈ N) and suppose that (3.41) holds. Let A be a densely deﬁned operator on X. Suppose that there exists ω ≥ 0 such that (ω, ∞) ⊂ ρ(A) and lim R(λ, An )x = R(λ, A)x

(3.42)

n→∞

for all x ∈ X, λ > ω. Then A is the generator of a C0 -semigroup T and T (t)x = lim Tn (t)x

(3.43)

n→∞

uniformly on [0, τ ] for all τ > 0 and all x ∈ X. Proof. a) Let x ∈ X. We show that the sequence (Tn (·)x)n∈N is equicontinuous on R+ . Since D(A) is dense in X, we can assume that x ∈ D(A). Let ε > 0. Let μ > ω, y := (μ − A)x. Choose n0 ∈ N such that M R(μ, A)y − R(μ, An )y ≤ ε/2 for all n ≥ n0 . Then (by Proposition 3.1.9 f)) for n ≥ n0 ,

Tn (t)x − Tn (s)x = Tn (t)R(μ, A)y − Tn (s)R(μ, A)y

≤

Tn (t)R(μ, An )y − Tn (s)R(μ, An )y + ε/2

t

= Tn (r)An R(μ, An )y dr

0

s

ε − Tn (r)An R(μ, An )y dr

+ 2 0 ≤ M |t − s| An R(μ, An )y + ε/2 = M |t − s| μR(μ, An )y − y + ε/2. Since supn∈N R(μ, An )y < ∞, there exists δ > 0 such that Tn (t)x−Tn (s)x ≤ ε whenever |t − s| ≤ δ and n ≥ n0 . Since Tn (·)x : R+ → X is continuous for n < n0 , this shows that the sequence is equicontinuous. b) Now it follows from Theorem 1.7.5 that Tn (t)x converges uniformly on [0, τ ] as n → ∞ for all x ∈ X and all τ > 0. It is clear that T (t)x := limn→∞ Tn (t)x (x ∈ X) deﬁnes a C0 -semigroup T on X. By the dominated convergence theorem one has ∞ ∞ e−λt T (t)x dt = lim e−λt Tn (t)x dt n→∞

0

=

0

lim R(λ, An )x = R(λ, A)x

n→∞

(x ∈ X, λ > ω).

By Deﬁnition 3.1.8 this means that A is the generator of T . In order to apply Theorem 3.6.1 it is useful to give other criteria equivalent to (3.42).

3.6. APPROXIMATION OF SEMIGROUPS

147

Proposition 3.6.2. Let ω ∈ R. Let A and An be operators such that (ω, ∞) ⊂ ρ(A) and (ω, ∞) ⊂ ρ(An ) for all n ∈ N. Assume that supn∈N R(λ, An ) < ∞ for all λ > ω. Then the following assertions are equivalent: (i) limn→∞ R(μ, An )x = R(μ, A)x for all x ∈ X and all μ > ω. (ii) limn→∞ R(μ, An )x = R(μ, A)x for all x ∈ X and some μ > ω. (iii) For all x ∈ D(A) there exist xn ∈ D(An ) such that limn→∞ xn = x and limn→∞ An xn = Ax. (iv) There exists a core D of A such that for all x ∈ D there exist xn ∈ D(An ) such that limn→∞ xn = x and limn→∞ An xn = Ax. Proof. (i) ⇒ (ii) is trivial. (ii) ⇒ (iii): Let x ∈ D(A). Then xn := R(μ, An )(μ − A)x ∈ D(An ), xn → x by hypothesis, and An xn = μxn − (μ − A)x → Ax as n → ∞. (iii) ⇒ (iv) is trivial. (iv) ⇒ (i): Let μ > ω. Let x ∈ D. By hypothesis, there exist xn ∈ D(An ) such that xn → x and An xn → Ax. Let yn := (μ − An )xn . Then yn → y := (μ − A)x. Hence, lim sup R(μ, An )y − R(μ, A)y

n→∞ ≤ lim sup R(μ, An )(y − yn ) + R(μ, An )yn − R(μ, A)y

n→∞

= lim sup R(μ, An )yn − R(μ, A)y

n→∞

= lim sup xn − x = 0. n→∞

Since D is a core and μ − A is surjective, {(μ − A)x : x ∈ D} is dense in X (see Appendix B), and (i) follows by approximation. Corollary 3.6.3. Let A be a densely deﬁned operator on X. Let An ∈ L(X) such that

etAn ≤ M (t ≥ 0, n ∈ N), where M ≥ 0. Assume that (ω, ∞) ⊂ ρ(A) and limn→∞ An x = Ax for all x ∈ D(A). Then A generates a C0 -semigroup T and for all x ∈ X, limn→∞ etAn x = T (t)x uniformly for t ∈ [0, τ ] for all τ > 0. It is easy to deduce the Hille-Yosida theorem (Corollary 3.3.5) from Corollary 3.6.3. In fact, let A be a densely deﬁned operator on X such that (0, ∞) ⊂ ρ(A) and λR(λ, A) ≤ 1 for all λ > 0. Denote by An := n2 R(n, A) − nI

(n ∈ N)

148

3. CAUCHY PROBLEMS

the Yosida approximation of A. Then

etAn = e−nt etn

2

R(n,A)

2

≤ e−nt etn

R(n,A)

≤1

(t > 0, n ∈ N).

Then limn→∞ An x = Ax for all x ∈ D(A). In fact, An x − Ax = nR(n, A)Ax − Ax → 0 by Lemma 3.3.12. Now it follows from Corollary 3.6.3 that A generates a contractive C0 -semigroup.

3.7

Holomorphic Semigroups

This section is devoted to the study of holomorphic semigroups. This class of semigroups plays an important role in the theory of evolution equations. Indeed, the modern treatment of linear and nonlinear parabolic problems is based on the theory of holomorphic semigroups. When compared with arbitrary C0 -semigroups, holomorphic C0 -semigroups show many special properties. We only mention here a) characterization results involving only a resolvent estimate (for example, Theorem 3.7.11 and Corollary 3.7.17), b) regularity properties of solutions of the Cauchy problem (Corollary 3.7.21 and applications in Chapters 6 and 7), c) determination of the asymptotic behaviour of the C0 -semigroup by spectral conditions on the generator (Theorem 5.1.12 and Theorem 5.6.5). Throughout this section, let Σθ := {z ∈ C \ {0} : | arg z| < θ} be the sector in the complex plane of angle θ ∈ (0, π]. Recall from Deﬁnition 3.2.5 that a semigroup T was deﬁned to be a strongly continuous mapping (0, ∞) → L(X) satisfying a) T (t + s) = T (t)T (s) for all s, t > 0; b) T (t) ≤ c for all t ∈ (0, 1] and some c > 0; and c) T (t)x = 0 for all t > 0 implies x = 0. Recall also from Corollary 3.3.11 that a semigroup is a C0 -semigroup if and only if its generator A is densely deﬁned. Deﬁnition 3.7.1. Let θ ∈ (0, π2 ]. A semigroup T on X is called holomorphic of angle θ if it has a holomorphic extension to Σθ which is bounded on Σθ ∩ {z ∈ C : |z| ≤ 1} for all θ ∈ (0, θ). If no confusion seems likely, we denote the extension of T to Σθ also by T . Also, if we do not want to specify the angle θ in Deﬁnition 3.7.1, we call a semigroup T holomorphic if it is holomorphic of angle θ for some θ ∈ (0, π2 ]. Proposition 3.7.2. Let θ ∈ (0, π2 ] and let T be a semigroup on X with generator A. Assume that T is holomorphic of angle θ. Then the following hold: a) T (z + z ) = T (z)T (z )

(z, z ∈ Σθ ).

b) For all θ ∈ (0, θ) there exist M ≥ 0, ω ≥ 0 such that T (z) ≤ M eω Re z for all z ∈ Σθ .

3.7. HOLOMORPHIC SEMIGROUPS

149

c) Let α ∈ (−θ, θ). Denote by Tα the semigroup given by Tα (t) := T (eiα t) (t ≥ 0). Then eiα A is the generator of Tα . d) If T is a C0 -semigroup, then lim

z→0,z∈Σθ

T (z)x = x

for all x ∈ X and all θ ∈ (0, θ). Proof. a) For ﬁxed z ∈ (0, ∞) consider the holomorphic functions z → T (z + z ) and z → T (z)T (z ) for z ∈ Σθ . Since the two functions coincide on (0, ∞), the identity theorem for holomorphic functions implies that T (z + z ) = T (z)T (z ) for z ∈ Σθ and z ∈ (0, ∞). For ﬁxed z ∈ Σθ the two holomorphic functions z → T (z + z ) and z → T (z)T (z ) coincide for z ∈ (0, ∞) and the assertion follows from the identity theorem for holomorphic functions. b) Let θ ∈ (0, θ), M :=

T (z) ,

sup

ω := max{log M, 0}.

z∈Σθ ,|z|≤1

Then T (z) ≤ M eω |z| for all z ∈ Σθ . In fact, let z = teiβ where |β| ≤ θ . Applying the proof of Theorem 3.1.7 a) to Tβ one obtains

T (z) = Tβ (t) ≤ M eω|z| . Since |z| ≤ Re z/ cos θ for z ∈ Σθ , the claim follows with ω := ω / cos θ. c) Let Aα be the generator of Tα . For R > 0, let ΓR be the contour consisting of the line segments {t : 0 ≤ t ≤ R} and {teiα : 0 ≤ t ≤ R} and the arc {Reiϕ : 0 ≤ ϕ ≤ α}. Cauchy’s theorem implies that ΓR exp(−λe−iα z)T (z)x dz = 0 for λ > 0 and x ∈ X. Letting R → ∞, we obtain ∞ ∞ iα iα −λt e R(λ, Aα )x = e e Tα (t)x dt = exp(−λe−iα t)T (t)x dt 0

0

= R(λe−iα , A)x, by Theorem 3.1.7. This implies that Aα = eiα A. d) Let θ ∈ (0, θ). By b) there exist ω ≥ 0 and M ≥ 0 such that e−ωz T (z) ≤ M for all z ∈ Σθ . It follows from Proposition 2.6.3 b) that lim

z→0,z∈Σθ

e−ωz T (z)x = x

for all x ∈ X. This implies the claim. We note that in the situation of Proposition 3.7.2, Tα is a C0 -semigroup for each α ∈ (−θ, θ) whenever T is a C0 -semigroup; i.e., if D(A) is dense. Next, we deﬁne bounded holomorphic semigroups.

150

3. CAUCHY PROBLEMS

Deﬁnition 3.7.3. Let θ ∈ (0, π2 ]. A semigroup T is called a bounded holomorphic semigroup of angle θ if T has a bounded holomorphic extension to Σθ for each θ ∈ (0, θ). We denote the extension of T to Σθ by T again. If we do not want to specify the angle, we call T a bounded holomorphic semigroup if T is a bounded holomorphic semigroup of angle θ for some θ ∈ (0, π2 ]. Some caution is required concerning this terminology. If T is a bounded semigroup which is holomorphic, then it is not necessarily a bounded holomorphic semigroup since it is just bounded on R+ and may not be bounded on a sector. For example, let X = C and T (t) = eit (t ≥ 0). The following result is an immediate consequence of Proposition 3.7.2 b). Proposition 3.7.4. An operator A generates a holomorphic semigroup if and only if there exists ω ≥ 0 such that A − ω generates a bounded holomorphic semigroup. Next we give some examples of holomorphic semigroups. Example 3.7.5 (Selfadjoint operators). Let A be a selfadjoint operator on a Hilbert space H. Assume that A is bounded above by ω. Then A generates a bounded holomorphic C0 -semigroup of angle π/2 satisfying

T (z) ≤ eω Re z

(Re z > 0).

Proof. By the Spectral Theorem B.13, we can assume that H = L2 (Ω, μ) and that A is given by D(A) = {f ∈ H : mf ∈ H}, Af = m · f, where m : Ω → (−∞, ω] is measurable. It is easy to see that (T (z)f )(x) := ezm(x) f (x)

(x ∈ Ω, Re z > 0),

deﬁnes a holomorphic C0 -semigroup on H, whose generator is A. Example 3.7.6 (Gaussian semigroup). Let X be one of the spaces Lp (Rn ) (1 ≤ p < ∞), C0 (Rn ) or BUC(Rn ). Then 2 (G(t)f )(x) := (4πt)−n/2 f (x − y)e−|y| /4t dy (t > 0, f ∈ X, x ∈ Rn ) Rn

deﬁnes a bounded holomorphic C0 -semigroup of angle π/2 on X. Its generator is the Laplacian ΔX on X with maximal domain; i.e., D(ΔX ) = {f ∈ X : Δf ∈ X}, ΔX f = Δf, where we identify X with a subspace of D(Rn ) , and Δf = pendix E).

n j=1

Dj2 f (see Ap-

3.7. HOLOMORPHIC SEMIGROUPS

151

Proof. a) Let kt ∈ S(Rn ) be given by kt (x) =

1 (4πt)

n/2

e−|x|

2

(t > 0, x ∈ Rn ).

/4t

2

Then G(t)f = kt ∗ f ∈ X. Note that F kt = ht , where ht (x) := e−t|x| . Hence, ht+s = ht hs . Recall that F is an isomorphism from S(Rn ) onto S(Rn ) such that F (ψ ∗ f ) = F ψ · F f for all f ∈ S(Rn ) , ψ ∈ S(Rn ) (see Appendix E). Thus, F (G(t)f ) = ht · Ff . It follows that G(t + s) = G(t)G(s) (t, s > 0). Since {kt : t > 0} is an approximate identity (see Lemma 1.3.3), it follows that

kt ∗ f − f X → 0 as t ↓ 0 for all f ∈ X. We have shown that G is a C0 -semigroup. b) The function kz is also deﬁned for Re z > 0 and z → kz : C+ → L1 (Rn ) is a holomorphic function satisfying supz∈Σθ kz L1 (Rn ) < ∞ for each 0 < θ < π/2. This shows that G(z)f := kz ∗ f deﬁnes a holomorphic extension of G to C+ with values in L(X) such that supz∈Σθ G(z) < ∞ for each 0 < θ < π/2. c) We identify the generator of G. First step: Let f ∈ X such that Δf ∈ X. We show that Δ(G(t)f ) = G(t)(Δf ). Let m(x) = −|x|2 . Then F(Δ(G(t)f )) = mF(G(t)f ) = mht Ff = ht mFf = ht F(Δf ) = F (G(t)Δf ). This proves the claim. Second step: Let ψ ∈ S(Rn ). Then 0

t

G1 (s)Δψ ds = G1 (t)ψ − ψ,

where G1 is the Gaussian semigroup on L1 (Rn ). In fact, t t F G1 (s)Δψ ds (x) = F (G1 (s)Δψ)(x) ds 0

0

=

t

2

e−s|x| (−|x|)2 (F ψ)(x) ds

0 −t|x|2

− 1)(Fψ)(x)

=

(e

=

(F(G1 (t)ψ − ψ))(x).

The claim follows from the uniqueness of Fourier transforms. Third step: Let f ∈ X, t > 0. We show that t Δ G(s)f ds = G(t)f − f. 0

Then it follows from Corollary 3.1.13 (using also the ﬁrst step) that the generator of G is the Laplacian with maximal domain. Let ψ ∈ S(Rn ). Then Fubini’s theorem

152

3. CAUCHY PROBLEMS

gives Δψ, G(s)f = G1 (s)Δψ, f and

ψ, Δ

t

G(s)f ds

=

0

Δψ,

G(s)f ds

0

t

=

t

0 t

= 0

Δψ, G(s)f ds G1 (s)Δψ, f ds

= 0

t

G1 (s)Δψ ds, f

= G1 (t)ψ − ψ, f = ψ, G(t)f − f , by the second step. This proves the claim. Remark 3.7.7. a) Let X = Lp (Rn ) (1 < p < ∞). Then D(ΔX ) = W 2,p (Rn ). In fact, R(1, ΔX ) is given by the Fourier multiplier x → (1 + |x|2 )−1 (see Appendix E). It is easy to see that the function mjk (x) := −xj xk (1 + |x|2 )−1 satisﬁes the condition of Mikhlin’s theorem (Theorem E.3), so it is a Fourier multiplier for Lp (Rn ) for j, k ∈ {1, . . . , n}. One veriﬁes easily that F (Dj Dk R(1, ΔX )f ) = mjk F f for all f ∈ S(Rn ). Thus Dj Dk R(1, ΔX ) : S(Rn ) → S(Rn ) has a bounded extension to Lp (Rn ). This means that D(ΔX ) = R(1, ΔX )Lp (Rn ) ⊂ W 2,p (Rn ). b) If X = L1 (Rn ), C0 (Rn ) or BUC(Rn ), then D(ΔX ) is not a classical function space. For example, if X = L1 (Rn ), then D(ΔX ) W 2,1 (Rn ). Similarly, if X = C0 (Rn ), then D(ΔX ) contains functions which are not in C 2 (Rn ). See [DL90, Chapter II, Section 3, Remark 5]. Modifying the Banach space in Example 3.7.6 we obtain an example of a holomorphic semigroup which is not a C0 -semigroup. Another example will be given in Chapter 6. Let Cb (Rn ) be the Banach space of all bounded continuous complex-valued functions on Rn with the supremum norm. Example 3.7.8. Let X = Cb (Rn ) or L∞ (Rn ). Deﬁne the Gaussian semigroup G on X as in Example 3.7.6. Then G is a bounded holomorphic semigroup which is not a C0 -semigroup. Its generator is the operator ΔX deﬁned as in Example 3.7.6. Example 3.7.9 (Poisson semigroup). Let X be one of the spaces considered in Example 3.7.6. Let pt (x) = cn

(t2

t + |x|2 )(n+1)/2

(x ∈ Rn , t > 0)

3.7. HOLOMORPHIC SEMIGROUPS

153

where cn := Γ((n + 1)/2)/π (n+1)/2 . Then pt ∈ L1 (Rn ) and (Fpt )(x) = e−t|x| (t > 0). Similarly to Example 3.7.6, one shows that T (t)f := pt ∗ f

(t > 0)

deﬁnes a C0 -semigroup on X, which is called the Poisson semigroup. It is again a bounded holomorphic C0 -semigroup of angle π/2 on X. Its holomorphic extension to the sector Σπ/2 is bounded on Σθ for θ < π/2 and is given by T (z)f := pz ∗ f, where pz (x) := cn

(z 2

z + |x|2 )(n+1)/2

for x ∈ Rn and Re z > 0. Its generator is the operator AX deﬁned by AX f D(AX )

:=

F −1 (−| · |Ff ),

:= {f ∈ X : F −1 (| · |Ff ) ∈ X},

where F now denotes the Fourier transform in S(Rn ) . A more explicit description of the operator ALp (R) is of particular interest. Observing that −(−i sign(ξ))(iξ) = −|ξ| (ξ ∈ R), it follows from (E.19) that ∂ , ∂x where H denotes the Hilbert transform deﬁned by 1 f (x − y) (Hf )(x) := lim dy. ε→0,R→∞ π ε≤|y|≤R y ALp (R) = −H

Since the Hilbert transform acts as a bounded operator on Lp (R) for 1 < p < ∞ by Proposition E.5, it follows that the domain D(ALp (R) ) of ALp (R) coincides with W 1,p (R). However, ALp (R) is not a ﬁrst-order diﬀerential operator. We shall return to the relationship between the Poisson and Gaussian semigroups in Example 3.8.5. Remark 3.7.10. Let G be the Gaussian semigroup on L1 (Rn ). It follows from the explicit formula for kz given in Example 3.7.6a) that for Re z > 0 n/2 |z|

G(z) L(L1 (Rn )) = kz L1 (Rn ) = . (3.44) Re z Thus, by Proposition 3.7.2c), Gα deﬁned by Gα (t) := G(eiα t) for |α| < π2 is an example of a bounded C0 -semigroup, which due to (3.44), is not a contraction semigroup.

154

3. CAUCHY PROBLEMS

The following characterization of a bounded holomorphic semigroup in terms of a single resolvent estimate for its generator is of fundamental importance. Theorem 3.7.11. Let A be an operator on X and θ ∈ (0, π2 ]. The following assertions are equivalent: (i) A generates a bounded holomorphic semigroup of angle θ. (ii) Σθ+ π2 ⊂ ρ(A) and sup λ∈Σθ+ π −ε

λR(λ, A) < ∞ for all ε > 0.

2

Proof. In order to prove the assertion (i) ⇒ (ii), note that if λ0 ∈ ρ(A) and λ → R(λ, A) has a holomorphic extension to some open connected set Ω containing λ0 , then by Proposition B.5, Ω ⊂ ρ(A) and the extension is the resolvent. Thus (i) ⇒ (ii) follows immediately from Theorem 2.6.1. In order to prove the converse implication (ii) ⇒ (i), note that by Theorem 2.6.1 there exists a holomorphic function T : Σθ → L(X) which is bounded on Σα for 0 < α < θ such that ∞ R(λ, A) = e−λt T (t) dt (Re λ > 0). (3.45) 0

The proof of Theorem 3.1.7 shows that T is a semigroup. This proves (i). In particular, a densely deﬁned operator A satisfying (ii) of Theorem 3.7.11 is a Hille-Yosida operator, by Example 3.5.9 c). Moreover, the bounded holomorphic semigroup T generated by A is a C0 -semigroup if and only if D(A) is dense in X, by Corollary 3.3.11. We note from (2.13) that the semigroup T generated by A is given by 1 T (z) = eλz R(λ, A) dλ (z ∈ Σα ), (3.46) 2πi Γ if 0 < α < θ, where the contour Γ consists of

Γ± := {re±γ : δ ≤ r} and Γ0 := {δeiθ : |θ | ≤ γ} with α +

π 2

<γ <θ+

π 2

and δ > 0.

If we do not want to specify the angle of holomorphy in the above theorem, then it suﬃces to verify condition (ii) above on a right half-plane. In fact, the following holds true. Corollary 3.7.12. For an operator A on X the following are equivalent: (i) A generates a bounded holomorphic semigroup on X.

3.7. HOLOMORPHIC SEMIGROUPS

155

(ii) {z ∈ C : Re z > 0} ⊂ ρ(A) and M := sup λR(λ, A) < ∞. Re λ>0

Proof. By Theorem 3.7.11, we only have to prove that (ii) implies the second 1 assertion of Theorem 3.7.11. Set c := 2M . For s ∈ R \ {0} and −c|s| < r ≤ 0, let λ := c|s| + r + is. Then |λ − (r + is)| = c|s| ≤ 12 R(λ, A) −1 . By Corollary B.3, r + is ∈ ρ(A) and

(r + is)R(r + is, A) ≤ 2M

|r + is| ≤ 2M (c + 1). |s|

Thus, (ii) is satisﬁed with θ = arctan c. Remark 3.7.13. Suppose that an operator A satisﬁes the equivalent conditions (i) and (ii) of the above corollary. For Y := D(A) let AY be the part of A deﬁned as in (3.27). It follows from Lemma 3.3.12 and Corollary 3.7.12 that in this case AY generates a bounded holomorphic C0 -semigroup TY on Y . Moreover, TY (t) = T (t)|Y for t ≥ 0, where T is the semigroup generated by A. Corollary 3.7.14. Let A be an operator on X such that σ(A) ⊂ iR. Assume that there exists a constant M > 0 such that

R(λ, A) ≤

M | Re λ|

(Re λ = 0).

(3.47)

Then A2 generates a bounded holomorphic semigroup of angle π/2 on X. Proof. Let θ ∈ (0, π/2] and λ ∈ Σθ+π/2 . Then there exist r > 0 and ϕ ∈ (0, π/2) such that λ = r 2 e2iϕ . Observe that λ − A2 = (reiϕ + A)(reiϕ − A). The assumption implies that λ ∈ ρ(A2 ) and that R(λ, A2 ) = −R(reiϕ , A)R(−reiϕ , A). The resolvent estimate (3.47) implies that

R(λ, A2 ) ≤

M2 M2 1 = (r cos ϕ)2 (cos ϕ)2 |λ|

(λ ∈ Σθ+π/2 ).

Hence, the assertion follows from Theorem 3.7.11. Applying Corollary 3.7.14 to the situation of generators of bounded C0 -groups we immediately obtain the following result.

156

3. CAUCHY PROBLEMS

Corollary 3.7.15. Let A be the generator of a bounded C0 -group U on X. Then A2 generates a bounded holomorphic C0 -semigroup T of angle π/2 on X. Moreover, for t > 0, T (t) = R

where kt (s) = (4πt)−1/2 e−|s|

2

/4t

kt (s)U (s) ds,

.

Proof. The fact that A2 generates a bounded holomorphic C0 -semigroup of angle π/2 is immediate from Corollary 3.7.14. Deﬁne T (0)x = x and √ T (t)x = kt (s)U (s)x ds = k1 (s)U (s t) ds R R∞ = kt (s)(U (s)x + U (−s)x) ds. 0

Then T is strongly continuous by the dominated convergence theorem. By Proposition 1.6.8, Tˆ (λ)

!√ " !√ " 1 1 √ R λ, A + √ R λ, −A 2 λ 2 λ = R(λ, A2 ) =

for λ > 0. Thus T is a C0 -semigroup generated by A2 . Let X be any of the spaces considered in Example 3.7.6 in the case n = 1, and let U be the shift group: (U (t)f )(x) = f (x − t) (see Example 3.3.10). Then the C0 -semigroup constructed in Corollary 3.7.15 is the Gaussian semigroup. The following proposition will be useful in Chapter 6 when we are dealing with the holomorphic semigroup generated by the Laplacian subject to Dirichlet boundary conditions on spaces of continuous functions. Note that A is not necessarily densely deﬁned. Proposition 3.7.16. Let A be a dissipative operator on X and assume that A generates a holomorphic semigroup T on X. Then T (t) ≤ 1 for all t > 0. Proof. By the remark after Theorem 3.7.11, A is a Hille-Yosida operator. Hence, there exists λ0 > 0 with λ0 ∈ ρ(A). By the proof of the Lumer-Phillips theorem, we have (0, ∞) ⊂ ρ(A) and ||λR(λ, A)|| ≤ 1 for all λ > 0. By the proof of Corollary 3.3.6, we have T (t)x = limn→∞ (I − nt A)−n x for t > 0 and x ∈ X. Hence,

! "−n

≤1

T (t) ≤ lim sup I − nt A n→∞

for t > 0. Applying Corollary 3.7.12 to the operator A − ω for suitable ω, in view of Proposition 3.7.4, we obtain the following characterization results for holomorphic semigroups.

3.7. HOLOMORPHIC SEMIGROUPS

157

Corollary 3.7.17. Let A be an operator on X, and a ∈ R. Then A generates a holomorphic semigroup if and only if there exists r > 0 such that {λ ∈ C : Re λ > a, |λ| > r} ⊂ ρ(A) and sup λR(λ, A) < ∞.

Re λ>a |λ|>r

Modifying the proof of Corollary 3.7.12 to the situation of holomorphic semigroups which are not necessarily bounded, we obtain the following result. Corollary 3.7.18. Let A be the generator of a semigroup T on X. Then T is holomorphic if and only if there exists r > 0 such that {is : s ∈ R, |s| > r} ⊂ ρ(A) and sup sR(is, A) < ∞. |s|>r

We now characterize bounded holomorphic semigroups in terms of the behaviour of tAT (t) for positive t. Theorem 3.7.19. Let T be a bounded semigroup on X with generator A. Then T is a bounded holomorphic semigroup if and only if T (t)x ∈ D(A) for all t > 0, x ∈ X, and sup tAT (t) < ∞. (3.48) t>0

Proof. Suppose that T is a bounded holomorphic semigroup of angle θ. Then T is norm-diﬀerentiable on (0, ∞), so T (t)x ∈ D(A) for t > 0 and x ∈ X. By Cauchy’s integral formula for the derivative, 1 T (z) AT (t) = T (t) = dz. 2πi |z−t|=t sin θ/2 (z − t)2 Hence,

tAT (t) ≤ (sin θ/2)−1 sup T (z) < ∞. z∈Σθ/2

Conversely, let M := supt>0 { T (t) , tAT (t) }. By assumption, T (t)x ∈ D(A) for all t > 0 and x ∈ X. Since An T (t) = (AT (t/n))n ∈ L(X) for n ∈ N, we have

n

n

A T (t) (AT (t/n))n ( nt M )n eM

=

≤ ≤ for all n ∈ N.

n!

n! n! t n ∞ T (t) t If |z| < 2eM , then T˜(t + z) := n=0 z n A n! converges in norm and T˜ (t + z) ≤ t ˜ 1 + M . Then for s ∈ [0, 2eM ), T (t + s) = T (t + s). In fact, let x ∈ D(A) and t u(s) := T˜(t + s)x. Then u(s) ∈ D(A) and u (s) = Au(s) (s ∈ [0, 2eM )). Since u(0) = T (t)x, it follows from Proposition 3.1.11 that u(s) = T (s)T (t)x = T (t+s)x. By uniqueness of analytic extensions, we obtain a bounded, holomorphic extension 1 T˜ of T to the sector Σθ where θ = arctan 2eM .

158

3. CAUCHY PROBLEMS

Remark 3.7.20. A slight modiﬁcation of the above proof implies the following assertion: Let A be the generator of a bounded holomorphic semigroup T on X. Then T (t)x ∈ D(An ) for all x ∈ X, t > 0 and n ∈ N, and we have sup ||tn An T (t)|| < ∞ t>0

(n ∈ N).

Theorem 3.7.19 and Remark 3.7.20 have several important consequences for the regularity of the solution of the associated Cauchy problem. In contrast to the situation of C0 -semigroups where we obtain a classical solution of the Cauchy problem only if the initial condition x belongs to the domain D(A) of the generator A, we see that if A generates a holomorphic C0 -semigroup, then for all x ∈ X we obtain a solution which is diﬀerentiable for t > 0. More precisely, the following holds. Corollary 3.7.21. Let x ∈ X and assume that A generates a holomorphic C0 semigroup on X. Then there exists a unique function u ∈ C ∞ ((0, ∞), X) ∩ C(R+ , X) ∩ C((0, ∞), D(A)) satisfying u (t) = Au(t) (t > 0), u(0) = x. The phenomenon described in Corollary 3.7.21 is frequently called the smoothing eﬀect of holomorphic C0 -semigroups. We now consider the inhomogeneous Cauchy problem u (t) = Au(t) + f (t) (t ∈ [0, τ ]), (ACPf ) u(0) = x, associated with the generator A of a holomorphic semigroup, and we establish that the variation of constants formula holds when x ∈ D(A). Proposition 3.7.22. Let A be the generator of a holomorphic semigroup T on X. Let f ∈ L1 ((0, τ ), X) and x ∈ D(A). Then (ACPf ) has a unique mild solution u which is given by

t

u(t) = T (t)x + 0

T (t − s)f (s) ds.

(t > 0)

Proof. By the remark following Theorem 3.7.11, A is a Hille-Yosida operator. Hence, the ﬁrst assertion follows from the Da Prato-Sinestrari Theorem 3.5.2. The formula for u follows from Lemma t 3.2.9, since the once integrated semigroup generated by A is given by S(t)x = 0 T (s)x ds and the derivative of S ∗ f is easily seen to be T ∗ f .

3.7. HOLOMORPHIC SEMIGROUPS

159

The following perturbation result for generators of holomorphic semigroups is particularly useful when dealing with lower order perturbations of diﬀerential operators. Consider a closed operator A on X. Then a mapping B : D(A) → X is continuous (with respect to the graph norm on D(A)) if and only if

Bx ≤ c Ax + b x

(x ∈ D(A)),

for suitable c, b ≥ 0. One frequently says that B is a relatively bounded perturbation of A in that case. Theorem 3.7.23. Let A be the generator of a holomorphic semigroup on X. Let B : D(A) → X be an operator such that for every ε > 0 there exists a constant b ≥ 0 such that

Bx ≤ ε Ax + b x

(x ∈ D(A)). (3.49) Then A + B generates a holomorphic semigroup. Proof. Assume ﬁrst that A generates a bounded holomorphic semigroup on X. Corollary 3.7.12 implies that there exists θ ∈ (0, π2 ] such that Σθ+π/2 ⊂ ρ(A) and

λR(λ, A) =: M < ∞.

sup λ∈Σθ+π/2

It follows from the assumption on B that given ε > 0, there exists b ≥ 0 such that for x ∈ X

BR(λ, A)x

≤ ε AR(λ, A)x + b R(λ, A)x

= ε λR(λ, A)x − x + b R(λ, A)x

bM ≤ ε(M + 1) x +

x

(λ ∈ Σθ+π/2 ). |λ|

Choosing ε := (2(M + 1))−1 , it follows that BR(λ, A) < 3/4 whenever |λ| > 4bM and hence that I − BR(λ, A) is invertible, with (I − BR(λ, A))−1 < 4. Since λ − (A + B) = (I − BR(λ, A))(λ − A) (λ ∈ Σθ+π/2 ), it follows that λ − (A + B) is invertible for λ ∈ Σθ+π/2 with |λ| > 4bM and that

R(λ, A + B) ≤

4M |λ|

(λ ∈ Σθ+π/2 , |λ| > 4bM ).

By Corollary 3.7.17, A + B generates a holomorphic semigroup. If A generates a holomorphic semigroup which is not bounded, choose ω ∈ R such that A−ω generates a bounded holomorphic semigroup on X (see Proposition 3.7.4). The ﬁrst part of the proof implies that A + B − ω, and therefore also A + B, generates a holomorphic semigroup on X. Note that in the situation of Theorem 3.7.23, A and A + B have the same domain. Thus, the semigroup generated by A is a C0 -semigroup if and only if the one generated by A + B is a C0 -semigroup.

160

3. CAUCHY PROBLEMS

Example 3.7.24 (First order perturbations of the Laplacian). Consider the Gaussian semigroup G with generator ΔX on any of the spaces X of Example 3.7.6. For j = 1, 2, . . . , n, let Uj be the C0 -group on X deﬁned by (Uj (t)f )(x) = f (x1 , . . . , xj − t, . . . , xn ) with generator −Dj , and let Tj be the holomorphic C0 -semigroup with generator Dj2 (see Corollary 3.7.15). Then T1 , . . . , Tn commute and G(t) = T1 (t) · · · Tn (t). Since (Tj (t)f )(x)

=

(Dj Tj (t)f )(x)

=

2

(4πt)−1/2 e−(xj −s) /4t f (x1 , . . . , s, . . . , xn ) ds, R (xj − s) −(xj −s)2 /4t √ 3/2 e − f (x1 , . . . , s, . . . , xn ) ds. 4 πt R

Hence by Young’s inequality (see Proposition 1.3.2),

Dj Tj (t) L(X)

≤ = =

2 |s| √ 3/2 e−s /4t ds R 4 πt ∞ 2 2 √ ue−u du πt 0 1 √ . πt

Since Tj (t) L(X) = 1, it follows that Dj G(t) L(X) ≤

Dj R(λ, ΔX ) L(X)

=

0

≤ =

1 √ π c √ λ

∞

√1 , πt

and hence

e−λt Dj G(t) dt

∞ 0

L(X)

e−λt √ dt t

for all λ > 0, for some constant c. Now let B be a ﬁrst-order diﬀerential operator of the form (Bf )(x) =

n

bj (x)(Dj f )(x) + b0 (x)f (x),

j=1

for some bj ∈ L∞ (Rn ) (j = 0, 1, . . . , n) (and bj continuous if X = C0 (Rn ); bj

3.7. HOLOMORPHIC SEMIGROUPS

161

uniformly continuous if X = BUC(Rn )). For f ∈ D(ΔX ),

Bf = BR(λ, ΔX )(λ − ΔX )f

n ≤

bj ∞ Dj R(λ, ΔX ) L(X) (λ − ΔX )f + b0 ∞ f

j=1

⎛ ⎞ n n √

bj ∞ c √ ≤

ΔX f + ⎝

bj ∞ c λ + b0 ∞ ⎠ f . λ j=1 j=1 Since λ may be chosen arbitrary large, this establishes (3.49), and Theorem 3.7.23 shows that ΔX + B generates a holomorphic C0 -semigroup on X. We shall extend Example 3.7.24 to more general diﬀerential operators in Section 7.2. We now prove a second perturbation theorem for holomorphic semigroups where the norm estimate (3.49) is replaced by compactness. Theorem 3.7.25 (Desch-Schappacher). Let A be the generator of a holomorphic C0 -semigroup T . Let B : D(A) → X be a compact linear operator where D(A) carries the graph norm. Then A + B generates a holomorphic C0 -semigroup S. Moreover, T (t) − S(t) is compact for each t > 0. Proof. By Corollary 3.7.17, there exist r > 0, M > 0 such that λ ∈ ρ(A) and

λR(λ, A) L(X) ≤ M whenever |λ| ≥ r, Re λ > 0. Since D(A) is dense in X, it follows that lim |λ|→∞ λR(λ, A)x = x for all x ∈ X (see Lemma 3.3.12). Since Re λ>0 λR(λ, A)x − x = AR(λ, A)x, it follows that lim |λ|→∞ R(λ, A)x = 0 in D(A) for all Re λ>0 x ∈ X. By Proposition B.15, the convergence is uniform on compact subsets of X. Since B : D(A) → X is compact, it follows that lim R(λ, A)B L(D(A)) = 0.

|λ|→∞ Re λ>0

Consequently, there exists r1 ≥ r such that R(λ, A)B L(D(A)) ≤ 12 whenever |λ| ≥ r1 , Re λ > 0. Denote by ID(A) the identity map on D(A). It follows that (ID(A) − R(λ, A)B)−1 exists in L(D(A)) and

(ID(A) − R(λ, A)B)−1 L(D(A)) ≤ 2

(|λ| ≥ r1 , Re λ > 0).

Thus (λ − (A + B)) = (λ − A)(I − R(λ, A)B) is invertible and R(λ, A + B) = (ID(A) − R(λ, A)B)−1 R(λ, A)

(Re λ > 0, |λ| ≥ r1 ).

Moreover, for |λ| ≥ r1 , Re λ > 0,

R(λ, A + B) L(X,D(A))

≤ ≤

2 R(λ, A) L(X,D(A)) M M1 := 2 +1+M , r1

162

3. CAUCHY PROBLEMS

since

R(λ, A)x D(A)

= R(λ, A)x X + AR(λ, A)x X M ≤

x X + λR(λ, A)x − x X |λ| M ≤

x X + (M + 1) x X . r1

Hence for x ∈ X, |λ| ≥ r1 , Re λ > 0,

λR(λ, A + B)x X

= (A + B)R(λ, A + B)x − x X ≤ R(λ, A + B)x D(A) + B L(D(A),X) R(λ, A + B)x D(A) + x X ! " ≤ M1 (1 + B L(D(A),X) ) + 1 x X .

Now it follows from Corollary 3.7.17 that A + B generates a holomorphic C0 semigroup. It remains to show the last assertion. Denote by K(X) the closed subspace of L(X) consisting of all compact operators and by q : L(X) → L(X)/K(X) the quotient mapping. For λ > λ0 := max{ω(T ), ω(S)}, we have

∞ 0

e−λt (S(t) − T (t)) dt = R(λ, A + B) − R(λ, A) = R(λ, A + B)[(λ − A) − (λ − A − B)]R(λ, A) = R(λ, A + B)BR(λ, A) ∈ K(X).

Since S and T are holomorphic, the function U := S − T is norm-continuous on (0, ∞). Since (q ◦ U )(λ) = q(U (λ)) = 0 for all λ > λ0 , it follows from the uniqueness theorem that q ◦ U ≡ 0; i.e., U (t) ∈ K(X) for all t > 0. We should point out that in the situation of Theorem 3.7.25 the norm estimate (3.49) is not true in general; see the Notes for further information.

3.8

Fractional Powers

A particularly interesting example of a holomorphic semigroup is the family of fractional powers of a sectorial operator. Consider an operator B on X for which (−∞, 0] ⊂ ρ(B) and supλ≤0 (1 − λ) R(λ, B) < ∞. It follows from Corollary B.3 that B is sectorial in the sense that there exist constants M > 0, ϕ ∈ (0, π) such that M σ(B) ⊂ Σϕ and R(λ, B) ≤ , λ ∈ C\Σϕ . (3.50) 1 + |λ|

3.8. FRACTIONAL POWERS

163

Let Γ be the downward path consisting of {se±iϕ : s ≥ r} and {reiθ : −ϕ ≤ θ ≤ ϕ}, where r > 0 is chosen so small that σ(B) is to the right of Γ. Then the fractional powers (B −z )Re z>0 of B are deﬁned by 1 B −z := λ−z R(λ, B) dλ, (Re z > 0). (3.51) 2πi Γ Here, λ−z = exp(−z(log |λ|+iθ)) if λ = |λ|eiθ , −π < θ < π. Note that the integral is absolutely convergent, uniformly for z in compact subsets of C+ , and therefore z → B −z is holomorphic from C+ to L(X). By Cauchy’s theorem, the deﬁnition of B −z is independent of the choices of ϕ and r. Moreover, when z = n ∈ N, Γ may be replaced by a closed contour around 0 and then the residue theorem shows that n−1 1 d B −z = − R(λ, B) = (−1)n R(0, B)n = B −n (n − 1)! dλ λ=0

in the usual sense. Now, assume for the time being that 0 < Re z < 1. Then 1 B −z = lim λ−z R(λ, B) dλ ϕ↑π 2πi Γ r↓0 e−iπz ∞ −z eiπz ∞ −z −1 = − s (s + B) ds + s (s + B)−1 ds 2πi 0 2πi 0 sin πz ∞ −z = s (s + B)−1 ds. (3.52) π 0 In the particular case when X = C and B = 1, (3.52) gives ∞ π s−z (s + 1)−1 ds = (0 < Re z < 1). sin πz 0

(3.53)

Hence,

B −z ≤

| sin πz| π

0

∞

s− Re z

M | sin πz| ds = M 1+s sin(π Re z)

(0 < Re z < 1).

(3.54)

Theorem 3.8.1. Let B be an operator on X such that (−∞, 0] ⊂ ρ(B) and supλ≤0 (1 − λ) R(λ, B) < ∞. Then the family (B −z )Re z>0 deﬁnes a holomorphic semigroup on X of angle π/2. If B is densely deﬁned, then (B −z )Re z>0 is a holomorphic C0 -semigroup. Proof. We have observed above that z → B −z is holomorphic for Re z > 0, and it follows from (3.54) that B −z is uniformly bounded in Σθ ∩ {z ∈ C : |z| < 1} for θ ∈ (0, π/2). To verify the semigroup property, let Γ and Γ be two contours as in

164

3. CAUCHY PROBLEMS

the deﬁnition of B −z , with Γ to the left of Γ . For Re z1 > 0 and Re z2 > 0, the resolvent identity and Fubini’s theorem give 1 B −z1 B −z2 = λ−z1 μ−z2 R(λ, B)R(μ, B) dμ dλ (2πi)2 Γ Γ 1 R(λ, B) − R(μ, B) −z1 −z2 = λ μ dμ dλ (2πi)2 Γ Γ μ−λ 1 1 μ−z2 = λ−z1 dμ R(λ, B) dλ 2πi Γ 2πi Γ μ − λ 1 1 λ−z1 −z2 + μ dλ R(μ, B) dμ 2πi Γ 2πi Γ λ − μ 1 = λ−z1 λ−z2 R(λ, B) dλ 2πi Γ = B −(z1 +z2 ) . Here, we have used Cauchy’s integral formula (after changing to a closed contour around 0) to see that 1 μ−z2 1 λ−z1 dμ = λ−z2 and dλ = 0. 2πi Γ μ − λ 2πi Γ λ − μ This proves the ﬁrst assertion. −z z→0 B Now suppose that B is densely deﬁned. We have to show that lim z∈Σ x= θ

x, for all θ ∈ (0, π/2) and all x ∈ X. Since B −z is uniformly bounded on Σθ ∩ {z ∈ C : |z| < 1}, we may assume that x ∈ D(B) (see Proposition B.15). For 0 < Re z < 1, it follows from (3.52) and (3.53) that " sin πz ∞ −z ! B −z x − x = s (s + B)−1 x − (s + 1)−1 x ds π 0 sin πz ∞ s−z = (s + B)−1 (I − B)x ds. π s + 1 0 Hence,

B −z x − x ≤ M

| sin πz| π

0

∞

s− Re z ds (I − B)x . (s + 1)2

−z z→0 B It follows that lim z∈Σ x − x = 0 for all θ ∈ (0, π/2). θ

Now suppose that B is an operator on X such that (−∞, 0) ⊂ ρ(B) and M := supλ<0 λR(λ, B) < ∞. We shall show that B has a special type of square root, which has interesting properties for semigroup generators. For ε > 0, (−∞, 0] ⊂ ρ(ε + B) and sup(1 − λ) R(λ, ε + B) = sup(1 − λ) R(λ − ε, B) ≤ M/ε. λ≤0

λ≤0

3.8. FRACTIONAL POWERS

165

Hence we can deﬁne (ε + B)−1/2 ∈ L(X) as above, and then ((ε + B)−1/2 )2 = (ε + B)−1 . In particular, (ε + B)−1/2 is injective. Let (ε + B)1/2 be the algebraic inverse of (ε + B)−1/2 , so D((ε + B)1/2 ) (ε + B)1/2 ((ε + B)−1/2 y)

Ran((ε + B)−1/2 ),

=

= y

(y ∈ X).

Then (ε+B)1/2 is a closed operator on X, ((ε+B)1/2 )2 = ε+B, and for x ∈ D(B), x = (ε + B)−1/2 ((ε + B)−1/2 (ε + B)x), so (ε + B)1/2 x = =

(ε + B)−1/2 (ε + B)x 1 ∞ −1/2 s (s + ε + B)−1 (ε + B)x ds. π 0

(3.55)

Proposition 3.8.2. Let B be a densely deﬁned operator on X such that (−∞, 0) ⊂ ρ(B) and supλ<0 λR(λ, B) < ∞. Then there is a unique closed operator B 1/2 such that a) (B 1/2 )2 = B, and b) For x ∈ D(B), B 1/2 x = lim(ε + B)1/2 x = ε↓0

1 π

∞

s−1/2 (s + B)−1 Bx ds.

0

Moreover, D(B) is a core for B 1/2 , and D(B 1/2 ) = D((ε + B)1/2 ) for all ε > 0. Proof. Since

s−1/2 (s + ε + B)−1 (ε + B)x ≤ M s−3/2 ( x + Bx )

(0 < ε < 1)

and

s−1/2 (s + ε + B)−1 (ε + B)x = s−1/2 (x − s(s + ε + B)−1 x)

≤

(M + 1) x s−1/2 ,

one can apply the dominated convergence theorem and take limits in (3.55) as ε ↓ 0. Thus, we let 1 ∞ −1/2 B 1/2 x := lim(ε + B)1/2 x = s (s + B)−1 Bx ds λ↓0 π 0

166

3. CAUCHY PROBLEMS

for x ∈ D(B). Moreover, for ε > 0, B 1/2 x − (ε + B)1/2 x " 1 ∞ −1/2 ! = s (s + B)−1 Bx − (s + ε + B)−1 (ε + B)x ds π 0 ε ∞ 1/2 = − s (s + ε + B)−1 (s + B)−1 x ds. π 0 Let Sε := −

ε π

∞

s1/2 (s + ε + B)−1 (s + B)−1 ds.

0

Then Sε ∈ L(X), ε

Sε ≤ π

∞

0

M2 ds = M 2 ε1/2 , s1/2 (s + ε)

by (3.53), and B 1/2 x − (ε + B)1/2 x = Sε x

(x ∈ D(B), ε > 0).

We deﬁne B 1/2 := (ε + B)1/2 + Sε

with

D(B 1/2 ) = D((ε + B)1/2 ).

Since (ε + B)1/2 is closed and Sε is bounded, B 1/2 is closed. Moreover, b) holds. Since (ε + B)1/2 is densely deﬁned and invertible, D(B) = D(((ε + B)1/2 )2 ) is a core for (ε + B)1/2 (see Appendix B) and hence for B 1/2 . It follows that the deﬁnition of B 1/2 is independent of ε. Let x ∈ D(B). Then (ε + B)1/2 x → B 1/2 x and B 1/2 (ε + B)1/2 x = (ε + B)x + Sε (ε + B)1/2 x → Bx as ε ↓ 0. Since B 1/2 is closed, it follows that B 1/2 x ∈ D(B 1/2 ) and (B 1/2 )2 x = Bx. Let y ∈ D((B 1/2 )2 ). By Lemma 3.3.12, ε(ε + B)−1 y → y and B(ε(ε + −1 B) y) = ε(ε + B)−1 (B 1/2 )2 y → (B 1/2 )2 y as ε → ∞. Since B is closed, y ∈ D(B). Thus B = (B 1/2 )2 . be any closed operator such that Bx = B 1/2 x for all x ∈ D(B) Finally, let B 2 2 +i and B = B. Then (B + i)(B − i) = B + I = B + I, which is invertible, so B 2 is invertible. Since B is densely deﬁned and ρ(B) is non-empty, D(B ) = D(B) (see Appendix B). Hence, B is the closure of B 1/2 |D(B) , and this is a core for B proves uniqueness. Now suppose that A is the generator of a bounded C0 -semigroup T on X. If 0 ∈ ρ(A), then the theory above can be applied to B := −A, so (−A)−z is deﬁned

3.8. FRACTIONAL POWERS

167

∞ for Re z > 0. Substituting (s − A)−1 = 0 e−st T (t) dt into (3.52), it is not diﬃcult to see that ∞ 1 (−A)−z = tz−1 T (t) dt (3.56) Γ(z) 0 for 0 < Re z < 1, and hence for Re z > 0 by uniqueness of holomorphic extensions. We shall not use this. Proposition 3.8.2 shows that (−A)1/2 is deﬁned whenever A generates a bounded C0 -semigroup. Theorem 3.8.3. Let A be the generator of a bounded C0 -semigroup T on X, and deﬁne ⎧ 2 ∞ ⎪ te−t /4s ⎨ √ T (s)x ds (t > 0), S(t)x = 2 πs3/2 0 ⎪ ⎩x (t = 0). Then S is a bounded holomorphic C0 -semigroup of angle π/4, and the generator of S is −(−A)1/2 . Furthermore, for x ∈ D(A), u(t) := S(t)x is the unique bounded classical solution of the second order Cauchy problem u (t) = −Au(t) (t ≥ 0), (3.57) u(0) = x. Moreover, if T is a bounded holomorphic C0 -semigroup of angle θ ∈ (0, π/2], then S is a bounded holomorphic C0 -semigroup of angle ( θ2 + π4 ). 2

ze−z /4s Proof. For z ∈ Σπ/4 , let ψz (s) = √ 3/2 (s > 0). Then ψz ∈ L1 (R+ ), ψz 1 = 2 πs ∞ ∞ |z| z∈Σθ , ψ (s) ds = 1, and lim |ψz (s)| ds = 0, for δ > 0 and 0 ≤ θ < z Re(z 2 ) 0 δ z→0 π/4. √ ∞ t (λ) = e−t λ (λ > 0) For t ∈ R+ , S(t)x = 0 ψt (s)T (s)x ds (x ∈ X) and ψ (Lemma 1.6.7). Hence (ψt ∗ ψt2 )(λ) = ψ t1 +t2 (λ). It follows from the uniqueness 1 theorem that ψt1 ∗ ψt2 = ψt1 +t2 (t1 , t2 ≥ 0). Now, ∞ ∞ S(t1 )S(t2 )x = ψt1 (s)ψt2 (r)T (s + r)x dr ds 0 0 ∞ t = ψt1 (s)ψt2 (t − s) ds T (t)x dt 0

0

= S(t1 + t2 )x. ∞ For z ∈ Σπ/4 , let S(z)x = 0 ψz (s)T (s)x ds. Then S(·) is holomorphic, S(z) ≤ |z| (Re z)2 sups≥0 T (s) and

∞ e−z 2 /4s

√ 3/2 (T (s)z − z) ds

S(z)x − x =

0 2 πs

→ 0

(z ∈ Σθ , z → 0).

168

3. CAUCHY PROBLEMS

Thus S is a bounded holomorphic C0 -semigroup of angle π/4. Let B be the generator of S and let x ∈ D(A). For t > 0, we can diﬀerentiate through the integral sign and obtain BS(t)x =

d (S(t)x) dt 0

∞

=

0 ∞

= 0

Hence

BS(t)x = 0

∞

2

e−t /4s √ 2 πs3/2

2

e−t /4s √ 2 πs3/2 2

e−t /4s √ 2 πs3/2

1−

t2 2s

t2 1− 2s

T (s)x ds, x ds.

t2 1− (T (s)x − x) ds. 2s

Since T (s)x − x ≤ s Ax supt≥0 T (t) , the dominated convergence theorem gives, on letting t ↓ 0, that x ∈ D(B) and ∞ T (s)x − x √ Bx = ds 2 πs3/2 0 ∞ ∞ 1 = λ1/2 e−sλ dλ (T (s)x − x) ds π 0 0 1 ∞ 1/2 = λ (R(λ, A)x − λ−1 x) dλ π 0 1 ∞ −1/2 = λ R(λ, A)Ax dλ π 0 = −(−A)1/2 x. Since D(A) is a core for −(−A)1/2 , it follows that B extends −(−A)1/2 . However, B + i is invertible (since B generates a bounded holomorphic semigroup) and i − (−A)1/2 is invertible (since I − A = ((−A)1/2 − i)((−A)1/2 + i)), so B = −(−A)1/2 . Let x ∈ D(A) = D(((−A)1/2 )2 ). Then it is immediate that u(t) := S(t)x is a bounded classical solution of (3.57). Let u1 be any bounded solution of (3.57). Take μ > 0, and let v(t) := R(μ, A)(u(t) − u1 (t))

(t ≥ 0).

Then v(0) = 0, v is bounded, and v (t) = −Av(t) = (I − μR(μ, A))(u(t) − u1 (t)), which is bounded. Since v (t) = v(t + 1) − v(t) −

1 2

t

t+1

(t + 1 − s)v (s) ds,

3.8. FRACTIONAL POWERS it follows that v is bounded. Let w(t) :=

v (−t) + (−A)1/2 v(−t) S(t)v (0)

169

(t ≤ 0), (t > 0).

Since (−A)1/2 R(μ, A) is bounded (by the closed graph theorem, or by direct calculation), w is bounded. For t ≤ 0, w (t)

= −v (−t) − (−A)1/2 v (−t) = Av(t) − (−A)1/2 v (−t) = −(−A)1/2 w(t).

Also, w (t) = −(−A)1/2 w(t) for all t ≥ 0, since −(−A)1/2 generates S. It follows from Proposition 3.1.11 that w(t + s) = S(t)w(s)

(t ≥ 0, s ∈ R).

Now we extend w to C by (λ ∈ Σπ/4 , s ∈ R).

w(λ + s) := S(λ)w(s)

This is well deﬁned since S(λ1 + λ2 ) = S(λ1 )S(λ2 ). Moreover, w is holomorphic, and bounded since λ may be chosen in Σπ/8 , where S is bounded. By Liouville’s theorem, w is constant, so S(t)v (0) = v (0) for all t ≥ 0. Now, v (t)

= −(−A)1/2 v(t) + w(−t)

v(0)

= −(−A)1/2 v(t) + v (0) = 0.

(t ≥ 0),

By Proposition 3.1.16, v(t) = 0

t

S(t − s)v (0) ds = tv (0).

But v is bounded, so v (0) = 0 and hence v(t) = 0. Since R(μ, A) is injective, it follows that u1 (t) = u(t). Now suppose that T is a bounded holomorphic C0 -semigroup of angle θ ∈ (0, π/2]. Let α ∈ (−θ, θ). An application of Cauchy’s theorem shows that ∞ 2 iα te−t /4re √ 3/2 iα/2 T (reiα ) dr S(t) = (t > 0). 2 πr e 0 Now let

∞

2

iα

ze−z /4re √ S(z) = T (reiα ) dr 2 πr3/2 eiα/2 0 for α2 − π4 < arg z < α2 + π4 . This deﬁnes a holomorphic extension of S to this sector, and it is bounded in each proper subsector. Varying α provides a holomorphic extension of S to Σ( θ + π ) which is bounded on Σθ for θ < ( θ2 + π4 ). 2

4

170

3. CAUCHY PROBLEMS

Example 3.8.4. Let (Ω, μ) be a measure space, X := L2 (Ω, μ), m : Ω → R+ be measurable, and A be deﬁned by D(A) := {f ∈ X : mf ∈ X} Af := mf. Then −A generates the C0 -semigroup T (t)f = e−tm f . The operator A1/2 of Proposition 3.8.2 is given by D(A1/2 ) = {f ∈ X : m1/2 f ∈ X} and A1/2 f = m1/2 f. The semigroup S generated by −A1/2 , as in Theorem 3.8.3, is given by S(t)f = 1/2 e−tm f . Example 3.8.5. Let X be any of the spaces of Example 3.7.6, and let T be the 2 Gaussian semigroup on X. Then T (s)f = ks ∗f , where ks (x) = (4πs)−n/2 e−|x| /4s . Hence, the holomorphic semigroup S of Theorem 3.8.3 is given by S(t)f = ht ∗ f , where ∞ −t2 /4s −|x|2 /4s te e √ 3/2 ht (x) = ds. 2 πs (4πs)n/2 0 Putting r = (t2 + |x|2 )/4s gives ht (x) =

Γ( n+1 2 )t . (n+1)/2 2 π (t + |x|2 )(n+1)/2

Thus, S is the Poisson semigroup considered in Example 3.7.9. Note that although the generator ΔX of T is a second order diﬀerential operator, the generator AX = −(−ΔX )1/2 of S is not a ﬁrst order diﬀerential operator. It should be mentioned that (3.57) is an abstract elliptic equation. For example, if T is the Gaussian semigroup on C0 (Rn ) (Example 3.8.5), then letting u(t, x) := (S(t)f )(x) for f ∈ C0 (Rn ), x ∈ Rn and t > 0, u is a solution of ∂ 2u ∂2u + =0 ∂t2 ∂x2j j=1 n

utt + Δu :=

on (0, ∞) × Rn ;

i.e., u is a solution of the Laplace equation. The wave equation utt = Δu will be treated in Sections 3.14–3.16 and Chapter 7.

3.9. BOUNDARY VALUES OF HOLOMORPHIC SEMIGROUPS

3.9

171

Boundary Values of Holomorphic Semigroups

Let T be a holomorphic C0 -semigroup on X of angle π2 . In this section we are interested in the behaviour of T (is + t) as t tends to 0 and we ask under what circumstances the “boundary value” T (is) of T (which will be deﬁned precisely below) exists and deﬁnes a C0 -group. We also address the converse problem: Which C0 -groups are obtained as boundary values of holomorphic C0 -semigroups? As in Section 3.7, we let Σϕ := {z ∈ C \ {0} : |argz| < ϕ} be the sector in the complex plane of angle ϕ ∈ (0, π). Furthermore, we set Σ+ ϕ := Σϕ ∩{z ∈ C : Im z ≥ 0}, Σ− := Σ ∩ {z ∈ C : Im z ≤ 0} and deﬁne D by D := {z ∈ Σπ/2 : |z| ≤ 1}. ϕ ϕ The following result gives an answer to our ﬁrst question above. Proposition 3.9.1. Let A be the generator of a holomorphic C0 -semigroup T on X of angle ϕ ∈ (0, π/2]. Then the following are equivalent: (i) eiϕ A generates a C0 -semigroup T (eiϕ ·) on X. (ii) supz∈Σ+ ||T (z)|| < ∞. ϕ ∩D In this case, the C0 -semigroup T (eiϕ ·) is given by T (eiϕ s)x = lim T (t + eiϕ s)x t↓0

(x ∈ X, s ≥ 0).

(3.58)

The C0 -semigroup S(s) := T (eiϕ s) deﬁned by (3.58) is called the boundary semigroup of T . The following lemma will be useful in the proof of Proposition 3.9.1. Lemma 3.9.2. Let A be the generator of a holomorphic C0 -semigroup T on X of angle ϕ ∈ (0, π/2]. Assume that eiϕ A generates a C0 -semigroup S on X. Then T (t + eiϕ s) = T (t)S(s)

for all

s, t ≥ 0.

Proof. Obviously, the resolvents of A and eiϕ A commute. By Proposition 3.1.5, S(s)T (t) = T (t)S(s) for all s, t ≥ 0. Fix a, b ≥ 0 and denote by B the generator of the C0 -semigroup V on X deﬁned by V (t) := S(bt)T (at). For x ∈ D(A) we have d d iϕ iϕ dt V (t)x = (a + be )AV (t)x. Hence dt V (t)x|t=0 = (a + be )Ax and therefore B extends (a + beiϕ )A. It follows that V (t) = T ((a + beiϕ )t) for all t ≥ 0 (see Proposition 3.7.2 c)). In particular, we have V (1) = T (a + beiϕ ) = S(b)T (a) = T (a)S(b) for all a, b ≥ 0. Proof of Proposition 3.9.1. Assume that (i) holds. Let z ∈ Σ+ ϕ ∩ D. There exist a, b ∈ [0, 1] such that z = a + beiϕ . It follows from Lemma 3.9.2 that T (z) = T (a + beiϕ ) = T (a)T (beiϕ ). Hence, T (z) ≤ T (a) T (beiϕ ) ≤ M for a suitable M and all z ∈ Σ+ ϕ ∩ D. This implies (ii). In order to prove the converse implication, ﬁx R > 0. Then there exists MR > 0 such that T (z) ≤ MR whenever z ∈ Σ+ ϕ and |z| ≤ R. For x ∈ X,

172

3. CAUCHY PROBLEMS

0 < t < t ≤ 1 and s ≥ 0 satisfying |t + eiϕ s| ≤ R, we have

! "

T (t + eiϕ s)x − T (t + eiϕ s)x ≤ T (t + eiϕ s) x − T (t − t)x ≤ MR x − T (t − t)x . It thus follows that T (eiϕ s)x := limt↓0 T (t + eiϕ s)x exists uniformly in s ∈ [0, R]. Consequently, the mapping R+ → L(X), s → T (eiϕ s) is strongly continuous. It is easy to see that T (eiϕ ·) is a C0 -semigroup. Denote by B its generator. Let x ∈ D(A) and τ > 0. Then τ τ T (eiϕ s)eiϕ Ax ds = lim eiϕ AT (t + eiϕ s)x ds t↓0 0 0 τ " d ! = lim T (t + eiϕ s)x ds t↓0 0 ds = lim(T (t + eiϕ τ )x − T (t)x) t↓0

= T (eiϕ τ )x − x. It follows from Proposition 3.1.9 f) that x ∈ D(B) and Bx = eiϕ Ax. We have shown that B is an extension of eiϕ A. Since ρ(B) ∩ ρ(eiϕ A) = ∅, it follows that both operators are equal. Remark 3.9.3. Note that the above result may be easily modiﬁed to the case where eiϕ A and e−iϕ A generate C0 -semigroups. Indeed, assuming that sup z∈Σϕ ∩D

||T (z)|| < ∞,

(3.59)

it follows that e±iϕ A generate C0 -semigroups on X which are given by T (e±iϕ s)x := limt↓0 T (t + e±iϕ s)x. In particular, if (3.59) is satisﬁed for ϕ = π2 , then iA and −iA generate C0 -semigroups T (±is) on X and we call S(s) := T (is) (s ∈ R), the boundary group of T . For 1 ≤ p ≤ ∞, let Δp be the Laplacian on Lp (Rn ) with maximal domain: Δp f D(Δp )

:=

Δf,

:= {f ∈ Lp (Rn ) : Δf ∈ Lp (Rn )},

(3.60)

where Δf is deﬁned in the distributional sense. We proved in Example 3.7.6 that for 1 ≤ p < ∞ the operator Δp is the generator of the Gaussian semigroup Tp on Lp (Rn ). Moreover, Δ∗p = Δp , where 1/p + 1/p = 1. Although Δ∞ does not generate a C0 -semigroup on L∞ (Rn ), it does generate a holomorphic semigroup T∞ given by T∞ (z)f := kz ∗ f (f ∈ L∞ (Rn )), (see Example 3.7.8).

3.9. BOUNDARY VALUES OF HOLOMORPHIC SEMIGROUPS

173

We wish to determine whether iΔp generates a C0 -semigroup. By duality, we may restrict ourselves in the following to the case when 1 ≤ p ≤ 2. Since FT2 (z)f = 2 e−z|·| F f and (2π)−n/2 F is unitary on L2 (Rn ), we have ||T2 (z)||L(L2 (Rn )) = 1 for all z ∈ C with Re z > 0. Hence, by Proposition 3.9.1 the operator iΔ2 generates a C0 -semigroup on L2 (Rn ). By Remark 3.7.10,

T1 (z) L(L1 (Rn )) =

|z| Re z

n/2 (Re z > 0).

Hence, by the Riesz-Thorin interpolation theorem (see [H¨ or83, Theorem 7.1.12])

Tp (z) L(Lp (Rn )) ≤

|z| Re z

n(1/p−1/2)

(1 ≤ p ≤ 2, Re z > 0).

(3.61)

In fact, we will now show that a multiple of the above upper bound will also serve as a lower bound for ||Tp (z)||L(Lp (Rn )) . Then we can apply Proposition 3.9.1 and obtain the following result. Theorem 3.9.4 (H¨ormander). Let 1 ≤ p < ∞. Then the operator iΔp generates a C0 -semigroup on Lp (Rn ) if and only if p = 2. Proof. By Proposition 3.9.1, it suﬃces to show that a multiple of the above upper bound (3.61) will also serve as a lower bound for ||Tp (z)||L(Lp (Rn )) . More precisely, we prove in the following that if 1 ≤ p < ∞, then

Tp (z) L(Lp (Rn )) ≥ 2−n/2p

|z| Re z

n|1/p−1/2| (Re z > 0).

(3.62)

We already observed that it suﬃces to consider the case where 1 < p ≤ 2. Fix z ∈ C+ and consider the function f : Rn → C deﬁned by f (x) := exp Taking p such that

1 p

+

1 p

−|x|2 z

.

= 1 we verify that

f p =

π p

n/2p

|z|2 Re z

n/2p .

(3.63)

Let x, y ∈ Rn and recall that z ∈ C+ . Then −

|x − y|2 |x|2 = −2 Re − z z

1 x − z

y 2 y 1 |y|2 1 y Im − Re . + 2i x − 2 2 z 2 z

174

3. CAUCHY PROBLEMS

Moreover, Tp (z/4)f (y) 1 |x − y|2 |x|2 = exp − − dx z z (πz)n/2 Rn 1 1 1 2 = exp −2|x| Re + 2ixy Im dx z z (πz)n/2 Rn |y|2 1 × exp − Re 2 z n/2 1 1 |x|2 Im(z −1 ) = exp − exp ixy dx 2 (πz)n/2 4 Re(z −1 ) (Re(z −1 ))1/2 Rn |y|2 1 × exp − Re 2 z n/2 1 1 = (2π)n/2 (πz)n/2 4 Re(z −1 ) |y|2 (Im(z −1 ))2 |y|2 1 × exp − exp − Re , 2 Re(z −1 ) 2 z where in the last step we used the fact that the function x → exp(−|x|2 /2) is an eigenvector of the Fourier transform. Hence, n/2 |z| |y|2 |Tp (z/4)f (y)| = exp − (y ∈ Rn ), 2 Re z 2 Re z and since Rn exp(−|x|2 /2) dx = (2π)n/2 , we obtain

Tp (z/4)f p =

|z| 2 Re z

n/2

2π p

n/2p

(Re z)n/2p .

Combining this equality with (3.63) we see that

Tp (z/4) L(Lq (Rn )) ≥

Tp (z/4)f p = 2−n/2p

f p

|z| Re z

n(1/p−1/2) .

Finally, since Tp (z) L(Lp (Rn )) = Tp (z) L(Lp (Rn )) , it follows that

Tp (z) L(Lp (Rn )) ≥ 2−n/2p

|z| Re z

n|1/p−1/2| (Re z > 0).

Interesting examples of boundary values of holomorphic C0 -semigroups occur also in connection with fractional powers of operators and the so-called RiemannLiouville semigroup. Indeed, consider in Lp (0, 1) the operator Au := u

with domain

D(A) := {u ∈ W 1,p (0, 1) : u(0) = 0}.

(3.64)

3.9. BOUNDARY VALUES OF HOLOMORPHIC SEMIGROUPS

175

We showed in Example 3.4.8 that −A generates a C0 -semigroup T on Lp (0, 1) (1 ≤ p < ∞), which may be represented by f (x − t) (t ≤ x), T (t)f (x) = (3.65) 0 (t > x). Since T (t) = 0 for t ≥ 1, we see that abs(T ) = −∞. Now inserting (3.65) in (3.56) we see that x 1 (x − y)z−1 f (y) dy (x ∈ (0, 1), Re z > 0, f ∈ Lp (0, 1)). A−z f (x) = Γ(z) 0 By Theorem 3.8.1, (A−z )Re z>0 is a holomorphic C0 -semigroup of angle π/2. This C0 -semigroup is called the Riemann-Liouville semigroup. By Proposition 3.9.1 and Remark 3.9.3, the question whether or not the Riemann-Liouville semigroup possesses a boundary group is equivalent to sup

z∈Σπ/2 ∩D

||A−z || < ∞.

(3.66)

When (3.66) holds true, we denote the boundary group by (S(s)) := (Ais )s∈R . In order to show the estimate (3.66) we make use of the transference principle due to Coifman and Weiss [CW77]. We will use it in the following form [Ama95, Chapter III, Example 4.7.3 c)]. For a measure space (Ω, μ) and 1 ≤ p ≤ ∞, we denote by Lp (Ω, μ) the usual Banach space of equivalence classes of p-integrable functions (bounded functions when p = ∞). Theorem 3.9.5 (Coifman-Weiss). Let (Ω, μ) be a σ-ﬁnite measure space and let 1 0 and s ∈ R let A−t+is be deﬁned as in (3.51). Then Ais f := limt↓0 A−t+is f ∈ Lp (Ω, μ) for f ∈ Lp (Ω, μ) and there exists a constant M , depending only on p, such that

Ais L(Lp (Ω,μ)) ≤ M (1 + s2 )eπ|s|/2

(s ∈ R).

Thus, for 1 < p < ∞, the Riemann-Liouville semigroup admits a boundary group. For p = 1 the situation is diﬀerent. Indeed, 1 1 −z ||A ||L(L1 (0,1)) = sup |(x − y)z−1 | dx y∈[0,1] |Γ(z)| y 1 1 1 1 = sup (x − y)Re z−1 dx = , |Γ(z)| y∈[0,1] y |Γ(z)| Re z which by Proposition 3.9.1 implies that we do not have a boundary value for p = 1. In summary, we have proved the following result for the Riemann-Liouville semigroup on Lp (0, 1).

176

3. CAUCHY PROBLEMS

Theorem 3.9.6. Let 1 ≤ p < ∞ and denote by G the generator of the RiemannLiouville semigroup on Lp (0, 1). Then iG is the generator of a C0 -group on Lp (0, 1) provided 1 < p < ∞. If p = 1, then iG does not generate a C0 -semigroup on L1 (0, 1). We now consider the converse to the situation described in Proposition 3.9.1; namely, we ask for conditions on the boundary group itself which imply that A generates a holomorphic C0 -semigroup. We begin with the following result. Theorem 3.9.7. Let A be an operator on X and let ϕ ∈ (0, π/2). Assume that e±iϕ A generate bounded C0 -semigroups on X. Then A generates a bounded holomorphic C0 -semigroup of angle ϕ. Our proof of Theorem 3.9.7 is based on the following version of the Phragm´enLindel¨ of theorem. Theorem 3.9.8 (Phragm´en-Lindel¨of ). Let ϕ ∈ (0, π/2] and let h : Σϕ → X be π continuous on Σϕ and holomorphic in Σϕ . Set α := 2ϕ . Assume that for all ε > 0 there exists a constant Cε > 0 such that α

h(z) ≤ Cε eε|z|

(z ∈ Σϕ ).

If h(re±iϕ ) ≤ M for all r > 0, then ||h(z)|| ≤ M for all z ∈ Σϕ . For a proof of Theorem 3.9.8 we refer to [Con73, Cor.6.4.4]. Proof of Theorem 3.9.7. Denote by T+ , T− the C0 -semigroups generated by A+ := eiϕ A and A− := e−iϕ A, respectively. Let M ≥ 0 such that ||T± (t)|| ≤ M for all t ≥ 0. Then

∞

M −λt

R(λ, A± ) = e T± (t) dt (Re λ > 0).

≤ Re λ 0 For λ ∈ Σ+ ϕ this implies that

R(λ, A) = R(λ, eiϕ A− ) = R(λe−iϕ , A− ) ≤ Similarly, R(λ, A) ≤

M |λ| cos ϕ

M M ≤ . −iϕ Re(λe ) |λ| cos ϕ

if λ ∈ Σ− ϕ . Thus we have

(I − zA)−1 = z −1 R(z −1 , A) ≤

M cos ϕ

(z ∈ Σϕ ).

For n ∈ N and z ∈ Σϕ set Tn (z) := (I − nz A)−n . Then Tn (z) ≤ M for z = re±iϕ and n M

Tn (z) ≤ for z = re±iα , r ≥ 0, |α| < ϕ. cos ϕ

3.9. BOUNDARY VALUES OF HOLOMORPHIC SEMIGROUPS

177

It follows from the Phragm´en-Lindel¨ of Principle 3.9.8 that Tn (z) ≤ M for all z ∈ Σϕ and all n ∈ N. The Hille-Yosida Theorem 3.3.4 implies now that A generates a C0 -semigroup T . By Corollary 3.3.6, we have limn→∞ Tn (t)x = T (t)x for t ≥ 0 and x ∈ X. It thus follows from Vitali’s theorem (see Theorem A.5 and Proposition A.3) that T has a holomorphic extension T˜ to Σϕ satisfying T˜ (z) ≤ M for all z ∈ Σϕ . Consider now the case where e±iϕ A generate C0 -semigroups T±ϕ which are not necessarily bounded. In this case, there exist constants M, ω ≥ 0 such that

T±ϕ (t) ≤ M eωt for t ≥ 0. It follows that e±iϕ (A − μ) generate bounded C0 semigroups for μ := cosω ϕ . Theorem 3.9.7 implies now that A − μ generates a bounded holomorphic C0 -semigroup of angle ϕ. We have thus proved the following result. Corollary 3.9.9. Let ϕ ∈ (0, π/2) and assume that e±iϕ A generate C0 -semigroups T±ϕ on X. Then A generates a holomorphic C0 -semigroup of angle ϕ with boundary semigroups T±ϕ . The following result is a consequence of the above Corollary 3.9.9 and Proposition 3.9.1. Corollary 3.9.10. Assume that A generates a C0 -semigroup T and that iA generates a C0 -group U . Then T has a holomorphic extension to Σπ/2 and U is the boundary group of T . Proof. By Corollary 3.9.9, the operator e±iπ/4 A generates a holomorphic C0 -semigroup of angle π/4. Thus eiθ A generates a C0 -semigroup for all θ ∈ (−π/2, π/2). It follows from Corollary 3.9.9 again that A generates a holomorphic C0 -semigroup of angle π/2. By Proposition 3.9.1, U is its boundary group. Next we consider spectral conditions on A which imply that A generates a holomorphic C0 -semigroup on X under the assumption that iA generates a C0 group U on X. An obvious necessary condition for this is that the spectrum of A is located in a left half-plane. However, this condition is not suﬃcient, in general. Indeed, consider the generator G of the Riemann-Liouville semigroup on Lp (0, 1) for 1 1, σ(A) is empty by Theorem 3.1.7. Moreover, iA generates a group by Theorem 3.9.6. However, A does not generate a C0 -semigroup. Nevertheless, if σ(A) is contained in some left half-plane, then A generates a holomorphic C0 -semigroup on X provided U satisﬁes a certain growth condition. Theorem 3.9.11. Let A be an operator on X such that iA generates a C0 -group U on X. Assume that there exists a dense subspace Y of X such that for all x ∈ Y there exist constants C ≥ 0 and k ∈ N (depending on x) such that

U (t)x ≤ C(1 + |t|)k

(t ∈ R).

178

3. CAUCHY PROBLEMS

If σ(A) ⊂ {λ ∈ C : Re λ ≤ b} for some b ∈ R, then A generates a holomorphic C0 -semigroup of angle π/2 on X (whose boundary group is U ). The key of the proof of Theorem 3.9.11 is the following result of Phragm´enLindel¨ of type. Proposition 3.9.12. Let r : Σπ/2 → X be continuous. Assume that r is holomorphic in Σπ/2 and that there exist constants C, M ≥ 0, R0 > 0, k ∈ N such that

r(λ)

≤

r(is)

≤

C (Re λ ≥ 0, Im λ = 0, |λ| ≥ R0 , arg λ = ϕ) | sin ϕ|k M (s ∈ R).

and

Then ||r(λ)|| ≤ M for all λ ∈ C with Re λ ≥ 0. Proof. For R ≥ R0 and k ∈ N consider the holomorphic function 0 Φ : DR := {z ∈ Σπ/2 : |z| < R} → X,

λ →

k λ2 1− 2 r(λ). R

Let λ := Reiϕ for ϕ ∈ (−π/2, π/2). If ϕ = 0, we obtain

Φ(λ) = |1 − ei2ϕ |k r(λ) = 2k | sin ϕ|k r(λ) ≤ 2k C. Moreover, Φ(R) = 0 and !

Φ(is) = 1 +

" s2 k R2

r(is) ≤ 2k M

(|s| ≤ R).

0. The maximum principle implies that Φ(λ) ≤ 2k max{C, M } for all λ ∈ DR k Letting R → ∞, we deduce that ||r(λ)|| ≤ 2 max{C, M } provided Re λ ≥ 0. Now the Phragm´en-Lindel¨of Principle 3.9.8 implies that

r(λ) ≤ M for all λ ∈ C with Re λ ≥ 0. Proof of Theorem 3.9.11. Replacing A by A − ω, we may assume that σ(A) ⊂ {λ ∈ C : Re λ < −δ} for some δ > 0. By assumption, iA generates a group and we therefore have sups∈R,|s|≥w ||sR(is, A)|| < ∞ for suitable w ≥ 0. It follows that M := sup ||sR(is, A)|| < ∞. s∈R

Thus the second assumption of Proposition 3.9.12 is satisﬁed for the function λ → λR(λ, A)x (x ∈ X). In order to verify the ﬁrst assumption in this proposition let x ∈ Y . Then, by hypothesis, there exist constants k ∈ N, C ≥ 0 such that

3.9. BOUNDARY VALUES OF HOLOMORPHIC SEMIGROUPS

179

U (t)x ≤ C(1 + |t|)k (t ∈ R). For λ of the form λ = reiϕ , where r ≥ 1 and ϕ ∈ (0, π/2] we therefore obtain

∞

∞

iλt

≤ |λ|C

λR(λ, A)x = λ e U (−t)x dt e−| Im λ|t (1 + tk ) dt

0 0 1 k! C(1 + k!) ≤ C|λ| + ≤ . | Im λ| | Im λ|k+1 | sin ϕ|k+1 Similarly, for λ = reiϕ , with r ≥ 1 and ϕ ∈ [−π/2, 0) we have

λR(λ, A)x ≤

C(1 + k!) . | sin ϕ|k+1

Hence both assumptions of Proposition 3.9.12 are satisﬁed for the function λ → λR(λ, A)x and it follows from that proposition that

λR(λ, A)x ≤ M x

(Re λ ≥ 0, x ∈ Y ).

Since Y is dense in X, we conclude that

λR(λ, A) ≤ M

(Re λ ≥ 0).

It follows from Corollary 3.7.12 that A generates a bounded holomorphic C0 semigroup and from Corollary 3.8.9 that the angle is π/2 and U is the boundary group. We ﬁnally turn our attention back to the question raised at the beginning of this section: under which conditions on the holomorphic C0 -semigroup T the “boundary value” of T exists and is again a C0 -semigroup. In the following, we weaken the sense of “boundary value” of T and also allow integrated semigroups as “boundary values” for holomorphic C0 -semigroups T . We say that an operator A on X generates a k-times integrated group on X for some k ∈ N0 if A and −A generate k-times integrated semigroups on X. A k-times integrated group is called exponentially bounded if the integrated semigroups generated by A and −A are exponentially bounded. Theorem 3.9.13. Let γ ≥ 0 and k ∈ N. Assume that A generates a holomorphic C0 -semigroup T of angle π/2. Then the following assertions hold: a) Assume that there exist constants M, ω ≥ 0 such that

T (z) ≤

M eω|z| (Re z)γ

(Re z > 0).

Then iA generates an exponentially bounded k-times integrated group provided k > γ.

180

3. CAUCHY PROBLEMS

b) Assume that iA generates an exponentially bounded k-times integrated group on X. Then there exist constants M, ω ≥ 0 such that

T (z) ≤

M eω|z| (Re z)k

(Re z > 0).

Proof. a) Let x ∈ X, k ∈ N and let λ0 > ω. Then k

R(λ0 , A) x =

∞

0

e−λ0 u

uk−1 T (u)x du. (k − 1)!

Hence, R(λ0 , A)k T (z)x =

∞

e−λ0 u

0

uk−1 T (u + z)x du (Re z > 0). (k − 1)!

Setting z = t + is the assumption implies that

R(λ0 , A)k T (z)

≤ ≤

∞ M eω|u+t+is| e−λ0 u uk−1 du (k − 1)! 0 (u + t)γ ∞ M e−λ0 u uk−1−γ eω|u+t+is| du < ∞, (3.67) (k − 1)! 0

provided k > γ. Since z → R(λ0 , A)k T (z) is holomorphic and bounded in a rectangle of the form {t + is : 0 < t < 1, −R < s < R} for some R > 0 it follows by dominated convergence that lim R(λ0 , A)k T (t + is)x t↓0

exists for all s ∈ R and all x ∈ X. For s ∈ R and x ∈ X set ⎧ ⎨i−k lim R(λ0 , A)k T (t + is)x (s ≥ 0), t↓0 S(s)x := ⎩(−i)−k lim R(λ0 , A)k T (t + is)x (s < 0). t↓0

In order to show that iA generates a k-times integrated semigroup it suﬃces by Proposition 3.2.7 to verify that R(·, iA)R(iλ0 , iA)k is a Laplace transform. Note ﬁrst that by the estimate (3.67) there exists a constant C ≥ 0 such that S(s) ≤ Ceω|s| for s ∈ R+ . We claim that ˆ S(λ) = R(λ, iA)R(iλ0 , iA)k

(λ > ω).

In fact, by Fubini’s theorem, the representation formula (3.46) for holomorphic

3.9. BOUNDARY VALUES OF HOLOMORPHIC SEMIGROUPS

181

C0 -semigroups and Cauchy’s theorem, we have ∞ ∞ ik e−λs S(s) ds = lim e−λs R(λ0 , A)k T (t + is) ds t↓0 0 0 ∞ = lim e−λs R(λ0 , A)k eμ(t+is) R(μ, A) dμ ds t↓0 0 Γ ∞ = lim e−λs eμis ds eμt R(λ0 , A)k R(μ, A) dμ t↓0 Γ 0 1 = lim eμt R(λ0 , A)k R(μ, A) dμ t↓0 Γ λ − iμ = lim R(λ, iA)R(λ0 , A)k T (t) t↓0

= R(λ, iA)R(λ0 , A)k

(λ > ω),

∞ where Γ denotes the path deﬁned in (3.46). Thus we have 0 e−λs S(s) ds = R(λ, iA)R(iλ0 , iA)k for λ > ω. The corresponding result for −iA is proved in exactly the same way. b) By rescaling, we may assume that σ(A) ⊂ {λ ∈ C : Re λ ≤ −1} (see Proposition 3.1.9 i) and Proposition 3.2.6). We subdivide the proof into two steps. Step 1: By assumption, A generates a holomorphic C0 -semigroup T of angle z (z−ξ) k−1 π/2. Hence, the function S : z → S(z) := 0 (k−1)! T (ξ) dξ is holomorphic in the open right half-plane. We claim that there exist constants M, ω ≥ 0 such that

S(z) ≤ M eω|z| Integrating by parts, using

dn −n dξ n T (ξ)A

S(z)x = T (z)A−k x −

(Re z > 0).

(3.68)

= T (ξ), gives

z k−1 A−1 x − · · · − A−k x (x ∈ X). (k − 1)!

Hence, in order to prove (3.68) it suﬃces to show that

T (z)A−k ≤ M eω|z|

(Re z > 0),

(3.69)

for suitable constants M, ω ≥ 0. Obviously (S(t))t≥0 is the k-times integrated semigroup generated by A. For z ∈ {μ ∈ C : Re μ > 0} set z = t + is for t > 0 and s ∈ R. Let (R(s))s∈R be the k-times integrated group generated by iA. For x ∈ D(Ak ) set T (is)x :=

k dk sn k R(s)x = R(s)(iA) x + (iA)n−1 x dsk n! n=1

(s ∈ R).

Then T (t + is)x = T (t)T (is)x = T (is)T (t)x (t > 0, s ∈ R),

(3.70)

182

3. CAUCHY PROBLEMS

because the function v given by v(s) := T (is)T (t)x − T (t + is)x is the unique mild solution of the problem u (s) = iAu(s), u(0) = 0, by Lemma 3.2.9 and Lemma 3.2.10. We consider ﬁrst the case where Re z ≥ 1. By (3.70) we have for z = t + is

T (z)A−k ≤ T (t − 1/2)A−k T (is)A−k Ak T (1/2) . Since t → T (t − 1/2)A−k and s → T (is)A−k are exponentially bounded we obtain (3.69) in the case where Re z ≥ 1. Next, we consider the case where 0 < Re z ≤ 1. For x ∈ X we have by (3.70): T (z)A−2k x = T (t)A−k T (is)A−k x. It then follows that there exist constants M1 , ω1 ≥ 0 such that

T (z)A−2k ≤ M1 eω1 |z|

and

T (is)A−k ≤ M1 eω1 |s|

(3.71)

for s ∈ R and z ∈ C satisfying Re z > 0. For z ∈ Ω := {μ ∈ C : 0 ≤ Re μ ≤ 1} and x ∈ X set f (z) := (cos z)−2ω1 T (z)A−2k x. Then f is holomorphic in the interior of Ω and continuous on Ω, by (3.70). For |s| ≥ 1 we have

f (z) ≤ 22ω1 T (z)A−2k x |es − e−s |−2ω1 , which by (3.71) implies that z → f (z) is bounded for z ∈ Ω. For z = is and z = 1 + is we have

f (is)

f (1 + is)

≤

22ω1

≤ 22ω1

T (is)A−k A−k x

≤ M2 A−k x

|es + e−s |2ω1

T (is)A−k T (1)A−k x

≤ M2 A−k x

|es + e−2i e−s |2ω1

for a suitable constant M2 ≥ 0. The three-lines lemma [Con73, Theorem VI.3.7] now implies that there exists a constant M ≥ 0 such that

T (z)A−2k x ≤ M e2ω1 |z| A−k x

(Re z > 0).

Since D(Ak ) is dense in X, the claim follows. Step 2: Set z Sk+1 (z) := S(ξ) dξ (Re z > 0). 0

Cauchy’s integral formula implies that (k + 1)! Sk+1 (ξ) T (z) = dξ k+2 2πi γz (ξ − z)

(Re z > 0),

3.9. BOUNDARY VALUES OF HOLOMORPHIC SEMIGROUPS where γz denotes the path deﬁned by the circle with centre z and radius r = It follows that (k + 1)! Sk+1 (ξ) − Sk+1 (z) T (z) = dξ 2πi (ξ − z)k+2 γz " (k + 1)! 2π ! = Sk+1 (z + reiϕ ) − Sk+1 (z) e−iϕ(k+1) dϕ. 2πrk+1 0

183 Re z 2 .

It follows from the deﬁnition of Sk+1 and (3.68) that

Sk+1 (z + reiϕ ) − Sk+1 (z) ≤ M3 reω3 |z|

(Re z > 0)

for suitable M3 , ω3 ≥ 0. Inserting this in the above representation of T (z), it follows that eω4 |z| 2k eω4 |z|

T (z) ≤ M4 k = M4 (Re z > 0) r (Re z)k for suitable constants M4 , ω4 ≥ 0. The proof is complete. As an application of Theorem 3.9.13 we consider once again “boundary values” of the Gaussian semigroup. More precisely, let 1 ≤ p < ∞ and let the operator Δp be deﬁned as in (3.60). Then the following corollary holds true. Corollary 3.9.14. Let 1 ≤ p < ∞ and let k ∈ N0 . Then iΔp generates an exponentially bounded k-times integrated group on Lp (Rn ) if k > n| 12 − p1 |. Moreover, the order of integration is optimal in the sense that iΔp does not generate a k-times integrated semigroup on Lp (Rn ) if k < n| 12 − 1p |. Proof. It follows from (3.61) and by duality that the assumption of Theorem 3.9.13 a) is satisﬁed for γ = n|1/2 − 1/p|. Hence, the ﬁrst assertion above follows from this theorem. Conversely, suppose that iΔp generates an exponentially bounded k-times integrated semigroup S on Lp (Rn ) for some 1 ≤ p < ∞ and k < n|1/2 − 1/p|. Let J be the conjugation on the complex space Lp (Rn ); Jf = f¯. Then −iΔp = J(iΔp )J, which generates the k-times integrated semigroup JS(·)J, so iΔp generates an exponentially bounded k-times integrated group. By Theorem 3.9.13 b), there exists a constant M > 0 such that Tp (z) ≤ M/(Re z)k for z ∈ Σπ/2 with |z| = 1, where Tp is the Gaussian semigroup as in Example 3.7.6. However, by (3.62) we have

Tp (z) ≥ 2−n/2p (Re z)−n|1/2−1/p|

(z ∈ Σπ/2 , |z| = 1),

which yields a contradiction. More general results for diﬀerential and pseudo-diﬀerential operators will be given in Section 8.3.

184

3. CAUCHY PROBLEMS

3.10 Intermediate Spaces It turns out that k-times integrated semigroups are the same as C0 -semigroups up to the choice of the underlying Banach space. This will be made precise in this section. Throughout this section, Z, X and Y are Banach spaces. We write Z → X if Z ⊂ X and there is a constant c such that x X ≤ c x Z for all x ∈ Z. If in d

addition Z is dense in X we write Z → X. The following lemma is a consequence of the closed graph theorem. Lemma 3.10.1. If Z → Y and Z ⊂ X → Y , then Z → X. Let A be an operator on X. If Z → X we denote by AZ the part of A in Z; i.e., D(AZ ) := {x ∈ D(A) ∩ Z : Ax ∈ Z}, AZ x := Ax. If A is closed, then AZ is closed. The following is easy to prove (see also Proposition B.8). Lemma 3.10.2. Let A be an operator on X, Z → X. Let μ ∈ ρ(A) such that R(μ, A)Z ⊂ Z. Let B be an operator on Z. Then B = AZ if and only if μ ∈ ρ(B) and R(μ, B) = R(μ, A)|Z . Let A be a closed operator on X and k ∈ N. Then D(Ak ) is a Banach space for the norm x Ak := x + Ax + . . . + Ak x . Moreover, D(Ak ) → X. We denote by Ak the part of A in D(Ak ); i.e., Ak is the operator on the Banach space D(Ak ) given by Ak x = Ax, D(Ak ) = D(Ak+1 ). If ρ(A) = ∅, then Ak and A are similar (see Section 3.5 for the deﬁnition). In fact, Ak = U −1 AU where U may be taken as U = (μ − A)k for any μ ∈ ρ(A). In particular, σ(Ak ) = σ(A) and Ak generates an (exponentially bounded) m-times integrated semigroup on D(Ak ) if and only if A generates an (exponentially bounded) m-times integrated semigroup on X. The following result on the spectrum of intermediate operators is of general interest. Proposition 3.10.3. Let A be an operator on X, Z → X. Assume that R(μ, A)Z ⊂ Z for some μ ∈ ρ(A) and that D(Ak ) ⊂ Z for some k ∈ N. Then σ(AZ ) = σ(A) and R(λ, AZ ) = R(λ, A)|Z for all λ ∈ ρ(A). Proof. a) Let B = AZ and λ ∈ ρ(A). Iterating the resolvent equation R(λ, A) = R(μ, A) + (μ − λ)R(μ, A)R(λ, A) gives R(λ, A) =

k (μ − λ)j−1 R(μ, A)j + (μ − λ)k R(μ, A)k R(λ, A).

(3.72)

j=1

Since R(μ, A)Z ⊂ Z and R(μ, A)k X = D(Ak ) ⊂ Z, it follows that R(λ, A)Z ⊂ Z. Hence by Lemma 3.10.2, λ ∈ ρ(B) and R(λ, B) = R(λ, A)|Z .

3.10. INTERMEDIATE SPACES

185

b) In order to prove the converse, we observe that ρ(A) = ρ(Ak ) since A and Ak are similar operators. Let Y := D(Ak ) with the graph norm. Then Y → Z by Lemma 3.10.1, D(B k ) ⊂ Y and Ak = BY . Moreover, R(μ, B)Y = R(μ, A)Y = D(Ak+1 ) ⊂ Y . It follows from a) (applied to B instead of A) that ρ(B) ⊂ ρ(Ak ) = ρ(A). Our aim is to prove the following result. Theorem 3.10.4 (Sandwich Theorem). Let A be an operator on X and let k ∈ N. The following assertions are equivalent: (i) The operator A generates a k-times integrated semigroup S such that ω(S) < ∞. (ii) There exists a Banach space Y and the generator B of a C0 -semigroup V on Y such that a) D(B k ) ⊂ X → Y , b) R(λ, B)X ⊂ X for some λ ∈ ρ(B), and c) A = BX . (iii) There exists a Banach space Z such that a) D(Ak ) ⊂ Z → X, b) R(λ, A)Z ⊂ Z for some λ ∈ ρ(A), and c) AZ generates a C0 -semigroup U on Z. Proof. (i) ⇒ (ii): Assume that A generates a k-times integrated semigroup S satisfying S(t) ≤ M eωt (t ≥ 0) where M, ω > 0. For x ∈ D(Ak ), let T (t)x := S(t)Ak x +

tk−1 Ak−1 x + . . . + tAx + x (t ≥ 0). (k − 1)!

(3.73)

By Lemma 3.2.2 (see also Lemma 3.2.10), v(t) := T (t)x is a mild solution of (ACP0 ). Hence, s → v(t + s) is a mild solution of (ACP0 ) with initial value T (t)x. By Theorem 3.1.3, ∞ e−λs T (t + s)x ds (t ≥ 0, λ > ω) (3.74) R(λ, A)T (t)x = 0

for all x ∈ D(A ). Fix μ0 > b > ω, and deﬁne a norm · Y on X by k

x Y := sup e−bt T (t)R(μ0 , A)k x X ,

(3.75)

t≥0

and let Y be the completion of (X, · Y ). We claim that

(λ − b)R(λ, A)x Y ≤ x Y

(λ > b, x ∈ X).

(3.76)

186

3. CAUCHY PROBLEMS

In fact,

e−bt T (t)R(μ0 , A)k R(λ, A)x X

−bt ∞ −λs

k

= e e T (s + t)R(μ0 , A) x ds

X

∞ 0

−(λ−b)s −b(t+s) k

= e e T (s + t)R(μ0 , A) x ds

0 X ∞

x

Y ≤ x Y e−(λ−b)s ds = . λ−b 0 Hence R(λ, A) has a unique extension R(λ) ∈ L(Y ) and (λ − b)R(λ) L(Y ) ≤ 1 (λ > b). Then (R(λ))λ>b is a pseudo-resolvent on Y . Next we show that lim λR(λ)y − y Y = 0

(3.77)

λ→∞

for all y ∈ Y . Since lim supλ→∞ λR(λ) L(Y ) < ∞, it suﬃces to prove (3.77) for y ∈ X, X being dense in Y . Let x := R(μ0 , A)k y. Let ε > 0. There exists M such that T (t)x X ≤ M eωt (t ≥ 0). Then

−bt ∞ −λs

e

λe (T (t + s)x − T (t)x) ds

0 X ∞ −bt −λs ω(t+s) ωt ≤ e λe M e +e ds 0 λ = M e−(b−ω)t +1 . λ−ω Hence, there exists t0 such that

−bt ∞ −λs

sup sup e λe (T (t + s)x − T (t)x) ds

λ>2ω t>t0

0

< ε.

(3.78)

X

Since t → T (t)x is uniformly continuous on [0, t0 + 1], there exists τ > 0 such that

T (t + s)x − T (t)x X < ε whenever s ∈ [0, τ ], t ∈ [0, t0 ]. For λ > 2ω and 0 ≤ t ≤ t0 ,

−bt ∞ −λs

e

λe (T (t + s)x − T (t)x) ds

0 X τ ∞ −bt −λs −bt −λs ≤ e λe ε ds + e λe M (eω(t+s) + eωt ) ds 0 τ λ ≤ ε + M e−(λ−ω)τ + e−λτ . (3.79) λ−ω

3.10. INTERMEDIATE SPACES

187

By (3.74), (3.75), (3.78) and (3.79), lim sup λR(λ, A)y − y Y λ→∞

= lim sup sup e−bt (λR(λ, A)T (t)x − T (t)x) X λ→∞ t≥0

−bt ∞ −λs

= lim sup sup e λe (T (t + s)x − T (t)x) ds

λ→∞ t≥0 0

≤

X

ε.

Since ε > 0 is arbitrary, the claim is proved. It follows from (3.77) and Proposition B.6 that there exists a densely deﬁned operator B on Y such that (b, ∞) ⊂ ρ(B) and R(λ, B) = R(λ) (λ > b). By the Hille-Yosida theorem, B generates a C0 -semigroup V on Y satisfying

V (t) L(Y ) ≤ ebt (t ≥ 0). It follows from Lemma 3.10.2 that A = BX . By deﬁnition, R(μ0 , A)k y X ≤ y Y (y ∈ X). Hence, R(μ0 , B)k Y ⊂ X; i.e., D(B k ) ⊂ X and the proof of (ii) is complete. ! " (ii) ⇒ (iii): Let Z := D(B k ), · Bk . Then AZ = Bk which is similar to B, so AZ generates a C0 -semigroup U on Z. By Proposition 3.10.3, ρ(A) = ρ(B) and R(λ, A) = R(λ, B)|X for all λ ∈ ρ(B). It follows that R(λ, A)Z ⊂ Z for all λ ∈ ρ(A). (iii) ⇒ (i): Note that D(Ak ) → Z by Lemma 3.10.1, and R(λ, AZ ) = R(λ, A)|Z for λ ∈ ρ(A) = ρ(AZ ), by Proposition 3.10.3. Let μ ∈ ρ(A). Then λ → R(λ, A)R(μ, A)k is the Laplace transform of t → U (t)R(μ, A)k . Now, (i) follows from Proposition 3.2.7. Corollary 3.10.5. Let A be the generator of an exponentially bounded k-times integrated semigroup T on X and let B ∈ L(X, D(Ak )). Then A + B generates a k-times integrated semigroup S on X satisfying ω(S) < ∞. Proof. We use the notation of Theorem 3.10.4 (iii). The operator B|Z is bounded. By Corollary 3.5.6, AZ + B|Z generates a C0 -semigroup on Z. It is clear that (A + B)Z = AZ + B|Z . So it will follow from Theorem 3.10.4 that A + B generates an exponentially bounded k-times integrated semigroup on X once we have proved that A + B satisﬁes conditions a) and b) of Theorem 3.10.4 (iii). It is easy to see that D((A + B)k ) ⊂ D(Ak ), so a) is satisﬁed. In order to show b), take μ ∈ ρ(A). Then C := (μ − A)k B ∈ L(X) and by (3.73) R(λ, A)B

=

R(λ, A)R(μ, A)k C

=

R(λ, A)C − (μ − λ)j−k−1 R(μ, A)j C. (μ − λ)k k

j=1

Since λ → λ−k R(λ, A) is the Laplace transform of an exponentially bounded function, lim supλ→∞ λ1−k R(λ, A) < ∞. This implies that limλ→∞ R(λ, A)B = 0.

188

3. CAUCHY PROBLEMS

Consequently, (I − R(λ, A)B) is invertible for large λ. Then by Lemma 3.5.8, (I − BR(λ, A)) is also invertible for large λ. Hence, (λ−(A+B)) = (I−BR(λ, A))(λ−A) is invertible for large λ. Hence, there exists λ ∈ ρ(A + B) ∩ ρ(AZ + B|Z ) ∩ ρ(A). Let y ∈ Z, x = R(λ, A + B)y. Then (λ − A)x = y + Bx ∈ Z. Hence x = R(λ, A)(y + Bx) ∈ Z by Proposition 3.10.3.

3.11 Resolvent Positive Operators In this section we assume that X is an ordered Banach space with normal cone X+ (see Appendix C). Deﬁnition 3.11.1. An operator A on X is called resolvent positive if there exists ω ∈ R such that (ω, ∞) ⊂ ρ(A) and R(λ, A) ≥ 0 for all λ > ω. If A generates a C0 -semigroup T , then A is resolvent positive if and only if T is positive (i.e. T (t)X+ ⊂ X+ for all t ≥ 0). In fact, if T is positive, then ∞ R(λ, A) = e−λt T (t) dt ≥ 0 0

for all λ > ω(T ). The converse follows from Euler’s formula ! "−n T (t)x = lim I − nt A x n→∞

(see Corollary 3.3.6). But there are interesting examples of resolvent positive operators which do not generate C0 -semigroups; see Section 6.1 for an example. Let A be a resolvent positive operator. Then for λ > ω one has (−1)n R(λ, A)(n) = n!R(λ, A)n+1 ≥ 0

(3.80)

for all n ∈ N (see Appendix B). Thus, the function R(·, A) is completely monotonic (cf. Section 2.7). We ﬁrst use Bernstein’s theorem for real-valued functions to prove some general properties of resolvent positive operators. Proposition 3.11.2. Let A be a resolvent positive operator. Denote by s(A) := sup{Re λ : λ ∈ σ(A)} the spectral bound of A. Then s(A) < ∞ and R(λ, A) ≥ R(μ, A) ≥ 0 whenever s(A) < λ < μ. Moreover, if λ ∈ R ∩ ρ(A) such that R(λ, A) ≥ 0, then λ > s(A). Finally, s(A) ∈ σ(A) if s(A) > −∞.

3.11. RESOLVENT POSITIVE OPERATORS

189

Proof. Let s := inf{ω : (ω, ∞) ⊂ ρ(A) and R(λ, A) ≥ 0 for all λ > ω}. By assumption, s < ∞. Replacing A by A − ω, we may assume that s ≤ 0. a) Let s < λ < μ. Then R(λ, A) − R(μ, A) = (μ − λ)R(λ, A)R(μ, A) ≥ 0. Thus, R(·, A) is a decreasing function on (s, ∞). b) Assume that s > −∞. Then s ∈ σ(A). In fact, if s ∈ ρ(A) then R(s, A) ≥ 0. Moreover, for μ < s suﬃciently close to μ one has R(μ, A) =

∞

(s − μ)n R(s, A)n+1 ≥ 0.

n=0

This contradicts the deﬁnition of s. c) Let Hs := {λ ∈ C : Re λ > s}. We claim that Hs ⊂ ρ(A); this and b) establish that s = s(A). Denote by Ω0 the connected component of Hs ∩ ρ(A) containing (s, ∞). If Hs ⊂ ρ(A), there exist μn ∈ Ω0 such that μ := limn→∞ μn ∈ Hs \ ρ(A). Then by Corollary B.3, sup R(μn , A) = ∞.

n∈N

By the uniform boundedness principle, there exist x ∈ X, x∗ ∈ X ∗ such that ∗ supn∈N |R(μn , A)x, x∗ | = ∞. Since X = span X+ and X ∗ = span X+ (Proposi∗ ∗ tion C.2), we can assume that x ∈ X+ , x ∈ X+ . By Bernstein’s Theorem 2.7.7, there exists an increasing function α : R+ → R such that α(0) = 0 and ∞ R(λ, A)x, x∗ = e−λt dα(t) = dα(λ) 0

for all λ > s. It follows from uniqueness of holomorphic extensions that n) R(μn , A)x, x∗ = dα(μ for all n ∈ N. Consequently, |R(μn , A)x, x∗ | ≤ dα(Re μn ) = R(Re μn , A)x, x∗ ≤ R(λ, A)x, x∗ , where λ := inf n∈N Re μn > s (since Re μ = limn→∞ Re μn > s). This is a contradiction. d) In order to prove the remaining assertion, assume that there exists λ ∈ ρ(A) such that λ < s(A) and R(λ, A) ≥ 0. Let μn ↓ s(A). Since s(A) ∈ σ(A),

R(μn , A) → ∞ as n → ∞.

190

3. CAUCHY PROBLEMS

As in a), we have R(λ, A) ≥ R(μn , A) ≥ 0

(n ∈ N).

This is impossible. From Theorem 3.11.2 and its proof, we note the following. Corollary 3.11.3. Let A be a resolvent positive operator. Then |R(λ, A)x, x∗ | ≤ R(ω, A)x, x∗

(3.81)

∗ whenever Re λ ≥ ω > s(A), x ∈ X+ and x∗ ∈ X+ . In particular, for each ω > s(A), sup R(λ, A) < ∞. Re λ≥ω

We need the following identity. Lemma 3.11.4. Let A be an operator and λ ∈ ρ(A). Then for all m ∈ N (−1)m λm+1 (R(λ, A)/λ)

(m)

/m! =

m

λk R(λ, A)k+1 .

(3.82)

k=0

Proof. This is immediate from Leibniz’s rule, since (−1)k R(λ, A)(k) /k! = R(λ, A)k+1 . Now we can prove the following generation theorem. Theorem 3.11.5. Let A be a resolvent positive operator. Then A generates a twice integrated semigroup which is Lipschitz continuous on bounded intervals. If D(A) is dense, then A generates a once integrated semigroup. Proof. Considering A − ω instead of A, we can assume that s(A) < 0 (see Proposition 3.2.6). Let m ∈ N, λ ≥ 0. Then m−1

λk R(λ, A)k+1 = R(0, A) − λm R(λ, A)m R(0, A).

(3.83)

k=0

For m = 1, this is just the resolvent equation. Then (3.83) follows by induction. Consequently, m−1 0≤ λk R(λ, A)k+1 ≤ R(0, A) k=0

for m ∈ N, λ ≥ 0. It follows from Lemma 3.11.4 that sup λ>0,m∈N0

λm+1 (R(λ, A)/λ)(m) /m! < ∞.

3.11. RESOLVENT POSITIVE OPERATORS

191

Now the claim follows from Theorem 3.3.1 and Theorem 3.3.2. More generally, if A is resolvent positive, it follows from Corollary 3.3.13 that the part of A in Y = D(A) generates a once integrated semigroup. The following example shows that a resolvent positive operator does not generate a once integrated semigroup in general. Example 3.11.6. Let X = C[−1, 0] × C and A be given by D(A) := C 1 [−1, 0] × {0},

A(f, 0) := (f , −f (0)).

Then ρ(A) = C and R(λ, A)(f, c) = (g, 0) with 0 λx −λy g(x) := e e f (y) dy + c x

for all λ ∈ C. Thus, A is resolvent positive. Let eλ ∈ C[−1, 0] be given by (λ > 0, x ∈ [−1, 0]). ∞ Then (eλ , 0) = R(λ, A)(0, 1). One has eλ = 0 λ2 e−λt kt dt where kt ∈ C[−1, 0] is given by kt (x) := 0 if x + t ≤ 0 and kt (x) := x + t otherwise. If A were the generator of a once integrated semigroup, then λ → eλ /λ would be a Laplace d transform. Hence, k : R+ → C[−1, 0] would be diﬀerentiable. But dt kt (x) does not exist at x = −t if t ∈ (0, 1). eλ (x) := eλx

The following result shows that the situation is diﬀerent if X has order continuous norm. Theorem 3.11.7. Let A be a resolvent positive operator and assume that X has order continuous norm. Then A generates a once integrated semigroup. Proof. Replacing A by A − ω, we may assume that s(A) ≤ 0. Let x ∈ X+ . Then by Theorem 2.7.18, there exists a unique normalized increasing function Fx : R+ → X such that Fx (0) = 0 and ∞ R(λ, A)x = e−λt dFx (t) (λ > s(A)). 0

It follows from uniqueness that Fx (t) is additive and positive homogeneous in x for all t ≥ 0. Hence, there exists a positive linear operator S1 (t) ∈ L(X) such that Fx (t) = S1 (t)x for all t ≥ 0, x ∈ X. Since S1 (·) is increasing, we can deﬁne the t Riemann integral S2 (t)x := 0 S1 (s)x ds for all x ∈ X (see Corollary 1.9.6). Then ∞ ∞ R(λ, A)x = λ e−λt S1 (t)x dt = λ2 e−λt S2 (t)x dt 0

0

for all x ∈ X, λ > max{0, s(A)}, where the ﬁrst integral is understood as an improper Riemann integral (see (1.22) and Proposition 1.10.2). We have to show that S1 is strongly continuous on R+ .

192

3. CAUCHY PROBLEMS

Since S2 (·)x is continuous, S2 is the twice integrated semigroup generated by A. Thus for x ∈ D(A), S2 (·)x is continuously diﬀerentiable by Lemma 3.2.2. Hence S1 (·)x is continuous if x ∈ D(A). Let x ∈ X+ and μ > s(A). Then R(μ, A)S1 (·)x is a normalized increasing function on R+ and ∞ ∞ e−λt d(R(μ, A)S1 (t)x) = R(μ, A) e−λt d(S1 (t)x) 0

0

= R(μ, A)R(λ, A)x = =

R(λ, A)R(μ, A)x ∞ e−λt d(S1 (t)R(μ, A)x) 0

for all λ > s(A). It follows from uniqueness of the representation (see Theorem 2.7.18) that S1 (t)R(μ, A)x = R(μ, A)S1 (t)x. Since the norm is order continuous, y+ := lims↓t S1 (s)x exists. Moreover, R(μ, A)y+

= lim R(μ, A)S1 (s)x s↓t

= lim S1 (s)R(μ, A)x s↓t

= S1 (t)R(μ, A)x = R(μ, A)S1 (t)x. Since R(μ, A) is injective, it follows that y+ = S1 (t)x. In the same way one shows that lims↑t S1 (s)x = S1 (t)x if t > 0. We have shown that S1 is strongly continuous on R+ . The closure of the domain of a resolvent positive operator is of a very special nature if the underlying space is a Banach lattice with order continuous norm. Let X be a complex Banach lattice; i.e., the complexiﬁcation of a real Banach lattice. A subspace J of X is called an ideal if a) x ∈ J implies Re x ∈ J; and b) if x, y ∈ X are real, |y| ≤ |x| and x ∈ J, then it follows that y ∈ J. In a space X := Lp (Ω, μ), where (Ω, μ) is a σ-ﬁnite measure space and 1 ≤ p < ∞, every closed ideal J is of the form J = {f ∈ Lp (Ω) : f |Ω0 = 0 a.e.} , where Ω0 is a measurable subset of Ω (see [Sch74, p.157]). Theorem 3.11.8. Let X be a Banach lattice with order continuous norm. If A is a resolvent positive operator, then D(A) is an ideal.

3.11. RESOLVENT POSITIVE OPERATORS

193

Proof. a) Note that by deﬁnition X is the complexiﬁcation of a real Banach lattice XR . Since the resolvent leaves XR invariant we have Re x ∈ D(A) whenever x ∈ D(A). Now observe that if J is a closed ideal of XR , then J ⊕ iJ is a closed ideal of X. These remarks show that we can assume that X is a real Banach lattice, which we do. b) We can also assume that s(A) < 0 (replacing A by A − ω otherwise). c) Let 0 ≤ y ≤ R(0, A)x where x ∈ X+ . We claim that y ∈ D(A). For λ > 0 we have 0 ≤ λR(λ, A)y ≤ λR(λ, A)R(0, A)x = R(0, A)x − R(λ, A)x ≤ R(0, A)x. Since the order interval [0, R(0, A)x] is weakly compact [AB85, Theorem 12.9], there exists a weak limit point z of λR(λ, A)y as λ → ∞. In particular, z ∈ D(A). Then R(0, A)y − R(λ, A)y = λR(0, A)R(λ, A)y has R(0, A)z as weak limit point. By the inequality above, lim R(λ, A)y = 0,

λ→∞

so R(0, A)y = R(0, A)z. Since R(0, A) is injective, it follows that y = z ∈ D(A). d) Let y ∈ D(A). Then |y| ∈ D(A). In fact, there exists x ∈ X such that y = R(0, A)x. Hence, |y| ≤ R(0, A)|x| and the claim follows from c). e) If y ∈ D(A), then |y| ∈ D(A). This follows from d) since the absolute value is a continuous mapping. f) Let 0 ≤ y ≤ x ∈ D(A). Let xn ∈ D(A) such that limn→∞ xn = x. It follows from e) that |xn | ∈ D(A). We have y ∧ |xn | ≤ |xn | = |R(0, A)Axn | ≤ R(0, A)|Axn |. It follows from c) that y ∧ |xn | ∈ D(A). Hence, y = limn→∞ (y ∧ |xn |) ∈ D(A). g) Let |y| ≤ |x|, where x ∈ D(A). Then 0 ≤ y + ≤ |x|, so y + ∈ D(A), by e) and f). Similarly, y − ∈ D(A), and therefore y = y + − y − ∈ D(A). In some special cases, densely deﬁned resolvent positive operators are automatically generators of C0 -semigroups. Theorem 3.11.9. Let X = C(K) where K is a compact space. Let A be a densely deﬁned resolvent positive operator. Then A generates a positive C0 -semigroup. Proof. Since D(A) is dense, there exists a strictly positive function u ∈ D(A); i.e., u(x) ≥ ε > 0 for all x ∈ K and some ε > 0. We can assume that s(A) < 0 (replacing A by A−ω if necessary). There exists v ∈ C(K) such that u = R(0, A)v.

194

3. CAUCHY PROBLEMS

Then for λ > 0, n ∈ N and f ∈ C(K) with f ∞ ≤ 1, |(λR(λ, A))n f |

≤

(λR(λ, A))n |f | 1 ≤ (λR(λ, A))n u ε 1 ≤ (λR(λ, A))n R(0, A)|v| ε n−1 1 k k+1 = R(0, A)|v| − λ R(λ, A) |v| ε k=0

≤

1 R(0, A)|v|, ε

using (3.83). It follows that

1

R(0, A)|v| ∞ ε for all λ > 0 and n ∈ N. Now the claim follows from the Hille-Yosida theorem.

(λR(λ, A))n ≤

Now we return to the case when X is an arbitrary ordered Banach space with normal cone. We consider the inhomogeneous Cauchy problem u (t) = Au(t) + f (t) (t ∈ [0, τ ]), (ACPf ) u(0) = u0 , where A is a closed operator on X, τ > 0 and f ∈ C([0, τ ], X). Recall from Section 3.1 that a mild solution of (ACPf ) is a function u ∈ C([0, τ ], X) such that t t t u(s) ds ∈ D(A) and u(t) − u0 = A u(s) ds + f (s) ds 0

0

0

for all t ∈ [0, τ ]. The function u is called a classical solution if in addition u ∈ C 1 ([0, τ ], X). In that case, since A is closed, it follows that u ∈ C([0, τ ], D(A)) and (ACPf ) is satisﬁed. The following result is a special case of a sharper version of Corollary 3.2.11 c) which is valid for generators of Lipschitz continuous integrated semigroups. Theorem 3.11.10. Let A be a resolvent positive operator. Let u0 ∈ D(A), f0 ∈ X t such that Au0 + f0 ∈ D(A). Let f (t) = f0 + 0 f (s) ds where f ∈ L1 ((0, τ ), X). Then (ACPf ) has a unique mild solution. Proof. Denote by S the twice integrated semigroup generated by A. Let v(t) = S(t)u0 + (S ∗ f )(t). By Lemmas 3.2.9 and 3.2.10, there exists a unique solution if and only if v ∈ C 2 ([0, τ ], X). By Proposition 1.3.6, one has S ∗ f ∈ C 1 ([0, τ ], X) and d (S ∗ f )(t) = (S ∗ f )(t) + S(t)f0 . dt

3.11. RESOLVENT POSITIVE OPERATORS

195

Now it follows from Lemma 3.2.2 c) that v ∈ C 1 ([0, τ ], X) and v (t) = S(t)(Au0 + f0 ) + tu0 + (S ∗ f )(t). Since S is Lipschitz continuous on [0, τ ] (by Theorem 3.11.5), it follows from Proposition 1.3.7 that S ∗ f ∈ C 1 ([0, τ ], X). Since Au0 + f0 ∈ D(A), it follows from Lemma 3.3.3 that S(·)(Au0 + f0 ) ∈ C 1 ([0, τ ], X). The proof is complete. Theorem 3.11.10 will be used in Section 6.2 to solve the heat equation with inhomogeneous boundary conditions. The following result shows that mild solutions of the inhomogeneous problem are positive if the initial value and the inhomogeneity are positive. In Section 6.2 this will be used to prove the parabolic maximum principle. Theorem 3.11.11. Let A be a resolvent positive operator, τ > 0, f ∈ C([0, τ ], X+ ), and u0 ∈ X+ . Let u be a mild solution of (ACPf ). Then u(t) ≥ 0 for all t ∈ [0, τ ]. Proof. Denote by S the twice integrated semigroup generated by A. It follows from Theorem 2.7.15 that S is an increasing convex function. Let t w(t) := S(t)u0 + S(t − r)f (r) dr. 0

It follows from Lemma suﬃces to show that w := S(t) for Deﬁne S(t) convex and S(0) = 0, it t 0

3.2.9 that w ∈ C 2 ([0, τ ], X) and u(t) = w (t). Thus, it is convex. We know this already for the ﬁrst term of w. = 0 for t < 0. Since S is increasing and t ≥ 0 and S(t) : R → L(X) is also convex. Hence, follows that S(t) S(t − r)f (r) dr =

∞

0

− r)f (r) dr S(t

is convex in t ≥ 0. Next we consider a simple perturbation result. Proposition 3.11.12. Let A be a resolvent positive operator. Let B : D(A) → X be linear and positive (i.e., Bx ≥ 0 if x ∈ D(A) ∩ X+ ). If the spectral radius r(BR(λ, A)) < 1 for some λ > s(A), then A + B is resolvent positive and s(A + B) < λ. Notice that BR(λ, A) is a linear, positive mapping on X and so it is automatically continuous. Proof. Let x ∈ D(A). Then (λ − (A + B))x = (I − BR(λ, A))(λ − A)x. Let Sλ := (I − BR(λ, A))−1 =

∞ n=0

(BR(λ, A))n .

196

3. CAUCHY PROBLEMS

Then Sλ is a bounded, positive operator on X and R(λ, A)Sλ (λ − (A + B))x = x for all x ∈ X and (λ − (A + B))R(λ, A)Sλ y = y

for all y ∈ X.

Thus, λ ∈ ρ(A + B) and R(λ, A + B) = R(λ, A)Sλ ≥ 0. If μ > λ, then by Proposition 3.11.2, μ ∈ ρ(A) and BR(μ, A) ≤ BR(λ, A) and so r(BR(μ, A)) ≤ r(BR(λ, A)) < 1. Replacing λ by μ, it follows that μ ∈ ρ(A+B) and R(μ, A+B) ≥ 0 for all μ ≥ λ. The following example shows that, in the theorem above, A + B may not be generator of a C0 -semigroup even if A generates a positive C0 -semigroup. Example 3.11.13. Let α ∈ (0, 1). Deﬁne the operator A by Af (x) := −f (x) +

α f (x) x

(x ∈ (0, 1])

on C0 (0, 1] := {f ∈ C[0, 1] : f (0) = 0} with domain D(A) := {f ∈ C 1 [0, 1] : f (0) = f (0) = 0}. Then A is resolvent positive but not the generator of a C0 semigroup. Moreover, s(A) = −∞. Proof. Let A0 f := −f with domain D(A0 ) = D(A). Then A0 is the generator of the C0 -semigroup (T (t))t≥0 given by f (x − t) (x ≥ t), (T (t)f )(x) = 0 (x < t). Moreover, σ(A0 ) = ∅ and (R(λ, A0 )f )(x) = e−λx

x

eλy f (y) dy

0

(λ ∈ C, f ∈ C0 (0, 1]).

Let B : D(A0 ) → C0 (0, 1] be given by (Bf )(x) :=

α f (x) x

(x > 0),

(Bf )(0) := 0.

Let g ∈ C0 (0, 1], f := R(0, A)g. Then x α |(Bf )(x)| = g(y) dy ≤ α g ∞ . x 0 Thus BR(0, A0 ) ≤ α < 1. Now Proposition 3.11.12 implies that A = A0 + B is resolvent positive.

3.12. COMPLEX INVERSION AND UMD-SPACES

197

It remains to show that A is not the generator of a C0 -semigroup. One can easily check that for all λ ∈ C one has λ ∈ ρ(A) and (R(λ, A)g)(x)

x e−λx xα y −α eλy g(y) dy 0 x = xα (x − t)−α g(x − t)e−λt dt

=

0

(g ∈ C0 (0, 1]).

Suppose that A generates a C0 -semigroup T . Then

∞

(R(λ, A)g)(x) =

e−λt (T (t)g)(x) dt

0

for suﬃciently large λ. It follows from the uniqueness theorem (Theorem 1.7.3) that xα (x − t)−α g(x − t) (x ≥ t), (T (t)g)(x) = 0 (x < t). This does not deﬁne a bounded operator on C0 (0, 1].

3.12 Complex Inversion and UMD-spaces In this section, we apply the complex inversion formula for Laplace transforms (Theorem 2.3.4) to orbits of C0 -semigroups; i.e., to solutions of well-posed Cauchy problems. In general this produces a representation only of classical solutions in terms of the resolvent of A, but we shall see in Theorem 3.12.2 that there is a class of Banach spaces where the representation holds for mild solutions. Proposition 3.12.1. Let T be a C0 -semigroup on a Banach space X with generator A, and let ω > ω(T ) and t ≥ 0. Then 1 k→∞ 2π

k

T (t)x = lim

e(ω+is)t R(ω + is, A)x ds

−k

for all x ∈ D(A). Proof. Replacing T (t) by e−αt T (t) where ω(T ) < α < ω, we may assume that ω(T ) < 0 < ω. Let x ∈ D(A) and deﬁne F (t) = T (t)x − x. Then F is diﬀerentiable with F (t) = T (t)Ax. Since F is bounded, F ∈ Lip0 (R+ , X). The Laplace-Stieltjes transform of F is (λ) = R(λ, A)Ax. dF

198

3. CAUCHY PROBLEMS

By Theorem 2.3.4, 1 k→∞ 2π

k

R(ω + is, A)Ax ds ω + is 1 x (ω+is)t = lim e R(ω + is, A)x − ds k→∞ 2π −k ω + is k 1 = lim e(ω+is)t R(ω + is, A)x ds − x, k→∞ 2π −k

T (t)x − x =

lim

e(ω+is)t

−k k

where we have used a standard contour integral. For f ∈ L2 (R, X) and 0 < ε < R, let (HεR f )(t) := where

1 π

ε≤|t−s|≤R

f (s) ds = (ψεR ∗ f )(t) t−s

⎧ ⎨1 ψεR (t) := πt ⎩0

(t ∈ R),

if ε ≤ |t| ≤ R, otherwise.

Then HεR ∈ L(L2 (R, X)), since ψεR ∈ L1 (R, X) (see Proposition 1.3.2). The Banach space X is said to be a UMD-space if Hf := lim HεR f ε↓0 R→∞

exists in L2 (R, X) for each f ∈ L2 (R, X). Then by the Banach-Steinhaus theorem, H is a bounded linear operator, known as the Hilbert transform, on L2 (R, X). When f (t) = χ[a,b] (t)x, then Hf exists in L2 (R, X). Since the step functions are dense in L2 (R, X), it follows that X is a UMD-space if sup0<ε ω(T ) and t > 0. Then 1 T (t)x = lim k→∞ 2π for all x ∈ X.

k

−k

e(ω+is)t R(ω + is, A)x ds

3.12. COMPLEX INVERSION AND UMD-SPACES

199

Proof. Replacing T (t) by e−ωt T (t), we may assume that ω(T ) < 0 = ω. For k > 0 and t ∈ R, deﬁne k 1 Tk (t) := eist R(is, A) ds, 2π −k k 1 Sk (t) := eist R(is, A)2 ds. 2π −k Since Tk (t)x → T (t)x for all x ∈ D(A) (Proposition 3.12.1) and D(A) is dense, it suﬃces to show that sup Tk (t) < ∞ k

for each t > 0. Integration by parts gives " 1 1 ! ikt e R(ik, A) − e−ikt R(−ik, A) + Sk (t). 2πit t ∞ Since R(±ik, A) ≤ 0 T (s) ds < ∞, it suﬃces to show that Tk (t) =

sup Sk (t) < ∞. k

∗

Since X is reﬂexive, T is strongly continuous (Corollary 3.3.9 and Proposition 3.3.14). Let x ∈ X, x∗ ∈ X ∗ and t ∈ R. Using Fubini’s theorem, x, Tk (t)∗ x∗ =

1 2π

=

1 2π

=

k

∞

eist e−isr x, T (r)∗ x∗ dr ds

−k 0 ∞ ik(t−r)

e

0

lim

ε↓0 R→∞

− e−ik(t−r) x, T (r)∗ x∗ dr i(t − r)

" 1 ! ikt e x, HεR (fk )(t) − e−ikt x, HεR (f−k )(t) , 2i

e−iar T (r)∗ x∗ fa (r) := 0

where

(r ≥ 0), (r < 0).

It follows that Tk (t)∗ x∗ =

" 1 ! ikt e H(fk )(t) − e−ikt H(f−k )(t) 2i

t-a.e.,

where H is the Hilbert transform on L2 (R, X ∗ ). Hence,

∞ −∞

∗ ∗ 2

Tk (t) x dt

1/2

H

( fk 2 + f−k 2 ) 2 ≤ M H x∗ , ≤

200

3. CAUCHY PROBLEMS

∞ where M := ( 0 T (s) 2 ds)1/2 < ∞. For x ∈ X and x∗ ∈ X ∗ , Fubini’s theorem gives k ∞ 1 Sk (t)x, x∗ = eist R(is, A)e−isr T (r)x, x∗ dr ds 2π −k 0 ∞1 k 2 1 = eis(t−r) R(is, A)T (r)x ds, x∗ dr 2π −k 0 ∞ = T (r)x, Tk (t − r)∗ x∗ dr. 0

Now the Cauchy-Schwarz inequality gives ∞ 1/2 ∗ 2 |Sk (t)x, x | ≤

T (r)x dr 0 2

∞ 0

∗ ∗ 2

1/2

Tk (t − r) x || dr

≤ M H x x∗ . Thus,

Sk (t) ≤ M 2 H .

Example 3.12.3. Theorem 3.12.2 is not valid if the assumption that X is a UMDspace is omitted. Let X := L1 (R) and T be the C0 -semigroup of invertible isometries on X deﬁned by (T (t)f )(r) := f (t + r) (t ≥ 0, r ∈ R). Let 1 Tk (t) := 2π

k

A routine calculation shows that (Tk (t)f ) (r) =

e(1+is)t R(1 + is, A) ds.

−k

1 π

∞

−∞

f (r + s + t)φk,t (s) ds,

⎧ −s ⎨ e sin(ks) φk,t (s) := s ⎩0

where

Hence,

Tk (t) = φk,t 1

∞

=

−s

e −t

≥ e−1

1 0

(s ≥ −t), (s < −t).

sin(ks) s ds

| sin(ks)| ds = e−1 s

k 0

| sin s| ds → ∞ s

as k → ∞. Thus, {Tk (t) : k ≥ 0} is not uniformly bounded and therefore not strongly convergent on X.

3.13. NORM-CONTINUOUS SEMIGROUPS AND HILBERT SPACES

201

3.13 Norm-continuous Semigroups and Hilbert Spaces In this section, we shall consider C0 -semigroups T = (T (t))t≥0 which are normcontinuous for t > 0 . This class contains all holomorphic C0 -semigroups, including many examples arising from diﬀerential operators. In general, there is no known characterization of such semigroups in terms of the generator and resolvent, but there is a simple characterization in the case of Hilbert spaces. Proposition 3.13.1. Let T be a C0 -semigroup on a Banach space X with generator A, and suppose that T is norm-continuous for t > 0. Let ω > ω(T ). Then R(ω + is, A) → 0 as |s| → ∞. Proof. Since T : (0, ∞) → L(X) is norm-continuous,

∞

R(ω + is, A) =

e−(ω+is)t T (t) dt

0

as an L(X)-valued Bochner integral. Let H(t) := e−ωt T (t) if t > 0, and H(t) := 0 if t < 0. Then H ∈ L1 (R, L(X)), and R(ω + is, A) = (FH)(s). The result follows from the Riemann-Lebesgue lemma (Theorem 1.8.1). It follows from Neumann series expansions (Corollary B.3) that if R(ω + is, A) → 0 as |s| → ∞, then for any real a, {λ ∈ σ(A) : Re λ ≥ a} is bounded, and R(α + is, A) → 0, uniformly for α > a, as |s| → ∞. The converse of Proposition 3.13.1 does not hold in general, but we will now prove that it is true when X is a Hilbert space. Theorem 3.13.2. Let T be a C0 -semigroup on a Hilbert space X with generator A. Let ω > ω(T ), and suppose that R(ω + is, A) → 0 as |s| → ∞. Then T is norm-continuous for t > 0. Proof. Replacing T (t) by e−ωt T (t), we may assume that ω(T ) < 0 = ω. Let x ∈ D(A) and F (t) := t2 T (t)x. Then F is diﬀerentiable and F (t) = 2tT (t)x + t2 T (t)Ax. Hence F ∈ Lip0 (R+ , X) with Laplace-Stieltjes transform (λ) dF

= = =

d d2 (R(λ, A)x) + 2 (R(λ, A)Ax) dλ dλ 2R(λ, A)2 x + 2R(λ, A)3 Ax 2λR(λ, A)3 x.

−2

By Theorem 2.3.4, 1 k→∞ π

t2 T (t)x = lim

k

−k

eist R(is, A)3 x ds.

(3.84)

202

3. CAUCHY PROBLEMS

! ∞ "1/2 Let M :=

T (t) 2 dt < ∞. Given ε > 0, there exists N such that 0

R(is, A) < ε/4M 2 whenever |s| > N . For k > N , t, t0 ≥ 0 and y ∈ X, ! ist " e − eist0 R(is, A)3 x dsy N ≤|s|≤k X ≤ 2

R(is, A)2 x R(is, A)∗ y ds N ≤|s|≤k

∞ 1/2 ∞ 1/2 ε 2 ∗ 2

R(is, A)x

ds

R(is, A) y

ds 2M 2 −∞ −∞ ∞ 1/2 ∞ 1/2 πε 2 ∗ 2 =

T (r)x

dr

T (r) y

dr M2 0 0 ≤ πε x y , ≤

where (·|·)X denotes the inner product on X and we have used the Cauchy-Schwarz inequality and Plancherel’s theorem for Hilbert spaces (Theorem 1.8.2). Hence,

! ist "

e − eist0 R(is, A)3 x ds ≤ πε x .

N ≤|s|≤k

It follows from (3.84) that

2

t T (t)x − t20 T (t0 )x ≤ ε x + 2N x sup eist − eist0 sup R(is, A) 3 , π s∈R |s|≤N so

2

2

t T (t) − t2 T (t0 ) ≤ ε + 2N |t − t0 | 0 π

This shows that

∞ 0

3

T (r) dr

.

lim sup t2 T (t) − t20 T (t0 ) ≤ ε. t→t0

Since ε > 0 is arbitrary, t → t2 T (t) is norm-continuous, and the result is proved.

3.14 The Second Order Cauchy Problem Let A be a closed operator on a Banach problem ⎧ ⎪ ⎨u (t) P 2 (x, y) u(0) ⎪ ⎩ u (0)

space X. Given x, y ∈ X we consider the = Au(t) (t ≥ 0) = x, = y.

3.14. THE SECOND ORDER CAUCHY PROBLEM

203

A classical solution of P 2 (x, y) is a function u ∈ C 2 (R+ , X) such that u(t) ∈ D(A) for all t ≥ 0 and P 2 (x, y) holds. A mild solution is a function u ∈ C(R+ , X) such that t

s

t

u(r) dr ds = 0

0

0

(t − s)u(s) ds ∈ D(A)

and

u(t) = x + ty + A 0

t

(t − s)u(s) ds

(3.85)

for all t ≥ 0. If u is a classical solution, then integrating P 2 (x, y) twice shows that u is a mild solution. Conversely, if u is a mild solution and u ∈ C 2 (R+ , X), then u is a classical solution. This follows from (3.85) and the fact that A is closed. Proposition 3.14.1. Let u ∈ C(R+ , X) with abs(u) < ∞. Let ω > max{abs(u), 0}. Then u is a mild solution of P 2 (x, y) if and only if u ˆ(λ) ∈ D(A) and λx + y = (λ2 − A)ˆ u(λ)

(3.86)

for all λ > ω. Proof. There exists M ≥ 0 such that v(t) ≤ M eωt (t ≥ 0), where v(t) := t u(s) ds. Taking Laplace transforms, Corollary 1.7.6 shows that (3.85) holds if 0 and only if ∞ t u ˆ(λ) −λt = e v(s) ds dt ∈ D(A) λ2 0 0 and u ˆ(λ) =

x y u ˆ(λ) + +A 2 λ λ2 λ

for all λ > ω.

Now let u ∈ C(R+ , X). Assume that abs(u) < ∞, ω > max{abs(u), 0} and (ω, ∞) ⊂ ρ(A). Then Proposition 3.14.1 shows that u is a mild solution of P 2 (x, y) if and only if u ˆ(λ) = λR(λ2 , A)x + R(λ2 , A)y (λ > ω). (3.87) This relation will lead us to consider operators A such that λR(λ2 , A) is a Laplace transform. Deﬁnition 3.14.2. A strongly continuous function Cos : R+ → L(X) is called a cosine function if Cos(0) = I and 2 Cos(t) Cos(s) = Cos(t + s) + Cos(t − s) Lemma 3.14.3. Let Cos be a cosine function. a) ω(Cos) < ∞. b) Cos(t) Cos(s) = Cos(s) Cos(t) for all s, t ≥ 0.

(t ≥ s ≥ 0).

(3.88)

204

3. CAUCHY PROBLEMS

c) Deﬁne Cos(−t) = Cos(t) for t ≥ 0. Then Cos is strongly continuous on R, and 2 Cos(t) Cos(s) = Cos(t + s) + Cos(t − s) (s, t ∈ R). (3.89) Proof. a) By the uniform boundedness principle, M := sup Cos(s) < ∞. 0≤s≤2

Choose ω > 0 such that 2 Cos(1) e−ω + e−2ω ≤ 1. We claim that Cos(t) ≤ M eωt (t ≥ 0). This is certainly true for t ∈ [0, 2]. Assume that it holds for t ∈ [0, n], where n ∈ N, n ≥ 2. We claim that it holds for t ∈ [0, n + 1]. Let t ∈ (n − 1, n]. Then

Cos(t + 1) = 2 Cos(t) Cos(1) − Cos(t − 1)

≤ 2 Cos(1) M eωt + M eω(t−1) ! " = 2 Cos(1) e−ω + e−2ω M eω(t+1) ≤ M eω(t+1) . b) Let t ≥ 0. Replacing s and t by t/2 in (3.88) gives Cos(t) = 2 Cos(t/2)2 −I. A simple induction shows that, for each n ≥ 1, Cos(t) is a polynomial in Cos(t/2n ), so they commute. Assume that Cos(t) commutes with Cos(rt/2n ) for r = 1, . . . , k, for some k ≥ 1. Since (k + 1)t kt t (k − 1)t Cos = 2 Cos Cos − Cos , 2n 2n 2n 2n it follows that Cos(t) commutes with Cos((k + 1)t/2n ). By induction, Cos(t) commutes with Cos(rt)/2n for all integers r, n ≥ 1. Now b) holds by strong continuity of Cos. c) This follows easily from b). We will frequently consider a cosine function to be extended to R as in Lemma 3.14.3 c) without further notice. In what follows, the Laplace integrals of operator-valued functions are interpreted as in Sections 1.4 and 1.5. Proposition 3.14.4. Let Cos : R+ → L(X) be strongly continuous. The following assertions are equivalent: (i) Cos is a cosine function. (ii) One has abs(Cos) < ∞, and there exist ω > max{abs(Cos), 0} and an operator A such that (ω 2 , ∞) ⊂ ρ(A) and ∞ λR(λ2 , A) = e−λt Cos(t) dt (λ > ω). (3.90) 0

3.14. THE SECOND ORDER CAUCHY PROBLEM

205

In that case, we call A the generator of the cosine function Cos. Proof. We extend Cos to an even function on R. Let ω > max{0, abs(Cos)}. Then for λ, μ > ω with λ = μ, one has ∞ ∞ ! " 2 e−λt e−μs (Cos(t + s) + Cos(t − s)) ds dt = 2 μQ(λ) − λQ(μ) , 2 μ −λ 0 0 (3.91) ∞ where Q(λ) := 0 e−λt Cos(t) dt. In fact, 0

∞

∞

e−λt e−μs (Cos(t + s) + Cos(t − s)) ds dt ∞ ∞ t −λt −μ(r−t) −μ(t−r) = e e Cos(r) dr + e Cos(r) dr dt 0

0

∞

= 0

e−μr

t r

0

∞

0

e(μ−λ)t dt Cos(r) dr +

−∞ ∞

e−λt

0

0

e−μ(t−r) Cos(r) dr dt

−∞

t

+ e−λt e−μ(t−r) Cos(r) dr dt 0 0 ∞ = e−μr (μ − λ)−1 (e(μ−λ)r − 1) Cos(r) dr 0

∞

∞

∞

Cos(r) dr + e−(λ+μ)t dt eμr Cos(r) dr −∞ 0 0 r ∞ " 1 ! 1 −μr Q(λ) − Q(μ) + e Cos(r) dr μ−λ λ+μ 0 ∞ 1 + e−(λ+μ)r eμr Cos(r) dr λ+μ 0 " " 1 ! 1 ! Q(λ) − Q(μ) + Q(μ) + Q(λ) μ−λ λ+μ ! " 2 μQ(λ) − λQ(μ) . 2 2 μ −λ +

=

= =

−(λ+μ)t

e

μr

dt e

Now assume that Q(λ) = λR(λ2 , A) (λ > ω). Then μ2

! " 2 R(λ2 , A) − R(μ2 , A) μQ(λ) − λQ(μ) = 2λμ 2 −λ μ2 − λ2 = 2λμR(λ2 , A)R(μ2 , A) ∞ ∞ −λt = 2 e Cos(t) dt e−μs Cos(s) ds. 0

0

Now (3.91) implies by the Uniqueness Theorem 1.7.3 that Cos(t + s) + Cos(t − s) = 2 Cos(t) Cos(s) for all s, t ∈ R+ .

(3.92)

206

3. CAUCHY PROBLEMS

Conversely, assume that Cos is a cosine function. By Lemma 3.14.3, (3.92) holds, and by (3.91), ∞ ∞ ! " 1 μQ(λ) − λQ(μ) = e−λt e−μs Cos(t) Cos(s) dt ds μ2 − λ2 0 0 = Q(λ)Q(μ). √ Let R(λ) := √1λ Q( λ) for λ > ω 2 . Then R(λ)R(μ)

= = =

√ 1 √ √ √ Q( λ)Q( μ) λ μ √ √ √ 1 1 √ √ √ μ Q( λ) − λ Q( μ) λ μ μ−λ " 1 ! R(λ) − R(μ) (λ, μ > ω 2 ). μ−λ

Thus, {R(λ) : λ > ω 2 } is a pseudo-resolvent. If R(λ)x = 0 for all λ > ω2 , then by the Uniqueness Theorem, Cos(t)x = 0 for all t ∈ R+ . Since Cos(0) = I, this implies that x = 0. By Proposition B.6, there exists an operator A such that (ω 2 , ∞) ⊂ ρ(A) and ∞ 2 2 λR(λ , A) = λR(λ ) = Q(λ) = e−λt Cos(t) dt 0

for all λ > ω. Let Cos be a cosine function on X with generator A. The sine function Sin : R → L(X) associated with Cos is deﬁned by t Sin(t) := Cos(s) ds (t ∈ R). (3.93) 0

t

This means that Sin(t)x = 0 Cos(s)x ds (x ∈ X), where the integral is a Bochner integral. Then Sin is an odd function satisfying the functional equation t+s 2 Sin(t) Sin(s) = Sin(r) dr (t, s ∈ R). (3.94) t−s

This follows from integrating (3.89) twice. The following functional equation is also useful: Sin(t + s) = Cos(s) Sin(t) + Sin(s) Cos(t)

(t, s ∈ R).

To see this, diﬀerentiate (3.94) to obtain 2 Sin(t) Cos(s) = Sin(t + s) + Sin(t − s).

(3.95)

3.14. THE SECOND ORDER CAUCHY PROBLEM

207

Interchanging s and t gives 2 Sin(s) Cos(t) = Sin(s + t) + Sin(s − t). Since Sin is odd, adding these two equations gives (3.95). Moreover, we deduce from (3.90) that ∞ e−λt Sin(t) dt (λ > max{ω(Cos), 0}). R(λ2 , A) = 0

We collect some further properties of a cosine function Cos and its relations with the generator A, and the associated sine function Sin. Proposition 3.14.5. The following assertions hold: t t a) 0 (t − s) Cos(s)x ds ∈ D(A) and A 0 (t − s) Cos(s)x ds = Cos(t)x − x for all x ∈ X, t ∈ R. b) If x ∈ D(A), then Cos(t)x, Sin(t)x ∈ D(A) and A Cos(t)x = Cos(t)Ax, A Sin(t)x = Sin(t)Ax for all t ∈ R. t c) Let x, y ∈ X. Then x ∈ D(A) and Ax = y if and only if 0 (t−s) Cos(s)y ds = Cos(t)x − x for all t ∈ R. ' ( d) D(A) = x ∈X : limt↓0 t22 (Cos(t)x−x) exists and Ax = limt↓0 t22 (Cos(t)x−x). e) A is densely deﬁned. f) For all x ∈ X, s, t ∈ R, one has Sin(t) Sin(s)x ∈ D(A) and A Sin(t) Sin(s)x =

1 2

(Cos(t + s)x − Cos(t − s)x) .

Proof. a) It follows from Proposition 3.14.1 that Cos(·)x is a mild solution of P 2 (x, 0). This implies a) for t ≥ 0. It follows for t < 0 since Cos is even. b) Let μ ∈ ρ(A). It follows from Proposition 3.1.5 that R(μ, A) Cos(t) = Cos(t)R(μ, A). This clearly implies b) (see Proposition B.7). t c) Assume that 0 (t − s) Cos(s)y ds = Cos(t)x − x (t ≥ 0). Taking Laplace transforms, we obtain 1 x R(λ2 , A)y = λR(λ2 , A)x − λ λ

(λ > max{ω(Cos), 0}).

Hence, x ∈ D(A) and Ax = y. The converse follows from a) and b). d) Let x ∈ D(A), Ax = y. It follows from c) that 2 2 t (Cos(t)x − x) − y = (t − s) Cos(s)y ds − y t2 t2 0 ! " 2 t = 2 (t − s) Cos(s)y − y ds → 0 as t ↓ 0, t 0

208

3. CAUCHY PROBLEMS

since Cos(t)y → y as t ↓ 0. Conversely, let x, y ∈ X with limt↓0 t22 (Cos(t)x−x) = y. Then by a), 2 t 2 A 2 (t − s) Cos(s)x ds = 2 (Cos(t)x − x) → y as t ↓ 0. t 0 t t Since t22 0 (t − s) Cos(s)x ds → x as t ↓ 0 and since A is closed, it follows that x ∈ D(A) and Ax = y. t e) Since x = limt↓0 t22 0 (t − s) Cos(s)x ds for all x ∈ X, it follows from a) that D(A) = X. f) This follows from (3.94) and a). Using the functional equation (3.88), one sees that a cosine function necessarily has non-negative exponential type. More precisely, the following holds. Proposition 3.14.6. a) Let Cos be a bounded cosine function and x ∈ X. If limt→∞ Cos(t)x = 0, then x = 0. b) In particular, ω(Cos) ≥ 0 for each cosine function Cos. Proof. a) Since Cos is bounded, it follows from the assumption that x = lim (Cos(2t)x + x) = lim 2(Cos(t))2 x = 0. t→∞

t→∞

It is natural to transform the second order problem into a ﬁrst order system. Let A be an operator on X. Consider the operator A on X × X given by D(A) := D(A) × X, x 0 I x y A := = . y A 0 y Ax

(3.96)

Here, X × X is considered with the norm (x, y) X×X = x X + y X . Let λ ∈ C such that λ2 ∈ ρ(A). Then one easily veriﬁes that λ ∈ ρ(A) and λ ∈ ρ(−A) and λR(λ2 , A) R(λ2 , A) R(λ, A) = , (3.97) AR(λ2 , A) λR(λ2 , A) λR(λ2 , A) −R(λ2 , A) R(λ, −A) = . (3.98) −AR(λ2 , A) λR(λ2 , A) Using this we can prove the following theorem. Theorem 3.14.7. The operator A generates a cosine function Cos on X if and only if A generates a once integrated semigroup S on X × X. In that case, S is given by t Sin(t) Sin(s) ds 0 S(t) = , (3.99) Cos(t) − I Sin(t) t where Sin(t) = 0 Cos(s) ds.

3.14. THE SECOND ORDER CAUCHY PROBLEM

209

Proof. Assume that A generates a cosine function Cos. Then ∞ 2 2 AR(λ , A)/λ = λR(λ , A) − I/λ = e−λt (Cos(t) − I) dt. 0

It follows from (3.97) that ∞ e−λt S(t) dt = R(λ, A)/λ

(λ > ω(Cos)).

0

Thus, S is a once integrated semigroup and A is its generator. Conversely, assume that A generates a once integrated semigroup. Then R(λ, A)/λ and AR(λ2 , A)/λ = λR(λ2 , A) − I/λ is a Laplace transform. ∞hence −λt Since 1/λ = 0 e dt, λR(λ2 , A) is also a Laplace transform; i.e., A is the generator of a cosine function. Corollary 3.14.8. Let A be the generator of a cosine function Cos and let Sin be the associated sine function. Let x, y ∈ X and u(t) := Cos(t)x + Sin(t)y

(t ≥ 0).

Then u is the unique mild solution of P 2 (x, y). Proof. It follows from (3.87) that u is a mild solution. Let v be another one and deﬁne t w(t) := (t − s)(u(s) − v(s)) ds. 0

w(t) Then w is a classical solution of P (0, 0). Consider the function φ(t) := . w (t) Then φ is a classical solution of the homogeneous Cauchy problem associated with A with initial value φ(0) = 0. It follows from Lemma 3.2.9 that φ ≡ 0. 2

Let A be a bounded operator. Then Cos(t) :=

∞ t2n n A (2n)! n=0

3 deﬁnes a continuous function from R+ into L(X), and for λ > A one has ∞ ∞ ∞ t2n e−λt Cos(t) dt = e−λt dt An (2n)! 0 0 n=0 ∞ An 2n+1 λ n=0 −1 1 A = I− 2 λ λ

=

= λR(λ2 , A).

210

3. CAUCHY PROBLEMS

Thus, Cos is a cosine function and A is its generator. As a consequence of Theorem 3.14.7, we may characterize cosine functions with bounded generators as follows. Corollary 3.14.9. The following assertions are equivalent: (i) The operator A generates a C0 -semigroup on X × X. (ii) A generates a cosine function Cos such that limt↓0 Cos(t) − I = 0. (iii) A is bounded. Proof. (i) ⇒ (ii): If A generates a C0 -semigroup, then limt↓0 S(t) = 0. Now (ii) follows from (3.99). (ii) ⇒ (iii): The Abelian Theorem 4.1.2 implies that limλ↓0 λ2 R(λ2 , A)−I = 0. Hence, λ2 R(λ2 , A) is invertible for large λ. Consequently, D(A) = X. (iii) ⇒ (i): If A is bounded, then A is also bounded. We now deduce a simple perturbation result which will be considerably improved in Corollary 3.14.13. Corollary 3.14.10. Let A be the generator of a cosine function and let B ∈ L(X). Then A + B generates a cosine function. Proof. Consider the operator

B :=

0 B

0 0

on X × X. Then B(X × X) ⊂ {0} × X ⊂ D(A). It follows from Corollary 3.10.5 0 I that A + B = generates a once integrated semigroup. Now the A+B 0 claim follows from Theorem 3.14.7. We have seen that A does not generate a C0 -semigroup on X × X unless A is bounded. However, a semigroup exists on a natural “phase space”. Theorem 3.14.11. The following assertions are equivalent: (i) The operator A generates a cosine function. (ii) There exists a Banach space V such that D(A) → V → X and such that the part B of A in V × X generates a C0 -semigroup. In that case, the Banach space V is uniquely determined by (ii). We call V × X the phase space associated with A. Moreover, one has Sin(·)y ∈ C(R, V ) for all y ∈ X, Cos(·)x ∈ C 1 (R, X) ∩ C(R, V ) for all x ∈ V , Sin(·)x ∈ C(R, D(A)) for all x ∈ V , and B generates a C0 -group J on V × X given by Cos(t) Sin(t) Cos(t) Sin(t) J (t) = = (t ∈ R), (3.100) Cos (t) Cos(t) A Sin(t) Cos(t) where Sin is the sine function associated with Cos, and Cos (t)x := A Sin(t)x (x ∈ V ).

d dt

Cos(t)x =

3.14. THE SECOND ORDER CAUCHY PROBLEM

211

Here, V × X is a Banach space for the norm (x, y) V ×X := x V + y X . Note that the operator B on V × X is deﬁned as follows: D(B) = D(A) × V, x 0 I x y B = = . y A 0 y Ax Proof of Theorem 3.14.11. (i) ⇒ (ii): Assume that A generates a cosine function. Then A generates a once integrated semigroup on X ×X (Theorem 3.14.7). By the Sandwich Theorem 3.10.4, there exists a Banach space Z such that D(A) → Z → X × X and such that the part B of A in Z generates a C0 -semigroup J . Deﬁne V := {x ∈ X : (x, 0) ∈ Z} with norm x V := (x, 0) Z . Then Z = V ×X. In fact, let (x, y) ∈ Z. Since (0, y) ∈ D(A) ⊂ Z, it follows that (x, 0) = (x, y) − (0, y) ∈ Z. Hence, x ∈ V and (x, y) ∈ V × X. The converse inclusion is obvious. Since Z is complete and Z → X × X, it follows that V is complete. It follows from the closed graph theorem that the embedding y → (0, y) from X into Z is continuous. So there exists β > 0 such that (0, y) Z ≤ β y X (y ∈ X). Thus,

(x, y) Z ≤ (x, 0) Z + (0, y) Z ≤ x V + β y X . Now it follows from the closed graph theorem that (x, y) V ×X := x V + y X deﬁnes a norm on V × X which is equivalent to · Z . Thus, (ii) is proved. (ii) ⇒ (i): Suppose that V is a Banach space such that D(A) → V → X and such that B generates a C0 -semigroup J on V × X. It follows from the Sandwich Theorem 3.10.4 that A generates a once integrated semigroup S. Moreover, S(t)z =

0

t

J (s)z ds for all z ∈ V × X, t ≥ 0.

In particular, S(·)z ∈ C(R+ , D(A) × V ) ∩ C 1 (R+ , V × X) for all z ∈ V × X. By Theorem 3.14.7, the operator A generates a cosine function Cos and S is given by (3.99). Since Sin is odd and Cos is even, this implies that Sin(·)y ∈ C(R, V ) for all y ∈ X, Cos(·)x ∈ C 1 (R, X) ∩ C(R, V ) and Sin(·)x ∈ C(R, D(A)) for all x ∈ V and that J is given by the ﬁrst matrix in (3.100) for t ≥ 0. In order to prove uniqueness of V , we show that V is equal to the space V := {x ∈ X : Sin(·)x ∈ C([0, 1], D(A))} . Note that V is a Banach space for the norm

x V = x X + sup A Sin(s)x X , 0≤s≤1

and we show that this norm is equivalent to · V . Since V is completely described by the operator A, this proves uniqueness.

212

3. CAUCHY PROBLEMS

As we noted above, one has V ⊂ V . It follows from the closed graph theorem that the injection is continuous. Next, we show that x V ≤ c x V for all x ∈ V and some constant c ≥ 0. First, since Sin(·)y ∈ C(R, V ) we note that, by the closed graph theorem again,

Sin(t)y V ≤ c1 y X

(3.101)

for all y ∈ X, 0 ≤ t ≤ 2 and some constant c1 ≥ 0. Let x ∈ V . Then by (3.101) and Proposition 3.14.5 f),

x V

2

1

1

= (x − Cos(t)x) dt + Sin(2)x

2 0 2 V

2

1 1

≤

Sin(2)x V + (x − Cos(t)x) dt

2 2 0

≤ c1 x X + sup x − Cos(t)x V

V

0≤t≤2

≤ c1 x X + sup Cos(t + s)x − Cos(t − s)x V 0≤t≤1 0≤s≤1

= c1 x X + 2 sup A Sin(t) Sin(s)x V 0≤t≤1 0≤s≤1

≤ c1 x X + 2c1 sup A Sin(s)x X 0≤s≤1

≤ 2c1 x V . We have shown that the norms · V and · V are equivalent on V . It remains to show that V ⊂ V . Let x ∈ V . Proposition 3.14.5 b) implies that Cos(1)x ∈ V . Since 2 Sin(t) Cos(1)x = Sin(1 + t)x − Sin(1 − t)x (by diﬀerentiation of (3.94)), it follows that Sin(·)x ∈ C([0, 2], D(A)). Since the function is odd, one has Sin(·)x ∈ C([−2, 2], D(A)). We have to show that x ∈ V . Since Sin(t)x ∈ V , it suﬃces to show that limt↓0 1t Sin(t)x − x V = 0. By (3.94), we have

A Sin(s) 1 Sin(t)x − x

t X

s+t

1

= (A Sin(r)x − A Sin(s)x) dr

→0 2t s−t X as t ↓ 0, uniformly for s ∈ [0, 1]. This completes the proof of uniqueness of V . Finally, we show that J extends to a C0 -group. We have seen that the semigroup J is given by (3.100), and we can extend J to R by the same formulae. For λ suﬃciently large, ∞ R(λ, A)z = e−λt J (t)z dt 0

3.14. THE SECOND ORDER CAUCHY PROBLEM

213

for z ∈ V × X. Since Sin is odd and Cos is even, it follows from (3.98) that R(λ, −A)z =

0

∞

e−λt J (−t)z dt.

This shows that J deﬁned by (3.100) is a C0 -group on V × X and its generator is the part B of A in V × X. The phase space can be computed in many concrete cases (see Section 7.2 and Examples 3.14.15 and 3.14.16). The phase space is also important in order to obtain classical solutions of the Cauchy problem of second order. Corollary 3.14.12. Let A be the generator of a cosine function Cos on X. Denote the associated phase space by V × X, and the corresponding sine function by Sin. Let x ∈ D(A), y ∈ V . Then u(t) := Cos(t)x + Sin(t)y deﬁnes a classical solution of P 2 (x, y). Proof. Since y ∈ V , one has Cos(·)y ∈ C 1 (R, X), and hence Sin(·)y ∈ C 2 (R, X). Since x ∈ D(A), it follows from Proposition 3.14.5 c) that Cos(·)x ∈ C 2 (R, X). Thus, u ∈ C 2 (R, X). Since u is a mild solution and A is closed, it follows that u is a classical solution. The following perturbation result improves Corollary 3.14.10. It will be most useful for applications to elliptic operators given in Chapter 7. Corollary 3.14.13. Let A be the generator of a cosine function with phase space V × X. Let B ∈ L(V, X). Then A + B generates a cosine function with the same phase space. 0 I Proof. The operator A = on V × X with domain D(A) × V generates A 0 0 0 a C0 -semigroup on V × X. Since B = ∈ L(V × X), it follows from B 0 0 I Corollary 3.5.6 that = A + B generates a C0 -semigroup on V × X. A+B 0 It follows from Theorem 3.14.11 that A + B generates a cosine function. The following corollary will have useful applications to hyperbolic equations in Chapter 7. Corollary 3.14.14. Let A be the generator of a cosine function on X with phase space V × X. Then the part AV of A in V generates a cosine function on V with phase space D(A) × V .

214

3. CAUCHY PROBLEMS

0 I Proof. The operator A = on V × X with domain D(A) = D(A) × V A 0 generates a C0 -semigroup on V × X. Hence the part G of A in D(A) also generates a C0 -semigroup on D(A) × V (see the remarks preceding Proposition 3.10.3). Note that D(G) = {(x, y) ∈ D(A) × V : (y, Ax) ∈ D(A) × V } = D(A) × D(AV ). Moreover, we have the continuous embeddings D(AV ) → D(A) → V . Now replacing X by V and V by D(A) in Theorem 3.14.11 we conclude that AV generates a cosine function on V with phase space D(A) × V . Next, we give two examples where the phase space can be determined easily. Example 3.14.15. Let B be the generator of a C0 -group U on X. Then A := B 2 generates a cosine function Cos on X given by Cos(t) :=

1 (U (t) + U (−t)) 2

(t ∈ R).

The phase space is given by D(B) × X, where D(B) carries the graph norm. Proof. Let M, ω ∈ R such that U (t) ≤ M eω|t| (t ∈ R). Let λ > ω. Then λ ∈ ρ(B) ∩ ρ(−B) and (λ2 − A) = (λ − B)(λ + B). Hence, λ2 ∈ ρ(A) and λR(λ2 , A)

−λR(λ, B)R(−λ, B) 1 = (R(λ, B) − R(−λ, B)) 2 1 ∞ −λt = e (U (t) + U (−t)) dt 2 ∞0 = e−λt Cos(t) dt.

=

0

This shows that Cos is a cosine function with generator A. For x ∈ D(B) one has Cos (t)x = 12 (BU (t)x − BU (−t)x). Thus, (3.100) deﬁnes a strongly continuous, exponentially bounded function J : R+ → L(D(B) × X) whose Laplace transform is the restriction of R(λ, A) to D(B) × X, by (3.97). Thus, J is a C0 -semigroup whose generator is the part of A in D(B) × X. So D(B) × X is the phase space, by Theorem 3.14.11. Example 3.14.16. Let A be a selfadjoint operator on a Hilbert space H. Assume that A is bounded above; i.e., (Ax|x)H ≤ ω x 2H for all x ∈ D(A) and some ω ∈ R. Then A generates a cosine function. Proof. By the Spectral Theorem B.13, we can assume that H = L2 (Ω, μ), Af = mf , D(A) = {f ∈ H : mf ∈ H}, where m : Ω → R is a measurable function. Moreover, the boundedness assumption implies that m(y) ≤ ω for μ-almost all y ∈ Ω.

3.14. THE SECOND ORDER CAUCHY PROBLEM

215

3 First case: Assume that m ≤ 0 a.e. Let q(y) := i −m(y), Bf := qf , D(B) := {f ∈ H : qf ∈ H}. Then B generates the C0 -group U given by U (t)f := etq f . Since B 2 = A, it follows from the preceding example that A generates a cosine function Cos. Moreover, (Cos(t)f )(y)

1 ((U (t) + U (−t)) f )(y) 2 √ 1 it√−m(y) = e + e−it −m(y) f (y) 2 3 = cos(t −m(y))f (y) (t ∈ R, f ∈ H, y ∈ Ω).

=

Second case: When m ≤ ω a.e., we can write m = m1 + m2 where m1 ≤ 0, 0 ≤ m2 ≤ ω. Then A = A1 + B where A1 f := m1 f, D(A1 ) = D(A) and Bf := m2 f deﬁnes a bounded operator. It follows from Corollary 3.14.10 and the ﬁrst case that A generates a cosine function. Next, we show that there is always a simple way to go from the second order equation to the ﬁrst order equation. Theorem 3.14.17. Let A be the generator of a cosine function Cos. Then A generates a holomorphic C0 -semigroup T of angle π/2. We give two diﬀerent proofs of this theorem. The ﬁrst uses the characterization theorem for holomorphic semigroups. However, it does not give the best possible angle. First proof. Let M, ω ≥ 0 such that Cos(t) ≤ M eωt (t ≥ 0). Then by holomorphic continuation (Proposition B.5) for Re λ > ω one has λ2 ∈ ρ(A) and

∞

M 2 −λt

λR(λ , A) = e Cos(t) dt

≤ Re λ − ω . 0 Let ω1 := 2ω 2 . Let√μ ∈ C such that Re μ > ω1 . Write μ = reiθ , where −π/2 < θ < π/2, and let λ := reiθ/2 . Then 4 4 4 √ θ √ cos θ + 1 r cos θ ω1 Re λ = r cos = r ≥ ≥ = ω. 2 2 2 2 Hence, μ = λ2 ∈ ρ(A) and

μR(μ, A) ≤ |λ|

√ M M r =√ Re λ − ω r cos(θ/2) − ω

= ≤

if

√1 2

−

ω √ r

≥

1 √ ; 2 2

i.e., if r ≥ 8ω 2 .

M √ cos(θ/2) − ω/ r √ M ≤ M2 2 1 ω √ − √ r 2

216

3. CAUCHY PROBLEMS

√ We have shown that μ ∈ ρ(A) and μR(μ, A) ≤ M 2 2 whenever Re μ > ω1 and |μ| ≥ 8ω 2 . By Corollary 3.7.17, this implies that A generates a holomorphic semigroup T . Since D(A) is dense, T is a C0 -semigroup. The second proof has the advantage of giving a formula which allows one to compute the semigroup T from the cosine function Cos. In fact, we will prove the Weierstrass formula

∞

T (t)x = 0

2

e−s /4t √ Cos(s)x ds πt

(t > 0).

(3.102)

In the context of Example 3.14.15, this formula was established in Corollary 3.7.15. Second proof of Theorem 3.14.17. Deﬁne T by (3.102). Then T (t) ∈ L(X) and T (·)x ∈ C ∞ ((0, ∞), X) √ for all x ∈ X. Let x ∈ X. We show that limt↓0 T (t)x = x. In fact, putting s = r t gives

∞

T (t)x = 0

e−r √

2

/4

π

√ Cos(r t)x dr → x

as t ↓ 0

√ √ by the dominated convergence theorem, since Cos(r t)x → x, Cos(r t) ≤ 2 ∞ M eωr for all 0 < t < 1, for some M and ω, and √1π 0 e−r /4 dr = 1. Let λ > ω. Then by Lemma 1.6.7,

∞

e−λt T (t)x dt =

0

∞

0 ∞ = =

0

∞

2

e−s /4t e−λt √ dt Cos(s)x ds πt

√ 1 √ e− λs Cos(s)x ds λ 0 R(λ, A)x.

It follows from Deﬁnition 3.1.8 that T is a C0 -semigroup and A is its generator. In order to show that T is holomorphic of angle π/2, we deﬁne ∞ 2 1 T (z)x = √ e−s /4z Cos(s)x ds (Re z > 0). πz 0 Then T : C+ → L(X) is holomorphic. Let θ ∈ (0, π2 ). According to Deﬁnition 3.7.1, it remains to show that sup T (z) < ∞.

z∈Σθ |z|≤1

Let z ∈ Σθ , |z| ≤ 1. Let u := Re z/|z|2 . Then

−s2 /4z

2 2 2

e Cos(s) ≤ M e−us /4 eωs = M e−u(s−2ω/u) /4 eω /u .

3.15. SINE FUNCTIONS AND REAL CHARACTERIZATION Hence,

217

1

−s2 /4z

T (z) = √ e Cos(s) ds

2 πz R 2 2 1 M √ ≤ 3 e−u(s−2ω/u) /4 ds eω /u 2 π |z| R 2 2 1 M 1 = 3 √ e−s ds √ eω /u u |z| π R = ≤

M |z| ω2 |z|2 / Re z 3 √ e |z| Re z 2 2 M M √ eω |z|/ cos θ ≤ √ eω / cos θ . cos θ cos θ

This proves the claim. The converse of Theorem 3.14.17 is not true: a generator of a holomorphic C0 -semigroup does not necessarily generate a cosine function. Indeed, generators of cosine functions satisfy a very restrictive spectral condition. Proposition 3.14.18. Let A be the generator of a cosine function Cos. Then there exists ω > 0 such that the spectrum σ(A) of A is contained in the parabola {ξ + iη : η ∈ R, ξ ≤ ω 2 − η 2 /4ω 2 }. Proof. There exist ω, M > 0 such that ∞

Cos(t) ≤ M eωt

(t ≥ 0).

Since λR(λ2 , A) = 0 e−λt Cos(t) dt for λ > ω, it follows from holomorphic continuation (Proposition B.5) that λ2 ∈ ρ(A) whenever Re λ > ω. It is easy to see that ' 2 ( η2 2 λ : λ ∈ C, Re λ > ω ⊃ ξ + iη : η ∈ R, ξ > ω − . 4ω 2 Example 3.14.19. Let H := L2 (R), m(s) := −|s| + is (s ∈ R), (Af )(s) := m(s)f (s),

D(A) := {f ∈ H : mf ∈ H}.

Then A generates a holomorphic C0 -semigroup on H. Since σ(A) = {−s ± is : s ∈ R+ }, the spectrum of A is not contained in any parabola as described in Proposition 3.14.18. Thus, A does not generate a cosine function.

3.15 Sine Functions and Real Characterization In this section, we consider sine functions. These include the integrals of cosine functions as in Section 3.14, and elementary properties of sine functions will be

218

3. CAUCHY PROBLEMS

used to prove the generation theorem for cosine functions which is analogous to the Hille-Yosida theorem. We shall also establish some useful perturbation results. Examples of sine functions occurring in applications will be given in the Notes of Chapter 7 and in Chapter 8. Deﬁnition 3.15.1. An operator A on X generates a sine function if there exist ω, M ≥ 0 and a strongly continuous function Sin : R+ → L(X) such that the following properties are satisﬁed:

t

a) 0 Sin(s) ds ≤ M eωt (t ≥ 0). b) λ2 ∈ ρ(A) whenever λ > ω. ∞ c) R(λ2 , A) = 0 e−λt Sin(t) dt (λ > ω). We then call Sin the sine function generated by A. If A generates a cosine function Cos, then ∞ t 1 ∞ −λt R(λ2 , A) = e Cos(t) dt = e−λt Cos(s) ds dt λ 0 0 0 for λ > ω(Cos). Thus A generates the sine function Sin given by Sin(t)x := t Cos(s)x ds. In other words, the deﬁnition is consistent with the previous notion 0 of Section 3.14. Next, we establish some relations between a sine function and its generator. Proposition 3.15.2. Let Sin be a sine function and A be its generator. Then the following hold: t a) 0 (t − s) Sin(s)x ds ∈ D(A) and A 0

t

(t − s) Sin(s)x ds = Sin(t)x − tx

(3.103)

for all x ∈ X. b) If x ∈ D(A), then Sin(t)x ∈ D(A) and A Sin(t)x = Sin(t)Ax for all t ≥ 0. c) Let x, y ∈ X. Then x ∈ D(A) and Ax = y if and only if

t 0

(t − s) Sin(s)y ds = Sin(t)x − tx

(t ≥ 0).

(3.104)

Proof. a) It follows from (3.87) that S(·)x is a mild solution of P 2 (0, x). This is precisely the claim. b) This follows from Proposition 3.1.5 and Proposition B.7.

3.15. SINE FUNCTIONS AND REAL CHARACTERIZATION

219

c) Let x, y ∈ X such that (3.104) holds. Taking Laplace transforms on both sides gives 1 x R(λ2 , A)y = R(λ2 , A)x − 2 . λ2 λ Hence, x ∈ D(A) and y = λ2 x − (λ2 − A)x = Ax. The converse implication follows from a) and b). We now prove the characterization theorem for generators of cosine functions. It is mainly of theoretical interest since the condition (3.105) seems to be diﬃcult to verify in concrete cases. Theorem 3.15.3. Let A be a densely deﬁned operator on a Banach space X. The following assertions are equivalent: (i) A is the generator of a cosine function. (ii) There exist ω, M ≥ 0 such that (ω 2 , ∞) ⊂ ρ(A) and ! "(k) 1

(λ − ω)k+1 λR(λ2 , A)

≤M k!

(3.105)

for all λ > ω and k ∈ N0 . Proof. (i) ⇒ (ii): Assume that A generates a cosine function Cos. There exist M ≥ ∞ 0, ω ≥ 0 such that Cos(t) ≤ M eωt . Since λR(λ2 , A)x = 0 e−λt Cos(t)x dt for all λ > ω and x ∈ X, the claim follows from Theorem 2.4.1. (ii) ⇒ (i): Assume that (ii) is satisﬁed. By Theorem 2.4.1, there exists a function S : R+ → L(X) satisfying

S(t + h) − S(t) ≤ M

t+h

(t, h ≥ 0)

eωs ds

(3.106)

t

such that R(λ2 , A) =

∞

e−λt S(t) dt

(λ > ω).

0

Thus, S is a sine function and A is its generator. Let x ∈ D(A). Then by Proposition 3.15.2, one has S(t)x − tx =

0

t

(t − s)S(s)Ax ds

(t ≥ 0).

It follows that S(·)x ∈ C 1 (R+ , X). Since D(A) is dense in X, it follows from d Lemma 3.3.3 that S(·)x ∈ C 1 (R+ , X) for all x ∈ X. Let C(t)x := dt S(t)x (x ∈ X, t ≥ 0). It follows that C(t) is linear and by (3.106) that

C(t) ≤ M eωt

(t ≥ 0).

220

3. CAUCHY PROBLEMS

Integration by parts shows that λR(λ2 , A)x =

∞

e−λt C(t)x dt

(λ > ω)

0

for all x ∈ X. Thus, C is a cosine function and A is its generator. Now we resume our investigation of sine functions. The following result parallels Example 3.14.15. Proposition 3.15.4. Let A be an operator such that A and −A generate once integrated semigroups. Then A2 generates a sine function. Moreover, the sine function is exponentially bounded if both integrated semigroups are. Proof. There exists ω ≥ 0 such that (ω, ∞) ⊂ ρ(A) ∩ ρ(−A) and λ1 (λ − A)−1 and 1 (λ + A)−1 are Laplace transforms (in the sense of Deﬁnition 3.1.4). Thus, λ (λ2 − A2 )−1 = (λ − A)−1 (λ + A)−1 exists for λ > ω, and by the resolvent equation we have (λ2 − A2 )−1 =

" 1 ! (λ + A)−1 + (λ − A)−1 . 2λ

Thus, (λ2 − A2 )−1 is a Laplace transform. This shows that A2 generates a sine function Sin given by Sin(t) = 12 (S+ (t) + S− (t)), where S+ and S− are the once integrated semigroups generated by A and −A, respectively. The next example shows that the generator of a sine function is not necessarily densely deﬁned. Example 3.15.5. Let B be the generator of the translation group U on L1 (R), given by (U (t)f )(x) = f (x − t). Then B ∗ and −B ∗ generate once integrated semigroups on L∞ (R) = L1 (R)∗ (see Corollary 3.3.7 and Example 3.3.10). By Proposition ! " 3.15.4, (B ∗ )2 generates a sine function. However, D (B ∗ )2 ⊂ D(B ∗ ) = W 1,∞ (R), which is not dense in L∞ (R) (cf. also Theorem 4.3.18). Next, we establish a perturbation result for sine functions. Our proof is completely diﬀerent from the corresponding results on cosine functions (Corollary 3.14.10 and Corollary 3.14.13). Theorem 3.15.6. Let A be the generator of an exponentially bounded sine function and let B ∈ L(D(A), X). Then A + B generates an exponentially bounded sine function. Proof. Denote by Sin the sine function generated by A and let M, ω ≥ 0 such that

Sin(t) ≤ M eωt (t ≥ 0). The idea of the proof is to solve the integral equation t S B (t) = Sin(t) + S B (s)B Sin(t − s) ds. (3.107) 0

3.15. SINE FUNCTIONS AND REAL CHARACTERIZATION

221

First, we remark that Sin(t)x ∈ D(A) for all t ≥ 0, x ∈ X. In fact by Proposition t 3.15.2, 0 (t − s) Sin(t)x ds ∈ D(A) for all t > 0. Diﬀerentiating twice we obtain that Sin(t)x ∈ D(A). Let α > 0. Multiplying (3.107) by e−(ω+α)t , we obtain the equivalent integral equation t U (t) = e−(ω+α)t Sin(t) + U (s)Be−(ω+α)(t−s) Sin(t − s) ds. (3.108) 0

Consider now the Banach space C := {V : R+ → L(X) : V is strongly continuous and bounded} with norm V := supt≥0 V (t) . Suppose from now on that α > M B . Then t (JV )(t) := V (s)Be−(ω+α)(t−s) Sin(t − s) ds 0

deﬁnes an operator J on C with norm J L(C) ≤ (3.108) can now be written in the form

M B α

< 1. The integral equation

(I − J)U = W with W (t) := e−(ω+α)t Sin(t). Hence, it has a unique solution U (given by (I − J)−1 W ). Therefore S B (t) := e(ω+α)t U (t) deﬁnes a solution of (3.107), which is exponentially bounded. ∞ For λ > ω + α, let Q(λ) := 0 e−λt S B (t) dt. Then ∞ Q(λ) − R(λ2 , A) = e−λt (S B (t) − Sin(t)) dt 0 ∞ t −λt = e S B (s)B Sin(t − s) ds dt 0 0 ∞ t = e−λs S B (s)Be−λ(t−s) Sin(t − s) ds dt 0 0 ∞ ∞ = e−λs S B (s)Be−λ(t−s) Sin(t − s) dt ds 0 ∞ s ∞ = e−λs S B (s)B e−λt Sin(t) dt ds 0

0

= Q(λ)BR(λ2 , A). Hence, Q(λ)(I − BR(λ2 , A)) = R(λ2 , A). The operator I − BR(λ2 , A) is invertible since

∞

2 −λt

BR(λ , A) ≤ B e Sin(t) dt

0

≤

B M (λ − ω)−1 < 1.

222

3. CAUCHY PROBLEMS

Since (λ2 − A − B) = (I − BR(λ2 , A))(λ2 − A), it follows that λ2 ∈ ρ(A + B) and R(λ2 , A + B)

= R(λ2 , A)(I − BR(λ2 , A))−1 = Q(λ) ∞ = e−λt S B (t) dt 0

for all λ > ω + α. This shows that S B is a sine function and A + B is its generator.

3.16 Square Root Reduction for Cosine Functions Let B be the generator of a C0 -group and let ω ∈ R. Then by Corollary 3.14.10 and Example 3.14.15, the operator A := B 2 + ω generates a cosine function. The aim of this section is to establish the following remarkable converse result: If A is the generator of a cosine function on a UMD-space X, then there exist a generator B of a C0 -group and a number ω ≥ 0 such that A = B 2 + ω (see Corollary 3.16.8). By the results of Section 3.8, there is a square root B of A − ω for ω suﬃciently large. We will show that B and −B always generate integrated semigroups, but the UMD-property is needed in order to obtain a C0 -group. We recall the following lemma which is easy to prove. Lemma 3.16.1. Let B be a closed operator on X, and A := B 2 . Let λ ∈ C. If λ2 ∈ ρ(A), then λ ∈ ρ(±B) and R(λ, B)

=

(λ + B)R(λ2 , A),

R(λ, −B)

=

(λ − B)R(λ2 , A).

Proposition 3.16.2. Let B be a closed operator. Assume that A := B 2 generates a cosine function. Then B and −B generate exponentially bounded once integrated semigroups. Proof. Denote by Cos and Sin the cosine and the sine functions associated with A. t Recall from Proposition 3.14.5 a) that 0 Sin(s)x ds ∈ D(A) and A 0

t

Sin(s)x ds = Cos(t)x − x

for all x ∈ X, t ≥ 0. Deﬁne V+ (t)x := Sin(t)x + B

t

Sin(s)x ds 0

(t ≥ 0).

(3.109)

3.16. SQUARE ROOT REDUCTION FOR COSINE FUNCTIONS Let λ ∈ ρ(A). Then t B Sin(s)x ds 0

223

t

= BR(λ, A)(λ − A) Sin(s)x ds 0 t = λBR(λ, A) Sin(s)x ds − BR(λ, A)(Cos(t)x − x) 0

for all x ∈ X. Since BR(λ, A) is bounded, it follows that V+ is strongly continuous and exponentially bounded. Moreover, for large λ > 0, ∞ 1 e−λt V+ (t)x dt = R(λ2 , A)x + BR(λ2 , A)x λ 0 1 1 = (λ + B)R(λ2 , A)x = R(λ, B)x, (3.110) λ λ by the preceding lemma. We have shown that V+ is a once integrated semigroup and B is its generator. Replacing B by −B shows that −B also generates a once integrated semigroup. Now we assume that B is a closed operator such that B 2 generates a cosine function. We want to investigate conditions under which this implies that B generates a C0 -group. Here is a characterization in terms of the behaviour of the sine function on D(B), which is considered as a Banach space for the graph norm. Proposition 3.16.3. Let A be the generator of a cosine function Cos with associated sine function Sin. Assume that B is a closed operator such that B 2 = A. Then the following are equivalent: (i) B generates a C0 -group U . (ii) For all x ∈ X, Sin(t)x ∈ D(B) for almost all t ∈ (0, ∞). (iii) Sin(·)x ∈ C(R, D(B)) for all x ∈ X. (iv) The phase space of Cos is D(B) × X. In that case, the group U is given by U (t)x = Cos(t)x + B Sin(t)x

(3.111)

for all t ∈ R, x ∈ X. For the proof we need the following results which are of independent interest. Lemma 3.16.4. Let U : R → L(X) be a mapping such that a) U (t + s) = U (t)U (s) (t, s ∈ R), b) U (0) = I, and

224

3. CAUCHY PROBLEMS

c) U (·)x is measurable for all x ∈ X. Then U is strongly continuous. Proof. a) We show that M := sups∈[1,2] U (s) < ∞. Otherwise, by the uniform boundedness principle, there exist x ∈ X, sn ∈ [1, 2] such that U (sn )x ≥ n (n ∈ N). Considering a subsequence if necessary, we can and do assume that limn→∞ sn =: γ exists. Since U (·)x is measurable, there exists a constant M1 ≥ 0 and a measurable set F ⊂ [0, γ] with Lebesgue measure m(F ) > γ/2 such that supt∈F U (t)x ≤ M1 . Let ' ( En := (sn − t) : t ∈ F ∩ [0, sn ] . Then m(En ) ≥ γ/2 for large n ∈ N. Now for t ∈ F ∩ [0, sn ] we have n ≤ U (sn )x

≤ U (sn − t) U (t)x

≤ U (sn − t) M1 . Hence, U (s) ≥ n/M1 for all s ∈ En . Let E = k∈N n≥k En . Then m(E) ≥ γ/2 and U (s) = ∞ for all s ∈ E. This is a contradiction. b) By the group property and a), one has for each t ∈ R,

U (s) =

sup s∈[t+1,t+2]

U (s − t)U (t) ≤ M U (t)

sup

(t ∈ R).

s∈[t+1,t+2]

Thus U is locally bounded. It follows that for each x ∈ X, U (·)x is locally Bochner integrable. c) Let x ∈ X, t0 ∈ R. Then U (t0 )x = U (t)U (t0 − t)x and U (t0 + h)x = U (t)U (t0 + h − t)x

(t ∈ [1, 2], h ∈ R).

Hence,

U (t0 )x − U (t0 + h)x

=

2

1

≤

M

→

0

! " U (t) U (t0 − t)x − U (t0 + h − t)x dt

1

2

U (t0 − t)x − U (t0 + h − t)x dt as h → 0,

using b) and the continuity of shifts on L1 (R, X). Lemma 3.16.5. Let F ⊂ R be a Lebesgue measurable set such that F + F ⊂ F . If R \ F is a null set, then F = R.

3.16. SQUARE ROOT REDUCTION FOR COSINE FUNCTIONS

225

Proof. Assume that N := R \ F = ∅. Let a ∈ N . Then for x ∈ F , the assumption implies that a − x ∈ N (otherwise, a = x + (a − x) ∈ F + F ⊂ F ). We have shown that F ⊂ a − N , which is impossible since a − N is a null set. Proof of Proposition 3.16.3. Assume that (i) holds. One has 1 (U (t)x + U (−t)x) 2

Cos(t)x = and Sin(t)x =

1 2

t

(U (s) + U (−s))x ds. 0

It follows from Proposition 3.1.9 e) that Sin(t)x ∈ D(B) and B Sin(t)x

= =

" 1! U (t)x − x − (U (−t)x − x) 2 1 (U (t)x − U (−t)x). 2

This proves (3.111). We have seen in Example 3.14.15 that (i) implies (iv). Theorem 3.14.11 shows that (iv) implies (iii). The implication (iii) ⇒ (ii) is trivial. It remains to show that (ii) ⇒ (i). It is obvious that for μ ∈ ρ(B) one has R(μ, B)R(λ2 , A) = R(λ2 , A)R(μ, B)

(λ > ω(Cos)).

Thus, by Proposition 3.1.5 and Proposition B.7, x ∈ D(B) implies that Sin(t)x, Cos(t)x ∈ D(B) and B Sin(t)x = Sin(t)Bx,

B Cos(t)x = Cos(t)Bx.

Recall from (3.95) that Sin(t + s) = Cos(s) Sin(t) + Cos(t) Sin(s)

(t, s ∈ R).

(3.112)

Let x ∈ X and F := {t ∈ R : Sin(t)x ∈ D(B)}. By hypothesis, R \ F has Lebesgue measure zero. It follows from (3.112) that F + F ⊂ F . Now we conclude from Lemma 3.16.5 that F = R. Deﬁne U (t)x := Cos(t)x + B Sin(t)x for all x ∈ X, t ∈ R. Since B is closed, U (t) is also closed. Hence, U (t) ∈ L(X). If x ∈ D(B), then U (·)x is continuous. Since D(B) ⊃ D(A) which is dense, it follows that U (·)x is measurable for all x ∈ X (in fact, let x ∈ X, xn ∈ D(B) such that limn→∞ xn = x; then U (t)x = limn→∞ U (t)xn ). It remains to show the group property. Recall from Proposition 3.14.5 f) that Sin(t) Sin(s)x ∈ D(A) and A Sin(t) Sin(s)x =

" 1! Cos(t + s)x − Cos(t − s)x 2

(3.113)

226

3. CAUCHY PROBLEMS

for all t, s ∈ R, x ∈ X. Hence by (3.112), U (t)U (s) = Cos(t) Cos(s) + Cos(t)B Sin(s) + B Sin(t) Cos(s) + B Sin(t)B Sin(s) = Cos(t) Cos(s) + B Sin(t + s) + A Sin(t) Sin(s) 1 1 = (Cos(t + s) + Cos(t − s)) + B Sin(t + s) + (Cos(t + s) − Cos(t − s)) 2 2 = Cos(t + s) + B Sin(t + s) = U (t + s). It follows from Lemma 3.16.4 that U is a C0 -group. Since V+ (t) = t ≥ 0, where V+ is given by (3.109), we have by (3.110), ∞ ∞ e−λt U (t)x dt = λ e−λt V+ (t)x dt = R(λ, B)x 0

t 0

U (s) ds for

0

for large λ. This shows that B is the generator of U . We will show that condition (ii) of Proposition 3.16.3 is automatically satisﬁed if X is a UMD-space (as deﬁned in Section 3.12), −A satisﬁes the hypotheses of Proposition 3.8.2, and B = i(−A)1/2 (as deﬁned in Section 3.8). For this we need some preparation concerning the Hilbert transform. For ω ≥ 0 we deﬁne ∞ 1 −ω|t| Lω (R, X) := f ∈ Lloc (R, X) : f ω,∞ := ess sup e f (t) < ∞ . t∈R

Then L∞ ω (R, X) is a Banach space for the norm · ω,∞ . Lemma 3.16.6. Let X be a UMD-space and let 0 ≤ ω < c. For f ∈ L∞ ω (R, X) and ε > 0 deﬁne the continuous function Hεc f : R → X by (Hεc f )(t) =

|s|≥ε

e−c|s| f (t − s) ds. s

Then for each τ > 0, limε↓0 Hεc f =: H c f exists in L2 ((−τ, τ ), X). Hence, H c is a 2 bounded operator from L∞ ω (R, X) into L ((−τ, τ ), X) for each τ > 0. Proof. Let τ ≥ 1, 0 < ε ≤ 1, |t| ≤ τ . Denote by χ the characteristic function of [−τ − 1, τ + 1]. Then (Hεc f )(t)

= |s−t|≥ε

e−c|t−s| f (s) ds t−s

= h1ε (t) + h2ε (t) + h3ε (t),

3.16. SQUARE ROOT REDUCTION FOR COSINE FUNCTIONS where

h1ε (t)

h2ε (t)

|s−t|≥ε

e−c|t−s| − 1 χ(s)f (s) ds, t−s

|s|≥τ +1

e−c|t−s| f (s) ds. t−s

:=

h3 (t)

|s−t|≥ε

1 χ(s)f (s) ds, t−s

:=

:=

227

It is clear that h3 (t) ≤ c1 f ω,∞ for |t| ≤ τ , where c1 ≥ 0 is a constant. Next, observe that χf ∈ L2 (R, X). If ψ(t) := 1t (e−c|t| − 1), then ψ ∈ L2 (R) and h2ε is the convolution ((1 − χ(−ε,ε) )ψ) ∗ (χf ). It follows from Proposition 1.3.2 that h2ε converges uniformly on [−τ, τ ] to ψ ∗ (χf ) as ε ↓ 0. Finally, let H be the Hilbert transform on L2 (R, X) and HεR be as in Section 3.12. Then h1ε = πHεR (χf )|[−τ,τ ] for R > 2τ + 1. Hence, h1ε → πH(χf)|[−τ,τ ] in L2 ((−τ, τ ), X) as ε ↓ 0. Thus the ﬁrst assertion is proved. The second now follows from the Banach-Steinhaus theorem. Now we are able to prove the main result. Theorem 3.16.7 (Fattorini). Let A be the generator of a cosine function on a UMDspace X. Assume that (0, ∞) ⊂ ρ(A) and supλ>0 λR(λ, A) < ∞. Deﬁne (−A)1/2 as in Proposition 3.8.2 and let B = i(−A)1/2 . Then B generates a C0 -group and B 2 = A. Proof. It is immediate from Proposition 3.8.2 that B 2 = A. We want to show condition (ii) of Proposition 3.16.3. Let c > ω > ω(Cos). For t > 0 and y ∈ D(A), Proposition 3.8.2 gives ∞ 1 B Sin(t)y = λ−1/2 R(λ, A) Sin(t)Ay dλ πi 0 = I1 (t)y + I2 (t)y, where I1 (t)y

:= =

1 πi 1 πi

c2

λ−1/2 R(λ, A) Sin(t)Ay dλ

0

0

c2

λ−1/2 (λR(λ, A) Sin(t)y − Sin(t)y) dλ

∞ 1 I2 (t)y := λ−1/2 R(λ, A) Sin(t)Ay dλ. πi c2 Since ω > ω(Cos), there exists M ≥ 0 such that Cos(s) ≤ M eωs (s ≥ 0). Consequently, Sin(s) ≤ M seωs (s ≥ 0) and

∞ √

∞ √

1 − λs

e Sin(s) Sin(t)Ay ds ≤ c1 e− λs seωs ds = c1 √

( λ − ω)2 ε 0 and

228

3. CAUCHY PROBLEMS

√ for all ε > 0, where c1 := M Sin(t)Ay . Since the function λ → λ−1/2 ( λ − ω)−2 is in L1 (c2 , ∞), and since ∞ √ R(λ, A) Sin(t)Ay = e− λs Sin(s) Sin(t)Ay ds, 0

we obtain from the dominated convergence theorem, ∞ ∞ √ 1 I2 (t)y = lim λ−1/2 e− λs Sin(s) Sin(t)Ay ds dλ. ε↓0 πi c2 ε Now we apply Fubini’s theorem and obtain ∞ −cs 2 e I2 (t)y = lim Sin(s) Sin(t)Ay ds. ε↓0 πi ε s By (3.113), this gives I2 (t)y

∞ −cs 1 e = lim (Cos(t + s) − Cos(t − s))y ds ε↓0 πi ε s i e−c|s| = lim Cos(t − s)y ds. ε↓0 π |s|≥ε s

At this point we use that X is a UMD-space, so that we can apply Lemma 3.16.6. We have shown that I2 (·)y = Hc (Cos(·)y) for y ∈ D(A). Let x ∈ X, and let yn ∈ D(A) such that limn→∞ yn = x. Then Cos(·)yn converges to Cos(·)x in 2 L∞ ω (R, X). It follows from Lemma 3.16.6 that I2 (·)yn converges in L ((−τ, τ ), X) as n → ∞ for all τ > 0. Considering a subsequence if necessary, we obtain that I2 (t)yn converges a.e. in X as n → ∞. Since I1 (t) is a bounded operator it follows that there exists a measurable subset F of R such that R\F has Lebesgue measure 0 and B Sin(t)yn converges in X as n → ∞ for all t ∈ F . Since B is closed, this implies that Sin(t)x ∈ D(B) for all t ∈ F . This ﬁnishes the proof. Now we obtain the following interesting characterization of generators of cosine functions on UMD-spaces. Corollary 3.16.8. Let A be an operator on a UMD-space. The following assertions are equivalent: (i) A generates a cosine function. (ii) There exist a generator B of a C0 -group and ω ≥ 0 such that A = B 2 + ω. Proof. (ii) ⇒ (i): This follows from Example 3.14.15 and Corollary 3.14.10. (i) ⇒ (ii): Assume that A generates a cosine function Cos. Then A generates a C0 -semigroup by Theorem 3.14.17. Hence, there exists ω ≥ 0 such that (ω, ∞) ⊂ ρ(A) and supλ>0 λR(λ, A − ω) < ∞. The operator A − ω is also the generator of

3.16. SQUARE ROOT REDUCTION FOR COSINE FUNCTIONS

229

a cosine function by Corollary 3.14.10. It follows from Theorem 3.16.7 that there exists a C0 -group with generator B such that A − ω = B 2 . We conclude this section with some remarks. First, we mention that, given an operator A, there may be inﬁnitely many generators of groups whose square is A. We give an example. Example 3.16.9. Let X = l2 , and deﬁne A by ' ( D(A) := x = (xn )n∈N ∈ l2 : (n2 xn )n∈N ∈ l2 , (Ax)n

:=

−n2 xn .

Let (n )n∈N be an arbitrary sequence in {−1, 1}. Deﬁne B on X by Bx := (in nxn )n∈N with domain D(B) := {x : (nxn )n∈N ∈ l2 }. Then B generates the C0 -group U given by U (t)x := (ein nt xn )n∈N and B 2 = A. It should be mentioned that even if the generator A of a cosine function has a square root which generates a C0 -group there may be other square roots which do not generate C0 -groups. We give an example. Example 3.16.10. Let A be the generator of a cosine function on a Banach space X. Assume that A is unbounded. Then we know from Theorem 3.14.7 that the operator A on X × X given by D(A)

=

A =

D(A) × X, 0 I , A 0

does not generate a C0 -semigroup. However, the operator A2 is given by D(A2 ) A2

D(A) × D(A); A 0 = . 0 A

=

Thus, A2 generates a cosine function. If A has a square root which generates a C0 -group, then so does A2 , but this square root is diﬀerent from A. There exists an example of a generator A of a cosine function on a Banach space X (which can even be chosen to be reﬂexive) such that A − ω = B 2 for each ω ≥ 0 and each generator B of a C0 -group. Of course in that case, X is not a UMD-space. We refer to [Fat85, Section III.8] for such examples.

230

3. CAUCHY PROBLEMS

It is interesting to observe that the question whether a square root reduction exists for a cosine function is equivalent to the existence of a boundary group for a certain holomorphic semigroup. To be more precise, let A be the generator of a cosine function. Then by Theorem 3.14.17, A generates a holomorphic C0 -semigroup T of angle π/2. Let ω > ω(T ). Then −(ω − A)1/2 generates a C0 -semigroup V (see Theorem 3.8.3). Now it follows from Corollary 3.9.10 that B := i(ω − A)1/2 generates a C0 -group if and only if V is a holomorphic semigroup of angle π/2 and V has a boundary group (in the sense of Proposition 3.9.1). This is not always the case (if X fails to have the UMD property) as we pointed out above. However, we know from Proposition 3.16.2 that V always has an integrated boundary group (in the sense explained in Theorem 3.9.13).

3.17 Notes Section 3.1 The characterization of mild solutions in terms of Laplace transforms given in Theorem 3.1.3 appeared in [Neu94]. It is at the heart of the theory and it may have been known for a long time. Another way to deﬁne mild solutions by approximation schemes (“bonnes solutions”) has been given by B´enilan, Crandall and Pazy [BCP88]; see also [BCP90]. This way is most important in non-linear theory where Laplace transforms have not yet been used eﬀectively. The idea of deﬁning semigroups and their generators directly by the property that the Laplace transform is a resolvent, is from [Are87b], but Theorem 3.1.7 is already contained in [DS59, Corollary VIII.1.16]. We should also mention the approach to semigroups by Laplace transforms given by Kisy´ nski [Kis76]. The characterization of generators in terms of well-posedness of the abstract Cauchy problem for classical solutions is contained in [Kre71]. Related results are contained in the lecture notes of van Casteren [Cas85]. Example 3.1.14 is taken from [Nag86, A-II, Example 1.4]. One may modify Theorem 3.1.12 by requiring a less restrictive existence assumption but a stronger uniqueness assumption by considering the Cauchy problem on a bounded interval [0, τ ]: u (t) = Au(t) (t ∈ [0, τ ]), C(τ ) u(0) = x. t As before, we call u ∈ C([0, τ ], X) a mild solution if 0 u(s) ds ∈ D(A) and u(t) − x = t A 0 u(s) ds for all t ∈ [0, τ ]. Then the following holds. Theorem 3.17.1. Let A be a closed operator, τ > 0. The following assertions are equivalent: (i) For all x ∈ X there exists a unique mild solution of C(τ ). (ii) The operator A generates a C0 -semigroup. Other results on the local problem were obtained by Lyubich [Lyu66], Sova [Sov68], Oharu [Oha71] and Sanekata [San75].

3.17. NOTES

231

Section 3.2 Integrated semigroups were introduced by Arendt in [Are84], [Are87a]. The systematic treatment based on techniques of Laplace transforms was developed by Arendt [Are87b], Neubrander [Neu88] and Kellermann [Kel86]. Theorem 3.2.13 is due to Arendt, Neubrander and Schlotterbeck [ANS92] and Lemma 3.2.14 to Arendt, El-Mennaoui and Keyantuo [AEK94]. The theory of integrated semigroups has been developed in many directions and we refer to the monographs of deLaubenfels [deL94] and Xiao and Liang [XL98] for further information. We do not aim to give an account of the many contributions, but we mention here a few extensions of the theory. First, one may consider α-times integrated semigroups for some α ∈ R+ ; i.e., the power k in Deﬁnition 3.2.1 is replaced by some α ≥ 0 (see [Hie91a]). This leads to sharper regularity results for the solution of the associated Cauchy problem than those which we have described. Moreover, a modiﬁcation of the Real Representation Theorem 2.2.1 shows that if a C ∞ -function r : (0, ∞) → X satisﬁes

sup λα+1 r(n) (λ)/n! : λ > 0, n ∈ N ∪ {0} < ∞ for some α > 0, then there exists a H¨ older continuous function F of exponent α satisfying F (0) = 0 such that ∞

r(λ) = λα

e−λt F (t) dt

(λ > 0)

0

(see [Hie91b]). Integrated Volterra equations were considered by Arendt and Kellermann [AK89] and deLaubenfels [deL90]. Second order problems in integrated form have been developed by Kellermann and Hieber [KH89], Arendt and Kellermann [AK89] and Neubrander [Neu89a]. For integrated solutions of implicit diﬀerential equations, see the papers of Arendt and Favini [AF93] and Knuckles and Neubrander [KN94]. Concerning the general theory of degenerate diﬀerential equations in Banach spaces we refer to the monograph by Favini and Yagi [FY99]. Many years before integrated semigroups were studied, Lions [Lio60] introduced distribution semigroups. They have been further developed by Chazarain [Cha71] and Beals [Bea72] who also give applications to partial diﬀerential equations. Further information can be found for example in the monograph of Fattorini [Fat83]. In terms of integrated semigroups, Lions’s concept may now be described as follows: Let S be a k-times integrated semigroup on X. Denote by D(R+ ) the test functions on R with support in R+ . For ϕ ∈ D(R+ ) deﬁne the operator T ∈ L(X) by ∞ T (ϕ) := (−1)k ϕ(k) (t)S(t) dt. 0

Then T : D(R+ ) → L(X) is a mapping satisfying the semigroup property T (ϕ ∗ ψ) = T (ϕ)T (ψ). It turns out that T deﬁned as above is a distribution semigroup of exponential type and that all distribution semigroups of exponential type as deﬁned by Lions in [Lio60] are of this form (see [AK89] for a proof). More generally, arbitrary distribution semigroups are deﬁned as mappings U : D(R+ ) → L(X) such that U (ϕ ∗ ψ) = U (ϕ)U (ψ)

232

3. CAUCHY PROBLEMS

and several other properties are satisﬁed. To each distribution semigroup one can associate a generator A. These distribution semigroups are equivalent to local integrated semigroups. More precisely, a closed operator A generates a distribution semigroup U if and only if there exists k ∈ N such that the integrated Cauchy problem (ACP )k+1 (see (3.20)) is well-posed on a bounded interval; i.e., A generates a local integrated semigroup of order k. The generators of local integrated semigroups can be completely characterized by spectral properties. See papers of Arendt, El-Mennaoui and Keyantuo [AEK94] or Okazawa and Tanaka [TanO90] for these and related results. These concepts have been extended further in a series of papers by Lumer [Lum90], [Lum92], [Lum94], [Lum97], Cioranescu and Lumer [CL94] and Lumer and Neubrander [LN99], [LN97], and in the monograph of Melnikova and Filinkov [MF01]. A further concept, namely regularized semigroups, had been developed by Da Prato [DaP66], and was rediscovered by Davies and Pang [DP87]. Regularized semigroups were extensively studied by deLaubenfels [deL94] and Miyadera and Tanaka [TM92], among many others and are now called C-semigroups, in general. Given a bounded operator C one says, in the language of Laplace transforms, that an operator A generates a Csemigroup if CR(λ, A) is a Laplace transform (see [HHN92]). Thus, by Proposition 3.2.7, the generator of a k-times integrated semigroup is the same as a R(μ, A)k -semigroup, where μ ∈ ρ(A). Note, however, that in contrast to the situation of integrated semigroups, the generator of a C-semigroup may have an empty resolvent set. Operators with empty resolvent set occur in particular in the context of Petrovskii correct systems of partial diﬀerential equations. Systems of this type can be treated by the theory of Csemigroups in a very eﬃcient manner (see [HHN92]). Again, we refer to the monographs of deLaubenfels [deL94] and Xiao and Liang [XL98] for further information. Another concept, which leads to a generalization of generators of bounded semigroups and groups is the following. A k-times integrated semigroup S is called tempered if S(t) ≤ ctk (t ≥ 0) for some c ≥ 0. Arendt and Kellermann [AK89] showed that a densely deﬁned operator A generates a k-times integrated tempered semigroup if and only if A generates a smooth distribution semigroup of order k as introduced by Balabane and Emami-Rad [BE79], [BE85]. Smooth distribution groups allow a spectral calculus similar to the one known for bounded groups, as shown by Balabane, Emami-Rad and Jazar [BEJ93] and Jazar [Jaz95]. It was established in [KH89] that perturbation theory for integrated semigroups is more complicated than for semigroups. We saw in Proposition 3.2.6 that perturbation by a scalar yields a quite complicated formula for the integrated semigroup. In general, generators of integrated semigroups are not even stable under bounded perturbations (in contrast to generators of C0 -semigoups, by Corollary 3.5.6). For instance, let A be the unbounded generator of a cosine function (see Section 3.14), A be the

operator given x −y by (3.96) and B be the bounded operator on X × X given by B = . Then y 0 A generates a once integrated semigroup, by Theorem 3.14.7. However, the range of μ − (A + B) is contained in D(A) × X for any μ ∈ C, so ρ(A + B) is empty and A + B does not generate a k-times integrated semigroup for any k ∈ N. One perturbation theorem for integrated semigroups can be found in Corollary 3.10.5, and we mention another one here. Using Theorem 3.2.8 and an argument similar to Proposition 3.11.12, Kaiser and Weis [KW03] have shown that A + B generates a

3.17. NOTES

233

(k + 2)-times integrated semigroup if A generates an exponentially bounded k-times integrated semigroup S and B : D(A) → X satisﬁes sup { BR(λ, A) : Re λ = ω} < 1 for some ω > ω(S). Finally, we mention that, more generally, instead of Cauchy problems involving a function and its derivatives at an instant of time, problems with memory may be considered. They lead to the theory of Volterra equations. Their solutions are governed by one-parameter families of operators, more general than semigroups or cosine functions. Vector-valued Laplace transforms also play a decisive role in this theory. We refer to the monograph of Pr¨ uss [Pr¨ u93] for a comprehensive presentation of Volterra equations. A study of regularized solutions of Volterra equations was initiated in [Liz00]. Section 3.3 The Hille-Yosida theorem (in its general form due to Hille, Yosida, Feller, Miyadera and Phillips) was the starting point of the subsequent development of the theory of semigroups. The classical approach of Yosida using what is now called the Yosida approximation and the classical method of Hille based on the convergence of the exponential formula T (t)x = limn→∞ (I − nt A)−n x can be found in many textbooks on semigroup theory (see, for example, those of Cl´ement et al. [CHA87], Davies [Dav80], [Dav07], Engel and Nagel [EN00], [EN06], Fattorini [Fat83], Goldstein [Gol85], Hille and Phillips [HP57], Kato [Kat66], Nagel et al. [Nag86], Pazy [Paz83] and Yosida [Yos80]). Hille and Phillips already mentioned explicitly on page 364 of [HP57] the problem of how to use Widder’s theorem in the proof of the Hille-Yosida theorem. The approach presented here based on the real representation theorem for Laplace transforms solves this problem. Our presentation follows the lines of Arendt [Are87b]. A diﬀerent condition which is suﬃcient for an operator to generate a C0 -semigroup has been given by Gomilko [Gom99] and Shi and Feng [SF00]. This condition has the advantage of involving only the square of the resolvent and not higher powers. Theorem 3.17.2. Let A be a densely deﬁned operator on a Banach space X and suppose that there exist K and ω such that σ(A) ⊂ {λ ∈ C : Re λ ≤ ω} and ∞ ∗ R(a + is, A)2 x, x∗ ds ≤ K x x

a−ω −∞ whenever a > ω, x ∈ X and x∗ ∈ X ∗ . Then A generates a C0 -semigroup T satisfying

T (t) ≤ M eωt (t ≥ 0) for some M . This theorem can be proved by using the Poisson representation of functions in a Hardy space on a half-plane to establish the conditions of the Hille-Yosida theorem. Alternatively, a complex inversion formula can be used to deﬁne tT (t)x when x ∈ D(A2 ) and then the operators T (t) extend by continuity to a C0 -semigroup. The converse of the theorem is true in Hilbert spaces. However, the derivative operator generates the C0 -group of translations on Lp (R), but it does not satisfy the hypotheses of Theorem 3.17.2 when p = 2. Proposition 3.3.8 appeared ﬁrst in a paper of Kato [Kat59]. We mention here that Proposition 3.3.8 is not true if merely the Radon-Nikodym property is assumed instead of reﬂexivity (see [Are87a]). However, we have the following result [Are87b]. Proposition 3.17.3. Let X be a Banach space with the Radon-Nikodym property. Let A be an operator on X satisfying the Hille-Yosida condition (3.23). Then A generates a semigroup in the sense of Deﬁnition 3.2.5.

234

3. CAUCHY PROBLEMS

The sun-dual T of a C0 -semigroup T was already introduced by Hille and Phillips in [HP57]. Dual semigroups have been investigated systematically by Cl´ement, Dieckmann, Gyllenberg, Thieme and van Neerven. For a comprehensive treatment and precise references, see the monograph by van Neerven [Nee92]. Section 3.4 Dissipative operators and the Lumer-Phillips theorem are classical objects in semigroup theory. Many further results in this direction can be found in the textbooks listed in the Notes of the previous section. It is also possible to characterize generators of positive C0 -semigroups of contractions by a similar notion, namely dispersiveness and a range condition. Even more generally, the norm may be replaced by a “half-norm” as studied by Arendt, Chernoﬀ and Kato [ACK82], Batty and Robinson [BR84] and Nagel et al. [Nag86, Section A-II.2]. If A is a densely deﬁned dissipative operator and x ∈ D(A), then ReAx, x∗ ≤ 0 for all x∗ ∈ dN(x) (instead of merely some x∗ ∈ dN(x) as required by the deﬁnition). This is easy to see when A generates a contraction semigroup. For densely deﬁned operators, it was proved in [Bat78]; see also [ACK82, Theorem 2.5]. A very interesting class of contraction semigroups is the class of Ornstein-Uhlenbeck semigroups which arises naturally in the study of stochastic processes. For a thorough study of this class of contraction semigroups, we refer to the book by Lorenzi and Bertoldi [LB07]. The Laplacian with Dirichlet boundary conditions is an easy example of an elliptic operator in divergence form. Generalizations of the arguments used in Example 3.4.7 lead to the theory of quadratic forms. For more information on this topic we refer to the books of Kato [Kat66], Dautray and Lions [DL90] and Davies [Dav80], [Dav95], and the survey article [Are04]. Section 3.5 Hille-Yosida operators were studied by Kato already in [Kat59]. Later, these operators were investigated systematically by Sinestrari [Sin85] and Da Prato and Sinestrari [DS87], where Theorem 3.5.2 was ﬁrst proved. A proof of the Da Prato-Sinestrari Theorem 3.5.2 based on the theory of integrated semigroups was given by Kellermann and Hieber [KH89]. Theorem 3.5.2 may also be proved via the theory of non-linear semigroups, as shown by B´enilan, Crandall and Pazy [BCP88]. In fact, operators which are not densely deﬁned are natural in the framework of the Crandall-Liggett-B´enilan theorem which corresponds to the Hille-Yosida theorem for non-linear operators. For a third, very diﬀerent, approach using “abstract Sobolev towers”, see the work of Nagel and Sinestrari [NS94]. The renorming Lemma 3.5.4 can be found in Pazy’s book [Paz83] and the bounded perturbation Theorem 3.5.5 for Hille-Yosida operators is due to Kellermann and Hieber [KH89]. Section 3.6 The Trotter-Kato theorem is a classical result in semigroup theory. It is frequently proved with the help of the Hille-Yosida theorem (see, for example, [Paz83]). The proof which we give here is based on the approximation theorem for Laplace transforms from Section 1.7, and is due to Jun Xiao and Liang [XL00] who also proved a general Trotter-Kato approximation theorem for integrated semigroups after previous work by Lizama [Liz94] and Nicaise [Nic93]. For other variants of the Trotter-Kato theorem, we refer to Bobrowski’s survey article [Bob97b].

3.17. NOTES

235

Section 3.7 The monographs by Amann [Ama95] and Lunardi [Lun95] are specialized texts on holomorphic semigroups and parabolic problems. Much further information, in particular on maximal regularity, interpolation and extrapolation scales of Banach spaces as well as many applications to non-linear problems can be found there. Our proof of the characterization Theorem 3.7.11 uses only Laplace transform theory. Corollary 3.7.14 is taken from the lecture notes of Nagel et al. [Nag86]. The perturbation Theorem 3.7.25 is due to Desch and Schappacher [DS88] with a diﬀerent proof. If A generates a C0 -semigroup on a reﬂexive space and B : D(A) → X is compact, then the estimate (3.49) holds (see [DS88]) or [EN00, p.179]). However, this is no longer true if X is not reﬂexive (see [Hes70] for an example). It was also shown in [DS88] (see also [AB06], [AB07], [DSS09]) that if A + B generates a C0 -semigroup for every B ∈ L(D(A), X) of rank 1, then A generates a holomorphic C0 -semigroup. Section 3.8 Fractional powers of sectorial operators are classical objects in semigroup theory, and they now have a very extensive theory (see [MS01] and [Haa06]). Theorem 3.8.1 and many further results in this direction can be found in the books of Amann [Ama95], Haase [Haa06] and others listed in the Notes on Section 3.3, and in the original papers of Balakrishnan [Bal60], Komatsu [Kom66] and others. The assumption that A is a generator is not essential in Theorem 3.8.3. If A is densely deﬁned and B = −A satisﬁes the assumptions of Proposition 3.8.2, then −(−A)1/2 generates a bounded holomorphic C0 -semigroup; see for example, the books of Fattorini [Fat83] or Mart´ınez and Sanz [MS01]. For a sectorial operator B and Re z > 0, one may deﬁne B z to be the algebraic inverse of B −z . If A generates a bounded C0 -semigroup and 0 < α < 1, then −(−A)α generates a bounded holomorphic C0 -semigroup. Section 3.9 Boundary values of holomorphic C0 -semigroups already appeared in the book of Hille and Phillips [HP57]. The method which we use here is based on Laplace transform techniques. Theorem 3.9.4 was proved by H¨ ormander [H¨ or60], but our approach follows the lines of Arendt, El-Mennaoui and Hieber [AEH97]. Basic information on the Riemann-Liouville semigroup and on fractional integration is in [HP57]; the idea of using the transference principle in this context is in [AEH97]. Results similar to Theorem 3.9.13 were ﬁrst obtained by Boyadzhiev and deLaubenfels [BdL93] and later improved by El-Mennaoui in [Elm92]. The proof of the implication (ii) ⇒ (i) of Theorem 3.9.13 follows the lines of [Elm92]. Corollary 3.9.14 is due to Hieber [Hie91a] with a diﬀerent proof using techniques from Fourier multipliers; see also [Lan68] and [Sjo70]. Further results in this direction such as behaviour of the critical exponent k = n|1/2 − 1/p| in Lp (Rn ) and L1 (Rn ), results on more general types of operators such as pseudo-diﬀerential operators with symbol a of the form a(ξ) = |ξ|α for some α > 0 can be found in [Sjo70], [Hie91a] and in Chapter 8. For example, if 1 < p < ∞, then iΔp generates a k-times integrated semigroup on Lp (Rn ) if and only if k ≥ n|1/2 − 1/p|. Moreover, iΔ1 and iΔ∞ generate a k-times integrated semigroup on L1 (Rn ) and L∞ (Rn ), respectively, if and only if k > n/2. Further applications of boundaries of holomorphic semigroups are given by El-Mennaoui and Keyantuo, to Schr¨ odinger operators in [EK96a] and to the wave equation in [EK96b]. In particular, they show the remarkable result that the Schr¨ odinger operator iΔ on Lp (Ω), where Ω = (−π, π)n , with Dirichlet or Neumann boundary conditions, generates a k-times integrated group for k > n2 | 12 − p1 |, and that this constant is optimal.

236

3. CAUCHY PROBLEMS

Section 3.10 In this section we follow Arendt, Neubrander and Schlotterbeck [ANS92]. Extensions to fractional integrated semigroups have been given by Keyantuo [Key95a]. The perturbation result Corollary 3.10.5 is due to Kellermann and Hieber [KH89]. Applications of the sandwich theorem to diﬀerential operators have been given in [Are91]. Section 3.11 Integrated semigroups were actually introduced ﬁrst in the context of resolvent positive operators in [Are84]. Theorem 3.11.7 and Proposition 3.11.12 as well as Example 3.11.13 are from [Are87a]. Theorem 3.11.5 and Example 3.11.6 are taken from [Are87b]. Theorem 3.11.9 was proved by Arendt, Chernoﬀ and Kato [ACK82]; it remains valid if X is an ordered Banach space with normal cone X+ which has non-empty interior. Theorem 3.11.10 is from [Are00], but our proof is diﬀerent from the original one which is based on the construction of an intermediate Banach lattice where Theorem 3.5.2 can be applied. Theorem 3.11.8 is due to Arendt and B´enilan [AB92a]. It is no longer true on arbitrary Banach lattices: Grabosch and Nagel [GN89] showed that there exists a generator A of a positive C0 -semigroup such that D(A∗ ) is not a sublattice of X ∗ . Example 3.11.13 also works on Lp (0, 1) if 1 < p < ∞, but not on L1 (0, 1). In fact, the following result is due to W. Desch (unpublished). It can also be seen in the framework of Miyadera–Voigt perturbation theory (see [Voi89]). Theorem 3.17.4. Let A be the generator of a positive C0 -semigroup on a space X of the form L1 (Ω, μ). Let B : D(A) → X be linear and positive. If A + B is resolvent positive, then A + B generates a C0 -semigroup. The analogous result also holds if X is an arbitrary Banach lattice, but A generates a holomorphic, positive C0 -semigroup, as shown by Arendt and Rhandi [AR91]. The theory of resolvent positive operators has been developed further by Thieme [Thi98a] and [Thi98b], where in particular spectral theory and perturbation theory are studied. One may also study asymptotic behaviour. Assume that A is a resolvent positive operator generating a once integrated semigroup S. Then lim sup λR(λ, A) < ∞ λ→∞

if and only if 1 lim sup S(t) < ∞. t t↓0 Moreover, limt↓0 1t S(t) = I strongly if and only if limλ→∞ λR(λ, A) = I strongly (we refer to [Are87a, Proposition 6.9]). Of course, an important class of resolvent positive operators are generators of positive C0 -semigroups. In the spirit of this book, we have concentrated in this section on those results which are related to Laplace transform techniques. We refer to the book [Nag86] edited by Nagel for the general theory of positive semigroups. Finally, we should mention that there exist natural examples of operators which are not resolvent positive even though the resolvent exists and is positive on some interval [λ1 , λ2 ] where −∞ < λ1 < λ2 < ∞. Examples have been given by Greiner, Voigt and Wolﬀ [GVW81] and Ulm [Ulm99]. Section 3.12 Theorem 3.12.2 and Example 3.12.3 are due to Driouich and El-Mennaoui [DE99]. Extensions to resolvent families associated with Volterra equations are given in [CL03] and [Haa08].

3.17. NOTES

237

The acronym UMD-space stands for “unconditional martingale diﬀerences”, reﬂecting the original probabilistic deﬁnition of this class of Banach spaces. Burkholder [Bur81] showed that the probabilistic property implies that the Hilbert transform is bounded on Lp (R, X) for 1 < p < ∞ and Bourgain [Bou83] established the converse. Every UMDspace is superreﬂexive (i.e., there is an equivalent norm which is uniformly convex), but there are uniformly convex spaces which are not UMD-spaces. These and other properties of UMD-spaces are discussed in the survey article [Fra86] by Rubio de Francia. The UMD-property has proved to be very important for the study of maximal regularity of solutions of inhomogeneous Cauchy problems. See, for example, [KW04], [Are04], [AB04]. Section 3.13 A C0 -semigroup T is said to be eventually norm-continuous if T is norm-continuous on (τ, ∞) for some τ ≥ 0. It was shown in the book of Hille and Phillips [HP57] that if T is eventually norm-continuous, then for any real a, {λ ∈ σ(A) : Re λ ≥ a} is compact. Theorem 3.13.2 was ﬁrst proved by You [You92], but the simple proof given here is due to El-Mennaoui and Engel [EE94]. With minor modiﬁcations, the proof shows that if X is a Hilbert space and T (τ )R(ω + is, A)n → 0 as |s| → ∞ for some n ≥ 1 and τ ≥ 0 then T is norm-continuous for t > τ . Other characterizations of immediately and eventually norm-continuous semigroups on Hilbert space have been given by Blasco and Martinez [BM96] and Blake [Bla01]. The latter characterization does not explicitly involve any decay of the resolvent of A (see also [Ile07]). Goersmeyer and Weis [GW99] have proved that if T is a positive semigroup on an Lp -space and R(ω +is, A) → 0 as |s| → ∞, then T is norm-continuous for t > 0. On the other hand, M´ atrai [Mat08], and Chill and Tomilov [CT09], have constructed examples showing that Theorem 3.13.2 does not extend to all Banach spaces. Section 3.14 For further literature on cosine functions we refer to the monographs of Fattorini [Fat85] and Goldstein [Gol85] and the survey article of Bobrowski [Bob97b], and the references given there. Kellermann and Hieber [KH89] proved the relation between cosine functions and integrated semigroups (Theorem 3.14.7). Uniqueness of the phase space (in Theorem 3.14.11) is due to Kisy´ nski [Kis72]. Further information about the phase space may be found in the Appendix of [Haa07a]. Corollary 3.14.13 seems to be a natural limit for perturbation results of that type. It is shown in [AB06, Theorem 2.2] (and stated in [AB07, Theorem 5.9]) that if A + B generates a cosine function for every bounded operator B : D((ω − A)γ ) → X of rank 1, where γ > 1/2, then A is bounded on X. More general versions of well-posedness of the Cauchy problem of second order (k-times integrated cosine functions) are considered in [AK89] and, using a diﬀerent approach, by Takenaka and Okazawa [TakO90]. Higher order problems are covered in [XL98]. Let X be a Hilbert space with inner product (·|·). The following result characterizes generators A of cosine functions X, up to choice of inner product, by a condition on the numerical range W (A) := {(Ax|x) : x ∈ X, x = 1} related to the spectral condition of Proposition 3.14.18. Here, part a) is due to Crouzeix [Cro04] who extended it to a remarkable piece of work on numerical range and functional calculus for matrices and operators on Hilbert space [Cro07], [Cro08]. Part b) is due to Haase [Haa06, Corollary 7.4.6].

238

3. CAUCHY PROBLEMS

Theorem 3.17.5. Let A be a densely deﬁned operator on a Hilbert space X. a) If W (A) is contained in the parabola Πω := {ξ + iη : η ∈ R, ξ ≤ ω 2 − η 2 /4ω 2 } for some ω > 0 and there is at least one point in ρ(A) ∩ (C \ Πω ), then A generates a cosine function. b) If A generates a cosine function, then there is an equivalent scalar product on X with respect to which the numerical range of A is contained in Πω for some ω > 0. Section 3.15 Sine functions were introduced by Arendt and Kellermann [AK89] where in particular the perturbation result Theorem 3.15.6 is proved. Moreover, the following holds. Theorem 3.17.6. Let A be a densely deﬁned operator which generates a sine function Sin satisfying lim supt↓0 1t Sin(t) < ∞. Then A generates a C0 -semigroup T . Moreover, T has a holomorphic extension to the right half-plane. The generation theorem for cosine functions (Theorem 3.15.3) is independently due to Da Prato and Giusti [DG67] and Sova [Sov66]. Section 3.16 Theorem 3.16.7 on square root reduction is due to Fattorini [Fat69]. At that time UMD-spaces had not been investigated and Fattorini formulated the result for Lp -spaces (1 < p < ∞). But he clearly pointed out that boundedness of the Hilbert transform is the crucial condition. Haase has given a diﬀerent approach to Fattorini’s theorem via functional calculus [Haa07a], and he has shown that i(−A)1/2 generates a bounded C0 -group if A generates a bounded cosine function on a UMD-space [Haa09].

Part II

Tauberian Theorems and Cauchy Problems

Part II

241

Again we consider the guide-line of this book, the characteristic equation u ˆ(λ) = R(λ, A)x, where u is a solution of the Cauchy problem u (t) = Au(t) (CP ) u(0) = x,

(t ≥ 0),

and R(λ, A) is the resolvent of the operator A. The aim of this part is to study the asymptotic behaviour of the solution u(t) as t → ∞. In applications, one typically has some information about the spectral behaviour of A. This means that, in many cases, we know the Laplace transform u ˆ(λ) for Re λ > 0. This part is devoted to the problem of determining the relation between the asymptotic behaviour of u(t) as t → ∞ and that of u ˆ(λ) as λ ↓ 0. It contains two chapters. The ﬁrst, Chapter 4, is devoted to the investigation of arbitrary functions; in Chapter 5 the asymptotic behaviour of solutions of (CP ) is investigated. The Abelian theorems given in Chapter 4 show that convergence of u(t) as t → ∞ implies convergence of λˆ u(λ) as λ ↓ 0. Much more interesting for the applications which we have in mind, are Tauberian theorems, which give statements about the converse implication. However, they do need additional hypotheses, socalled Tauberian conditions. Of particular importance are complex Tauberian theorems where the assumptions involve the behaviour of u ˆ(λ) for λ close to the imaginary axis. Indeed, assumptions of this kind are directly related to spectral properties of the operator via the characteristic equation. We not only consider convergence of functions but we also investigate their periodic behaviour. In fact, Chapter 4 contains an introduction to the theory of almost periodic and asymptotically almost periodic functions. The main results show that under suitable assumptions, countability of the spectrum implies asymptotic almost periodicity. In the second chapter of this part, Chapter 5, the results of Chapter 4 are applied to solutions of the Cauchy problem, and also a variety of new and special results are obtained. The most important framework is given by generators of C0 semigroups, but we also consider individual solutions in more general situations. This will be useful, for example, for the investigation of the heat equation with inhomogeneous boundary conditions which we give in Chapter 6. In the ﬁrst three sections of Chapter 5 the diverse abscissas of Laplace transform of the semigroup are related to spectral properties. Indeed, it turns out that the relation is not simple, in general. However there are very satisfying results in special cases; for example on Hilbert spaces (Section 5.2) or for positive semigroups (Section 5.3); both are important for applications. The complex Tauberian theorems from Chapter 4 are applied to solutions of (CP ) in many diﬀerent situations. Asymptotically almost periodic solutions

242

Part II

lead to splitting theorems which are considered in Section 5.4. Countability of the spectrum is a criterion for asymptotic periodicity. This is shown in Section 5.5 by exploiting the results of Chapter 4. Typically, one has to assume boundedness of the solution for these applications. This may be a diﬃcult assumption to verify when the inhomogeneous Cauchy problem is considered, but we show that this is automatic if the underlying semigroup is holomorphic and a certain spectral condition of non-resonance is satisﬁed. This last topic of Chapter 5 concludes our investigation of the interesting interplay of spectrum and asymptotics.

Chapter 4

Asymptotics of Laplace Transforms Frequently, convergence of a function f : R+ → X for t → ∞ implies convergence of an average of this function. Assertions of this type are called Abelian theorems. A theorem is called Tauberian if, conversely, convergence of the function is deduced from the convergence of an average. The Abelian theorems which we present in Section 4.1 are quite easy to prove. However, the Tauberian theorems corresponding to their converse versions are much more delicate. They need additional hypotheses, so-called Tauberian conditions. Section 4.2 is devoted to Tauberian conditions of real type (for example, boundedness or positivity of f ). Interesting applications of these Abelian and real Tauberian theorems to semigroups are given in Section 4.3 where mean ergodicity is discussed. This interrupts the general theme of this chapter, but the results will be useful in the subsequent sections where the notion of mean ergodicity is needed. In Section 4.4 a complex Tauberian theorem is proved with the help of an elegant contour argument. Here we make assumptions on holomorphic extensions of fˆ on the imaginary axis. We restrict ourselves to the case of one singularity in order to keep the ideas more transparent, but this case is already of special interest. For example, an immediate consequence is Gelfand’s theorem, saying that a bounded C0 -group is trivial (i.e., the identity) if and only if the spectrum of its generator is reduced to {0}. One interesting type of asymptotic behaviour for large time is almost periodicity. The concept is introduced in Section 4.5 where elementary properties are proved for functions on R. In Section 4.6, Loomis’s theorem and its vectorvalued version are proved by an elegant quotient method which allows one to apply Gelfand’s theorem. The basic notion is the Carleman spectrum for a bounded measurable function deﬁned on the line, and Loomis’s theorem states that any

W. Arendt et al., Vector-valued Laplace Transforms and Cauchy Problems: Second Edition, Monographs in Mathematics 96, DOI 10.1007/978-3-0348-0087-7_4, © Springer Basel AG 2011

243

244

4. ASYMPTOTICS OF LAPLACE TRANSFORMS

bounded uniformly continuous function f : R → C with countable spectrum is almost periodic. There is one vector-valued version of Loomis’s theorem, which is valid for every Banach space and involves an ergodicity condition. The other vector-valued version holds without further assumptions on the function but a geometric condition on the Banach space is needed. Functions on the half-line are considered in Section 4.7. The naturally associated “half-line” spectrum is discussed and the main theorem is a complex Tauberian theorem for functions with countable spectrum, which is proved by the same technique as we proved Loomis’s theorem. In Section 4.8 we come back to functions deﬁned on the line showing that the Carleman spectrum (deﬁned by holomorphy) and the Beurling spectrum (deﬁned by the Fourier transform) coincide. This allows us to prove a very general complex Tauberian theorem for functions on the half-line in Section 4.9. Here we use Fourier transform methods which allow one to reduce the problem to an application of Loomis’s theorem (in the scalar case). The structure of this chapter needs some explanation in view of our main purpose, namely the proof of a complex Tauberian theorem on the half-line. We present three diﬀerent methods by which we prove the result in increasing generality (Theorems 4.4.8, 4.7.7 and 4.9.7); namely, the contour method, the quotient method and the Fourier method. If Loomis’s theorem (Corollary 4.6.4) is accepted, the Fourier method of Section 4.9 is the most general. The contour method, presented as the ﬁrst approach in Section 4.4, is the most elementary. It gives us Gelfand’s theorem (Corollary 4.4.12) and other interesting consequences. The quotient method uses Gelfand’s theorem. It gives us Loomis’s theorem on the line (Section 4.6), and in an elegant way the fairly general complex Tauberian Theorem 4.7.7 on the half-line.

4.1

Abelian Theorems

Throughout this section, f denotes a function in L1loc (R+ , X), where X is a Banach space and R+ := [0, ∞) is the right half-line. We consider the following three types of averages. Deﬁnition 4.1.1. Let f∞ ∈ X. We say that a) f (t) converges to f∞ in the sense of Abel as t → ∞ if abs(f ) ≤ 0 and A- limt→∞ f (t) := limλ↓0 λfˆ(λ) = f∞ ; b) f (t) is B-convergent to f∞ as t → ∞, or simply write B- limt→∞ f (t) = f∞ , t+δ if for every δ > 0, limt→∞ 1δ t f (s) ds = f∞ ; c) f (t) converges to f∞ in the sense of Ces` aro as t → ∞ if 1 t C- lim f (t) := lim f (s) ds = f∞ . t→∞ t→∞ t 0

4.1. ABELIAN THEOREMS

245

It will t be convenient to consider the antiderivative of a function f given by F (t) := 0 f (s) ds. Then the following Abelian theorem holds. Theorem 4.1.2. Let f∞ , F∞ ∈ X. a) If limt→∞ f (t) = f∞ , then B- limt→∞ f (t) = f∞ . b) If B- limt→∞ f (t) = f∞ , then C- limt→∞ f (t) = f∞ . c) If C- limt→∞ f (t) = f∞ , then A- limt→∞ f (t) = f∞ . d) If limt→∞ F (t) = F∞ , then limλ↓0 fˆ(λ) = F∞ . Proof. a) is obvious. b) Replacing f by f − f∞ we can assume that f∞ = 0. Then (taking δ = 1) t+1 we have by assumption that lim

t→∞ t f (s) ds = 0. Let ε > 0. There exists

t+1

t0 ≥ 0 such that t f (s) ds ≤ 2ε for all t ≥ t0 . Moreover, there exists t1 ≥ t0 t +1 such that 1t 0 0 f (s) ds ≤ 2ε for all t ≥ t1 . Let t ≥ t1 . Take n ∈ N such that t − n ≤ t0 ≤ t − n + 1. Then

t

t−n+1

t

1

1

≤ 1

+

f (s) ds f (s) ds f (s) ds

t

t

t

0 0 t−n+1

t−n+k+1 n−1

1 t0 +1

1

≤

f (s) ds + f (s) ds

t t−n+k

t 0 ≤

ε n−1ε + ≤ ε. 2 t 2

k=1

This proves the claim. c) Assume that C- limt→∞ f (t) = f∞ . Then F (t) = O(t) as t → ∞, so fˆ(λ) exists for Re λ > 0 by Theorem 1.4.3. Let ε > 0. There exists τ > 0 such that 1t F (t) − f∞ ≤ ε (t ≥ τ ). Hence by integration by parts, lim sup λfˆ(λ) − f∞

λ↓0

∞

−λt

= lim sup λ e (f (t) − f∞ ) dt

λ↓0 0

∞

2

−λt

= lim sup λ e (F (t) − tf ) dt ∞

λ↓0 0

∞

2

1 = lim sup te−λt F (t) − f∞ dt

λ

t λ↓0 0

τ ∞

1

dt + lim ελ2 ≤ lim sup λ2 te−λt F (t) − f te−λt dt ∞

t

λ↓0 λ↓0 0 τ ∞ = lim ελ2 te−λt dt = ε. λ↓0

0

246

4. ASYMPTOTICS OF LAPLACE TRANSFORMS

(λ) = λFˆ (λ) (λ > 0) if abs(f ) ≤ 0, this follows from the d) Since fˆ(λ) = F previous parts since convergence implies A-convergence. Concerning the behaviour for t → 0, we note: Proposition 4.1.3. Let x ∈ X. Assume that abs(f ) < ∞. Then lim sup λfˆ(λ) − x ≤ lim sup f (t) − x . λ→∞

t↓0

In particular, limt↓0 f (t) = x implies limλ→∞ λfˆ(λ) = x. Proof. Since abs(f ) < ∞, there exist M, ω > 0 such that F (t) ≤ M eωt (t ≥ 0). Let b > 0, λ > ω. Then

∞

∞

−λt −λt

= −λe−λb F (b) + λ2

λe f (t) dt e F (t) dt

b

b

≤

−λb

λe

M −(λ−ω)b

F (b) + λ e . λ−ω 2

Hence, lim sup λfˆ(λ) − x

λ→∞

∞

−λt

= lim sup λe (f (t) − x) dt

λ→∞ 0

∞

−λt

≤ lim sup f (t) − x + λe f (t) dt

+ λ→∞

=

0≤t≤b

b

∞

λe

−λt

dt x

b

sup f (t) − x .

0≤t≤b

∞ Here, we used throughout that λ 0 e−λt dt = 1 (λ > 0). More generally, for α > −1 one has ∞ λα+1 e−λt tα dt = 1 (λ > 0). Γ(α + 1) 0 One shows the following in a similar way as above. Proposition 4.1.4. Let f∞ ∈ X, f0 ∈ X, α > −1. a) If C- limt→∞ f (t) = f∞ , then limλ↓0 b) If limt↓0 f (t) = f0 , then limλ→∞

λα+1 Γ(α+1)

λα+1 Γ(α+1)

∞ 0

∞ 0

e−λt tα f (t) dt = f∞ .

e−λt tα f (t) dt = f0 .

This allows one to compute the derivative of f at 0 from its Laplace transform: Corollary 4.1.5. Let f ∈ L1loc (R+ , X) such that abs(f ) < ∞. Assume that f ∈ C 1 ([0, τ ], X) for some τ > 0. Then lim λ2 fˆ(λ) − λf (0) = f (0). λ→∞

4.2. REAL TAUBERIAN THEOREMS

247

Proof. Let α = 1. Applying Proposition 4.1.4 b) to g(t) := 1t (f (t)−f (0)), we obtain ∞ that limλ→∞ (λ2 fˆ(λ)−λf (0)) = limλ→∞ λ2 0 e−λt t 1t (f (t) −f (0)) dt = f (0). The following theorem gave the name to this type of result, and to this section. We deduce it from the corresponding result for Laplace transforms. Usually, one uses Abel’s “summation by parts” for power series. ∞ Theorem 4.1.6 (Abel’s Continuity Theorem). Let p(z) := n=0 an z n be a power ∞ series, where an ∈ X. If n=0 an = b, then limz↑1 p(z) = b. Proof. Let f (t) = an if t ∈ [n, n + 1). Then limt→∞ F (t) = b. Hence, −λ ∞ −λ n abs(f ) ≤ 0 and fˆ(λ) = 1−eλ ) for all λ > 0. By Theorem 4.1.2 n=0 an (e d), lim p(z) z↑1

= lim z↑1

∞

an z n = lim

n=0

λ↓0

∞

an (e−λ )n

n=0

λ = lim fˆ(λ) = lim fˆ(λ) λ↓0 1 − e−λ λ↓0 = lim F (t) = b. t→∞

4.2

Real Tauberian Theorems

Let f ∈ L1loc (R+ , X). In the preceding section we established four notions of convergence for f (t) (as t → ∞) of increasing generality: convergence in the usual sense, B-convergence, convergence in the sense of Ces`aro, and convergence in the sense of Abel. None of the implications established in Theorem 4.1.2 is reversible. An additional condition, which allows one to reverse one of the implications is called a Tauberian condition, and the corresponding result a Tauberian theorem. Of particular interest are those results which allow one to deduce convergence in the usual sense from convergence of a mean. In this section we consider assumptions on f (t), i.e. real Tauberian conditions, in contrast to conditions on fˆ(λ), known as complex Tauberian conditions, which will be the subject of the next section. We ﬁrst consider the reverse implication of Theorem 4.1.2 a). A. Conditions under which B- limt→∞ f (t) = f∞ implies limt→∞ f (t) = f∞ . Let f ∈ L1loc (R+ , X). In general, if B- limt→∞ f (t) = f∞ , one cannot conclude that limt→∞ f (t) = f∞ even if f is bounded. A simple example is the function ∞ f = n=1 χ[n,n+1/n] which has B-limit 0 (as t → ∞) but does not converge. Deﬁnition 4.2.1. A function g : R+ → X is called slowly oscillating if for all ε > 0 there exist t0 ≥ 0, δ > 0 such that g(t) − g(s) ≤ ε whenever s, t ≥ t0 , |t − s| ≤ δ.

248

4. ASYMPTOTICS OF LAPLACE TRANSFORMS

A bounded continuous function is slowly oscillating if and only if it is uniformly continuous. Indeed, one may think of Deﬁnition 4.2.1 as expressing that the function g is uniformly continuous at inﬁnity. This is made more precise by the following characterization. Proposition 4.2.2. A function g : R+ → X is slowly oscillating if and only if g = g0 + g1 where g1 : R+ → X is uniformly continuous and g0 : R → X satisﬁes limt→∞ g0 (t) = 0. Proof. It is clear that the condition is suﬃcient. In order to prove the converse, let g be slowly oscillating. Then we ﬁnd a decreasing sequence (δn )n∈N ⊂ (0, ∞) such that limn→∞ δn = 0 and an increasing sequence (tn )n∈N ⊂ [0, ∞) such that limn→∞ tn = ∞ and f (t + δ) − f (t) ≤ n1 whenever t ≥ tn , 0 < δ ≤ δn . For each n ∈ N, choose a partition of the interval [tn , tn+1 ] so that every subinterval has length smaller than δn . Deﬁne g1 to be linear on each subinterval and equal to f at the endpoints of the subintervals. Extend g1 continuously to [0, t1 ]. Then g1 is uniformly continuous and limt→∞ (f (t) − g1 (t)) = 0. We put g0 := f − g1 . The following Tauberian theorem is easy to prove. Theorem 4.2.3. Let f ∈ L1loc (R+ , X) be slowly oscillating and f∞ ∈ X. If B- lim f (t) = f∞ , then lim f (t) = f∞ . t→∞

t→∞

Proof. For δ > 0 one has lim sup f (t) − f∞

t→∞

≤ = ≤

1 t+δ

lim sup f (t) − f (s) ds

δ t→∞ t

1 t+δ

lim sup (f (t) − f (s)) ds

t→∞ δ t lim sup sup f (t) − f (s)

t→∞

t≤s≤t+δ

The last expression tends to 0 as δ ↓ 0 since f is slowly oscillating. Next, we consider: B. Conditions under which C- limt→∞ f (t) = f∞ implies limt→∞ f (t) = f∞ . First, we notice that, in general, convergence in the sense of Ces` aro does not imply B-convergence even if f is bounded. For example, let f (t) := sin t. Then t t+δ t C- limt→∞ f (t) = limt→∞ 1t 0 sin s ds = limt→∞ 1−cos = 0. But 1δ t sin s ds t = 1δ (cos(t + δ) − cos t), and the B-limit does not exist.

4.2. REAL TAUBERIAN THEOREMS

249

A function f : R+ → X is called feebly oscillating (when t → ∞) if lim f (t) − f (s) = 0.

t,s→∞ t/s→1

It is clear that every feebly oscillating function is slowly oscillating. Example 4.2.4. Assume that there exist τ > 0, M ≥ 0 such that tf (t) ≤ M t (t ≥ τ ). Then F (t) := 0 f (s) ds is feebly oscillating. In fact, F (t) − F (s) ≤ t !t" M s dr r = M log s for t ≥ s ≥ τ . Theorem 4.2.5. Let f ∈ L1loc (R+ , X) and f∞ ∈ X. Assume that f is feebly oscillating. If C- limt→∞ f (t) = f∞ , then limt→∞ f (t) = f∞ . Proof. Let ε > 0. There exist δ > 0, t0 > 0 such that f (s) − f (t) < ε whenever t ≥ t0 , s ∈ [t − δt, t + δt]. Hence,

t(1+δ)

1 t(1+δ)

1

f (s) ds = (f (t) − f (s)) ds ≤ ε if t ≥ t0 .

f (t) −

2δt t(1−δ)

2δt t(1−δ) The assumption implies that t(1+δ) 1 f (s) ds 2δt t(1−δ) t(1+δ) t(1−δ) 1+δ 1 1−δ 1 = f (s) ds − f (s) ds 2δ t(1 + δ) 0 2δ t(1 − δ) 0 → f∞ as t → ∞. It follows that limt→∞ f (t) = f∞ . In the next theorem the Tauberian condition is of order-theoretic nature. Theorem 4.2.6. Let X be an ordered Banach space with normal cone. Furthermore let f : R+ → X+ be a function such that for some k ∈ N, t → tk f (t) is increasing. Then C- lim f (t) = f∞ implies lim f (t) = f∞ . t→∞

t→∞

Proof. Since the cone is normal, there exists c ≥ 0 such that u ≤ x ≤ v implies

x ≤ c( u + v ) for u, x, v ∈ X (see Appendix C). Let ρ > 1. The assumption implies that ρt 1 ρ 1 ρt 1 1 t f (s) ds = f (s) ds − f (s) ds (ρ − 1)t t ρ − 1 ρt 0 ρ−1 t 0 → f∞ (t → ∞) t ρ and, similarly, (ρ−1)t f (s) ds → f∞ (t → ∞). t/ρ

250

4. ASYMPTOTICS OF LAPLACE TRANSFORMS Since tk f (t) is increasing, one has

ρt

f (s) ds = t

≥ and

t

ρ

t/ρ −k

ρt

sk f (s)s−k ds

t k

t f (t)(ρt)−k (ρt − t) = f (t)tρ−k (ρ − 1)

f (s) ds ≤ f (t)tρk−1 (ρ − 1). Thus,

ρ (ρ − 1)t

t

1 ≤ρ (ρ − 1)t

f (s) ds − f∞ ≤ f (t) − f∞

ρt

k

t/ρ

f (s) ds − f∞ .

t

Consequently, lim sup f (t) − f∞

t→∞

≤

t

ρ

−k

c lim sup ρ f (s) ds − f∞

(ρ − 1)t t/ρ t→∞

ρt

k

1 + f (s) ds − f∞

ρ (ρ − 1)t

t

≤ c f∞ (1 − ρ−k + ρk − 1). Letting ρ ↓ 1 yields the claim. C. Conditions under which A- limt→∞ f (t) = f∞ implies C- limt→∞ f (t) = f∞ . In general, Abel convergence does not imply Ces` aro convergence. For example, let f (t) := t sin t (t ≥ 0). Then it is easy to see that A- lim f (t) = lim 2λ2 (1 + λ2 )−2 = 0 t→∞

λ↓0

t but 1t 0 f (s) ds = sint t − cos t does not converge. Thus additional hypotheses are needed. First, we consider a boundedness condition on f . Theorem 4.2.7. Let f∞ ∈ X. Assume that supt≥τ f (t) < ∞ for some τ ≥ 0. If A- limt→∞ f (t) = f∞ , then C- lim∞ f (t) = f∞ . Proof. a) We ﬁrst assume that τ = 0. For β > 0 let eβ (t) := e−βt (t > 0). Then span{eβ : β > 0} is dense in L1 (R+ ) (by Lemma 1.7.1). By hypothesis, lim

α→∞

0

∞

e−s f (αs) ds = lim λ↓0

∞

0

= lim λ λ↓0

0

e−s f

∞

s λ

ds

e−λs f (s) ds = f∞ .

4.2. REAL TAUBERIAN THEOREMS Hence,

251

∞

e−βs f (αs) ds 0 ∞ α e−s f s ds α→∞ 0 β = f∞ = βeβ , f∞ for all β > 0. ∞ It follows that limα→∞ h, f (α ·) = f∞ 0 h(t) dt for all h ∈ L1 (R+ ). Letting h = χ(0,1] , we obtain 1 1 α lim f (s) ds = lim f (αs) ds = lim h, f (α ·) = f∞ . α→∞ α 0 α→∞ 0 α→∞ lim βeβ , f (α ·) =

lim β α→∞ = lim

α→∞

b) If τ > 0 the result follows by applying a) to g(t) = f (t + τ ). Next, we show that positivity is a Tauberian condition. Theorem 4.2.8 (Karamata). Let X be an ordered Banach space with normal cone and let β > −1. Let f∞ ∈ X and f ∈ L1loc (R+ , X) such that f (t) ≥ 0 (t ∈ R+ ). Suppose that ∞ a) 0 e−λt tβ f (t) dt exists for all λ > 0; and ∞ −λt β λβ+1 b) limλ↓0 Γ(β+1) e t f (t) dt = f∞ . 0 Then C- limt→∞ f (t) = f∞ . This is a converse of Proposition 4.1.4 a). In particular, it follows from Theorem 4.2.8 that for positive f ∈ L1loc (R+ , X), A- lim f (t) = f∞ implies C- lim f (t) = f∞ . t→∞

t→∞

Proof. It follows from the assumption that for n ∈ N0 , ∞ λβ+1 lim e−λt (e−λt )n f (t)tβ dt λ↓0 Γ(β + 1) 0 1 (λ(n + 1))β+1 ∞ −λ(n+1)t = lim e f (t)tβ dt (n + 1)β+1 λ↓0 Γ(β + 1) 0 ∞ 1 1 = f∞ = f∞ tβ e−t (e−t )n dt. (n + 1)β+1 Γ(β + 1) 0 Consequently, for every polynomial p, ∞ ∞ λβ+1 1 lim e−λt p(e−λt )tβ f (t) dt = f∞ tβ e−t p(e−t ) dt. λ↓0 Γ(β + 1) 0 Γ(β + 1) 0 Let q : [0, 1] → R+ be given by q(x) := 0 if x < e−1 , q(x) := x−1 if x ≥ e−1 . Let δ > 0 and let q1 , q2 be continuous functions such that 0 ≤ q1 ≤ q ≤ q2 on [0, 1] and

252

4. ASYMPTOTICS OF LAPLACE TRANSFORMS

a) q1 (x) = q(x) if x ≤ e−1 or x > e−1 + δ, b) q2 (x) = q(x) if x ≤ e−1 − δ or x > e−1 , c) sup0≤x≤1 (q2 (x) − q1 (x)) ≤ e. t2 β −t e Now, let ε > 0. Choose δ > 0 such that Γ(β+1) t e dt < ε, where e−t2 = e−1 −δ t1 −t1 −1 and e = e + δ. By the Stone-Weierstrass theorem, there exist polynomials p1 and p2 such that q1 − ε ≤ p1 ≤ q1 ≤ q ≤ q2 ≤ p2 ≤ q2 + ε on [0, 1]. Deﬁne kj : R+ → X (j = 1, 2) and h : (0, ∞) → X by: kj (λ)

:=

kj (0)

:=

h(λ)

:=

∞ λβ+1 tβ e−λt pj (e−λt )f (t) dt (λ > 0), Γ(β + 1) 0 ∞ f∞ e−t pj (e−t )tβ dt, Γ(β + 1) 0 ∞ λβ+1 e−λt q(e−λt )tβ f (t) dt (λ > 0). Γ(β + 1) 0

Since 0 ≤ p2 − p1 ≤ q2 − q1 + 2ε, one has 0 ≤ k2 (0) − k1 (0)

≤ ≤

f∞ e Γ(β + 1) 3 εf∞ .

t2 t1

e−t tβ dt + 2ε

f∞ Γ(β + 1)

∞

e−t tβ dt

0

The ﬁrst part of the proof shows that limλ↓0 kj (λ) = kj (0). Let λ0 > 0 such that ˆ ≤ λ0 . Then

kj (λ) − kj (0) < ε for 0 < λ ≤ λ0 , j = 1, 2. Let 0 < λ, λ ˆ ≤ h(λ) − h(λ) ˆ ≤ k2 (λ) − k1 (λ). ˆ k1 (λ) − k2 (λ) Hence, for a ﬁxed constant c > 0, ˆ

h(λ) − h(λ)

ˆ + k1 (λ) − k2 (λ) ˆ

≤ c k2 (λ) − k1 (λ) ˆ

≤ c k2 (λ) − k2 (0) + k2 (0) − k1 (0) + k1 (0) − k1 (λ) ˆ

+ k1 (λ) − k1 (0) + k1 (0) − k2 (0) + k2 (0) − k2 (λ)

ˆ ≤ λ0 . ≤ 4 c ε + 2 · 3 c ε f∞ whenever 0 < λ, λ This shows that h(λ) converges as λ ↓ 0. Since h(λ) =

λβ+1 Γ(β+1)

1/λ 0

sβ f (s) ds, it

4.2. REAL TAUBERIAN THEOREMS follows that limt→∞ t−(β+1) f∞

= lim λ↓0

t

sβ f (s) ds =: g exists. Hence,

λβ+1 Γ(β + 1)

= lim(β + 1) λ↓0

=

0

253

∞

0 β+2

e−λt f (t)tβ dt

λ Γ(β + 2)

∞ 0

e−λt tβ+1

1 tβ+1

t

f (s)sβ ds dt

0

(β + 1)g,

f∞ by the Abelian theorem mentioned above (Proposition 4.1.4 a)). Thus, g = β+1 . t f∞ −(β+1) β We have proved that t s f (s) ds → β+1 as t → ∞. Since convergence 1 implies Ces` aro-convergence, we also have s 1 t 1 f∞ lim rβ f (r) dr ds = . t→∞ t 1 sβ+1 1 β+1

Hence, integration by parts yields s 1 t 1 t 1 d f (s) ds = rβ f (r) dr ds t 1 t 1 sβ ds 1 t 1 β t −(β+1) s β β = r f (r) dr + s r f (r) dr ds tβ+1 1 t 1 1 f∞ f∞ → +β = f∞ . β+1 β+1 D. Conditions under which limλ↓0 fˆ(λ) = F∞ implies limt→∞ F (t) = F∞ . The following theorem is due to Hardy t and Littlewood in the scalar-valued case. Let f ∈ L1loc (R+ , X) and let F (t) := 0 f (s) ds (t ≥ 0). Theorem 4.2.9 (Hardy-Littlewood). Assume that M := supt≥τ t f (t) < ∞ for some τ ≥ 0 and let F∞ ∈ X. If limλ↓0 fˆ(λ) = F∞ , then limt→∞ F (t) = F∞ . Proof. Replacing f by f (· + τ ) we can assume that τ = 0. For t > 0 we have

t

∞

!1"

−s/t −s/t ˆ

f (s)(1 − e ) ds − f (s)e ds

F (t) − f t =

0

t

∞ 1 − e−s/t −r dr ≤ M sup t + e s r 0<s
Since limt→∞ fˆ( 1t ) = F∞ , it follows that F is bounded. But A- limt→∞ F (t) = limλ↓0 fˆ(λ) = F∞ . It follows from Theorem 4.2.7 that C- limt→∞ F (t) = F∞ . By

254

4. ASYMPTOTICS OF LAPLACE TRANSFORMS

Example 4.2.4, F is feebly oscillating. Now Theorem 4.2.5 implies that limt→∞ F (t) = F∞ . The following classical example is a beautiful application of the HardyLittlewood theorem. Example 4.2.10. Let f (t) := sint t . Then fˆ(λ) = − arctan λ + π2 . In fact, ∞ ∞ d ˆ f (λ) = e−λt (−t)f (t) dt = − e−λt sin t dt dλ 0 0 ∞ 1 = − e−λt (eit − e−it ) dt 2i 0 1 1 1 −1 = − − = 2i λ − i λ + i 1 + λ2 d = − arctan λ. dλ Since limλ→∞ fˆ(λ) = 0, the claim follows. Now by Theorem 4.2.9, it follows that ∞ sin t t sin s dt := limt→∞ ds = limλ↓0 fˆ(λ) = π . 0

0

t

2

s

If f is positive, then fˆ(λ) is decreasing. So in the real-valued case it is clear that

lim fˆ(λ) λ↓0

=

λ>0

=

t

sup fˆ(λ) = sup sup

sup sup t>0 λ>0

λ>0 t>0 t

e−λs f (s) ds

0

e−λs f (s) ds = sup F (t) = lim F (t).

0

t>0

t→∞

If X is an ordered Banach space we cannot argue like this (unless the norm is order continuous). However, using the preceding results we obtain: Theorem 4.2.11. Let X be an ordered Banach space with normal cone. Let f ∈ L1loc (R+ , X) and assume that f (t) ≥ 0 (t ≥ 0) and abs(f ) ≤ 0. If limλ↓0 fˆ(λ) = F∞ then limt→∞ F (t) = F∞ . Proof. We have A- limt→∞ F (t) = limλ↓0 λFˆ (λ) = limλ↓0 fˆ(λ) = F∞ . It follows from Karamata’s theorem (Theorem 4.2.8) that C- limt→∞ F (t) = F∞ . Since F is increasing, the claim follows from Theorem 4.2.6. E. Conditions under which A- limt→∞ f (t) = f∞ implies limt→∞ f (t) = f∞ . t Let f ∈ L1loc (R+ , X) and F (t) = 0 f (s) ds (t ≥ 0). Assume that abs(f ) ≤ 0. Since λFˆ (λ) = fˆ(λ), every Tauberian theorem of type E yields one of type D. In order to go the other way around, we apply Tauberian theorems of type D to the function fδ deﬁned by fδ (t) :=

1 (f (t + δ) − f (t)) δ

(t ≥ 0).

4.2. REAL TAUBERIAN THEOREMS

255

Lemma 4.2.12. Let f∞ ∈ X, δ > 0. Consider the following assertions: (i) limt→∞ f (t) = f∞ . (ii) limt→∞ (iii) limt→∞

1 δ

t+δ

t 0

t

f (s) ds = f∞ .

fδ (s) ds = f∞ −

(iv) limλ↓0 fδ (λ) = f∞ −

1 δ

δ 0

1 δ

δ 0

f (s) ds.

f (s) ds.

(v) A- limt→∞ f (t) = f∞ . Then (i) ⇒ (ii) ⇔ (iii) ⇒ (iv) ⇔ (v). Proof. The implication (i) ⇒ (ii) is obvious, and (ii) is equivalent to (iii) since 0

t

1 fδ (s) ds = δ

t+δ t

1 f (s) ds − δ

Since fδ (λ) =

1 λδ eλδ (e − 1)λfˆ(λ) − λδ δ

δ

f (s) ds. 0

δ

e−λs f (s) ds,

(4.1)

0

(iv) is equivalent to (v). By Theorem 4.1.2, (iii) implies (iv). Remark 4.2.13. It is easy to see that f is B-convergent if (ii) or (iii) of Lemma 4.2.12 holds for all δ ∈ (0, δ0 ] for some δ0 > 0. See also the Notes for more general results. Now we can obtain the following Tauberian theorem by applying the HardyLittlewood Tauberian theorem to fδ . Theorem 4.2.14. Let f ∈ L1loc (R+ , X), f∞ ∈ X, δ0 > 0, τ ≥ 0, M ≥ 0. Assume that t f (t) − f (s) ≤ M

(4.2)

whenever t ≥ τ, |s − t| ≤ δ0 . If A- limt→∞ f (t) = f∞ , then limt→∞ f (t) = f∞ . Proof. By the assumption, lim supt→∞ t fδ (t) < ∞ for all δ ∈ (0, δ0 ). Moreover, f is slowly oscillating. Since A- limt→∞ f (t) = f∞ , Lemma 4.2.12 gives us δ t limλ↓0 fδ (λ) = f∞ − 1δ 0 f (s) ds. Theorem 4.2.9 implies that limt→∞ 0 fδ (s) ds = δ f∞ − 1δ 0 f (s) ds for all δ ∈ (0, δ0 ). Hence, B- limt→∞ f (t) = f∞ by Lemma 4.2.12 and Remark 4.2.13. Now the claim follows from Theorem 4.2.3. Now we can actually use Theorem 4.2.14 to prove a slight improvement of the Hardy-Littlewood theorem; i.e., we deduce a Tauberian theorem of type D from a result of type E as indicated above.

256

4. ASYMPTOTICS OF LAPLACE TRANSFORMS

Theorem 4.2.15. Let f ∈ L1loc (R+ , X), F∞ ∈ X. Assume that for some δ > 0,

t+δ

lim sup t→∞

r f (r) dr < ∞.

t

If limλ↓0 fˆ(λ) = F∞ , then limt→∞ F (t) = F∞ , where F (t) :=

t 0

f (s) ds.

Proof. We have lim sup

sup

t→∞

t≤s≤t+δ

t F (t) − F (s)

t+δ

≤ lim sup t t→∞

≤ lim sup t→∞

f (s) ds

t t+δ

s f (s) ds < ∞.

t

Thus, F satisﬁes (4.2). Since A- limt→∞ F (t) = limλ↓0 fˆ(λ) = F∞ , it follows from Theorem 4.2.14 that limt→∞ F (t) = F∞ . Next, we consider an order condition. Theorem 4.2.16. Let X be an ordered Banach space with normal cone, let f ∈ L1loc (R+ , X) such that tk f (t) is positive and increasing on [τ, ∞) for some τ ≥ 0 and some k ∈ N. If A- limt→∞ f (t) = f∞ , then limt→∞ f (t) = f∞ . Proof. Replacing f (t) by f (t + τ ) we can assume that τ = 0. Since f is positive, it follows from Karamata’s Theorem 4.2.8 that C- limt→∞ f (t) = f∞ . Now Theorem 4.2.6 implies that limt→∞ f (t) = f∞ . F. Power series. ∞ n Let p(z) := be a power series which converges for |z| < 1, where n=0 an z an ∈ X. Deﬁning f ∈ L1loc (R+ , X) by f (t) = an

if

t ∈ [n, n + 1),

(4.3)

the preceding results yield Tauberian theorems for p. In fact, one has abs(f ) ≤ 0 and ∞ 1 − e−λ ˆ f (λ) = an e−λn (Re λ > 0). (4.4) λ n=0 Thus, from Theorem 4.2.9 we obtain the following Tauberian counterpart of Theorem 4.1.6. Theorem 4.2.17 (Hardy). ∞Assume that supn∈N0 n an < ∞, and let b∞ ∈ X. If limz↑1 p(z) = b∞ , then n=0 an = b∞ .

4.2. REAL TAUBERIAN THEOREMS

257

The special case where limn→∞ nan = 0 had been proved by Tauber in 1897 and was the starting point of Tauberian theory. In the case nof power series, theorems of types D and E are equivalent. In fact, let bn := k=0 ak , or equivalently, a0 = b0 , an = bn − bn−1 (n ∈ N). Then ∞ q(z) := n=0 bn z n also converges for |z| < 1. Moreover, ∞

ak z k = (1 − z)

k=0

∞

bk z k

(|z| < 1).

k=0

∞ ∞ Thus, A- limn→∞ bn := limz↑1 (1 − z) k=0 bk z k = limz↑1 k=0 ak z k whenever one of the limits exists. So we obtain the following Corollary 4.2.18. Let bn ∈ X such that sup n bn −bn−1 < ∞. If A- lim bn = b∞ , n→∞

n∈N

then lim bn = b∞ . n→∞

G. Fourier series. Let Y be a Banach space. Let f : R → Y be a continuous 2π-periodic function. π 1 −ikx By ck := 2π f (x)e dx ∈ Y (k ∈ Z), we denote the Fourier coeﬃcients of f −π and by m sm (x) := ck eikx (x ∈ R) k=−m

the Fourier sums. It is well known that, in general, sm (x) does not converge as m → ∞. However, it converges in the sense of Ces`aro. Theorem 4.2.19 (Fej´er). One has 1 n→∞ n+1

lim

n

sm (x) = f (x)

m=0

uniformly in x ∈ R.

m Proof. a) We show that for m ∈ N, y ∈ R, k=−m eiky = Dm (y), where ⎧ ⎨ cos my − cos(m + 1)y (y ∈ 2πZ) 1 − cos y Dm (y) := ⎩ 2m + 1 (y ∈ 2πZ) m m (the so-called Dirichlet kernel). In fact, k=−m eiky = 1 + k=1 (eiky + e−iky ) is real. Thus, m m (1 − cos y) eiky = Re (1 − eiy ) eiky k=−m

k=−m

= =

Re(e−imy − ei(m+1)y ) cos my − cos(m + 1)y.

258

4. ASYMPTOTICS OF LAPLACE TRANSFORMS b) We obtain from a) and periodicity, sm (x)

m

=

k=−m

1 2π

=

1 2π

π

1 2π

f (t)e−ikt dt eikx =

−π

π

1 f (t)Dm (x − t) dt = 2π −π

π −π

π

f (t) −π

m

eik(x−t) dt

k=−m

f (x − t)Dm (t) dt.

c) Let σn (x)

:=

= = = where

n−1 1 sm (x) n m=0 π n−1 1 1 f (x − t) (cos mt − cos(m + 1)t) dt 2πn −π 1 − cos t m=0 π 1 1 − cos nt f (x − t) dt 2πn −π 1 − cos t π 1 f (x − t)Fn (t) dt, 2π −π

⎧ nt 2 ⎪ ⎨ 1 sin 2 Fn (t) = n sin 2t ⎪ ⎩n

(t ∈ 2πZ) (t ∈ 2πZ)

(the so-called Fej´er kernel). Here, we used that cos a = 1 − 2(sin a2 )2 (a ∈ R). d) We have to show that σn (x) converges π uniformly to f (x). If f ≡ 1, then 1 clearly σn (x) = 1. It follows from c) that 2π F (t) dt = 1 (n ∈ N). Moreover, −π n Fn (t) ≥ 0 (t ∈ (−π, π)). Since Fn (t) = Fn (−t), we have π 1 σn (x) = f (x + t)Fn (t) dt 2π −π π 1 f (x + t) + f (x − t) = Fn (t) dt. 2π −π 2 π 1 Thus, σn (x)−f (x) = 2π F (t)g(t) dt, where g(t) := 12 (f (x+t)+f (x−t))−f (x). −π n Let ε > 0. Since f is uniformly continuous, there exists δ > 0 such that g(t) ≤ ε whenever |t| ≤ δ. Thus,

δ

δ

Fn (t)g(t) dt ≤ Fn (t) g(t) dt

−δ

−δ δ ≤ ε Fn (t) dt ≤ ε · 2π −δ

4.2. REAL TAUBERIAN THEOREMS and

δ

π

259

1 1 π −2 Fn (t)g(t) dt

≤ g ∞ (π − δ) n (sin δ/2)2 < g ∞ n (sin δ/2) .

−δ Similarly, −π Fn (t)g(t) dt ≤ g ∞ πn (sin δ/2)−2 . Hence, σn (x) − f (x) ≤ ε + n−1 g ∞ (sin δ/2)−2 . Consequently, lim supn→∞ σn − f ≤ ε. Now we deduce the following from the Tauberian theorem Corollary 4.2.18. Theorem 4.2.20. Assume that f : R → Y is a continuous 2π-periodic function which is of bounded semivariation on [−π, π]. Then limn→∞ sn (x) = f (x) uniformly in x ∈ R. π −ikx 1 Proof. Let k ∈ Z, k = 0. Then ck = 2πik e df (x) by integration by parts. −π Hence, there exists a constant c ≥ 0 such that |k| ck ≤ c (k ∈ Z). Consequently, m sm −sm−1 ∞ ≤ m( cm + c−m ) ≤ 2c. Now the claim follows from Theorem 4.2.9 and Corollary 4.2.18 by choosing as X the Banach space of all continuous 2π-periodic functions on R with values in Y with the uniform norm · ∞ . H. Inversion of Laplace transforms. Now we give a result for Laplace transforms which corresponds to Theorem 4.2.19 and which leads to a proof of the Complex Inversion Theorem 2.3.4. Here X is any Banach space. Theorem 4.2.21. Let f ∈ L1loc (R+ , X) such that abs( f ) < ∞. a) If ω > abs( f ), then

a R ω+ir

1

1 λt ˆ

lim e f (λ) dλ − f (t)

dt = 0 R→∞ 0 R 0 2πi ω−ir for all a > 0. b) If f is continuous and exponentially bounded, f (0) = 0, and ω > ω(f ), then 1 r→∞ 2πi

ω+ir

C- lim

eλt fˆ(λ) dλ = f (t)

ω−ir

uniformly for t ∈ [0, a], for all a ≥ 0. 2 sin(t/2) 1 Proof. Let Φ(t) := 2π (t ∈ R). Then Φ ∈ L1 (R), and the inverse Fourier t/2 transform of Φ is given by 1 (1 − |s|) (|s| ≤ 1), −1 (F Φ)(s) = 2π 0 (|s| > 1).

260

4. ASYMPTOTICS OF LAPLACE TRANSFORMS

Let ΦR (t) := RΦ(Rt), so (F −1 ΦR )(s) = (F −1 Φ)(s/R), and {ΦR } acts an approximate unit on L1 (R, X) and C0 (R, X) as R → ∞ (see Lemma 1.3.3). Consider f as a function on R with f (s) = 0 (s < 0). Let ω > abs( f ), t ≥ 0. Put g(s) := e−ω(s+t) f (s + t) (s ∈ R). Then g ∈ L1 (R, X) and

(F g)(s) = eist fˆ(ω + is).

By Fubini’s theorem and Theorem 1.8.1 b), ω+ir 1 R 1 eλt fˆ(λ) dλ dr R 0 2πi ω−ir 1 ωt 1 R R ist ˆ = e e f (ω + is) dr ds 2π R −R |s| R 1 |s| ωt = e 1− (F g)(s) ds R −R 2π ∞ = eωt ΦR (s)g(s) ds −∞

eωt (ΦR ∗ fω )(t),

=

where fω (s) := e−ωs f (s) (note that ΦR (s) = ΦR (−s)). Now a) and b) follow, since

ΦR ∗fω −fω 1 → 0 if ω > abs( f ) and ΦR ∗fω −fω ∞ → 0 if fω ∈ C0 (R, X). Now we are able to give the following proof of Theorem 2.3.4. Proof of Theorem 2.3.4. Let F ∈ Lip0 (R+ , X). For Re λ = ω > 0, we have Fˆ (λ) = (λ)/λ and dF ∞

dF (λ) ≤ e−ωt F Lip0 (R+ ,X) dt = F Lip0 (R+ ,X) /ω. 0

For 0 ≤ t ≤ a and S > R > 0,

ω+iS

1 ω+iR

1

λt ˆ λt ˆ e F (λ) dλ − e F (λ) dλ

2πi ω−iR

2πi ω−iS eωa S F Lip0 (R+ ,X) ≤ ds π R ω|ω + is| eωa ≤

F Lip0 (R+ ,X) log(S/R). πω ω+iR λ· 1 Hence, R → 2πi e Fˆ (λ) dλ is feebly oscillating as a function from R+ to ω−iR C([0, a], X). It follows from Theorem 4.2.21 and Theorem 4.2.5 that ω+iR 1 eλt Fˆ (λ) dλ → F (t) 2πi ω−iR uniformly for t ∈ [0, a].

4.3. ERGODIC SEMIGROUPS

4.3

261

Ergodic Semigroups

This section interrupts the general theme of this chapter: we consider convergence in mean of semigroups. This is an interesting illustration of some of the results in Sections 4.1 and 4.2. Moreover, the notions introduced here will be useful in the context of almost periodic functions which form the subject of Section 4.5. We also prove a striking result due to Lotz: every C0 -semigroup on an L∞ -space has a bounded generator (Corollary 4.3.19). Let A be an operator on a Banach space X such that (0, λ0 ) ⊂ ρ(A) for some λ0 > 0 and M := sup λR(λ, A) < ∞. (4.5) 0<λ<λ0

Note that (4.5) ∞ is satisﬁed if A generates a bounded C0 -semigroup T . In fact, then R(λ, A) = 0 e−λt T (t) dt (λ > 0) and (4.5) holds for M = supt≥0 T (t) . Denote by Ker A := {x ∈ D(A) : Ax = 0} the kernel of A and by Ran A := {Ax : x ∈ D(A)} the range of A. Let x ∈ X. Since AR(λ, A)x = λR(λ, A)x − x

(0 < λ < λ0 ),

it follows from (4.5) that x ∈ Ran A if and only if lim λR(λ, A)x = 0.

(4.6)

λR(λ, A)x = x (0 < λ < λ0 ) if and only if x ∈ Ker A.

(4.7)

λ↓0

Moreover,

In particular, Ker A ∩ Ran A = {0}.

(4.8)

In what follows, as elsewhere, limits in Banach spaces are norm-limits unless speciﬁed otherwise. Proposition 4.3.1. Let A be an operator satisfying (4.5) and let x ∈ X. a) The following assertions are equivalent: (i) There exist λn ↓ 0 such that λn R(λn , A)x converges weakly as n → ∞. (ii) x0 := limλ↓0 λR(λ, A)x exists. (iii) x ∈ Ker A + Ran A. In that case, x0 ∈ Ker A and x − x0 ∈ Ran A. b) If A generates a bounded C0 -semigroup, then (i)–(iii) are equivalent to t (iv) x1 := limt→∞ 1t 0 T (s)x ds exists.

262

4. ASYMPTOTICS OF LAPLACE TRANSFORMS In that case, x1 = x0 .

Proof. a) (i) ⇒ (iii): Assume that λn R(λn , A)x converges weakly to y as n → ∞. By the resolvent equation, we have μR(μ, A)λn R(λn , A)x =

μ μλn (λn R(λn , A)x) − R(μ, A)x. μ − λn μ − λn

for all μ ∈ (0, λ0 ). Taking weak limits gives μR(μ, A)y = y. It follows from (4.7) that y ∈ Ker A. Since λn R(λn , A)x − x = AR(λn , A)x ∈ Ran A, it follows that y − x is in the weak closure of Ran A, which coincides with the norm closure. (iii) ⇒ (ii): This follows from (4.6) and (4.7). (ii) ⇒ (i): This is trivial. b) Since the Laplace transform of u(t) := T (t)x is given by u ˆ(λ) = R(λ, A)x, it follows from the Abelian Theorem 4.1.2 that (iv) implies (ii), and from the Tauberian Theorem 4.2.7 that (ii) implies (iv). Corollary 4.3.2. Let A be an operator satisfying (4.5). a) The following assertions are equivalent: (i) P x := limλ↓0 λR(λ, A)x exists for all x ∈ X. (ii) X = Ker A ⊕ Ran A. In that case, P is the projection onto Ker A along Ran A. b) If D(A) is dense, then (i) and (ii) are equivalent to (iii) Ker A separates Ker A∗ . Note that (iii) means, by deﬁnition, that for all x∗ ∈ Ker A∗ such that x∗ = 0 there exists x ∈ Ker A such that x, x∗ = 0. Proof. a) follows directly from Proposition 4.3.1 and (4.8). b) Assume that D(A) is dense. (ii) ⇒ (iii): Let x∗ ∈ Ker A∗ , x∗ = 0. Then Ax, x∗ = 0 for all x ∈ D(A). Hence y, x∗ = 0 for all y ∈ Ran A. Since X = Ker A ⊕ Ran A, it follows that x, x∗ = 0 for some x ∈ Ker A. (iii) ⇒ (i): We ﬁrst show that Ker A + Ran A is dense in X. In fact, otherwise there exists x∗ ∈ X ∗ \ {0} vanishing on Ker A + Ran A. Since x∗ vanishes on Ran A, one has x∗ ∈ Ker A∗ . Thus, condition (iii) is violated, and the claim is proved. It follows from (4.6) and (4.7) that (λR(λ, A)x) converges as λ ↓ 0 for x ∈ Ker A + Ran A. This implies convergence for all x ∈ X by density. Deﬁnition 4.3.3. A C0 -semigroup T on X is called Ces` aro-ergodic (or mean-ergodic) if 1 t Qx := lim T (s)x ds (4.9) t→∞ t 0

4.3. ERGODIC SEMIGROUPS

263

exists for all x ∈ X. The semigroup is called Abel-ergodic if its generator A satisﬁes (4.5) and the equivalent conditions (i), (ii), (iii) of Corollary 4.3.2 are satisﬁed. Proposition 4.3.4. Let T be a C0 -semigroup with generator A. a) If T is Ces` aro-ergodic, then T is Abel-ergodic and Q given by (4.9) is the projection onto Ker A along Ran A. b) Assume that T is bounded and Abel-ergodic. Then T is Ces` aro-ergodic. Proof. a) Assume that T is Ces` aro-ergodic. It follows from

t the uniform

boundedness principle that there exists M ≥ 0 such that 1t 0 T (s)x ds ≤ M x

for all t > 0, x ∈ X. It follows from Theorem 3.1.7 that (0, ∞) ⊂ ρ(A) and ∞ R(λ, A)x = 0 e−λt T (t)x dt for all λ > 0, x ∈ X. Integration by parts yields

∞

t

2 −λt

λR(λ, A)x = λ e T (s)x ds dt

0 0 ∞ ≤ M λ2 e−λt t dt x

0

= M x (λ > 0). Thus, condition (4.5) is satisﬁed. Since u(t) := T (t)x has Laplace tranform u ˆ(λ) = R(λ, A)x, the claim now follows from Theorem 4.1.2 and Proposition 4.3.1 a). b) If T is bounded by M , then condition (4.5) is satisﬁed. So the claim follows from Proposition 4.3.1. Corollary 4.3.5. Every bounded C0 -semigroup on a reﬂexive Banach space X is Ces` aro-ergodic. Proof. Let x ∈ X. It follows from reﬂexivity that condition (i) of Proposition 4.3.1 is satisﬁed. Thus, T is Abel-ergodic and the claim follows from Proposition 4.3.4. Condition (iii) of Corollary 4.3.2 is frequently the most convenient way to verify Ces`aro-ergodicity of a bounded C0 -semigroup. Note that the dual assertion is always true: the space Ker A∗ separates Ker A whenever A generates a bounded C0 -semigroup. More generally, the following holds. Proposition 4.3.6. Let A be a densely deﬁned operator satisfying (4.5). Let 0 = x ∈ Ker A. Then there exists x∗ ∈ Ker A∗ such that x, x∗ = 0. Proof. Let y ∗ ∈ X ∗ such that x, y ∗ = 1. Let x∗ be a weak* limit point of λR(λ, A)∗ y ∗ as λ ↓ 0. Since λR(λ, A)x = x, it follows that x, x∗ = 1. Let y ∈ D(A). Then lim λR(λ, A)Ay = lim λ(λR(λ, A)y − y) = 0. λ↓0

λ↓0

264

4. ASYMPTOTICS OF LAPLACE TRANSFORMS

Hence,

Ay, x∗ = limλR(λ, A)Ay, y ∗ = 0. λ↓0

∗

∗

Thus, x ∈ Ker A . We have shown that for bounded C0 -semigroups Abel- and Ces`aro-ergodicity are equivalent. This is not always the case for arbitrary C0 -semigroups as the 1 t 2 it following easy example shows: Let X := C , T (t) := e . Then T is 0 1 Abel-ergodic but not Ces`aro-ergodic. However, by Karamata’s theorem (Theorem 4.2.8) both notions do coincide for positive semigroups. Theorem 4.3.7. Let T be a positive C0 -semigroup on an ordered Banach space with normal cone. Then T is Abel-ergodic if and only if T is Ces` aro-ergodic. Next, we extend the results on Ces`aro-ergodicity to Lipschitz continuous integrated semigroups. Let A be an operator on X satisfying the Hille-Yosida condition (0, ∞) ⊂ ρ(A) and (λR(λ, A))n ≤ M

(4.10)

for all λ > 0, n ∈ N0 and some M ≥ 0. Then A generates a Lipschitz continuous once integrated semigroup S. More precisely, there exists S : R+ → L(X) satisfying S(0) = 0, S(t) − S(s) ≤ M |t − s| (s, t > 0)

and R(λ, A) = λ

∞

e−λt S(t) dt

(λ > 0);

0

see Theorem 3.3.1 and Section 3.5. If D(A) is dense, then A generates a bounded C0 -semigroup T and 1t S(t) = t 1 aro t 0 T (s) ds. If D(A) is not dense, then T does not exist on X, but the Ces` means 1t S(t) still make sense. Thus, the following theorem generalizes Proposition 4.3.4, but is more complicated to prove in this more general context. Theorem 4.3.8. Let A be the generator of a Lipschitz continuous once integrated semigroup S. The following conditions are equivalent: (i) X = Ker A ⊕ Ran A. (ii) P := limt→∞ 1t S(t) exists in the strong operator topology. In that case, P is the projection onto Ker A along Ran A. Proof. (i) ⇒ (ii): If x ∈ Ker A, then S(t)x = tx (t > 0). Next, let x = Ay ∈ Ran A. We show that 1 lim S(t)x = 0. t→∞ t

4.3. ERGODIC SEMIGROUPS

265

Note that 1t S(t) ≤ M . Since by Lemma 3.2.2,

t

S(t)y = ty +

S(s)Ay ds, 0

one has 1 lim t→∞ t2

t

S(s)Ay ds = 0. 0

By the Abelian Theorem 4.1.2, it follows that 1 t→∞ t

t

lim

0

1 s2

s

1 t→∞ t2

S(r)Ay dr ds = C- lim 0

t

S(s)Ay ds = 0. 0

Integration by parts yields 1 S(t)Ay t→∞ t

C- lim

1 t1 S(s)Ay ds t→∞ t 0 s s 1 t1 d = lim S(r)Ay dr ds t→∞ t 0 s ds 0 t 1 1 t 1 s = lim S(r)Ay dr + S(r)Ay dr ds t→∞ t2 0 t 0 s2 0 = 0. =

lim

Now observe that

1

S(t)x − 1 S(s)x

t

s

1

1 1

≤ (S(t)x − S(s)x) + − S(s)x

t t s 1 ≤ |t − s|2M x . t

Thus, the function 1t S(t)Ay is feebly oscillating. It follows from the Tauberian Theorem 4.2.5 that lim

t→∞

1 1 S(t)Ay = C- lim S(t)Ay = 0. t→∞ t t

We have shown that 1t S(t)x converges to P x as t → ∞ for x ∈ Ker A + Ran A, where P is the projection onto Ker A along Ran A. Since 1t S(t) ≤ M , it follows that limt→∞ 1t S(t)x = P x for all x ∈ X. (ii) ⇒ (i): This is an Abelian theorem whose proof is analogous to Theorem 4.1.2 c): Let x ∈ X. By Corollary 4.3.2, we have to show that lim λR(λ, A)x = lim λ2 λ↓0

λ↓0

∞ 0

e−λt S(t)x dt = P x.

266

4. ASYMPTOTICS OF LAPLACE TRANSFORMS

Let ε > 0. There exists t0 such that 1t S(t)x − P x ≤ ε for all t ≥ t0 . Hence lim sup λR(λ, A)x − P x

λ↓0

∞

1 2 −λt

= lim sup λ e t S(t)x − P x dt

t λ↓0 0

t0

1

≤ lim sup λ2 e−λt t

t S(t)x − P x dt + ε λ↓0 0 = ε.

Example 4.3.9. Consider the operator A on X := C[0, 1] given by ' ( D(A) := f ∈ C 2 [0, 1] : f (0) = f (1), f (0) = 0 , Af := f . It is not diﬃcult to see that A is dissipative and (I − A)D(A) = X. By Corollary 3.4.6, A generates a once integrated semigroup S satisfying

S(t) − S(s) ≤ |t − s|. Let f ∈ X. Then the constant function f (0) ∈ Ker A and f − f (0) = Ag ∈ Ran A, where x t 1 t (f (s) − f (0)) ds dt − x (f (t) − f (0)) ds dt. g(x) := 0

0

0

0

It follows from Theorem 4.3.8 that lim

t→∞

1 (S(t)f )(x) = f (0) t

uniformly for all x ∈ [0, 1], for all f ∈ X. Next, we ﬁx a bounded C0 -semigroup T with generator A. Recall that the operator A − iη generates the C0 -semigroup (e−iηt T (t))t≥0 , and we may consider ergodicity for the rescaled semigroup. Deﬁnition 4.3.10. a) Let η ∈ R. A vector x ∈ X is called ergodic at η (with respect to T ) if the mean 1 t −iηs Mη x := lim e T (s)x ds t→∞ t 0 converges in norm. b) A vector x is called totally ergodic (with respect to T ) if x is ergodic at η for all η ∈ R. c) The semigroup T is called totally ergodic if each vector x is totally ergodic; i.e., if the semigroup (e−iηt T (t))t≥0 is Ces` aro-ergodic for all η ∈ R.

4.3. ERGODIC SEMIGROUPS

267

By Xe we denote the space of all totally ergodic vectors in X and by Xe0 the space of all vectors x ∈ Xe such that Mη x = 0 for all η ∈ R. Both spaces Xe and Xe0 are closed and invariant under the semigroup. It follows from Proposition 4.3.1 that a vector x ∈ X is totally ergodic if and only if xη := lim αR(α + iη, A)x α↓0

exists for all η ∈ R. In that case, xη = Mη x. Moreover, Mη x ∈ Ker (A − iη) and x − Mη x ∈ Ran(A − iη). In particular, T (t)Mη x = eiηt Mη x (t ≥ 0), 1 R(λ, A)Mη x = Mη x (λ ∈ ρ(A), λ = iη). λ − iη If x ∈ X is totally ergodic we denote by Freq(x) := {η ∈ R : Mη x = 0} the set of all frequencies of X. Proposition 4.3.11. Let x be totally ergodic (with respect to T ). Then Freq(x) is countable. Proof. We can assume that X is separable, as we may replace X by span{T (t)x : t ≥ 0}. Let M := supt≥0 T (t) . Then

αR(α + iη, A) ≤ M (α > 0, η ∈ R). t Let xη := Mη x = limt→∞ 1t 0 e−iηs T (s)x ds (η ∈ R). For η ∈ Freq(x), let yη := xη / xη . Then yη = 1 and αR(α + iη, A)yη = yη (α > 0) and for μ ∈ Freq(x), μ = η, αR(α + iη, A)yμ =

α yμ α + iη − iμ

(α > 0).

Let μ, η ∈ Freq(x) such that η > μ. Then for α > 0, M yη − yμ

Choosing α = η − μ, we obtain

≥

αR(α + iη, A)(yη − yμ )

α

= yη − yμ α + iη − iμ α ≥ 1− α + iη − iμ 1 . = 1− 1 + i(η − μ)/α

1 1 − i = 1 − √1 > 0. M yη − yμ ≥ 1 − =1− 1+i 2 2

268

4. ASYMPTOTICS OF LAPLACE TRANSFORMS

If Freq(x) is uncountable this contradicts the fact that X is separable. It follows from Corollary 4.3.5 that every bounded C0 -semigroup on a reﬂexive Banach space is totally ergodic. This can be generalized as follows. Proposition 4.3.12. Assume that the vector x ∈ X has relatively weakly compact orbit {T (t)x : t ≥ 0}. Then x is totally ergodic with respect to T . Proof. It follows from the assumption that {e−iηt T (t)x : t ≥ 0} is relatively weakly compact for all η ∈ R. By Krein’s theorem [Meg98, Theorem 2.8.14], the closed convex K := co{e−iηt T (t)x : t ≥ 0} is weakly compact. Since αR(α + ∞ hull −αt −iηt iη, A)x = 0 αe e T (t)x dt ∈ K for all α > 0, condition (i) of Proposition t 4.3.1 is satisﬁed and so 1t 0 e−iηt T (t)x dt converges as t → ∞, by Proposition 4.3.1. Next we consider positive C0 -semigroups on L1 -spaces. Proposition 4.3.13. Let X = L1 (Ω, μ) where (Ω, μ) is a σ-ﬁnite measure space. Let T be a bounded positive C0 -semigroup on X. If f ∈ X+ is ergodic at 0, then f is totally ergodic. Proof. Since αR(α, A)f converges as α ↓ 0, the set K := {αR(α, A)f : 0 < α ≤ 1} is relatively compact. It follows that the solid hull so(K) := {g ∈ X : |g| ≤ k for some k ∈ K} is relatively weakly compact (by [AB85, Theorem 13.8]). Let η ∈ R. Then ∞ −t(α+iη) |αR(α + iη, A)f | = αe T (t)f dt 0∞ ≤ αe−αt T (t)f dt = αR(α, A)f. 0

Thus, αR(α + iη, A)f ∈ so(K) for 0 < α ≤ 1. Consequently, condition (i) of Proposition 4.3.1 is satisﬁed. In particular, if T is a positive bounded C0 -semigroup on L1 (Ω, μ), then T is totally ergodic whenever T is Ces`aro-ergodic. Here is a criterion which is sometimes convenient for verifying total ergodicity. Proposition 4.3.14. Let X := L1 (Ω, μ), where (Ω, μ) is a σ-ﬁnite measure space. Let T be a positive bounded C0 -semigroup on X. Assume that there exists a function u ∈ X such that u > 0 μ-a.e. and T (t)u ≤ u for all t ≥ 0. Then T is totally ergodic. Proof. Suppose that f ∈ X and 0 ≤ f ≤ u. For α > 0, 0 ≤ αR(α, A)f ≤ αR(α, A)u ≤ u. Since {g ∈ X : 0 ≤ g ≤ u} is weakly compact (see [Sch74, Theorem II.5.10 and Proposition II.8.3]), it follows from Proposition 4.3.1 that limα↓0 αR(α, A)f exists.

4.3. ERGODIC SEMIGROUPS

269

Since {f ∈ X : 0 ≤ f ≤ u} is total in X, it follows that T is Abel-ergodic, and hence totally ergodic by Proposition 4.3.4 and Proposition 4.3.13. In Proposition 4.3.14, the condition of a subinvariant strictly positive function cannot be omitted. For example, the shift semigroup T on L1 (R+ ) given by (T (t)f )(x) = f (x + t) is not Ces`aro-ergodic since Ker A = {0}, but Ker A∗ = C · 1 (where 1 denotes the constant 1 function). Next, we return to general Banach spaces, but we consider the special case where the Abel means in Proposition 4.3.1 converge with respect to the operator norm. Let A be an operator on X. We say that 0 is a simple pole of the resolvent if {λ ∈ C : 0 < |λ| < ε} ⊂ ρ(A) for some ε > 0 and there exists 0 = P ∈ L(X) such that P R(λ, A) − (4.11) λ has a holomorphic extension to the disc B(0, ε) := {λ ∈ C : |λ| < ε}. For example, if A generates a bounded C0 -semigroup T , 0 ∈ σ(A) and A has compact resolvent, then 0 is a simple pole. Indeed, if P is the spectral projection of A associated with {0}, then Am P = 0 for some m ∈ N (see Proposition B.9 and the subsequent remarks). It follows that T (t)P =

m−1 n n=0

t An P n!

(t ≥ 0).

Since T is bounded, AP = 0. Hence, R(λ, A) = R(λ, AZ )(I − P ) + P/λ, where Z = (I − P )(X) and AZ is the part of A in Z. Since 0 ∈ σ(AZ ), the claim follows. Proposition 4.3.15. Let A be an operator on X. The following assertions are equivalent: (i) There exists λ0 > 0 such that (0, λ0 ) ⊂ ρ(A) and P := limλ↓0 λR(λ, A) converges in the operator norm. (ii) 0 is a simple pole of the resolvent, or 0 ∈ ρ(A). (iii) The range Ran A is closed, (0, λ0 ) ⊂ ρ(A) for some λ0 > 0 and λR(λ, A) converges in the strong operator topology as λ ↓ 0.

270

4. ASYMPTOTICS OF LAPLACE TRANSFORMS

Proof. (i) ⇒ (ii): It follows from Proposition 4.3.2 that X = Ker A⊕Ran A and that P is the projection onto Ker A along Y := Ran A. Since P commutes with R(λ, A), we may consider the part AY of A in Y . Then (0, λ0 ) ⊂ ρ(AY ) and R(λ, AY ) = R(λ, A)|Y for λ ∈ (0, λ0 ) (see Proposition B.8). Since P = 0 on Y , it follows that limλ↓0 λR(λ, AY ) = 0. So there exists λ > 0 such that λR(λ, AY ) ≤ 12 . Hence, dist(λ, σ(AY )) ≥ R(λ, AY ) −1 ≥ 2λ (see Corollary B.3). Thus 0 ∈ ρ(AY ), and so R(λ, A)|Y has a holomorphic extension to B(0, ε) for some ε > 0. Since R(λ, A) = P/λ + R(λ, AY )(I − P ) (λ ∈ ρ(AY ), λ = 0), this implies (ii). It also shows that Ran A = Ran(AY ) = Y is closed. (ii) ⇒ (i): It follows from (4.11) that limλ↓0 λR(λ, A) = P in L(X). (i) ⇒ (iii): Strong convergence is trivial from (i). It has been shown above that Ran A is closed. (iii) ⇒ (ii): By assumption, Y := Ran A is closed and X = Ker A ⊕ Y by Corollary 4.3.2. Thus, the part AY in Y is invertible. Let P be the projection onto Ker A along Y . Since R(λ, A) = P/λ + R(λ, AY )(I − P ) (λ ∈ ρ(AY ), λ = 0), the result follows. Next, we show a very peculiar phenomenon on the space X := L∞ (Ω, μ), where (Ω, μ) is a measure space. Such a space has two remarkable properties concerning convergence of sequences, namely xn → 0 weakly in X and x∗n → 0 weakly in X ∗ (DP ) implies xn , x∗n → 0 as n → ∞; and x∗n → 0 weak* in X ∗ implies (G) x∗n → 0 weakly in X ∗ as n → ∞. The ﬁrst property is called the Dunford-Pettis property. A space having the second property is called a Grothendieck space. It is obvious that the properties (DP) and (G) are inherited by complemented subspaces. We refer to [Sch74, Theorems II.9.7 and II.10.4] for a proof that L∞ (Ω, μ) has these properties. The key argument involving these two properties is expressed in the following lemma. Lemma 4.3.16. Let X be a Banach space such that (G) and (DP) are satisﬁed. Suppose that Tn ∈ L(X) (n ∈ N) such that lim Tn x = 0 for all x ∈ X

n→∞

and lim Tn∗ x∗ = 0 for all x∗ ∈ X ∗ .

n→∞

Then lim Tn2 = 0.

n→∞

4.3. ERGODIC SEMIGROUPS

271

Proof. Assume that Tn2 does not converge to 0. Then there exist ε > 0 and a subsequence such that Tn2k ≥ 2ε. By the Hahn-Banach theorem, we ﬁnd xk ∈ X, x∗k ∈ X ∗ such that xk = x∗k = 1 but Tn xk , T ∗ x∗ = T 2 xk , x∗ ≥ ε. (4.12) nk k nk k k Let yk := Tnk xk , yk∗ := Tn∗k x∗k . Then for x∗ ∈ X ∗ , |yk , x∗ | = |xk , Tn∗k x∗ | ≤ Tn∗k x∗ → 0

as k → ∞.

Thus, yk → 0 weakly. Let x ∈ X. Then |x, yk∗ | = |Tnk x, x∗k | ≤ Tnk x → 0

as k → ∞.

Thus, yk∗ → 0 weak*. It follows from (G) that yk∗ → 0 weakly. Now, (DP) implies that yk , yk∗ → 0 as k → ∞. This contradicts (4.12). The following surprising result holds in particular on a space X := L∞ (Ω, μ) for any measure space (Ω, μ). Theorem 4.3.17. Let X be a Banach space satisfying (DP) and (G). Let A be a densely deﬁned operator on X such that (0, λ0 ) ⊂ ρ(A) for some λ0 > 0. If λR(λ, A)x converges weakly as λ ↓ 0 for all x ∈ X, then λR(λ, A) converges in L(X) as λ ↓ 0. Proof. It follows from Corollary 4.3.2 that P x := limλ↓0 λR(λ, A)x converges in norm for all x ∈ X, and that X = Ker A ⊕ Ran A. Replacing A by its part in Ran A, we can assume that Ker A = 0 and so P = 0. Let x∗ ∈ X ∗ . Then for x ∈ X we have x, λR(λ, A)∗ x∗ = λR(λ, A)x, x∗ → 0

as λ ↓ 0.

Hence, λR(λ, A)∗ x∗ → 0 weak* for all x∗ ∈ X ∗ . It follows from (G) that λn R(λn , A)∗ x∗ → 0 weakly whenever λn ↓ 0, for all x∗ ∈ X ∗ . Now Corollary 4.3.2 implies that λR(λ, A)∗ x∗ → 0 as λ ↓ 0, for all x∗ ∈ X ∗ . We have shown that Tn := λn R(λn , A) satisﬁes the hypotheses of Lemma 4.3.16 whenever λn ↓ 0. It follows that (λR(λ, A))2 → 0 as λ ↓ 0. Since (λR(λ, A))2 = (I + AR(λ, A))2 , it follows that r(I + AR(λ, A)) ≤ (I + AR(λ, A))2 1/2 < 1 for λ > 0 small (where r(I + AR(λ, A)) denotes the spectral radius). This implies that AR(λ, A) is invertible. In particular, Ran A = X. Now the claim follows from Proposition 4.3.15. It is also interesting to consider convergence of λR(λ, A) for λ → ∞. Again, the following result holds in particular on a space X = L∞ (Ω, μ), where (Ω, μ) is a measure space.

272

4. ASYMPTOTICS OF LAPLACE TRANSFORMS

Theorem 4.3.18. Let X be a Banach space satisfying (DP) and (G). Let A be a densely deﬁned operator on X such that (λ0 , ∞) ⊂ ρ(A) and supλ>λ0 λR(λ, A) < ∞ for some λ0 . Then A is bounded. Proof. By Lemma 3.3.12, limn→∞ (nR(n, A)x − x) = 0 for all x ∈ X. Let x∗ ∈ X ∗ . Then it follows that x, nR(n, A)∗ x∗ − x∗ = nR(n, A)x − x, x∗ → 0

as n → ∞,

for all x ∈ X. Thus, (nR(n, A)∗ − I)x∗ → 0 weak* as n → ∞. It follows from (G) that (nR(n, A)∗ − I)x∗ → 0 weakly as n → ∞, for all x∗ ∈ X ∗ . But this implies that x∗ is in the weak closure of D(A∗ ) = R(λ, A∗ )X ∗ (λ ≥ λ0 ) for all x∗ ∈ X ∗ . Since the weak and norm closures coincide, A∗ is densely deﬁned. It follows as above that (nR(n, A)∗ − I)x∗ → 0 as n → ∞, for all x∗ ∈ X ∗ . Now Lemma 4.3.16 implies that (nR(n, A) − I)2 → 0 as n → ∞. In particular, r((nR(n, A) − I)) ≤ (nR(n, A) − I)2 1/2 < 1 for n suﬃciently large. This implies that nR(n, A) is invertible. Hence, D(A) = X, so A is bounded by the closed graph theorem. We deduce from Theorem 4.3.18 the following surprising and important result. Corollary 4.3.19 (Lotz). Let T be a C0 -semigroup on the Banach space L∞ (Ω, μ), where (Ω, μ) is a measure space. Then T has a bounded generator. In other words, if a semigroup T deﬁned on X := L∞ (Ω, μ) converges strongly to the identity as t ↓ 0, it converges already in the operator norm.

4.4 Complex Tauberian Theorems: the Contour Method ∞ Let f ∈ L∞ (R+ , X) and let fˆ(λ) = 0 e−λt f (t) dt (Re λ > 0) be its Laplace transform. We deﬁne the half-line spectrum sp(f ) of f by sp(f ) := η ∈ R : fˆ does not have a holomorphic extension to an open neighbourhood of iη in C .

It turns out that countability of the spectrum with certain growth conditions is a Tauberian hypothesis. Here we prove a Tauberian theorem of type D by completely elementary contour arguments. For simplicity, we consider ﬁrst the simplest case where the spectrum is empty, giving a qualitative result (Theorem 4.4.1) and then a quantiﬁed version (Theorem 4.4.6). Then we adapt the argument to the case when the spectrum consists of one point (Theorem 4.4.8). This suﬃces for many interesting applications (see Corollaries 4.4.12 and 4.4.13 and Theorems 4.4.14 and 4.4.16) but we shall prove some more general results in Sections 4.7 and 4.9 by other means. In Theorem 4.4.18 we give a related result under diﬀerent assumptions on f and fˆ, and using a diﬀerent contour method.

4.4. THE CONTOUR METHOD

273

Theorem 4.4.1. Let f ∈ L∞ (R+ , X), and assume that sp(f ) is empty. Then t lim f (s) ds = fˆ(0). t→∞

0

Here, fˆ denotes the holomorphic extension of fˆ to a neighbourhood of 0. As usual, we will denote the open right half-plane {λ ∈ C : Re λ > 0} by C+ . Proof. We can assume that fˆ(0) = 0. Otherwise, we replace f by f − χ(0,1) fˆ(0). There is a simply connected open set Ω containing C+ such that fˆ has a holomorphic extension (also denoted by fˆ) to Ω. t Let gt (λ) := 0 e−λs f (s) ds (λ ∈ C, t ≥ 0). Let R > 1, and γ be a contour consisting of the semi-circle {λ ∈ C : |λ| = R, Re λ > 0} and a path γ connecting iR with −iR and lying entirely in Ω ∩ {Re λ < 0} (except at the endpoints). Let λ2 . R2 Since h(0) = 1, it follows from Cauchy’s theorem that 1 dλ eλt fˆ(λ)h(λ) ; 0 = −fˆ(0) = − 2πi γ λ t 1 dλ f (s) ds = gt (0) = eλt gt (λ)h(λ) . 2πi λ 0 |λ|=R h(λ) := 1 +

Adding up, we have t f (s) ds 0

=

1 2πi

|λ|=R Re λ>0

dλ eλt (gt (λ) − fˆ(λ))h(λ) λ

1 dλ eλt fˆ(λ)h(λ) 2πi γ λ 1 dλ + eλt gt (λ)h(λ) |λ|=R 2πi Re λ<0 λ −

=: I1 (t) + I2 (t) + I3 (t).

(4.13)

We estimate these three integrals separately. I1 (t): Let λ := Reiθ , θ ∈ (− π2 , π2 ). Then

∞

λ(t−s)

eλt (gt (λ) − fˆ(λ)) = e f (s) ds

t

∞

−λs

= e f (s + t) ds

0

h(λ) λ

≤

f ∞

f ∞ = ; Re λ R cos θ

≤

|1 + e2iθ |

1 2 cos θ = . R R

274

4. ASYMPTOTICS OF LAPLACE TRANSFORMS

Thus,

1 f ∞ 2

f ∞ πR = (t ≥ 0). 2π R R R I2 (t): Note that limt→∞ I2 (t) = 0 by the dominated convergence theorem. I3 (t): Let λ := Reiθ , θ ∈ ( π2 , 3π ). Then 2

I1 (t) ≤

t

λ(t−s)

e gt (λ) = e f (s) ds

λt

0

h(λ) λ

Hence,

I3 (t) ≤

t

≤

f ∞

≤

2| cos θ| . R

0

eRs cos θ ds ≤

f ∞ ; R | cos θ|

1 f ∞ 2

f ∞ πR = . 2π R R R

These estimates give

t

2 f ∞

lim sup f (s) ds

≤ R . t→∞ 0

(4.14)

Letting R → ∞ proves the claim. From the proof above we deduce the following. Proposition 4.4.2. Let f ∈ L∞ (R+ , X). If R > 0 such that sp(f ) ∩ [−R, R] = ∅, then

t

2 ˆ

lim sup f (s) ds − f (0) ≤ lim sup f (t) .

R t→∞ t→∞ 0 Proof. Let c > lim supt→∞ f (t) . Choose τ ≥ 0 such that f (t+ τ ) ≤ c for all τ t ≥ 0 and apply the estimate (4.14) to f (· + τ ) − χ(0,1) fˆ(0) − 0 f (s) ds . The next two corollaries follow from Proposition 4.4.2 by replacing f (t) by e−iηt f (t). Corollary 4.4.3. Let f ∈ L∞ (R+ , X). If η ∈ sp(f ), then

t

−iηs

sup e f (s) ds

< ∞. t≥0

0

Corollary 4.4.4. Let f ∈ L∞ (R+ , X) such that limt→∞ f (t) = 0. Then

t

fˆ(iη) = lim

t→∞

for all η ∈ R\sp(f ).

0

e−iηs f (s) ds

4.4. THE CONTOUR METHOD

275

Next we give a quantiﬁed version of Theorem 4.4.1. When sp(f ) is empty, there is a continuous increasing function M : R+ → (0, ∞) such that fˆ has a holomorphic extension (also denoted by fˆ) to 1 ΩM := λ ∈ C : Re λ > − M (| Im λ|) and fˆ(λ) ≤ M (| Im λ|) for all λ ∈ ΩM . Deﬁne Mlog (s) = M (s) [log(1 + M (s)) + log(1 + s)] −1 for s ∈ R+ . The inverse Mlog of Mlog is an increasing function of (Mlog (0), ∞) onto (0, ∞). Although we do not require M to be strictly increasing, we shall let M −1 denote any increasing function from the range of M to R+ such that M (M −1 (t)) = t for all t in the range of M . −1 Example 4.4.5. a) If M is bounded, then Mlog (t) ∼ Cet as t → ∞, for some constant C. b) If M (s) = β(1 + s)α where α, β > 0, then 1/α t −1 Mlog (t) ∼ Cα,β as t → ∞, log t

for some constant Cα,β . −1 c) If M (s) = βeαs where α, β > 0, then Mlog (t) ∼

1 α

log t as t → ∞.

∞

Theorem 4.4.6. Let f ∈ L (R+ , X), and assume that sp(f ) is empty. Let M and Mlog be as above, and c ∈ (0, 1). Then there exist constants C and t0 , depending only on f ∞ , M and c, such that

t

C ˆ

f (s) ds − f (0) (4.15)

≤ M −1 (ct) for all t ≥ t0 . 0 log −1 Remark 4.4.7. In (4.15), the function Mlog cannot be replaced by M −1 in general. For example, if X = C and f (t) = e−t , then the assumptions of Theorem 4.4.6 are satisﬁed for any continuous increasing function M with M (0) = 1. So M −1 may increase arbitrarily fast, but t f (s) ds − fˆ(0) = −e−t . 0

−1 On the other hand, M −1 and Mlog behave similarly if M increases rapidly and consistently (see Example 4.4.5). We shall see in Theorem 4.4.14 that, in the −1 context of C0 -semigroups, Mlog (ct) can never be replaced in (4.15) by a function −1 increasing faster than M (Ct) if M is chosen in the optimal way. Example 4.4.15 −1 will show that it may not be possible to replace Mlog by M −1 in (4.15) when M increases arbitrarily rapidly but inconsistently.

276

4. ASYMPTOTICS OF LAPLACE TRANSFORMS

Proof of Theorem 4.4.6. We use the same method and notation as in the proof of Theorem 4.4.1, taking Ω = ΩM and choosing γ and R as follows. Let a = c−1/2 > 1 and take γ to be the union of γ0 , γ+ and γ− , where γ0 (s) γ± (τ )

= −

1 aM (|s|)

= τ ± iR

(−R ≤ s ≤ R) , (−(aM (R))−1 ≤ τ ≤ 0) .

On γ± , |h(λ)| ≤ C/R, so we can estimate the norm of the integral I2 (t) over γ± by

(aM (R))−1

dλ 1 e−tτ CM (R)

λt ˆ e f (λ)h(λ) M (R) dτ ≤ .

≤C

γ± λ R R R2 t 0 Here and subsequently, C denotes a constant which may depend only on M , c and f ∞ , although it may vary from place to place. On γ0 , h(λ) is bounded independently of R, and |etλ | ≤ e−t/aM (R) . Hence, we can estimate the norm of the integral over γ0 by

dλ

eλt fˆ(λ)h(λ) dλ ≤ CM (R)e−t/aM (R) ≤ CM (R)(1+log R)e−t/aM (R) .

λ γ0 γ0 λ With these estimates for I2 (t) and those in the proof of Theorem 4.4.1 for I1 (t) and I3 (t), we obtain

t

2 f ∞ M (R) −t/aM (R)

f (s) ds ≤ +C + M (R)(1 + log R)e .

R R2 t 0 −1 Given t > c−1 Mlog (1), choose R = Mlog (ct) > 1. Then

M (R) c C = ≤ , 2 2 R t 2R log ((1 + M (R))(1 + R)) R (1 + M (R))a (1 + R)a C M (R)(1 + log R) ≤ C = et/aM(R) . R R Hence

t

C C

f (s) ds

≤ R = M −1 (ct) . 0 log

Now we consider the case when sp(f ) is a single point, stating only the qualitative result. Theorem 4.4.8. Let f ∈ L∞ (R+ , X), and assume that sp(f ) = {η} and

t

−iηs

sup e f (s) ds

< ∞, t≥0

where η ∈ R\{0}. Then limt→∞

0

t 0

f (s) ds = fˆ(0).

(4.16)

4.4. THE CONTOUR METHOD

277

Remark 4.4.9. The growth condition (4.16) in Theorem 4.4.8 cannot be omitted. For example, let f (t) := eit . Then fˆ(λ) = (λ − i)−1 , so sp(f ) = {1}. However, t f (s) ds = 1i (eit − 1) does not converge as t → ∞. 0 Proof of Theorem 4.4.8. The proof is similar to Theorem 4.4.1, with the following changes. a) We take R > |η| and 0 < ε < min{|η|, R −|η|}. The path γ now consists of two separate paths connecting iR with i(η + ε) and connecting i(η − ε) with −iR. It lies entirely in a simply connected domain where fˆ is deﬁned holomorphically, and also in {Re λ < 0} (except at the four endpoints). The contour γ includes also the semi-circle {λ ∈ C : |λ − iη| = ε, Re λ > 0}. b) Let ε2 η2 λ2 h(λ) := 1 + 1+ 2 . (λ − iη)2 (η 2 − ε2 ) R For λ = Reiθ , we have h(λ) λ

ε2 η2 1 1+ |1 + e2iθ | 2 2 2 (R − |η|) η −ε R 2 2 ε η 2| cos θ| = 1+ . (R − |η|)2 η 2 − ε2 R ≤

(4.17)

c) In (4.13) we have two additional integrals: 1 dλ I4 (t) := eλt (gt (λ) − fˆ(λ))h(λ) ; |λ−iη|=ε 2πi Re λ>0 λ 1 dλ 1 dλ I5 (t) := − eλt gt (λ)h(λ) = eλt gt (λ)h(λ) . |λ−iη|=ε 2πi |λ−iη|=ε λ 2πi λ Re λ>0 Re λ<0 We estimate these integrals as follows: t I4 (t): Let λ := iη + εeiθ , θ ∈ (− π2 , π2 ). Let F1 (t) := 0 e−iηs f (s) ds. Then by assumption (4.16), K := supt≥0 F1 (t) < ∞. We have

λt ∞ −λs

eλt (gt (λ) − fˆ(λ)) = e e f (s) ds

t

∞

λt

−εseiθ d

= e e F (s) ds 1

ds t

∞

λt −εteiθ

λt iθ −εseiθ

= −e e F1 (t) + e εe e F1 (s) ds

t

∞

iθ ≤ K +ε e−εe (s−t) F1 (s) ds

t K 1 ≤ K +ε =K 1+ ; ε cos θ cos θ

278

4. ASYMPTOTICS OF LAPLACE TRANSFORMS h(λ) λ

≤

|1 + e−2iθ |

η2 1 η2 1 2 = 2 cos θ 2 . η 2 − ε2 |η| − ε η 2 − ε2 |η| − ε

Consequently,

I4 (t) ≤

1 η2 1 η2 1 K2 · 2 2 2 επ = 4Kε . 2π η − ε2 |η| − ε η 2 − ε2 |η| − ε

I5 (t): Let λ := iη + εeiθ , θ ∈ ( π2 , 3π 2 ). Then

e gt (λ)

λt

t

λt −λs

= e e f (s) ds

0

t

λt −εseiθ d

= e e F1 (s) ds

ds 0

t

λt −εteiθ

iθ λt −εseiθ

= e e F1 (t) + εe e e F1 (s) ds

≤ K + εK ≤ K 1+

0

t

e−ε(s−t) cos θ ds 1 . | cos θ| 0

Thus, as for I4 (t) one obtains

I5 (t) ≤ 4Kε

η2

η2 1 . − ε2 |η| − ε

d) The estimates for I1 (t) and I3 (t) now include additional factors appearing in (4.17). The dominated convergence theorem shows that limt→∞ I2 (t) = 0. Thus, we obtain that

t

lim sup f (s) ds

t→∞ 0 2 f ∞ ε2 η2 η2 1 ≤ 1+ + 8Kε . R (R − |η|)2 η 2 − ε2 η 2 − ε2 |η| − ε Letting ε ↓ 0, we obtain

t

2 f ∞

lim sup f (s) ds

≤ R . t→∞ 0 Finally, letting R → ∞ proves the claim. We derive from Theorem 4.4.8 a Tauberian theorem of type E for slowly oscillating functions.

4.4. THE CONTOUR METHOD

279

Corollary 4.4.10. Let f ∈ L∞ (R+ , X) be slowly oscillating. Assume that a) sp(f ) = ∅, or b) sp(f ) = {η} and supt≥0

t 0

e−iηs f (s) ds < ∞, where η ∈ R.

Then limt→∞ f (t) = 0. Proof. We give the proof in the case when sp(f ) = {η}. a) Assume that η = 0. For δ > 0, consider fδ (t) := before. It follows from (4.1) that sp(fδ ) ⊂ {η}. Moreover, t

−iηs

e 0

fδ (s) ds = δ

δ+t

−1

e

iη(δ−s)

f (s) ds −

δ

1 (f (δ δ

+ t) − f (t)) as

t

e

−iηs

f (s) ds ,

0

t which is bounded. It follows from Theorem 4.4.8 that limt→∞ 0 fδ (s) ds = fδ (0) = δ − 1δ 0 f (s) ds (by (4.1)). Hence, B- limt→∞ f (t) = 0 by Lemma 4.2.12. Theorem 4.2.3 implies that limt→∞ f (t) = 0. b) Assume that η = 0. Let g(t) := eit f (t). g is bounded and slowly

Then

t −is

oscillating. Moreover, sp(g) ⊂ {i} and supt≥0 0 e g(s) ds < ∞. It follows from a) that limt→∞ g(t) = 0. Hence, limt→∞ f (t) = 0. Corollary 4.4.11. Let f : R+ → X be Lipschitz continuous such that

t

sup f (s) ds

< ∞. t≥0

(4.18)

0

Assume that sp(f ) ⊂ {0}. Then limt→∞ f (t) = 0. Note that (4.18) implies that C- limt→∞ f (t) = 0. Thus, Corollary 4.4.11 is a Tauberian theorem of type A. Proof of Corollary 4.4.11. We show that f is bounded. Then we can apply Corolt lary 4.4.10. Let F (t) := 0 f (s) ds. By assumption, M := supt≥0 F (t) < ∞. Moreover, there exists L ≥ 0 such that

f (t) − f (s) ≤ L|t − s| (t, s ≥ 0). Let x∗ ∈ X ∗ such that x∗ ≤ 1. Then x∗ ◦ f is diﬀerentiable a.e. and d ∗ (x ◦ f ) (t) ≤ L (t ≥ 0). dt The Taylor expansion for x∗ ◦ F yields for s ≥ 0, s+1 d F (s + 1), x∗ = F (s), x∗ + f (s), x∗ + (s + 1 − t) f (t), x∗ dt. dt s

280

4. ASYMPTOTICS OF LAPLACE TRANSFORMS

Hence, ∗

|f (s), x |

≤ 2M +

s+1

(s + 1 − t) · L dt

s

=

2M +

L . 2

The Hahn-Banach theorem implies that f (s) ≤ 2M + 12 L (s ≥ 0). Corollary 4.4.10 and 4.4.11 are ﬁrst versions of a complex Tauberian theorem where the main hypothesis says that the spectrum is suﬃciently small. Generalizations will be given in Section 4.7 (Corollary 4.7.8) and in Section 4.9 (Theorem 4.9.7). The proof of both generalizations will make use of Gelfand’s theorem which we obtain now as another immediate consequence of the contour method. In fact, applying Corollary 4.4.11 to bounded groups we obtain the following result. Corollary 4.4.12 (Gelfand). Let A be the generator of a bounded C0 -group U = (U (t))t∈R on X. If σ(A) ⊂ {0}, then U (t) = I for all t ∈ R. In particular, if σ(A) is empty, then X = {0}. t Proof. Let x ∈ D(A2 ), f (t) := AU (t)x. Then 0 f (s) ds = U (t)x − x, which d is bounded. Since fˆ(λ) = R(λ, A)Ax, we have sp(f ) ⊂ {0}. Since dt f (t) = 2 U (t)A x, the function f is Lipschitz continuous. It follows from Corollary 4.4.11 that limt→∞ f (t) = 0. Let M := supt∈R U (t) . Since Ax = U (−t)U (t)Ax ≤ M U (t)Ax , it follows that Ax = 0. Hence, U (t)x = x for all x ∈ D(A2 ), t ∈ R. Since R(1, A) has dense range D(A), the range D(A2 ) of R(1, A)2 is also dense. It follows that U (t) = I (t ∈ R). When σ(A) is empty, one can also apply the previous case to A − iη for any η ∈ R. Corollary 4.4.13. Let A be the generator of a bounded C0 -group U . Then each isolated point in σ(A) is an eigenvalue. Proof. Let iη ∈ σ(A) be isolated. Consider the spectral projection P associated with iη, and let Y := P X = {0} (see Proposition B.9). Then the group leaves the space Y invariant. Consider the restricted group. The spectrum of its generator is reduced to {iη}. Applying Corollary 4.4.12 to e−iηt U (t)|Y , one obtains that e−iηt U (t)y = y for all y ∈ Y . Next, we prove a result on the asymptotic behaviour of a bounded C0 semigroup T with generator A, speciﬁcally the possible decay of T (t)R(μ, A)

for large t, where μ ∈ ρ(A) is ﬁxed. Note that the rate of any such decay is independent of the choice of μ up to multiplicative constants, since (μ − A)R(μ, A) is always a bounded operator. Let m : R+ → (0, ∞) be a continuous decreasing function such that m(t) → 0 as t → ∞. We let m−1 denote any decreasing function from (0, ∞) to R+ such that m(m−1 (t)) = t for all t in the range of m.

4.4. THE CONTOUR METHOD

281

Theorem 4.4.14. Let A be the generator of a bounded C0 -semigroup T , and let μ ∈ ρ(A). The following are equivalent: (i) σ(A) ∩ iR is empty. (ii) limt→∞ T (t)R(μ, A) = 0. More precisely: a) Assume that σ(A) ∩ iR is empty, and let M : R+ → (0, ∞) be a continuous increasing function such that R(is, A) ≤ M (|s|) for all s ∈ R, and let c ∈ (0, 1). Then there exist C and t0 such that

T (t)R(μ, A) ≤

C −1 Mlog (ct)

for all t ≥ t0 .

(4.19)

b) Assume that limt→∞ T (t)R(μ, A) = 0, and let m : R+ → (0, ∞) be a continuous decreasing function such that T (t)R(μ, A) ≤ m(t) for t ≥ 0 and limt→∞ m(t) = 0. Then σ(A) ∩ iR is empty, and for each c ∈ (0, 1) there exist C and s0 such that c

R(is, A) ≤ Cm−1 whenever |s| ≥ s0 . |s| Proof. The implication from (1) to (2) can be seen from Theorem 4.4.1 with f (t) = T (t). Although that function is not measurable from R+ to L(X), its Laplace transform fˆ(λ) = R(λ, A) is holomorphic, and the proof of Theorem 4.4.1 remains valid. Alternatively, we give the following more precise argument for a). We let K = supt≥0 T (t) . a) First, by Corollary B.3, for any a > 1, ΩaM ⊂ ρ(A) and R(λ, A) ≤ a M (| Im λ|) for all λ ∈ ΩaM . a−1 Let x ∈ X with x ≤ (a − 1)/a, and let f (t) = T (t)x (t ≥ 0). Then

f ∞ ≤ K and fˆ(λ) = R(λ, A)x, so fˆ(λ) ≤ M (| Im λ|) for λ ∈ ΩaM . By Proposition 3.1.9 e), t T (t)R(μ, A)x = R(μ, A)x + AR(μ, A) f (s) ds 0 t = AR(μ, A) f (s) ds − fˆ(0) . 0

Since AR(μ, A) is a bounded operator, the result follows from Theorem 4.4.8 and its proof, noting that C and t0 are independent of x for x ≤ (a − 1)/a. b) Take s ∈ R, and consider x∗ ∈ D(A∗ ) with A∗ x∗ = isx∗ . Let x ∈ D(A), and put g(t) = e−ist T (t)x, x∗ (t ≥ 0). Then g (t) = e−ist AT (t)x − isT (t)x, x∗ = 0,

282

4. ASYMPTOTICS OF LAPLACE TRANSFORMS

so g is constant. By the assumption (ii), limt→∞ T (t)x = 0, so limt→∞ g(t) = 0. Hence 0 = g(0) = x, x∗ . Since D(A) is dense, it follows that x∗ = 0. By the Hahn-Banach theorem, A − is has dense range. Let c ∈ (c, 1), and x ∈ D(A). By Proposition 3.1.9 f) applied to the rescaled semigroup, t

0

e−isτ T (τ )(A − is)x dτ = e−ist T (t)x − x

for any t ≥ 0. Hence, Kt (A − is)x

t

−isτ

≥ e T (τ )(A − is)x dτ

0

≥ ≥ ≥

x − T (t)x

x − m(t) (A − μ)x

! "

x − m(t) (A − is)x + (|μ| + |s|) x .

Thus,

(A − is)x ≥

1 − m(t)(|μ| + |s|)

x . Kt + m(t)

Let t = m−1 (c /(|μ| + |s|)), so m(t) ≤ c /(|μ| + |s|). Thus,

(A − is)x ≥

(1 − c ) x

. Km−1 (c /(|μ| + |s|)) + c /(|μ| + |s|)

This implies that A − is has closed range, and hence it is invertible. Moreover, K c c c −1 −1

R(is, A) ≤ m + ≤ Cm 1 − c |μ| + |s| (1 − c )(|μ| + |s|) |s| whenever |s| is suﬃciently large. Part a) of Theorem 4.4.14 gives the sharpest result when M is chosen as small as possible, that is, M (s) = sup { R(is , A) : |s | ≤ s} . Then part b) provides the lower bound

T (t)R(μ, A) ≥

c M −1 (Ct)

for t suﬃciently large and some constants c > 0 and C, and the sharpest estimate is given by taking M −1 (t) = inf{|s| : R(is, A) ≥ t}. Then the gap between this lower bound and the upper bound in (4.19) is small if M grows rapidly. To understand this more explicitly, we consider diagonal semigroups on Hilbert space.

4.4. THE CONTOUR METHOD

283

Example 4.4.15. Let X = 2 , (αn )n∈N be a strictly decreasing sequence in (0, 1], and λn = −αn + in. A C0 -semigroup of contractions on 2 is deﬁned by ! " T (t)x = eλn t xn (x = (xn ) ∈ 2 ). The generator A is given by ' ( ' ( D(A) = x ∈ 2 : (λn xn ) ∈ 2 = x ∈ 2 : (nxn ) ∈ 2 , Ax = (λn xn ). Moreover, {λn : n ∈ N}, ' (

R(is, A) = sup |λn − is|−1 : n ∈ N . σ(A)

=

Take the optimal deﬁnitions of M and M −1 as above. Then M (k) = R(ik, A) = 1/αk (k ∈ N). Let tk = 1/αk . Then λ t −1 e n k e−1 e−1 −1 ≥ e

T (tk )A = sup ∼ = as k → ∞. λn |λk | k M −1 (tk ) n∈N When (αn ) decays regularly and not too slowly, we may have

T (t)A−1 ∼

c M −1 (t)

as t → ∞.

(4.20)

For example, when αn = n−γ where γ > 0,

T (t)A−1 ∼

cγ cγ ∼ −1 . M (t) t1/γ

On the other hand, (4.20) may fail if (αn ) has relatively long periods of slow decay. Suppose for example that (αn ) is strictly decreasing and α(k+1)! ≥

αk!+1 2

(k ≥ 1).

Given c > 0, let tk = 1/(cα(k+1)! ). Then M −1 (ctk ) = (k + 1)!, λ −2/c e k!+1 tk ke−2/c −1 ≥ e

T (tk )A ≥ ∼ . λk!+1 |λk!+1 | M −1 (ctk ) Thus there is no upper bound of the form T (t)A−1 ≤ C/M −1 (ct). By arranging that the subsequence (αk! ) decreases arbitrarily fast, this phenomenon can be found even when M increases arbitrarily fast.

284

4. ASYMPTOTICS OF LAPLACE TRANSFORMS

The next result on asymptotic behaviour of C0 -semigroups is analogous to the Katznelson-Tzafriri theorem for contractions (see the Notes of this chapter). A C0 -semigroup T = (T (t))t≥0 is called eventually diﬀerentiable if there exists τ ≥ 0 such that T (τ )X ⊂ D(A). Semigroup properties imply that T (t)X ⊂ D(A) for all t ≥ τ and T (t)X ⊂ D(A2 ) for all t ≥ 2τ . By the closed graph theorem, AT (t) ∈ L(X) for t ≥ τ and A2 T (t) ∈ L(X) for t ≥ 2τ . Theorem 4.4.16. Let A be the generator of an eventually diﬀerentiable, bounded C0 -semigroup. The following are equivalent: (i) σ(A) ∩ iR ⊂ {0}. (ii) limt→∞ AT (t) = 0. Proof. Let K := supt≥0 T (t) . Assume that (i) holds. Let f : R+ → L(X) be deﬁned by f (t) := AT (t + 2τ ). For x ∈ X,

s

d

f (t)x − f (s)x = (f (r)x) dr

t dr

s

2

= A T (r + 2τ )x dr

t

s

2

= T (r)A T (2τ )x dr

t

≤ K A2 T (2τ ) |t − s| x

(s, t ≥ 0).

Thus f is Lipschitz continuous. Moreover, fˆ(λ) = R(λ, A)AT (2τ ). Hence, we have t sp(f ) ⊂ {0}. Finally, 0 f (s) ds = T (t + 2τ ) − T (2τ ) ≤ 2K (t ≥ 0). It follows from Corollary 4.4.11 that limt→∞ f (t) = 0. Conversely, assume that (ii) holds. Let iη ∈ σ(A). By Proposition B.2 d), there exist xn ∈ D(A) with xn = 1 such that limn→∞ (A − iη)xn = 0. Then ! iηt " ! " iηe − AT (t) xn = iη eiηt − T (t) xn + T (t) (iη − A) xn t = iηeiηt e−iηs T (s)(iη − A)xn ds + T (t)(iη − A)xn → 0

0

as n → ∞.

Thus, iηeiηt ∈ σ(AT (t)). Hence, |η| = |iηeiηt | ≤ AT (t) for all t ≥ τ . Since limt→∞ AT (t) = 0, we conclude that η = 0. t Theorem 4.4.1 shows that limt→∞ 0 f (s) ds = fˆ(0) if f ∈ L∞ (R+ , X) and sp(f ) is empty. This is not true if the assumption that f is bounded is omitted (see Example 1.5.2). In the ﬁnal part of this section we shall show that it is true if f is exponentially bounded and fˆ has a bounded holomorphic extension to a half-plane {λ : Re λ > −ε} for some ε > 0. Indeed, we shall show that abs(f ) ≤ hol0 (fˆ)

4.4. THE CONTOUR METHOD

285

whenever f is exponentially bounded, where hol0 (f ) := inf{ω ∈ R : fˆ has a bounded holomorphic extension for Re λ > ω} is the abscissa of boundedness of fˆ. We begin with a general estimate. Proposition 4.4.17. Let f ∈ L1loc (R+ , X) be exponentially bounded and suppose that hol(fˆ) ≤ 0 and fˆ is bounded on C+ . Then there is a constant C such that

∞

φ(t)f (t) dt

≤ C F φ 1 0

for all functions φ ∈ L1 (R+ ) such that F φ ∈ L1 (R) and φf ∈ L1 (R+ , X). Proof. First, we assume that φ has compact support and Fφ ∈ L1 (R). The ˆ Laplace transform φˆ is deﬁned on C, and (F φ)(s) = φ(is) = (F φ)(−is). Take −ωt ω > max(0, ω(f )) and 0 < α < ω. The function t → e f (t) belongs to L1 (R+ , X) and its Fourier transform is s → fˆ(ω + is). Let ψ ∈ Cc∞ (R) with ψ(t) = e(ω−α)t whenever t ∈ supp φ. Then (F ψ) ∗ (Fφ) ∈ L1 (R). By the Fourier Inversion Theorem 1.8.1 d), F((Fψ) ∗ (Fφ))(t) = 4π 2 ψ(t)φ(t) = 4π 2 e(ω−α)t φ(t) for all t, and ((F ψ) ∗ (Fφ))(s) = 2π φ(α − ω − is) for all s. By Theorem 1.8.1 b), ∞ ∞ −αt e f (t)φ(t) dt = e−ωt f (t)e(ω−α)t φ(t) dt 0 0 ∞ 1 ˆ − ω − is) ds. = fˆ(ω + is)φ(α (4.21) 2π −∞ Now consider the contour integral ˆ − z) dz fˆ(z)φ(α around the rectangle with vertices α ± ir, ω ± ir, where r > 0. The integral along the bottom edge is ω ˆ − ξ + ir) dξ. fˆ(ξ − ir)φ(α α

For α < ξ < ω, ˆ − ξ + ir) = φ(α

∞

0

e−(α−ξ)t φ(t)e−irt dt → 0

as r → ∞, by the Riemann-Lebesgue lemma. Moreover, ∞ 1/2 1 ˆ 2ωt 2 √ φ(α − ξ + ir) ≤ e |φ(t)| dt , 2α 0

ˆ

f (ξ − ir) ≤ sup fˆ(z) , z∈C+

286

4. ASYMPTOTICS OF LAPLACE TRANSFORMS

whenever r > 0, α < ξ < ω. By the dominated convergence theorem, ω ˆ − ξ + ir) dξ = 0. lim fˆ(ξ − ir)φ(α r→∞

α

A similar argument shows that the integral along the top edge of the rectangle tends to 0 as r → ∞. By Cauchy’s theorem, r r ˆ ˆ ˆ ˆ lim f (ω + is)φ(α − ω − is) ds − f (α + is)φ(−is) ds = 0. r→∞

−r

−r

From (4.21),

∞

1 r→∞ 2π

e−αt f (t)φ(t) dt = lim

0

so

∞

0

r

fˆ(α + is)(F φ)(−s) ds,

−r

1

ˆ e−αt φ(t)f (t) dt ≤ sup f (z)

Fφ 1 .

2π z∈C +

Letting α ↓ 0 gives

0

∞

1 φ(t)f (t) dt sup fˆ(z) Fφ 1 .

≤ 2π z∈C +

Now, consider the case when φ is any function in L1 (R+ ) such that Fφ ∈ ∞ L (R) and 0 |φ(t)| f (t) dt < ∞. Let ψ ∈ Cc∞ (R) be any function satisfying ∞ 0 ≤ ψ ≤ 1, ψ(0) = 1 and −∞ ψ(t) dt = 1. Let ψn (t) = ψ(t/n) (t ∈ R) and φn (t) = φ(t)ψn (t). Then (2π)−1 (Fψn )(s) = (2π)−1 n(Fψ)(ns), which forms a molliﬁer. Hence, Fφn = (2π)−1 F φ ∗ Fψn → F φ in L1 (R) (see Lemma 1.3.3). Applying the previous result to the functions φn and taking the limit provides the result. 1

Next, we show that the antiderivative can grow at most linearly if fˆ is bounded on C+ and f is exponentially bounded. We shall apply this result to C0 -semigroups in Theorem 5.1.8. Theorem 4.4.18. Let f ∈ L1loc (R+ , X) be exponentially bounded and suppose that hol(fˆ) ≤ 0 and fˆ is bounded on C+ . Then there is a constant c such that

t

f (s) ds

≤ c(1 + t) 0

for all t ≥ 0.

Proof. Let C be as in Proposition 4.4.17 and let ω > max(0, ω(f )), so there exists M such that f (s) ≤ M eωs for all s ≥ 0. Take t > 0 and let α := 2t e−ωt and

4.4. THE CONTOUR METHOD 1 α χ(0,α)

φ :=

287

∗ χ(0,t) , so ⎧ s/α ⎪ ⎪ ⎪ ⎪ ⎨1 φ(s) = t + α − s ⎪ ⎪ ⎪ α ⎪ ⎩ 0

(0 ≤ s < α), (α ≤ s < t), (t ≤ s < t + α), (t + α ≤ s).

Then α ≤ 1/(2ωe) and 1 1 − e−iαs 1 − e−ist 4e−isα/2 e−ist/2 αs st (F φ)(s) = = sin sin . 2 α is is αs 2 2 Hence,

Fφ 1

= ≤ = ≤ = =

Also, t

so

∞

f (s) ds = 0

α

φ(s)f (s) ds + 0

t

f (s) ds

0

st 4 ∞ | sin αs 2 sin 2 | ds 2 α 0 s 1/α | sin st 4 ∞ ds 2| 2 ds + s α 1/α s2 0 t/2α | sin u| 2 du + 4 u 0 t/2α du +6 u 1 log (t/2α) + 6 ωt + 6.

0

≤

0

∞

s 1− f (s) ds − α

φ(s)f (s) ds

+

α

0

t+α t

t+α−s f (s) ds, α

t+α

f (s) ds +

f (s) ds

t

≤ C(ωt + 6) + M αeωα (1 + eωt ) t ≤ C(ωt + 6) + M e−ωt e1/2e 2eωt 2 =

6C + Cω + M e1/2e t.

Finally, we prove the result comparing abs(f ) and hol0 (fˆ). Theorem 4.4.19. Let f ∈ L1loc (R+ , X) be exponentially bounded. Then abs(f ) ≤ hol0 (fˆ).

288

4. ASYMPTOTICS OF LAPLACE TRANSFORMS

ˆ Proof. Let ω > hol0 (fˆ) and let fω (t) := e−ωt f (t). Then f ω (λ) = f (λ + ω), so hol(fω ) < 0 and fω is bounded on C+ . By Theorem 4.4.18 and Theorem 1.4.3, abs(fω ) ≤ 0. Hence, abs(f ) = abs(fω ) + ω ≤ ω whenever ω > hol0 (fˆ). Corollary 4.4.20. Let f ∈ L1loc (R+ , X) be exponentially bounded. Assume that fˆ has a bounded holomorphic extension to a half-plane {λ ∈ C : Re λ > −ε} for some ε > 0. Then t lim f (s) ds = fˆ(0). t→∞

4.5

0

Almost Periodic Functions

This section is divided into two parts: in the ﬁrst we describe relatively compact orbits of C0 -groups. The abstract results which we obtain are applied in the second part to characterize almost periodic functions on the real line. A subset Q of R is called relatively dense if there exists a length l > 0 such that [a, a + l] ∩ Q = ∅ for all a ∈ R. Let U = (U (t))t∈R be a C0 -group of isometries on a Banach space Y . Our aim is to prove the following. Theorem 4.5.1. Let y ∈ Y . The following assertions are equivalent: (i) The set {U (t)y : t ∈ R} is relatively compact in Y . (ii) The set {U (t)y : t ≥ 0} is relatively compact in Y . (iii) For all ε > 0, the set Qε := {τ ∈ R : U (τ )y − y ≤ ε} is relatively dense in R. ' ( (iv) y ∈ span x ∈ Y : there exists η ∈ R such that U (t)x = eiηt x for all t ∈ R . If these four equivalent conditions are satisﬁed we say that y has a relatively compact orbit. Since Y is complete, a subset K of Y is relatively compact if and only if K is precompact (i.e., for every ε > 0 it can be covered by a ﬁnite number of εballs). We shall use this equivalence frequently without comment, and we shall use the terminology “relatively compact” except in some proofs where precompactness is more relevant. The equivalence of the conditions (i), (ii) and (iii) will be proved by simple direct arguments. In order to show that they imply (iv) we need the following basic result of harmonic analysis (see [Rud62, Section 1.5.2]). Proposition 4.5.2. Let G be a compact abelian group with Haar measure dx. Let f : G → C be continuous such that f (x)γ(x) dx = 0 G

4.5. ALMOST PERIODIC FUNCTIONS

289

for every character γ (i.e., every continuous homomorphism γ : G → C \ {0}). Then f (x) = 0 for all x ∈ G. Proof of Theorem 4.5.1. (i) ⇒ (ii): This is trivial. (ii) ⇒ (iii). Let ε > 0. By assumption, there exist t1 , . . . , tm ≥ 0 such that for all t ≥ 0 there exists j ∈ {1, . . . , m} such that U (t)y − U (tj )y ≤ ε. Let l := maxj=1,...,m tj . We show that [t, t + l] ∩ Qε = ∅ for all t ∈ R. Let t ≥ 0. Choose j ∈ {1, . . . , m} such that U (t)y − U (tj )y ≤ ε. Let τ := t − tj . Then

U (τ )y − y = U (−tj )(U (t)y − U (tj )y) = U (t)y − U (tj )y ≤ ε. Thus, τ ∈ Qε ∩ [t − l, t]. Let t < 0. Then there exists j ∈ {1, . . . , m} such that U (−t)y−U (tj )y ≤ ε. Let τ := t + tj . Then U (τ )y − y = U (t)(U (tj )y − U (−t)y) ≤ ε. Thus, τ ∈ Qε and τ ∈ [t, t + l]. We have shown that [t, t + l] ∩ Qε = ∅ for all t ∈ R. (iii) ⇒ (i): Let ε > 0. By assumption, there exists l > 0 such that for all n ∈ Z, [ln, ln + l] ∩ Qε = ∅. Since {U (t)y : t ∈ [0, 2l]} is compact, there exist t1 , . . . , tm ∈ [0, 2l] such that for all s ∈ [0, 2l] there exists j ∈ {1, . . . , m} such that

U (s)y − U (tj )y ≤ ε. Let t ∈ R. Take n ∈ Z such that t ∈ [ln, ln + l] and choose τ ∈ Qε ∩ [−ln, −ln + l]. Then t + τ ∈ [0, 2l]. There exists j ∈ {1, . . . , m} such that

U (t + τ )y − U (tj )y ≤ ε. Thus,

U (t)y − U (tj )y ≤ U (t)y − U (t)U (τ )y + U (t + τ )y − U (tj )y ≤ 2ε. We have shown that the orbit {U (t)y : t ∈ R} is covered by balls B(U (tj )y, 2ε) = {z ∈ Y : U (tj )y − z ≤ 2ε} (j = 1, . . . , m). Thus, (i) holds. (i) ⇒ (iv): Replacing Y by span{U (t)y : t ∈ R}, we can assume that every orbit {U (t)x : t ∈ R} (x ∈ Y ) is relatively compact. Then U is bounded, by the uniform boundedness principle. Denote by G the closure of {U (t) : t ∈ R} in Ls (Y ), the space L(Y ) with respect to the strong operator topology. Then Gx ⊂ K(x) := {U (t)x : t ∈ R}− for all x ∈ Y . By hypothesis, K(x) is compact. We consider the strong operator topology on G, and all limits in this5proof will be in that topology. Then G can be identiﬁed with a closed subset of x∈Y K(x) via S ∈ G → {Sx : x ∈ Y }. Hence, G is compact by Tychonov’s theorem. Since multiplication is jointly continuous on bounded subsets of Ls (X), S, T ∈ G implies ST ∈ G. Let T ∈ G. There exists a net (ti )i∈I in R such that T = limi∈I U (ti ). Considering a subnet if necessary, we can assume that S := limi∈I U (−ti ) exists as well. Then T S = ST = I. We show that inversion is continuous. In fact, let limi∈I Ti = T in G. It follows from compactness that every subnet of (Ti )i∈I has a subnet (Tj )j∈J such that limj Tj−1 exists and joint continuity of multiplication implies that limj Tj−1 = T −1 . This implies that limi Ti−1 = T −1 . Thus, G is a compact abelian group with continuous multiplication. ˆ the dual group of G and by dS the Haar measure on G. Denote by G ˆ x ∈ Y deﬁne Pγ x := For γ ∈ G, γ(S)Sx dS ∈ Y . Then Pγ ∈ L(Y ) and G T Pγ x = G γ(S)T Sx dS = γ(T ) G γ(T S)T Sx dS = γ(T )Pγ x (T ∈ G). The mapˆ there ping φ(t) := γ(U (t)) is a continuous character on R. Hence, for each γ ∈ G,

290

4. ASYMPTOTICS OF LAPLACE TRANSFORMS

exists η ∈ R such that γ(U (t)) = eiηt (t ∈ R). Thus, F

:= ⊂

ˆ x∈Y} {Pγ x : γ ∈ G, ' ( z ∈ Y : there exists η ∈ R such that U (t)z = eiηt y for all t ∈ R .

∗ It remains to show that span F is dense in Y . Let ψ ∈ Y such that z, ψ = 0 for ˆ Since S → Sx, ψ all z ∈ F . Let x ∈ Y . Then G γ(S)Sx, ψ dS = 0 for all γ ∈ G. is a continuous mapping, it follows from Proposition 4.5.2 that Sx, ψ = 0 for all S ∈ G. In particular, x, ψ = 0. Thus, ψ = 0. It follows from the Hahn-Banach theorem that spanF = Y . (iv) ⇒ (i): It is obvious that the set

Y1 := {x ∈ Y : x has precompact orbit} is a closed subspace of Y . Since x ∈ Y1 whenever U (t)x = eiηt x for all t ∈ R, the implication follows. We recall the following facts from Section 4.3. An element x ∈ Y is called totally ergodic if 1 t −iηs Mη x := lim e U (s)x ds t→∞ t 0 converges for all η ∈ R. In that case, U (t)Mη x = eiηt Mη x (t ∈ R). Moreover, the set of all frequencies Freq(x) := {η ∈ R : Mη x = 0} is countable (by Proposition 4.3.11). It is obvious that the set Ye of all totally ergodic vectors is a closed subspace of Y . We introduce the space Yap of all almost periodic vectors (with respect to U ) deﬁned by Yap

:= {x ∈ Y : x has relatively compact orbit} = span{x ∈ Y : there exists η ∈ R such that U (t)x = eiηt x for all t ∈ R}.

Then Yap is a closed subspace of Y which is invariant under U . If x ∈ Y is a periodic vector, i.e., U (t)x = eiξt x (t ∈ R) for some ξ ∈ R, then Freq(x) ⊂ {ξ} and x Mη x = 0

if η = ξ, if η = ξ.

(4.22)

It follows that all almost periodic vectors are totally ergodic. The following approximation result shows that every x ∈ Yap can be approximated by linear combinations of eigenvectors associated with frequencies of x.

4.5. ALMOST PERIODIC FUNCTIONS

291

Proposition 4.5.3 (Spectral synthesis). Let x ∈ Yap . Then x ∈ span{y ∈ Y : there exists η ∈ Freq(x) such that U (t)y = eiηt y for all t ∈ R}. Proof. The space Z := {y ∈ Yap : Mη y = 0 for all η ∈ R\Freq(x)} is closed and invariant under the group. If η ∈ R\Freq(x) and y ∈ Z such that U (t)y = eiηt y (t ∈ R), then y = Mη y = 0. Now the claim follows from Theorem 4.5.1, applied to the restriction of U to Z. Corollary 4.5.4. Let x ∈ Yap . Then a) x = 0 if and only if Freq(x) = ∅. b) Freq(x) ⊂ {η} if and only if U (t)x = eiηt x (t ∈ R). m c) Freq(x) ⊂ {η1 , . . . , ηm } if and only if x = j=1 xj with U (t)xj = eiηj t xj for all t ∈ R. d) Let τ > 0. Then U (t+τ )x = U (t)x for all t ∈ R if and only if Freq(x) ⊂

2π Z. τ

Proof. a) and b) follow directly from Proposition 4.5.3.If Freq(x) ⊂ {η1 , . . . , ηm }, m then by Proposition 4.5.3, x = limn→∞ xn , where xn = j=1 xnj for some xnj ∈ X such that U (t)xnj = eiηj t xnj (j = 1, . . . , m). We can assume that ηj = ηk for k = j. Then Mηj x = limn→∞ Mηj xn = limn→∞ xnj by (4.22). It follows that m x = j=1 Mηj x. This proves one implication of c). The other follows from (4.22). We prove d). Assume that U (t + τ )x = U (t)x (t ∈ R). Recall that for z ∈ C, |z| = 1, one has n−1 1 if z = 1, 1 k lim z = n→∞ n 0 if z = 1. k=0 Let η ∈ R. Then 1 Mη x = lim n→∞ τ n =

lim

n→∞

1 τn

τn

U (s)xe−iηs ds

0 n−1 (k+1)τ k=0

U (s)xe−iηs ds

kτ

n−1 τ

1 U (s)xe−iηs ds e−ikητ τn k=0 0 τ 1/τ 0 U (s)xe−iηs ds if ητ ∈ 2πZ, = 0 if ητ ∈ 2πZ. =

lim

n→∞

This proves one implication. The other follows directly from Proposition 4.5.3. Now we come to the second part of this section where we consider a special group of operators. Let X be a Banach space. By BUC(R, X) we denote the space

292

4. ASYMPTOTICS OF LAPLACE TRANSFORMS

of all bounded uniformly continuous functions on R with values in X. It is a Banach space for the uniform norm

f ∞ = sup f (t) . t∈R

The shift group on BUC(R, X) deﬁned by (S(t)f )(r) := f (r + t)

(r ∈ R, t ∈ R)

is a C0 -group whose generator is denoted by B. Lemma 4.5.5. For f ∈ BUC(R, X) one has f ∈ D(B) if and only if f is diﬀerentiable and f ∈ BUC(R, X). In that case, Bf = f . t Proof. Let f ∈ D(B), g := Bf . Then S(t)f − f = 0 S(s)g ds for all t ≥ 0. In t particular, f (t) = (S(t)f )(0) = f (0) + 0 g(s) ds. This shows one implication of the claim. Conversely, assume that f is diﬀerentiable and g := f ∈ BUC(R, X). r+t t Then (S(t)f − f )(r) = f (r + t) − f (r) = r g(s) ds = 0 (S(s)g)(r) ds. Thus, t S(t)f − f = 0 S(s)g ds. Hence, f ∈ D(B) and Bf = g. Let η ∈ R, x ∈ X. By eiη ⊗ x we denote the periodic function given by (eiη ⊗ x)(t) = eiηt x

(t ∈ R).

Linear combinations of functions of the form eiη ⊗ x with η ∈ R, x ∈ X are called trigonometric polynomials. Deﬁnition 4.5.6. A function f : R → X is called almost periodic if it can be approximated uniformly on R by trigonometric polynomials. By AP(R, X) := span{eiη ⊗ x : η ∈ R, x ∈ X} we denote the space of all almost periodic functions on R with values in X. Let η ∈ R. A function f ∈ BUC(R, X) satisﬁes S(t)f = eiηt f for all t ∈ R if and only if f = eiη ⊗ f (0). Thus, considering the group S on Y := BUC(R, X), we have Yap = AP(R, X). Now we can reformulate the results of the ﬁrst part of this section for this special case. Let f ∈ BUC(R, X). Let ε > 0. A real number τ > 0 is called an ε-period of f if f (τ + s) − f (s) ≤ ε for all s ∈ R. Theorem 4.5.7. Let f ∈ BUC(R, X). The following are equivalent: (i) f is almost periodic.

4.5. ALMOST PERIODIC FUNCTIONS

293

(ii) For every ε > 0 the set of all ε-periods is relatively dense in R. (iii) The orbit {S(t)f : t ∈ R} is relatively compact in BUC(R, X). This is an immediate consequence of Theorem 4.5.1. It follows in particular that for every f ∈ AP(R, X) there exist tn ∈ R such that limn→∞ tn = ∞ and

f (tn + s) − f (s) ≤

1 n

for all s ∈ R.

(4.23)

In particular, if f ∈ AP(R, X), then

f ∞ = sup f (t)

(4.24)

t≥τ

for all τ ∈ R. For f ∈ AP(R, X), η ∈ R, we deﬁne the mean Mη f

:= =

1 t −iηs lim e S(s)f ds t→∞ t 0 lim αR(α + iη, B)f. α↓0

These limits exist in BUC(R, X) (i.e., with respect to the uniform norm) by the remarks preceding Proposition 4.5.3 and by Proposition 4.3.1. Moreover, S(t)Mη f = eiηt Mη f for all t ∈ R. Evaluating at 0, we deduce that Mη f = eiη ⊗ (Mη f )(0).

(4.25)

As before in the abstract setting, we let Freq(f ) := {η ∈ R : Mη f = 0} be the set of all frequencies of f . By Proposition 4.3.11, this is a countable set. Moreover the following property of spectral synthesis holds. Proposition 4.5.8 (Spectral synthesis). Let f ∈ AP(R, X). Then f ∈ span{eiη ⊗ x : η ∈ Freq(f ), x ∈ f (R), } where the closure is taken in BUC(R, X). Proof. Let X0 := span{f (t) : t ∈ R} ⊂ X. Then by Theorem 4.5.7, f is also almost periodic when it is considered as a function with values in X0 . So we can assume that X0 = X. Now the claim follows from Proposition 4.5.3. From Corollary 4.5.4 we see the following.

294

4. ASYMPTOTICS OF LAPLACE TRANSFORMS

Corollary 4.5.9. Let f ∈ AP(R, X). Then a) f ≡ 0 if and only if Freq(f ) = ∅. b) Freq(f ) ⊂ {η} if and only if f = eiη ⊗ f (0). c) f is a trigonometric polynomial if and only if Freq(f ) is ﬁnite. d) f is τ -periodic if and only if Freq(f ) ⊂

2π τ Z

(where τ > 0).

We call assertion a) the uniqueness theorem for almost periodic functions. We also obtain from Theorem 4.5.7 that almost periodic functions have relatively compact range. Corollary 4.5.10. Let f ∈ AP(R, X). Then the set {f (t) : t ∈ R}− is compact. Proof. By hypothesis, the set {S(t)f : t ∈ R} is relatively compact in BUC(R, X). Since evaluation at 0 is a continuous operator from BUC(R, X) into X, it follows that the set {f (t) : t ∈ R} = {(S(t)f )(0) : t ∈ R} is relatively compact in X. A function f ∈ BUC(R, X) is called weakly almost periodic if x∗ ◦ f is almost periodic for all x∗ ∈ X ∗ . Remark 4.5.11. There are various diﬀerent deﬁnitions of weak almost periodicity in the literature. In particular, a function f ∈ BUC(R, X) is called weakly almost periodic in the sense of Eberlein if the set {S(t)f : t ∈ R} is relatively weakly compact in the Banach space BUC(R, X). This property is independent of weak almost periodicity, in general. We refer to the Notes. Proposition 4.5.12. Assume that f ∈ BUC(R, X) has relatively compact range. If f is weakly almost periodic, then f is almost periodic. Proof. Since f is separably valued, we can assume that X is separable. Then we can consider X as a closed subspace of C[0, 1] (see [Woj91, p.36]). Let (Pn )n∈N be a bounded sequence of ﬁnite rank operators on C[0, 1] such that limn→∞ Pn g = g for all g ∈ C[0, 1] (see [Woj91, pp.37,40]). Then Pn ◦ f is weakly almost periodic, hence almost periodic since it has values in a ﬁnite dimensional space. We have limn→∞ Pn (f (t)) = f (t) for all t ∈ R. Since f has relatively compact range, this convergence is uniform in t ∈ R (see Proposition B.15). Thus, Pn ◦ f converges to f in BUC(R, C[0, 1]). Consequently, f is almost periodic. Remark 4.5.13 (Almost periodic orbits). Let U be a bounded C0 -group on a Banach space Y and let x ∈ Y . Then the following are equivalent: (i) x ∈ Yap (i.e., {U (t)x : t ∈ R} is relatively compact in Y ). (ii) U (·)x ∈ AP(R, Y ).

4.6. COUNTABLE SPECTRUM AND ALMOST PERIODICITY

295

In fact, if U (·)x ∈ AP(R, Y ), then {U (t)x : t ∈ R} is relatively compact, by Corollary 4.5.10. Conversely, let x ∈ Yap . Let tn ∈ R. Then there exists a subsequence such that limk→∞ U (tnk )x := y exists. This implies that U (tnk + ·)x converges to U (·)y in BUC(R, Y ). Thus, {U (t + ·)y : t ∈ R} is relatively compact in BUC(R, Y ).

4.6

Countable Spectrum and Almost Periodicity

In this section we deﬁne the spectrum of a function f ∈ BUC(R, X) with the help of the Laplace transform (or more precisely, the Carleman transform). The main result of this section says that, under suitable conditions on the space X, a function f ∈ BUC(R, X) with countable spectrum is almost periodic. Let f ∈ BUC(R, X). The Carleman transform fˆ of f is deﬁned by ∞ −λt e f (t) dt (Re λ > 0), 0 ˆ f (λ) := ∞ λt − 0 e f (−t) dt (Re λ < 0). Thus, fˆ is a holomorphic function deﬁned on C\iR. Remark 4.6.1. Let f+ ∈ BUC(R+ , X) be the restriction of f to R+ and f− ∈ BUC(R+ , X) be given by f− (t) = f (−t) (t ∈ R+ ). Then fˆ+ (λ) (Re λ > 0), fˆ(λ) = −fˆ− (−λ) (Re λ < 0), where fˆ+ and fˆ− are the Laplace transforms of f+ and f− , respectively. We use the same symbol for the Carleman transform and the Laplace transform. This will not lead to confusion. A point iη ∈ iR is called regular for fˆ if fˆ has a holomorphic extension to a neighbourhood of iη (i.e., iη is regular if there exists an open neighbourhood V of iη and a holomorphic function h : V → X such that h(λ) = fˆ(λ) for all λ ∈ V \iR). The Carleman spectrum spC (f ) is deﬁned by spC (f ) = {η ∈ R : iη is not regular for fˆ}.

(4.26)

The following remark explains why this notion of spectrum is well adapted to spectral theory of C0 -groups. Remark 4.6.2 (Carleman spectrum and C0 -groups). Let Y be a Banach space and U be a bounded C0 -group on Y with generator A. Then σ(A) ⊂ iR, since A and −A generate bounded C0 -semigroups. Let x ∈ Y and f (t) := U (t)x. Then the Carleman transform fˆ of f is given by fˆ(λ) = R(λ, A)x (λ ∈ C\iR).

(4.27)

296

4. ASYMPTOTICS OF LAPLACE TRANSFORMS

In particular, i spC (f ) ⊂ σ(A).

(4.28)

Infact, (4.27) is clear for Re λ > 0. If Re λ < 0, then fˆ(λ) = −fˆ− (−λ) = ∞ − 0 eλt U (−t)x dt = −R(−λ, −A)x = R(λ, A)x. As in Section 4.5, we denote by S the shift group on BUC(R, X) and by B its generator. For f ∈ BUC(R, X) and s ∈ R, let fs := S(s)f . An easy calculation (see Remark 4.6.2) shows that s λs −λt ˆ (R(λ, B)f )(s) = fs (λ) = e f (λ) − e f (t) dt 0

for λ ∈ C \ iR. We shall see in Lemma 4.6.8 that the singularities of R(·, B) coincide with the singularities of fˆ; i.e., that the Carleman spectra of S(·)f and f (·) coincide. Let η ∈ R. A function f ∈ BUC(R, X) is called uniformly ergodic at η if it is ergodic at η with respect to S; i.e., if 1 t −iηs Mη f := lim e S(s)f ds (4.29) t→∞ t 0 exists in BUC(R, X). By Section 4.3, this is equivalent to saying that Mη f = lim αR(α + iη, B)f α↓0

(4.30)

exists in BUC(R, X). Since (R(α + iη, B)f )(s) = fs (α + iη), this can be reformulated by saying that αfs (α + iη) converges as α ↓ 0 uniformly in s ∈ R. We say that f is totally ergodic if f is uniformly ergodic at each η ∈ R; i.e., if f is totally ergodic with respect to S in the sense of Sections 4.3 and 4.5. Next, we introduce a geometric condition of Banach spaces. Let c0 be the Banach space of all complex sequences converging to 0 with the supremum norm (as in Example 1.1.5 b)). We say that X contains c0 , and write brieﬂy c0 ⊂ X, if there exists a closed subspace of X which is isomorphic to c0 . Since closed subspaces of reﬂexive spaces are reﬂexive, no reﬂexive Banach space contains c0 . Also, any space of the form L1 (Ω, μ) does not contain c0 . See Appendix D for further information. Now we can formulate the main theorem of this section. Theorem 4.6.3. Let f ∈ BUC(R, X) have countable Carleman spectrum. Assume that one of the following conditions is satisﬁed: a) f is totally ergodic, or b) f has relatively compact range, or c) c0 ⊂ X.

4.6. COUNTABLE SPECTRUM AND ALMOST PERIODICITY

297

Then f is almost periodic. As a corollary, we note the important scalar case which is due to Loomis. Here, we let BUC(R) := BUC(R, C) and AP(R) := AP(R, C). Corollary 4.6.4 (Loomis’s Theorem). Let f ∈ BUC(R) have countable Carleman spectrum. Then f is almost periodic. We have seen in the preceding section that every almost periodic function is totally ergodic and has relatively compact range. Thus, conditions a) and b) in Theorem 4.6.3 are necessary for the conclusion to hold. If the Banach space X contains c0 , countability of the Carleman spectrum alone does not imply almost periodicity. This is shown by the following example. Example 4.6.5. Let X := c, the space of all convergent complex sequences x = (xn )n∈N with the supremum norm x = supn |xn |. Then X is isomorphic to c0 . Let f (t) := eit/n =: (fn (t))n∈N . n∈N

Since fn (t) = ni eit/n , the function f is Lipschitz continuous and thus f∈ BUC(R, c). The Carleman transform fˆ of f is given by 1 ˆ f (λ) = λ − i/n n∈N for all λ ∈ C \iR. Consequently, spC (f ) = {1/n : n ∈ N} ∪ {0}, which is countable. However, f ∈ AP(R, c). In fact, f does not have relatively compact range. To see this, consider φn ∈ c∗ given by x, φn := xn for x = (xk )k∈N ∈ c, and φ∞ ∈ c∗ given by x, φ∞ := limk→∞ xk . Then φn ≤ 1 (n ∈ N ∪ {∞}) and limn→∞ x, φn = x, φ∞ for all x ∈ c. Suppose that K := {f (t) : t ∈ R} is relatively compact in c. Then x, φn converges to x, φ∞ uniformly on K (see Proposition B.15). In particular, limn→∞ Imf (t), φn = limn→∞ sin nt = 0 uniformly in t ∈ R, which is absurd. We need several auxiliary results for the proof of Theorem 4.6.3. The following is a special kind of maximum principle for holomorphic functions. Lemma 4.6.6. Let V be an open neighbourhood of iη such that V contains the closed disc B(iη, 2r) = {z ∈ C : |z − iη| ≤ 2r}. Let h : V → X be holomorphic and c ≥ 0, k ∈ N0 such that

h(z) ≤

c | Re z|k

if |z − iη| = 2r, Re z = 0.

Then h(z) ≤ (4/3) cr−k for all z ∈ B(iη, r). k

298

4. ASYMPTOTICS OF LAPLACE TRANSFORMS

Proof. We can assume that η = 0 (replacing h(z) by h(z + iη) otherwise). Deﬁne k z2 g : V → X by g(z) := 1 + h(z). Let |z| = 2r, z = 2reiθ . Then (2r)2

g(z) = |1 + ei2θ |k h(z) = |eiθ (e−iθ + eiθ )|k h(z)

c = 2k | cos θ|k h(z) ≤ k . r −k It follows from themaximum principle that g(z) ≤ cr if |z| ≤ 2r. Let |z| ≤ r. −1 2 2 2 z 4 (2r) ≤ 4r Then 1 + = . Hence, = (2r)2 (2r)2 + z 2 4r2 − r 2 3

−k z2

h(z) = 1 + g(z) ≤ (4/3)k cr−k (2r)2

(|z| ≤ r).

We now describe the local spectrum associated with a bounded C0 -group and an individual vector. Let U be an isometric C0 -group on a Banach space Y with generator A. For x ∈ Y we deﬁne the space Yx := span{U (t)x : t ∈ R}. It is invariant under the group U . We denote by Ax the generator of this restriction group. Then σ(Ax ) ⊂ σ(A) ⊂ iR, and R(λ, Ax ) = R(λ, A)|Yx (λ ∈ ρ(A)). As before, C+ := {z ∈ C : Re z > 0} denotes the right half plane. Lemma 4.6.7. Let x ∈ Y, η ∈ R. The following assertions are equivalent: (i) iη ∈ ρ(Ax ). (ii) There exists an open neighbourhood V of iη and a holomorphic function h : V → X such that h(λ) = R(λ, A)x for all λ ∈ C+ ∩ V . In that case, h(λ) = R(λ, A)x for all λ ∈ V \iR. Proof. (i) ⇒ (ii): Let V := ρ(Ax ), h(λ) := R(λ, Ax )x. (ii) ⇒ (i): Assume (ii) and assume that V is connected. We ﬁrst show that h(λ) = R(λ, A)x for all λ ∈ V \iR. Let μ ∈ ρ(A). Then k(λ) := (λ − A)R(μ, A)h(λ) deﬁnes a holomorphic function on V such that k(λ) = R(μ, A)x for all λ ∈ C+ ∩ V . It follows from the uniqueness theorem that (λ − A)R(μ, A)h(λ) = k(λ) = R(μ, A)x for all λ ∈ V . This implies that R(μ, A)h(λ) = R(λ, A)R(μ, A)x = R(μ, A)R(λ, A)x for all λ ∈ V \iR. Since R(μ, A) is injective, the

∞claim follows. We now show that iη ∈ ρ(Ax ). We have R(λ, A) = 0 e−λt U (t) dt ≤ 1/(Re λ) (Re λ > 0), and similarly, R(λ, A) = R(−λ, −A) ≤ −1/(Re λ) for Re λ < 0. Thus R(λ, A) ≤ 1/| Re λ| for all λ ∈ V \iR. Choose r > 0 such that B(iη, 2r) ⊂ V . It follows from the assumption that R(·, A)z has a holomorphic extension to V for all z ∈ span{U (t)x : t ∈ R}. Now it follows from Lemma 4.6.6 that 4

R(λ, A)z ≤

z for all λ ∈ B(iη, r)\iR. 3r

4.6. COUNTABLE SPECTRUM AND ALMOST PERIODICITY Hence, R(λ, Ax ) ≤

4 3r

299

for all λ ∈ B(iη, r)\iR. This implies that iη ∈ ρ(Ax ).

We deﬁne the local spectrum of A at x by σ(A, x) := σ(Ax ) = C\ρ(Ax ). Thus, iη ∈ σ(A, x) if and only if the equivalent conditions (i) and (ii) of Lemma 4.6.7 are not satisﬁed. Now we consider the shift group S on BUC(R, X) with generator B, as before. We show that the Carleman spectrum coincides with the local spectrum with respect to B. Lemma 4.6.8. Let f ∈ BUC(R, X). Then i spC (f ) = σ(B, f ). Proof. a) Let iη ∈ iR \ σ(B, f ). There exists an open neighbourhood V of iη and a holomorphic function h : V → BUC(R, X) such that h(λ) = R(λ, B)f (λ ∈ V \iR). Let k(λ) := h(λ)(0). Then k : V → X is holomorphic, k(λ) = (R(λ, B)f )(0) = ∞ −λt e (S(t)f )(0) dt = fˆ(λ) for λ ∈ V ∩ C+ , and k(λ) = (R(λ, B)f )(0) = 0 ∞ ∞ −(R(−λ, −B)f )(0) = − 0 eλt (S(−t)f )(0) dt = − 0 eλt f (−t) dt for λ ∈ V , Re λ < 0. Thus, η ∈ spC (f ). b) Let η ∈ R\spC (f ). By assumption, there exists an open neighbourhood V of iη and a holomorphic function h : V → X such that ∞ −λt e f (t) dt (λ ∈ V ∩ C+ ) 0 h(λ) = ∞ λt − 0 e f (−t) dt (λ ∈ V, Re λ < 0). For λ ∈ V, s ∈ R, deﬁne

H(λ, s) := e Then for λ ∈ V, Re λ > 0,

(R(λ, B)f )(s) =

λs

∞

h(λ) −

s

e

−λt

f (t) dt .

0

e−λt f (t + s) dt = H(λ, s)

0

and for λ ∈ V, Re λ < 0, (R(λ, B)f )(s) = −(R(−λ, −B)f )(s) = −

0

∞

(s ∈ R)

eλt f (s − t) dt = H(λ, s).

Thus, H(λ, s) ≤ f ∞ /| Re λ| for λ ∈ V \ iR. By Lemma 4.6.6, this 6 implies that for r > 0 such that B(iη, 2r) ⊂ V , one has H(λ, s) ≤ 4 f ∞ 3r for all λ ∈ B(iη, r), s ∈ R. We know that H(λ, ·) = R(λ, B)f ∈ BUC(R, X) if Re λ > 0. Since H(·, s) is holomorphic on B(iη, r) and bounded uniformly in s, it follows

300

4. ASYMPTOTICS OF LAPLACE TRANSFORMS

from Corollary A.4 that λ → H(λ, ·) is holomorphic on B(iη, r) with values in BUC(R, X). Here, we choose linear functionals on BUC(R, X) of the form g → g(s), x∗ for s ∈ R, x∗ ∈ X ∗ , x∗ ≤ 1. Now, iη ∈ σ(B, f ) by Lemma 4.6.7. We deduce from Lemma 4.6.8 the following interesting observation. Proposition 4.6.9. Let f ∈ BUC(R, X) and let η ∈ R \spC (f ). Then f is uniformly ergodic at η and Mη f = 0. Proof. It was shown in the proof of Lemma 4.6.8 that λ → R(λ, B)f has a holomorphic extension to a neighbourhood of iη. It follows in particular that limα↓0 αR(α + iη, B)f = 0 in BUC(R+ , X). Proposition 4.6.10. Let f ∈ AP(R, X). Then spC (f ) = Freq(f ). Proof. The inclusion Freq(f ) ⊂ spC (f ) is immediate from Proposition 4.6.9, although in this case the fact that f is uniformly ergodic at each point of R is already known (see Section 4.5). Since spC (f ) is closed, it follows that Freq(f ) ⊂ spC (f ). For the converse, let Y0 = span {eiη ⊗ x : η ∈ Freq(f ), x ∈ X} and Y := Y0 . By Proposition 4.5.8, f ∈ Y . For g ∈ Y0 , gˆ has a holomorphic extension g (λ) ≤ g ∞ /| Re λ|. By Lemma 4.6.6, ˆ g (iη) ≤ 6! to C \ i Freq(f ),"and ˆ 8 g ∞ 3 dist(η, Freq(f )) if η ∈ R \ Freq(f ). If (gn )n∈N is a sequence in Y0 such that gn − f ∞ → 0, it follows that limn→∞ g n (λ) exists for λ ∈ C \ i Freq(f ) and by Vitali’s Theorem A.5, this gives a holomorphic extension of fˆ to C \ i Freq(f ). Hence, spC (f ) ⊂ Freq(f ). The condition that c0 ⊂ X will enter the proof of Theorem 4.6.3 in form of the following theorem due to Kadets. Theorem 4.6.11 (Kadets). Assume that c0 ⊂ Xand let f ∈ BUC(R, X). If furthermore S(t)f − f ∈ AP(R, X) for all t ∈ R, then f ∈ AP(R, X). We will give the proof of Kadets’s theorem at the end of this section. For X = C, Theorem 4.6.11 is due to H. Bohr. We will give a separate direct proof for this easier case. For Theorem 4.6.3 we use the following spectral-theoretic reformulation of Kadets’s theorem. Denote by π : BUC(R, X) → BUC(R, X)/ AP(R, X) the quotient map. Recall that the quotient space is a Banach space for the norm

π(f ) := inf{ f + g : g ∈ AP(R, X)} and π is contractive for this norm. Since S(t) leaves the space AP(R, X) invariant, there exists a C0 -group S on the quotient BUC(R, X)/ AP(R, X) given by S(t)π(f ) := π(S(t)f ). Now Kadets’s theorem can be rephrased in the following We call its generator B. form.

4.6. COUNTABLE SPECTRUM AND ALMOST PERIODICITY

301

Corollary 4.6.12. The following assertions are equivalent: (i) c0 ⊂ X. has empty point spectrum. (ii) B So it suﬃces to show Proof. (i) ⇒ (ii): Theorem 4.6.11 says that 0 ∈ σp (B). iηr that σp (B) = σp (B) + iR. Let η ∈ R, eiη (r) := e . Then Qη f := eiη · f deﬁnes an isomorphism Qη on BUC(R, X) which leaves AP(R, X) invariant. Moreη on BUC(R, X)/ AP(R, X) by Q η (π(f )) := over, Q−η S(t)Qη = eiηt S(t). Deﬁne Q iηt π(Qη f ). Then Q−η S(t)Qη = e S(t). This implies the claim. (ii) ⇒ (i): We have to show that if c0 ⊂ X then there exists f ∈ BUC(R, X) such that f ∈ AP(R, X) but S(t)f − f ∈ AP(R, X) for all t ≥ 0. Since the space c0 is isomorphic to the space c, it suﬃces to" consider X = c. Consider ! the function f ∈ BUC(R, c) given by f (t) := eit/n n∈N . Then f ∈ AP(R, c) by ! " Example 4.6.5. Let t ∈ R, g := S(t)f − f . Then g(s) = (eit/n − 1)eis/n n∈N =: (gn (s))n∈N . Let hn (s) := (g1 (s), . . . , gn (s), 0, 0, . . .). Since limn→∞ (eit/n − 1) = 0, we have limn→∞ hn = g in BUC(R, c). But hn ∈ AP(R, c) for all n ∈ N. Thus, g ∈ AP(R, c). Proof of Theorem 4.6.3. Let f ∈ BUC(R, X) such that spC (f ) is countable. Case c): We assume that c0 ⊂ X. Let Y := BUC(R, X)/ AP(R, X) and on Y . Assume that f ∈ AP(R, X); consider the shift group S with generator B f : t ∈ R} and the i.e., f := π(f ) = 0. We consider the space Yf := span{S(t) (see the discussion before Lemma 4.6.7). Let η ∈ R\spC (f ). induced operator B f

Then by Lemma 4.6.8, the function λ → R(λ, B)f has a holomorphic extension h : V → BUC(R, X) where V is an open neighbourhood of iη. Then π ◦ h is a f = π(R(λ, B)f ). It follows from Lemma holomorphic extension of λ → R(λ, B) ). We have proved that σ(B ) ⊂ spC (f ). Thus, σ(B ) is 4.6.7 that iη ∈ σ(B f f f countable, closed and non-empty (by Corollary 4.4.12), so it contains an isolated has non-empty point spectrum. point. It follows from Corollary 4.4.13 that B f also has non-empty point spectrum. This contradicts Corollary 4.6.12. Hence, B The proof is ﬁnished in this case. Case a): Now assume that f is totally ergodic. Consider the space E := {g ∈ BUC(R, X) : g is totally ergodic} which is closed and invariant under the shift group. We denote the shift group on E also by S and its generator by B. Consider the quotient space E := E/ AP(R, X), the quotient map π : E → E and the induced It suﬃces to show group S on E given by S(t)π(g) := π(S(t)g) with generator B. has empty point spectrum. Then the proof given in Case a) carries over. that B Assume that g ∈ E, η ∈ R such that S(t)π(g) = eiηt π(g) (t ∈ R). Then −iηt e S(t)g − g ∈ AP(R, X) (t ∈ R). It follows that " 1 t ! −iηs Mη g − g = lim e S(s)g − g ds ∈ AP(R, X). t→∞ t 0

302

4. ASYMPTOTICS OF LAPLACE TRANSFORMS

But Mη g = eiη ⊗ (Mη g)(0) ∈ AP(R, X). Thus g ∈ AP(R, X). Case b): Assume that f has relatively compact range. It follows from Case c) applied in the scalar case that f is weakly almost periodic. Hence, f ∈ AP(R, X) by Proposition 4.5.12. Now Theorem 4.6.3 is completely proved in the case a) where f is assumed to be uniformly ergodic at each η ∈ spC (f ). The proof in the other cases is complete admitting Kadets’s theorem. Note however that for the case b), where f is assumed to have relatively compact range, we merely need Kadets’s theorem in the scalar case. The scalar case is much easier to prove and will be particularly important for the general complex Tauberian theorem presented in Section 4.9. For this reason, we ﬁrst give a direct proof of Kadets’s theorem in the scalar case. The following reformulation will be useful. Lemma 4.6.13. Let X be a Banach space. The following assertions are equivalent: t (i) If f ∈ AP(R, X) such that F (t) := 0 f (s) ds (t ∈ R) is bounded, then F ∈ AP(R, X). (ii) If f ∈ BUC(R, X) such that S(t)f − f ∈ AP(R, X) for all t ∈ R, then f ∈ AP(R, X). t Proof. (ii) ⇒ (i): Let f ∈ AP(R, X) such that F (t) = 0 f (s) ds is bounded. Then t F ∈ BUC(R, X) and S(t)F − F = 0 S(s)f ds ∈ AP(R, X) for all t ∈ R. Hence, F ∈ AP(R, X) by (ii). (i) ⇒ (ii): Let f ∈ BUC(R, X) such that S(t)f − f ∈ AP(R, X) for all t ∈ R. Let λ > 0 and g = R(λ, B)f . Then S(t)g − g ∈ AP(R, X) and g ∈ D(B). It follows that g = lim 1/t (S(t)g − g) ∈ AP(R, X). t↓0

t

Thus by (i), g(t) − g(0) = 0 g (s) ds deﬁnes an almost periodic function. We have shown that R(λ, B)f ∈ AP(R, X) for all λ > 0. Since f = limλ→∞ λR(λ, B)f in BUC(R, X) (see Proposition 3.1.9 a)), it follows that f ∈ AP(R, X). Proof of Theorem 4.6.11 in the scalar case. Let f ∈ AP(R) such that F (t) := t f (s) ds is bounded. We show that F ∈ AP(R). Then the result follows from 0 Lemma 4.6.13. It follows easily from Deﬁnition 4.5.6 that we can assume that f is real-valued. It is clear that F ∈ BUC(R). Let sk ∈ R (k ∈ N). There exists a subsequence tm := skm such that g := limm→∞ S(tm )f exists in BUC(R). We have to show that S(tm )F has a convergent subsequence. Then F ∈ AP(R) by Theorem 4.5.7. t +t t a) Since (S(tm )F ) (t) = 0 m f (s) ds = 0 S(tm )f (s) ds + F (tm ), and (F (tm ))m∈N is bounded, it follows that S(tm )F has a limit point G for compact convergence (by which we mean the topology of uniform convergence on bounded t intervals), where G(t) = 0 g(s) ds + d for some d ∈ R. Clearly, G ∈ BUC(R) and sup G := supt∈R G(t) ≤ sup F, inf G ≥ inf F . Since S(−tm )g → f as m → ∞,

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303

for the same reason S(−tm )G has a limit point H for compact convergence, H is diﬀerentiable, H = f, sup H ≤ sup G, inf H ≥ inf G. Hence, H(t) = F (t) + c for some c ∈ R and c + sup F ≤ sup G ≤ sup F and inf F + c ≥ inf G ≥ inf F . Hence, c = 0. Thus, sup G = sup F and inf G = inf F . This determines d uniquely. So the limit point G is unique. Hence, S(tm )F converges to G ∈ BUC(R) uniformly on bounded intervals. b) Now it remains to show that S(tm )F converges uniformly to G. If not, passing to a subsequence we can assume that there exist ε > 0 and sm ∈ R such that |F (tm + sm ) − G(sm )| ≥ ε > 0 (m ∈ N). Taking subsequences, we can assume that h := limm→∞ S(tm + sm )f and h1 := limm→∞ S(sm )g = h1 exist in BUC(R). Then h = h1 , since |h(t) − h1 (t)|

Let H(t) :=

t 0

≤

|h(t) − f (t + tm + sm )| + |f (t + tm + sm ) − g(sm + t)|

+ |g(sm + t) − h1 (t)| → 0 as m → ∞. h(s) ds. By a), there exist constants c1 , c2 such that

lim (S(tm + sm )F ) (t) = H(t) + c1 and

m→∞

lim (S(sm )G) (t) = H(t) + c2

m→∞

uniformly on bounded intervals, and inf F = inf H + c1 , and inf G = inf H + c2 . Since inf F = inf G, it follows that c1 = c2 . Thus ε ≤ limm→∞ |F (tm + sm ) − G(sm )| = |c1 − c2 | = 0, a contradiction. Now we have proved Kadets’s theorem in the scalar case, and hence also the proof of Theorem 4.6.3 in the case b), and in particular of Loomis’s theorem, is complete. Before proving Kadets’s theorem in full generality, we consider another special case which can be deduced from the scalar case by a short but clever argument. Recall that a Banach space X is called uniformly convex if for all ε > 0 there exists δ > 0 such that for x ≤ 1, y ≤ 1,

x + y

x − y ≥ ε ⇒ (4.31)

2 ≤ 1 − δ. Every uniformly convex space is reﬂexive, and Lp -spaces are uniformly convex for 1 < p < ∞. Proof of Theorem 4.6.11 for uniformly convex spaces. t Let f ∈ AP(R, X) such that F (t) = 0 f (s) ds is bounded. We claim that F is almost periodic. We can assume that

F ∞ := sup F (t) = 1. t∈R

Since by the scalar case, F is weakly almost periodic, in view of Proposition 4.5.12 it suﬃces to show that F has precompact range.

304

4. ASYMPTOTICS OF LAPLACE TRANSFORMS Assume that this is false. Then there exist ε > 0, tn ∈ R such that

F (tn ) − F (tm ) ≥ ε (n = m).

Choose δ > 0 according to (4.31). Let x∗ ∈ X ∗ such that x∗ = 1 and x∗ ◦F ∞ > 1 − 4δ . Replacing (tn )n∈N by a subsequence if necessary, we can assume that S(tn )f S(tn )(x∗ ◦ F )

→ f in BUC(R, X) → g in BUC(R).

Then g ∞ ≥ 1 − 4δ , since S(−tn )g → x∗ ◦ F in BUC(R). Let t ∈ R such that |g(t)| ≥ 1 − δ2 . Then lim sup F (tn + t) − F (tm + t)

n,m→∞ n=m

t

= lim sup F (tn ) − F (tm ) + (f (tn + s) − f (tm + s)) ds

n,m→∞ 0

n=m

= lim sup F (tn ) − F (tm ) ≥ ε. n,m→∞ n=m

Hence by (4.31), 1−

δ ≤ |g(t)| = 2

lim n,m→∞ n=m

≤ lim sup n,m→∞ n=m

|F (t + tn ) + F (t + tm ), x∗ | 2

F (t + tn ) + F (t + tm )

2

≤ 1 − δ. This is a contradiction. For the proof of Kadets’s theorem in the general case we use the following characterization which is proved in Appendix D. Theorem 4.6.14. Let X be a Banach space. The following are equivalent: (i) c0 ⊂ X.

∞ (ii) there is a divergent series n=1 xn in X which is unconditionally bounded; i.e., there exists M ≥ 0 such that

m

x nj ≤ M

j=1

whenever nj ∈ N (j = 1, 2, . . . , m) such that n1 < n2 < . . . < nm .

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305

Proof of Kadets’s Theorem 4.6.11. t Assume that there exists f ∈ AP(R, X) such that F (t) = 0 f (s) ds is bounded, but F ∈ AP(R, X). We show that this implies that c0 ⊂ X. In view of Lemma 4.6.13, this proves Kadets’s theorem. By the scalar case already proved, F is weakly almost periodic. a) We show that there exists ε > 0 such that the set U (ε) := {t ∈ R : F (t) < ε} is not relatively dense. Assume that, on the contrary, U (ε) is relatively dense for all ε > 0. n Let ε > 0. We show that R = j=0 (U (ε) + sj ) for suitable s0 , . . . , sn ∈ R. In fact, there exists l > 0 such that U (ε/2) ∩ [a − l, a] = ∅ for all a ∈ R. Since F is Lipschitz continuous, there exists δ > 0 such that F (z) ≤ ε/2 implies F (r) ≤ ε for all r ∈ [z, z + δ]. Let s0 := 0, s1 := δ, s2 := 2δ, . . . , sn := nδ, where nδ > l. Let t ∈ R. Take s ∈ [t − l, t] ∩ U (ε/2). Then [s, s + δ] ⊂ U (ε) = U (ε) + s0 , [s + δ, s + 2δ] = [s, s + δ] + δ ⊂ U (ε) + s1 , .. . [s + (n − 1)δ, s + nδ] ⊂ U (ε) + sn . n Thus, t ∈ [s, s + l] ⊂ j=0 (U (ε) + sj ). t Since f ∈ AP(R, X), the function S(t)F − F = 0 S(r)f dr ∈ AP(R, X) for all t ∈ R. In particular, S(sj )F − F has precompact range Kj . Now let t ∈ R. Then t = s + sj for some s ∈ U (ε), j ∈ {0, 1, 2, . . . , n}. Thus, F (t) = F (s + sj ) = (S(s n j )F − F ) (s) + F (s) ∈ Kj + B(0, ε). Thus, the range of F is contained in j=0 Kj + B(0, ε). Since ε > 0 is arbitrary, this implies that F has precompact range. This is impossible by Proposition 4.5.12. b) For γ ∈ R, δ > 0, let ' ( Vγ (δ) := t ∈ R : F (t + γ) − F (t) − F (γ) < δ . n We show that j=1 Vγj (δ) is relatively dense in R for all δ > 0, γ1 , γ2 , . . . , γn ∈ R. In fact, let γ := maxj=1,...,n |γj |. Then

γj

F (t + γj ) − F (t) − F (γj ) = (f (t + s) − f (s)) ds

0

≤ γ. sup f (t + s) − f (s) . 0≤s≤γ

Thus, n

j=1

Vγj (δ) ⊃

δ t ∈ R : f (t + s) − f (s) ≤ for all s ∈ [0, γ] . γ

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4. ASYMPTOTICS OF LAPLACE TRANSFORMS

The last set is relatively dense since f ∈ AP(R, X). c) By a) and b), there exists ε > 0 such that the following holds: If δ > 0, γ1 , γ2 , . . . , γr ∈ R, then U (ε) ⊃ ∩rj=1 Vγj (δ).

(4.32)

We construct tn ∈ R such that F (tn ) ≥ ε (n ∈ N) and

F (tn + tj1 + . . . + tjr ) − F (tn ) − F (tj1 + . . . + tjr ) ≤ 2−n

(4.33)

whenever j1 , j2 , . . . , jr ∈ N, 1 ≤ j1 < j2 < . . . < jr ≤ n − 1. Choose t1 ∈ R \ U (ε). Assume that t1 , . . . , tn−1 are constructed. By (4.32), there exists tn ∈ R\U (ε) such that tn ∈ Vγ (2−n ) whenever γ = tj1 + . . . + tjr with 1 ≤ j1 < j2 < . . . < jr ≤ n − 1. This means that (4.33) holds. ∞ d) Let xn := F (tn ). Then the series n=1 xn diverges. We show that

m

xik ≤ F ∞ + 1 (4.34)

k=1

whenever i1 < i2 < . . . < im . In fact,

m−1 m m m

tik − F (tik ) ≤ F tik − F (tim ) − F tik

F

k=1 k=1 k=1 k=1

m−1 m−2

+ F tik − F (tim−1 ) − F tik

k=1 k=1

+ . . . + F (ti2 + ti1 ) − F (ti2 ) − F (ti1 ) ≤ 2−im + . . . + 2−i2 ≤ 1. Thus,

m

m

xik = F (tik ) ≤ 1 + F ∞ .

k=1

k=1

Now it follows from Theorem 4.6.14 that c0 ⊂ X.

4.7

Asymptotically Almost Periodic Functions

In this section we study bounded uniformly continuous functions on the half-line. The main result is a Tauberian theorem (Theorem 4.7.7) which says that, under additional assumptions, such a function with countable spectrum is asymptotically almost periodic. Here we again use a quotient method similar to the preceding section. A similar result, in the case when the spectrum has at most one point, was given in Theorem 4.4.8 with a proof by elementary contour integrals.

4.7. ASYMPTOTICALLY ALMOST PERIODIC FUNCTIONS

307

Let X be a Banach space and denote by BUC(R+ , X) the space of all bounded uniformly continuous functions deﬁned on R+ with values in X. It is a Banach space for the uniform norm

f ∞ = sup f (t) (f ∈ BUC(R+ , X)). t≥0

By C0 (R+ , X) we denote the closed subspace consisting of all f ∈ BUC(R+ , X) such that limt→∞ f (t) = 0. For x ∈ X, η ∈ R, we now consider the function eiη ⊗ x : t → eiηt x on R+ and denote by AP(R+ , X) := span{eiη ⊗ x : η ∈ R, x ∈ X} the space of all almost periodic functions on the half-line (the closure being taken in BUC(R+ , X)). Proposition 4.7.1. Every f ∈ AP(R+ , X) has a unique extension f ∈ AP(R, X) and f ∞ = f ∞ . Moreover,

f ∞ = sup f (t)

for all τ ≥ 0.

(4.35)

t≥τ

Proof. Let f ∈ AP(R+ , X). There exist trigonometric polynomials fn ∈ span{eiη ⊗ x : η ∈ R, x ∈ X} such that fn → f in BUC(R+ , X) as n → ∞. It follows from (4.24) that (fn )n∈N is a Cauchy sequence in BUC(R, X). Let f be the limit of (fn )n∈N in BUC(R, X). Then f ∈ AP(R, X) and (4.35) follows from (4.24). It follows from (4.35) that AP(R+ , X) ∩ C0 (R+ , X) = {0}.

(4.36)

By AAP(R+ , X) := C0 (R+ , X) ⊕ AP(R+ , X), we denote the space of all asymptotically almost periodic functions on the half-line. It follows from (4.35) that for f = f0 + f1 with f0 ∈ C0 (R+ , X), f1 ∈ AP(R+ , X), one has f1 ∞ ≤ f ∞ . Thus, AAP(R+ , X) is a closed subspace of BUC(R+ , X). The following observation is useful for later purposes. By (4.23), there exist tn ∈ R+ such that limn→∞ tn = ∞ and

f (tn + s) − f1 (s) ≤

1 n

for all s ∈ R+ .

(4.37)

In the following, we consider the shift semigroup S on BUC(R+ , X) given by (S(t)f ) (s) = f (t + s)

(s, t ≥ 0, f ∈ BUC(R+ , X)).

It is a C0 -semigroup of contractions, and its generator will be denoted by B.

308

4. ASYMPTOTICS OF LAPLACE TRANSFORMS Similarly to Section 4.6, for f ∈ BUC(R+ , X), Re λ > 0 and s ≥ 0, s λs −λt ˆ (R(λ, B)f )(s) = fs (λ) = e f (λ) − e f (t) dt , 0

where fs := S(s)f . We shall see in Lemma 4.7.9 that the singularities of R(·, B)f and fˆ(·) in iR coincide, and we shall exploit this to prove an analogue of part of Theorem 4.6.3 (Theorem 4.7.7). Let f ∈ BUC(R+ , X). Let η ∈ R. We say that f is uniformly ergodic at η if f is ergodic at η with respect to S; i.e., if 1 t −iηs Mη f := lim e S(s)f ds t→∞ t 0 converges in BUC(R+ , X). This is equivalent to the convergence of Mη f = lim αR(α + iη, B)f. α↓0

In that case, Mη f = eiη ⊗ (Mη f )(0).

(4.38)

A function f ∈ BUC(R+ , X) is called totally ergodic if f is ergodic at all η ∈ R. From now on, we denote by E(R+ , X) the set of all totally ergodic functions in BUC(R+ , X). This is a closed subspace of BUC(R+ , X) containing AAP(R+ , X). In the following we will consider the quotient space E := E(R+ , X)/ AAP(R+ , X) Then E is a Banach space for the norm with quotient map π : E(R+ , X) → E.

π(f ) := inf{ f − g ∞ : g ∈ AAP(R+ , X)}

(f ∈ E(R+ , X)).

Since E(R+ , X) and AAP(R+ , X) are invariant under the shift semigroup, we can deﬁne a C0 -semigroup S on E by S(t)π(f ) := π (S(t)f )

(t ≥ 0, f ∈ E(R+ , X)).

the generator of S. We denote by B The interesting fact about this construction is the following. Proposition 4.7.2. Each S(t) is isometric and surjective. Thus, S extends to an ˜ ˜ has empty point spectrum. isometric C0 -group on E. Moreover, B Proof. It is immediate that S(t)

≤ S(t) = 1 for all t ≥ 0. We show that S(t) is isometric. Let f ∈ E(R+ , X), t ≥ 0 and g ∈ AAP(R+ , X). Deﬁne h : R+ → X by g(s − t) (s ≥ t) h(s) := g(0) + f (s) − f (t) (s < t).

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309

Then h ∈ AAP(R+ , X) and

f − h ∞

=

sup f (s) − g(s − t) = sup f (s + t) − g(s)

s≥t

s≥0

= S(t)f − g ∞ . Hence, π(f ) ≤ π(S(t)f ) = S(t)π(f ) . ˜ Since S(t) maps E(R+ , X) onto E(R+ , X), it follows that S(t) is surjective. By Proposition 3.1.23, S extends to a C0 -group on E. is empty. Let f ∈ E(R+ , X) It remains to show that the point spectrum of B t iηt and η ∈ R such that S(t)π(f ) = e π(f ) (t ≥ 0). Then 1t 0 e−iηs S(s)π(f ) ds = 1 t −iηs π(f ) (t ≥ 0). On the other hand, limt→∞ t 0 e S(s)f ds = Mη f ∈ AP(R+ , X). t Applying π on both sides, we conclude that limt→∞ 1t 0 e−iηs S(s)π(f ) ds = π(Mη f ) = 0. Hence, π(f ) = 0. The following simple result is essentially a reformulation of part of Proposition 4.7.2 (see Lemma 4.6.13). t Proposition 4.7.3. Let f ∈ AAP(R+ , X) and F (t) := 0 f (s) ds. Suppose that F is bounded and uniformly ergodic at 0. Then F ∈ AAP(R+ , X). Proof. Note ﬁrst that S(s)F − F = Hence, 1 t→∞ t

M0 F − F = lim

s

0

0

t

S(r)f dr ∈ AAP(R+ , X).

(S(s)F − F ) ds ∈ AAP(R+ , X).

Since M0 F is a constant function, it follows that F ∈ AAP(R+ , X). Next, we characterize asymptotically almost periodic functions by relative compactness of the orbits under S. This is analogous to the characterization of almost periodic functions on the line proved in Section 4.5 (see Theorem 4.5.7). Theorem 4.7.4. Let f ∈ BUC(R+ , X). The following are equivalent: (i) f ∈ AAP(R+ , X). (ii) The orbit {S(t)f : t ≥ 0} is relatively compact in BUC(R+ , X). Proof. (i) ⇒ (ii): If h ∈ C0 (R+ , X), then limt→∞ S(t)h = 0. Thus, {S(t)h : t ≥ 0} is precompact. If f = eiη ⊗x, then S(t)f = eiηt f , so f has precompact orbit. It follows that every f ∈ span ({eiη ⊗ x : η ∈ R, x ∈ X} ∪ C0 (R+ , X)) = AAP(R+ , X) has precompact orbit. (ii) ⇒ (i): Assume that Of := {S(t)f : t ≥ 0} is relatively compact in BUC(R+ , X). Then f ∈ E(R+ , X) by Proposition 4.3.12, and Oπ(f ) := π(Of ) =

310

4. ASYMPTOTICS OF LAPLACE TRANSFORMS

It follows from Theorem 4.5.1 that {S(t)π(f ) : t ≥ 0} is relatively compact in E. ' ( iηt π(f ) ∈ span g ∈ E : S(t) g = e g (t ∈ R) for some η ∈ R . By Proposition 4.7.2, this set is reduced to {0}. Thus, π(f ) = 0; i.e., f ∈ AAP(R+ , X). Similarly to the results on the line in Section 4.5, one can also characterize asymptotically almost periodic functions by “the relative density of asymptotic ε-periods”, but the deﬁnition on the half-line is slightly diﬀerent. Let f ∈ BUC(R+ , X). For ε > 0, λ > 0, we let ' ( Qε,λ (f ) := τ ∈ R+ : f (t + τ ) − f (t) ≤ ε whenever t ≥ λ . We say that a subset Q of R+ is relatively dense in R+ if there exists a length l > 0 such that [a, a + l] ∩ Q = ∅ for all a ∈ R+ . Theorem 4.7.5. Let f ∈ BUC(R+ , X). The following assertions are equivalent: (i) f ∈ AAP(R+ , X). (ii) For all ε > 0 there exists a λ > 0 such that Qε,λ (f ) is relatively dense in R+ . Proof. (ii) ⇒ (i): By Theorem 4.7.4, we have to show that the set {S(t)f : t ≥ 0} is precompact in BUC(R+ , X). Let ε > 0. By assumption, there exists λ > 0 such that Qε,λ (f ) is relatively dense in R+ . Choose a length l such that [a, a + l] ∩ Qε,λ (f ) = ∅ for all a ≥ 0. Since {S(t)f : 0 ≤ t ≤ 2λ + l} is precompact, it suﬃces to cover {S(t)f : t ≥ 2λ + l} by ﬁnitely many balls of radius 2ε. There exist t1 , . . . , tm ∈ [λ, λ + l] such that for all t ∈ [λ, λ + l] there exists j ∈ {1, . . . , m} such that

S(t)f − S(tj )f ∞ ≤ ε. Now let t ≥ 2λ+l. There exists τ ∈ Qε,λ (f )∩[t−λ−l, t−λ]. Then t−τ ∈ [λ, λ+l]. There exists j ∈ {1, . . . , m} such that

S(t − τ )f − S(tj )f ∞ ≤ ε. Hence,

S(t)f − S(tj )f ∞ ≤ S(t)f − S(t − τ )f ∞ + S(t − τ )f − S(tj )f ∞ ≤ 2ε since τ ∈ Qε,λ (f ) and t − τ ≥ λ. (i) ⇒ (ii): Let f ∈ AAP(R+ , X). Let ε > 0. There exist λ ≥ 0 and h ∈ AP(R, X) such that f (t) − h(t) ≤ ε/3 for all t ≥ λ. Let τ > 0 be an ε/3-period for h. Then τ ∈ Qε,λ (f ). Since ε/3-periods for h are relatively dense in R, Qε, λ (f ) is relatively dense in R+ . Recall from Section 4.4 that the half-line spectrum sp(f ) of a function f ∈ BUC(R+ , X) is deﬁned in the following way: call iη ∈ iR regular for fˆ if fˆ has a holomorphic extension to a neighbourhood of iη. Then we set ' ( sp(f ) = η ∈ R : iη is not regular for fˆ (4.39)

4.7. ASYMPTOTICALLY ALMOST PERIODIC FUNCTIONS

311

The half-line spectrum should not be confused with the Carleman spectrum introduced in the preceding section. Indeed, if f is the restriction of a function g ∈ BUC(R, X), then the half-line spectrum of f is much smaller than the Carleman spectrum of g in general. We make this more precise in the following remark. Remark 4.7.6 (Comparison of half-line spectrum and Carleman spectrum). a) Let f ∈ BUC(R+ , X). Assume that there exists η ∈ R \ sp(f ). Then f has at most one extension g ∈ BUC(R, X) such that η ∈ spC (g). In fact, assume that there are two extensions g, h ∈ BUC(R, X) such that η ∈ spC (g) ∪ spC (h). Consider k := g − h. Then k|R+ = 0. It follows that the Carleman transform kˆ of k is 0 on the right half-plane. Since kˆ is a holomorphic function on C \ iR with a holomorphic extension to a neighbourhood of iη, it ˆ follows from the uniqueness theorem for holomorphic functions that k(λ) = 0 also on the left half-plane. This implies that g − h = 0 by the uniqueness theorem for Laplace transforms. b) Let f ∈ BUC(R+ , X) be exponentially decreasing, but f = 0. Then spC (g) = R for all extensions g ∈ BUC(R, X) of f , but sp(f ) = ∅. In fact, there exist M ≥ 0, ε > 0 such that f (t) ≤ M e−εt . Thus the Laplace transform fˆ(λ) of f has a holomorphic extension to the region {Re λ > −ε} and fˆ(λ) is bounded for Re λ ≥ −ε/2. Now assume that g ∈ BUC(R, X) is an extension of f such that spC (g) = R. Then the Carleman transform gˆ(λ) of g agrees with fˆ(λ) for Re λ > 0, and hence for Re λ > −ε by the uniqueness theorem for holomorphic functions. Since gˆ(λ) is bounded for Re λ ≤ −ε/2, it follows that gˆ extends to a bounded entire function. So gˆ is constant by Liouville’s theorem. Since limλ→∞ fˆ(λ) = 0, it follows that fˆ = 0. Hence, f = 0. It is clear from the preceding remark that the assumption that the half-line spectrum be countable is much less restrictive than countability of the Carleman spectrum. In addition, we have the following observation about countability of a part of the spectrum. Let f ∈ E(R+ , X). As in Section 4.3, we deﬁne the set of all frequencies of f by Freq(f ) := {η ∈ R : Mη f = 0} . Since (Mη f )(0) = limα↓0 αfˆ(α + iη), it follows from (4.38) that Freq(f ) ⊂ sp(f ). It follows from Proposition 4.3.11 that Freq(f ) is countable. Now we can formulate the main result of this section which is a complex Tauberian theorem. With the help of Proposition 4.7.2, we are able to reduce the proof to an application of Gelfand’s theorem (Corollary 4.4.12). Theorem 4.7.7. Let f be a totally ergodic function in BUC(R+ , X) with countable half-line spectrum sp(f ). Then f is asymptotically almost periodic. The following corollary illustrates particularly well the Tauberian character of this theorem.

312

4. ASYMPTOTICS OF LAPLACE TRANSFORMS

Corollary 4.7.8. Let f ∈ E(R+ , X) with countable half-line spectrum. a) If Freq(f ) = ∅, then f ∈ C0 (R+ , X). b) If Freq(f ) = {0}, then limt→∞ f (t) exists. c) If Freq(f ) ⊂ 2π Z, where τ > 0, then there exists a τ -periodic, continuous τ function g : R+ → X such that limt→∞ f (t) − g(t) = 0. Proof. By Theorem 4.7.7, one has f = g + h ∈ AP(R+ , X) + C0 (R+ , X). Since Freq(g) = Freq(f ), the claim follows from Corollary 4.5.9. For the proof of Theorem 4.7.7 we need the following lemma which is analogous to Lemma 4.6.8. Lemma 4.7.9. Let f ∈ BUC(R+ , X). Assume that the Laplace transform fˆ of f has a holomorphic extension to a neighbourhood of iη, where η ∈ R. Then the function R(·, B)f : C+ → BUC(R+ , X) has a holomorphic extension to a neighbourhood of iη. Proof. We can assume that η = 0. Let V be a connected neighbourhood of 0 and g : V → X be holomorphic such that g(λ) = fˆ(λ) for λ ∈ V ∩ C+ . For λ ∈ V let s λs −λt g(λ) − e f (t) dt (s ∈ R+ ). G(λ, s) := e 0

Then for Re λ > 0,

∞

G(λ, s) =

e−λt f (t + s) dt = (R(λ, B)f )(s).

0

It is clear that G(·, s) : V → X is holomorphic for all s ∈ R+ . Choose r > 0 such that B(0, 2r) ⊂ V . We show that sup s∈R+ , |λ|≤r

G(λ, s) < ∞.

In fact, let M := sup|λ|≤2r g(λ) . If Re λ > 0, then

G(λ, s) = (R(λ, B)f )(s) ≤ R(λ, B)f ∞ ≤

f ∞ . Re λ

If Re λ < 0, |λ| = 2r, then |eλs | ≤ 1 and so s

f ∞

G(λ, s) ≤ M + e− Re λ (t−s) dt f ∞ ≤ M + | Re λ| 0 2rM + f ∞ ≤ . | Re λ|

4.7. ASYMPTOTICALLY ALMOST PERIODIC FUNCTIONS It follows from Lemma 4.6.6 that G(λ, s) ≤

4 3

2rM + f ∞ r

313

=: c for all λ ∈

B(0, r), s ∈ R+ . Now the claim follows from Corollary A.4 if we choose linear functionals ψ on BUC(R+ , X) which are of the form f, ψ := f (s), x∗ where s ∈ R+ , x∗ ∈ X ∗ , x∗ ≤ 1. Proof of Theorem 4.7.7. We keep the notation introduced before Proposition 4.7.2. Let f ∈ E(R+ , X) have countable spectrum. Assume that f := π(f ) = 0. Then f : t ∈ R} = {0}. Ef := span{S(t) the generator of the group S restricted to E. Let η ∈ R \ sp(f ). Denote by B f f By Lemma 4.7.9, there exists an open neighbourhood V of iη and a holomorphic function H : V → BUC(R+ , X) such that H(λ) = R(λ, B)f for λ ∈ V, Re λ > 0. Since H(λ) ∈ E(R+ , X) for Re λ > 0, it follows from the identity theorem (Proposition A.2) that H(λ) ∈ E(R+ , X) for all λ ∈ V . The function π ◦H : V → E is holomorphic and for λ ∈ V ∩ C+ , (π ◦ H)(λ) = R(λ, B)π(f ). It follows from ) contains Lemma 4.6.7 that iη ∈ σ(Bf). Thus, σ(Bf) is countable. But then σ(B f by Corollary 4.4.13. This contradicts an isolated point which is an eigenvalue of B f

Proposition 4.7.2. Thus, π(f ) = 0; i.e., f ∈ AAP(R+ , X).

An even more general version of Theorem 4.7.7 will be given in Section 4.9. However, before that some further preparation concerning harmonic analysis is given in the following section. We note an immediate corollary of Lemma 4.7.9. Corollary 4.7.10. Let f ∈ BUC(R+ , X). If η ∈ R \ sp(f ), then f is uniformly ergodic at η and Mη f = 0. We now give an example which shows that the condition of total ergodicity is crucial in Theorem 4.7.7, even in the scalar case. This contrasts dramatically with the situation of the entire line where total ergodicity is a consequence of countability of the Carleman spectrum if c0 ⊂ X (Theorem 4.6.3. The example also shows that ergodicity has to be uniform with respect to translates of the function. Recall that total ergodicity means by deﬁnition that the function is uniformly ergodic at each η ∈ R. √ Example 4.7.11. The function f (t) := sin t has half-line spectrum sp(f ) = {0} and 0 ∈ Freq(f ). Since f (t) does not converge to 0 as t → ∞, it follows from Thet orem 4.7.7 that f is not uniformly ergodic at 0. However, limt→∞ 1t 0 f (s) ds = 0. Proof. a) We show that √ fˆ(λ) =

πe−1/4λ 2λ3/2

(Re λ > 0).

314

4. ASYMPTOTICS OF LAPLACE TRANSFORMS

In fact, f (t) = sin term, we obtain

√

t=

∞

fˆ(λ)

(−1)n tn+1/2 (2n+1)! .

n=0 ∞

=

Thus, for λ > 0, integrating term by

√ sin t e−λt dt

0

∞ ∞ (−1)n = tn+1/2 e−λt dt (2n + 1)! 0 n=0 ∞ ∞ (−1)n 1 = un+1/2 e−u du n+3/2 (2n + 1)! λ 0 n=0 ∞ (−1)n −n−3/2 λ Γ(n + 3/2). (2n + 1)! n=0

=

Standard formulas for the Gamma function give 1 Γ(n + 3/2) = (n + 1/2)(n − 1/2) . . . Γ(1/2) 2 (2n + 1)(2n − 1) . . . 1 √ (2n + 1)! √ = π= π. 2n+1 n!22n+1 Thus, fˆ(λ)

=

∞

λ−n−3/2

n=0

(−1)n √ π n!22n+1

√

=

√ ∞ π 1 (−1)n π = 3/2 e−1/4λ . n 3/2 n! (4λ) 2λ 2λ n=0

By uniqueness of holomorphic extensions, this formula is true for Re λ > 0. It follows that sp(f ) = {0}. b) For t > 0, 1 t

t

f (s) ds = 0

1 t

√

t

2u sin u du 0

√t √ 1 2 t = [−2u cos u]0 + cos u du t t 0 √ √ 2 cos t 2 = − √ + sin t → 0 t t

as t → ∞. c) One can directly see that f is not uniformly ergodic at 0. In fact, given t > 0 we may choose s such that √ √ (2n + 1/4)π < s < s + t < (2n + 3/4)π

4.7. ASYMPTOTICALLY ALMOST PERIODIC FUNCTIONS

315

√ √ for some integer n (since s + t − s → 0 as s → ∞). Then f (u) > √12 whenever s+t s ≤ u ≤ s + t, so 1t s f (u) du > √12 . Thus, f is not uniformly ergodic at 0. We want to extend Theorem 4.7.7 to bounded measurable functions. Then we can only obtain assertions for the means of f , for example B-convergence, but this is suﬃcient for interesting applications to power series. Let f ∈ L∞ (R+ , X). We deﬁne the half-line spectrum sp(f ) as in Section 4.4; viz., sp(f ) := η ∈ R : fˆ(λ) does not have a holomorphic extension to a neighbourhood of iη . Moreover, we say that f is totally ergodic if for each η ∈ R, ∞ (Mη f )(t) := lim α e−(α+iη)s f (t + s) ds α↓0

0

converges uniformly in t ∈ R+ . In that case, it follows as in (4.25) that Mη f = eiη ⊗ x, where eiη (t) = eiηt , x = (Mη f )(0) ∈ X. We set Freq(f ) := {η ∈ R : Mη f = 0}. The following is a Tauberian theorem where B-convergence is deduced. Theorem 4.7.12. Let f ∈ L∞ (R+ , X) be totally ergodic. Assume that sp(f ) is countable and Freq(f ) ⊂ {0}. Then B-limt→∞ f (t) = (M0 f )(0). t+δ Proof. Let δ > 0, g(t) := 1δ t f (s) ds. Then g ∈ BUC(R+ , X). Integrating by parts we obtain for Re λ > 0, ∞ 1 d ! −λt " t+δ gˆ(λ) = − e f (s) ds dt δλ 0 dt t ∞ δ 1 −λt = f (s) ds + e (f (t + δ) − f (t)) dt δλ 0 0 ∞ δ 1 λδ −λt = f (s) ds + e e f (t) dt − fˆ(λ) δλ 0 δ δ δ 1 λδ 1 λδ −λs ˆ = (e − 1)f (λ) + f (s) ds − e e f (s) ds δλ δλ 0 0 δ λδ δ −λs 1 λδ 1 − e 1 − e λδ = (e − 1)fˆ(λ) + f (s) ds + e f (s) ds . δλ λδ λδ 0 0

316

4. ASYMPTOTICS OF LAPLACE TRANSFORMS

1 The right-hand summand deﬁnes an entire function of λ. Since δλ (eλδ −1) is entire, it follows that sp(g) ⊂ sp(f ). Next, we show that g is totally ergodic. Let r ∈ R, gr (s) := g(s + r), fr (s) := t+δ f (s + r). Then gr (t) = 1δ t fr (s) ds. By the computation above, we have for Re λ > 0, δ 1 λδ 1 − eλδ δ 1 − e−λδ gr (λ) = (e − 1)fr (λ) + fr (s) ds + eλδ fr (s) ds. δλ λδ λδ 0 0

Let λ = α + iη (α > 0). Then αfr (α + iη) converges uniformly in r ∈ R+ as α ↓ 0. Consequently, so does αgr (α + iη). We have shown that g is totally ergodic. Moreover, (M0 g)(0) = limα↓0 αˆ g (α) = limα↓0 αfˆ(α) = (M0 f )(0), whereas for η = 0, (Mη g)(0) = 0. It follows from Corollary 4.7.8 that g(t) converges as t → ∞; i.e., f (t) is B-convergent as t → ∞. It follows from Theorem 4.1.2 that B- lim f (t) = lim g(t) = (M0 g)(0) = (M0 f )(0). t→∞

t→∞

Lemma 4.7.13. Let f ∈ L∞ (R+ , X). Let η ∈ R such that

t

−iηs

sup e f (s) ds

< ∞. 0

t≥0

Then f is uniformly ergodic at η and Mη f = 0.

t

Proof. Let M be such that 0 e−iηs f (s) ds ≤ M (t ≥ 0). Let r ≥ 0. Then

t

iηr t+r −iηs

−iηs

e f (s + r) ds = e e f (s) ds

≤ 2M. 0

t

As an application of Theorem 4.7.12 we obtain the following result on power series. We let T := {z ∈ C : |z| = 1}. m Proposition 4.7.14. Let an ∈ X such that M := supm∈N n=0 an < ∞. Consider the power series ∞ an z n (|z| < 1). p(z) := n=0

Assume that for each z ∈ T \ {1} the function p has a holomorphic extension to a neighbourhood of z. Then limn→∞ an = 0. Proof. Let f (t) := an for t ∈ [n, n + 1), n ∈ N0 . Then f ∈ L∞ (R+ , X) and

f ∞ ≤ 2M . For Re λ > 0, one has m+1 1 ∞ ∞ fˆ(λ) = am e−λt dt = am e−λm e−λt dt m=0

=

1−e λ

m ∞ −λ m=0

m=0

am e−λm .

0

4.7. ASYMPTOTICALLY ALMOST PERIODIC FUNCTIONS

317

t

Thus, sp(f ) ⊂ 2πZ. We will show that supt≥0 0 e−iηs f (s) ds < ∞ for all η ∈ 2πZ. Then Lemma 4.7.13 and Corollary 4.7.10 imply that f is totally ergodic and Mη f = 0 for all η ∈ R. Finally, Theorem 4.7.12 implies that

n+1

lim an = lim

n→∞

n→∞

f (s) ds = B- lim f (t) = 0. n→∞

n

Let η = 0. For t ≥ 0, let n ∈ N such that t ∈ [n, n + 1). Then

t

f (s) ds = 0

n−1

am + an (t − n).

m=0

t

Thus, 0 f (s) ds ≤ 3M . Let η = 2πk, k ∈ Z \ {0}. Let t ≥ 0, t ∈ [n, n + 1). Then

t

e−iηs f (s) ds

0

=

n−1

t

= an

t

Hence, 0 e−iηs f (s) ds ≤ 2M .

e−i2πkt dt + an

m

m=0

m+1

am

t

e−i2πks ds

n

e−i2πks ds.

n

Corollary 4.7.15 (Katznelson-Tzafriri). Let T ∈ L(X) be an operator such that M := supn∈N T n < ∞. Then σ(T ) ∩ T ⊂ {1} if and only if limn→∞ T n (I − T ) = 0. Proof. a) Assume that σ(T ) ∩ T ⊂ {1}. Let p(z) :=

∞

(T − I)T n z n = (T − I)(I − zT )−1

for |z| < 1.

n=0

Then

m

(T − I)T n = T m+1 − I ≤ M + 1.

n=0

Thus the claim follows from Proposition 4.7.14. b) Suppose that limn→∞ T n (I − T ) = 0 and μ ∈ σ(T ) ∩ T. By the spectral mapping theorem, μn (1 − μ) ∈ σ(T n (I − T )). Then |1 − μ| = |μn (1 − μ)| ≤ T n (I − T ) → 0. Hence, μ = 1.

318

4.8

4. ASYMPTOTICS OF LAPLACE TRANSFORMS

Carleman Spectrum and Fourier Transform

For functions on the line, the Carleman spectrum is the natural notion if we are interested in the properties of the Laplace transform (or more precisely, the Carleman transform). However, the Fourier transform is also a powerful tool. In order to make it available we show in this section that the Carleman spectrum coincides with the support of the Fourier transform of the given function and also with the Beurling spectrum. As applications we give several results which allow us to deduce asymptotic properties of a function from the nature of its spectrum. For example, we show that a function f ∈ BUC(R, X) is almost periodic whenever it has discrete spectrum. A function f ∈ L∞ (R, X) is τ -periodic if and only if its spectrum is contained in 2π Z. τ Let f ∈ L1 (R). As in Section 1.8 and Appendix E, we denote by Ff ∈ C0 (R) the Fourier transform given by Ff (t) := e−ist f (s) ds (t ∈ R) (4.40) R

and we let Ff (t) :=

R

eits f (s) ds (t ∈ R).

(4.41)

By S(R) we denote the Schwartz space; i.e., the space of all inﬁnitely differentiable functions f : R → C such that supt∈R |f (m) (t)|(1 + |t|)k < ∞ for all m, k ∈ N0 . Then F is a bijective mapping from S(R) into S(R) with inverse (2π)−1 F . See Appendix E for further information. −k Let X be a Banach space. By L1 (R, we denote the space of (1 + |t|) dt; X) 1 all functions f ∈ Lloc (R, X) such that R f (t) (1 + |t|)−k dt < ∞, where k ∈ N0 . Note that L1 (R, X) ⊂ L1 (R, (1 + |t|)−k dt; X) for all k ∈ N0 . Let f ∈ L1 (R, (1 + |t|)−k dt; X), where k ∈ N0 . Then we deﬁne Ff as a linear mapping from S(R) into X by ϕ, Ff = f (t)(Fϕ)(t) dt (ϕ ∈ S(R)). R

The support of F f is deﬁned by supp F f := η ∈ R : for all ε > 0 there exists ϕ ∈ S(R) such that

supp ϕ ⊂ (η − ε, η + ε) and ϕ, F f , = 0 .

(4.42)

In the particular when k = 0, i.e. f ∈ L1 (R, X), Fubini’s theorem shows that case −ist ϕ, Ff = R R f (t)e dt ϕ(s) ds. Hence, supp Ff = {s ∈ R : (Ff )(s) = 0}− ,

(4.43)

4.8. CARLEMAN SPECTRUM AND FOURIER TRANSFORM

319

and our notation is consistent with (4.40) and the identiﬁcation of absolutely regular functions with distributions when X = C (see Theorem 1.8.1 b) and Appendix E). If f ∈ L1 (R, (1 + |t|)−k dt; X), then one can deﬁne the Carleman transform ˆ f : C \ (iR) → X and the Carleman spectrum spC (f ) exactly as in Section 4.6. We are going to show that supp Ff and spC (f ) coincide. We recall the notion of molliﬁer (ρn )n∈N from Section 1.3, but here we shall assume that ρ1 ∈ S(R). The function ρ1 ∈ S(R) satisﬁes R ρ1 (t) dt = 1. Then ρn ∈ S(R) is given by ρn (t) = nρ1 (nt) (t ∈ R, n ∈ N). Thus, F ρn ∈ S(R)

(n ∈ N),

(4.44)

and F ρn (t) = Fρ1 ( nt ), so lim (Fρn )(t) = 1 (t ∈ R); |(Fρn )(t)| ≤ |ρ1 (s)| ds (t ∈ R, n ∈ N). n→∞

(4.45) (4.46)

R

Moreover, if supp ρ1 ⊂ (−1, 1), then

! " supp ρn ⊂ − n1 , n1

(n ∈ N);

(4.47)

(n ∈ N).

(4.48)

and if supp Fρ1 ⊂ (−1, 1), then supp Fρn ⊂ (−n, n)

Theorem 4.8.1. Let k ∈ N0 , f ∈ L1 (R, (1 + |t|)−k dt; X). Then spC (f ) = supp F f . Proof. Let η ∈ R. Then by the deﬁnition of the Carleman transform, for α > 0, one has ∞ ˆ f (α + iη) = e−αt e−iηt f (t) dt and 0

fˆ(−α + iη)

=

−

0

eαt e−iηt f (t) dt.

−∞

Hence,

fˆ(α + iη) − fˆ(−α + iη) =

∞

e−α|t| e−iηt f (t) dt.

−∞

Let ϕ ∈ S(R). Then by Fubini’s theorem, ϕ, F f = f (t)(Fϕ)(t) dt R = lim e−α|t| f (t)(F ϕ)(t) dt α↓0 R = lim e−α|t| f (t) e−ist ϕ(s) ds dt α↓0 R R −α|t| −ist = lim e e f (t) dt ϕ(s) ds. α↓0

R

R

(4.49)

320

4. ASYMPTOTICS OF LAPLACE TRANSFORMS

Thus by (4.49), we obtain ϕ, F f = lim

α↓0

R

fˆ(α + is) − fˆ(−α + is) ϕ(s) ds

(4.50)

for all ϕ ∈ S(R). From this, the inclusion supp F f ⊂ spC (f ) follows immediately: Assume that η ∈ spC (f ). Then there exists ε > 0 such !that fˆ has a bounded holomorphic " extension to B(iη, ε). This implies that limα↓0 fˆ(α + is) − fˆ(−α + is) = 0 whenever s ∈ (η − ε, η + ε). Thus, by (4.50) and the dominated convergence theorem, ϕ, Ff = 0 whenever ϕ ∈ S(R) and supp ϕ ⊂ (η − ε, η + ε). Next, we prove the other inclusion: spC (f ) ⊂ supp Ff . a) We ﬁrst deal with the special case where k = 0; i.e., f ∈ L1 (R). Then F f is a continuous function. Let η ∈ R \ supp Ff . Then there exists ε > 0 such that Ff vanishes on (η − ε, η + ε). Putting fˆ(iβ) :=

∞

−iβt

e 0

f (t) dt = −

0

e−iβt f (t) dt

−∞

for β ∈ (η − ε, η + ε), fˆ has a continuous extension to a neighbourhood of iη. It follows from Morera’s theorem that this extension is holomorphic. Thus, η ∈ spC (f ). b) Now let k ∈ N0 be arbitrary. We ﬁrst give an estimate of fˆ. Let M := (1 + |t|)−k f (t) dt. Then R

fˆ(λ) ≤ M kk e1−k | Re λ|−k

(4.51)

whenever 0 < | Re λ| < 1. In fact, if 0 < Re λ < 1, then ∞

fˆ(λ) ≤ e− Re λt f (t) dt 0 ! " ≤ M sup (1 + t)k e− Re λt t>0

≤ M kk e1−k (Re λ)−k . The estimation is similar for −1 < Re λ < 0. Now let η ∈ R \ supp F f . Then there exists ε > 0 such that ϕ, F f = 0 for all ϕ ∈ S(R) with supp ϕ ⊂ (η − 2ε, η + 2ε). Let (ρn )n∈N be a molliﬁer in S(R), such that supp ρn ⊂ (− n1 , n1 ). Let fn := (Fρn ) · f . Since F ρn ∈ S(R), we have fn ∈ L1 (R, X). For n > 1/ε, we have supp Ffn ∩ (η − ε, η + ε) = ∅. In fact, let ϕ ∈ S(R) such that supp ϕ ⊂ (η − ε, η + ε). Then F f , ϕ = f F ϕ = f F ρn · F ϕ = n R n R f F (ρn ∗ ϕ) = 0 since ρn ∗ ϕ ∈ S(R) and supp ρn ∗ ϕ ⊂ supp ρn + supp ϕ ⊂ (− n1 , n1 ) + (η − ε, η + ε) ⊂ (η − 2ε, η + 2ε).

4.8. CARLEMAN SPECTRUM AND FOURIER TRANSFORM

321

From part a) of the proof we conclude that fˆn has a holomorphic extension to B(iη, ε). Since fn (t) ≤ f (t) ρ1 1 , we obtain from (4.51) (replacing f by fn ) that fˆn (λ) ≤ M ρ 1 k k e1−k | Re λ|−k whenever 0 < | Re λ| < 1. Now it follows from Lemma 4.6.6 that

fˆn (λ) ≤ c

(λ ∈ B(iη, ε/2))

(4.52)

for all n ∈ N, where c is a constant independent of n ∈ N. Since fn (t) → f (t) as n → ∞ for all t ∈ R (see Lemma 1.3.3), it follows from the dominated convergence theorem that limn→∞ fˆn (λ) = fˆ(λ) if Re λ > 0. It follows from Vitali’s theorem (Theorem A.5) that fˆ has a holomorphic extension to B(iη, ε/2). Thus, η ∈ spC (f ). The following are consequences of Theorem 4.8.1, and extensions of parts of Corollary 4.5.9 (see Proposition 4.6.10). Theorem 4.8.2. Let f ∈ L1 (R, (1 + |t|)−k dt; X) for some k ∈ N0 . a) spC (f ) = ∅ if and only if f (t) = 0 a.e. b) spC (f ) = {0} if and only if f is a polynomial. c) spC (f ) ⊂ {η 1 . . . ηm } if and only if there exist polynomials p1 , . . . , pm such m that f (t) = j=1 pj (t)eiηj t (t ∈ R). Proof. a) Assume that spC (f ) = ∅. Then it follows from the proof of Theorem 4.8.1 that R f (t)Fϕ(t) dt = 0 for all ϕ ∈ Cc∞ (R). Since Cc∞ (R) is dense in S(R) for the canonical topology of S(R) and since F is continuous(see Appendix E), it follows that R f (t)(Fϕ)(t) dt = 0 for all ϕ ∈ S(R). Hence, R f (t)ψ(t) dt = 0 for all ψ ∈ S(R). This implies that f (t) = 0 a.e. b) Assume that spC (f ) = {0}. Then the Laurent series fˆ(λ) =

∞

an λn

n=−∞

converges for λ ∈ C \ {0}, where 1 an := f (z)z −n−1 dz 2πi |z|=r

(n ∈ Z)

independently of r > 0. From (4.51) we obtain a constant c > 0 such that

fˆ(λ) ≤ c| Re λ|−k for 0 < | Re λ| < 1. This implies that

1

z2 k ˆ

n+k−1 z (1 + 2 ) f (z) dz ≤ c 2k r n

2πi |z|=r

r

(4.53)

(n ∈ Z, r ∈ (0, 1))

322

4. ASYMPTOTICS OF LAPLACE TRANSFORMS 2

(observe that for z = reiθ , (1 + zr2 )k fˆ(z) ≤ |1 + ei2θ |k c |r cos θ|−k = 2k cr−k ). Hence,

k

k

1 −2j n+k−1+2j ˆ

c 2k r n ≥ r · z f (z) dz

2πi |z|=r

j=0 j

k k −2j

= r a −n−k−2j

j=0 j

for all r ∈ (0, 1). Multiplying by r 2k , we obtain

k

k

k n+2k 2k−2j

c2 r ≥ r a−n−k−2j

. j

j=0

Now let n ≥ 1−2k. Then the left-hand term, as well as all terms on the right except for j = k, converge to 0 as r ↓ 0. Consequently, a−n−3k = 0 whenever n ≥ 1 − 2k, ∞ k i.e., am = 0 if m < −k, so fˆ(λ) = n=−k an λn . Let f0 (t) := j=0 a−j−1 tj /j!. −1 Then fˆ0 (λ) = n=−k an λn . Thus fˆ− fˆ0 is entire and so spC (f −f0 ) = ∅. It follows from a) that f (t) = f0 (t) a.e. c) We prove the assertion by induction. Let m = 1. Then spC (f ) = {η1 }. Let g(t) := e−iη1 t f (t). Then spC (g) = 0. It follows from b) that g is a polynomial. Now assume that the assertion holds for m and that spC (f ) = {η1 , . . . , ηm+1 }, where ηj = ηl if j = l. Replacing f (t) by e−iηm+1 t f (t) if necessary, we can assume that ηm+1 = 0. It follows from the proof of b) that 0 is a pole of fˆ. Moreover, there exists a polynomial pm+1 such that (f − pm+1 ) has a holomorphic extension to a neighbourhood of 0. Thus, spC (f − pm+1 ) ⊂ {η 1m, . . . , ηm }. Now it follows from the inductive assumption that (f − pm+1 )(t) = n=1 pn (t)eiηn t (t ∈ R). Corollary 4.8.3. Let f ∈ L∞ (R, X). a) If spC (f ) = {0}, then f is constant. b) If spC (f ) is ﬁnite, then f is a trigonometric polynomial. If f ∈ L∞ (R, X), there is an alternative way to describe the spectrum of f . We deﬁne the Beurling spectrum by spB (f ) := η ∈ R : for all ε > 0 there exists g ∈ L1 (R) such that supp Fg ⊂ (η − ε, η + ε) and f ∗ g = 0 . Proposition 4.8.4. Let f ∈ L∞ (R, X). Then spC (f ) = spB (f ).

4.8. CARLEMAN SPECTRUM AND FOURIER TRANSFORM

323

Proof. By Theorem 4.8.1, we have spC (f ) = supp F f = η ∈ R : for all ε > 0 there exists ϕ ∈ S(R) such that supp ϕ ⊂ (η − ε, η + ε) and =

η ∈ R : for all ε > 0 there exists h ∈ S(R) such that supp F h ⊂ (η − ε, η + ε) and

=

=

R

f (t)h(t) dt = 0

η ∈ R : for all ε > 0 there exists h ∈ S(R) such that supp F h ⊂ (η − ε, η + ε) and

R

f (t)(F ϕ)(t) dt = 0

R

f (t)h(−t) dt = 0

η ∈ R : for all ε > 0 there exists h ∈ S(R) such that supp F h ⊂ (η − ε, η + ε) and (f ∗ h)(0) = 0 .

This shows that spC (f ) ⊂ spB (f ). In order to show the converse, let η ∈ spB (f ). Let ε > 0. Then there exists g ∈ L1 (R) such that supp F g ⊂ (η − ε, η + ε) and f ∗ g = 0. Replacing g by g(· − t) if necessary, we can assume that (f ∗ g)(0) = 0. Take a molliﬁer (ρn )n∈N in S(R) such that supp ρn ⊂ (− n1 , n1 ), and deﬁne gn := F ρn · g. By (4.45), (4.46) and the dominated convergence theorem, limn→∞ gn = g in L1 (R). It follows from Proposition 1.3.2 that limn→∞ (gn ∗ f )(0) = (g ∗ f )(0) = 0. We have ρn ∗ Fg ∈ C ∞ (R) (see Proposition 1.3.6) and supp(ρn ∗ Fg) ⊂ 1 1 (− n , n ) + (η − ε, η + ε) ⊂ (η − 2ε, η + 2ε) if n > 1/ε. Moreover, (ρn ∗ Fg)(s) = ρn (s − r)Fg(r) dr = ρn (s − r) g(t)e−irt dt dr = ρn (r) g(t)e−i(s−r)t dt dr = F((Fρn )·g)(s). Hence, Fgn = F((Fρn )·g) ∈ Cc∞ (R) ⊂ S(R). Since F is a bijection from S(R) onto S(R), it follows that gn ∈ S(R). We have shown that there exists n > 1/ε such that gn ∈ S(R), supp F gn ⊂ (η − 2ε, η + 2ε) and (f ∗ gn )(0) = 0. Since ε > 0 is arbitrary, it follows that η ∈ supp F f = spC (f ). So far, we have established the identity of three diﬀerent notions of spectrum of a function f ∈ L∞ (R, X): the Carleman spectrum, the support of its Fourier transform, and the Beurling spectrum. Next, we consider two situations in which a special form of the spectrum of a function tells us a lot about the nature of the function itself (Theorems 4.8.7 and 4.8.8). We need the following auxiliary result. Lemma 4.8.5. Let f ∈ L∞ (R, X) and g ∈ L1 (R). Then spC (f ∗ g) ⊂ spC (f ) ∩ supp Fg. Proof. a) Let η ∈ R \ spC (f ). By Proposition 4.8.4, there exists ε > 0 such that

324

4. ASYMPTOTICS OF LAPLACE TRANSFORMS

f ∗ h = 0 whenever h ∈ L1 (R), supp Fh ⊂ (η − ε, η + ε). Hence, (f ∗ g) ∗ h = (f ∗ h) ∗ g = 0. Thus, η ∈ spB (f ∗ g) = spC (f ∗ g). b) Let η ∈ supp F g. We show that η ∈ spB (f ∗ g). There exists ε > 0 such that (Fg)(r) = 0 for all r ∈ (η − ε, η + ε). Let h ∈ L1 (R) such that supp Fh ⊂ (η − ε, η + ε). Then F(g ∗ h) = F g · Fh = 0. Thus, g ∗ h = 0. Hence, (f ∗ g) ∗ h = 0. This proves the claim. Remark 4.8.6. The same property is true if f ∈ L1 (R, (1 + |t|)−k dt; X) for some k ∈ N0 and g ∈ S(R). Now we can prove that a function f ∈ BUC(R, X) is almost periodic whenever it has discrete Carleman spectrum. Thus, the geometric condition “c0 ⊂ X”, which appeared in Theorem 4.6.3, is not needed if the spectrum does not have any accumulation point. Theorem 4.8.7. Let f ∈ BUC(R, X). If the spectrum spC (f ) of f is discrete, then f is almost periodic. Proof. Let (ρn )n∈N be a molliﬁer in S(R) such that supp F ρ1 ⊂ (−1, 1). By Proposition 1.3.2 and Lemma 4.8.5, f ∗ ρn ∈ BUC(R, X) and spC (f ∗ ρn ) ⊂ spC (f ) ∩ supp F ρn ⊂ spC (f ) ∩ (−n, n). Thus, f ∗ ρn has ﬁnite spectrum, and it follows from Corollary 4.8.3 that f ∗ ρn ∈ AP(R, X). By Lemma 1.3.3, it follows that f = limn→∞ f ∗ ρn ∈ AP(R, X). Next, we give a spectral characterization of periodic functions extending part of Corollary 4.5.9. If g is a function deﬁned on R we let gˇ(t) = g(−t) (t ∈ R). Theorem 4.8.8. Let f ∈ L∞ (R, X) and let τ > 0. Then f is τ -periodic (i.e., f (t + τ ) = f (t) t-a.e.) if and only if spC (f ) ⊂ 2π τ Z. Proof. a) Let f be τ -periodic. Then fˆ(λ) = (1 − e−λτ )−1

τ

e−λt f (t) dt

0

for λ ∈ C \ iR. Thus, fˆ extends to a meromorphic function with at most simple poles at λn := 2πin/τ (n ∈ Z). The residue at λn is given by 1 τ −2πint/τ cn := e f (t) dt (n ∈ Z). τ 0 Therefore we have spC (f ) = {2πn/τ : n ∈ Z, cn = 0} . b) Assume that spC (f ) ⊂ 2π τ Z. Let (ρn )n∈N be a molliﬁer in S(R) such that supp Fρn ⊂ (−n, n) and ρˇn = ρn (n ∈ N). Then ρn ∗ f ∈ BUC(R, X) and spC (ρn ∗ f ) ⊂ supp Fρn ∩ spC (f ) ⊂ (−n, n) ∩ spC (f ) ⊂ (−n, n) ∩ 2π τ Z. It follows

4.9. COMPLEX TAUBERIAN THEOREMS: THE FOURIER METHOD 325 from Corollary 4.8.3 and Proposition 4.6.10 that ρn ∗f is a τ -periodic trigonometric polynomial. Letϕ ∈ Cc (R). Then limn→∞ ρn ∗ ϕ = ϕ in L1 (R). Thus, f (t + τ ) ϕ(t)dt = limn→∞ f (t + τ ) (ρn ∗ ϕ)(t) dt = limn→∞ (f ∗ρn )(t + τ ) ϕ(t) dt = limn→∞ (f ∗ ρn )(t) ϕ(t) dt = limn→∞ f (t) (ρn ∗ ϕ)(t) dt = f (t) ϕ(t) dt. Since ϕ ∈ Cc (R) is arbitrary, it follows that f (t + τ ) = f (t) t-a.e.

4.9

Complex Tauberian Theorems: the Fourier Method

In this section we present an approach via Fourier transforms to complex Tauberian theorems for Laplace transforms. This method was already used in a restricted form by Ingham in 1935 (see [Ing35]). It will eventually lead to the most general complex Tauberian theorem presented in this book (Theorem 4.9.7). It is not such a great diﬀerence to consider bounded measurable functions which are slowly oscillating, instead of uniformly continuous functions as we did before, but the point is that the notion of spectrum is changed. Instead of considering a point iη as regular if the Laplace transform has a holomorphic extension in a neighbourhood of iη, we merely assume that a locally integrable extension to the imaginary axis exists. This leads to a smaller spectrum which we call the weak half-line spectrum. Thus, asking that this small spectrum be countable is a weaker hypothesis. It turns out that this weaker hypothesis is more natural or easier to verify in some applications (see the Notes of this section). In the case where the weak half-line spectrum is empty, the proof is completely elementary and leads to a slight generalization of Ingham’s Tauberian theorem. We consider this case ﬁrst (Theorem 4.9.5). The general case will be proved by a Hahn-Banach argument which allows us to apply Loomis’s theorem in a quite tricky way. Let f ∈ L∞ (R+ , X). We deﬁne the weak half-line spectrum spw (f ) of f as follows. Deﬁnition 4.9.1. Let η ∈ R. We say that iη is a weakly regular point for fˆ if there exist ε > 0 and h ∈ L1 ((η − ε, η + ε), X) such that fˆ(α + i·) → h in the distributional sense on (η − ε, η + ε) as α ↓ 0.

(4.54)

Then the weak half-line spectrum spw (f ) of f is deﬁned as the set of all real numbers which are not weakly regular for fˆ. Of course, (4.54) means by deﬁnition that lim α↓0

fˆ(α + is)ϕ(s) ds =

R

η+ε

h(s)ϕ(s) ds η−ε

for all test functions ϕ ∈ D(η − ε, η − ε) = Cc∞ (η − ε, η − ε).

(4.55)

326

4. ASYMPTOTICS OF LAPLACE TRANSFORMS

It is clear that spw (f ) is a closed subset of R. Moreover, if fˆ(λ) has a continuous extension to C+ ∪ (i(η − ε), i(η + ε)), where η ∈ R, ε > 0, then clearly η ∈ spw (f ). In particular, we have spw (f ) ⊂ sp(f ). The inclusion is strict in general as the following example shows. Example 4.9.2. Let X := l2 , f (t) := n−1 e−t/n

. n∈N

Then fˆ has a continuous extension to C+ ∪ iR, but not a holomorphic extension to a neighbourhood of 0. Lemma 4.9.3. Let f ∈ L∞ (R+ , X). Then there exists h ∈ L1loc (R \ spw (f ), X) such that fˆ(α + i·) → h in the distributional sense on R \ spw (f ) as α ↓ 0.

(4.56)

We set fˆ(is) := h(s) (s ∈ R \ spw (f )). Proof. For all x ∈ R \ spw (f ), we ﬁnd an open neighbourhood Ux of x and hx ∈ L1 (Ux , X) such that fˆ(α + i·) → hx in the distributional sense on Ux as α ↓ 0. Clearly, hx (t) = hy (t) almost everywhere on Ux ∩ Uy whenever x, y ∈ R \ spw (f ). Hence, there exists a function h ∈ L1loc (R \ spw (f ), X) such that h|Ux = hx

(x ∈ R \ spw (f )).

It remains to show (4.56). Let ϕ ∈ D(R \ spw (f )). Let K := supp ϕ. There exist x1 , x2 , . . . , xn such that the sets Uj := Uxj (j = 1, 2, . . . , n) cover K. Let ϕj ∈ D(R \ spw (f )) (j = 1, 2, . . . , n) be a partition of unity nsubordinate to this covering; i.e., 0 ≤ ϕj ≤ 1, supp ϕj ⊂ Uj (j = 1, 2, . . . , n), j=1 ϕj (x) = 1 for all x ∈ K. n Then ϕ = j=1 ψj , where ψj := ϕϕj . Since supp ψj ⊂ Uj , lim fˆ(α + is)ψj (s) ds = h(s)ψj (s) ds α↓0

R

Uj

for all j = 1, 2, . . . , n. Hence, lim fˆ(α + is)ϕ(s) ds = α↓0

R

h(s)ϕ(s) ds.

(4.57)

R\spw (f )

Since ϕ ∈ D(R \ spw (f )) is arbitrary, this is precisely the meaning of (4.56). Lemma 4.9.4. Let f ∈ L∞ (R+ , X). Then ϕ ∗ f ∈ C0 (R+ , X) for all ϕ ∈ S(R) such that Fϕ ∈ Cc∞ (R) and supp F ϕ ∩ spw (f ) = ∅.

4.9. COMPLEX TAUBERIAN THEOREMS: THE FOURIER METHOD 327 Here as elsewhere, we identify a function deﬁned on R+ with its extension by 0 on R. In particular, ∞ (ϕ ∗ f )(t) = f (s)ϕ(t − s) ds (t ∈ R). 0

Proof of Lemma 4.9.4. Let ϕ ∈ S(R) such that F ϕ ∈ Cc∞ (R) and supp Fϕ ∩ spw (f ) = ∅. Then Fϕ · fˆ(i·) ∈ L1 (R, X), where fˆ(i·) was deﬁned in Lemma 4.9.3. Let t ≥ 0. Then the inverse Fourier transform of the function s → ϕ(t − s) is the function η → (2π)−1 eiηt Fϕ(η). Thus by Theorem 1.8.1 b), ∞ (ϕ ∗ f )(t) = f (s)ϕ(t − s) ds 0 ∞ = lim e−αs f (s)ϕ(t − s) ds α↓0 0 −1 = lim (2π) fˆ(α + iη)eiηt Fϕ(η) dη α↓0 R −1 = (2π) fˆ(iη)Fϕ(η)eiηt dη. R

It follows from the Riemann-Lebesgue lemma (Theorem 1.8.1) that limt→∞ (ϕ ∗ f )(t) = 0. Now we prove the complex Tauberian theorem in the case where the weak half-line spectrum is empty. Theorem 4.9.5 (Ingham). Let f ∈ L∞ (R+ , X) be slowly oscillating. If spw (f ) = ∅, then lim f (t) = 0. t→∞

Proof. a) We show that g ∗ f ∈ C0 (R+ , X) for all g ∈ L1 (R). By Proposition 1.3.2, T g := (g ∗ f )|R+ deﬁnes a bounded linear operator from L1 (R) into BUC(R+ , X). Let φ be a continuous linear functional on BUC(R+ , X) vanishing on C0 (R+ , X). By the Hahn-Banach theorem, it suﬃces to show that T g, φ = 0 for all g ∈ L1 (R). Let h := T ∗ φ ∈ L∞ (R). Then g ∗ f, φ = hg dt R

for all g ∈ L1 (R). It follows from Lemma 4.9.4 that R hϕ dt = 0 if ϕ ∈ S(R) and Fϕ ∈ Cc∞ (R). Hence, supp F h = ∅ (see (4.42)). Thus, spC (h) = ∅ by Theorem 4.8.1 which implies that h = 0 by Theorem 4.8.2. This proves the claim. We have shown that g ∗ f ∈ C0 (R+ , X) for all g ∈ L1 (R). b) Taking in particular g := 1δ χ[0,δ] , it follows from a) that B-limt→∞ f (t) = 0. Now it follows from Theorem 4.2.3 that limt→∞ f (t) = 0.

328

4. ASYMPTOTICS OF LAPLACE TRANSFORMS

We will see in Example 5.5.7 that in general the hypothesis that g be slowly oscillating cannot be omitted. Next, we consider the case of countable weak spectrum and we extend Theorem 4.7.12. We need the notion of uniform ergodicity on a subset of R, for bounded measurable functions, similar to the total ergodicity of Section 4.7. Deﬁnition 4.9.6. Let f ∈ L∞ (R+ , X) and let E be a subset of R. We say that f is uniformly ergodic on E if for all η ∈ E the limit ∞ (Mη f ) (t) = lim α e−(α+iη)s f (t + s) ds α↓0

0

exists uniformly in t ∈ R+ . As in (4.25) it follows that Mη f = eiη ⊗ x where eiη (t) = e

iηt

, x = (Mη f ) (0) ∈ X.

The following complex Tauberian theorem is the main result of this section. Its proof is based on the same idea as the one of Theorem 4.9.5, but it is less elementary since Loomis’s theorem is used. Theorem 4.9.7. Let f ∈ L∞ (R+ , X) be slowly oscillating. Assume that spw (f ) is countable and that f is uniformly ergodic on spw (f ). Then f = f0 + f1 , where f1 ∈ AP(R+ , X) and limt→∞ f0 (t) = 0. In particular, if f ∈ BUC(R+ , X) then f ∈ AAP(R+ , X). Proof. a) We show that g ∗ f ∈ AAP(R+ , X) for all g ∈ L1 (R) (and for this we do not need the assumption that f is slowly oscillating). As in the proof of Theorem 4.9.5, we consider the operator T : L1 (R) → BUC(R+ , X) given by T g := (g ∗ f )|R+ . Let φ ∈ BUC(R+ , X)∗ be a continuous linear functional on BUC(R+ , X) which vanishes on AAP(R+ , X). We have to show that g ∗ f, φ = 0 for all g ∈ L1 (R). Then the claim follows from the Hahn-Banach theorem. Let h := T ∗ φ. As in the proof of Theorem 4.9.5 it follows from Lemma 4.9.4 that spC (h) ⊂ spw (f ). Hence, spC (h) is countable. Let ϕ ∈ L1 (R). Since spC (ϕ ∗ h) ⊂ spC (h) (by Lemma 4.8.5), spC (ϕ ∗ h) is also countable. Moreover, ϕ ∗ h ∈ BUC(R) by Proposition 1.3.2. It follows from Loomis’s theorem (Corollary 4.6.4) that ϕ ∗ h ∈ AP(R) for all ϕ ∈ L1 (R). Next, let ϕ ∈ Cc∞ (0, ∞). We show that Mη (ϕ ∗ h) = 0 for all η ∈ R in order to conclude that ϕ ∗ h = 0 by Corollary 4.5.9. First, since Freq(ϕ ∗ h) ⊂ spC (ϕ ∗ h) ⊂ spw (f ), it is clear that Mη (ϕ ∗ h) = 0 for all η ∈ R \ spw (f ). Now, let η ∈ spw (f ). Let cη := Mη (ϕ ∗ h)(0). Then Mη (ϕ ∗ h)(t) = cη eiηt

(see (4.25)).

4.10. NOTES

329

We have to show that cη = 0. For λ ∈ R, we deﬁne eλ (t) := eλt (t ∈ R). Note that ∞ cη = lim α e−(α+iη)t (ϕ ∗ h) (t) dt α↓0

0

= lim αχR+ e−(α+iη) , ϕ ∗ h α↓0

= lim αh, χR− eα+iη ∗ ϕ α↓0

= lim α(f ∗ χR− eα+iη ∗ ϕ)|R+ , φ . α↓0

However, kα (t) := α

0

−∞

f (t − s)ei(α+iη)s ds = α

∞

f (t + s)e−i(α+iη)s ds

0

converges to the function eiη ⊗ x as α ↓ 0 uniformly in t ≥ 0. Since ϕ ∈ Cc∞ (0, ∞), this implies that (kα ∗ ϕ)(t) converges to ∞ eiηt e−iηs ϕ(s) ds x =: k(t) 0

uniformly in t ≥ 0 as α ↓ 0. Thus, cη = k, φ . Since k ∈ AP(R+ , X), it follows that cη = 0. We have shown that h ∗ ϕ = 0 for all ϕ ∈ Cc∞ (0, ∞). Choosing a molliﬁer (ρn )n∈N ⊂ Cc∞ (−∞, 0), we deduce that for all k ∈ L1 (R), k, h = lim k ∗ ρn , h = lim k, ρˇn ∗ h = 0. n→∞

n→∞

Thus, T ∗ φ = h = 0. Here, for v : R → X we let vˇ(t) = v(−t). It follows that g ∗ f, φ = 0 for all φ ∈ BUC(R+ , X)∗ vanishing on AAP(R+ , X). Hence, g ∗ f ∈ AAP(R+ , X). b) Since f is slowly oscillating, f = g0 + g1 where g1 ∈ BUC(R+ , X) and limt→∞ g0 (t) = 0, by Proposition 4.2.2. Let (ρn )n∈N ⊂ Cc∞ (0, ∞) be a molliﬁer with supp ρn ⊂ (0, 1/n). Then ρn ∗ g0 ∈ C0 (R+ , X), and ρn ∗ f ∈ AAP(R+ , X) by a). Hence, ρn ∗ g1 = ρn ∗ f − ρn ∗ g0 ∈ AAP(R+ , X). Since g1 ∈ BUC(R+ , X), limn→∞ ρn ∗ g1 = g1 in BUC(R+ , X). Thus, g1 ∈ AAP(R+ , X). Now, g1 = f1 + f2 with f1 ∈ AP(R+ , X), f2 ∈ C0 (R+ , X). Thus, f = f1 + (f2 + g0 ).

4.10 Notes Sections 4.1 and 4.2 The prototype for Abelian theorems was Abel’s classical continuity theorem (Theorem 4.1.6) which was proved by N. Abel [Abe26] in 1826. Tauberian theorems form their counterpart and the ﬁrst result of this type is Tauber’s classical theorem from 1897

330

4. ASYMPTOTICS OF LAPLACE TRANSFORMS

[Tau97] which is mentioned in connection with Hardy’s Theorem 4.2.17. There is an enormous amount of literature on Tauberian theorems. We refer to the monographs by Widder [Wid71] and by Doetsch [Doe50, Volume I] for the classical results. The monographs of Pitt [Pit58] and van de Lune [Lun86], and Chatterji’s historical account [Cha84], cover some of the subsequent developments, and Korevaar’s book [Kor04] is a wide-ranging account which provides much historical information up to recent times. The notion of B-limit was introduced by Arendt and Pr¨ uss [AP92] following ideas of Batty [Bat90] showing how to pass from theorem of type D to results of type E. Theorem 4.2.15 illustrates in a surprising way how this strategy can be used. The presentation given here is close to [AP92]. With the help of Wiener’s Tauberian theorem the following clariﬁcation of Bconvergence was given by Arendt and Batty [AB00]. Theorem 4.10.1. Let u ∈ L∞ (R, X), u∞ ∈ X, and suppose that 1 t→∞ δ

lim

t

t+δ

u(s) ds = u∞

holds for δ = δ1 and δ = δ2 , where δ1 and δ2 are rationally independent. Then

lim (ρ ∗ u)(t) = ρ u∞ t→∞

R

for all ρ ∈ L1 (R). In particular, B-limt→∞ u(t) = u∞ . Karamata’s theorem has important applications to the study of eigenvalue distributions (see e.g. [Sim79, Theorem 10.6]). An elegant short proof of Karamata’s theorem in the scalar case has been given by K¨ onig [K¨ on60]. Further Abelian and Tauberian theorems for positive vector-valued functions were given by El-Mennaoui [Elm94]. Another direction of Tauberian theorems for Laplace transforms occurs in the framework of limitation theory. A ﬁnal result in this direction has been obtained by Stadtm¨ uller and Trautner [ST81] and by Kratz and Stadtm¨ uller [KS90] in the discrete case. Section 4.3 The results at the beginning of the section are quite standard. A systematic study of mean convergence of integrated semigroups was carried out by El-Mennaoui [Elm92]. In particular, Theorem 4.3.7, saying that for positive C0 semigroups Ces` aro-ergodicity and Abel-ergodicity are equivalent, can be extended to integrated semigroups. We state a special case explicitly as follows: Let E be a Banach lattice with order continuous norm. Let A be a resolvent positive operator, so A generates a once integrated semigroup S (by Theorem 3.11.7). Then X = Ker A ⊕ Ran A if and only if P x = lim

t→∞

1 S(t)x t

exists for all x ∈ E. In that case, P is the projection onto Ker A along Ran A. See [Elm92, Theorem 4.1].

4.10. NOTES

331

Theorem 4.3.7 and Theorem 4.3.8 are explicitly proved in Arendt and Pr¨ uss [AP92]. More generally, El-Mennaoui [Elm92] studied strong convergence of t1k S(t) (as t → ∞), where S is a k-times integrated semigroup and k ∈ N. This is a very natural problem. In fact, if A is the generator of S, then for x ∈ D(Ak+1 ), S(·)x ∈ C k+1 ((0, ∞), X) and u(t) := S (k) (t)x deﬁnes a classical solution of the Cauchy problem u (t) = Au(t) (t ≥ 0), u(0) = x. t k−1 Thus, t1k S(t)x = t1k 0 (t−s) u(s) ds is the kth Ces` aro mean of the solution u and (k−1)! describes its asymptotic behaviour for t → ∞. Proposition 4.3.13 is due to Arendt and Batty [AB92a] who also showed that the result does not hold on every Banach lattice. In fact, an example is given in [AB92a] which shows that a positive C0 -semigroup on a space C(K), where K is compact, may well be Ces` aro-ergodic without being totally ergodic. However, in Proposition 4.3.13 the L1 -space may be replaced by any Banach lattice with order continuous norm. The striking properties concerning C0 -semigroups on L∞ -spaces are due to Lotz [Lot85]. In fact, it had been proved before by Kishimoto and Robinson [KR81] that every generator of a positive C0 -semigroup on an L∞ -space is bounded, and independently of Lotz, Coulhon [Cou84] had proved Corollary 4.3.19 for contraction semigroups. However, it was Lotz who discovered the interesting interplay of the geometric properties (G) and (DP). He proved in particular Lemma 4.3.16 and also proved ergodic theorems in the discrete case (i.e., for power bounded operators). Other examples of Banach spaces having both properties (DP) and (G) are the following: a) C(K), where K is an F -space (cf. the Notes of Section 2.7). b) H ∞ (D), the space of all bounded holomorphic functions on the unit disc D := {z ∈ C : |z| < 1} with the supremum norm. We refer to [Nag86, Section A-II.3] and the references given there. Section 4.4. A version of the complex Tauberian Theorem 4.4.1 was proved by Ingham [Ing35]. The ingenious contour argument proof of a case of Ingham’s theorem was given in Korevaar’s beautiful article [Kor82] which also gives an elementary proof of the prime number theorem based on Ingham’s Tauberian theorem. Korevaar was inspired by Newman [New80] who proved the corresponding result for Dirichlet series; see also the book by Newman [New98]. The quantiﬁed version presented in Theorem 4.4.6 is due to Batty and Duyckaerts [BD08]. Remark 4.4.7 shows that the estimate (4.15) is quite sharp when M is bounded or grows very slowly, and Borichev and Tomilov [BT10] have shown that it is optimal for polynomial growth. Theorem 4.4.8, as stated here, is due to Arendt and Batty [AB88], who also gave other versions allowing a countable number or even a null set of singularities. Here, merely the case of one singularity is presented which allows several interesting applications including those given in this section and later in the chapter. A quantiﬁed version is given in [BD08]. Further versions are contained in the work of Arendt and Pr¨ uss [AP92]. Ces` aro convergence is investigated in [AB95] with the help of contour arguments similar to those of Theorem 4.4.8. For example, it is shown that a function f ∈ L1loc (R+ , X) is Ces´ aro

332

4. ASYMPTOTICS OF LAPLACE TRANSFORMS

convergent if f (t) = O(t) as t → ∞ and every point of iR is either regular for fˆ or a pole of order 1. Corollary 4.4.12 is contained in Arveson’s work on spectral subspaces [Arv82], but it is usually associated with Gelfand’s name. More precisely, it is the corresponding result on bounded operators which is due to Gelfand (saying that an isometry whose spectrum is {1} is necessarily the identity). Since the weak spectral mapping theorem holds for bounded C0 -groups [Nag86, A-III, Theorem 7.4], Corollary 4.4.12 follows immediately from Gelfand’s classical theorem on isometries. Another elegant proof of Gelfand’s theorem is due to Allan and Ransford [AR89], see also the survey article of Zemanek [Zem94]. Extensions of Gelfand’s theorem have been obtained by Zarrabi [Zar93]. He proved in particular that an invertible contraction T on a Banach space X with countable spectrum is already an isometry if it satisﬁes the growth condition log T −n

√ = 0. n→∞ n lim

Theorem 4.4.14 is taken from [BD08]. Remarkably the implication (ii) ⇒ (i) does not seem to have been in the earlier literature of operator semigroups. The implication (i) ⇒ (ii) was already implicit in [AB88] (see [Bat94, p.41]). Quantiﬁed versions were obtained for polynomial growth of M on arbitrary Banach spaces [BEPS06], and for polynomial or exponential growth of M [Bur98], [Leb96], [LR05], using methods which generally give slightly slower rates of decay than (4.19). For M (s) = β(1 + s)α , it is −1 shown in [BT10] that the rate in (4.19) is optimal on general Banach spaces, but Mlog can be replaced by M −1 in the case of Hilbert spaces. A version of Theorem 4.4.14 when σ(A) ∩ iR is ﬁnite is given in [Mar11]. These results have applications to energy decay of damped wave equations of the form ⎧ 2 ∂ u ∂u ⎪ ⎪ (t > 0, x ∈ Ω), ⎪ ⎨ ∂t2 = Δu − 2a(x) ∂t (4.58) u(x, t) = 0 (t > 0, x ∈ ∂Ω), ⎪ ⎪ ⎪ ∂u 1 2 ⎩ u(·, 0) = u0 ∈ H0 (Ω), (·, 0) = u1 ∈ L (Ω). ∂t Here, Ω is a smooth or convex open subset of Rn with boundary ∂Ω, and a : Ω → [0, ∞) is a bounded continuous function representing the damping at each point. Let X be the Hilbert space H01 (Ω) × L2 (Ω), and let A be the operator on X deﬁned by 2 D(A) := H (Ω) ∩ H01 (Ω) × H01 (Ω),

0 I A := . Δ −2a(x) Then A generates a C0 -semigroup of contractions on X, and the energy of the solution u of (4.58) is given by E(u, t) := 12 T (t)(u0 , u1 ) 2X . The following result was proved by Lebeau [Leb96] with a slightly slower rate of decay in b), and the optimal rate given here was established in [Bur98]. Once a) is proved, b) follows directly from Theorem 4.4.14 a) with M (s) = βeαs . Theorem 4.10.2. Consider the damped wave equation (4.58), where Ω is a bounded open set and a is not identically zero. We assume that Ω is smooth or convex. Let A and E(u, t) be as above. Then the following hold.

4.10. NOTES

333

a) σ(A) ⊂ {λ ∈ C : −2 a ∞ ≤ Re λ < 0}, and there exist α, β > 0 such that

R(is, A) ≤ βeα|s| for all s ∈ R. b) There is a constant C such that E(u, t) ≤ H01 (Ω), u1 ∈ H01 (Ω), and t ≥ 0.

C (u0 , u1 ) 2H 2 ×H 1 for all u0 ∈ H 2 (Ω) ∩ (log(2 + t))2

When the data is suﬃciently smooth that (u0, u1 ) ∈ D(Ak ), the rate of energy decay in b) can readily be improved to O (log t)−k . In many cases, the rate is much faster than logarithmic. When the domain of damping {x ∈ Ω : a(x) > 0} satisﬁes the “geometric optics condition”, the energy decay is exponential (and uniform with respect to (u0 , u1 ) X ), i.e., ω(T ) < 0 and R(is, A) is bounded [BLR89]. When Ω is a rectangle in R2 and the domain of damping is a strip across Ω, the geometric optics condition is not satisﬁed but R(is, A) = O(s2 ) as |s| → ∞ [LR05, Example 3]. In this case, the optimal rate of decay E(u, t) = O(t−1 ) follows from [BT10, Theorem 2.4]. Laplace transform methods can also be applied to local energy decay of wave equations on exterior domains, a subject which already featured prominently in the 1967 monograph by Lax and Phillips [LP67]. In this context, Ω = Rn \ K where K is a compact obstacle, and a = 0 in (4.58). The quantity of study is now the local energy of the solutions, and one restricts attention to initial data supported in a ﬁxed compact set. Thus one is interested in the asymptotic behaviour of P1 T (t)R(μ, A)P2 where P1 and P2 are multiplications by characteristic functions of balls. The semigroup property does not hold for P1 T (t)P2 , but Laplace transform results are still applicable. However P1 R(λ, A)P2 has a singularity at the origin when n is even, so a quantiﬁed version of Theorem 4.4.8 is used (see [BD08]). The general result, originally obtained in [Bur98], is that the local energy decays at least logarithmically. For the case when the obstacle is “not trapping”, earlier results had shown that the decay occurs at an exponential rate if n is odd and at a polynomial rate if n is even (see [MRS77], [MS78]). Theorem 4.4.16 is due to Arendt and Pr¨ uss [AP92]; it is a continuous analogue of the Katznelson-Tzafriri theorem [KT86] (see Corollary 4.7.15). Most of these results were originally proved by means of harmonic analysis; the link with Korevaar’s contour methods for complex Tauberian theorems was ﬁrst established by Allan, Ransford and O’Farrell [AOR87] whose work inspired [AB88]. Theorem 4.4.18 and Theorem 4.4.19 were discovered by Blake [Bla99] (see also [BB00]) who developed a method used by van Neerven [Nee96b] in the case of orbits of C0 -semigroups (see the Notes on Section 5.1). These results are valid not only when f is exponentially bounded, but even when e−ωt f (t) ∈ Lp (R+ , X) for some ω and some p > 1. However, they are not valid in the case p = 1. Bloch [Blo49] gave an example where e−ωt f (t) ∈ L1 (R+ ) whenever ω > 0, abs(f ) = 0 and hol0 (fˆ) = −∞. His example can be adapted to show that the estimate c(1 + t) in the conclusion of Theorem 4.4.18 is sharp for the class of exponentially bounded functions f [Bat03]. Section 4.5 The material presented here is standard. Almost periodic functions were introduced by Harald Bohr [Boh25] in 1925, and the ﬁrst edition of his book [Boh47] was published in 1934. Further textbooks are those of Levitan and Zhikov [LZ82], Fink [Fin74] and Amerio and Prouse [AP71]. Concerning the role of almost periodic functions for dynamical systems governed by partial diﬀerential equations, we refer to Haraux [Har91].

334

4. ASYMPTOTICS OF LAPLACE TRANSFORMS

Some authors use the terminology “scalarly almost periodic” functions instead of weakly almost periodic functions in order to distinguish from the notion of “weakly almost periodic in the sense of Eberlein”. We refer to Milnes [Mil80] and to the Notes of Section 4.7 for more information and comparison of these diﬀerent notions. Section 4.6 It was Loomis [Loo60] who proved that bounded uniformly continuous functions with countable spectrum are almost periodic in the scalar case. The extension to the vectorvalued case when c0 ⊂ X was included in the book of Levitan and Zhikov [LZ82] after Kadets had proved his striking result (Theorem 4.6.11) in [Kad69]. For the proof of Kadets’s theorem we follow [LZ82]. The condition of total ergodicity was used by Levitan [Lev66] in the context of antiderivatives of almost periodic functions. Theorem 4.6.3 in case a) is due to Ruess and V˜ u [RV95], but with a diﬀerent proof. The condition of total ergodicity turned out to be crucial in later developments on the half-line (see Sections 4.7, 4.9). The proof of Theorem 4.6.3 which we give here, is due to Arendt and Batty [AB97], but Lemma 4.6.6 and Lemma 4.6.7 are taken from Batty, van Neerven and R¨ abiger [BNR98a]. Several results presented here can be extended to measurable functions. In particular, a version of Theorem 4.6.3 remains true if f ∈ L∞ (R, X) is slowly oscillating at inﬁnity and a priori not continuous. Then the result says that f is equal almost everywhere to an almost periodic function (see the paper of Arendt and Batty [AB00, Corollary 3.3]). A harmonic analytic approach to countable spectrum on the line and almost periodicity was taken by Baskakov [Bas78], [Bas85] and Basit [Bas95], [Bas97], using the Beurling spectrum instead of the Carleman spectrum (c.f. Section 4.8). Section 4.7 The main result of this section, Theorem 4.7.7 is due to Batty, van Neerven and R¨ abiger [BNR98a], [BNR98b] who also proved Lemma 4.7.9. However, their proof is more complicated and based on the countable spectrum theorem (Theorem 5.5.4). The direct proof via the quotient method given here is due to Arendt and Batty [AB99] who proved Proposition 4.7.2 in particular. Example 4.7.11 was considered by Staﬀans [Sta81, p.608], Ruess and V˜ u [RV95, Example 3.12] and Batty, van Neerven and R¨ abiger [BNR98b]. The proof of the Katznelson-Tzafriri theorem (Corollary 4.7.15) presented here is similar to the one given by Arendt and Pr¨ uss [AP92]. Here our emphasis is on (strong) asymptotic almost periodicity, but there are many interesting results on weak versions of these notions. A function f ∈ BUC(R+ , X) is called weakly asymptotically almost periodic in the sense of Eberlein (in short, Eberleinw.a.a.p.) if the set {S(t)f : t ≥ 0} is relatively weakly compact in BUC(R+ , X). Ruess and Summers have investigated this notion in a series of articles [RS86], [RS87], [RS88a], [RS88b], [RS89], [RS90a], [RS90b], [RS92a], [RS92b]; see also papers of Ruess [Rue91], [Rue95] and Rosenblatt, Ruess and Sentilles [RRS91]. They show in a convincing way that this is the right notion for evolution equations. It is diﬀerent from weak asymptotic almost periodicity (w.a.a.p.) in the sense that x∗ ◦ f is asymptotically almost periodic for all x∗ ∈ X ∗ . An example of an Eberlein-w.a.a.p. orbit of a bounded C0 -semigroup, which is not w.a.a.p. is given in [RS90b, p.180]. The notion of Eberlein-w.a.a.p. functions is particularly useful in the context of the mean ergodic theorem for non-linear semigroups; see [RS87], [RS88a], [RS90a] and [RS92a]. An Eberlein-w.a.a.p. function f splits, f = f0 + f1 , where f1 is almost periodic and f0 is such that 0 is in the weak closure of {S(t)f0 : t ≥ 0} in BUC(R+ , X) (see

4.10. NOTES

335

Theorem 5.4.11). This implies that Mη f0 = 0 for all η ∈ R, but otherwise the asymptotic behaviour as t → ∞ of the function f0 is very weak. It can still happen that for some sequence tn → ∞, S(tn )f0 − f0 ∞ → 0 (see [RRS91, Section 3]). Section 4.8 In this section we closely follow the book of Pr¨ uss [Pr¨ u93] where in particular Proposition 4.8.4 is proved with the help of our favourite fudge factor. A diﬀerent proof is given in Davies’s book [Dav80, Chapter 8]. Theorem 4.8.7 on discrete spectrum is contained in a paper of Arendt and Schweiker [AS99] with a slightly diﬀerent proof. In more abstract contexts the result appeared already in work of Baskakov [Bas85], Beurling [Beu47] and Reiter [Rei52]. Section 4.9 In this section we follow closely Chill’s thesis [Chi98a] (see also [Chi98b]). In particular, Theorem 4.9.7 is due to Chill with the proof which we give here. This theorem seems to be a deﬁnitive complex Tauberian theorem involving countable spectrum. It is worth mentioning that uniform ergodicity is automatic outside the weak half-line spectrum. More precisely, Chill [Chi98a, Lemma 1.16] proved the following. Lemma 4.10.3. Let f ∈ L∞ (R+ , X) and let η ∈ R \ spw (f ). Then f is uniformly ergodic at η. The more general weak half-line spectrum seems to be more natural in the context of Volterra equations. A ﬁrst investigation of asymptotic behaviour of the corresponding solution operators (which are called resolvents in the theory of Volterra equations) has been carried out by Arendt and Pr¨ uss [AP92] (see also [Pr¨ u93]). Theorem 4.9.7 can now be more directly applied, as shown by Chill and Pr¨ uss [CP01] and Fasangova and Pr¨ uss [FP01].

Chapter 5

Asymptotics of Solutions of Cauchy Problems In this chapter, we give various results concerning the long-time asymptotic behaviour of mild solutions of homogeneous and inhomogeneous Cauchy problems on R+ (see Section 3.1 for the deﬁnitions and basic properties). For the most part, we shall assume that the homogeneous problem is well-posed, so that the operator A generates a C0 -semigroup T , mild solutions of the homogeneous problem (ACP0 ) are given by u(t) = T (t)x =: ux (t) (Theorem 3.1.12), and mild solutions of the inhomogeneous problem (ACPf ) are given by u(t) = T (t)x + (T ∗ f )(t), where T ∗ f is the convolution of T and f (Proposition 3.1.16). In typical applications, the operator A and its spectral properties will be known, but solutions u will not be known explicitly, so the objective is to obtain information about the behaviour of u from the spectral properties of A. To achieve this, we shall apply the results of earlier chapters, making use of the fact that the Laplace transform of u can easily be described in terms of the resolvent of A. In Section 5.1, we obtain general relations between spectral bounds of A, abscissas associated with the Laplace transform of T , growth bounds of T and its associated integrated semigroup, and the behaviour of convolutions T ∗ f for general f . In Sections 5.2 and 5.3, we give more precise relations between spectral and growth bounds in the cases of semigroups on Hilbert spaces and positive semigroups on Banach lattices. In Section 5.4, we use the general theory of ergodicity and asymptotic almost periodicity (Sections 4.3 and 4.7) to obtain splitting theorems (Glicksberg-deLeeuw theorems) for C0 -semigroups with relatively (weakly) compact orbits. In Section 5.5, we apply the complex Tauberian theorem (Theorem 4.7.7 or Theorem 4.9.7) to the case when the imaginary part of the spectrum of A is countable.

W. Arendt et al., Vector-valued Laplace Transforms and Cauchy Problems: Second Edition, Monographs in Mathematics 96, DOI 10.1007/978-3-0348-0087-7_5, © Springer Basel AG 2011

337

338

5. ASYMPTOTICS OF SOLUTIONS OF CAUCHY PROBLEMS

In Section 5.6, we consider the asymptotic behaviour of T ∗ f , showing in particular that T ∗ f is bounded when T is a bounded holomorphic C0 -semigroup, f is bounded, and T and f are out of phase in a sense described by their Laplace transforms.

5.1

Growth Bounds and Spectral Bounds

Let T be a C0 -semigroup on X with generator A. Recall from Section 3.1 that T is exponentially bounded, and the exponential growth bound ω(T ) is deﬁned by: ωt ω(T ) = inf ω ∈ R : there exists Mω such that T (t) ≤ Mω e for all t ≥ 0 . By the uniform boundedness principle applied to {e−ωt T (t) : t ≥ 0}, ω(T ) = inf ω ∈ R : for each x ∈ X, there exists Mω,x such that T (t)x ≤ Mω,x e

ωt

=

sup ω(ux ),

for all t ≥ 0 (5.1)

x∈X

where ux (t) := T (t)x. This suggests several possible ways of deﬁning other bounds, for example by replacing ω(ux ) by hol( ux ) or abs(ux ), and/or taking the supremum in (5.1) not over all x ∈ X, but only over x ∈ D(A) (i.e., considering classical solutions ux of the homogeneous Cauchy problem rather than mild solutions). Later in this section, we shall consider such bounds, and also bounds associated with the spectrum and resolvent of the generator A, but ﬁrst we establish some properties of ω(T ). The ﬁrst elementary result exploits the semigroup property of T to obtain some simple properties of the growth bound. Proposition 5.1.1. Let T be a C0 -semigroup on X. Then a) ω(T ) = lim t−1 log T (t) = inf t−1 log T (t) . t→∞

t>0

b) The spectral radius r(T (t)) of T (t) is etω(T ) . ∞ c) Let x ∈ X and ω ∈ R, and suppose that 0 e−ωt T (t)x dt < ∞. Then ω(ux ) ≤ ω. Proof. a) For ω > ω(T ), log T (t)

log Mω ≤ + ω, t t so

lim sup t−1 log T (t) ≤ ω. t→∞

5.1. GROWTH BOUNDS AND SPECTRAL BOUNDS Hence,

339

lim sup t−1 log T (t) ≤ ω(T ). t→∞

For the reverse inequality we may assume that T (t) > 0 for all t ≥ 0. For τ > 0 and nτ ≤ t < (n + 1)τ ,

T (t)

= T (τ )n T (t − nτ ) ≤ Cτ T (τ ) n ! " ≤ Cτ T (τ ) t/τ = Cτ exp (τ −1 log T (τ ) )t ,

where Cτ =

Cτ = sup T (s) , 0≤s≤τ

Thus,

Cτ

if T (τ ) ≥ 1, if T (τ ) < 1.

Cτ T (τ )

ω(T ) ≤ τ −1 log T (τ )

for all τ > 0. This proves a). b) By the spectral radius formula and a), n 1/n

r(T (t)) = lim T (t)

n→∞

1/n

= lim T (nt)

n→∞

=

lim exp

n→∞

t log T (nt)

nt

= etω(T ) . c) Take τ ≥ 1. For τ − 1 ≤ t ≤ τ ,

ux (τ ) = T (τ )x = T (τ − t)T (t)x ≤ C1 T (t)x = C1 ux (t) . Hence, e−ωτ ux (τ ) ≤ C1

τ

τ −1

This shows that τ → e ω(ux ) ≤ ω.

−ωτ

e−ωτ ux (t) dt ≤ C1 e|ω|

∞ 0

e−ωt ux (t) dt < ∞.

ux (τ ) is bounded on [1, ∞) and hence on R+ , so

It follows from Proposition 5.1.1 c) that ∞ −ωt ω(T ) = inf ω ∈ R : e

T (t)x dt < ∞ for all x ∈ X

(5.2)

0

=

sup {abs( ux ) : x ∈ X}

=

abs( T ).

The following result (which we call Datko’s theorem, although there were other contributions; see the Notes) shows that the inﬁmum in (5.2) is never attained. Indeed, applied to e−ωt T (t) with p = 1, condition b) shows that ∞ when −ωt ω(T ) < ω if 0 e

T (t)x dt < ∞ for all x ∈ X.

340

5. ASYMPTOTICS OF SOLUTIONS OF CAUCHY PROBLEMS Recall from Section 1.3 that T ∗ f denotes the convolution of T with f , so (T ∗ f )(t) :=

t 0

T (t − s)f (s) ds =

t 0

T (s)f (t − s) ds

when f ∈ L1loc (R+ , X). Theorem 5.1.2 (Datko’s Theorem). Let T be a C0 -semigroup on X, and let 1 ≤ p < ∞. The following are equivalent: (i) ω(T ) < 0. (ii) For all x ∈ X, ux ∈ Lp (R+ , X). (iii) For all f ∈ Lp (R+ , X), T ∗ f ∈ Lp (R+ , X). (iv) For all f ∈ L∞ (R+ , X), T ∗ f ∈ L∞ (R+ , X). (v) For all f ∈ C0 (R+ , X), T ∗ f ∈ C0 (R+ , X). (vi) For all f ∈ C0 (R+ , X),

t

< ∞. sup T (s)f (s) ds

t≥0

0

(vii) For all f ∈ AP(R+ , X),

t

< ∞. sup T (s)f (s) ds

t≥0

0

(viii) There is a constant C such that

t

≤ C sup f (s)

T (s)f (s) ds

0≤s≤t

(5.3)

0

for all f ∈ C([0, t], X) and all t ≥ 0. Proof. First, assume that (i) holds. Then there exist M and α > 0 such that

T (t) ≤ M e−αt for all t ≥ 0. Since ux (t) ≤ M e−αt x , ux ∈ Lp (R+ , X), so (ii) holds. Proposition 1.3.5 shows that (iii), (iv) and (v) hold. The proofs of (vi), (vii) and (viii) all follow from the estimate

t

t

M

T (s)f (s) ds M e−αs sup f (r) ds ≤ sup f (s) .

≤ α 0≤s≤t 0≤r≤t 0 0

5.1. GROWTH BOUNDS AND SPECTRAL BOUNDS

341

(ii) ⇒ (i): By hypothesis, x → ux maps X into Lp (R+ , X), and it is easy to check that this map has closed graph. Hence, there is a constant C such that ∞

T (t)x p dt ≤ C x p (5.4) 0

for all x ∈ X. Suppose that ω(T ) ≥ 0. By Proposition 5.1.1 b), there exists λ ∈ σ(T (1)) with |λ| = eω(T ) ≥ 1, and λ is in the topological boundary of σ(T (1)). Now λ is an approximate eigenvalue of T (1) (Proposition B.2), so there is a sequence (xk ) in X such that xk = 1 and limk→∞ T (1)xk − λxk = 0. Then limk→∞ T (1)n xk − λn xk = 0 for n ∈ N. Passing to a subsequence of (xk ), we may assume that

xk = 1, Hence, T (n)xk ≥

1 2

T (1)n xk − λn xk ≤

1 2

(n = 1, 2, . . . , k).

(n = 1, 2, . . . , k). If n − 1 ≤ t ≤ n ≤ k, then

1 ≤ T (n)xk = T (n − t)T (t)xk ≤ C1 T (t)xk , 2 where C1 = sup0≤s≤1 T (s) . Thus, T (t)xk ≥ 1/(2C1 ) whenever 0 ≤ t ≤ k, so p ∞ 1 p

T (t)xk dt ≥ k xk p (k = 1, 2, . . .). 2C1 0 This contradicts (5.4). It follows that ω(T ) < 0. (iii) ⇒ (ii): Choose ω > max(0, ω(T )). Take x ∈ X, and let f (t) := e−ωt T (t)x. Then f ∈ Lp (R+ , X), so (iii) implies that T ∗ f ∈ Lp (R+ , X). But t 1 − e−ωt (T ∗ f )(t) = T (s) e−ω(t−s) T (t − s)x ds = ux (t). ω 0 ω Thus, ux (t) ≤

(T ∗ f )(t) for t ≥ 1, and ux is bounded on [0, 1], so 1 − e−ω p ux ∈ L (R+ , X). (iv) or (v) ⇒ (viii): Deﬁne Vt : C0 (R+ , X) → X by Vt g := (T ∗ g)(t). Either (iv) or (v) implies that supt≥0 Vt g < ∞ for each g ∈ C0 (R+ , X). By the uniform boundedness principle, there is a constant C such that

t

≤ C g ∞ T (s)g(t − s) ds

0

for all g ∈ C0 (R+ , X). Given t ≥ 0 and f ∈ C([0, t], X), choose g ∈ C0 (R+ , X) such that g(s) = f (t − s) whenever 0 ≤ s ≤ t and g ∞ = sup0≤s≤t f (s) . Then

t

t

T (s)f (s) ds = T (s)g(t − s) ds

≤ C g ∞ = C sup f (s) . 0

0

0≤s≤t

342

5. ASYMPTOTICS OF SOLUTIONS OF CAUCHY PROBLEMS

(vi) or (vii) ⇒ (viii): As in the proof of (iv) or (v) ⇒ (viii), (vi) (respectively, (vii)) implies that there is a constant C such that

t

T (s)g(s) ds

≤ C g ∞ 0

for all g ∈ C0 (R+ , X) (respectively, g ∈ AP(R+ , X)). Given t ≥ 0 and f ∈ C([0, t], X) there exists an extension g ∈ C0 (R+ , X) (respectively, a periodic extension g) such that g ∞ = sup0≤s≤t f (s) . It follows that

t

T (s)f (s) ds sup f (s) .

≤ C 0≤s≤t 0

(viii) ⇒ (i): Take ω > max(0, ω(T )), so there exists M such that T (s) ≤ M eωs for all s ≥ 0. For t ≥ 0 and x ∈ X, let f (s) := eωs T (t − s)x (0 ≤ s ≤ t). Then (5.3) gives ωt e −1

T (t)x ≤ CM eωt x . ω Thus,

T (t)x ≤

CM ω x

1 − e−ω

for all t ≥ 1. It follows that M0 := supt≥0 T (t) < ∞. Putting f (s) := T (t − s)x in (5.3) gives t T (t)x ≤ CM0 x

for all t ≥ 0 and all x ∈ X. Thus, T (t) ≤ CM0 /t < 1 for suﬃciently large t, so ω(T ) = inf t>0 t−1 log T (t) < 0. The following corollary of Datko’s theorem can also be proved by a variation of the method of Lemma 3.2.14, without assuming that T is a semigroup. Corollary 5.1.3. Let T be a C0 -semigroup on X. Then there exists x ∈ X such that ω(ux ) = ω(T ). In particular, if for each x ∈ X there exist Mx and αx > 0 such that T (t)x ≤ Mx e−αx t for all t ≥ 0, then there exist M and α > 0 such that T (t) ≤ M e−αt for all t ≥ 0. Proof. The result is trivial if ω(T ) = −∞, so we assume that ω(T ) > −∞. Replacing T (t) by e−ω(T )t T (t), we may assume that ω(T ) = 0. Then the result follows immediately from Theorem 5.1.2, (ii) ⇒ (i). Recall from Sections 1.4 and 1.5 that abs(T )

:= =

sup {abs(ux ) : x ∈ X} inf ω ∈ R : for all x ∈ X, lim τ →∞

τ 0

e−ωt T (t)x dt exists ,

5.1. GROWTH BOUNDS AND SPECTRAL BOUNDS

343

t lim e−λs T (s)x ds = u x (λ) (λ > abs(T ), x ∈ X), t→∞ 0 := inf ω ∈ R : Tˆ extends to a holomorphic

Tˆ(λ)x := hol(Tˆ )

function from {Re λ > ω} to L(X) = =

sup {hol(ux ) : x ∈ X} inf ω ∈ R : for all x ∈ X, ux has a holomorphic extension to {λ ∈ C : Re λ > ω} .

We now deﬁne ω1 (T )

:= = =

=

sup {ω(ux ) : x ∈ D(A)} ∞ inf ω ∈ R : e−ωt T (t)x dt < ∞ for all x ∈ D(A) 0 inf ω ∈ R : for all x ∈ D(A), there exists Mω,x such that T (t)x ≤ Mω,x eωt for all t ≥ 0 inf ω ∈ R : there exists Mω such that

T (t)R(λ, A) ≤ Mω eωt for all t ≥ 0 .

Here, λ is any point in ρ(A). The equalities follow from the deﬁnition of ω(ux ), Proposition 5.1.1 c) and the uniform boundedness principle. It is clear from (1.10), (1.14) and the deﬁnitions that hol(T ) ≤ abs(T ) ≤ ω(T ) and ω1 (T ) ≤ ω(T ). The spectral bound s(A) of the generator A of T is deﬁned by s(A) := sup {Re λ : λ ∈ σ(A)} , with the convention that s(A) = −∞ if σ(A) is empty. The following results make precise the relation between the spectrum and resolvent of A and abscissas associated with the Laplace transform of T . In particular, Proposition 5.1.4 shows that s(A) < ∞. Proposition 5.1.4. Let T be a C0 -semigroup on X with generator A. Then Moreover, for x ∈ X,

s(A) = hol(Tˆ ).

(5.5)

u x (λ) = R(λ, A)x

(5.6)

whenever Re λ > s(A),

R(λ, A)x = lim

τ →∞

0

τ

e−λt T (t)x dt

(5.7)

344

5. ASYMPTOTICS OF SOLUTIONS OF CAUCHY PROBLEMS

whenever Re λ > abs(T ), and sup { R(λ, A) : Re λ > ω} < ∞

(5.8)

whenever ω > ω(T ). Proof. The functions Tˆ (λ) and R(λ, A) are holomorphic for Re λ > hol(Tˆ) and for λ ∈ ρ(A), respectively, and they coincide for λ > ω(T ). Thus Tˆ has a holomorphic extension for Re λ > min(s(A), hol(Tˆ)) and this implies that hol(Tˆ) ≤ s(A) and (5.6) and (5.7) hold. The equality (5.5) now follows from Theorem 3.1.7. For ω > ω > ω(T ), there exists M such that T (t) ≤ M eω t for all t. For Re λ > ω, ∞ M x

R(λ, A)x ≤ e− Re λt T (t)x dt ≤ , Re λ − ω 0 so M sup { R(λ, A) : Re λ > ω} ≤ < ∞. ω − ω When Re λ = abs(T ), the existence of the limit in (5.7) for all x ∈ X does not correspond exactly to the existence of R(λ, A) (consider, for example, T (t) = I with λ ∈ iR, λ = 0). The following result describes the relation between these two properties and stability of classical solutions of the homogeneous problem. Proposition 5.1.5. Let T be a C0 -semigroup on X, and let λ ∈ C. The following are equivalent: t (i) lim e−λs T (s)x ds exists for all x ∈ X. t→∞

0

(ii) λ ∈ ρ(A) and lim e−λt T (t)x = 0 for all x ∈ D(A). t→∞

In that case,

t

R(λ, A)x = lim

t→∞

e−λs T (s)x ds

0

for all x ∈ X. Proof. Replacing T (t) by e−λt T (t), we may assume that λ = 0. t (i) ⇒ (ii): Suppose that Bx := limt→∞ 0 T (s)x ds exists for all x ∈ X. Then 1 1 (T (h)Bx − Bx) = − h h

0

h

T (s)x ds → −x

as h ↓ 0. Thus, Bx ∈ D(A) and ABx = −x for all x ∈ X. Now suppose that x ∈ D(A). Then t BAx = lim T (s)Ax ds = lim T (t)x − x, t→∞

0

t→∞

5.1. GROWTH BOUNDS AND SPECTRAL BOUNDS

345

t by Proposition 3.1.9. Thus, y := limt→∞ T (t)x exists. Since limt→∞ 0 T (s)x ds also exists, it follows that y = 0 and BAx = −x (for all x ∈ D(A)). Thus A has an algebraic inverse −B. By Proposition B.1, 0 ∈ ρ(A) and B = −A−1 = R(0, A). (ii) ⇒ (i): Suppose that 0 ∈ ρ(A) and limt→∞ T (t)y = 0 for all y ∈ D(A). Then, for x ∈ X, t t T (s)x ds = − T (s)AR(0, A)x ds 0

0

R(0, A)x − T (t)R(0, A)x

=

→ R(0, A)x as t → ∞. The following result is the analogue of Theorem 1.4.3 for semigroups. Proposition 5.1.6. Let T be a C0 -semigroup on X with generator A, let S be the associated integrated semigroup: t S(t)x := T (s)x ds (x ∈ X), 0

and let := S(t) Then

S(t) − R(0, A) S(t)

if 0 ∈ ρ(A), if 0 ∈ σ(A).

abs(T ) = ω1 (T ) = ω(S).

follows from Proposition 1.4.5, Remark 1.4.6 Proof. The fact that abs(T ) = ω(S) and Proposition 5.1.5. To prove that abs(T ) = ω1 (T ), we may assume that ω(T ) < 0 (replacing T (t) by e−ωt T (t)). Then 0 ∈ ρ(A). By Proposition 3.1.9, S(t)x ∈ D(A) and AS(t)x = T (t)x − x for all x ∈ X. Hence, S(t)x = S(t)x − R(0, A)x = T (t)A−1 x. Thus,

= sup {ω(uA−1 x ) : x ∈ X} = ω1 (T ). abs(T ) = ω(S)

It turns out that the spectral bound s(A) is of limited use in the study of asymptotic behaviour—the spectrum of an operator may be unstable under small perturbations. However, such instability can only occur when the norm of the resolvent is large, so it is more useful to consider the pseudo-spectral bound deﬁned by s0 (A) := inf ω > s(A) : there exists Cω such that

R(λ, A) ≤ Cω whenever Re λ > ω .

346

5. ASYMPTOTICS OF SOLUTIONS OF CAUCHY PROBLEMS

It is clear from (5.8) that s(A) ≤ s0 (A) ≤ ω(T ). It follows from (5.5) and the uniform boundedness principle that s0 (A) = sup {hol0 ( ux ) : x ∈ X} , where hol0 (fˆ) is the abscissa of boundedness of fˆ deﬁned in Sections 1.5 and 4.4. Theorem 4.4.19 provides the following result. Theorem 5.1.7. Let T be a C0 -semigroup on X with generator A. Then abs(T ) ≤ s0 (A). Proof. By Theorem 4.4.19, abs(ux ) ≤ hol0 ( ux ) for each x ∈ X. Hence, abs(T ) = supx abs(ux ) ≤ supx hol0 ( ux ) = s0 (A). Proceeding as in the proof of Theorem 4.4.14 a), we can deduce from Theorem 4.4.18 the following more precise information about asymptotic behaviour for individual vectors. Theorem 5.1.8. Let T be a C0 -semigroup on X with generator A, and let S be the associated integrated semigroup. Let x ∈ X, and suppose that hol( ux ) ≤ 0 and u x is bounded on C+ . Then there is a constant c (depending on x) such that

S(t)x ≤ c(1 + t) for all t ≥ 0. Moreover, for each μ ∈ ρ(A), there is a constant cμ (depending on x and μ) such that

T (t)R(μ, A)x ≤ cμ (1 + t) for all t ≥ 0. Proof. The ﬁrst statement is immediate from Theorem 4.4.18. By Proposition 3.1.9, t T (t)R(μ, A)x = R(μ, A)x + T (s)AR(μ, A)x ds 0

= R(μ, A)x + (μR(μ, A) − I)S(t)x, and the second statement follows. We now summarise the general relations between spectral bounds, abscissas and growth bounds associated with semigroups, obtained in Proposition 5.1.4, Proposition 5.1.6 and Theorem 5.1.7. Theorem 5.1.9. Let T be a C0 -semigroup on X with generator A. Then s(A) = hol(Tˆ ) ≤ ω1 (T ) = abs(T ) ≤ s0 (A) ≤ ω(T ).

5.1. GROWTH BOUNDS AND SPECTRAL BOUNDS

347

Now, we give two examples which show that none of the inequalities in Theorem 5.1.9 can be replaced by an equality, and we shall give a further example in Section 5.3. In Theorem 5.1.12 and the next two sections of this chapter, we shall see that further equalities are valid under various additional assumptions on X and/or T . Example 5.1.10. There is a C0 -semigroup T on a Hilbert space X such that s(A) < ω1 (T ) < s0 (A) = ω(T ). Let X be the Hilbert space X

x = (xn )n∈N : xn ∈ C , n

:=

x :=

∞

2

xn < ∞ ,

n=1

1/2

xn 2

∞

,

n=1 (n)

where the norm on Cn is the Euclidean norm. Let Bn := (βi,j )1≤i,j≤n be the (n)

(n)

n × n complex matrix with βi,i+1 = 1 for 1 ≤ i < n, βi,j = 0 otherwise, and let An := i2n In + Bn . Let A be the operator on X deﬁned by D(A)

:=

x∈X :

∞

22n xn 2 < ∞ ,

n=1

Ax

:= (An xn )n∈N .

Since Bn = 1 and Bnn = 0, tj

tA tB n−1

e n = e n ≤ ≤ et . j! j=0

On the other hand, if xn := n−1/2 (1, 1, 1, . . . , 1)T ∈ Cn , then xn = 1 and ⎛ ⎞2 n−1 m j

tB

t

e n xn − et xn 2 = 1 ⎝ − et ⎠ → 0 n m=0 j=0 j!

as n → ∞. Thus, supn etAn = et . We may deﬁne T (t) : X → X by ! " T (t)x := etAn xn n∈N . Then T (t) = et , and T is a C0 -semigroup with generator A and ω(T ) = 1.

348

5. ASYMPTOTICS OF SOLUTIONS OF CAUCHY PROBLEMS For x ∈ D(A),

∞

tA

e n xn 2

T (t)x 2 =

≤

n=1

∞ n=1

≤

∞

⎛

⎞2 t ⎠

xn 2 j!

n−1 j

⎝ ⎛

j=0

n−1

⎝

n=1

j=0

⎞2 ∞ tj ⎠ 2n 2 t 2

x

≤ e 22n xn 2 . n 2j j! n=1

Thus, ω1 (T ) ≤ 12 . On the other hand, if 0 < α < 1/2 and xn = αn (1, 1, . . . , 1)T ∈ Cn , then x = (xn ) ∈ D(A) and

T (t)x 2

⎛ ⎞2 m j t ⎝ ⎠ α2n = j! n=1 m=0 j=0 ⎛ ⎞ ∞ 2n−2 ⎜ r! ⎟ tr 2n ≥ ⎝ ⎠ α j!k! r! j+k=r n=1 r=0 ∞ n−1

0≤j,k≤n−1

∞ 2n−2 2 r tr ≥ α2n r + 1 r! n=1 r=0

=

∞

r=0 n≥ r2 +1

≥

α2n

2 r tr (r + 1)!

∞ αr+3 2r tr α2 e2αt − 1 = . 1 − α2 (r + 1)! 1 − α2 2t r=0

Thus, ω(ux ) ≥ α. It follows that ω1 (T ) = 12 . To calculate the spectral bounds, note that σ(An ) = {i2n } and

R(λ, An ) = R(λ − i2n , Bn ) ≤

1 |λ − i2n | − 1

(5.9)

if |λ − i2n | > 1. It follows that supn R(λ, An ) < ∞ whenever λ ∈ / {i2n : n ∈ N}. n Hence, σ(A) = {i2 : n = 1, 2, . . .}, so s(A) = 0. It also follows from (5.9) that s0 (A) ≤ 1. On the other hand,

n−1

j 1/2

R(1 + i2 , A) ≥ R(1, Bn ) = Bn

≥n . n

j=0

Hence, s0 (A) = 1.

5.1. GROWTH BOUNDS AND SPECTRAL BOUNDS

349

We shall see in Section 5.2 that the equality s0 (A) = ω(T ) holds for all C0 -semigroups on Hilbert space. Example 5.1.11. There is a positive C0 -semigroup on a (reﬂexive) Banach lattice X such that s(A) = ω1 (T ) = s0 (A) < ω(T ). Let X := Lp (1, ∞) ∩ Lq (1, ∞), where 1 ≤ p ≤ q < ∞. Then X is a Banach lattice with the natural ordering and norm:

f := max( f p , f q ). Let Tp be the C0 -semigroup on Lp (1, ∞) deﬁned by (Tp (t)g) (s) := g(set ), and let T be the positive C0 -semigroup on X obtained by restricting Tp to X. Let A and Ap be the generators of T and Tp , respectively. For f ∈ X, 1/p ∞ 1/q ∞ t p t q

T (t)f = max |f (se )| ds , |f (se )| ds 1

=

max e−t/p

1

∞

et

1/p |f (r)|p dr

, e−t/q

≤ max e−t/p f p , e−t/q f q

∞

et

1/q |f (r)|q dr

≤ e−t/q f . On the other hand, if 1 (et ≤ s ≤ et + 1), f (s) := 0 otherwise, then f = 1 and T (t)f = e−t/q . Thus, T (t) = e−t/q and ω(T ) = −1/q. For Re λ < −1/p, let fλ (s) := sλ . Then fλ ∈ X and T (t)fλ = eλt fλ , so fλ ∈ D(A) and Afλ = λfλ . Hence, σ(A) ⊇ λ ∈ C : Re λ ≤ − p1 , and s(A) ≥ −1/p. In the case p = q, we now know that s(Ap ) = ω(Tp ) = −1/p, and for Re λ := α > −1/p and f ∈ Lp (1, ∞), ∞ ∞ f (r) (R(λ, Ap )f ) (s) = e−λt f (set ) dt = sλ dr. rλ+1 0 s

350

5. ASYMPTOTICS OF SOLUTIONS OF CAUCHY PROBLEMS

For 1 < p < q < ∞ and p such that 1

∞

1 p

+

≤ = =

= 1, we have

q f (r) dr ds r λ+1 1 s ∞ q/p ∞ dr αq q s f p ds r (α+1)p 1 s ∞

f qp s−q/p ds ((α + 1)p − 1)q/p 1

f qp p . ((α + 1)p − 1)q/p (q − p)

q

|(R(λ, Ap )f )(s)| ds =

1 p

∞

sαq

∞

If 1 = p < q < ∞, then

∞ 1

q

|(R(λ, Ap )f )(s)| ds ≤

f q1 . q−1

In each case, R(λ, Ap ) maps Lp (1, ∞) into X, so D(Ap ) ⊂ X. Since p (λ)|X = R(λ, Ap )|X R(λ, A) = Tˆ (λ) = T for λ > −1/q, it follows that A is the part of Ap in X, σ(A) ⊂ σ(Ap ) ⊂ {Re λ ≤ −1/p}, and R(λ, A) = R(λ, Ap )|X for Re λ > −1/p (see Proposition B.8). For f ∈ X and Re λ = α > −1/p,

f p p1/q

R(λ, A)f ≤ max R(λ, A)f p , ((α + 1)p − 1)1/p (q − p)1/q p p1/q ≤ max ,

f

αp + 1 ((α + 1)p − 1)1/p (q − p)1/q if 1 < p < q < ∞;

R(λ, A)f ≤ max

1 1 , α + 1 (q − 1)1/q

f

if 1 = p < q < ∞. Thus, s0 (A) = −1/p. It follows from Theorem 5.1.9 that ω1 (T ) = −1/p. We shall see in Section 5.3 that the equality s(A) = s0 (A) holds for all positive semigroups on Banach lattices, while the equality s(A) = ω(T ) holds for all positive semigroups on Lp -spaces. We conclude this section by showing that s(A) = ω(T ) for all holomorphic semigroups. Theorem 5.1.12. Let T be a holomorphic C0 -semigroup on X with generator A. Then ω(T ) = s(A). Moreover, there exists λ ∈ σ(A) such that Re λ = ω(T ).

5.2. SEMIGROUPS ON HILBERT SPACES

351

Proof. For each x ∈ X, ux has a holomorphic extension to a sector Σθ , given by ux (z) = T (z)x. By Theorem 2.6.2, ω(ux ) = inf ω ∈ R : λ → R(λ, A)x has a holomorphic extension to {λ : Re λ > ω} ≤ s(A). Hence, ω(T ) = sup ω(ux ) ≤ s(A). x∈X

The ﬁnal statement follows from the fact that {λ ∈ σ(A) : Re λ ≥ s(A) − 1} is nonempty and compact (see Theorem 3.7.11 and Corollary 3.7.17).

5.2

Semigroups on Hilbert Spaces

Example 5.1.10 shows that there are C0 -semigroups on Hilbert spaces such that s(A) < ω1 (T ) < s0 (A). In that example, s0 (A) = ω(T ), and we now show that this equality always holds on Hilbert spaces. Theorem 5.2.1. Let T be a C0 -semigroup on a Hilbert space X with generator A. Then s0 (A) = ω(T ). Proof. Let x ∈ X. For ω > ω(T ), the function s → R(ω + is, A)x on R is the Fourier transform of the function t → e−ωt T (t)x on R+ . By Plancherel’s Theorem 1.8.2, ∞ ∞ 1 e−2ωt T (t)x 2 dt =

R(ω + is, A)x 2 ds. (5.10) 2π −∞ 0 Suppose that s0 (A) < ω(T ) and let C := supRe λ>ω(T ) R(λ, A) < ∞. For ω(T ) < ω1 < ω2 , R(ω1 + is, A)x = R(ω2 + is, A)x + (ω2 − ω1 )R(ω1 + is, A)R(ω2 + is, A)x, so

R(ω1 + is, A)x ≤ (1 + C(ω2 − ω1 )) R(ω2 + is, A)x . By (5.10), ∞ 0

2

e−2ω1 t T (t)x 2 dt ≤ (1 + C(ω2 − ω1 ))

Letting ω1 ↓ ω(T ) gives ∞ 2 e−2ω(T )t T (t)x 2 dt ≤ (1 + C(ω2 − ω(T ))) 0

∞

0

∞ 0

e−2ω2 t T (t)x 2 dt.

e−2ω2 t T (t)x 2 dt < ∞

352

5. ASYMPTOTICS OF SOLUTIONS OF CAUCHY PROBLEMS

for all x ∈ X. The implication (ii) ⇒ (i) of Theorem 5.1.2, with p = 2 and T (t) replaced by e−ω(T )t T (t), gives a contradiction. Theorem 5.2.1 is not valid for X = Lp (0, 1) (1 < p < ∞, p = 2). Example 5.2.2. Let 2 < q < ∞. Example 5.1.11 shows that there is a C0 -semigroup Tq on L2 (1, ∞) ∩ Lq (1, ∞) whose generator Aq satisﬁes s0 (Aq ) = − 12 < − 1q = ω(Tq ). There is a linear homeomorphism Jq of Lq (0, 1) onto L2 (1, ∞) ∩ Lq (1, ∞) [LT77, Corollary II.2.e.8]. Let Sq (t) := Jq−1 Tq (t)Jq . Then Sq is a C0 -semigroup on Lq (0, 1) whose generator Bq satisﬁes s0 (Bq ) = − 12 < − 1q = ω(Sq ). Let 1 < p < 2, and let q be the conjugate index, so that p1 + 1q = 1 and Lp (0, 1) = Lq (0, 1)∗ . Let Sp (t) := Sq (t)∗ . By Corollary 3.3.9, Sp is a C0 -semigroup on Lp (0, 1), whose generator Bp = Bq∗ satisﬁes s0 (Bp ) = − 12 < − 1q = ω(Sp ). We shall see in the next section that ω(T ) = s0 (A) = s(A) for all positive semigroups on Lp -spaces. The analogue of Theorem 5.2.1 for individual orbits is not true. Example 5.2.3. There is a C0 -semigroup T on a Hilbert space X with a vector x ∈ X and a real number a < ω(ux ) such that u x = R(·, A)x has a bounded holomorphic extension to the half-plane {λ ∈ C : Re λ > a}. Let X := L2 (1, ∞) and (T (t)f )(s) := f (set ). By Example 5.1.11, s(A) = ω(T ) = ω(A) = −1/2. Let A1 be the generator of the C0 -semigroup on L1 (1, ∞) ∩ L2 (1, ∞) obtained by restricting T , so s0 (A1 ) = −1, again by Example 5.1.11. For f ∈ L1 (1, ∞) ∩ L2 (1, ∞), R(·, A1 )f has an extension to a bounded holomorphic map of {λ ∈ C : Re λ > a} into L1 (1, ∞) ∩ L2 (1, ∞) whenever a > −1. But R(λ, A)f = R(λ, A1 )f when Re λ is large, so R(·, A)f has an extension to a bounded holomorphic map of {λ ∈ C : Re λ > a} into L2 (1, ∞). However, it is possible to choose f ∈ L1 (1, ∞) ∩ L2 (1, ∞) such that ω(uf ) = −1/2 (where ω(uf ) is calculated in L2 (1, ∞)). For example, let 1 (en ≤ s ≤ en + n−2 ; n ∈ N), f (s) = 0 otherwise. Then (T (n)f )(s) = 1 for 1 ≤ s ≤ 1 + n−2 e−n , so T (n)f 2 > n−1 e−n/2 . Hence, ω(uf ) ≥ −1/2. On the other hand, ω(uf ) ≤ ω(T ) = −1/2.

5.3

Positive Semigroups

Let T be a C0 -semigroup on an ordered Banach space X, with generator A. We recall from Section 3.11 that T is positive if and only if A is resolvent positive. Example 5.1.11 shows that there are positive C0 -semigroups T on spaces of the form Lp (Ω, μ)∩Lq (Ω, μ) (1 ≤ p < q < ∞) such that s(A) < ω(T ). On the other hand, we shall show in this section that s(A) = abs(T ) = s0 (A) for all positive

5.3. POSITIVE SEMIGROUPS

353

semigroups on any ordered Banach space with normal cone, and that s(A) = ω(T ) for all positive semigroups on Lp (Ω, μ). Note that Proposition 3.11.2 shows that s(A) ∈ σ(A) if A generates a positive semigroup and σ(A) is non-empty. The following result makes this more precise. We give a proof that abs(T ) ∈ σ(A) based on Theorem 1.5.3, whereas Proposition 3.11.2 was proved by means of Bernstein’s Theorem 2.7.7. Theorem 5.3.1. Let X be an ordered Banach space with normal cone and let T be a positive C0 -semigroup on X with generator A. Then s(A) = ω1 (T ) = abs(T ) = s0 (A). Moreover, s(A) ∈ σ(A) if s(A) > −∞. Proof. By Theorem 5.1.9, s(A) ≤ ω1 (T ) = abs(T ) ≤ s0 (A). It suﬃces to prove that abs(T ) ∈ σ(A) if abs(T ) > −∞, and that supRe λ>ω R(λ, A) < ∞ whenever ω > abs(T ), so abs(T ) = s0 (A). Suppose that abs(T ) > −∞ and abs(T ) ∈ ρ(A). Let ε > 0 such that the ball B(abs(T ), ε) ⊂ ρ(A). For x ∈ X+ , the function ux is positive with Laplace transform R(λ, A)x (Re λ > ω1 (T )), which has a holomorphic extension to B(abs(T ), ε). By Theorem 1.5.3, abs(ux ) ≤ abs(T ) − ε. By linearity, abs(ux ) ≤ abs(T ) − ε for all x ∈ X. But this contradicts the deﬁnition of abs(T ). This proves that abs(T ) ∈ σ(A) if abs(T ) > −∞, and hence s(A) = abs(T ). By Corollary 3.11.3, supRe λ>ω R(λ, A) < ∞. Example 5.3.2. There exist a Hilbert space X which is a vector lattice with continuous lattice operations, and a positive C0 -semigroup T on X such that s(A) = ω1 (T ) < s0 (A) = ω(T ). Let X be the Sobolev space H 1 (1, ∞) := {f ∈ L2 (1, ∞) : f ∈ L2 (1, ∞)}, (see Appendix E) with ! "1/2

f H 1 (1,∞) := f 22 + f 22 . Then

X is a Hilbert space, and it is a vector lattice with the properties that

|f | = f [DL90, Chapter IV, Section 7, Proposition 6] and lattice operations are continuous (see [BY84, p.219]), but it is not a Banach lattice and the positive cone is not normal. Let T be the C0 -semigroup on H 1 (1, ∞) given by (T (t)f )(s) := f (set ). The generator A of T is given by ' ( D(A) = f ∈ H 1 (1, ∞) : s → sf (s) ∈ H 1 (1, ∞) , (Af )(s) = sf (s).

354

5. ASYMPTOTICS OF SOLUTIONS OF CAUCHY PROBLEMS

The semigroup governs the following very natural partial diﬀerential equation: ∂u ∂u = s , ∂t ∂s u(0, s) = u0 (s),

(t > 0, s > 1), (s > 1),

where u(t, s) := (T (t)u0 )(s). For α < −1/2, the function fα (s) := sα lies in X and T (t)fα = eαt fα . Hence, s(A) ≥ −1/2. For f ∈ H 1 (1, ∞), ∞ ∞ 2 2 t 2

T (t)f H 1 (1,∞) = f (se ) ds + e2t f (set ) ds 1 1 ∞ ∞ 2 −t 2 = e |f (r)| dr + et |f (r)| dr ≤ e

t

et

f 2H 1 (1,∞) .

et

Thus, ω(T ) ≤ 1/2. Choose non-zero g ∈ Cc∞ (R) with support in R+" . Given ∞! t ≥ 0, let f (s) := g(s − et ). Then f 2H 1 (1,∞) = 0 |g(s)|2 + |g (s)|2 ds and ∞ 2

T (t)f 2H 1 (1,∞) ≥ et 1 |g (r)| dr. It follows that ω(T ) ≥ 1/2, so s0 (A) = 1/2 by Theorem 5.2.1. We shall show that abs(T ) ≤ −1/2. It then follows from Theorem 5.1.9 that s(A) = ω1 (T ) = −1/2. Let S be the corresponding C0 -semigroup on L2 (1, ∞) with generator B, so abs(S) = ω1 (S) = ω(S) = −1/2 (Example 5.1.11). Let ω > −1/2 and f ∈ H 1 (1, ∞). For t > 0, let t gt := e−ωr T (r)f dr ∈ H 1 (1, ∞), 0

g

:=

R(ω, B)f ∈ L2 (1, ∞).

Then limt→∞ ||gt − g 2 = 0, since ω > abs(S). We have to show that limt→∞ ||gt − g H 1 (1,∞) = 0. By Proposition 3.1.9, gt ∈ D(A − ω) = D(A) ⊂ D(B), and (A − ω)gt = (B − ω)gt = e−ωt S(t)f − f. But (Agt )(s) = sgt (s), so

! " gt = h · ωgt − f + e−ωt S(t)f ,

where h(s) := s−1 . Since |h(s)| ≤ 1 for all s ∈ (1, ∞), it is clear that lim gt − h(ωg − f ) 2 = 0,

t→∞

so limt→∞ gt exists in H 1 (1, ∞). Thus, abs(uf ) ≤ −1/2 for all f ∈ H 1 (1, ∞), so abs(T ) ≤ −1/2.

5.3. POSITIVE SEMIGROUPS

355

Now we consider positive semigroups on Lp (Ω, μ), where we aim to show that s(A) = ω(T ). For p = 2, this result is immediate from Theorems 5.3.1 and 5.2.1. There is also an easy proof for p = 1, which we present in Proposition 5.3.7. The general case needs some preliminaries. We work with the product space R × Ω, and we use a vector-valued norm on Lp (R × Ω). We begin by deﬁning this norm, and establishing its properties. Let (Ω, μ) be a σ-ﬁnite measure space, and consider R×Ω to be equipped with the product of Lebesgue measure m on R and the given measure μ on Ω. Let 1 ≤ p < ∞. We write Lp (Ω) for Lp (Ω, μ) and we identify Lp (R×Ω) with Lp (R, Lp (Ω)), so that the notations g(t, y) and g(t)(y) (t ∈ R, y ∈ Ω) are interchangeable. We consider the non-linear map Φ : Lp (R, Lp (Ω)) → Lp (Ω) given by

∞

Φ(g) := −∞

1/p |g(t)| dt p

.

This is the composition of three maps: Φ1 : Lp (R × Ω) → L1 (R × Ω), Φ2 : L1 (R × Ω) → L1 (Ω), 1

Φ1 (g) := |g|p , ∞ Φ2 (h)(y) := h(t, y) dt, −∞ 1/p

Φ3 : L (Ω) → L (Ω),

Φ3 (k) := |k|

p

.

This makes it clear that Φ is well deﬁned. Lemma 5.3.3. Let g, h ∈ Lp (R, Lp (Ω)), f ∈ L∞ (Ω), s ∈ R. Then a) Φ(g) Lp (Ω) = g Lp (R×Ω) . b) Φ(gs ) = Φ(g), where gs (t) = g(t + s). c) Φ(f · g) = |f |Φ(g), where (f · g)(t, y) = f (y)g(t, y). d) Φ(g + h) ≤ Φ(g) + Φ(h) in Lp (Ω). e) Φ is continuous. Proof. a), b) and c) are all trivial. To prove d), let Gy (t) := g(t, y), Hy (t) := h(t, y) (t ∈ R, y ∈ Ω). For μ-almost all y, Gy ∈ Lp (R) and Hy ∈ Lp (R), so Minkowski’s inequality gives

Gy + Hy Lp (R) ≤ Gy Lp (R) + Hy Lp (R) . But

Gy Lp (R) =

∞

−∞

1/p |g(t, y)|p dt

= Φ(g)(y),

etc., so (5.11) gives Φ(g + h)(y) ≤ Φ(g)(y) + Φ(h)(y).

(5.11)

356

5. ASYMPTOTICS OF SOLUTIONS OF CAUCHY PROBLEMS

This holds μ-a.e., so Φ(g + h) ≤ Φ(g) + Φ(h) in Lp (Ω). Now, e) follows from a) and d). By d), Φ(g) ≤ Φ(g − h) + Φ(h), so Φ(g) − Φ(h) ≤ Φ(g − h). Similarly, Φ(h) − Φ(g) ≤ Φ(h − g) = Φ(g − h). Since Φ(g) etc. are real-valued, this shows that |Φ(g) − Φ(h)| ≤ Φ(g − h). Hence,

Φ(g) − Φ(h) Lp (Ω) ≤ Φ(g − h) Lp (Ω) = g − h Lp (R,Lp (Ω)) . Lemma 5.3.3 shows that Φ is a convex, vector-valued function. The next lemma is a vector-valued instance of Jensen’s inequality. Lemma 5.3.4. Let G : [a, b] → Lp (R, Lp (Ω)) be continuous. Then b

Φ

G(t) dt a

b

≤

Φ(G(t)) dt

in Lp (Ω).

a

Proof. By Lemma 5.3.3, c) and d), Φ

2n −1 2n −1 b−a rb + (2n − r)a b−a rb + (2n − 1)a G ≤ n Φ G . 2n r=0 2n 2 2n r=0

Letting n → ∞ and using the continuity of Φ (Lemma 5.3.3 e)) gives the result. For g ∈ Lp (R, Lp (Ω)) and a bounded operator T on Lp (Ω), we may deﬁne T ◦ g ∈ Lp (R, Lp (Ω)) by (T ◦ g)(t) = T (g(t)). Proposition 5.3.5. Let T be a positive bounded linear operator on Lp (Ω), and g ∈ Lp (R, Lp (Ω)). Then Φ(T ◦ g) ≤ T (Φ(g)). (5.12) Proof. Both sides of (5.12) depend continuously on g, so it suﬃces to assume that g is a simple function n g(t)(y) = χAk (t)gk (y), k=1

5.3. POSITIVE SEMIGROUPS

357

where A1 , . . . , An are disjoint Borel subsets of R, and g1 , . . . , gn ∈ Lp (Ω). Let hk := m(Ak )1/p gk (k = 1, . . . , n). Then Φ(T ◦ g) =

n

1/p m(Ak )|T gk |

p

k=1

T (Φ(g)) = T

n

=

m(Ak )|gk |

1/p |T hk |

p

k=1

1/p p

n

=T

k=1

n

, 1/p

|hk |

p

.

k=1

n Take αk ∈ Q + iQ with k=1 |αk |p ≤ 1, where p is the conjugate index of p (maxk |αk | ≤ 1 if p = 1). By H¨ older’s inequality, Re

n

≤

αk hk

k=1

n

1/p |hk |

p

= Φ(g).

k=1

Applying T gives Re

n

αk T hk

=T

Re

k=1

n

αk h k

≤ T (Φ(g)).

k=1

Now

n k=1

=

1/p |(T hk )(y)| sup Re

p

n

αk (T hk )(y)

: αk ∈ Q + iQ,

k=1

n

|αk |p ≤ 1

k=1

≤ T (Φ(g))(y) μ-a.e. Thus, Φ(T ◦ g) ≤ T (Φ(g)). Theorem 5.3.6. Let (Ω, μ) be a σ-ﬁnite measure space, 1 ≤ p < ∞, and T be a positive C0 -semigroup on Lp (Ω), with generator A. Then s(A) = ω(T ). Proof. First, assume that s(A) < 0. Fix α > max(0, ω(T )). Take f ∈ Lp (Ω) and deﬁne g ∈ Lp (R, Lp (Ω)) by e−αt T (t)f (t ≥ 0), g(t) := 0 (t < 0). Deﬁne G : R+ → Lp (R, Lp (Ω)) by G(s) := T (s) ◦ g−s ,

358

5. ASYMPTOTICS OF SOLUTIONS OF CAUCHY PROBLEMS

where g−s (t) := g(t − s). Thus, G(s)(t) =

e−α(t−s) T (t)f 0

Then Φ

m

G(s) ds

∞

=

0

0

1 α

=

(0 ≤ s ≤ t), (−∞ < t < s).

p 1/p min(m,t) −α(t−s) e T (t)f ds dt 0 ∞

−α max(0,t−m)

e 0

−αt

−e

p

1/p

|T (t)f | dt p

.

Thus, 1 α

0 ≤ =

−α max(0,t−m)

e 0m

Φ m

≤

∞

−e

−αt

p

1/p |T (t)f | dt p

G(s) ds

0

Φ(G(s)) ds 0 m

=

0

0

≤

Φ (T (s) ◦ g−s ) ds

m

T (s) (Φ(g−s )) ds m

=

T (s)(Φ(g)) ds, 0

where we have used Lemma 5.3.4, Proposition 5.3.5 and Lemma 5.3.3 b) in the third, ﬁfth and sixth lines respectively. Since abs(T ) = s(A) < 0 (Theorem 5.3.1), m T (s)(Φ(g)) ds → R(0, A)(Φ(g)) in Lp (Ω), 0

as m → ∞ (Proposition 5.1.4). By the monotone convergence theorem, ∞ 1/p ! "p 1 0≤ 1 − e−αt |T (t)f |p dt ≤ R(0, A)(Φ(g)). α 0 Hence,

1 − e−α α

1

∞

1/p |T (t)f |p dt

Taking norms in Lp (Ω) gives ∞ p |(T (t)f )(y)| dt dμ(y) ≤ Ω

1

α 1 − e−α

≤ R(0, A)(Φ(g)). p

R(0, A) p Φ(g) pLp (Ω) ,

5.3. POSITIVE SEMIGROUPS

so

∞

1

359

T (t)f pLp (Ω) dt < ∞.

It follows from Theorem 5.1.2 (ii) ⇒ (i), that ω(T ) < 0. In the general case, we may apply the case above to the semigroup e−ωt T (t) for ω > s(A), and we deduce that ω(T ) < ω whenever ω > s(A). Thus ω(T ) ≤ s(A). For p = 1, there is a much simpler proof of Theorem 5.3.6. A Banach lattice X is said to be an L-space if

x + y = x + y

for all x, y ∈ X+ .

(5.13)

If (Ω, μ) is any measure space, then L1 (Ω, μ) is an L-space. If Ω is a locally compact, Hausdorﬀ space, then C0 (Ω)∗ (which can be identiﬁed with the space of all regular Borel measures on Ω) is an L-space. On the other hand, any L-space is isomorphic as a Banach lattice to a space of the form L1 (Ω, μ) [Sch74, Theorem II.8.5]. Proposition 5.3.7. Let T be a positive C0 -semigroup on an L-space X, with generator A. Then s(A) = ω(T ). ∗ Proof. By (5.13), there exists x∗ ∈ X+ such that x, x∗ = x for all x ∈ X+ . Let ω > abs(T ) = s(A) (Theorem 5.3.1). For x ∈ X+ and τ ≥ 0, τ 1 τ 2 e−ωt T (t)x dt = e−ωt T (t)x dt, x∗ ≤ R(ω, A)x, x∗ . 0

Hence,

0

∞ 0

e−ωt T (t)x dt < ∞

for all x ∈ X+ , and so for all x ∈ X. It follows from Datko’s theorem (Theorem 5.1.2) that ω(T ) < ω whenever ω > s(A), and therefore ω(T ) = s(A). Theorem 5.3.6 is also true in the case p = ∞ (even without the assumption that the semigroup is positive), but for the trivial reason that the semigroup is norm-continuous (Corollary 4.3.19). A more interesting case is that of spaces of the form C0 (Ω), and this can be deduced from Proposition 5.3.7 by duality. Theorem 5.3.8. Let Ω be a locally compact, Hausdorﬀ space, and T be a positive C0 -semigroup on C0 (Ω), with generator A. Then s(A) = ω(T ). Proof. Let X := C0 (Ω)∗ , which is an L-space and therefore has order-continuous norm. Let Y := D(A∗ ). By Theorem 3.11.8, Y is a closed ideal in X, so Y is also an L-space. Let T (t) := T (t)∗ |Y . By Proposition 3.3.14, T is a C0 -semigroup on Y whose generator A is the part of A∗ in Y . Moreover, σ(A ) = σ(A) (see

360

5. ASYMPTOTICS OF SOLUTIONS OF CAUCHY PROBLEMS

Propositions B.8 and B.11). Let ω > s(A) = s(A ). By Proposition 5.3.7, there exists M such that T (t) ≤ M eωt for all t ≥ 0. For x ∈ X and y ∗ ∈ Y , |T (t)x, y ∗ | ≤ M eωt x y ∗ . For x∗ ∈ X ∗ and λ > ω(T ), R(λ, A)∗ x∗ ∈ D(A∗ ) ⊂ Y . Furthermore, x = limλ→∞ λR(λ, A)x (Proposition 3.1.9), and c := lim supλ→∞ λ R(λ, A) < ∞ since T is exponentially bounded. Hence, |T (t)x, x∗ |

=

lim |T (t)x, λR(λ, A)∗ x∗ |

λ→∞

≤ lim sup M eωt x λ R(λ, A)∗ x∗

λ→∞

≤ cM eωt x x∗ . It follows that T (t) ≤ cM eωt , so ω(T ) < ω whenever ω > s(A).

5.4

Splitting Theorems

Let T be a bounded C0 -semigroup on X, and let x ∈ X. In this section, we shall apply the theory of asymptotic behaviour of functions on R+ , as developed in Chapter 4, to the special case of the orbit ux , where ux (t) := T (t)x. We shall see that ergodicity and (asymptotic) almost periodicity of ux correspond to natural semigroup properties of x, and also to compactness properties of the orbit. In particular, the main results will be two splitting theorems. We saw already in Proposition 4.3.12 that a vector x is totally ergodic with respect to T if the orbit {T (t)x : t ≥ 0} is relatively weakly compact. The weak splitting theorem shows that x can be uniquely decomposed as x = x0 + x1 , where x0 is totally ergodic with all means Mη x0 = 0, and x1 is in the closed linear span of the unimodular eigenvectors of T . The strong splitting theorem states that if the orbit of x is relatively compact in the norm topology, then limt→∞ T (t)x0 = 0. As in Section 4.3, we let Xe denote the space of all vectors x ∈ X which are totally ergodic with respect to T . Thus, x ∈ Xe if and only if 1 t −iηs Mη x := lim e T (s)x ds t→∞ t 0 exists (in the norm topology of X), for each η ∈ R. We also let Xe0 X0

:= {x ∈ Xe : Mη x = 0 for all η ∈ R} , := {x ∈ X : T (t)x → 0 as t → ∞} .

Since T is bounded, all these are closed T -invariant subspaces of X, and X0 ⊂ Xe0 ⊂ Xe .

5.4. SPLITTING THEOREMS

361

In Example 5.4.3, we shall exhibit these subspaces in a very fundamental example of multiplier semigroups, but we ﬁrst give two very simple general results. Proposition 5.4.1. Let T be a bounded C0 -semigroup on X. Then ux ∈ BUC(R+ , X) for each x ∈ X. Moreover, the map x → ux is bounded and linear from X into BUC(R+ , X). Proof. Uniform continuity of ux follows from the strong continuity of T and the estimate

ux (t + h) − ux (t) = T (t)(T (h)x − x) ≤ M T (h)x − x , where M = sups≥0 T (s) . The other properties are immediate. Recall from Section 4.7 that a function f ∈ BUC(R+ , X) is totally ergodic if it is totally ergodic with respect to the shift semigroup on BUC(R+ , X); i.e., for each η ∈ R, 1 τ −iηs (Mη f )(t) := lim e f (t + s) ds τ →∞ τ 0 exists in X, uniformly for t ≥ 0. The space E(R+ , X) of all totally ergodic functions is a closed subspace of BUC(R+ , X). Proposition 5.4.2. Let T be a bounded C0 -semigroup on X, and let x ∈ X. Then ux ∈ E(R+ , X) if and only if x ∈ Xe . In that case, Mη (ux )(t) = T (t)Mη x = eiηt Mη x for all t ≥ 0. Proof. Observe that τ 1 τ −iηs 1 −iηs e ux (t + s) ds = T (t) e T (s)x ds . τ 0 τ 0 It follows easily that ux ∈ E(R+ , X) if and only if x ∈ Xe , and that then Mη (ux )(t) = T (t)Mη x. By (4.38), Mη (ux )(t) = eiηt Mη (ux )(0) = eiηt Mη x. Let T be a bounded C0 -semigroup on X with generator A. A vector x ∈ X is a unimodular eigenvector of T if there exists η ∈ R such that T (t)x = eiηt x for all t ≥ 0 (equivalently, x ∈ D(A) and Ax = iηx). Let Xap = span {unimodular eigenvectors of T } be the space of almost periodic vectors of T . Then Xap is a closed T -invariant subspace of X. Proposition 5.4.2 shows that Mη x ∈ Xap whenever x ∈ Xe and η ∈ R. When T is the restriction of a C0 -group of isometries on X, this deﬁnition of Xap is consistent with the deﬁnition given in Section 4.5.

362

5. ASYMPTOTICS OF SOLUTIONS OF CAUCHY PROBLEMS

Example 5.4.3. Let μ be a Borel measure on C− := {λ ∈ C : Re λ ≤ 0}, and let X := Lp (μ) := Lp (C− , μ) for some 1 ≤ p < ∞. Let T be the multiplier semigroup given by (T (t)f )(λ) := eλt f (λ) (f ∈ X, λ ∈ C− , t ≥ 0). The generator A is given by D(A) (Af )(λ)

f ∈X:

= =

|λf (λ)| dμ(λ) < ∞ , p

C−

λf (λ).

Furthermore, σ(A) = supp μ, σp (A) = σp (A∗ )

= {λ : λ atom of μ}.

The measure μ can be decomposed as μ = μ− + ν = μ− + νa + νn , where μ− is the restriction of μ to C− := {λ ∈ C : Re λ < 0}, ν is the restriction of μ to iR, and νa and νn are the atomic and non-atomic parts of ν, respectively. Since μ− , νa and νn are carried by disjoint sets, X splits in a natural way as X = Lp (μ− ) ⊕ Lp (ν) = Lp (μ− ) ⊕ Lp (νa ) ⊕ Lp (νn ). It is easy to verify that T is totally ergodic (this is automatic for 1 < p < ∞, since X is then reﬂexive), and Xe

= X,

Xe0 X0

= Lp (μ− ) ⊕ Lp (νn ), = Lp (μ− ),

Xap

= Lp (νa ).

For f ∈ X, uf is asymptotically almost periodic if and only if f ∈ Lp (μ− )⊕Lp (νa ). Moreover, there is a bounded C0 -group U on Lp (ν) such that U (t)f = T (t)f for all f ∈ Lp (ν) and all t ≥ 0. Although multiplier semigroups are very special in some ways, we shall see in the results which follow that some of the features of Example 5.4.3 hold very generally. Proposition 5.4.4. Let T be a bounded C0 -semigroup on X. Then a) Xap ⊂ Xe . b) Xap = {x ∈ X : ux ∈ AP(R+ , X)}. c) Xe0 ∩ Xap = {0}.

5.4. SPLITTING THEOREMS

363

d) There is a bounded C0 -group U on Xap such that T (t)x = U (t)x for all x ∈ Xap and all t ≥ 0. Proof. Suppose that y is a unimodular eigenvector, with uy (t) = T (t)y = eiηt y for all t ≥ 0. Then uy ∈ AP(R+ , X) and y = T (t)(e−iηt y) ∈ T (t)(Xap ). By linearity and continuity (Proposition 5.4.1), ux ∈ AP(R+ , X) for all x ∈ Xap , and T (t)(Xap ) is a dense subspace of Xap . Let x ∈ Xap . Then ux ∈ E(R+ , X), so x ∈ Xe by Proposition 5.4.2. Let t ≥ 0. By (4.23), there exist sn ∈ R+ such that lim x − T (sn + t)x = lim ux (0) − ux (sn + t) = 0.

n→∞

n→∞

Hence,

x = lim T (sn )T (t)x ≤ M T (t)x , n→∞

where M := sups≥0 T (s) . Thus, M −1 x ≤ T (t)x ≤ M x ,

(5.14)

for all x ∈ Xap and all t ≥ 0. This implies that T (t)(Xap ) is closed, and we saw above that it is a dense subspace of Xap . It follows from this and (5.14) that T |Xap extends to a bounded C0 -group U on Xap . If x ∈ Xe0 ∩Xap , then Mη (ux )(0) = Mη x = 0 for all η, so x = 0 by Proposition 4.7.1 and Corollary 4.5.9. be the Finally, suppose that ux ∈ AP(R+ , X) and let π : X → X/Xap =: X and for each η ∈ R, quotient map. Then π ◦ ux ∈ AP(R+ , X) Mη (π ◦ ux )(0) = π(Mη (ux )(0)) = π(Mη x) = 0, since Mη x is a unimodular eigenvector (Proposition 5.4.2). By Proposition 4.7.1 and Corollary 4.5.9, π ◦ ux = 0, so ux (t) ∈ Xap for all t ≥ 0. In particular, x = ux (0) ∈ Xap . In Proposition 5.4.15, we shall extend part b) of Proposition 5.4.4 to individual solutions of homogeneous Cauchy problems, and in particular to individual orbits of (unbounded) semigroups. On the other hand, the following example shows that the assumption that T is bounded is important for many aspects of Proposition 5.4.4. Example 5.4.5. There is an (unbounded) C0 -semigroup T on a Banach space X such that ω(T ) = 0, the set of vectors x such that limt→∞ T (t)x = 0 is dense in X, and the span of the unimodular eigenvectors of T is dense in X. Let w : R+ → R+ be a continuous function with the following properties: a) w(0) = 1, b) w is strictly decreasing,

364

5. ASYMPTOTICS OF SOLUTIONS OF CAUCHY PROBLEMS

c) limt→∞ w(t) = 0, d) w(s + t) ≥ w(s)w(t) (s, t ≥ 0). Let X

:=

f :=

f : R+ → C : f continuous, lim f (t)w(t) = 0 , t→∞

sup |f (t)|w(t), t≥0

(T (t)f ) (s)

:= f (s + t).

Then T is a C0 -semigroup on X, with T (t) = 1/w(t), and we may choose w such that ω(T ) = 0 (for example, w(t) = (1 + t)−1 ). For η ∈ R, eiη is a unimodular eigenvalue of T , where eiη (t) := eiηt . Suppose that f ∈ X has compact support. Then limt→∞ T (t)f = 0. Given ! "−1 ε > 0, choose τ such that supp f ⊂ [0, τ ] and w(τ ) < ε supt≥0 |f (t)| + ε . By Fej´er’s Theorem (Theorem 4.2.19), there is a trigonometric polynomial p of period τ such that |f (t) − p(t)| < ε whenever 0 ≤ t ≤ τ . For t > τ , f (t) = 0 and there exists t ∈ [0, τ ] such that p(t) = p(t ), so |f (t) − p(t)|w(t) = |p(t )|w(t) ≤ (|f (t )| + ε)w(τ ) < ε. Thus, f − p < ε. Since the functions of compact support are dense in X, it follows that the unimodular eigenvectors span a dense subspace of X, and also that the vectors f with limt→∞ T (t)f = 0 are dense. The following theorem is one of the main results of this section. Theorem 5.4.6 (Strong Splitting Theorem). Let T be a bounded C0 -semigroup on X, and let x ∈ X. The following are equivalent: (i) x ∈ X0 ⊕ Xap . (ii) ux is asymptotically almost periodic. (iii) {T (t)x : t ≥ 0} is relatively compact. Moreover, if x = x0 + x1 where x0 ∈ X0 and x1 ∈ Xap , then x1 ≤ M x , where M = supt≥0 T (t) . Proof. The implication (i) ⇒ (ii) follows immediately from the fact that AAP(R+ , X) = C0 (R+ , X) ⊕ AP(R+ , X), together with Proposition 5.4.4 b). Suppose that ux is asymptotically almost periodic. Then ux = f + g for some f ∈ C0 (R+ , X) and g ∈ AP(R+ , X). By (4.37), there is a sequence (tn ) in R+ such that lim ux (tn + s) − g(s) = 0, n→∞

5.4. SPLITTING THEOREMS

365

uniformly for s ≥ 0. Let x1 := g(0) = limn→∞ ux (tn ). For s ≥ 0, g(s) = lim ux (tn + s) = lim T (s)ux (tn ) = T (s)x1 . n→∞

n→∞

Thus, g = ux1 . By Proposition 5.4.4 b), x1 ∈ Xap . Moreover, ux−x1 = f ∈ C0 (R+ , X), so x − x1 ∈ X0 . By Proposition 5.4.4 c), x1 is unique. Moreover,

x1 = limn→∞ T (tn )x ≤ M x . This proves the implication (ii) ⇒ (i) and the ﬁnal statement of the theorem. The implication (ii) ⇒ (iii) follows from the fact that any asymptotically almost periodic function has relatively compact range (see Theorem 4.7.4). Now suppose that {T (t)x : t ≥ 0} is relatively compact. Let (tn ) be a sequence in R+ . By assumption, there is a subsequence (tnk ) such that T (tnk )x converges to a limit x1 in X. Then (S(tnk )ux ) (t) = T (t + tnk )x → T (t)x1 uniformly for t ≥ 0. Thus, {S(t)ux : t ≥ 0} is relatively compact in BUC(R+ , X), so Theorem 4.7.4 shows that ux is asymptotically almost periodic. A bounded C0 -semigroup T on X is said to be asymptotically almost periodic if the equivalent conditions of Theorem 5.4.6 are satisﬁed for every x ∈ X. In other words, T is asymptotically almost periodic if and only if X = X0 ⊕ Xap (as a topological direct sum). In the literature, such semigroups are often called “almost periodic”, but we will not use this loose terminology. We shall see in Section 5.5 that a totally ergodic semigroup with generator A is asymptotically almost periodic if σ(A) ∩ iR is countable. We can see this immediately in the special case when A has compact resolvent, i.e., R(λ, A) is compact for λ ∈ ρ(A) (see Appendix B). Proposition 5.4.7. Let T be a bounded C0 -semigroup on X such that the generator A of T has compact resolvent. Then X = X0 ⊕ Xap . Proof. Choose λ ∈ ρ(A). For x ∈ D(A), {T (t)x : t ≥ 0} = {R(λ, A)T (t)(λI − A)x : t ≥ 0} , which is relatively compact. It follows by density of D(A) that {T (t)x : t ≥ 0} is relatively compact for all x ∈ X, so the result follows from Theorem 5.4.6. Now we turn towards the weak splitting theorem. First, we require some preliminary results of a very classical nature. Recall from Section 4.5 that a complexvalued trigonometric polynomial is a function p : R → C of the form p=

m

λj eiηj

j=1

for some m ∈ N, λj ∈ C and distinct ηj ∈ R, where eiη (t) = eiηt . Then p is totally ergodic with means λj if η = ηj for some j, Mη (p)(0) = 0 otherwise.

366

5. ASYMPTOTICS OF SOLUTIONS OF CAUCHY PROBLEMS

We shall write μη (p) for Mη (p)(0). Proposition 5.4.8. Let η1 , η2 , . . . , ηk ∈ R, and ε > 0. There is a trigonometric polynomial p : R → C such that a) p(t) ≥ 0 for all t, b) μ0 (p) = 1, c) 1 ≥ μηj (p) > 1 − ε for j = 1, 2, . . . , k. Proof. Let {η1 , η2 , . . . , ηm } be a basis over Q of the Q-linear span of {η1 , η2 , . . . , ηk }, m so that ηj = r=1 βjr ηr for some unique βjr ∈ Q. Replacing ηr by ηr /nr and βjr by βjr nr where nr is a common multiple of the denominators of β1r , β2r , . . . , βmr , we may assume that βjr ∈ Z. For j = 1, 2, . . . , m and N ∈ N, let FjN be the Fej´er kernel corresponding to the frequency ηj (see the proof of Theorem 4.2.19):

⎛ ⎞2 N ηj t N |n| iηj t 1 ⎝ sin 2 ⎠ FjN (t) := 1− e = . η t N N sin 2j n=−N Let pN (t) :=

m 9 j=1

=

FjN (t)

1−

1≤n1 ,...,nm ≤N

|n1 | N

|nm | ... 1 − exp (i(n1 η1 + . . . + nm ηm )t) . N

Then pN (t) ≥ 0 and the Q-independence of {η1 , . . . , ηm } implies that

μ0 (pN ) μηj (pN )

=

1, |βj1 | |βjm | = 1− ... 1− N N

for j = 1, 2, . . . , k. The result follows by choosing N suﬃciently large. Corollary 5.4.9. There is a net (pα )α∈Λ of complex-valued trigonometric polynomials such that a) pα (t) ≥ 0 for all t ∈ R and all α ∈ Λ; b) μ0 (pα ) = 1 for all α ∈ Λ; and c) limα μη (pα ) = 1 for all η ∈ R.

5.4. SPLITTING THEOREMS

367

Proof. Let P be the set of all trigonometric polynomials p such that p(t) ≥ 0 for all t and μ0 (p) = 1. For p ∈ P, deﬁne νp : R → C by νp (η) := μη (p). By Proposition 5.4.8, the constant function 1 is in the closure of {νp : p ∈ P} in CR for the topology of pointwise convergence, and the result follows. Proposition 5.4.10. Let T be a bounded C0 -semigroup on X, and let x ∈ Xe . Then there is a net (xα )α∈Λ in X such that a) For each α, xα ∈ Xap ; b) For each α, xα is in the closed convex hull of {T (t)x : t ≥ 0}; and c) limα Mη (xα ) = Mη (x) for all η ∈ R. Proof. Given a complex-valued trigonometric polynomial p(t) = (where μη (p) = 0 for all except ﬁnitely many η), let p · x := μ−η (p)Mη (x) η

= =

1 t→∞ t lim lim

t→∞

1 t

t

0

η

μη (p)eiηt

μ−η (p)e−iηs T (s)x ds

η t

p(s)T (s)x ds. 0

Since Mη (x) is a unimodular eigenvector of T , p · x ∈ Xap and Mη (p · x) = μ−η (p)Mη (x). Suppose that p(t) ≥ 0 for all t and μ0 (p) = 1. Then 1 t p(s)T (s)x ds t 0 p · x = lim . 1 t t→∞ t 0 p(s) ds For each t > 0, 1 t

t 0 1 t

p(s)T (s)x ds t p(s) ds 0

is the mean value of ux with respect to a probability measure on [0, t], and therefore it is in the closed convex hull of the orbit of x. It follows that p · x also lies in this closed set. The result now follows by taking (pα )α∈Λ as in Corollary 5.4.9 and putting xα := pα · x. Now we can give the second main theorem of this section, which is the analogue of Theorem 5.4.6 for the weak topology.

368

5. ASYMPTOTICS OF SOLUTIONS OF CAUCHY PROBLEMS

Theorem 5.4.11 (Weak Splitting Theorem). Let T be a bounded C0 -semigroup on X, let x ∈ X, and suppose that {T (t)x : t ≥ 0} is relatively weakly compact. Then x = x0 + x1 for some unique x0 ∈ Xe0 and x1 ∈ Xap . Moreover, x1 ≤ M x , where M := supt≥0 T (t) . Proof. The uniqueness follows from Proposition 5.4.4. For existence, we may replace X by the closed linear span of the orbit of x, so we may assume that every vector in X is totally ergodic with respect to T . Let (xα )α∈Λ be as in Proposition 5.4.10. The property b) of Proposition 5.4.10 shows that xα ≤ M x and {xα : α ∈ Λ} is relatively weakly compact, by Krein’s theorem [Meg98, Theorem 2.8.14]. Hence, there is a subnet which is weakly convergent to a limit x1 ∈ X with x1 ≤ M x . Since {xα } ⊂ Xap and Xap is norm closed and hence weakly closed, x1 ∈ Xap . Let x0 := x − x1 . Since limα Mη (x − xα ) = 0 and Mη is bounded, hence weakly continuous on X, Mη (x0 ) = 0. Corollary 5.4.12. Let T be a bounded C0 -semigroup on a reﬂexive space X. Then X = Xe0 ⊕ Xap as a topological direct sum. Comparison of Theorems 5.4.11 and 5.4.6 provides the following corollary. Corollary 5.4.13. Let T be a bounded C0 -semigroup on X, let x ∈ X, and suppose that x is totally ergodic with respect to T with all means Mη x = 0 and that the orbit {T (t)x : t ≥ 0} is relatively compact. Then limt→∞ T (t)x = 0. Corollary 5.4.14. Let T be a bounded C0 -semigroup on X, let x ∈ X, and suppose that x is totally ergodic with respect to T and that the orbit {T (t)x : t ≥ 0} is relatively compact. If Mη x = 0 for all η ∈ R \ {0}, then limt→∞ T (t)x exists. Proof. This follows by applying Corollary 5.4.13 with x replaced by x − M0 x. Next, we consider mild solutions of Cauchy problems, without assuming the existence of a semigroup. For some purposes, it is not natural to specify an initial value, and we therefore extend our previous terminology by saying that u ∈ C(R+ , X) is a mild solution of the abstract Cauchy problem (ACPf )

u (t) = Au(t) + f (t) t

(t ≥ 0),

t where f ∈ L1loc (R+ , X), if 0 u(r) dr ∈ D(A) and u(t) = u(0) + A 0 u(r) dr + t f (r) dr for all t ≥ 0. It is easy to see that this implies that u(t) = u(s) + 0 t t A s u(r) dr + s f (r) dr for t ≥ s ≥ 0. We begin by describing the almost periodic solutions of the homogeneous Cauchy problem, thereby extending Proposition 5.4.4 c). Proposition 5.4.15. Let A be a closed linear operator on X, and let u ∈ AP(R+ , X) be a mild solution of (ACP0 ). For each ε > 0, there exist n ∈ N, x1 , . . . , xn ∈ D(A), and η1 , . . . , ηn ∈ R such that Axj = iηj xj and

n

iηj t

u(t) − e xj < ε (5.15)

j=1

5.4. SPLITTING THEOREMS

369

for all t ≥ 0. Proof. Let Z be the set of all u ∈ BUC(R+ , X) which are mild solutions of (ACP0 ). Let (un ) be a sequence in Z, u ∈ BUC(R+ , X) and suppose that un − u ∞ → 0. t For t ≥ 0, 0 un (s) ds ∈ D(A) and t un (t) = un (0) + A un (s) ds. 0

Since A is closed, it follows on letting n → ∞ that t u(t) = u(0) + A u(s) ds. 0

Thus, Z is a closed subspace of BUC(R+ , X). Moreover, if S is the shift semigroup on BUC(R+ , X) and u ∈ Z, then S(t)u ∈ Z for all t ≥ 0. Now, suppose that u ∈ Z ∩ AP(R+ , X). By Proposition 4.7.1, u = g|R+ for some g ∈ AP(R, X). Let Y := span{SR (t)g : t ∈ R} ⊂ BUC(R, X), where SR is the shift group on BUC(R, X). Applying Theorem 4.5.1 to the restriction of SR to Y shows that g ∈ span {eiη ⊗ x : η ∈ R, x ∈ X, eiη ⊗ x ∈ Y } .

(5.16)

Let t ∈ R. By (4.23) there exist τn → ∞ such that lim SR (t + τn )g − SR (t)g ∞ = 0.

n→∞

(5.17)

For t + τn > 0, (SR (t + τn )g)|R+ = S(t + τn )u, so it follows from (5.17) and the ﬁrst paragraph that (SR (t)g)|R+ ∈ Z for every t ∈ R, and hence that h|R+ ∈ Z for every h ∈ Y . Now suppose that h := eiη ⊗ x ∈ Y . Take t > 0 and let ⎧ iηt ⎨ e − 1 if η = 0, ζ := iη ⎩ t if η = 0. We can choose t in such a way that ζ = 0. Then 1 t x= h(s) ds ∈ D(A), ζ 0 and

e

iηt

x = h(t) = h(0) + A

t

h(s) ds = x + ζAx. 0

Hence, Ax = iηx. The result follows from this and (5.16). Now we show that the splitting of AAP(R+ , X) as C0 (R+ , X) ⊕ AP(R+ , X) respects mild solutions of inhomogeneous Cauchy problems.

370

5. ASYMPTOTICS OF SOLUTIONS OF CAUCHY PROBLEMS

Proposition 5.4.16. Let A be a closed linear operator on X, let u0 ∈ C0 (R+ , X), u1 ∈ AP(R+ , X), f0 ∈ C0 (R+ , X), f1 ∈ AP(R+ , X), u = u0 + u1 , and f = f0 + f1 . Suppose that u is a mild solution of (ACPf ). Then u0 and u1 are mild solutions of (ACPf0 ) and (ACPf1 ), respectively. Proof. It is easy to verify that the function t → (u1 (t), f1 (t)) is almost periodic (with values in X × X), and hence that t → (u(t), f (t)) is asymptotically almost periodic. By (4.37), there is a sequence (τn )n≥1 in R+ such that τn → ∞ and sup u(t + τn ) − u1 (t) → 0,

sup f (t + τn ) − f1 (t) → 0

t≥0

t≥0

as n → ∞. Since u is a mild solution of (ACPf ), u(t + τn ) = A

t u(s + τn ) ds + f (s + τn ) ds + u(τn ).

t 0

0

Since A is closed, it follows on letting n → ∞ that u1 (t) = A

t

0

u1 (s) ds +

t 0

f1 (s) ds + u1 (0),

so u1 is a mild solution of (ACPf1 ). By linearity, u0 = u − u1 is a mild solution of (ACPf0 ). The following corollary is a generalization of part of the strong splitting theorem (Theorem 5.4.6, (i) ⇔ (ii)). Corollary 5.4.17. Let T be a C0 -semigroup on X, let x ∈ X, and suppose that ux is asymptotically almost periodic. Then there exist unique x0 and x1 in X such that a) x = x0 + x1 ; b) limt→∞ T (t)x0 = 0; and c) There is a sequence (yn ) in the linear span of the unimodular eigenvectors of T such that limn→∞ T (t)yn = T (t)x1 uniformly for t ≥ 0. Conversely, if a), b) and c) hold, then ux is asymptotically almost periodic. Proof. Let ux = v0 + v1 , where v0 ∈ C0 (R+ , X) and v1 ∈ AP(R+ , X). By Proposition 5.4.16 (with A equal to the generator of T and f0 = f1 = 0), v0 = ux0 and v1 = ux1 for some x0 , x1 ∈ X. Now, a) and b) are immediate, c) follows from Proposition 5.4.15, and the uniqueness follows from (4.36). The converse statement is immediate, as in Theorem 5.4.6.

5.5. COUNTABLE SPECTRAL CONDITIONS

5.5

371

Countable Spectral Conditions

In this section, we shall give various results showing that solutions of well-posed homogeneous Cauchy problems are asymptotically almost periodic under assumptions including boundedness of the solution and countability of the purely imaginary part of the (local) spectrum of A, using the results and methods of Chapter 4. In Section 5.6, we shall extend some of the results to inhomogeneous Cauchy problems which are not necessarily associated with C0 -semigroups. Suppose that A generates a C0 -semigroup T on a complex Banach space X. For x ∈ X, we again put ux (t) := T (t)x (t ≥ 0). Then ux is an exponentially bounded function and its Laplace transform u x (λ) coincides with R(λ, A)x for large real λ. We shall assume that ux is bounded, and our ﬁrst step is to relate the half-line spectrum sp(ux ) of ux in the sense of Section 4.7 to the operator A. The imaginary local resolvent set ρu (A, x) of A at x is deﬁned to be the set of all points iη ∈ iR such that there exist an open set U containing C+ ∪ {iη} and a holomorphic function g : U → X such that g(λ) ∈ D(A) and (λ − A)g(λ) = x for all λ ∈ C+ . We shall see in Proposition 5.5.1 that g(λ) = R(λ, A)x whenever λ ∈ C+ ∩ ρ(A), so g is uniquely determined if U is connected. The imaginary local spectrum σu (A, x) of A at x is: σu (A, x) := iR \ ρu (A, x). It is clear from the deﬁnition that ρu (A, x) is open in iR, so σu (A, x) is closed. When A generates a bounded C0 -group, Lemma 4.6.7 shows that σu (A, x) coincides with the local spectrum σ(A, x) deﬁned in Section 4.6. The following is a local version of Proposition B.5. Proposition 5.5.1. Let T be a C0 -semigroup on X with generator A, and let x ∈ X. a) Let V be a connected open subset of C, let g : V → X be holomorphic, and suppose that there is a subset U of V , with a limit point in V , such that g(λ) ∈ D(A) and (λ − A)g(λ) = x whenever λ ∈ U . Then g(λ) ∈ D(A) and (λ − A)g(λ) = x whenever λ ∈ V . b) If Re λ > hol( ux ), then u x (λ) ∈ D(A) and (λ − A) ux (λ) = x. c) If Re λ > hol( ux ) and λ ∈ ρ(A), then u x (λ) = R(λ, A)x. Proof. a) Fix μ ∈ ρ(A). For λ ∈ U , (λ − A)R(μ, A)g(λ) = R(μ, A)x. Since AR(μ, A) = μR(μ, A) − I is a bounded operator on X, the left-hand side is holomorphic on V . By uniqueness of holomorphic extensions, this formula is true whenever λ ∈ V . Hence, g(λ) = R(μ, A)x − (λ − μ)R(μ, A)g(λ) ∈ D(A),

372

5. ASYMPTOTICS OF SOLUTIONS OF CAUCHY PROBLEMS

and (λ − A)g(λ) = x, since R(μ, A) is injective. Statement b) follows by applying a) with g := u x , and c) is then immediate. The following result identiﬁes σu (A, x) with sp(ux ) when ux is bounded. Proposition 5.5.2. Let T be a C0 -semigroup on X with generator A. Let x ∈ X, and suppose that ux is bounded. Then a) σu (A, x) ⊂ σ(A) ∩ iR. b) sp(ux ) = {η ∈ R : iη ∈ σu (A, x)}. Proof. a) By Proposition 5.5.1 c), u x (λ) = R(λ, A)x whenever λ ∈ C+ ∩ ρ(A). Thus, we may deﬁne a holomorphic function g : C+ ∪ ρ(A) → X by u x (λ) (λ ∈ C+ ), g(λ) := R(λ, A)x (λ ∈ ρ(A)). This shows that σu (A, x) ⊂ σ(A) ∩ iR. b) Suppose that iη ∈ / σu (A, x), so there exist an open set U containing C+ ∪ {iη} and a holomorphic function g : U → X such that g(λ) ∈ D(A) and (λ − A)g(λ) = x for all λ ∈ C+ . Then g(λ) = R(λ, A)x = u x (λ) whenever λ ∈ ρ(A) ∩ C+ . By uniqueness of holomorphic extensions, g(λ) = u x (λ) whenever λ ∈ C+ , so η∈ / sp(ux ). Conversely, suppose that η ∈ / sp(ux ), so there exists a connected open set V containing iη and a holomorphic function g : V → X such that g(λ) = u x (λ) for all λ ∈ V ∩ C+ . By Proposition 5.5.1 b), g(λ) ∈ D(A) and (λ − A)g(λ) = x for all x ∈ V ∩ C+ . By Proposition 5.5.1 a), this holds for all λ ∈ V . Thus, iη ∈ / σu (A, x). The next four results appear in decreasing order of generality, with the assumptions becoming stronger but less technical. Theorem 5.5.3. Let T be a C0 -semigroup on X with generator A. Let x ∈ X, and suppose that the following conditions are satisﬁed: a) ux : t → T (t)x is bounded and uniformly continuous; b) σu (A, x) is countable; and c) For each iη ∈ σu (A, x), limα↓0 αT (s) ux (α + iη) exists, uniformly for s ≥ 0. Then ux is asymptotically almost periodic. If all the limits in c) are zero, then

T (t)x → 0 as t → ∞.

5.5. COUNTABLE SPECTRAL CONDITIONS

373

Proof. By Proposition 5.5.2, condition b) is equivalent to sp(ux ) being countable. Moreover, c) is equivalent to total ergodicity of ux . Indeed, if S denotes the shift semigroup on BUC(R+ , X), then x (λ) = T (s) (S(s)ux ) (t) = T (s)ux (t), so S(s)u ux (λ). The result follows from Theorem 4.7.7 or 4.9.7. Theorem 5.5.4. Let T be a bounded C0 -semigroup on X, with generator A. Let x ∈ X, and suppose that the following conditions are satisﬁed: a) σu (A, x) is countable; and b) x ∈ Ker(A − iη) + Ran(A − iη) for each iη ∈ σu (A, x). Then x ∈ X0 ⊕ Xap . If x ∈ Ran(A − iη) for each iη ∈ σu (A, x), then x ∈ X0 . Proof. By Proposition 5.4.1, ux is uniformly continuous and bounded. Moreover, u x (α + iη) = R(α + iη, A)x for α > 0, limα↓0 αT (s) ux (α + iη) exists, uniformly for s ≥ 0, by Proposition 4.3.1 and the boundedness of T , and the limit is 0 if x ∈ Ran(A − iη). Now the results follow from Theorem 5.5.3. The results of Section 4.3 (applied with T (t) replaced by e−iηt T (t)) show that condition b) of Theorem 5.5.4 is equivalent to any of the following: (i) If x∗ ∈ D(A∗ ), A∗ x∗ = iηx∗ for some iη ∈ σu (A, x) and y, x∗ = 0 for all y ∈ Ker(A − iη), then x, x∗ = 0; (ii) limα↓0 αR(α + iη, A)x exists in X whenever iη ∈ σu (A, x); t (iii) limt→∞ t−1 0 e−iηs T (s)x ds exists whenever iη ∈ σu (A, x). Since the Abel limit exists (and equals 0) when iη ∈ ρu (A, x), the restriction in each condition that iη ∈ σu (A, x) is redundant. Recall from Deﬁnition 4.3.10 that a bounded C0 -semigroup T on X is said to be totally ergodic if the condition (iii) above is satisﬁed for all x ∈ X (and all η ∈ R). By Proposition 5.4.2, T is totally ergodic if and only if each orbit ux is a totally ergodic function. As in the case of individual η and x discussed above, this property can be characterized in several ways in terms of the generator A. Recall also from Section 5.4 that T is said to be asymptotically almost periodic if each orbit ux is asymptotically almost periodic (see Theorem 5.4.6 for equivalent properties). Thus, every asymptotically almost periodic semigroup is totally ergodic, but the converse is not true (see Example 5.5.9). The following result shows that the converse is true under a countable spectral condition. Note that Ran(A − iη) is dense in X if and only if iη does not belong to the point spectrum σp (A∗ ) of A∗ , and that σp (A) ∩ iR ⊂ σp (A∗ ) when A generates a bounded semigroup (see Proposition 4.3.6).

374

5. ASYMPTOTICS OF SOLUTIONS OF CAUCHY PROBLEMS

Theorem 5.5.5 (Countable Spectrum). Let T be a bounded C0 -semigroup on X with generator A, and suppose that σ(A) ∩ iR is countable. a) If T is totally ergodic, then T is asymptotically almost periodic. b) If σp (A∗ ) ∩ iR is empty, then T (t)x → 0 as t → ∞, for each x ∈ X. Proof. This is immediate from Proposition 5.5.2, Theorem 5.5.4 and the remarks above. Theorem 5.5.6. Let T be a bounded C0 -semigroup on a reﬂexive space X with generator A, and suppose that σ(A) ∩ iR is countable. Then the following hold: a) T is asymptotically almost periodic. b) If σp (A) ∩ iR ⊂ {0}, then there is a projection P such that limt→∞ T (t) = P in the strong operator topology. Proof. a) This is immediate from Corollary 4.3.5 and Theorem 5.5.5 a). b) By a), X = X0 ⊕ Xap . Let P be the projection onto Xap along X0 . The assumption on σp (A) ∩ iR implies that Xap = {x ∈ X : T (t)x = x (t ≥ 0)}, and the result follows. Theorem 5.5.6 is also true if T is a positive, bounded, Ces` aro-ergodic C0 semigroup on L1 (Ω, μ) and σ(A) ∩ iR is countable. This follows from Proposition 4.3.14. We saw in Proposition 5.4.7 that Theorem 5.5.5 can be proved more directly when A has compact resolvent. There is also a simple proof when T is a bounded holomorphic C0 -semigroup in the sense of Deﬁnition 3.7.1. Then σ(A) ∩ iR ⊂ {0} by Theorem 3.7.11, and there is a constant c such that AT (t) ≤ c/t for all t > 0, by Theorem 3.7.19. Hence, limt→∞ T (t)Ay = 0 for all y ∈ D(A), and limt→∞ T (t)(x1 + x2 ) = x1 whenever x1 ∈ Ker A and x2 ∈ Ran A. If A is totally ergodic, then Ker A + Ran A is dense in X, so limt→∞ T (t)x exists for all x ∈ X. The following example shows that the assumption of uniform continuity cannot be omitted from Theorem 5.5.3. Example 5.5.7. There is a C0 -semigroup T on a Hilbert space X and a vector x ∈ X such that σ(A)∩iR is empty and ux is bounded, but ux is not asymptotically almost periodic (and not uniformly continuous, by Theorem 5.5.3). Let X := 2 and T be the C0 -semigroup deﬁned by: (T (t)x)2n−1

:=

eλn t (x2n−1 + tx2n ),

(T (t)x)2n

:=

eλn t x2n ,

where λn := in − 1/n. The generator A is given by: ' ( D(A) = x ∈ 2 : (nxn ) ∈ 2 , (Ax)2n−1 = x2n + λn x2n−1 , (Ax)2n = λn x2n , σ(A) = {λn : n ≥ 1}.

5.5. COUNTABLE SPECTRAL CONDITIONS

375

Now take x ∈ 2 given by: x2n := n−3/2 .

x2n−1 := 0, Then (T (t)x)2n−1

=

(T (t)x)2n

=

teλn t , n3/2 eλn t . n3/2

A simple argument involving Riemann sums of s−3 e−2/s shows that ∞ ∞ ∞ 1 + t2 −2t/n 1 −3 −2/s

T (t)x 2 = e → s e ds = ue−2u du = 3 n 4 0 0 n=1 as t → ∞. Thus, all the assumptions of Theorem 5.5.3, except uniform continuity, are satisﬁed, with c) vacuous, and T has no unimodular eigenvectors since A has no imaginary eigenvalues. However, limt→∞ T (t)x = 0, so ux is not asymptotically almost periodic, by Corollary 5.4.17. Theorem 5.5.4 was obtained from Theorem 5.5.3 by observing that the convergence in condition c) of Theorem 5.5.3 is automatically uniform when T is uniformly bounded. The next example shows that this may not be valid for individual bounded orbits of unbounded semigroups. Example 5.5.8. There is a (norm-continuous, unbounded) C0 -semigroup T on a Hilbert space X and a vector x ∈ X such that σ(A)∩iR = {0}, 0 ∈ / σp (A)∪σp (A∗ ), and ux is bounded and uniformly continuous, but ux is not asymptotically almost periodic (and not totally ergodic). Let X := 2 , and T be the norm-continuous semigroup given by (T (t)x)2n−1

:=

e−t/n (x2n−1 + tx2n ),

(T (t)x)2n

:=

e−t/n x2n .

The generator A is the bounded operator given by: (Ax)2n−1

= x2n −

x2n−1 , n

x2n = − , n 1 σ(A) = − : n ≥ 1 ∪ {0}. n

(Ax)2n

Moreover, 0 ∈ / σp (A) ∪ σp (A∗ ), so Ran A = X, and Xap = {0}. Now take x2n−1 := 0, x2n := n−3/2 .

376

5. ASYMPTOTICS OF SOLUTIONS OF CAUCHY PROBLEMS

As in Example 5.5.7, T (t)x → 12 as t → ∞. Thus, ux : t → T (t)x is bounded. Since A is bounded, it follows that t → AT (t)x = ux (t) is bounded, so ux is uniformly continuous. Since T has no unimodular eigenvectors and limt→∞ T (t)x

= 0, it follows from Corollary 5.4.17 that ux is not asymptotically almost periodic. By Theorem 4.7.7 or 4.9.7 (see also Theorem 5.5.3), ux is not totally ergodic. It is not diﬃcult to verify directly that limα↓0 αR(α, A)x does not exist, which also shows that ux is not totally ergodic, and therefore not asymptotically almost periodic. The next example shows that the countability condition in Theorem 5.5.5 (and hence in other results of this section) is best possible in a certain sense. Example 5.5.9. Let E be any uncountable closed subset of R. There is a C0 -group T of isometries on a Banach space X (even a Hilbert space) such that σ(A) ⊂ iE and σp (A∗ ) is empty, but T is not (asymptotically) almost periodic. Choose [a, b] so that E ∩ [a, b] is uncountable, and let E := {x ∈ E ∩ [a, b] : for all ε > 0, E ∩ [a, b] ∩ (x − ε, x + ε) is uncountable}. Then (E ∩ [a, b]) \ E is countable, and E is compact with no isolated points. If E contains no interval, then E is homeomorphic to the Cantor set [Wil70, Theorem 30.3]. If E contains an interval, then E clearly contains a subset homeomorphic to the Cantor set. There is a non-zero non-atomic Borel measure on the Cantor set (for example, the Lebesgue-Stieltjes measure associated with the LebesgueCantor function [Tay73, Sections 2.7,4.5]). Hence, there is a non-zero non-atomic measure μ supported by iE. Let T be the multiplier (semi)group on Lp (μ), where 1 ≤ p < ∞ (see Example 5.4.3). Then σ(A) = supp μ ⊂ iE,

σp (A∗ ) = ∅,

but T is not asymptotically almost periodic. Suppose that T is a bounded, totally ergodic semigroup on X and there is a dense subspace Y of X such that σu (A, y) is countable for each y ∈ Y . From Theorem 5.5.4 and the fact that X0 ⊕ Xap is closed, it follows that T is asymptotically almost periodic. The following example shows that the converse does not hold. Example 5.5.10. There is a C0 -semigroup T on a Banach space X such that limt→∞ T (t)x = 0 for all x ∈ X, but σu (A, x) = iR whenever x = 0. Let X := L1 (R+ , w(t)dt), where w : R+ → R+ satisﬁes: a) w is non-increasing, b) limt→∞ w(t) = 0, c) For each a > 0, there is a constant ca > 0 such that w(t) ≥ ca e−at for all t ≥ 0.

5.5. COUNTABLE SPECTRAL CONDITIONS

377

Let T be the C0 -semigroup of contractions on X deﬁned by f (s − t) (s ≥ t ≥ 0), (T (t)f ) (s) := 0 (t > s ≥ 0), and let A be the generator. For f ∈ X, ∞

T (t)f || = |f (s − t)|w(s) ds =

∞

0

t

|f (s)|w(s + t) ds → 0

as t → ∞. For Re λ < 0, let gλ (s) :=

eλs w(s)

(s ≥ 0).

Then gλ is bounded, by c), so eλ (s) := eλs deﬁnes an element of X ∗ . For f ∈ X and t ≥ 0, ∞ ∞ T (t)f, eλ = f (s − t)eλs ds = f (s)eλ(s+t) ds t 0 ∞ = eλt f (s)eλs ds = eλt f, eλ . 0

Thus,

T ∗ (t)eλ = eλt eλ

(t ≥ 0),

so eλ ∈ D(A∗ ) and A∗ eλ = λeλ . Suppose that f ∈ X and λ → R(λ, A)f has a holomorphic extension F to a connected neighbourhood V of some point iη0 in iR. We shall show that f = 0. For λ ∈ V ∩ C+ , R(1, A)f = F (λ) + (λ − 1)R(1, A)F (λ). By uniqueness of holomorphic extensions, R(1, A)f = F (λ) + (λ − 1)R(1, A)F (λ) = (λ − A)R(1, A)F (λ) for all λ ∈ V . Hence, for λ ∈ V ∩ C− , ∞ eλs (R(1, A)f ) (s) ds = R(1, A)f, hλ = R(1, A)F (λ), (λ − A∗ )hλ = 0. 0

As a function of λ, the left-hand side is holomorphic on C− and vanishes on C− ∩V . Therefore, ∞

0

eλs (R(1, A)f ) (s) ds = 0

(λ ∈ C− ).

By uniqueness of Laplace transforms, this implies that R(1, A)f = 0. Hence, f = 0, since R(1, A) is injective.

378

5.6

5. ASYMPTOTICS OF SOLUTIONS OF CAUCHY PROBLEMS

Solutions of Inhomogeneous Cauchy Problems

In this section, we consider the asymptotic behaviour of mild solutions of inhomogeneous Cauchy problems. In practice, once one knows that the solution is bounded and uniformly continuous, it often follows that further asymptotic properties are inherited by u from asymptotic properties of a semigroup generated by A (or spectral properties of A) and of the inhomogeneity f . For example, Theorems 5.6.6 and 5.6.8 are results of this type under assumptions of non-resonance and countable spectrum, respectively. Thus, we are interested ﬁrst in conditions which ensure that a mild solution is bounded and uniformly continuous. We ﬁrst assume that A generates a bounded C0 -semigroup T on X, and that f ∈ L1loc (R+ , X) is given. Recall from Proposition 3.1.16 that the unique solution of the inhomogeneous Cauchy problem u (t) = Au(t) + f (t), (ACPf ) u(0) = x, is given by u(t) = T (t)x + (T ∗ f )(t), where

(T ∗ f )(t) =

t 0

T (t − s)f (s) ds.

Thus, we are seeking conditions which ensure that T ∗ f is bounded and uniformly continuous. We consider ﬁrst the simple case when ω(T ) < 0. Proposition 5.6.1. Let T be a semigroup on X with generator A, and suppose that ω(T ) < 0. a) If f ∈ L∞ (R+ , X), then T ∗ f is bounded. b) If f ∈ BUC(R+ , X), then T ∗ f ∈ BUC(R+ , X). c) If f ∈ AAP(R+ , X), then T ∗ f ∈ AAP(R+ , X). d) If f∞ := limt→∞ f (t) exists, then limt→∞ (T ∗ f )(t) = R(0, A)f∞ . e) If f ∈ AP(R+ , X), then there exist unique x ∈ X and g ∈ AP(R+ , X) such that (T ∗ f )(t) = T (t)x + g(t) for all t ≥ 0. If f is τ -periodic, then g is τ -periodic. Proof. Parts a), b) and the case f∞ = 0 of d) all follow from Proposition 1.3.5. Moreover, the map f → T ∗ f is bounded on BUC(R+ , X). Let Y := {ux : x ∈ X}, where ux (t) = T (t)x. Since ω(T ) < 0, Y is a closed subspace of C0 (R+ , X). Let Z := Y ⊕ AP(R+ , X), a closed subspace of

5.6. SOLUTIONS OF INHOMOGENEOUS CAUCHY PROBLEMS

379

AAP(R+ , X). Suppose that f = eiη ⊗ y for some η ∈ R and y ∈ X, so f (t) = eiηt y. Then t iηt (T ∗ f )(t) = e e−iηs T (s)y ds 0 t " d ! −iηs = eiηt −e T (s)R(iη, A)y ds ds 0 = T (t) (−R(iη, A)y) + eiηt R(iη, A)y. (5.18) Thus, T ∗ f ∈ Z, and the almost periodic part of T ∗ f has the same period as f . By linearity and continuity, T ∗ f ∈ Z for all f ∈ AP(R+ , X), and the almost periodic part of T ∗ f is τ -periodic when f is τ -periodic. This proves e), and d) follows from the case f∞ = 0 and the case η = 0 of (5.18). Finally, c) follows from d) and e). In the context of Proposition 5.6.1 a), we cannot conclude that T ∗ f is uniformly continuous. Example 5.6.2. There exist a C0 -semigroup S on a Hilbert space X with ω(S) < 0 and a function f ∈ L∞ (R+ , X) such that S ∗ f is not uniformly continuous. Let T , X and x be as in Example 5.5.7. Take ω > ω(T ), and let S(t) := e−ωt T (t) and f (t) := T (t)x. Then f is bounded and continuous, and t 1 − e−ωt (S ∗ f )(t) = e−ωs T (t)x ds = T (t)x. ω 0 Since T (·)x is not uniformly continuous, S ∗ f is not uniformly continuous. Remark 5.6.3. In part e) of Proposition 5.6.1, T ∗ f is an asymptotically almost periodic mild solution of the inhomogeneous Cauchy problem (ACPf ), with initial value 0. The decompositions f = 0 + f and T ∗ f = ux + g, where ux (t) = T (t)x, correspond to the splitting AAP(R+ , X) = C0 (R+ , X) ⊕ AP(R+ , X). As predicted by Proposition 5.4.16, ux is a mild solution of (ACP )0 with initial value x, and g is a mild solution of (ACP )f with initial value −x. When f is τ -periodic, x = R(1, T (τ ))((T ∗ f )(τ )) and it is easy to verify directly that T ∗ f − ux is τ -periodic. Now suppose that T is bounded but ω(T ) = 0. Then Datko’s theorem (Theorem 5.1.2) shows that there exist bounded f such that T ∗ f is unbounded. The simplest examples in which T ∗ f is unbounded arise from eigenvalues. If f (t) = T (t)x = eiηt x for all t ≥ 0, then (T ∗ f )(t) = teiηt x, and this is unbounded if x = 0. Note that there is resonance between T and f , reﬂected in the fact that iη ∈ σ(A) ∩ i sp(f ). In order to obtain positive results showing that T ∗f is bounded when ω(T ) = 0, we have to place some constraints on f and we may also impose assumptions on T . The ﬁrst possibility (which is somewhat dual to the case considered in Proposition 5.6.1) is to assume that f ∈ L1 (R+ , X).

380

5. ASYMPTOTICS OF SOLUTIONS OF CAUCHY PROBLEMS

Proposition 5.6.4. Let T be a bounded C0 -semigroup on X, and let f ∈ L1 (R+ , X). Then a) T ∗ f ∈ BUC(R+ , X). b) If T is asymptotically almost periodic, then T ∗ f ∈ AAP(R+ , X). c) If limt→∞ T (t) exists in the strong operator topology, then limt→∞ (T ∗ f )(t) exists. d) If limt→∞ T (t) = 0 in the strong operator topology, then limt→∞ (T ∗ f )(t) = 0. Proof. First, note that (T ∗ f )(t) ≤ M f 1 , where M := supt≥0 T (t) . Thus, the map f → T ∗ f is bounded, and by density it suﬃces to prove the results when f has support in [0, τ ] for some τ . But then (T ∗ f )(t) = T (t − τ )((T ∗ f )(τ )) for t ≥ τ , and all the results are immediate. Another way to obtain results that T ∗ f is bounded is to impose a condition of non-resonance by assuming that σ(A) ∩ i sp(f ) is empty. However, there are examples (see Examples 5.1.10 and 5.1.11 after rescaling) of bounded semigroups where σ(A) ∩ iR is empty (so non-resonance occurs for all f ), but ω(T ) = 0, and then by Datko’s theorem there exist bounded f such that T ∗ f is unbounded. Thus, non-resonance is not suﬃcient on its own to obtain positive results about boundedness of T ∗ f ; we need to impose further assumptions on T , or f , or both. If T is holomorphic and ω(T ) = 0, then σ(A) ∩ iR is nonempty (Theorem 5.1.12), and we now establish that non-resonance implies boundedness of T ∗ f when T is holomorphic. Theorem 5.6.5 (Non-resonance Theorem). Let T be a holomorphic C0 -semigroup on X with generator A such that supt≥0 T (t) < ∞. Let f ∈ L∞ (R+ , X), and suppose that σ(A) ∩ i sp(f ) is empty. Then T ∗ f ∈ BUC(R+ , X). Proof. Note ﬁrst that T is norm-continuous on (0, ∞), and that −iσ(A) ∩ R is compact (by Corollary 3.7.18) and disjoint from sp(f ) by assumption. Let ψ ∈ Cc∞ (R) be such that ψ = 1 on a neighbourhood of −iσ(A) ∩ R in R and ψ = 0 on a neighbourhood of sp(f ). Deﬁne G, H : R → L(X) by ∞ T (t) − 0 (F −1 ψ)(t − s)T (s) ds (t ≥ 0), ∞ G(t) := − 0 (F −1 ψ)(t − s)T (s) ds (t < 0), (1 − ψ(η))R(iη, A) (iη ∈ ρ(A)), H(η) := 0 (iη ∈ σ(A)).

5.6. SOLUTIONS OF INHOMOGENEOUS CAUCHY PROBLEMS

381

Then G ∈ L∞ (R, L(X)) and G is continuous on R \ {0}. Also, H ∈ C ∞ (R, L(X)) and, for all large |η|, H(η) = R(iη, A), so H (η) = −2R(iη, A)3 . By Corollary 3.7.18, there is a constant C such that H (η) ≤ C|η|−3 for all large |η|. Hence, H ∈ L1 (R, L(X)) and F −1 H : R → L(X) is bounded. We shall use this to show that G ∈ L1 (R, L(X)). Let ρ ∈ Cc∞ (R). By two applications of Theorem 1.8.1 b), the dominated convergence theorem and integration by parts, ∞ ∞ (Fρ)(t)(F −1 H )(t) dt = ρ(η)H (η) dη −∞ −∞ ∞ = ρ (η)H(η) dη −∞ ∞ = lim ρ (η)(1 − ψ(η))R(ξ + iη, A) dη ξ↓0 −∞ ∞ = lim (F ρ )(t)Gξ (t) dt ξ↓0 −∞ ∞ = lim (−t2 )(Fρ)(t)Gξ (t) dt, ξ↓0

where

Gξ (t) :=

−∞

∞ e−ξt T (t) − 0 (F −1 ψ)(t − s)e−ξs T (s) ds (t ≥ 0), ∞ −1 − 0 (F ψ)(t − s)e−ξs T (s) ds (t < 0).

By the dominated convergence theorem, Gξ (t) − G(t) → 0 as ξ ↓ 0. Moreover, ! "

Gξ (t) ≤ 1 + F −1 ψ 1 M for all ξ > 0 and t ∈ R, where M := supt≥0 T (t) . By the dominated convergence theorem again, ∞ ∞ (F ρ)(t)(F −1 H )(t) dt = (Fρ)(t)(−t2 )G(t) dt. −∞

Since this holds for all ρ ∈

−∞

Cc∞ (R),

it follows that

(F −1 H )(t) = −t2 G(t) a.e. (in fact, everywhere, since both sides are continuous). Hence, G ∈ L1 (R, L(X)). Now deﬁne g, h : R → X by ∞ g(t) := (F −1 ψ)(t − s)f (s) ds, 0 ψ(η)fˆ(iη) (η ∈ / sp(f )), h(η) := 0 (η ∈ sp(f )).

382

5. ASYMPTOTICS OF SOLUTIONS OF CAUCHY PROBLEMS

Then g ∈ BUC(R, X) (by Proposition 1.3.2 c)) and h ∈ Cc∞ (R, X). A similar argument to the previous paragraph shows that (F −1 h )(t) = −t2 g(t), so that g ∈ L1 (R, X). For t ≥ 0, ∞ ∞ ∞ (T ∗ f )(t) = G(t − s)f (s) ds + (F −1 ψ)(t − s − r)T (r)f (s) dr ds 0 ∞ 0 ∞ 0 = G(t − s)f (s) ds + T (r)g(t − r) dr. 0

0

Both of these terms may be regarded as being convolutions of functions on R, where f (s) = 0 for s < 0 and T (r) = 0 for r < 0. Since T is bounded and g ∈ L1 (R, X), and f is bounded and G ∈ L1 (R, L(X)), both terms are bounded and uniformly continuous on R and hence on R+ (see Propositions 1.3.2, 1.3.5 and Remark 1.3.8). Next, we give a result which shows that asymptotic properties of T ∗ f generally follow from those of T and f when non-resonance holds and T ∗ f is bounded and uniformly continuous. Theorem 5.6.6. Let T be a bounded C0 -semigroup on X, and let f ∈ BUC(R+ , X). Suppose that σ(A) ∩ i sp(f ) is empty and T ∗ f ∈ BUC(R+ , X). a) If T is asymptotically almost periodic and f ∈ AAP(R+ , X), then T ∗ f ∈ AAP(R+ , X). b) If limt→∞ T (t) exists in the strong operator topology and limt→∞ f (t) exists, then limt→∞ (T ∗ f )(t) exists. c) If limt→∞ T (t) = 0 in the strong operator topology and limt→∞ f (t) = 0, then limt→∞ (T ∗ f )(t) = 0. Proof. First, let Y be any of the spaces BUC(R+ , X), AAP(R+ , X), C0 (R+ , X) or the space of continuous functions f : R+ → X such that limt→∞ f (t) exists. We assume that Y contains all orbits ux of T (x ∈ X), f ∈ Y and σ(A) ∩ i sp(f ) is empty, and we shall prove that (T ∗ ϕ ∗ f )|R+ ∈ Y whenever ϕ ∈ L1 (R) and Fϕ ∈ Cc∞ (R). Here we are regarding T and f as being deﬁned on R with T (t) = 0 and f (t) = 0 for t < 0, and the convolutions are deﬁned accordingly (see Section 1.3). We shall also consider the convolution T ∗ ϕ : R → L(X) deﬁned by ∞ (T ∗ ϕ)(t)x = ϕ(t − s)T (s)x ds. 0

Note that T ∗ ϕ is bounded and uniformly norm-continuous (see Proposition 1.3.5 c)).

5.6. SOLUTIONS OF INHOMOGENEOUS CAUCHY PROBLEMS

383

We proceed in a similar way as in Theorem 5.6.5. Let ψ ∈ Cc∞ (R) be such that ψ = 1 on a neighbourhood of −iσ(A) ∩ supp(Fϕ) and ψ = 0 on a neighbourhood of sp(f ). Deﬁne G, H : R → L(X) by: G H(η)

T ∗ ϕ − T ∗ ϕ ∗ F −1 ψ, (F ϕ)(η)(1 − ψ(η))R(iη, A) := 0 :=

(iη ∈ ρ(A)), (iη ∈ σ(A)).

Then G ∈ BUC(R, X), H ∈ Cc∞ (R, L(X)), and as in the proof of Theorem 5.6.5, (F −1 H )(t) = −t2 G(t). Hence, G ∈ L1 (R, L(X)). It follows that (G ∗ f )|R+ ∈ Y , (see Remark 5.6.3 b)). Let g := ϕ ∗ F −1 ψ ∗ f , and let (Fϕ)(η)ψ(η)fˆ(iη) (η ∈ sp(f )), h(η) := 0 (η ∈ sp(f )). Then g ∈ BUC(R, X), h ∈ Cc∞ (R, X) and (F −1 h )(t) = −t2 g(t). Hence, g ∈ L1 (R, X). We claim that this implies that (T ∗g)|R+ ∈ Y . Note that T ∗(χ(a,b) ⊗x) = χ(a,b) ∗ux ∈ Y (see Remark 5.6.3 b)). Since the step functions are dense in L1 (R, X) and convolution with T is a bounded map from L1 (R, X) into L∞ (R, X), it follows that (T ∗ g)|R+ ∈ Y , as claimed. Since T ∗ϕ∗f

= G ∗ f + T ∗ ϕ ∗ F −1 ψ ∗ f = G ∗ f + T ∗ g,

it follows that (T ∗ ϕ ∗ f )|R+ ∈ Y , as claimed. Now suppose that T ∗ f ∈ BUC(R+ , X). Let (ϕn ) be an approximate unit in L1 (R) such that F ϕn ∈ Cc∞ (R) for every n (see Lemma 1.3.3). Since (T ∗f )(0) = 0, we can consider T ∗ f as an element of BUC(R, X) and we obtain that T ∗ ϕn ∗ f = ϕn ∗ (T ∗ f ) → T ∗ f uniformly as n → ∞. It follows that (T ∗ f )|R+ ∈ Y , and this completes the proof. Finally in this section, we consider the situation when the assumption of non-resonance is replaced by a countable spectral condition. As in Theorem 5.6.6, we assume that the inhomogeneity f and the given mild solution u of (ACPf ) are bounded and uniformly continuous (and totally ergodic), and we aim to show that asymptotic properties of f are transferred to u. In contrast to Section 5.5 and the earlier part of this section, we consider problems which may not be associated with C0 -semigroups; i.e., A is not assumed to be a generator. Thus we shall extend Theorem 5.5.3 in several directions. Let f, u ∈ BUC(R+ , X). Recall from Section 5.4 that u is said to be a mild solution of the inhomogeneous Cauchy problem (ACPf )

u (t) = Au(t) + f (t)

(t ≥ 0),

384 if

t 0

5. ASYMPTOTICS OF SOLUTIONS OF CAUCHY PROBLEMS u(s) ds ∈ D(A) and u(t) = u(0) + A

t

t

u(s) ds + 0

f (s) ds 0

for all t ≥ 0. As in Section 4.7, we let S be the shift semigroup on BUC(R+ , X), (S(t)f )(s) := f (s + t), and B be the generator of S, so ∞ (R(λ, B)f )(t) = e−λs f (s + t) ds = ft (λ) (λ ∈ C+ , t ≥ 0), 0

where ft (s) := f (s + t). The following proposition relates the Laplace transform, spectrum and ergodicity of u to the properties of f . Proposition 5.6.7. Let A be a closed operator on X, let f ∈ BUC(R+ , X), and let u be a bounded, uniformly continuous, mild solution of the inhomogeneous Cauchy problem (ACPf ). Then a) For t ≥ 0 and λ ∈ C+ ∩ ρ(A), (R(λ, B)u)(t) = R(λ, A) ((R(λ, B)f )(t)) + R(λ, A)(u(t)). b) sp(u) ⊂ sp(f ) ∪ {η ∈ R : iη ∈ σ(A)}. c) If f is uniformly ergodic at η where iη ∈ ρ(A) ∩ iR, then u is uniformly ergodic at η and Mη u(t) = R(iη, A)(Mη f (t)). Proof. Note ﬁrst that

where vt (s) :=

u(s + t) = u(t) + A(vt (s)) + gt (s), (5.19) s+t u(r) dr and gt (s) := t f (r) dr. Moreover, for Re λ > 0, ∞ (R(λ, B)u)(t) = e−λs u(s + t) ds = λvt (λ),

s+t t

0

and similarly (R(λ, B)f )(t) = λgt (λ). ∞ −λs For λ ∈ C+ , the integral 0 e vt (s) ds is absolutely convergent, and it follows ∞ from (5.19) that 0 e−λs A(vt (s)) ds is also absolutely convergent. By Proposition 1.6.3, vt (λ) ∈ D(A) and ∞ A(vt (λ)) = e−λs u(s + t) ds − gt (λ) − λ−1 u(t). 0

Hence, (λ − A)((R(λ, B)u)(t)) = (R(λ, B)f )(t) + u(t),

5.6. SOLUTIONS OF INHOMOGENEOUS CAUCHY PROBLEMS

385

and a) follows. Since fˆ(λ) = (R(λ, B)f )(0) and u ˆ(λ) = (R(λ, B)u)(0), it follows from a) that ! " u ˆ(λ) = R(λ, A) fˆ(λ) + R(λ, A)(u(0)) for λ ∈ C+ ∩ ρ(A). If iη ∈ ρ(A) and η ∈ / sp(f ), then fˆ has a holomorphic extension near iη, and then u ˆ also has a holomorphic extension. This proves b). Since Mη f = limα↓0 αR(α + iη, B)f , c) follows from a). We now give the extension of Theorem 5.5.3 to inhomogeneous Cauchy problems. Although this case is not a corollary of the Tauberian theorem (Theorem 4.7.7), the idea of the proof is similar. Theorem 5.6.8. Let A be a closed operator on X such that σ(A) ∩ iR is countable. Let f : R+ → X be asymptotically almost periodic, and u be a bounded, uniformly continuous, mild solution of the inhomogeneous Cauchy problem (ACPf ), and suppose that u is uniformly ergodic at η whenever iη ∈ σ(A) ∩ iR. Then a) u is asymptotically almost periodic, and i Freq(u) ∩ ρ(A) = i Freq(f ) ∩ ρ(A). ! " b) If τ > 0, f is τ -periodic and i Freq(u) ⊂ ρ(A) ∪ 2πi Z, then u = u0 + u1 , τ where u0 is a mild solution of (ACP0 ), limt→∞ u0 (t) = 0, and u1 is a τ periodic solution of (ACPf ). c) If i Freq(f ) ⊂ σ(A) ∪ {0} and i Freq(u) ⊂ ρ(A) ∪ {0}, then limt→∞ u(t) = R(0, A)(M0 f (0)). d) If i Freq(f ) ⊂ σ(A) and i Freq(u) ⊂ ρ(A), then limt→∞ u(t) = 0. Proof. By Proposition 5.6.7 c), u is uniformly ergodic at η whenever iη ∈ ρ(A) ∩ iR so u is totally ergodic. Let B be the generator of the shift semigroup on the space E(R+ , X) (see Section 4.7). By Proposition 5.6.7 a), (R(λ, B)u)(t) = (R(λ, B)(R(λ, A) ◦ f ))(t) + R(λ, A)(u(t)) for λ ∈ C+ ∩ ρ(A). The space AAP(R+ , X) is invariant under the operation of composition with a ﬁxed member of L(X) and under R(λ, B), so R(λ, B)(R(λ, A)◦ f ) ∈ AAP(R+ , X). Hence, R(λ, B)π(u) = π(R(λ, B)u) = π(R(λ, A) ◦ u) whenever λ ∈ C+ ∩ ρ(A), where + , X)/ AAP(R+ , X) π : E(R+ , X) → Y := E(R

386

5. ASYMPTOTICS OF SOLUTIONS OF CAUCHY PROBLEMS

is the generator of the C0 -group S on E induced by the is the quotient map and B shift semigroup (see Proposition 4.7.2). This shows that the map λ → R(λ, B)π(u) has a holomorphic extension to a map h : ρ(A) → E, given by h(λ) = π(R(λ, A)◦u). It now follows as in the proof of Theorem 4.7.7 that π(u) = 0, so u is asymptotically almost periodic. The fact that i Freq(u) ∩ ρ(A) = i Freq(f ) ∩ ρ(A) follows from Proposition 5.6.7 c), and the remaining statements follow from Corollary 4.7.8 and Proposition 5.4.16. The following corollary generalizes several parts of Proposition 5.6.1. Corollary 5.6.9. Let A be a closed operator on X such that σ(A) ∩ iR is empty. Let f ∈ BUC(R+ , X), and u be a bounded, uniformly continuous, mild solution of the inhomogeneous Cauchy problem (ACPf ). a) If f is totally ergodic, then u is totally ergodic. b) If f is asymptotically almost periodic, then u is asymptotically almost periodic, and Freq(u) = Freq(f ). c) If f is τ -periodic, then u = u0 + u1 , where u0 is a mild solution of (ACP0 ), limt→∞ u0 (t) = 0, and u1 is a τ -periodic solution of (ACPf ). d) If f∞ := limt→∞ f (t) exists, then limt→∞ u(t) = R(0, A)f∞ . e) If limt→∞ f (t) = 0, then limt→∞ u(t) = 0. Proof. a) follows from Proposition 5.6.7 c), and the remaining statements from Theorem 5.6.8.

5.7

Notes

Accounts of the asymptotic behaviour of C0 -semigroups have appeared in the books of Daletskii and Krein [DK74], Levitan and Zhikov [LZ82], Nagel et al. [Nag86], van Neerven [Nee96c], Chicone and Latushkin [CL99] and Eisner [Eis10]. Section 5.1 The growth bound ω(T ) appeared in the book of Hille and Phillips [HP57], while ω1 (T ) arose in papers of D’Jacenko [Jac76] and Zabczyk [Zab79]. It is possible to deﬁne higher order and fractional growth bounds in the following way. For μ > ω(T ), μ − A is a sectorial operator, and the fractional powers (μ − A)α and R(μ, A)α are deﬁned whenever α ≥ 0 (see the Notes on Section 3.8). Deﬁne ωα (T )

:=

ω( T (·)R(μ, A)α )

=

inf {ω(ux ) : x ∈ D((μ − A)α )} .

This is independent of μ > ω(T ), since D((μ − A)α ) is independent of μ (see Proposition 3.8.2). With only minor modiﬁcations, the proof of Proposition 5.1.5 shows that ωα+1 (T ) = sup {abs(ux ) : x ∈ D((μ − A)α )} .

5.7. NOTES

387

Moreover, the resolvent identity may be used to show that hol( uR(μ,A)x ) = hol( ux ), from which it follows that s(A) = inf {hol( ux ) : x ∈ D(An )} ≤ ωn (T ) for all positive integers n, and hence that s(A) ≤ ωα (T ) for all α ≥ 0. Theorem 5.1.2 is mostly due to Datko [Dat70], [Dat72], with contributions also from Pazy [Paz72], van Neerven [Nee96a] and Sch¨ uler and V˜ u [SV98]. The equality of ω1 (T ) and abs(T ) (Proposition 5.1.6) and Proposition 5.1.5 were established by Neubrander [Neu86]. Theorem 5.1.7 was ﬁrst proved by Weis and Wrobel [WW96], following preliminary results of Slemrod [Sle76] (showing that ω2 (T ) ≤ s0 (A)) and van Neerven, Straub and Weis [NSW95] (showing that ωα (T ) ≤ s0 (A) whenever α > 1). In [WW96], interpolation theory was used to show that ωα (T ) is a convex function of α and therefore it is continuous for α > 0. Thus, Theorem 5.1.7 followed from the result of [NSW95]. A Banach space X is said to have Fourier type p (where 1 ≤ p ≤ 2) if the Fourier trans form deﬁnes a bounded linear map from Lp (R, X) into Lp (R, X), where 1/p + 1/p = 1. Every Banach space has Fourier type 1; every superreﬂexive space has Fourier type p for some p > 1; X has Fourier type 2 if and only if X is (isomorphic to) a Hilbert space (see also the Notes on Section 1.8). van Neerven, Straub and Weis [NSW95] showed that ωα (T ) ≤ s0 (A) if X has Fourier p and α > 1/p − 1/p , and Weis and Wrobel [WW96] extended this to the case when α = 1/p−1/p and p < 2 (for the case p = 2, see the notes on Section 5.2). van Neerven [Nee09] has given another variation of the result. Assume that X has (Rademacher) type q1 and cotype q2 , where 1 ≤ q1 ≤ 2 and q2 ≥ 2, and let α = 1/q1 − 1/2; for example, X may be any Lp -space (1 ≤ p < ∞) and then α = p1 − 12 . Then ωα (T ) is bounded above by an abscissa for the resolvent of A at least as large as, but often equal to, s0 (A). For the deﬁnitions and properties of type and cotype, see [Woj91] for example. Trefethen and Embree [TE05] provide many examples of the behaviour of the pseudo-spectrum (sets where the norm of the resolvent of an operator is large) including strong evidence that the pseudo-spectrum is much more stable than the spectrum under perturbations. For α ≥ 0, there is an associated pseudo-spectral bound deﬁned by sα (A) = inf ω > s(A) : there exists Cω such that

R(λ, A) ≤ Cω (1 + |λ|)α whenever Re λ > ω .

The proofs of Theorem 5.1.7 can be modiﬁed to show that ωα+1 (T ) ≤ sα (A) for any semigroup on any Banach space (earlier, Slemrod [Sle76] proved that ωn+2 (T ) ≤ sn (A), and Wrobel [Wro89] showed that ωn+1 (T ) ≤ sn (A) if X has non-trivial Fourier type). van Neerven [Nee96b] proved Theorem 5.1.8 by means of Laplace inversion along a well-chosen contour. Both conclusions that growth is at most linear are sharp. Suppose that x ∈ X, and u x is deﬁned and bounded on C+ , and let α > 1. Under certain additional assumptions, T (t)R(μ, A)α x → 0 as t → ∞. This was established by Huang and van Neerven [HN99] when X has the analytic Radon-Nikodym property, and by Batty, Chill and van Neerven [BCN98] when T is sun-reﬂexive in the sense of [Nee92]; i.e., when R(μ, A) is weakly compact, by a theorem of de Pagter [Pag89]. In particular,

388

5. ASYMPTOTICS OF SOLUTIONS OF CAUCHY PROBLEMS

it is true if X is reﬂexive (the Hilbert space case was ﬁrst covered by Huang [Hua99]). However, if X = C0 (R), T is the shift group: (T (t)f )(s) = f (s + t), and g ∈ Cc (R), then u g exists and is bounded on C+ , but T (t)R(μ, A)α g = R(μ, A)α g for all t ≥ 0. Blake [Bla99] has obtained further results related to Theorem 5.1.8. A detailed account of many of these reﬁnements of Theorems 5.1.7 and 5.1.8 is given in the book of van Neerven [Nee96c, Chapter 4]. Example 5.1.10 is a modiﬁcation, due to Wrobel [Wro89], of an example of Zabczyk [Zab75]. Example 5.1.11 is due to Arendt [Are94b], and van Neerven [Nee96d] has analyzed it for many rearrangement-invariant function spaces. The ﬁrst example of a positive C0 -semigroup with s(A) < ω(T ) was given by Greiner, Voigt and Wolﬀ [GVW81], and the ﬁrst example of a positive C0 -group by Wolﬀ [Wol81]. Theorem 5.1.12 is a consequence of the spectral mapping theorem: σ(T (t)) \ {0} = eμt : μ ∈ σ(A) , which is valid for eventually norm-continuous semigroups. It appeared in the book of Hille and Phillips [HP57] and it can be proved either by product space techniques (see [Nag86]) or by Banach algebra methods (see [HP57] or [Dav80]), but we do not know of a proof by Laplace transform methods of either the spectral mapping theorem for holomorphic semigroups, or Theorem 5.1.12 for eventually norm-continuous semigroups. Greiner and M¨ uller [GM93] obtained a spectral mapping theorem for all exponentially bounded integrated semigroups. Martinez and Mazon [MM96], Blake [Bla01] and Nagel and Poland [NP00] have established a version of the spectral mapping theorem for asymptotically norm-continuous semigroups (see the Notes on Section 5.2). Section 5.2 Theorem 5.2.1 originated in the work of Gearhart [Gea78] who considered the case when T is a contraction semigroup on a Hilbert space X. He showed that eμt ∈ ρ(T (t)) if and only μ + (2πi/t)Z ⊂ ρ(A) and supn∈Z R(A, μ + 2πin/t) < ∞. This was extended to the non-contractive case with new and simpler proofs, independently by Herbst [Her83], Howland [How84], Huang [Hua85] and Pr¨ uss [Pr¨ u84]. A consequence of these results is that if the resolvent of A is bounded on the imaginary axis, then T is “hyperbolic”; i.e., X splits as a topological direct sum of closed, T -invariant subspaces, X = X− ⊕ X+ , such that ω(T |X− ) < 0 and there is a C0 -group U on X+ with T (t)|X+ = U (−t) and ω(U ) < 0. A particularly beautiful proof of Theorem 5.2.1 was given by Latushkin and Montgomery-Smith [LM95] which is based on the theory of “evolution semigroups”. See also the monograph by Chicone and Latushkin [CL99] where many far-reaching consequences are given. Our short proof of Theorem 5.2.1 is a simpliﬁed version of a proof given by Weiss [Wei88]. For higher order growth and spectral bounds, Weiss [Wei90] and Wrobel [Wro89] showed that ωn (T ) = sn (A) for semigroups on Hilbert space. For Banach spaces, Greiner [Gre84] gave a characterisation of σ(T (t)) based on Fej´er’s theorem, from which Theorem 5.2.1 can be deduced in the case of Hilbert spaces. Examples 5.2.2 and 5.2.3 are due to Weis [Wei98] and Arendt [Are94b], respectively. Building on preliminary work of Martinez and Mazon [MM96], Blake [Bla01] proved the following variant of Theorem 5.2.1, showing that the absence of norm-continuity for large t is reﬂected in a rather precise way in the shape of the spectrum and the growth of the resolvent (i.e., the shape of the pseudo-spectrum).

5.7. NOTES

389

For a C0 -semigroup T with generator A, let s∞ := inf ω ∈ R : there exist bω > 0 and Cω > 0 such that λ ∈ ρ(A) and 0 (A)

R(μ, A) ≤ Cω whenever Re λ > ω and | Im λ| > bω , δ(T ) := inf ω ∈ R : there exists Mω > 0 such that lim sup T (t + h) − T (t) ≤ Mω eωt for all t ≥ 0 . h→0

We say that T is asymptotically norm-continuous if δ(T ) < ω(T ). Theorem 5.7.1. Let T be a C0 -semigroup on a Hilbert space, with generator A. Then s∞ 0 (A) = δ(T ). Section 5.3 Part of Theorem 5.3.1 was proved by Greiner, Voigt and Wolﬀ [GVW81], and the remainder by Neubrander [Neu86]. Example 5.3.2 is due to Arendt [Are94b]. Theorem 5.3.6 was ﬁrst proved by Weis [Wei95], answering a question which had been open for some time. The proof in [Wei95] used interpolation theory and the theory of evolution semigroups developed by Latushkin and Montgomery-Smith [LM95]. The simpliﬁed proof given here follows a later method of Weis [Wei96]. Proposition 5.3.5 appeared in a proof given by Montgomery-Smith [Mon96] but it can alternatively be established by methods shown in [Haa07b]. Another variant of the proof of Theorem 5.3.6 is given in [Wei98]. The special cases of p = 1 (Proposition 5.3.7) and p = 2 had been proved by Derndinger [Der80] and Greiner and Nagel [GN83], respectively. Our proof of Proposition 5.3.7 is taken from [Der80]. Theorem 5.3.8 was proved by Derndinger [Der80] for compact Ω and by Batty and Davies [BD82] for locally compact Ω. Note that the proofs of Proposition 5.3.7 and Theorem 5.3.8 do not use the lattice properties, and the equality s(A) = ω(T ) holds for positive semigroups on ordered Banach spaces where either the norm on X+ or the norm ∗ on X+ is additive in the sense of (5.13) (see [BD82]). In particular, this is true for C ∗ algebras (the result for positive semigroups on unital C ∗ -algebras was ﬁrst established by Groh and Neubrander [GN81]). For a positive semigroup on a Banach lattice with (Rademacher) type p and cotype q, one has ωα (T ) ≤ s0 (A) for α = 1/p − 1/q [Nee09] (see the Notes on Section 5.1). There are many aspects of asymptotic behaviour of positive semigroups which are not covered in this book. An early account of the theory was given in [Nag86]. See also the Notes on Section 5.5. Section 5.4 Splitting theorems for relatively compact orbits of semigroups, such as Theorems 5.4.6 and 5.4.11, are often associated with the names of Glicksberg and de Leeuw. In [GL61], they were the ﬁrst to obtain such a theorem for general Banach spaces, following special cases due to Jacobs [Jac56, etc]. In those papers, the splitting theory was carried out for very general semigroups of operators, and the methods were algebraic and topological. For one-parameter semigroups, the Glicksberg-deLeeuw theorem is the following variant of Theorem 5.4.11. Theorem 5.7.2 (Glicksberg-deLeeuw Theorem). Let T be a bounded C0 -semigroup on X, and suppose that {T (t)x : t ≥ 0} is weakly relatively compact for each x ∈ X. Then X = Xw0 ⊕ Xap ,

390

5. ASYMPTOTICS OF SOLUTIONS OF CAUCHY PROBLEMS

where Xw0 := {x ∈ X : 0 is in the weak closure of {T (t)x : t ≥ 0}} . Accounts of the general Glicksberg-deLeeuw theory can be found in the books of Krengel [Kre85, Section 2.4], van Neerven [Nee96c, Section 5.7], Engel and Nagel [EN00, Section V.2] and Eisner [Eis10, Section I.1]. It is easy to deduce the strong splitting theorem (Theorem 5.4.6) from Theorem 5.7.2, but comparison with the weak splitting theorem (Theorem 5.4.11) is more delicate. A priori, it is not clear that the spaces Xe0 and Xw0 are contained in each other. However, it is easy to verify that the following properties are equivalent for a totally ergodic semigroup T on X: (i) There is a closed T -invariant subspace Y of X such that X = Y ⊕ Xap as a topological direct sum. (ii) There is a bounded projection P of X onto Xap such that P T (t) = T (t)P for all t ≥ 0. (iii) X = Xe0 ⊕ Xap . When these properties hold, Y = Ker P = Xe0 . The weak and strong splitting theorems and the Glicksberg-deLeeuw theorem provide important special cases when the properties hold, and they show in particular that Xe0 = Xw0 if the orbits of T are weakly relatively compact. There are some totally ergodic C0 -semigroups for which the properties (i), (ii) and (iii) above do not hold. In particular, Woodward [Woo74] showed that the space E(R) of all totally ergodic functions in BUC(R) is strictly larger than E0 (R) ⊕ AP(R), where E0 (R) is the space of totally ergodic functions whose means are all zero, and therefore (iii) does not hold for the C0 -group of shifts on E(R). The weak splitting theorem shows that there is such a splitting of the space of all Eberlein-w.a.a.p. functions (see the Notes on Section 4.7). The descriptions of the spaces Xe0 and Xw0 are both rather weak, and the strongest result has been obtained by Ruess and Summers [RS90b]. Theorem 5.7.3. Let T be a bounded C0 -semigroup on X, let x ∈ X, and suppose that {T (t)x : t ≥ 0} is relatively weakly compact. Then ux is Eberlein-w.a.a.p. Moreover, x = x0 + x1 , where x1 ∈ Xap and ux0 is Eberlein-w.a.a.p. with means Mη ux0 = 0 for all η ∈ R. Ruess and Summers [RS87], [RS88a], [RS90a], [RS92a], [RS92b] have also carried out detailed investigations of orbits, and almost-orbits, of non-linear semigroups and solutions of non-autonomous Cauchy problems. Their results show that Eberlein’s notion of weak asymptotic almost periodicity is important even in those cases. The construction of trigonometric polynomials as in Proposition 5.4.8 by means of Fej´er kernels occurred in Bohr’s book [Boh47] where it was used in a proof (attributed to De La Vall´ee Poussin) of a fundamental property of almost periodic functions. Other techniques in the proof of Theorem 5.4.11 appear in the work of Datry and Muraz [DM95], [DM96], who have considered splittings in a very abstract situation. Section 5.5 There is a very large literature on the subject of the local spectrum for a bounded operator, or a commuting family of bounded operators. The book of Erdelyi and Wang

5.7. NOTES

391

[EW85] includes an account of the theory for unbounded operators. In the literature, the operators are usually assumed to satisfy the “single-valued extension property”, but in the context of Section 5.5, we consider only a peripheral part of the local spectrum, for which we have chosen a deﬁnition which makes this property hold automatically (see Proposition 5.5.1) and which is consistent with the notion of spectrum for functions (see Proposition 5.5.2). An alternative notion of imaginary local spectrum, σ ˜u (T, x), of a C0 -semigroup T has been introduced by Batty and Yeates [BY00], using ideas of Albrecht [Alb81]. When T is bounded, the deﬁnition of σ ˜u (T, x) is as follows. 1 )x = ∞ For a bounded semigroup T and f ∈ L (R+ ), deﬁne f (T ) ∈ L(X) by f (T 1 f (t)T (t)x dt. A point iη ∈ iR is in ρ ˜ (T, x) if there exist n ∈ N, f , . . . , f ∈ L (R u 1 n + ), 0 a neighbourhood V of the point ((F f1 )(−η), . . . , (F fn )(−η)) in Cn and holomorphic functions gi : V → X for each i = 1, . . . , n such that n

(zi − fi (T )) gi (z) = x

(z = (z1 , . . . , zn ) ∈ V ).

i=1

Then σ ˜u (T, x) := iR \ ρ˜u (T, x). While it is easy to see that σ ˜u (T, x) is contained in σu (A, x), it remains open whether equality holds. Theorem 5.5.3 remains valid when σu (A, x) is replaced by σ ˜u (T, x) (see [BY00]). Part b) of Theorem 5.5.5 was proved independently by Arendt and Batty [AB88] and Lyubich and V˜ u [LV88]. The proof in [AB88] was based on the contour integral method of Section 4.4, combined with an unusual argument by transﬁnite induction. The functional analytic proof in [LV88] is related to the quotient method of Sections 4.7 and 4.8. Part a) of Theorem 5.5.5 was proved by Lyubich and V˜ u [LV90a]. Prototypes of Theorem 5.5.5 for norm-continuous semigroups, some discrete semigroups, and the case of empty peripheral spectrum had been obtained by Sklyar and Shirman [SS82], Atzmon [Atz84], and Huang [Hua83] (see also [Hua93a], [Hua93b]), respectively. A diﬀerent proof of Theorem 5.5.5 b) was subsequently given by Esterle, Strouse and Zouakia [ESZ92]. We summarize their method in the next few paragraphs. Let E be a closed subset of R. A function f ∈ L1 (R+ ) is said to be of spectral synthesis with respect to E if there is a sequence (gn ) in L1 (R) such that limn→∞ gn − f 1 = 0 and, for each n, F gn vanishes on a neighbourhood of E. (Here, we are regarding f as a member of L1 (R) with f (t) = 0 for t < 0.) If f is of spectral synthesis with respect to E, then F f vanishes on E. If the boundary of E is countable and F f vanishes on E, then f is of spectral synthesis with respect to E [Kat68, p.230]. For a bounded semigroup T such that σ(A) ∩ iR is countable, it was shown by Esterle, Strouse, and Zouakia [ESZ92], using an abstract Mittag-Leﬄer theorem (a generalization of Baire’s category theorem), that the linear span of the union of the ranges of f (T ) for all f which are of spectral synthesis with respect to iσ(A) ∩ R is dense in X. Then, Theorem 5.5.5 b) follows from the following analogue of the Katznelson-Tzafriri theorem [KT86]. Theorem 5.7.4. Let T be a bounded C0 -semigroup on X and let f ∈ L1 (R+ ) be of spectral synthesis with respect to iσ(A) ∩ R. Then T (t)f (T ) → 0 as t → ∞. Theorem 5.7.4 was proved in [ESZ92] using methods of harmonic analysis, and in [Vu92] using a functional analytic method. In the next paragraph, we sketch a proof

392

5. ASYMPTOTICS OF SOLUTIONS OF CAUCHY PROBLEMS

which uses the ideas of Section 4.9, in particular a version of Ingham’s Theorem 4.9.5 for functions on R. Let g ∈ L1 (R) be such that F g vanishes in a neighbourhood of iσ(A) ∩ R, and let ∞ (ˇ g ∗ T )(t)x = 0 g(s − t)T (s)x ds. Then gˇ ∗ T ∈ BUC(R, X), and it is not diﬃcult to show that gˇ ∗ T has distributional Fourier transform (F g)(−s)R(is, A) (is ∈ ρ(A)), H(s) = 0 (is ∈ σ(A)), in the sense that

∞

−∞

(ˇ g ∗ T )(t)(F ϕ)(t) dt =

∞

H(s)ϕ(s) ds −∞

for all ϕ ∈ Cc∞ (R). If ρ ∈ S(R) is such that F ρ has compact support, then (F ρ) · H ∈ L1 (R, L(X)) with F −1 ((F ρ) · H) = ρ ∗ (ˇ g ∗ T ) ∈ C0 (R, L(X)) by the RiemannLebesgue lemma. Using a molliﬁer (ρn ), it follows that gˇ ∗ T ∈ C0 (R, L(X)). Choosing g to approximate f , it follows that T (t)f (T ) = (fˇ ∗ T )(t) → 0 as t → ∞. Another proof of Theorem 5.5.5 b) using the abstract Mittag-Leﬄer theorem has been given in [BCT02]. This was one of a series of papers, also including [Tom01], [CT03], [CT04] and [BCT07], in which Chill and Tomilov investigated stability of semigroups in the sense of strong operator convergence to 0. An excellent survey is given in [CT07]. For general Banach spaces the main results are as follows. Theorem 5.7.5. Let T be a bounded C0 -semigroup T on X, with generator A. Then limt→∞ T (t)x = 0 for all x ∈ X if any of the following conditions holds: a) x ∈ X : lim αR(α + is, A)2 x = 0 for all s ∈ R is dense in X. α↓0 γ−1 γ b) For some γ > 1, x ∈ X : lim α R(α + is, A) x ds = 0 is dense in X. α↓0

R

c) For each s0 ∈ R, there exists a neighbourhood U of s0 in R such that αR(α + is, A)2 x ds = 0 x ∈ X : lim α↓0

U

is dense in X. The assumptions of Theorem 5.5.5 b) imply that s∈R Ran(A−is) is dense in X,by the abstract Mittag-Leﬄer theorem. If x ∈ Ran(A−is), then limα↓0 αR(α + is, A)2 x = 0. So Theorem 5.5.5 b) follows from case a) of Theorem 5.7.5, which is proved by means of an edge-of-the-wedge theorem. Theorem 5.7.5 can be improved when X has non-trivial Fourier type. For example, for a Hilbert space X, density of √ x ∈ X : lim αR(α + is, A)x = 0 for all s ∈ R α↓0

is suﬃcient for stability. Moreover there is a variant of the integral condition b) of Theorem 5.7.5 which is both necessary and suﬃcient on Hilbert spaces. We refer the reader to the survey [CT07], or the original articles, for the details. Greenﬁeld [Gre94] (see also [BBG96]) proved the following quantitative version of Theorem 5.5.5 b).

5.7. NOTES

393

Theorem 5.7.6. Let T be a C0 -semigroup of contractions on X, and suppose that σ(A)∩iR is countable. Let X1∗ be the weak*-closed linear span of the unimodular eigenvectors of the dual semigroup T ∗ on X ∗ . Then lim T (t)x

t→∞

=

inf { x − y : y ∈ X0 }

=

sup {|x, x∗ | : x∗ ∈ X1∗ , x∗ ≤ 1}

for each x ∈ X. A consequence of Theorem 5.5.6 is that if X is superreﬂexive (i.e., if there is an equivalent uniformly convex norm on X), and A generates a bounded C0 -semigroup T on X and σ(A) ∩ iR is countable, then every ultrapower of T is asymptotically almost periodic. Huang and R¨ abiger [HR94] proved a converse result, that σ(A)∩iR is countable if every ultrapower of T is asymptotically almost periodic (the discrete version was given earlier by Nagel and R¨ abiger [NR93]; see also the work of R¨ abiger and Wolﬀ [RW95]). There was a considerable interval before the global Theorem 5.5.5 was improved to the local Theorem 5.5.3. Batty and V˜ u [BV90] gave a few results on individual orbits, and an intermediate stage between the global and the local was considered by Huang [Hua93a], [Hua93b] and Batty [Bat96]. Theorem 5.5.3 was ﬁrst proved by Batty, van Neerven and R¨ abiger [BNR98b], and the method given here is from [AB99]. Theorems 5.5.5 and 5.5.3, and their discrete analogues, have been extended in various directions: to weighted results (showing that T (t)x /w(t) → 0 where w is a weight on R+ and T (t) ≤ w(t)), by Allan and Ransford [AR89], V˜ u [Vu93], K´erchy [K´er97] and Batty and Yeates [BY00]; to representations of subsemigroups of locally compact abelian groups, by Lyubich and V˜ u [LV90b], Batty and V˜ u [BV92], Batty and Yeates [BY00] and K´erchy [K´er99]; to once integrated semigroups, by El-Mennaoui [Elm94] (using Theorem 5.5.5 and the extrapolation construction of Section 3.10); and to Volterra equations, by Arendt and Pr¨ uss [AP92]. Examples 5.5.7–5.5.10 are taken from papers of Arendt and Batty [AB88], Batty and Vu [BV90], and Batty, van Neerven and R¨ abiger [BNR98b]. Using a direct sum of weighted shifts, van Neerven [Nee00] has extended Example 4.7.11 to give an example of a C0 -semigroup T with a vector x such that ux is bounded and uniformly continuous, σu (A, x) = {0}, and limα↓0 αT (s) ux (α) exists for each s ≥ 0, but ux is not asymptotically almost periodic. Thus, the assumption of uniform convergence cannot be omitted from condition c) of Theorem 5.5.3. Perron-Frobenius theory provides a special interplay between spectral properties of the generator and asymptotic behaviour of a positive semigroup. The lecture notes [Nag86] give a complete account of the state of the art in 1986. They were written before the results of Section 5.5 on countable spectrum were discovered. Consequently some results can now be formulated or proved more easily, and we describe some of them here. A basic result in Perron-Frobenius theory is the cyclicity of the boundary spectrum. If A generates a bounded positive C0 -semigroup and is ∈ σ(A) for some s ∈ R, then ins ∈ σ(A) for all n ∈ Z. If the semigroup is eventually norm-continuous (i.e., norm-continuous on (t0 , ∞) for some t0 ≥ 0), then it follows that σ(A) ∩ iR ⊂ {0}. The following result was proved in [Nag86] by means of the discrete Katznelson-Tzafriri theorem (Corollary 4.7.15), but it now has the same proof as Theorem 5.5.6 b).

394

5. ASYMPTOTICS OF SOLUTIONS OF CAUCHY PROBLEMS

Theorem 5.7.7. Let T be a bounded, positive, eventually norm-continuous, Ces` aro-ergodic C0 -semigroup on a Banach lattice. Then there is a projection P such that limt→∞ T (t) = P in the strong operator topology. A positive C0 -semigroup on a Banach lattice X is said to be irreducible if the only closed T –invariant ideals of X are {0} and X (see Appendix C for the deﬁnition of ideals). If A generates a bounded, positive, irreducible, C0 -semigroup and A has compact resolvent, then Perron-Frobenius theory establishes that σ(A) ∩ iR is either empty or equal to 2πiτ Z for some τ ≥ 0. The following is then an easy consequence of Proposition 5.4.7 (see [Are04, Sections 3.4, 3.5]). Theorem 5.7.8. Let T be a bounded, positive, irreducible C0 -semigroup on a Banach lattice X, and assume that the generator A has compact resolvent. Then there is a periodic C0 group U on a closed subspace Y of X such that, for each x ∈ X there exists y ∈ Y such that limt→∞ T (t)x − U (t)y = 0. Let X = Lp (Ω, μ), where (Ω, μ) is a σ-ﬁnite measure space and 1 ≤ p < ∞. A bounded operator S on X is a kernel operator if there is a measurable function k : Ω×Ω → C such that (Sf )(y) = Ω k(y, z) dμ(z) for μ-almost all y ∈ Ω and all f ∈ X. For abstract characterizations of kernel operators, see [Sch74, Proposition IV.9.8] or [Mey91, Theorem 3.3.11]. All bounded operators on p are kernel operators. Many elliptic operators generate irreducible C0 -semigroups of kernel operators, and the properties of the kernels have been much studied (see [Dav90], [Ouh05], [Rob91]). The following result is due to Greiner [Gre82] (see also [Are08]). Theorem 5.7.9. Let T be a positive, bounded, irreducible C0 -semigroup on Lp (Ω, μ), with generator A. Suppose that T (t0 ) is a kernel operator for some t0 > 0 and that 0 ∈ σp (A). Then there is a projection P of rank one such that limt→∞ T (t) = P in the strong operator topology. It is remarkable that any positive, bounded C0 -semigroup of kernel operators on Lp (Ω, μ) satisﬁes σp (A) ∩ iR ⊂ {0}. This is also due to Greiner [Gre82] and it was discussed more recently in [Dav05], [Kei06], [Wol07] and [Are08]. The Gaussian semigroup on L1 (Rn ) is positive, contractive and irreducible, and it consists of kernel operators (see Example 3.7.6). However 0 is not an eigenvalue of its generator, and the semigroup does not converge. It should be mentioned that for cosine functions countable spectrum also gives the expected asymptotic behaviour. The following result is due to Arendt and Batty [AB97, Proposition 4.9] (see also [Bas85, Theorem 10]). Theorem 5.7.10. Let Cos be a bounded cosine function on a Banach space X with generator A. Assume that the following conditions hold: a) c0 ⊂ X; b) σ(A) is countable; and c) 0 ∈ σ(A). Then for each x, y ∈ X the function u(t) := Cos(t)x + Sin(t)y is almost periodic.

(t ∈ R)

5.7. NOTES

395

Note that u is the unique mild solution of P 2 (x, y) as deﬁned in Section 3.14. Here, t Sin(t)y := Cos(s)y ds (t ∈ R). 0

Since Cos is bounded, one knows that σ(A) ⊂ (−∞, 0]. An equivalent formulation of the theorem is to say that, under the conditions a), b) and c), one has X = span x ∈ D(A) : there exists η ∈ R such that Ax = −η 2 x . Surveys of the topics of this section have previously been written by Batty [Bat94], V˜ u [Vu97], van Neerven [Nee96c, Chapter 5], and Chill and Tomilov [CT07]. Section 5.6 A version of Theorem 5.6.5 was ﬁrst given by Basit [Bas97] under the stronger assumption that f has an extension g ∈ BUC(R, X) such that σ(A) ∩ i spC (g) is empty (see Section 4.6). Basit’s method originated in Lyapunov’s ﬁnite-dimensional theory, and it was developed and applied to inﬁnite-dimensional Cauchy problems on the line by V˜ u [Vu91], Ruess and V˜ u [RV95] and Sch¨ uler and V˜ u [SV98]. It involves solving operator equations of the form AY − Y B = C (Lyapunov equations) for an operator Y from a subspace of BUC(R, X) into X. The proof of Theorem 5.6.5 given here is due to Batty and Chill [BC99] who carried out the basic argument in a more general situation in which T is not necessarily a semigroup. It shows that the conclusion of boundedness in Theorem 5.6.5 is valid, not only for holomorphic semigroups, but also for many eventually norm-continuous semigroups and for asymptotically norm-continuous semigroups on Hilbert space. Blake [Bla99] has further reﬁned the method and extended Theorem 5.6.5 to various classes of asymptotically norm-continuous semigroups on Banach spaces, including eventually diﬀerentiable semigroups (for the deﬁnition of asymptotically normcontinuous semigroups, see the Notes on Section 5.2 above). Theorem 5.6.6 also appeared in [BC99]. The role of smooth functions in the proof also leads to another result of [BC99] that T ∗ f is bounded and uniformly continuous if T is bounded, R(iη, A) exists and is bounded for large |η|, and f is bounded with bounded derivatives of ﬁrst and second order. Proposition 5.6.7 and Theorem 5.6.8 are due to Arendt and Batty [AB99], following results of Ruess and V˜ u [RV95] for inhomogeneous Cauchy problems on R. Batty, Hutter and R¨ abiger [BHR99] obtained a version for periodic Cauchy problems. Applications to inhomogeneous Volterra equations have been given by Arendt and Batty [AB00], Chill and Pr¨ uss [CP01] and Fasangova and Pr¨ uss [FP01].

Part III

Applications and Examples

Part III

399

In this part of the book we present some applications and examples which illustrate how the theory developed in Parts I and II can be used. There are three chapters which are independent of each other; they all use basic concepts from distribution theory which can be found in Appendix E. In Chapter 6 the heat equation with inhomogeneous boundary conditions is investigated. The idea of the approach presented here is to work entirely in spaces of continuous functions. We assume that Ω is a bounded open set on which the Dirichlet problem is well-posed. This is a very weak regularity assumption on the boundary, and it is well known from potential theory. Based on this assumption, the results of this chapter rely on the methods developed in Parts I and II and do not use complicated results of partial diﬀerential equations. Resolvent positive operators (as developed in Section 3.11) play an important role giving the transition from the elliptic problem to a parabolic problem. Results of Part II will be used to show how the asymptotic behaviour of the given function on the boundary determines the asymptotic behaviour of the solution. In the approach of Chapter 6 we do not use Hilbert space techniques at all. This is diﬀerent in Chapter 7 where we prove well-posedness of a fairly general hyperbolic equation in L2 (Ω). The results are based on the theory of cosine functions as they are presented in Section 3.14. Most important is the role of the phase space introduced there. We need a brief introduction to quadratic form methods which is given in Section 7.1. Here we only consider the most simple case; the spectral theorem for selfadjoint operators (as stated in Appendix B) plays a major role in this introduction. Then the results of Chapter 7 follow from the general theory of cosine functions given in Section 3.14. In Chapter 8 diﬀerential operators with constant coeﬃcients on Rn , and more generally pseudo-diﬀerential operators, are considered. With the help of the notion of integrated semigroups, precise results on well-posedness and regularity of the corresponding parabolic problem in Lp -spaces are obtained. In particular, it will be shown that the wave equation is not well-posed in the semigroup sense on Lp (Rn ) for p = 2. This explains why Chapter 7 is restricted to L2 (Ω). However, it will be shown that the wave equation as well as some other equations from mathematical physics lead to k-times integrated semigroups on Lp -spaces. An important issue is to determine the best possible value of k depending on p. This tells us something about the regularity properties of the equations which we consider. A principal tool in Chapter 8 is the theory of Fourier multipliers. A resum´e of some of the required results, including Mikhlin’s theorem about Fourier multipliers on Lp (Rn ) for 1 < p < ∞, is given in Appendix E without proofs. Some further results on Fourier multipliers are proved in Section 8.2, including a weak form of Mikhlin’s theorem which is valid for L1 (Rn ).

Chapter 6

The Heat Equation In this chapter we consider the Laplacian on spaces of continuous functions. If Ω ⊂ Rn is an open, bounded set with boundary ∂Ω which is Dirichlet regular, we will show that the Laplacian generates a holomorphic semigroup on the space C0 (Ω) := {u ∈ C(Ω) : u|∂Ω = 0}. Furthermore, using the theory of resolvent positive operators developed in Section 3.11 we show that the heat equation with inhomogeneous boundary conditions is well-posed. We use the results of Chapter 5 to study the asymptotic behaviour of its solutions.

6.1

The Laplacian with Dirichlet Boundary Conditions

Let Ω ⊂ RN be an open, bounded set with boundary ∂Ω =: Γ. Given ϕ ∈ C(Γ), we consider the Dirichlet problem ⎧ ⎪ ⎨u ∈ C(Ω), D(ϕ) u|Γ = ϕ, ⎪ ⎩ Δu = 0 in D(Ω) . Here and throughout this chapter, D(Ω) is the space of all distributions on Ω, and we identify functions in C(Ω) with their restrictions to Ω and locally integrable 1 functions on Ω with distributions on Ω. Thus, C(Ω) ⊂ C(Ω) ⊂ Lloc (Ω) ⊂ D(Ω) , and the third line of D(ϕ) says that Ω uΔψ dx = 0 for all ψ ∈ D(Ω) (see Appendix E). Although we allow complex-valued functions ϕ and u, D(ϕ) is essentially a real problem; u is a solution of D(ϕ) if and only if Re u and Im u are solutions of D(Re ϕ) and D(Im ϕ) respectively. It is well known that a function satisfying D(ϕ) is in C ∞ (Ω) (see [Rud91, p.220], for example). So we look for harmonic functions in Ω having a continuous extension to the boundary with prescribed boundary

W. Arendt et al., Vector-valued Laplace Transforms and Cauchy Problems: Second Edition, Monographs in Mathematics 96, DOI 10.1007/978-3-0348-0087-7_6, © Springer Basel AG 2011

401

402

6. THE HEAT EQUATION

values. We will not talk about methods to solve the Dirichlet problem. For us, it serves as a reference problem. In fact, this problem is well studied in potential theory (see [DL90, Chapter II], [Hel69], [GT83], [Kel67], [Lan72]). Many geometric properties of the boundary are known to be suﬃcient for well-posedness of the Dirichlet problem. Deﬁnition 6.1.1. The set Ω is called Dirichlet regular if for all ϕ ∈ C(Γ) there exists a solution of D(ϕ). Examples 6.1.2. a) If n = 1, then each bounded open set is Dirichlet regular (see [DL90, Chapter II, Section 4, Example 6]). On the other hand, if n ≥ 2 and Ω ⊂ Rn is open, then Ω \ {z} is not Dirichlet regular for any z ∈ Ω (see [DL90, Chapter II, Section 4, Remark 1]). b) If the boundary of Ω is C 1 , or more generally, Lipschitz continuous, then Ω is Dirichlet regular (see [DL90, Chapter II, Section 4, Proposition 4]). c) For n = 2, any simply connected Ω ⊂ R2 is Dirichlet regular (see [Con73, Chapter X, Corollary 4.18]). d) For n = 3, Lebesgue’s cusp is a simply connected set which is not Dirichlet regular (see [Lan72, p.287], [AD08]). Next, we establish the elliptic maximum principle. It will be important for us to consider distributional inequalities which are easy to deﬁne. Let f ∈ D(Ω) . We write f ≥ 0 if ϕ, f ≥ 0 for all ϕ ∈ D(Ω)+ , (6.1) where D(Ω)+ := {ϕ ∈ D(Ω) : ϕ(x) ≥ 0 for all x ∈ Ω}. If f ∈ L1loc (Ω) is identiﬁed with a distribution in D(Ω) , then f ≥ 0 as a distribution if and only if f (x) ≥ 0 a.e. in Ω. Theorem 6.1.3 (Elliptic maximum principle). Let M ≥ 0, λ ≥ 0, u ∈ C(Ω) such that a) λu − Δu ≤ 0 in D(Ω) ; and b) u|Γ ≤ M . Then u ≤ M on Ω. Proof. By considering the real and imaginary parts of u separately, we may assume that u is real-valued. Let c := maxx∈Ω u(x). √ First case: We assume that u ∈ C 2 (Ω). Assume that c > M . Let γ > λ and δ := supx∈Ω eγx1 (where x = (x1 , . . . , xn )). Choose ε > 0 such that M + εδ < c. Let v(x) := u(x) + εeγx1 . Then v ∈ C 2 (Ω) ∩ C(Ω) and v ≤ M + εδ < c on Γ, but maxx∈Ω v(x) ≥ c. Thus, there exists x0 ∈ Ω such that v(x0 ) = maxΩ v(x). It

6.1. THE LAPLACIAN WITH DIRICHLET BOUNDARY CONDITIONS 403 follows that Dj2 v(x0 ) = so Δv(x0 ) ≤ 0. Hence,

d2 v(x0 dt2

0 ≤ λv(x0 ) − Δv(x0 )

+ tej ) ≤ 0 (where ej = (0, 0, . . . , 1, 0, . . . , 0)), and = λu(x0 ) − Δu(x0 ) + εeγx01 (λ − γ 2 ) ≤ εeγx01 (λ − γ 2 ),

which is a contradiction. Second case: Now let u be arbitrary and assume that c > M . Let K := {x ∈ Ω : u(x) = c}. Then assumption b) implies that K is a non-empty compact subset of Ω. Let Ω ⊂ Ω be open such that K ⊂ Ω ⊂ Ω ⊂ Ω. Then c1 := sup∂Ω u(x) < c. Let c1 < c2 < c, c2 > M . Denote by (ρk )k∈N a molliﬁer in Cc∞ (Rn ) with ρk ≥ 0 and supp ρk ⊂ {y ∈ Rn : |y| < 1/k} (see Section 1.3). Then vk (x) := (ρk ∗ u)(x) = u(x − y)ρk (y) dy |y|<1/k

is deﬁned for x ∈ Ωk := {y ∈ Ω : dist(y, Γ) > 1/k} and vk ∈ C ∞ (Ωk ). Moreover, vk converges to u uniformly on compact subsets of Ω as k → ∞. Hence, there exists k ∈ N such that Ω ⊂ Ωk , supΩ vk > c2 and sup∂Ω vk < c2 . But λvk (x) − Δvk (x) = ρk (x − ·), λu − Δu ≤ 0 for all x ∈ Ω . This is impossible by the ﬁrst case. It follows immediately from Theorem 6.1.3 that D(ϕ) has at most one solution for all each ϕ ∈ C(Γ) and the solution is real if ϕ is real-valued. Next, we consider the space X := C(Ω) × C(Γ) which is a Banach lattice for the ordering (u, ϕ) ≥ 0 ⇐⇒ u ≥ 0 and ϕ ≥ 0 (u ∈ C(Ω), ϕ ∈ C(Γ)) and the norm

' (

(u, ϕ) := max u C(Ω) , ϕ C(Γ) ,

with

u C(Ω)

:=

max |u(x)|,

ϕ C(Γ)

:=

max |ϕ(z)|.

x∈Ω z∈Γ

On C(Ω) we consider the Laplacian Δmax with maximal domain; i.e., D(Δmax ) := {u ∈ C(Ω) : Δu ∈ C(Ω)}, Δmax u := Δu in D(Ω) . It is obvious that Δmax is a closed operator.

404

6. THE HEAT EQUATION

Remark 6.1.4. It is known that D(Δmax ) ⊂ C 2 (Ω) whenever Ω is a non-empty open set in Rn (n ≥ 2). However, we will see in Lemma 6.1.5 that D(Δmax ) is contained in C 1 (Ω) (cf. Remark 3.7.7 b)). Thus, there always exist functions u ∈ C 1 (Ω) such that Δu ∈ C(Ω) but for some i, j, the distribution Di Dj u is not a function in C(Ω). This fact may be considered as unpleasant. However, for our purposes it does not matter. We will see that solutions of the heat equation are always of class C ∞ . We consider the operator A on X given by D(A) := D(Δmax ) × {0}, A(u, 0) := (Δu, −u|Γ ). Thus, for u ∈ D(Δmax ), f ∈ C(Ω), ϕ ∈ C(Γ), we have −A(u, 0) = (f, ϕ) if and only if −Δu = f in D(Ω) , (6.2) u|Γ = ϕ; i.e., if and only if u solves Poisson’s equation. For this reason we call A the Poisson operator. Since Δmax is closed, it follows that A is also closed. By En we denote the Newtonian potential; i.e., En : Rn \ {0} → R is given by ⎧ |x|/2 if n = 1, ⎪ ⎪ ⎪ ⎨ log |x| if n = 2, En (x) := 2π ⎪ ⎪ 1 1 ⎪ ⎩− if n ≥ 3, n(n − 2)ωn |x|n−2 where ωn := |B(0, 1)| is the volume of the unit ball in Rn . Then En ∈ C ∞ (Rn \{0}) and En , Dj En ∈ L1loc (Rn ) (j = 1, . . . , n), as is easy to see. Let f ∈ Cc (Rn ). Then one has v := En ∗ f ∈ C 1 (Rn ). Moreover, Δv = f in D(Rn ) .

(6.3)

We refer to ([DL90, Chapter II, Section 3]) for this standard fact of distribution theory. Frequently, v = En ∗f is called the Newtonian potential of f . Note however, that v ∈ C 2 (Rn ) in general. We deduce the following regularity result which will be useful. In the proof and elsewhere in this chapter, we do not distinguish notationally between functions on Ω and their restrictions to Ω ⊂ Ω. Lemma 6.1.5. Let u, f ∈ C(Ω) such that Δu = f in D(Ω) . Then u ∈ C 1 (Ω). If f ∈ C k (Ω) for some k ∈ N, then u ∈ C k+1 (Ω). Proof. a) Let Ω be open such that Ω ⊂ Ω. Let ρ ∈ D(Ω) such that ρ(x) = 1 on Ω . Consider ρf ∈ Cc (Rn ) and v := En ∗ (ρf ) ∈ C 1 (Rn ). Then Δv = ρf in D(Rn ) (see (6.3)). Hence, Δ(u − v) = f − ρf = 0 in D(Ω ) .

6.1. THE LAPLACIAN WITH DIRICHLET BOUNDARY CONDITIONS 405 Thus, u− v is harmonic and so in C ∞ (Ω ). Consequently, u = (u− v)+v ∈ C 1 (Ω ). Since Ω is arbitrary, the ﬁrst assertion is proved. b) We prove the second assertion. It is true for k = 0 by a). Assume that it holds for k ∈ N0 . Assume that f ∈ C k+1 (Ω). Then u ∈ C 1 (Ω) by a) and ΔDj u = Dj Δu = Dj f in D(Ω) . Since Dj f ∈ C k (Ω), it follows that Dj u ∈ C k+1 (Ω) (j = 1, . . . , n) by the inductive hypothesis. Hence, u ∈ C k+2 (Ω). Theorem 6.1.6. Assume that Ω is Dirichlet regular. The Poisson operator A is resolvent positive and s(A) < 0. Proof. a) Let λ ≥ 0 and suppose that λ ∈ ρ(A). Then R(λ, A) ≥ 0. In fact, let f ∈ C(Ω), ϕ ∈ C(Γ), (u, 0) = R(λ, A)(f, ϕ). Then λu − Δu = f in D(Ω) and u|Γ := ϕ. If f ≤ 0 and ϕ ≤ 0, it follows from Theorem 6.1.3 that u ≤ 0 in Ω. b) We show that 0 ∈ ρ(A). Let f ∈ C(Ω) and ϕ ∈ C(Γ). Let f˜ ∈ Cc (Rn ) be an extension of f . Let w := −En ∗ f˜. Then w ∈ C(Rn ) and −Δw = f in D(Ω) . Let v be the solution of the Dirichlet problem D(ϕ − ψ), where ψ = w|Γ . Then u = v + w ∈ C(Ω), u|Γ = ϕ and Δu = Δv + Δw = Δw = −f in D(Ω) . We have shown that A is surjective. It follows from Theorem 6.1.3 that A is injective. Hence, 0 ∈ ρ(A) since A is closed. c) Let Q := R+ ∩ ρ(A). Then by a), R(λ, A) ≥ 0 for all λ ∈ Q and R(0, A) − R(λ, A) = λR(λ, A)R(0, A) ≥ 0, hence 0 ≤ R(λ, A) ≤ R(0, A) for all λ ∈ Q. Thus, R(λ, A) ≤ R(0, A) for all λ ∈ Q. By Corollary B.3, it follows that |λ − μ| ≥ R(λ, A) −1 ≥ R(0, A) −1 for all λ ∈ Q, μ ∈ σ(A). Since 0 ∈ Q, this implies that Q = R+ . It follows from Proposition 3.11.2 that s(A) < 0. In the remainder of this chapter we assume that Ω is Dirichlet regular. The Poisson operator is not densely deﬁned and is not a Hille-Yosida operator since

λR(λ, A) ≥ λ

(λ > 0).

In fact, let (u, 0) = R(λ, A)(0, 1Γ ). Then u|Γ = 1Γ . Hence by Theorem 6.1.3,

λR(λ, A) ≥ λ(u, 0) = λ. Moreover, since the polynomials are dense in C(Ω) by the Stone-Weierstrass theorem, it follows that D(A) = C(Ω) × {0}. (6.4) If we consider the part Ac of A in D(A) = C(Ω) × {0}, then we obtain a Hille-Yosida operator as we shall see in the next theorem. The operator Ac is given by D(Ac ) Ac (u, 0)

= {(u, 0) : u ∈ D(Δmax ), u|Γ = 0} , = (Δmax u, 0).

406

6. THE HEAT EQUATION

Since R(λ, A)X ⊂ D(A), it follows from Proposition B.8 that (s(A), ∞) ⊂ ρ(Ac ) and R(λ, Ac ) = R(λ, A)|D(A) for all λ > s(A). Thus, Ac is a resolvent positive operator and s(Ac ) ≤ s(A) < 0. Identifying C(Ω) × {0} with C(Ω), Ac is identiﬁed with the operator Δc on C(Ω) given by D(Δc )

=

C0 (Ω) ∩ D(Δmax ),

Δc u

=

Δu in D(Ω) .

Here, C0 (Ω) = {u ∈ C(Ω) : u|Γ = 0}. Theorem 6.1.7. Assume that Ω is Dirichlet regular. Then the operator Δc on C(Ω) is dissipative and resolvent positive, and s(Δc ) < 0. Proof. We have established above that Δc is resolvent positive and s(Δc ) < 0. It remains to show that Δc is dissipative. Let t > 0, u ∈ D(Δc ), u − tΔu = f . We have to show that u C(Ω) ≤ f C(Ω) . Let M := f C(Ω) . Let θ ∈ [0, 2π] and v := Re(eiθ u). Then (v − M ) − tΔ(v − M ) = Re(eiθ f ) − M ≤ 0 in D(Ω) . It follows from Theorem 6.1.3 that v − M ≤ 0; i.e., Re(eiθ u) ≤ M for all θ. We deduce that

u C(Ω) ≤ M . Note that the operator Δc is also not densely deﬁned. So we consider the part Δ0 of Δc in C0 (Ω) = D(Δc ). Then Δ0 is given by D(Δ0 )

=

{u ∈ C0 (Ω) : Δu ∈ C0 (Ω)} ,

Δ0 u

=

Δu in D(Ω) .

Since D(Ω) ⊂ D(Δ0 ), the operator Δ0 is densely deﬁned. Moreover, for λ > s(Δc ), R(λ, Δc )C0 (Ω) ⊂ D(Δc ) ⊂ C0 (Ω). Consequently, (s(Δc ), ∞) ⊂ ρ(Δ0 ) and R(λ, Δ0 ) = R(λ, Δc )|C0 (Ω) for all λ > s(Δc ). Hence, λR(λ, Δ0 ) ≤ λR(λ, Δc )

≤ 1 for λ > 0. Applying the Hille-Yosida theorem we obtain the following result. Theorem 6.1.8. Assume that Ω is Dirichlet regular. Then the operator Δ0 generates a positive contractive C0 -semigroup T0 on C0 (Ω). Next, we prove holomorphy of T0 . More generally, the following holds. Theorem 6.1.9. Assume that Ω is Dirichlet regular. Then Δc generates a bounded holomorphic semigroup Tc on C(Ω). The operator Δ0 generates a bounded holomorphic C0 -semigroup on C0 (Ω). Proof. We recall from Example 3.7.6 that the Laplacian generates a bounded holomorphic C0 -semigroup on C0 (Rn ); i.e., deﬁning the operator L := ΔC0 (Rn ) on C0 (Rn ) by D(L) := {f ∈ C0 (Rn ) : Δf ∈ C0 (Rn )} , Lu := Δu in D(Rn ) ,

6.1. THE LAPLACIAN WITH DIRICHLET BOUNDARY CONDITIONS 407 there exists M ≥ 0 such that λ ∈ ρ(L) and λR(λ, L) ≤ M

(6.5)

whenever Re λ > 0. We show a similar estimate for Δc . Let f ∈ C(Ω), Re λ > 0, g := R(λ, Δc )f . Let f˜ ∈ C0 (Rn ) be an extension of f with f˜ C0 (Rn ) = f C(Ω) and let g˜ := R(λ, L)f˜. Let ϕ := g˜|Γ and (w, 0) = R(λ, A)(0, ϕ), where A is the Poisson operator on X = C(Ω) × C(Γ). Then w ∈ C(Ω), λw − Δw = 0 in D(Ω) and w|Γ = ϕ = g˜|Γ . Thus, h := g˜ − w ∈ C0 (Ω) and λh − Δh = λ˜ g − Δ˜ g = f in D(Ω) . It follows that h = g. Observe that

R(λ, A) ≤ R(0, A)

by Corollary 3.11.3 (since A is resolvent positive and s(A) < 0). Hence,

w C(Ω) ≤ R(0, A) ϕ C(Γ) ≤ R(0, A) ˜ g C(Ω) . Hence, setting c := 1 + R(0, A) we have,

g C(Ω) = ˜ g − w C(Ω) ≤ c ˜ g C(Ω) ≤

cM ˜ cM

f C0 (Rn ) =

f C(Ω) |λ| |λ|

by (6.5). We have shown that

λR(λ, Δc ) ≤ cM

(Re λ > 0).

(6.6)

It follows from Corollary 3.7.12 that Δc generates a bounded holomorphic semigroup Tc on C(Ω). Since Δ0 is the part of Δc in D(Δc ) = C0 (Ω), the second assertion is an immediate consequence (see Remark 3.7.13). We should mention that

Tc (t) ≤ 1

for all t > 0.

(6.7)

This follows from Proposition 3.7.16. In fact, one can show that Δc is dissipative (regularizing as in Theorem 6.1.3). We will not use (6.7) and we omit the proof. Moreover, (t > 0). (6.8) T0 (t) = Tc (t)|C0 (Ω) This follows from (3.46). It follows from Theorem 5.1.12 or Theorem 5.3.8 that s(Δ0 ) = ω(T0 ). Since s(Δ0 ) < 0, we conclude that ω(T0 ) < 0. Since Tc is holomorphic we have Tc (t)C(Ω) ⊂ D(Δc ) ⊂ C0 (Ω) for all t > 0, hence Tc (t + s) = T0 (t)Tc (s) for all t > 0, s > 0. We conclude that

Tc (t) ≤ M e−εt

(t ≥ 0)

(6.9)

for some M > 0 and ε > 0. Finally, as a consequence of Theorem 6.1.9 we note the following regularity result.

408

6. THE HEAT EQUATION

Proposition 6.1.10. Assume that Ω is Dirichlet regular. Let u0 ∈ C0 (Ω), and let u(t, x) = (T0 (t)u0 )(x) (t ≥ 0, x ∈ Ω). Then ⎧ ∞ ⎪ ⎨u ∈ C ((0, ∞) × Ω) ∩ C(R+ × Ω), (6.10) ut (t, x) = Δu(t, x) (t > 0, x ∈ Ω), ⎪ ⎩ u(0, x) = u0 (x) (x ∈ Ω). Proof. Since T0 is holomorphic, for f ∈ C0 (Ω) we have T0 (·)f ∈ C ∞ ((0, ∞), D(Δk0 )) for all k ∈ N. It follows from Lemma 6.1.5 that D(Δk0 ) ⊂ C k (Ω). The closed graph theorem implies that for each open set Ω such that Ω ⊂ Ω the restriction map of D(Δk0 ) into C k (Ω ) is continuous where D(Δk0 ) carries the graph norm and C k (Ω ) the norm

u C k (Ω ) = max{ D α u C(Ω ) : |α| ≤ k}, n where α = (α1 , . . . , αn ) is a multi-index, |α| = j=1 αi , Dα = D1α1 . . . Dnαn and D0 u = u. In particular, T (·)f ∈ C ∞ ((0, ∞), C k (Ω )) for all k ∈ N. This implies that the function satisﬁes (6.10).

6.2

Inhomogeneous Boundary Conditions

In Section 6.1 we solved the elliptic problem showing that the Poisson operator is resolvent positive. Now we prove well-posedness of an evolutionary problem with time-dependent boundary conditions, by converting it into an inhomogeneous Cauchy problem and using the results of Chapter 3. We keep the notation of Section 6.1. Let τ > 0. Given u0 ∈ C(Ω) and ϕ ∈ C([0, τ ], C(Γ)), we consider the parabolic problem ⎧ ⎪ = Δu(t) (t ∈ [0, τ ]), ⎨u (t) Pτ (u0 , ϕ) u(t)|Γ = ϕ(t) (t ∈ [0, τ ]), ⎪ ⎩ u(0) = u0 . Remark 6.2.1. Let n = 3. Then Ω is a solid body and the solution u of Pτ (u0 , ϕ) describes the heat ﬂow in Ω. More precisely, for x ∈ Ω, u0 (x) is the given initial temperature at the point x ∈ Ω. For t ∈ [0, τ ], z ∈ Γ, the quantity ϕ(t)(z) is the given temperature at z at the time t. We may imagine that the boundary is heated by some resistance surrounding Ω. Then the solution u(t)(x) is the temperature at the point x ∈ Ω at the time t ∈ [0, τ ]. Deﬁnition 6.2.2. A mild solution of Pτ (u0 , ϕ) is a function u ∈ C([0, τ ], C(Ω)) such that t Δ u(s) ds = u(t) − u0 in D(Ω) (6.11) 0

6.2. INHOMOGENEOUS BOUNDARY CONDITIONS

409

and u(t)|Γ = ϕ(t)

(6.12)

for all t ∈ [0, τ ].

t Note that (6.11) implies in particular that 0 u(s) ds ∈ D(Δmax ) for all t ∈ [0, τ ]. Consider the Poisson operator A on X = C(Ω) × C(Γ) given by A(u, 0) = (Δu, −u|Γ ) on D(A) = D(Δmax ) × {0}. Given U0 ∈ X and Φ ∈ C([0, τ ], X), we consider the Cauchy problem U (t) = AU (t) + Φ(t) (t ∈ [0, τ ]), (6.13) U (0) = U0 . Proposition 6.2.3. Let u0 ∈ C(Ω), U0 := (u0 , 0), and let ϕ ∈ C([0, τ ], C(Γ)), Φ(t) := (0, ϕ(t)). Let U ∈ C([0, τ ], X). Then U is a mild solution of (6.13) if and only if U is of the form U (t) = (u(t), 0) (t ∈ [0, τ ]) where u is a mild solution of Pτ (u0 , ϕ). Proof. If U is a mild solution of (6.13), then d t U (t) = U (s) ds ∈ D(A) = C(Ω) × {0} dt 0 for all t ∈ [0, τ ]. Thus, U (t) = (u(t), 0) for some u ∈ C([0, τ ], C(Ω)). Now the claim is immediate from the deﬁnition of A and Deﬁnition 6.2.2. Now let u0 ∈ D(Δmax ), U0 = (u0 , 0) ∈ D(A). Let ϕ ∈ C([0, τ ], C(Γ)), Φ(t) = (0, ϕ(t)). Then AU0 + Φ(0) = (Δu0 , −u0 |Γ + ϕ(0)). Thus, the consistency condition AU0 + Φ(0) ∈ D(A) = C(Ω) × {0}

(6.14)

from Theorem 3.11.10 becomes u0 |Γ = ϕ(0).

(6.15)

This is obviously a necessary condition for the existence of a mild solution of Pτ (u0 , ϕ). Now we obtain the following from Theorem 3.11.10. Proposition 6.2.4. Assume that Ω is Dirichlet regular. Let ϕ0 ∈ C(Γ), ϕ ∈ t L1 ((0, τ ), C(Γ)), ϕ(t) := ϕ0 + 0 ϕ (s) ds (t ∈ [0, τ ]). Let u0 ∈ D(Δmax ). If condition (6.15) is satisﬁed, then there exists a unique mild solution of Pτ (u0 , ϕ). Next, we obtain the weak parabolic maximum principle as a direct consequence of Theorem 3.11.11. It will serve as an a priori estimate for solutions of Pτ (u0 , ϕ).

410

6. THE HEAT EQUATION

Proposition 6.2.5 (Parabolic maximum principle). Assume that Ω is Dirichlet regular. Let u be a mild solution of Pτ (u0 , ϕ), where u0 and ϕ are real-valued. Let c+ , c− ∈ R be constants such that c−

≤ u0

≤ c+

c−

≤ ϕ(t) ≤ c+

and (t ∈ [0, τ ]).

Then c− ≤ u(t) ≤ c+ (t ∈ [0, τ ]). Proof. Note that e(t) := c+ deﬁnes a mild solution of Pτ (c+ , c+ ). Let v(t) := c+ − u(t). Then v is a mild solution of Pτ (c+ − u0 , c+ − ϕ). Since c+ − u0 ≥ 0 and c+ − ϕ(t) ≥ 0, it follows from Theorem 3.11.11, applied to (6.13) with U (t) = (v(t), 0), U0 = (c+ − u0 , 0), Φ(t) = (0, c+ − ϕ(t)), that c+ − u(t) = v(t) ≥ 0 for all t ∈ [0, τ ]. The other inequality is proved in a similar way. Let u be a mild solution of Pτ (u0 , ϕ), where u0 and ϕ may be complex-valued. By considering Re(eiθ u), which is a mild solution of Pτ (Re(eiθ u0 ), Re(eiθ ϕ)), it follows from Proposition 6.2.5 that ' (

u C([0,τ ],C(Ω)) ≤ max ϕ C([0,τ ],C(Γ)) , u0 C(Ω) . (6.16) Here, we consider C([0, τ ], C(Ω)) and C([0, τ ], C(Γ)) as Banach spaces for the norms

u C([0,τ ],C(Ω)) = sup u(t) C(Ω) and ϕ C([0,τ ],C(Γ)) = sup ϕ(t) C(Γ) , 0≤t≤τ

0≤t≤τ

respectively. Now we can prove well-posedness of Pτ (u0 , ϕ). Theorem 6.2.6. Assume that Ω is Dirichlet regular. Let u0 ∈ C(Ω) and ϕ ∈ C([0, τ ], C(Γ)) such that u0 |Γ = ϕ(0). Then there exists a unique mild solution of Pτ (u0 , ϕ). Proof. Uniqueness follows from Lemma 3.2.9. For existence, choose u0n ∈ D(Δmax ) such that limn→∞ u0n = u0 in C(Ω). Choose ϕn ∈ C 1 ([0, τ ], C(Γ)) such that ϕn (0) = u0n |Γ and ϕn → ϕ as n → ∞ in C([0, τ ], C(Γ)). For example, one may let ϕn (t) := (1 − λ(nt))ψn (t) + λ(nt)u0n |Γ , where λ(s) := (1 − min(s, 1))2 , ψn ∈ C 1 ([0, τ ], C(Γ)) and ψn − ϕ C([0,τ ],C(Γ)) < 1/n. By Proposition 6.2.4, there exists a unique mild solution un of Pτ (u0n , ϕn ). By (6.16), we have ' (

un − um C([0,τ ],C(Ω)) ≤ max ϕn − ϕm C([0,τ ],C(Γ)) , u0n − u0m C(Ω) . Hence, (un )n∈N is a Cauchy sequence in C([0, τ ], C(Ω)). Let u := limn→∞ un in t C([0, τ ], C(Ω)). Then u(t)|Γ = limn→∞ ϕn (t) = ϕ(t). Since Δmax 0 un (s) ds = un (t) − u0n and Δmax is closed, it follows that t t t u(s) ds = lim un (s) ds ∈ D(Δmax ) and Δmax u(s) ds = u(t) − u0 0

n→∞

0

0

6.2. INHOMOGENEOUS BOUNDARY CONDITIONS

411

for all t ∈ [0, τ ]. We have shown that u is a mild solution of Pτ (u0 , ϕ). So far, we have seen that for each u0 ∈ C(Ω) and ϕ ∈ C([0, τ ], C(Γ)) satisfying ϕ(0) = u0 |Γ there exists a unique mild solution. We now show that the mild solution u is always of class C ∞ on (0, τ ] × Ω. In fact, we may identify u with a continuous function deﬁned on [0, τ ] × Ω with values in R by letting u(t, x) := u(t)(x) (t ∈ [0, τ ], x ∈ Ω). Then the following holds. Theorem 6.2.7. Assume that Ω is Dirichlet regular. Let u0 ∈ C(Ω) and ϕ ∈ C([0, τ ], C(Γ)) such that u0 |Γ = ϕ(0). Let u be the mild solution of Pτ (u0 , ϕ). Then u ∈ C ∞ ((0, τ ] × Ω). t Proof. a) Assume that u0 = 0. Let v(t) := 0 u(s) ds. Then v ∈ C 1 ([0, τ ], C(Ω)), v(t) ∈ D(Δmax ) and v (t) = Δv(t) for t ∈ [0, τ ]. Let 0 < t0 ≤ τ, x0 ∈ Ω. Choose r > 0 such that B(x0 , r) ⊂ Ω, and let C := [0, τ ] × B(x0 , r) and C := [t0 /2, τ ] × B(x0 , r/2). Choose ξ ∈ C ∞ ([0, τ ] × Rn ) such that ξ ≡ 1 on C , ξ ≡ 0 on ([0, τ ] × Rn ) \ C, and ξ ≡ 0 on [0, t0 /4] × Rn . Let w := ξ · v on [0, τ ] × Ω, w := 0 on [0, τ ] × (Rn \ Ω). Then w ∈ C 1 ([0, τ ], C0 (Rn )), and w (t) = ξ (t)v(t) + ξ(t)v (t) on [0, τ ] × Ω, with w (t) = 0 on [0, τ ] × (Rn \ Ω). It follows from Lemma 6.1.5 and the closed graph theorem that D(Δmax ) → C 1 (Ω) when D(Δmax ) carries the graph norm and C 1 (Ω) has the natural Fr´echet topology. In particular, ∇v is continuous on [0, τ ] × C. We have Δw(t) = ξ(t)Δv(t) + 2∇ξ(t) · ∇v(t) + Δξ(t)v(t) on [0, τ ] × Ω. Let

ξ (t)v(t) − 2∇ξ(t) · ∇v(t) − Δξ(t)v(t) f (t) := 0

on [0, τ ] × Ω, on [0, τ ] × (Rn \ Ω).

Then f ∈ C([0, τ ], C0 (Rn )) and w (t) = Δw(t) + f (t) on [0, τ ] × Rn . Denote by G the Gaussian semigroup on C0 (Rn ), i.e. G(t)g := kt ∗ g,

where kt (x) := (4πt)−n/2 e−|x|

2

/4t

.

Since w(0) = 0, it follows from Proposition 3.1.16 that t w(t) = G(t − s)f (s) ds, i.e., 0 t 2 w(t, x) = (4π(t − s))−n/2 e−|x−y| /4(t−s) f (s, y) dy ds 0

Rn

412

6. THE HEAT EQUATION

for all 0 < t ≤ t0 , x ∈ Rn . Since f ≡ 0 on C and outside C, the integrand has no singularities for (t, x) in the interior of C . Thus, w is of class C ∞ in a neighbourhood of (t0 , x0 ) in (0, τ ] × Ω. Since v = w in C and (t0 , x0 ) is arbitrary, it follows that v, and hence also u, belong to C ∞ ((0, τ ] × Ω). b) Now consider the general case when u0 |Γ = ϕ(0). Let w0 be the solution of the Dirichlet problem D(ϕ(0)). Consider v(t) := u(t) − w0 . Then v is a mild solution of v (t) = Δv(t) (t ∈ [0, τ ]) and v(0)|Γ = 0. Denote by T0 the C0 -semigroup generated by Δ0 on C0 (Ω). Let w(t) := v(t) − T0 (t)v(0). Then w is a mild solution of Pτ (0, ϕ − ϕ(0)). Hence, w ∈ C ∞ ((0, τ ] × Ω) by a). Since T0 (·)v(0) ∈ C ∞ ((0, ∞) × Ω) by Proposition 6.1.10, the proof is complete. Now we can reformulate the results. For this, we consider the parabolic domain Ωτ := (0, τ ] × Ω with parabolic boundary Γτ = ({0} × Ω) ∪ ((0, τ ] × Γ), where τ > 0. Thus, Ωτ is a cylinder and Γτ is the topological boundary without the top. Theorem 6.2.8. Assume that Ω is Dirichlet regular. Then for every ψ ∈ C(Γτ ) there exists a unique function u ∈ C(Ωτ ) ∩ C ∞ (Ωτ ) such that ut − Δu u|Γτ

= 0 in Ωτ , and = ψ.

(6.17)

Thus, (6.17) is formulated exactly as the Dirichlet problem, the Laplacian d being replaced by the parabolic operator dt − Δ, Ω by the parabolic domain Ωτ and Γ by the parabolic boundary Γτ .

6.3

Asymptotic Behaviour

We keep the notation of the preceding section. But now we consider the problem on the half-line ⎧ ⎪ = Δu(t) (t ≥ 0), ⎨u (t) P∞ (u0 , ϕ) u(t)|Γ = ϕ(t) (t ≥ 0), ⎪ ⎩ u(0) = u0 ,

6.3. ASYMPTOTIC BEHAVIOUR

413

where u0 ∈ C(Ω) and ϕ ∈ C(R+ , C(Γ)) are given functions. As before, by a mild solution of P∞ (u0 , ϕ) we understand a function u ∈ C(R+ , C(Ω)) such that Δ 0

t

u(s) ds = u(t) − u0 in D(Ω) and u(t)|Γ

= ϕ(t)

for all t ∈ R+ . We assume throughout this section that Ω is Dirichlet regular. It follows from Theorem 6.2.6 that for each u0 ∈ C(Ω) and ϕ ∈ C(R+ , C(Γ)) such that u0 |Γ = ϕ(0) there exists a unique mild solution u of P∞ (u0 , ϕ). In this section we study the asymptotic behaviour of u(t) as t → ∞. The results are analogous to (and in some cases, consequences of) abstract results given in Sections 5.4 and 5.6. We start with Ces`aro convergence. Proposition 6.3.1. Let ϕ : R+ → C(Γ) be continuous and bounded. Assume that 1 lim t→∞ t

0

t

ϕ(s) ds = ϕ∞ in C(Γ).

Let u0 ∈ C(Ω) satisfying u0 |Γ = ϕ(0) and let u be the mild solution of P∞ (u0 , ϕ). Then 1 t lim u(s) ds = u∞ exists in C(Ω). t→∞ t 0 Moreover, Δu∞ = 0 in D(Ω) and u∞ |Γ = ϕ∞ . Proof. By (6.16), u is bounded. Taking Laplace transforms, we have λˆ u(λ) − Δˆ u(λ) = u0 and u ˆ(λ)|Γ = ϕ(λ) ˆ (λ > 0). Denote by w(λ) the solution of the Dirichlet problem D(ϕ(λ)). ˆ Then u ˆ(λ) − w(λ) ∈ C0 (Ω) and λ(ˆ u(λ) − w(λ)) − Δ(ˆ u(λ) − w(λ)) = u0 − λw(λ). Thus, u ˆ(λ) − w(λ) = R(λ, Δc )(u0 − λw(λ)). Let u∞ be the solution of D(ϕ∞ ). By Theorem 4.1.2, limλ↓0 λϕ(λ) ˆ = C − limt→∞ ϕ(t) = ϕ∞ . It follows from the maximum principle (Theorem 6.1.3) that λw(λ) converges to a function u∞ in C(Ω) as λ ↓ 0. Clearly, u∞ solves D(ϕ∞ ). Thus, u ˆ(λ) − w(λ) → R(0, Δc )(u0 − u∞ ) in C(Ω) as λ ↓ 0. Consequently, λ(ˆ u(λ) − w(λ)) → 0 in C(Ω) as λ ↓ 0. This implies that limλ↓0 λˆ u(λ) = u∞ in C(Ω). By Theorem 4.2.7, this implies the claim. Next we consider uniform continuity. Proposition 6.3.2. Let ϕ ∈ BUC(R+ , C(Γ)) and u0 ∈ C(Ω) such that u0 |Γ = ϕ(0). Let u be the mild solution of P∞ (u0 , ϕ). Then u ∈ BUC(R+ , C(Ω)). Proof. By (6.16), u is bounded. For δ > 0 let uδ (t) := u(t + δ) − u(t), ϕδ (t) := ϕ(t + δ) − ϕ(t) (t ≥ 0). Then uδ is the mild solution of P∞ (uδ (0), ϕδ ). Since

414

6. THE HEAT EQUATION

ϕδ → 0 in BUC(R+ , C(Γ)) and uδ → 0 in C(Ω) as δ ↓ 0, it follows from (6.16) that uδ (t) → 0 as δ ↓ 0 uniformly on R+ . This means that u is uniformly continuous. Using Propositions 6.2.3 and 6.3.2, the results of Chapter 5 on inhomogeneous Cauchy problems give the following. Theorem 6.3.3. Let ϕ ∈ AAP(R+ , C(Γ)) and u0 ∈ C(Ω) such that u0 |Γ = ϕ(0). Denote by u the mild solution of P∞ (u0 , ϕ). Then a) u ∈ AAP(R+ , C(Ω)) and Freq(u) = Freq(ϕ). b) If ϕ = ϕ1 + ϕ2 where ϕ1 ∈ AP(R+ , C(Γ)), ϕ2 ∈ C0 (R+ , C(Γ)), and u = u1 + u2 where u1 ∈ AP(R+ , C(Ω)) and u2 ∈ C0 (R+ , C(Ω)), then u1 is the mild solution of P∞ (u1 (0), ϕ1 ) and u2 is the mild solution of P∞ (u2 (0), ϕ2 ). c) If limt→∞ ϕ(t) = ϕ∞ exists in C(Γ), then limt→∞ u(t) = u∞ where u∞ is the solution of the Dirichlet problem D(ϕ∞ ). Proof. a) Let X := C(Ω) × C(Γ). Consider the function U : R+ → X deﬁned by U (t) := (u(t), 0). Then by Proposition 6.3.2, U ∈ BUC(R+ , X). Let Φ(t) := (0, ϕ(t)). Then Φ ∈ AAP(R+ , X) and Proposition 6.3.2 shows that U is a mild solution of U (t) = AU (t) + Φ(t) (t ≥ 0), (6.18) U (0) = (u0 , 0), where A is the Poisson operator. Since s(A) < 0, it follows from Corollary 5.6.9 that U ∈ AAP(R+ , X), hence u ∈ AAP(R+ , C(Ω)). Moreover, it also follows that Freq(u) = Freq(U ) = Freq(Φ) = Freq(ϕ). b) This is a direct consequence of Propositions 5.4.16 and 6.2.3. c) By Corollary 5.6.9, limt→∞ U (t) exists and equals R(0, A)(0, ϕ∞ ) = (u∞ , 0), so that u∞ is the solution of D(ϕ∞ ). Corollary 6.3.4. Let ϕ ∈ AP(R+ , C(Γ)). Then there exists a unique u0 ∈ C(Ω) satisfying u0 |Γ = ϕ(0) such that the mild solution u of P∞ (u0 , ϕ) is almost periodic. Proof. Existence: Let v0 ∈ C(Ω) such that v0 |Γ = ϕ(0). Let v be the mild solution of P∞ (v0 , ϕ). Then v = v1 + v2 where v1 ∈ AP(R+ , C(Ω)) and v2 ∈ C0 (R+ , C(Ω)), by Theorem 6.3.3. Moreover, v1 is the mild solution of P∞ (v1 (0), ϕ). So we may choose u0 = v1 (0). Uniqueness: Assume that the mild solution u ˜ of P∞ (˜ u0 , ϕ) is almost periodic. Then v = u − u ˜ ∈ AP(R+ , C(Ω)) and v is the mild solution of P∞ (u0 − u ˜0 , 0). It follows from Theorem 6.3.3 c) that limt→∞ v(t) = 0. Hence, v(t) ≡ 0 by (4.36). In the situation of Corollary 6.3.4, if ϕ is τ -periodic, then u is also τ -periodic. This follows from Corollary 4.5.4, since Freq(u) = Freq(ϕ) ⊂ 2π Z (see also Corolτ lary 5.6.9 c). Finally, we consider the inhomogeneous heat equation with inhomogeneous boundary conditions.

6.4. NOTES

415

Given u0 ∈ C(Ω), ϕ ∈ C(R+ , C(Γ)) such that u0 |Γ = ϕ(0) and f ∈ C(R+ , C(Ω)), we consider the problem ⎧ ⎪ Δu(t) + f (t) (t ≥ 0), ⎨u (t) = P∞ (u0 , ϕ, f ) u(t)|Γ = ϕ(t) (t ≥ 0), ⎪ ⎩ u(0) = u0 . A mild solution is a continuous function u : R+ → C(Ω) such that u(0) = t u0 , u(t)|Γ = ϕ(t), 0 u(s) ds ∈ D(Δmax ) and Δ

t

t

u(s) ds + 0

0

f (s) ds = u(t) − u0 in D(Ω)

(6.19)

for all t ≥ 0. By Theorem 6.2.6, there is at most one mild solution of P∞ (u0 , ϕ, f ). By Proposition 3.7.22, the function t v(t) := Tc (t − s)f (s) ds (t ≥ 0) (6.20) 0

deﬁnes a mild solution of P∞ (0, 0, f ). Since Ω is Dirichlet regular, by Theorem 6.2.6, there exists a unique mild solution w of P∞ (u0 , ϕ, 0). Hence, u = v + w is a mild solution of P∞ (u0 , ϕ, f ). We have shown the following. Theorem 6.3.5. Let f ∈ C(R+ , C(Ω)), ϕ ∈ C(R+ , C(Γ)) and u0 ∈ C(Ω) such that u0 |Γ = ϕ(0). Then there exists a unique mild solution of P∞ (u0 , ϕ, f ). Recall that ω(Tc ) < ∞. Thus, v = Tc ∗ f ∈ BUC(R+ , C(Ω)) (respectively, AAP(R+ , C(Ω))) if f ∈ BUC(R+ , C(Ω)) (respectively, f ∈ AAP(R+ , C(Ω))), by Proposition 5.6.1. So we obtain the following from Theorem 6.3.3. Theorem 6.3.6. Let f ∈ AAP(R+ , C(Ω)), ϕ ∈ AAP(R+ , C(Γ)) and u0 ∈ C(Ω). Assume that u0 |Γ = ϕ(0). Let u be the mild solution of P∞ (u0 , ϕ, f ). Then u ∈ AAP(R+ , C(Ω)). Corollary 6.3.7. Let f : R+ → C(Ω) and ϕ : R+ → C(Γ) be continuous. Assume that limt→∞ f (t) = f∞ exists in C(Ω) and limt→∞ ϕ(t) = ϕ∞ in C(Γ). Let u0 ∈ C(Ω) such that u0 |Γ = ϕ(0). Let u be the mild solution of P∞ (u0 , ϕ, f ). Then limt→∞ u(t) = u∞ exists in C(Ω) and u∞ | Γ = ϕ∞ , (6.21) −Δu∞ = f∞ in D(Ω) .

6.4

Notes

The approach to solving the heat equation with the help of the Poisson operator is taken from [Are00], where general strongly elliptic operators in divergence form with

416

6. THE HEAT EQUATION

bounded measurable coeﬃcients are also considered. Greiner [Gre87] developed an abstract perturbation theory for boundary conditions. Theorems 6.1.7 and 6.1.8 are proved in [AB98], where it is also shown that Dirichlet regularity is a necessary condition. Lumer and Schnaubelt [LS99] consider also non-cylindrical domains. Lumer gave a proof of the holomorphy of the semigroup generated by Δ0 which is based on the maximum principle (cf. [LP79]). In Theorem 6.1.9 we use the properties of resolvent positive operators to prove holomorphy. Theorem 6.2.7 is an adaptation of Evans’s proof [Eva98, Section 2.3, Theorem 8] to the solutions deﬁned here. Theorem 6.2.8 was probably ﬁrst proved by Tychonoﬀ [Tyc38] in 1938 with the help of integral equations. Other proofs were given by Fulks [Ful56], [Ful57] and Babuˆska and V´ yborn´ y [BV62] (see also the work of Lumer [Lum75]). Concerning an Lp -approach to boundary value problems via holomorphic semigroups we refer to the monograph by Taira [Tai95].

Chapter 7

The Wave Equation In this chapter we study the wave equation utt = Δu on an open subset Ω of Rn . We will use the theory of cosine functions and work on L2 (Ω). We ﬁrst consider the Laplacian with Dirichlet boundary conditions. This is a selfadjoint operator and well-posedness is a consequence of the spectral theorem. A further aim is to replace the Laplace operator by a general elliptic operator. This will be done by a perturbation theorem for selfadjoint operators which we prove in Section 7.1. We give a brief introduction to symmetric sesquilinear forms which are the natural tool for proving selfadjointness. However, we restrict ourselves to the minimum needed to show that quite general equations can be solved in a simple way by functional analytical methods. The restriction to Dirichlet boundary conditions and to second order operators is not essential; we choose these in order to present the simplest case.

7.1

Perturbation of Selfadjoint Operators

A selfadjoint operator which is bounded above generates a cosine function (see Example 3.14.16). We will present a perturbation result in terms of the form domain which again leads to a generator of a cosine function with the same phase space. It is important to know this since the phase space yields the natural domain for initial data for classical solutions (Corollary 3.14.12). We will see in the following section that the abstract setting which we present here is very well adapted to elliptic operators. Let H be a Hilbert space with scalar product (·|·)H and norm · H . We consider another Hilbert space V with scalar product (·|·)V and norm · V . Moreover, we assume that V is continuously embedded into H with dense image.

W. Arendt et al., Vector-valued Laplace Transforms and Cauchy Problems: Second Edition, Monographs in Mathematics 96, DOI 10.1007/978-3-0348-0087-7_7, © Springer Basel AG 2011

417

418

7. THE WAVE EQUATION

This means, we assume that V ⊂ H, that V is dense in H and that there exists a constant ω > 0 such that ω u 2H ≤ u 2V (7.1) for all u ∈ V . We use the abbreviation d

V → H for these three properties. In this situation, we associate to V an operator AH on H by postulating: D(AH )

:=

u ∈ V : there exists f ∈ H such that

(u|v)V = −(f |v)H for all v ∈ V , AH u :=

f.

Note that f is uniquely determined by u since V is dense in H. We call AH the operator on H associated with V . Proposition 7.1.1. The operator AH is selfadjoint and bounded above by −ω. In particular, AH generates a holomorphic C0 -semigroup T on H of angle π/2 satisfying

T (z) ≤ e−ω Re z (Re z > 0). Moreover, AH generates a cosine function. Proof. a) AH is symmetric. In fact, let u, v ∈ D(AH ). Then by the deﬁnition of AH , (AH u|v)H = −(u|v)V = −(v|u)V = (AH v|u)H = (u|AH v)H . b) For u ∈ D(AH ) one has (AH u|u)H = −(u|u)V ≤ −ω u 2H , by (7.1). Thus, AH is bounded above by −ω. c) We show that −AH is surjective. Let f ∈ H. Then F (v) := (v|f )H deﬁnes a continuous linear form on V . By the Riesz-Fr´echet lemma, there exists a unique u ∈ V such that (v|f )H = (v|u)V for all v ∈ V . Hence by the deﬁnition of AH , one has u ∈ D(AH ) and −AH u = f . It follows from Theorem B.14 that AH is selfadjoint. The remaining two assertions follow from Example 3.7.5 and Example 3.14.16. Let us consider multiplication operators as a ﬁrst example. It is very simple; nevertheless, it is a generic example by the spectral theorem (Theorem B.13).

7.1. PERTURBATION OF SELFADJOINT OPERATORS

419

Example 7.1.2 (Multiplication operators). Let (Y, μ) be a measure space, ω > 0 and m : Y → [ω, ∞) be measurable. Let H := L2 (Y, μ) and let V := L2 (Y, m dμ) := u ∈ H : |u|2 m dμ < ∞ . Y

Then V is a Hilbert space for the scalar product (u|v)V := u(x)v(x)m(x) dμ(x) Y d

and V → H. It is easy to see that the operator AH on H associated to V is the multiplication operator given by D(AH )

=

AH u =

{u ∈ H : mu ∈ H}, −mu.

(cf. Example B.12). The cosine function Cos generated by AH is given by 3 (Cos(t)f )(x) = (cos t m(x))f (x) (cf. Example 3.14.16). Thus, Cos(t) =

1 (U (t) + U (−t)), 2

where U is the C0 -group on H given by √ (U (t)f )(x) = eit m(x) f (x) (cf. Example 3.14.16). The generator B of U is given by √ D(B) = {f ∈ H : m · f ∈ H}, √ Bf = i m · f. Thus, V = D(B). It follows from Example 3.14.15 that the phase space of Cos is V × H. In Example 7.1.2, the phase space of the cosine function associated with d

V is V × H. In fact, this is the case whenever V → H. The spectral theorem applied to the selfadjoint operator AH shows that there is a measure space (Y, μ), a measurable function m : Y → [ω, ∞) and a unitary equivalence U taking H onto L2 (Y, μ) and AH onto the operator appearing in Example 7.1.2. The next result, which is the converse of Proposition 7.1.1, shows that V is uniquely determined by AH . Hence, it follows from Example 7.1.2 that U takes V onto L2 (Y, m dμ) and that the phase space is V × H.

420

7. THE WAVE EQUATION

Proposition 7.1.3. Let B be a selfadjoint operator on H which is bounded above by d

−ω, where ω > 0. Then there exists a unique Hilbert space V such that V → H and such that B is the operator associated with V . Moreover, the phase space associated with the cosine function generated by B is V × H. d

Proof. Uniqueness: Assume that B is associated with V where V → H. We show that D(B) is dense in V . In fact, let v ∈ V such that (u|v)V = 0 for all u ∈ D(B). Then (Bu|v)H = −(u|v)V = 0 for all u ∈ D(B). Since B is selfadjoint, it follows that v ∈ D(B) and Bv = 0. Since B is invertible, we conclude that v = 0. This proves the claim. Now we observe that

u 2V = (u|u)V = −(Bu|u)H for 3 all u ∈ D(B). Thus, V is the completion of D(B) for the norm u V := −(Bu|u)H . Moreover, (u|v)V = −(Bu|v)H for all u, v ∈ D(B). Thus, the scalar product is also determined by B. Existence: Using the spectral theorem, we may assume that B is a multiplication operator. Then the assertion is proved in Example 7.1.2. Propositions 7.1.1 and 7.1.3 establish a bijective correspondence between selfadjoint operators B on H which are bounded above by −ω (where ω > 0) and d

Hilbert spaces V such that V → H and (7.1) is satisﬁed. One frequently calls V the form domain of B, and (·|·)V the sesquilinear form associated with B. From Corollary 3.14.13 we now deduce our ﬁrst perturbation result for selfadjoint operators. d

Corollary 7.1.4. Let V be a Hilbert space such that V → H and let AH be the operator associated with V on H. Let C ∈ L(V, H). Then AH + C generates a cosine function on H with phase space V × H. Our next aim is to introduce another kind of perturbation. For this we need some preparation. d

Let V → H. A mapping ϕ : V → C is called antilinear if ϕ(u + v) ϕ(λu) By

= ϕ(u) + ϕ(v) (u, v ∈ V ), ¯ = λϕ(u) (u ∈ V, λ ∈ C).

and

V := {ϕ : V → C : ϕ antilinear and continuous}

we denote the antidual of V . It is a Banach space for the norm

ϕ V := sup |ϕ(u)|. u V ≤1

7.1. PERTURBATION OF SELFADJOINT OPERATORS

421

We embed H into V in the following way. For f ∈ H we deﬁne ϕf ∈ V by ϕf (u) := (f |u)H . It is clear that the mapping f → ϕf : H → V is linear, injective and continuous. This is the desired embedding. For ϕ ∈ V we use the notation (ϕ|u) := ϕ(u)

(u ∈ V ).

Thus, if f ∈ H we have (ϕf |u) = (f |u)H

(u ∈ V ).

By the Riesz-Fr´echet lemma, the mapping A : V → V given by (Au|v) := −(u|v)V

(u, v ∈ V )

is an isometric isomorphism from V onto V . In particular, V is itself a Hilbert space for the scalar product (f |g)V := (A−1 f |A−1 g)V

(f, g ∈ V ).

It follows from (7.1) that ω ϕf 2V ≤ f 2H

(7.2)

for all f ∈ H. In fact, ω ϕf 2V

= ω sup |(f |u)H |2 u∈V uV ≤1

≤ ≤

sup √ u∈V ωuH ≤1

√ |(f | ωu)H |2

f 2H ,

by the Cauchy-Schwarz inequality. Now we identify H with a subspace of V , and we identify f and ϕf for f ∈ H. In particular, for f ∈ H we write (f |v) = (f |v)H

(v ∈ V ).

Having this in mind, the proof of the following proposition is easy. Proposition 7.1.5. The operator A on V is selfadjoint with upper bound −ω. Moreover, H × V is the phase space associated with the cosine function generated by A on V . The operator AH associated with V is the part of A in H.

422

7. THE WAVE EQUATION

Proof. We ﬁrst prove that (Au|f )V = −(u|f )H

(7.3)

for all u ∈ V, f ∈ H. In fact, let w = A−1 f ∈ V . Then (w|u)V = −(Aw|u) = −(f |u)H . Hence,

(Au|f )V = (u|A−1 f )V = (u|w)V = −(u|f )H .

It follows from (7.3) that for all u, v ∈ V , (Au|v)V = −(u|v)H = −(v|u)H = (Av|u)V = (u|Av)V . Thus, A is symmetric. Moreover, (Au|u)V = − u 2H ≤ −ω u 2V for all u ∈ V by (7.2). Thus, A is bounded above by −ω. Since A is surjective, it follows from Theorem B.14 that A is selfadjoint. In particular, V = D(A) is dense d

in V . Thus, H → V . Let B be the operator on V associated with H. It follows from (7.3) that B is an extension of A. Since both operators are invertible, they are equal. It follows from Proposition 7.1.3 that the phase space associated with the cosine function generated by B on V is H × V . The ﬁnal assertion is easy to verify. We will illustrate Proposition 7.1.5 by considering multiplication operators in the following example. The discussion before Proposition 7.1.3 shows that they describe the most general situation; in fact, the example gives an alternative proof of Proposition 7.1.5. Example 7.1.6 (Antidual associated with multiplication operator). Let H := L2 (Y, μ), V := L2 (Y, m dμ) where m : Y → [ω, ∞) is measurable, ω > 0. Then we 1 may identify V with L2 (Y, m dμ) by letting (w|v) := w¯ v dμ Y 1 for all w ∈ L2 (Y, m dμ), v ∈ V . Since for f ∈ H, one has (ϕf |v) = f v¯ dμ, Y

the embedding H → V : f → ϕf corresponds to the identity mapping L2 (Y, μ) → 1 L2 (Y, m dμ). Now the mapping A : V → V is given by Au = −mu. In fact, for u ∈ V one has (Au|v) = − u¯ v m dμ Y

7.2. THE WAVE EQUATION IN L2 (Ω)

423

for all v ∈ V . Note that A, considered as an operator on V , is associated to the subspace 1 H = L2 (Y, dμ) of V = L2 (Y, m dμ). Thus, A generates a cosine function on V and the associated phase space is H × V . Now we are in the position to prove the following general perturbation result. d

Theorem 7.1.7. Let V → H and identify H with a subspace of V in the canonical way. Denote by A : V → V the isomorphism given by the Riesz-Fr´echet lemma. Let C ∈ L(V, H), B ∈ L(H, V ). Then the part (A + B + C)H of A + B + C in H generates a cosine function Cos on H whose phase space is V × H. Note that D((A + B + C)H ) = {u ∈ V : Au + Bu + Cu ∈ H}. Proof. We consider A as an unbounded operator on V with domain V . Then we know from Proposition 7.1.5 that A generates a cosine function on V with associated phase space H × V . Since B ∈ L(H, V ), A + B also generates a cosine function on V with phase space H × V (by Corollary 3.14.13). Now we apply Corollary 3.14.14 to deduce that the part (A + B)H of A + B in H generates a cosine function with associated phase space D(A + B) × H = V × H. Since C ∈ L(V, H), one has (A + B + C)H = (A + B)H + C. Now the claim follows by another application of Corollary 3.14.13. Corollary 7.1.8. Under the conditions of Theorem 7.1.7, the operator (A+B +C)H generates a holomorphic C0 -semigroup on H of angle π/2. Proof. This follows from Theorem 3.14.17.

7.2

The Wave Equation in L2 (Ω)

Let Ω ⊂ Rn be an open set. In this section we will consider the wave equation on Ω with Dirichlet boundary conditions. We consider ﬁrst the Laplacian and then more general elliptic operators. For this, we recall some distributional notions (see Appendix E). We denote the ﬁrst Sobolev space in L2 (Ω) by H 1 (Ω), i.e., H 1 (Ω) = W 1,2 (Ω), and we deﬁne H 1 (Ω)

H01 (Ω) := D(Ω) . This allows us to give a meaning to Dirichlet boundary conditions: for f ∈ H 1 (Ω) we say that f |∂Ω = 0 weakly if f ∈ H01 (Ω). As usual, we consider L2 (Ω) as a subspace of D(Ω) . In particular, if f ∈ 1 H (Ω), then Δf ∈ D(Ω) is deﬁned by ϕ, Δf = Δϕ, f =

n j=1

Ω

f Dj2 ϕ dx

=−

n j=1

Ω

Dj f Dj ϕ dx

424

7. THE WAVE EQUATION

for all ϕ ∈ D(Ω). Thus, to say that Δf ∈ L2 (Ω) means that there exists a function g ∈ L2 (Ω) such that −

n

Ω

j=1

Dj f Dj ϕ dx =

gϕ dx Ω

for all ϕ ∈ D(Ω). We then identify g and Δf . Next, we deﬁne the Laplacian with Dirichlet boundary conditions. This example has already been given in Chapter 3 (Example 3.4.7). Here we show how it ﬁts into the setting of the preceding section. After that, it will be easy to investigate more general elliptic operators than the Laplacian. Example 7.2.1 (The Dirichlet Laplacian). Deﬁne the operator ΔL2 (Ω) on L2 (Ω) by ' ( D(ΔL2 (Ω) ) := f ∈ H01 (Ω) : Δf ∈ L2 (Ω) , ΔL2 (Ω) f

:=

Δf.

Then ΔL2 (Ω) is selfadjoint and bounded above by 0. Moreover, ΔL2 (Ω) generates a cosine function with phase space H01 (Ω) × L2 (Ω). Proof. Let H := L2 (Ω) and V := H01 (Ω) with scalar product (u|v)V :=

u¯ v dx + Ω

n j=1

Ω

Dj uDj v¯ dx.

d

Then clearly, V → H. Let B be the operator on H which is associated with V . We show that ΔL2 (Ω) = B + I. In fact, let u ∈ D(B), Bu =: f . Then u ∈ H01 (Ω) and n − Dj uDj ϕ dx − uϕ dx = −(u|ϕ) ¯V = f ϕ dx j=1

Ω

Ω

Ω

for all ϕ ∈ H01 (Ω). Taking ϕ ∈ D(Ω), we obtain ϕ, Δu = Δϕ, u

=

n j=1

= −

Ω

uDj2 ϕ dx

n j=1

Ω

Dj uDj ϕ dx =

f ϕ dx +

uϕ dx.

Ω

Ω

Hence, Δu = f + u. This shows that u ∈ D(ΔL2 (Ω) ) and Bu = ΔL2 (Ω) u − u. Conversely, let u ∈ D(ΔL2 (Ω) ). Then u ∈ H01 (Ω) and for ϕ ∈ D(Ω), −(u|ϕ)V = −

n j=1

Ω

Dj uDj ϕ¯ dx −

Ω

uϕ¯ dx = ϕ, ¯ Δu −

uϕ¯ dx. Ω

7.2. THE WAVE EQUATION IN L2 (Ω)

425

Since D(Ω) is dense in H01 (Ω), it follows that −(u|ϕ)V = (Δu − u | ϕ)L2 (Ω) for all ϕ ∈ H01 (Ω). Thus, u ∈ D(B) and Bu = Δu − u = ΔL2 (Ω) u − u. We have shown that B = ΔL2 (Ω) −I. Hence, ΔL2 (Ω) = B+I is also selfadjoint. Since B generates a cosine function with phase space H01 (Ω) × L2 (Ω), so does ΔL2 (Ω) , by Corollary 3.14.13. Now we obtain the following well-posedness result for the wave equation. Theorem 7.2.2 (Wave equation). Let f ∈ H01 (Ω) such that Δf ∈ L2 (Ω). Let g ∈ H01 (Ω). Then there exists a unique function u ∈ C 2 (R+ , L2 (Ω)) such that a) Δu(t) ∈ L2 (Ω) for t ≥ 0; b) u(t) ∈ H01 (Ω) for t ≥ 0; c) u (t) = Δu(t) for t ≥ 0; d) u(0) = f, u (0) = g. Proof. Denote by ΔL2 (Ω) the Dirichlet Laplacian and by Cos the cosine function generated by ΔL2 (Ω) on L2 (Ω) (see Example 7.2.1). Let Sin be the associated sine function. Then u(t) = Cos(t)f + Sin(t)g is a solution of ⎧ u ∈ C 2 (R+ , L2 (Ω)), ⎪ ⎪ ⎪ ⎨u(t) ∈ D(Δ 2 ) (t ≥ 0) L (Ω) ⎪u (t) = ΔL2 (Ω) u(t) (t ≥ 0) ⎪ ⎪ ⎩ u(0) = f, u (0) = g, by Corollary 3.14.12. Uniqueness follows from Corollary 3.14.8. Next, we consider general uniformly elliptic operators of second order. Let aij ∈ L∞ (Ω) be complex-valued coeﬃcients such that aij = aji and n

aij (x)ξi ξj ≥ α|ξ|2

i,j=1

for all ξ ∈ Rn , x-a.e. on Ω, where α > 0 is ﬁxed. This last condition is called uniform ellipticity and is equivalent to saying that the smallest eigenvalue of the hermitian matrix (aij (x)) is at least α for almost all x ∈ Ω. Let bi , ci , d ∈ L∞ (Ω) be complex-valued functions (i = 1, 2, . . . , n). We consider the formal elliptic second order operator Lu :=

n i,j=1

Di (aij Dj u) +

n

(Dj (bj u) + cj Dj u) + du.

(7.4)

j=1

It is possible to give a sense to L by multiplying (7.4) by a test function and integration by parts. More precisely, we deﬁne L as follows.

426

7. THE WAVE EQUATION

Let u ∈ H01 (Ω). Deﬁne the distribution Lu ∈ D(Ω) by n ϕ, Lu = − aij (x)(Dj u)(x)(Di ϕ)(x) dx i,j=1 Ω n

−

j=1

+

Ω

bj (x)u(x)(Dj ϕ)(x) dx +

n j=1

Ω

cj (x)(Dj u)(x)ϕ(x) dx

d(x)u(x)ϕ(x) dx Ω

for all ϕ ∈ D(Ω). Then L : H01 (Ω) → D(Ω) is linear. Consider the part LH of L in H := L2 (Ω); i.e., LH is the operator on H given by D(LH ) = {u ∈ H01 (Ω) : Lu ∈ L2 (Ω)} LH u = Lu. Then the following holds. Theorem 7.2.3. The operator LH generates a cosine function on L2 (Ω) with phase space H01 (Ω) × L2 (Ω). Proof. Let α > 0 and consider V := H01 (Ω) with the scalar product n (u|v)V = aij (x)Di u(x)Dj v¯(x) dx + α u(x)¯ v (x) dx. Ω i,j=1

Then u V =

3

Ω

(u|u)V is equivalent to the given norm on H01 (Ω). Thus, V is a d

Hilbert space and V → L2 (Ω). We identify L2 (Ω) with a subspace of V . Deﬁne B ∈ L(H, V ) by n (Bu|ϕ) := − bj uDj ϕ dx (ϕ ∈ V = H01 (Ω)) Ω j=1

and C ∈ L(V, H) by Cu :=

n

cj Dj u + du − αu.

j=1

Let A : V → V be the isomorphism given by the Riesz-Fr´echet lemma. It follows from Theorem 7.1.7 that the operator (A + B + C)H given by ' ( D((A + B + C)H ) := u ∈ H01 (Ω) : Au + Bu + Cu ∈ L2 (Ω) , (A + B + C)H u :=

Au + Bu + Cu,

generates a cosine function on L2 (Ω) with associated phase space H01 (Ω) × L2 (Ω). Since D(Ω) is dense in H01 (Ω), the restriction mapping V → D(Ω) is injective;

7.3. NOTES

427

thus, we may identify V with a subspace of D(Ω) . With this identiﬁcation one has LH = (A + B + C)H . Now we deduce from Corollary 3.14.12 well-posedness of the following hyperbolic problem. As before, L : H01 (Ω) → D(Ω) denotes the elliptic operator associated with the coeﬃcients aij , bi , ci , d. Corollary 7.2.4 (Hyperbolic equation). Let f ∈ H01 (Ω) such that Lf ∈ L2 (Ω) and let g ∈ H01 (Ω). Then there exists a unique function u ∈ C 2 (R+ , L2 (Ω)) satisfying a) u(t) ∈ H01 (Ω), Lu(t) ∈ L2 (Ω) b) u (t) = Lu(t)

(t ≥ 0);

(t ≥ 0);

c) u(0) = f, u (0) = g. Proof. This follows from Theorem 7.2.3 and Corollary 3.14.12 as in the proof of Theorem 7.2.2. Since each generator of a cosine function is also the generator of a holomorphic C0 -semigroup (by Theorem 3.14.17), we also obtain well-posedness of the corresponding parabolic equation. The result is an extension of Example 3.7.24. Corollary 7.2.5 (Parabolic equation). Let f ∈ L2 (Ω). Then there exists a unique function u ∈ C ∞ ((0, ∞), L2 (Ω)) ∩ C(R+ , L2 (Ω)) such that a) u(t) ∈ H01 (Ω), Lu(t) ∈ L2 (Ω) b) u (t) = Lu(t)

(t > 0);

(t > 0);

c) u(0) = f . Proof. Denote by T the holomorphic C0 -semigroup generated by LH . Then u(t) = T (t)f is the unique solution of a), b) and c), by Theorem 3.7.19.

7.3

Notes

Section 7.1 merely contains a direct approach to constructing selfadjoint operators by scalar products. We refer to the textbooks [Dav80], [Dav95], [Kat66] and [RS72] for a systematic treatment of quadratic form methods. Elliptic operators generating cosine functions are described in the monographs by Fattorini [Fat83], [Fat85] and Goldstein [Gol85], but the perturbation arguments leading to Theorem 7.1.7 and Corollary 7.2.4 may be new (in the case B = 0; i.e., when the coeﬃcients bj do not vanish). Section 7.2 gives a fairly general well-posedness result on L2 (Ω), and the restriction to Dirichlet boundary conditions has been chosen merely for convenience. However, these results are deﬁnitely restricted to L2 (Ω) and no longer valid on Lp (Ω) (p = 2). This will be made precise in Example 8.4.9. On L1 (Rn ) or C0 (Rn ) the Laplacian does not generate a cosine function. However, the following holds for n = 3.

428

7. THE WAVE EQUATION

Theorem 7.3.1. Let X := L1 (R3 ) or C0 (R3 ), and D(ΔX ) := {f ∈ X : Δf ∈ X}, ΔX f := Δf. Then ΔX generates a sine function Sin on X given by 1 (Sin(t)f )(x) = f (z) dσ(z) tσ2 ∂B(x,t) where σ denotes the surface measure on ∂B(x, t) := {z ∈ R3 : |x − z| = t} and σ2 := 2|B(0, 1)| is the surface area of the 2-dimensional sphere. Thus, (Sin(t)f )(x) is t-times the mean of f over the sphere ∂B(x, t). This can be seen by inspecting the proofs given in [Eva98, Section 2.4]. Generalizing the method of spherical means, a systematic treatment of an “abstract Laplacian” (given as the closure of the sum of n generators of commuting cosine functions) is given by Keyantuo [Key95b].

Chapter 8

Translation Invariant Operators on Lp(Rn) In this chapter we consider diﬀerential operators with constant coeﬃcients, and more generally pseudo-diﬀerential operators on Lp (Rn ). The realization Opp (a) in Lp (Rn ) of such an operator is translation invariant. We assume that the “symbol” a satisﬁes certain smoothness and growth assumptions. In particular, when a is a polynomial, then Opp (a) is a diﬀerential operator. In the following sections we investigate the question under which conditions on the symbol a the operator Opp (a) generates a C0 -semigroup or an integrated semigroup on Lp (Rn ). This t k−1 question is closely related to the problem whether eta or 0 (t−s) esa ds is a k! p n Fourier multiplier for L (R ). In Section 8.2 (see also Appendix E) we consider Fourier multipliers in some detail. Since we are interested in the case p = 1 as well as 1 < p < ∞, we need to include Fourier multipliers on L1 (Rn ). Bernstein’s lemma and a partition of unity argument are our main tools. Proposition 8.2.3 gives a simple criterion for a function to belong to the Fourier algebra F L1 (Rn ) and hence to be a Fourier multiplier on L1 (Rn ). These techniques are essential for our main results. Assuming a suitable growth condition on the symbol a, which is in particular fulﬁlled for elliptic and even hypoelliptic polynomials, we prove in Section 8.3 that the operator Opp (a) associated to a generates a k-times integrated semigroup on Lp (Rn ) for some k ∈ N provided ρ(Opp (a)) = ∅ and the range of a lies in a left half-plane. Observe that the result covers the case of the operator iΔ which has already been considered in Section 3.9 (Theorem 3.9.4 and Corollary 3.9.14). It is interesting to note that the order of integration stated in Theorem 8.3.6 is in fact optimal for homogeneous symbols of the form a(ξ) = i|ξ|m . These results are also closely related to the existence of the boundary group of the Poisson semigroup on Lp (Rn ) (see Corollary 8.3.11) and to Littman’s result on the cosine function generated by the Laplacian on Lp (Rn ) (see Theorem 8.3.12).

W. Arendt et al., Vector-valued Laplace Transforms and Cauchy Problems: Second Edition, Monographs in Mathematics 96, DOI 10.1007/978-3-0348-0087-7_8, © Springer Basel AG 2011

429

430

8. TRANSLATION INVARIANT OPERATORS ON LP (RN )

In the ﬁnal section of this chapter, we consider systems of diﬀerential operators with constant coeﬃcients on Lp -spaces. Brenner’s result (Theorem 8.4.3) states that ﬁrst order symmetric, hyperbolic systems generate C0 -semigroups on Lp (Rn )N for p = 2 if and only if the matrices commute. This means that the solutions of the wave equation, Maxwell’s equation or Dirac’s equation are not governed by a C0 -semigroup on Lp -spaces if p = 2. We prove, however, that the solutions are given by integrated semigroups on these spaces.

8.1

Translation Invariant Operators and C0-semigroups

In this section we consider Cauchy problems u (t) = Au(t) (t ≥ 0), u(0) = u0 , where A is the realization of a pseudo-diﬀerential operator in a function space X of the form X = Lp (Rn ) (1 ≤ p < ∞) or C0 (Rn ), and u0 ∈ X. More precisely, let m m > 0 and ρ ∈ [0, 1]. We deﬁne Sρ,0 to be the set of all functions a ∈ C ∞ (Rn ) n such that for each multi-index α ∈ N0 there exists a constant Cα such that |Dα a(ξ)| ≤ Cα (1 + |ξ|)m−ρ|α|

(ξ ∈ Rn ).

m Obviously, a polynomial of order m belongs to S1,0 . We call a ∈ C ∞ (Rn ) a symbol m if a ∈ Sρ,0 for some m > 0 and some ρ ∈ [0, 1]. For a symbol a we deﬁne the pseudo-diﬀerential operator Op(a) associated to a by Op(a)u(x) := eix·ξ a(ξ)Fu(ξ)dξ (x ∈ Rn , u ∈ S(Rn )), Rn

where x · ξ is the scalar product of x and ξ, and Fu denotes the Fourier transform of u. The operator OpX (a) deﬁned by OpX (a)f

:=

D(OpX (a)) :=

F −1 (aF f ), {f ∈ X : F −1 (aFf ) ∈ X},

(8.1)

is called the realization of Op(a) in X, or the X-realization of Op(a). When X = Lp (Rn ), we may write Opp (a) for OpX (a). Here, F −1 (aFf ) is interpreted in the sense of distributions (see Appendix E) as follows: as usual, we identify f ∈ X with Tf ∈ S(Rn ) given by ϕ, Tf = fϕ (ϕ ∈ S(Rn )). Rn

The Fourier transform is an isomorphism of S(Rn ) (see (E.10)) which implies that Ff ∈ S(Rn ) . Since a and all its derivatives are polynomially bounded, a · F f is a

8.1. TRANSLATION INVARIANT OPERATORS AND C0 -SEMIGROUPS 431 well deﬁned element of S(Rn ) (see (E.3)). Hence, F −1 (aFf ) ∈ S(Rn ) since F is an isomorphism of S(Rn ) . It is not diﬃcult to verify that OpX (a) is a closed operator in X whenever a is a symbol. In addition, OpX (a) is densely deﬁned since S(Rn ) ⊂ D(OpX (a)). Moreover, by (E.12), OpX (a) is a diﬀerential operator of order m with constant coeﬃcients aα ∈ C, i.e., OpX (a)f = aα D α f (f ∈ D(OpX (a))), (8.2) |α|≤m

when a is the polynomial of order m of the form a(ξ) = aα (iξ)α (ξ ∈ Rn ).

(8.3)

|α|≤m

A polynomial a of the form (8.3) is called elliptic if its principal part am , deﬁned by am (ξ) := aα (iξ)α (ξ ∈ Rn ), |α|=m

vanishes only at ξ = 0. We call OpX (a) an elliptic operator on X if a is an elliptic polynomial. Moreover, a polynomial a is called hypoelliptic if Dα a(ξ) →0 a(ξ)

as |ξ| → ∞

whenever |α| = 0. Our ﬁrst lemma in this section shows that for operators OpX (a) under consideration there is a close relationship between the resolvent set of OpX (a) and Fourier multipliers for X. For the deﬁnition of Fourier multipliers and the space MX (Rn ) we refer to Appendix E, but we note that a symbol a is a Fourier multiplier for X if and only if OpX (a) is a bounded operator on X. In order to simplify our notation, we also write AX for OpX (a) if no confusion seems likely. Finally, given a symbol a we set a(Rn ) := {a(ξ) : ξ ∈ Rn }. Lemma 8.1.1. Let X be one of the spaces Lp (Rn ) (1 ≤ p < ∞) or C0 (Rn ). Let a be a symbol and let λ ∈ C. Then λ ∈ ρ(AX ) if and only if (λ − a) is nowhere zero and (λ − a)−1 ∈ MX (Rn ). In particular, a(Rn ) ⊂ σ(AX ). Proof. Assume that (λ − a)−1 ∈ MX (Rn ). For f ∈ X set Trλ f := F −1 ((λ − a)−1 F f ). If f ∈ D(AX ), then Trλ (λ − AX )f = F −1 ((λ − a)−1 (λ − a)F f ) = f . Moreover, if f ∈ X, then Trλ f ∈ D(AX ) since F −1 ((λ − a)(λ − a)−1 Ff ) = f ∈ X. Hence, (λ − AX )Trλ f = F −1 ((λ − a)(λ − a)−1 F f ) = f.

432

8. TRANSLATION INVARIANT OPERATORS ON LP (RN )

We have therefore proved that (λ − AX ) is invertible with inverse operator Trλ . Conversely, let λ ∈ ρ(AX ). If f ∈ D(AX ), then τa f ∈ D(AX ) and AX τa f = τa AX f , where (τa f )(x) := f (x−a). Hence, R(λ, AX )τa = τa R(λ, AX ). Proposition E.1 shows that there exists rλ ∈ MX (Rn ) satisfying R(λ, AX )f = F −1 (rλ F f ) (f ∈ X). Therefore, f = (λ − AX )R(λ, AX )f = F −1 ((λ − a)rλ F f )

(f ∈ X),

which implies by the uniqueness theorem for Fourier transforms that (λ − a)rλ = 1 a.e. Since a is continuous and rλ is bounded, it follows that λ ∈ a(Rn ) and rλ = (λ − a)−1 a.e. Remark 8.1.2. The above proof shows that, given λ ∈ ρ(AX ), we have R(λ, AX )f = F −1 (rλ F f )

(f ∈ X),

where rλ = (λ − a)−1 . Proposition 8.1.3. Let X be one of the spaces Lp (Rn ) (1 ≤ p < ∞) or C0 (Rn ). Let a be a symbol. Then the following assertions are equivalent: (i) eta ∈ MX (Rn ) for all t ≥ 0 and there exist constants M, ω ≥ 0 such that

eta MX (Rn ) ≤ M eωt

(t ≥ 0).

(ii) AX generates a C0 -semigroup on X. Proof. (i) ⇒ (ii): By replacing a(ξ) by a(ξ) − ω, we may assume without loss of generality that ω = 0. It follows from Proposition E.2 and the assumptions that sup et Re a(ξ) = eta M2 (Rn ) ≤ eta MX (Rn ) ≤ M

ξ∈Rn

(t ≥ 0).

(8.4)

∞ Hence, Re a ≤ 0. For λ > 0 and f ∈ X, the integral 0 e−λt F −1 (eta F)f dt converges in X and it is easy to see that it coincides in S(Rn ) with F −1 ((λ − a)−1 )Ff . Hence, (λ − a)−1 ∈ MX (Rn ) and ∞ ∞ M

(λ − a)−1 MX (Rn ) ≤ e−λt eta MX (Rn ) dt ≤ M e−λt dt ≤ . λ 0 0 By Lemma 8.1.1, we conclude that (0, ∞) ⊂ ρ(AX ). For f ∈ X set F −1 (eta Ff ) (t > 0), T (t)f := f (t = 0). By assumption, T (t) ∈ L(X) for all t ≥ 0. In order to prove that the mapping T : R+ → L(X) is strongly continuous, assume, for the time being, that f ∈ D(AX ). Since t

a 0

esa ds = eta − 1

(t ≥ 0)

8.1. TRANSLATION INVARIANT OPERATORS AND C0 -SEMIGROUPS 433 it follows easily that

t

−1 sa

T (t)f − f X = F (e F )A f ds X

0

≤ M t AX f X

(t ≥ 0)

X

for f ∈ D(AX ). Since D(AX ) is dense in X and since T (t) L(X) ≤ M by assumption, it follows that T is strongly continuous. Finally, let f ∈ S(Rn ). By Fubini’s theorem, ∞ ∞ e−λt F −1 (eta F f ) dt = F −1 e−λt eta Ff dt 0

0

= F −1 ((λ − a)−1 F f )

(λ > 0).

Since (λ − a)−1 ∈ MX (Rn ), it follows from Remark 8.1.2 that ∞ R(λ, AX ) = e−λt T (t) dt (λ > 0). 0

Thus, the assertion follows from Theorem 3.1.7. (ii) ⇒ (i): Denote by TX the C0 -semigroup generated by AX . Since TX (t) commutes with translations for all t ≥ 0, it follows from Proposition E.1 that there exists ut ∈ MX (Rn ) such that TX (t)f = F −1 (ut F f )

(f ∈ S(Rn ), t ≥ 0).

By Theorem 3.1.7, we have TX (t) L(X) ≤ M eωt (t ≥ 0) for some M, ω ≥ 0. Let f ∈ S(Rn ), ϕ ∈ Cc∞ (Rn ). For λ suﬃciently large, we have by Remark 8.1.2, Deﬁnition 3.1.6 and Fubini’s theorem that ∞ e−λt ϕ, eta F f dt = ϕ, (λ − a)−1 F f 0

= Fϕ, F −1 (λ − a)−1 F f = F ϕ, R(λ, AX )f ∞ = e−λt Fϕ, F −1 ut F f dt. 0

The uniqueness theorem for Laplace transforms implies that ϕ, eta F f = ϕ, ut F f for all t ≥ 0, ϕ ∈ Cc∞ (Rn ) and f ∈ S(Rn ). This implies that ut = eta a.e. for each t ≥ 0. Thus, eta ∈ MX (Rn ) and

eta MX (Rn ) = ut MX (Rn ) = TX (t) L(X) ≤ M eωt

(t ≥ 0).

Since M2 (Rn ) = L∞ (Rn ) (see Proposition E.2 b)), the following corollary is obvious.

434

8. TRANSLATION INVARIANT OPERATORS ON LP (RN )

Corollary 8.1.4. Let a be a symbol. Then AL2 (Rn ) generates a C0 -semigroup on L2 (Rn ) if and only if there exists ω ∈ R such that Re a(ξ) ≤ ω for all ξ ∈ Rn . A necessary condition for eta to belong to MX (Rn ) for t > 0 is given in the next lemma. Lemma 8.1.5. Let X be one of the spaces Lp (Rn ) (1 ≤ p < ∞) or C0 (Rn ). Consider a polynomial a of order m with principal part am . Suppose that eta ∈ MX (Rn ) for all t ≥ 0 and that eta MX (Rn ) ≤ M eωt (t ≥ 0) for suitable constants M, ω ≥ 0. Then eam ∈ MX (Rn ). Proof. Let a = am + am−1 + · · · + a0 , where each term aj is homogeneous of degree j (j = 0, . . . , m). The change of variables ξ → t−1/m ξ implies by Proposition E.2 e) that ut , deﬁned by −1/m

ut (ξ) := eam (ξ) etam−1 (t

ξ)

. . . eta0 ,

belongs to MX (Rn ) and eta MX (Rn ) = ut MX (Rn ) for all t > 0. By assumption, there exists C > 0 such that ut MX (Rn ) ≤ C for all t ∈ (0, 1). Since limt↓0 ut (ξ) = eam (ξ) for all ξ ∈ Rn , it follows from Proposition E.2 f) that eam ∈ MX (Rn ). When AX is a ﬁrst order diﬀerential operator of the form AX f =

n

aj Dj f + a0 f,

j=1

where aj ∈ C (j = 0, 1, . . . , n), then AX generates a C0 -semigroup given by (T (t)f )(x) = ea0 t f (x + ta), where a = (a1 , . . . , an ), X = Lp (Rn ) (1 ≤ p < ∞) or X = C0 (Rn ). In the following proposition a converse assertion is proved. Proposition 8.1.6. Let X be one of the spaces Lp (Rn ) (1 ≤ p < ∞, p = 2) or C0 (Rn ). Assume that AX is a diﬀerential operator of the form (8.2) on X such that the symbol of the principal part am satisﬁes Re am = 0. Then AX generates a C0 -semigroup on X if and only if the order m of AX is 1. Proof. It follows from Proposition 8.1.3 and Lemma 8.1.5 that eam ∈ Mp (Rn ) for X = Lp (Rn ) and eam ∈ M∞ (Rn ) for X = C0 (Rn ). Since Re am = 0 and p = 2 by assumption, it follows from Theorem E.4 a) that m = 1. Conversely, if m = 1, then eta MX (Rn ) = eta0 for all t ≥ 0 and the assertion follows from Proposition 8.1.3. Note that the special case of the symbol a(ξ) = −i|ξ|2 was already considered in Theorem 3.9.4.

8.2. FOURIER MULTIPLIERS

8.2

435

Fourier Multipliers

In this section on Fourier multipliers we give several suﬃcient conditions for a function to be a Fourier multiplier for Lp (Rn ). The results presented in the following are the basis of our subsequent analysis of Cauchy problems in Lp (Rn ) corresponding to operators of the form (8.1). We start with a classical result due to Bernstein. Recall from Appendix E that for j ∈ N0 the space H j (Rn ) is deﬁned to be the space of all functions f ∈ L2 (Rn ) whose distributional derivatives Dα f belong to L2 (Rn ) for |α| ≤ j. Plancherel’s theorem implies that f ∈ H j (Rn ) if and only if ξ → ξ α F f (ξ) belong to L2 (Rn ) for all |α| ≤ j. It is not hard to verify that there exist constants C1 , C2 > 0 such that C1 (1 + |ξ|2 )j ≤ |ξ α |2 ≤ C2 (1 + |ξ|2 )j (ξ ∈ Rn ), |α|≤j

from which it follows that f ∈ H j (Rn ) if and only if ξ → (1 + |ξ|2 )j/2 F f (ξ) ! "1/2 α 2 is equivalent to belongs to L2 (Rn ), and that the norm |α|≤j D f L2 (Rn )

(1 + | · |2 )j/2 F f (·) L2 (Rn ) . Thus, H j (Rn ) = f ∈ L2 (Rn ) : F −1 ((1 + | · |2 )j/2 F f (·)) ∈ L2 (Rn ) . Lemma 8.2.1 (Bernstein). Let u ∈ H j (Rn ) for some j > n2 . Then Fu ∈ L1 (Rn ) and there exists a constant C (depending only on n and j) such that n/2j 1−(n/2j) α

F u L1 (Rn ) ≤ C u L2 (Rn )

D u L2 (Rn ) (u ∈ H j (Rn )). |α|=j

Proof. For R > 0, we obtain by the Cauchy-Schwarz inequality and Plancherel’s theorem,

Fu L1 (Rn ) = 1 · |F u(ξ)| dξ + |ξ|−j |ξ|j |F u(ξ)| dξ |ξ|≤R

≤

|ξ|≥R

1/2

u L2 (Rn )

1 dξ |ξ|≤R

+

|ξ|≥R

|ξ|

−2j

1/2

1/2 2j

dξ |ξ|≥R

≤ CRn/2 u L2 (Rn ) + CR(n/2)−j

2

|ξ| |Fu(ξ)| dξ

Dα u L2 (Rn )

|α|=j

for some constant C (depending on n and j). The assertion follows by choosing 1/j −1/j α R := u L2 (Rn ) . |α|=j D u L2 (Rn ) Our next result on a “partition of unity” will be very useful in the sequel.

436

8. TRANSLATION INVARIANT OPERATORS ON LP (RN )

∞ n n Lemma 8.2.2. ( There exists ϕ ∈ Cc (R ) satisfying ϕ ≥ 0, supp ϕ ⊂ {ξ ∈ R : 1 < |ξ| < 2 , and 2 ϕ(2−k ξ) = 1 (ξ = 0). k∈Z

Proof. Choose f ∈ Cc∞ (Rn ) such that supp f ⊂ {ξ ∈ Rn : 12 < |ξ| < 2}, f ≥ 0 √ and f (ξ) > 0 if √12 ≤ |ξ| ≤ 2. For ξ ∈ Rn , set f0 (ξ) := k∈Z f (2−k ξ). Then f0 ∈ C ∞ (Rn \{0}), f0 (ξ) > 0 for all ξ ∈ Rn \ {0} and f0 (2−k ξ) = f0 (ξ) for all k ∈ Z and all ξ = 0. Hence, the function ϕ deﬁned by ⎧ ⎨0 (ξ = 0), ϕ(ξ) := f (ξ) ⎩ (ξ = 0), f0 (ξ) satisﬁes the desired assertions. A very eﬃcient suﬃcient condition for a function to belong to Mp (Rn ) is given by Mikhlin’s theorem (see Theorem E.3). In fact, let j := min{k ∈ N : k > n } and deﬁne MM as 2 ' ( MM := m : Rn → C : m ∈ C j (Rn \{0}), |m|M < ∞ , where the norm | · |M is deﬁned as |m|M := max

sup

|α|≤j ξ∈Rn \{0}

|ξ||α| |Dα m(ξ)|.

Mikhlin’s theorem then states that MM → MLp (Rn ) provided 1 < p < ∞ (see Theorem E.3). Note that Mikhlin’s theorem does not hold for p = 1. In the following we give a simple criterion for a function to belong to ML1 (Rn ) . We set FL1 (Rn ) := {Fg : g ∈ L1 (Rn )} = {f ∈ C0 (Rn ) : Ff ∈ L1 (Rn )}. This space is a Banach space for the norm inherited from L1 (Rn ); i.e.,

f FL1 (Rn ) := F −1 f L1 (Rn ) = (2π)−n F f L1 (Rn ) . The convolution theorem for Fourier transforms shows that F L1 (Rn ) ⊂ M1 (Rn ) isometrically; i.e., f FL1 (Rn ) = f M1 (Rn ) for f ∈ FL1 (Rn ). Moreover, by Proposition E.2, FL1 (Rn ) → Mp (Rn ) for 1 ≤ p ≤ ∞. Let ε > 0 and put j := min{k ∈ N : k > n2 }. Deﬁne Mε := {m ∈ C j (Rn ) : |m|Mε < ∞}, where

|m|Mε := max sup |ξ||α|+ε |D α m(ξ)|. |α|≤j ξ∈Rn

Then (Mε , | · |Mε ) is a Banach space and the following holds true.

8.2. FOURIER MULTIPLIERS

437

Proposition 8.2.3. Let ε > 0. Then Mε → F L1 (Rn ). Proof. Let m ∈ Mε . Choose ψ ∈ Cc∞ (Rn ) such that ψ(ξ) = 1 whenever |ξ| ≤ 2 and write m = ψm + (1 − ψ)m. It follows from Bernstein’s Lemma 8.2.1 that F (ψm) ∈ L1 (Rn ). We may therefore assume that m(ξ) = 0 whenever |ξ| ≤ 2. Let ϕ be a function as in Lemma 8.2.2, and for k ∈ Z set

Then m =

mk := mϕk ,

∞

ϕk (ξ) := ϕ(2−k ξ)

(ξ ∈ Rn ).

mk . By Leibniz’s rule, we obtain for α with |α| ≤ j, α α α−β −k|β| β −k |D mk (ξ)| = D m(ξ)2 (D ψ)(2 ξ) β β≤α α ≤ C |ξ|−(|α−β|+ε) 2−k|β| D β ψ ∞ β β≤α α 2−k|β| ≤ C β 2k(|α−β|+ε) β≤α k=1

≤ C2−k(|α|+ε) , where C denotes a constant (which may diﬀer from line to line). Here, we used the fact that 2k−1 < |ξ| for ξ ∈ supp mk . The L2 -norm of Dα mk may hence be estimated as follows: 1/2 α −k(|α|+ε)

D mk L2 (Rn ) ≤ C2 1 dξ 2k−1 <|ξ|<2k+1

! "1/2 ≤ C2−k(|α|+ε) 2kn

(|α| ≤ j).

This implies that

Dα mk L2 (Rn ) ≤ C2−k(|α|+ε−(n/2))

(k ≥ 1, |α| ≤ j).

Hence, mk ∈ H (R ) and it follows from Lemma 8.2.1 that 1−(n/2j) n/2j

Fmk L1 (Rn ) ≤ C 2−k(ε−(n/2)) 2−k(j+ε−(n/2)) j

n

= C2−kε . ∞ Therefore, Fm L1 (Rn ) ≤ k=1 Fmk L1 (Rn ) < ∞ and it follows that m ∈ FL1 (Rn ). The closed graph theorem implies that the embedding Mε → F L1 (Rn ) is continuous (in fact, the constants above are proportional to |m|Mε ). Lemma 8.2.4. Let a : Rn → C be continuous such that a ∈ C j (Rn \{0}), where j = min{k ∈ N : k > n2 }. Assume that there exist constants m > 0 and Cα > 0 such that |Dα a(ξ)| ≤ Cα |ξ|m−|α|

(0 < |α| ≤ j, |ξ| ≤ 1, ξ = 0).

438

8. TRANSLATION INVARIANT OPERATORS ON LP (RN )

Let ψ ∈ Cc∞ (Rn ) such that ψ(ξ) = 1 for all ξ ∈ Rn with |ξ| ≤ 1. Then aψ ∈ FL1 (Rn ). Proof. Replacing a(ξ) by a(ξ) − a(0), we may assume that a(0) = 0. Then 1 |a(ξ)| = ξ · ∇a(tξ) dt ≤ C0 |ξ|m 0

for some constant C0 . For k ∈ Z− , set vk (ξ) := a(ξ)ψ(ξ)ϕ(2−k ξ), where ϕ is a function as in Lemma 8.2.2. The L2 -norm of Dα vk may be estimated exactly as in the proof of Proposition 8.2.3, giving

Dα vk L2 (Rn ) ≤ C2k(m−|α|+(n/2))

(k ≤ −1, |α| ≤ j).

It follows from Lemma 8.2.1 that

Fvk L1 (Rn ) ≤ C2kmn/2j . Thus, −1

Fvk L1 (Rn ) ≤ C

k=−∞

∞

2−kmn/2j < ∞,

k=1

−1

1

and it follows that k=−∞ vk ∈ FL (R ). Since aψ − Lemma 8.2.1 gives the result. n

−1

k=−∞ vk

∈ H j (Rn ),

Examples 8.2.5. The assumptions of Lemma 8.2.4 are in particular satisﬁed for the functions a : Rn → C deﬁned by a) a(ξ) := c|ξ|m

(m > 0, c ∈ C);

m

b) a(ξ) := ei|ξ| (m > 0); 1 (1 − s)k−1 is|ξ|m c) a(ξ) := e ds (m > 0, k ∈ N). (k − 1)! 0

Proposition 8.2.6. Let 1 ≤ p ≤ ∞ and let m ∈ C j (Rn ) for some j > n2 . Suppose that m(ξ) = 0 whenever |ξ| ≤ 1. Let ε > 0 and ρ ∈ (−∞, 1]. Assume that there exists a constant M ≥ 1 such that sup 0<|α|≤j

1/|α| sup |D m(ξ)| |ξ| α

ε+ρ|α|

|ξ|≥1

sup |m(ξ) ξ|ε

|ξ|≥1

≤ M, ≤ M.

8.2. FOURIER MULTIPLIERS

439

If ε > n 12 − p1 (1 − ρ), then m ∈ Mp (Rn ) and there exists a constant C (depending on n, p, ρ and ε but otherwise independent of m and M ) such that

m Mp (Rn ) ≤ CM 1+n| 2 − p | . 1

1

Proof. By Proposition E.2 c), we may assume without loss of generality that 1 ≤ p ≤ 2. Let ϕ be a function as in Lemma 8.2.2. For k ∈ Z, put mk := mϕk , where ϕk (ξ) = ϕ(2−k ξ) for ξ ∈ Rn . We claim that

m Mp (Rn ) ≤

∞

mk Mp (Rn ) < ∞.

k=0

Observe that the ﬁrst inequality follows from the assumption that m(ξ) = 0 for |ξ| ≤ 1. In order to estimate mk Mp (Rn ) , note that |ξ| > 2k−1 for ξ ∈ supp mk . By Leibniz’s rule, we have α α α−β −k|β| β −k |D mk (ξ)| = D m(ξ)2 (D ϕ)(2 ξ) β β≤α C0 M 2−kε (|α| = 0), ≤ |α| k(−ε−ρ|α|) Cα M 2 (|α| = 0), for suitable constants C0 , Cα > 0. Consequently, there exist constants Cαn such that C0n M 2−kε 2kn/2 (|α| = 0),

Dα mk L2 (Rn ) ≤ |α| k(−ε−ρ|α|) kn/2 Cαn M 2 2 (|α| = 0). Choosing now j >

mk M1 (Rn )

n , 2

we conclude by Lemma 8.2.1 that

= mk FL1 (Rn ) 1−(n/2j) n/2j ≤ C M 2−kε 2kn/2 M j 2k(−ε−ρj) 2kn/2

≤ CM 1−(n/2j) M n/2 2k(−ε+n(1−ρ)/2) , ! " for a suitable constant C > 0. Setting θ := 2 1 − 1p for p ∈ (1, 2), it follows from Proposition E.2 d) that

mk Mp (Rn )

Thus,

∞ k=0

≤

θ

mk 1−θ M1 (Rn ) mk M2 (Rn )

≤

CM 1+n| 2 − p | 2k(−ε+(1−ρ)n| 2 − p |) . 1

1

1

1

mk Mp (Rn ) < ∞ and the proof is complete.

m For a symbol a ∈ Sρ,0 and r > 0, we consider the following growth hypothesis:

440

8. TRANSLATION INVARIANT OPERATORS ON LP (RN ) (Hr ): There exist constants C, L > 0 such that |a(ξ)| ≥ C|ξ|r for all ξ ∈ Rn with |ξ| ≥ L. m It is clear that if a ∈ Sρ,0 satisﬁes (Hr ) then r ≤ m.

Remark 8.2.7. We note that by the Seidenberg-Tarski theorem (see [H¨ or83, Theorem 11.1.3]), Hypothesis (Hr ) is in particular satisﬁed for all polynomials a satisfying |a(ξ)| → ∞ as |ξ| → ∞. Hence, assumption (Hr ) holds for hypoelliptic polynomials. If a is an elliptic polynomial of order m, then (Hr ) is satisﬁed with r = m. Lemma 8.2.8. Let 1 ≤ p ≤ ∞, N ∈ N, m ∈ (0, ∞), ρ ∈ [0, 1] and ! r > 0. Suppose " m that a ∈ Sρ,0 satisﬁes (Hr ) and that 0 ∈ a(Rn ). If N > n 12 − p1 m−ρ−r+1 , then r −N n a ∈ Mp (R ). Proof. Let ψ ∈ Cc∞ (Rn ) such that 1 ψ(ξ) := 0

(|ξ| ≤ max (L, 1)), (|ξ| ≥ L + 1),

where L is the constant arising in Hypothesis (Hr ). Then, writing a−N = ψa−N + (1−ψ)a−N , we conclude by Lemma 8.2.1 that it suﬃces to prove that (1−ψ)a−N ∈ Mp (Rn ). Now, Dα ((1 − ψ)a−N )(ξ) = Dα (a−N )(ξ) for |ξ| ≥ L + 1. Using the m assumption that a ∈ Sρ,0 and a satisﬁes (Hr ), and noting that r ≤ m, one sees that |Dα (a−N )(ξ)| ≤ Cα |ξ|−rN+(m−r−ρ)|α| (|ξ| ≥ L + 1), for suitable constants Cα . Hence, the assertion follows from Proposition 8.2.6. Lemma 8.2.9. Let 1 ≤ p ≤ ∞, N ∈ N, m ∈ (0, ∞), ρ ∈ [0, 1], r > 0 and let m a ∈ Sρ,0 . Assume that supξ∈Rn Re a(ξ) ≤ −1 and that Hypothesis (Hr ) is satisﬁed. ! " If N > n 12 − p1 1+m−ρ , then eta a−N ∈ Mp (Rn ) and there exists a constant C r (depending on N, n, ρ, p, m and r but otherwise independent of a) such that

eta a−N Mp (Rn ) ≤ C(1 + t)n| 2 − p | 1

1

(t ≥ 0).

Proof. By Proposition E.2 c), we may restrict ourselves to the case 1 ≤ p ≤ 2. Let ψ ∈ Cc∞ (Rn ) such that 0 ≤ ψ ≤ 1 and ψ(ξ) :=

1 0

(|ξ| ≤ L1 ), (|ξ| ≥ L1 + 1),

8.3. LP -SPECTRA AND INTEGRATED SEMIGROUPS

441

where L1 := max(L, C −1/r , 1) and C, L are the constants appearing in (Hr ). For t ≥ 0, we set ut := eta a−N . By Lemma 8.2.1, we conclude that ψut ∈ M1 (Rn ) and that

ψut M1 (Rn ) ≤ Cn (1 + t)n/2 (t ≥ 0), for some constant Cn . Since

ψut M2 (Rn ) = ψut L∞ (Rn ) ≤ 1 for all t ≥ 0, it follows from Proposition E.2 d) that

ψut Mp (Rn ) ≤ Cn (1 + t)n| 2 − p | 1

1

(t ≥ 0),

for a suitable constant Cn . Writing ut = ψut + (1 − ψ)ut , we conclude that it remains to prove the assertion for (1 − ψ)ut instead of ut . Now, by Leibniz’s rule, D α ut =

β+γ=α

α! β ta γ −N D (e )D (a ) β!γ!

(t ≥ 0).

m Since a ∈ Sρ,0 , we have

|(Dβ eta )(ξ)| ≤ Cβ (1 + t)|β| |ξ||β|(m−ρ)

(|ξ| ≥ L).

As in the proof of Lemma 8.2.8, |(Dγ a−N )(ξ)| ≤ Cγ |ξ|−rN+|γ|(m−r−ρ)

(|ξ| ≥ L),

and it follows that there exists a constant C > 1 such that 1/|α| sup sup |Dα [(1 − ψ)(ξ)ut (ξ)] ||ξ|rN+|α|(ρ−m) ≤ C(1 + t)

(t > 0).

0<|α|≤j |ξ|≥1

Since L ≥ C −1/r , we see that there exists a constant C > 1 such that sup ((1 − ψ)(ξ) ut )(ξ)|ξ|rN ≤ C

|ξ|≥1

for t ≥ 0. Hence, the assertion follows from Proposition 8.2.6.

8.3

Lp -spectra and Integrated Semigroups

m For a symbol a ∈ Sρ,0 and r > 0, consider again the Hypothesis (Hr ) introduced in the previous Section 8.2:

(Hr ): There exist constants C, L > 0 such that |a(ξ)| ≥ C|ξ|r for all ξ ∈ Rn with |ξ| ≥ L.

442

8. TRANSLATION INVARIANT OPERATORS ON LP (RN )

In order to obtain a precise description of σ(Opp (a)), Lemma 8.1.1 shows that we need to decide whether or not the function (λ − a)−1 is an Lp -multiplier. In general, this is a diﬃcult matter. However, if the symbol a satisﬁes Hypothesis (Hr ), the situation is much simpler. Indeed, we have the following result. Recall that a(Rn ) was deﬁned as a(Rn ) = {a(ξ) : ξ ∈ Rn }. Proposition 8.3.1. Let 1 ≤ p < ∞, m ∈ (0, ∞), ρ ∈ [0, 1] and r > 0. Suppose that m a ∈ Sρ,0 satisﬁes (Hr ). If ρ(Opp (a)) = ∅, then σ(Opp (a)) = σ(Op2 (a)) = a(Rn ). The following lemma will be useful in the proof of Proposition 8.3.1. m Lemma 8.3.2. Let 1 ≤ p < ∞, m ∈ (0, ∞) and ρ ∈ [0, 1]. Suppose that a ∈ Sρ,0 k and ρ(Opp (a)) = ∅. Let q be a polynomial of order k of the form q(t) = ck t + . . . + c0 (t ∈ R), with coeﬃcients c0 , . . . , ck ∈ C. Then

Opp (q(a)) = q(Opp (a)). Proof. It is clear that Opp (q(a)) is an extension of q(Opp (a)). Moreover, we have D(q(Opp (a))) = D(Opp (a)k ). We are claiming that D(Opp (q(a))) = D(Opp (a)k ), and we shall prove this by induction on k. For k = 1, this is trivial. Let μ ∈ ρ(Opp (a)). Then, there exist d0 ∈ C, dk ∈ C\{0} and polynomials q1 , q2 of degree k − 1 such that q(t) = (μ − t)q1 (t) + d0 = dk tk + q2 (t)

(t ∈ R).

For f ∈ D(Opp (q(a))) ⊂ Lp (Rn ), we have F −1 ((μ − a)q1 (a)F f ) + d0 f ∈ Lp (Rn ). Hence, F −1 ((μ − a)q1 (a)Ff ) ∈ Lp (Rn ). Since μ ∈ ρ(Opp (a)), we have (μ − a)−1 ∈ Mp (Rn ) by Lemma 8.1.1. Thus, F −1 (q1 (a)F f ) ∈ Lp (Rn ). Therefore, we have f ∈ D(Opp (q1 (a))) = D(Opp (a)k−1 ) = D(Opp (q2 (a))) by the induction hypothesis. Moreover, dk F −1 (ak F f ) = F −1 (q(a)F f ) − F −1 (q2 (a)Ff ) ∈ Lp (Rn ). Thus, Opp (a)k−1 f ∈ D(Opp (a)) and f ∈ D(Opp (a)k ) as required. Proof of Proposition 8.3.1. Note ﬁrst that Lemma 8.1.1 together with the fact that M2 (Rn ) = L∞ (Rn ) implies that σ(Op2 (a)) coincides with a(Rn ). Since a(Rn ) ⊂ σ(Opp (a)) by Lemma 8.1.1, we only need to prove that σ(Opp (a)) ⊂ a(Rn ). Choose λ ∈ C\a(Rn ). The assumption (Hr ) and Lemma 8.2.8 imply that (λ − a)−N ∈ Mp (Rn ) if N is suﬃciently large. Therefore, by Lemma 8.1.1, 0 ∈ ρ(Opp ((λ−a)N )). It follows from Lemma 8.3.2 that ! "N ! "N Opp ((λ − a)N ) = Opp (λ − a) = λ − Opp (a) . (8.5) This implies that (λ−Opp (a))N is invertible. It follows that λ−Opp (a) is invertible with inverse (λ − Opp (a))N −1 ((λ − Opp (a))N )−1 , which is bounded by the closed graph theorem. A quantitative version of Proposition 8.3.1 is given in the following theorem.

8.3. LP -SPECTRA AND INTEGRATED SEMIGROUPS

443

Theorem 8.3.3. Let 1 ≤ p < ∞, m ∈ (0, ∞), ρ ∈ [0, 1] and r > 0. Suppose that m a ∈ Sρ,0 satisﬁes (Hr ). Then the following assertions hold true: ! " a) If n 12 − 1p m−ρ−r+1 < 1, then σ(Opp (a)) = σ(Op2 (a)). r ! " b) If ρ = 1, then the bound given in a) is optimal; i.e., if n 12 − 1p 1−ρ > 1, m m there exists a ∈ Sρ,0 , satisfying (Hr ) with r = m, such that σ(Opp (a)) = σ(Op2 (a)). Proof. The assertion a) follows by combining Proposition 8.3.1 with Lemma 8.2.8 n(1−ρ) and Lemma 8.1.1. In order to prove assertion b), let m ∈ 0, 2 and let a : Rn → C be a C ∞ -function such that 1−ρ |ξ|m ei|ξ| (|ξ| ≥ 2), a(ξ) := 1 (|ξ| ≤ 1), m and |a(ξ)| ≥ 1 for all ξ ∈ Rn . Then a ∈ Sρ,0 and (Hr ) is satisﬁed with r = m. It 1 1 m −1 n follows from Theorem E.4 b) that a ∈ Mp (R ) only if n 2 − p ≤ 1−ρ . Therefore, 1 1 1−ρ n by Lemma 8.1.1, 0 ∈ σ(Opp (a)) if n 2 − p m > 1. Since 0 ∈ a(R ) = σ(Op2 (a)), the assertion follows.

Observing that for elliptic polynomials of degree m we have ρ = 1 and (Hr ) is satisﬁed with r = m, we immediately have the following corollary. Corollary 8.3.4. Let 1 ≤ p < ∞ and let a be an elliptic polynomial. Then σ(Opp (a)) = a(Rn ). Remark 8.3.5. It is worthwhile noticing that assertion a) of Theorem 8.3.3 is no longer true if Hypothesis (Hr ) is not satisﬁed. In fact, consider the symbol a given by a(ξ) := −i(ξ1 + ξ22 + ξ32 − i) (ξ ∈ R3 ). Then σ(Op2 (a)) = {z ∈ C : Re z = −1}, but, by Theorem E.4 c)(i), we have a−1 ∈ Mp (Rn ) if p = 2. Hence, by Lemma 8.1.1, 0 ∈ σ(Opp (a)) whenever p = 2. We now consider the question whether operators associated to symbols a ∈ m Sρ,0 satisfying Hypothesis (Hr ) are generators of integrated semigroups on Lp spaces. To this end, let Np be the smallest integer such that 1 1 1 + m − ρ Np > n − (1 ≤ p < ∞). 2 p r We then have the following result (see Corollary 3.9.14 for the special case where a(ξ) = −i|ξ|2 ).

444

8. TRANSLATION INVARIANT OPERATORS ON LP (RN )

Theorem 8.3.6. Let 1 ≤ p < ∞, m ∈ (0, ∞), ρ ∈ [0, 1] and r > 0. Suppose that m a ∈ Sρ,0 satisﬁes (Hr ). Then the following assertions are equivalent: (i) ρ(Opp (a)) = ∅ and supξ∈Rn Re a(ξ) ≤ ω for some ω ∈ R. (ii) The operator Opp (a) generates an Np -times integrated semigroup on Lp (Rn ). (iii) σ(Opp (a)) ⊂ {z ∈ C : Re z ≤ ω} for some ω ∈ R. Proof. (i) ⇒ (ii): By rescaling we may assume that ω = −1 (see Proposition 3.2.6). It follows from Proposition 8.3.1 that 0 ∈ ρ(Opp (a)). For t ≥ 0 and k ∈ N deﬁne the function ukt : Rn → C by ukt

(t − s)k−1 sa e ds. (k − 1)!

t

:= 0

(8.6)

Integrating by parts we obtain eta 1 tk−j = k − . a (k − j)! aj j=1 k

ukt

(8.7)

We conclude from Lemma 8.1.1 and from the fact that Mp (Rn ) is a Banach algebra that there exists a constant C such that

k

1 tk−j

(k − j)! aj j=1

Mp

(Rn )

≤ C(1 + t)k−1

(t ≥ 0).

(8.8)

By assumption, the symbol a satisﬁes (Hr ). It thus follows from Lemma 8.2.9 that

ta

e 1 1

≤ C(1 + t)n| 2 − p | (t ≥ 0), (8.9)

aNp Mp (Rn ) Np

for some constant C. Combining (8.8) with (8.9) it follows that ut for all t ≥ 0 and that N

ut p Mp (Rn ) ≤ C(1 + t)α

(t ≥ 0),

for some constants C, α. For f ∈ Lp (Rn ) and t ≥ 0 set S(t)f := F −1 (ut p Ff ). N

Since

a 0

t

Np

p uN s ds = ut

−

tNp (Np )!

(t ≥ 0),

∈ Mp (Rn )

8.3. LP -SPECTRA AND INTEGRATED SEMIGROUPS

445

it follows that for f ∈ S(Rn ) and r > t ≥ 0 we have

S(t)f − S(r)f Lp (Rn ) ≤ Opp (a)f Lp (Rn )

t

r

Np p

uN − tNp ) f Lp (Rn ) . s Mp (Rn ) ds + (r

Thus, S(·)f : R+ → Lp (Rn ) is continuous. Since S(Rn ) is dense in Lp (Rn ) and S is locally bounded, it follows that S : R+ → L(Lp (Rn )) is strongly continuous. It remains to show that the generator of S is Opp (a). To this end, ﬁx λ > 0 and let f ∈ S(Rn ). It follows from Fubini’s theorem that ∞ ∞ N −λt e S(t)f dt = e−λt F −1 (ut p Ff ) dt 0 0 ∞ −1 −λt Np = F e ut dt F f 0 1 1 −1 −1 = F (λ − a) F f = Np R(λ, Opp (a))f. λNp λ Since S(Rn ) is dense in Lp (Rn ), the assertion follows by Deﬁnition 3.2.1. (ii) ⇒ (iii): This follows from the deﬁnition of an integrated semigroup. (iii) ⇒ (i): This is a consequence of Proposition 8.3.1. The following is immediate from Theorem 8.3.3 and Theorem 8.3.6. Corollary 8.3.7. Let 1 ≤ p < ∞, m ∈ (0, ∞), ρ ∈ [0, 1] and r > 0. Assume that m a ∈ Sρ,0 satisﬁes (Hr ) and supξ∈Rn Re a(ξ) ≤ ω for some ω ∈ R. For N ∈ N, there exists a constant δN > 0 such that Opp (a) generates an N -times integrated semigroup on Lp (Rn ) provided 12 − 1p < δN . Example 8.3.8. The example of the symbol a given by a(ξ) := (−i)(ξ1 − ξ22 − ξ32 − i)(ξ1 + ξ22 + ξ32 + i)

(ξ ∈ R3 )

shows that Opp (a) generates an integrated semigroup on Lp (R3 ) only for certain values of p. Indeed, we verify that supξ Re a(ξ) = 0 and (Hr ) is satisﬁed with 1 1 r = 1. Hence, by Theorem 8.3.3, we see that ρ(Opp (a)) = ∅ provided 2 − p < 1 Opp (a) generates a once integrated semigroup on Lp (R 3 ) provided 91. Therefore, 1 1 − < . However, by Theorem E.4 c)(ii), σ(Opp (a)) = a(Rn ) if 1 − 1 > 3 . 2 p 12 2 p 8 Proposition 8.3.1 implies that σ(Opp (a)) = C if 12 − 1p > 38 . Thus, Opp (a) does not generate an N -times integrated semigroup on Lp (R3 ) for any N for those values of p. m We consider now the case where a is no longer a symbol belonging to Sρ,0 but a is a homogeneous function of the form

a(ξ) = i|ξ|m or a(ξ) = −i|ξ|m

(ξ ∈ Rn )

(8.10)

446

8. TRANSLATION INVARIANT OPERATORS ON LP (RN )

for some m > 0. In that case, the realization of the pseudo-diﬀerential operator associated to a in function spaces X of the form Lp (Rn ) (1 ≤ p < ∞) or C0 (Rn ) is deﬁned as follows. For f ∈ X and a of the form (8.10), deﬁne aF f ∈ S(Rn ) as the mapping ϕ →

Rn

f F(aϕ) dx (ϕ ∈ S(Rn )).

(8.11)

Notice that F (aϕ) ∈ L2 (Rn ) ∩ L∞ (Rn ) by Plancherel’s theorem and the RiemannLebesgue lemma. We even have F (aϕ) ∈ L1 (Rn ) by Lemma 8.2.4 and Example 8.2.5. Together with the inequality

f q ≤ f θ1 f 1−θ ∞

(1 < q < ∞, θ := 1/q),

this implies that F(aϕ) ∈ Lq (Rn ) for 1 ≤ q ≤ ∞. It follows that the mapping given in (8.11) is well deﬁned. It is also not diﬃcult to verify that aF f ∈ S(Rn ) . We thus deﬁne, for a of the form (8.10), := F −1 (aF f ),

OpX (a)

D(OpX (a)) :=

{f ∈ X : F −1 (aFf ) ∈ X}.

(8.12)

We note that the assertions of Lemma 8.1.1 and Remark 8.1.2 remain true if the symbol a is replaced by a homogeneous function a of the form (8.10). By the proof of Theorem 8.3.6, the operator Opp (a) := OpLp (Rn ) (a), deﬁned as in (8.12), generates a k-times integrated semigroup on Lp (Rn ) (1 ≤ p < ∞) for some k > 0 if and only if ukt ∈ Mp (Rn ), where ukt is deﬁned by ukt (ξ) :=

t

(t − s)k−1 sa(ξ) e ds (k − 1!)

1

(1 − s)k−1 sta(ξ) e ds (k − 1)!

0

(ξ ∈ Rn ).

Notice ﬁrst that ukt (ξ) = tk

0

(ξ ∈ Rn ).

The change of variables ξ → t−1/m ξ implies by Proposition E.2 e) that ukt ∈ Mp (Rn ) if and only if uk1 ∈ Mp (Rn ), and then

ukt Mp (Rn ) = tk uk1 Mp (Rn ) . In order to determine whether uk1 ∈ Mp (Rn ), let ψ ∈ Cc∞ (Rn ) such that ψ(ξ) = 1 for |ξ| ≤ 1. It follows from Lemma 8.2.4 and from Example 8.2.5 that ψuk1 ∈ FL1 (Rn ) ⊂ Mp (Rn ) (1 ≤ p < ∞). Furthermore, (1 − ψ)uk1 = (1 − ψ)

k 1 ea 1 − (1 − ψ) . k a (k − j)! aj j=1

8.3. LP -SPECTRA AND INTEGRATED SEMIGROUPS

447

k Cα 1 1 n Now, since Dα ((1 − ψ) j=1 (k−j)! aj )(ξ) ≤ |ξ|m+|α| for |α| ≤ l with l > 2 , it follows from Proposition 8.2.3 that the second term on the right-hand side above k n belongs to Mp (Rn ). By Theorem E.4 b), we conclude 1 that u1 ∈ Mp (R ) when 1 1 n2 ) in the case m = 1; and uk1 ∈ Mp (Rn ) when 1 n−1 2 ) for the case m = 1. Consider now the case k = 0. It follows from the proof of Proposition 8.1.3 that Opp (a) := OpLp (Rn ) (a), deﬁned as in (8.12), generates a C0 -semigroup on m Lp (Rn ) (1 ≤ p < ∞) if and only if u0t : ξ → eit|ξ| ∈ Mp (Rn ) and u0t Mp (Rn ) is exponentially bounded in t. The change of variables ξ → t−1/m ξ implies by Proposition E.2 e) that u0t Mp (Rn ) = u01 Mp (Rn ) for all t > 0. Let now ψ ∈ Cc∞ (Rn ) such that ψ(ξ) = 1 for ξ ∈ Rn with |ξ| ≤ 1, and write u01 = u01 ψ+u01 (1−ψ). Then u01 ψ ∈ Mp (Rn ) by Lemma 8.2.4 and Example 8.2.5. It follows from Theorem E.4 u01 ∈ Mp (Rn ) when 1 < p < ∞ (respectively, p = 1) if and only if 1 b) 1that n 2 − p ≤ 0 (respectively, n2 < 0) in the case m = 1; and u01 ∈ Mp (Rn ) when 1 0. Deﬁne a : Rn → C by a(ξ) := i|ξ|m . a) If m = 1 and 1 n2 ). b) If m = 1 and 1 n−1 ). 2 Remark 8.3.10. a) We note that the assertions of Theorem 8.3.9 remain true for the homogeneous function a given by a(ξ) := −i|ξ|m . b) Let OpC0 (a) be the operator on C0 (Rn ) deﬁned as in (8.12) with Lp (Rn ) replaced by C0 (Rn ). Then the assertions of Theorem 8.3.9 remain true if Opp (a) in Theorem 8.3.9 is replaced by OpC0 (a) and 1/p by 0. Theorem 8.3.9 has interesting consequences for boundary values of holomorphic semigroups as discussed in Section 3.9. In fact, consider the Poisson semigroup T deﬁned as in Example 3.7.9. There we proved that T is a bounded holomorphic C0 -semigroup of angle π/2 on Lp (Rn ) for 1 ≤ p < ∞. Its generator is given by Ap f = F −1 (−| · |Ff ) for f ∈ D(Ap ) = {f ∈ Lp (Rn ) : F −1 (−| · |Ff ) ∈ Lp (Rn )}. Theorem 8.3.9 and Remark 8.3.10 imply the following corollary. Corollary 8.3.11. a) Let 1 < p < ∞ and let Tp be the Poisson semigroup on p n Lp (Rn ). Then Tp admits a boundary 1 group on L (R ) in the sense of Propo1 sition 3.9.1 if and only if (n − 1) 2 − p ≤ 0, i.e., if and only if n = 1 or p = 2.

448

8. TRANSLATION INVARIANT OPERATORS ON LP (RN )

b) Let T1 be the Poisson semigroup on L1 (Rn ). Then T1 does not admit a boundary group on L1 (Rn ) in the sense of Proposition 3.9.1. Another interesting consequence of Theorem 8.3.9 concerns the question whether the Laplacian Δp generates a cosine function on Lp (Rn ) (see Example 3.7.6, Section 3.9 and Section 3.14). Indeed, let 1 ≤ p < ∞ and recall that Δp on Lp (Rn ) may be written as Δp f

=

D(Δp )

=

F −1 (−| · |2 F f ), ' ( f ∈ Lp (Rn ) : F −1 (−| · |2 F f ) ∈ Lp (Rn ) .

Since Δp generates a bounded C0 -semigroup on Lp (Rn ) (see Example 3.7.6), it follows that (0, ∞) ⊂ ρ(Δp ) and supλ>0 λR(λ, Δp ) < ∞. Moreover, by Remark 8.1.2 we have 1 −1 R(λ, Δp )f = F Ff (λ > 0, f ∈ S(Rn ). λ + | · |2 For the time being, let n = 1. Then Δp = A2p , where Ap is the generator of the C0 group Tp of shifts considered in Example 3.3.10. By Example 3.14.15, Δp generates a cosine function Cos on Lp (R) given by 1 (Tp (t) + Tp (−t)) (t ∈ R). 2 For n > 1, the situation is diﬀerent. Indeed, suppose that Δp generates a cosine function Cos on Lp (Rn ) for n > 1. For the time being, let 1 < p < ∞. Then Lp (Rn ) is a UMD-space and it follows from Theorem 3.16.7 that i(−Δp )1/2 , deﬁned as in Proposition 3.8.2, generates a C0 -group U on Lp (Rn ). By Example ∞ 1 |ξ| n 1/2 3.8.5 or by noting that |ξ| = 0 λ− 2 λ+|ξ| 2 dλ (ξ ∈ R ), we see that −(−Δp ) coincides with the generator of the Poisson semigroup T on Lp (Rn ); i.e., Cos(t) =

−(−Δp )1/2 f = F −1 (−|ξ|Ff )

(f ∈ S(Rn )).

Since the Poisson semigroup T is a bounded holomorphic C0 -semigroup of angle π/2 on Lp (Rn ), i(−Δp )1/2 generates the boundary semigroup of T in the sense of Proposition 3.9.1. By Corollary 8.3.11, we conclude that this implies that p = 2, since we assumed that n > 1. Finally, consider the case p = 1 and assume that Δ1 generates a cosine function Cos on L1 (Rn ) for n > 1. Observe that Δ2 generates a cosine function Cos on L2 (Rn ) given by Cos(t)f = F −1 (cos(t| · |)Ff )

(t ∈ R, f ∈ L2 (Rn )).

The Riesz-Thorin interpolation theorem [H¨ or83, Theorem 7.1.12] implies that Δp generates a cosine function on Lp (Rn ) for 1 < p < 2. This contradicts the assertion proved above and we have therefore proved the following result. Theorem 8.3.12. Let 1 ≤ p < ∞ and assume that the Laplacian Δp generates a cosine function on Lp (Rn ). Then n = 1 or p = 2.

8.4. SYSTEMS OF DIFFERENTIAL OPERATORS ON LP -SPACES

8.4

449

Systems of Diﬀerential Operators on Lp -spaces

In this section we consider initial value problems for systems of the form ⎧ ∂u ⎨ = Au (t ≥ 0, x ∈ Rn ), ∂t ⎩u(0, x) = u0 (x) (x ∈ Rn ), where u : R+ × Rn → CN and A is an N × N -matrix whose entries (Aij )1≤i,j≤N are diﬀerential operators with constant coeﬃcients of order mij in the sense of (8.2). The realization of A in Lp (Rn )N (1 ≤ p < ∞) is deﬁned as follows: let a : Rn → L(CN ) be of the form ⎛ ⎞ a11 (ξ) . . . a1N (ξ) ⎟ ⎜ .. .. a(ξ) := ⎝ (ξ ∈ Rn ), (8.13) ⎠ . . aN1 (ξ) . . .

aN N (ξ)

where aij (ξ) := |α|≤mij aijα (iξ)α . Let m := max{mi,j : 1 ≤ i, j ≤ N }. Then a(ξ) = a0 (ξ)+a1 (ξ)+· · ·+am (ξ), where each term aj (0 ≤ j ≤ m), is homogeneous of degree j. The term am is called the principal part of a. For 1 ≤ p < ∞ we deﬁne Ap f D(Ap )

F −1 (aFf ), ' ( := f ∈ Lp (Rn )N : F −1 (aFf ) ∈ Lp (Rn )N , :=

(8.14)

where the Fourier transform of vector-valued functions is deﬁned by applying the transform elementwise. The proof of Lemma 8.1.1 and Proposition 8.1.3 imply the following result. Lemma 8.4.1. The operator A2 generates a C0 -semigroup on L2 (Rn )N if and only if there exists ω ∈ R such that

sup et(a(ξ)−ωI) : ξ ∈ Rn , t ≥ 0 < ∞. (8.15) Matrix-valued symbols a satisfying (8.15) have been completely characterized by Kreiss. The following consequence of his result will be very useful in the sequel. Proposition 8.4.2. Suppose that a of the form (8.13) satisﬁes (8.15). Assume that σ(am (ξ)) ⊂ iR for all ξ ∈ Rn . Then there exists S ∈ L∞ (Rn , L(CN )) such that S(ξ) is invertible and S(ξ)−1 a(ξ)S(ξ) is diagonal for all ξ ∈ Rn . For a proof of Proposition 8.4.2 we refer to [Kre59]. In the following we examine in detail the special case of symmetric hyperbolic systems on Lp (Rn )N . To this end, let a : Rn → L(CN ) be of the form a(ξ) :=

n

Mj (iξj ),

j=1

where M1 , . . . , Mn are hermitian N × N -matrices. Then the following holds true.

450

8. TRANSLATION INVARIANT OPERATORS ON LP (RN )

Theorem 8.4.3 (Brenner). Let 1 ≤ p < ∞ such that p = 2. Let a : Rn → L(CN ) be given by n a(ξ) = Mj (iξj ) (8.16) j=1

where M1 , . . . , Mn are hermitian matrices. Then Ap generates a C0 -semigroup on Lp (Rn )N if and only if the matrices M1 , . . . , Mn commute. We base the proof of Theorem 8.4.3 on the following two lemmas. Here, n n MN p (R ) is the space of all (N × N )-matrices m = (mij ) where mij ∈ Mp (R ) for i, j = 1, 2, . . . , N (see Appendix E). Lemma 8.4.4. Let 1 ≤ p < ∞ such that p = 2. Let a be of the form (8.16) and set n b = −ia. Suppose that ea ∈ MN p (R ). Then the eigenvalues λk (·) of b(·) can be chosen in such a way that λk (ξ) =

n

λkj ξj

(ξ ∈ Rn ),

(8.17)

j=1

where λkj ∈ R for 1 ≤ j ≤ n, 1 ≤ k ≤ N . Proof. The implicit function theorem implies that there is an open ball U ⊂ Rn and C ∞ -functions λk : U → R and uk : U → CN (k = 1, . . . , N ) such that, for all ξ ∈ U, uk (ξ) ≥ 1, b(ξ)uk (ξ) = λk (ξ)uk (ξ) and {uk (ξ) : k = 1, . . . , N } is a basis of CN . For the time being, we ﬁx k and write λ and u for λk and uk . Let ξ0 ∈ U and let ψ ∈ Cc∞ (U ) such that ψ(ξ0 ) = 1. Choose v ∈ Cc∞ (Rn , CN ) such that u(ξ) · v(ξ) = 1 (ξ ∈ U ). For t > 0, we have (ξ ∈ U ),

ψ(ξ)eitλ(ξ) u(ξ) = ψ(ξ)eta(ξ) u(ξ) and therefore ψ(ξ)eitλ(ξ) = eta(ξ) ψ(ξ)u(ξ) · v(ξ)

(ξ ∈ U ).

Since ψu, v ∈ Cc∞ (Rn , CN ), each of their coordinates belongs to Mp (Rn ). It follows from the homogeneity of a and Proposition E.2 e) that there exists a constant C such that a

ψeitλ Mp (Rn ) ≤ C eta MN n = C e MN (Rn ) . (8.18) p (R ) p Deﬁne now μ(ξ) := λ(ξ0 + ξ) − λ(ξ0 ) − ξ · ∇λ(ξ0 ) if ξ0 + ξ ∈ U , and set −1/2 ξ) ψ(ξ0 + t−1/2 ξ)eitμ(t ft (ξ) := 0

if ξ0 + t−1/2 ξ ∈ U, otherwise.

8.4. SYSTEMS OF DIFFERENTIAL OPERATORS ON LP -SPACES

451

For g ∈ Mp (Rn ) and x ∈ Rn , let (τx g)(ξ) := g(ξ − x) and let hx (ξ) := eiξ·x . Then F −1 (τx g · F)ϕ = hx · F −1 (gF )(h−x · ϕ) and F −1 ((hx · g)F)ϕ = τ−x (F −1 (gF)ϕ) for all ϕ ∈ S(Rn ). It follows that τx g, hx · g ∈ Mp (Rn ) and

τx g Mp (Rn ) = hx · g Mp (Rn ) = g Mp (Rn ) . Using these relations, (8.18) and Proposition E.2 e), we have that ft Mp (Rn ) ≤ iP (ξ) n for all t > 0. Moreover, limt→∞ ft (ξ) = e C ea MN uniformly on compact p (R ) n 1 n subsets of R , where P (ξ) := 2 i,j=1 Di Dj λ(ξ0 )ξi ξj . Proposition E.2 f) implies that eiP ∈ Mp (Rn ). However, since we assumed that p = 2, Theorem E.4 a) implies that P ≡ 0. Hence, all the second derivatives of λ vanish at an arbitrary point ξ0 ∈ U , which implies that λ is linear on U . We have shown that λk (ξ) = λk0 +

n

(ξ ∈ U, k = 1, . . . , N ),

λkj ξj

j=1

where λk0 , λkj ∈ R (k = 1, . . . , N, j = 1, . . . , n). It follows that det(zI − b(ξ)) =

N n 9 z − λk0 − λkj ξj

(ξ ∈ U, z ∈ C).

(8.19)

j=1

k=1

Since both sides of the equation above are polynomials in ξ, it follows that (8.19) holds for ξ ∈ Rn and z ∈ C. By homogeneity, λk0 = 0 for all k = 1, . . . , N . Thus, we can choose n λk (ξ) = λkj ξj (ξ ∈ Rn ). j=1

Lemma 8.4.5. Let a : Rn → L(CN ) be of the form (8.16) and let b = −ia. Assume that the eigenvalues λ(·) of b(·) are of the form (8.17). Then the matrices M1 , . . . , Mn commute. Proof. Let λj (·), j = 1, . . . , r be the distinct linear functions representing the eigenvalues of b(·) for ξ ∈ Rn . Denote by V the set where two or more eigenvalues coincide. Then r b(ξ) = λj (ξ)Pj (ξ) (ξ ∈ Rn \V ), j=1

where Pj (ξ) are orthogonal projections given by 5 Nj (ξ) k=j (b(ξ) − λk (ξ)) Pj (ξ) = 5 =: . Dj (ξ) k=j (λj (ξ) − λk (ξ)) Since Pj (ξ) = 1 for ξ ∈ Rn \V , we have Nj (ξ) = |Dj (ξ)| for ξ ∈ Rn \ V , and hence for all ξ ∈ Rn by continuity. The entries of Nj (ξ) are polynomials, so this

452

8. TRANSLATION INVARIANT OPERATORS ON LP (RN )

implies that they are divisible by each of the linear factors λj (ξ) − λk (ξ) of Dj (ξ). Since Nj (ξ) and Dj (ξ) both have degree r − 1, it follows that Pj (ξ) is constant for all ξ ∈ Rn \V . Set Pj := Pj (ξ). Then b(ξ) =

r

λj (ξ)Pj

(ξ ∈ Rn \V ),

(8.20)

j=1

and by continuity (8.20) holds for all ξ ∈ Rn . Obviously, the projections Pj commute and it thus follows that the matrices M1 , . . . , Mn also commute. Proof of Theorem 8.4.3. The proof of Theorem 8.4.3 may now be completed as follows. Suppose that Ap generates a C0 -semigroup on Lp (Rn ). Then, by the proof of n ta a Proposition 8.1.3, eta ∈ MN n = e MN (Rn ) by Proposition p (R ) and e MN p (R ) p E.2 e). Lemma 8.4.4 and Lemma 8.4.5 imply that the matrices commute. The converse implication is easy to prove. In fact, if M1 , . . . , Mn commute, then the Mj may be simultaneously diagonalised by a unitary matrix U so that Dj = U Mj U ∗ , where Dj = diag(λj1 , . . . , λjn ). Hence, n n eta(ξ) = exp it U ∗ Dj U ξj = U ∗ exp it Dj ξj U j=1

j=1

n ta and it follows that eta ∈ MN n < ∞. The assertion p (R ) and supt≥0 e MN p (R ) now follows as in the proof of Proposition 8.1.3. The C0 -semigroup generated by Ap may be written explicitly in terms of U and N translation semigroups on Lp (Rn ).

The following result describes the generalisation of Theorem 8.4.3 to the situation of systems of arbitrary order m. Theorem 8.4.6 (Brenner). Let 1 ≤ p < ∞ such that p = 2. Assume that a : Rn → L(CN ) of the form (8.13) satisﬁes σ(am (ξ)) ⊂ iR for all ξ ∈ Rn . Then Ap (deﬁned as in (8.14)) generates a C0 -semigroup on Lp (Rn )N if and only if there exist commuting diagonalisable matrices M1 , . . . , Mn , with real eigenvalues such that n am (ξ) = Mj (iξj ) (ξ ∈ Rn ). (8.21) j=1

We do not aim to give here a detailed proof of Theorem 8.4.6. For this we refer to [Bre73]. We only notice that Proposition 8.1.5 generalizes to the situation of systems discussed in Theorem 8.4.6. Hence, if Ap generates a C0 -semigroup n am n on Lp (Rn )N , then eta ∈ MN ∈ MN p (R ) which implies that e p (R ), where am denotes the principal part of a. One can now show that the order m is necessarily 1 and that am is of the form (8.21). Starting from this situation, we show in the following that the operator Ap generates a k-times integrated semigroup on Lp (Rn )N for suitable k > 0 provided

8.4. SYSTEMS OF DIFFERENTIAL OPERATORS ON LP -SPACES

453

A2 generates a C0 -semigroup on L2 (Rn )N and further additional assumptions on the symbol a are satisﬁed. More precisely, the following holds true. Theorem 8.4.7. Let 1 < p < ∞. Assume that a of the form (8.13) is homogeneous of degree m for some m ≥ 1. Suppose that σ(a(ξ)) ⊂ iR for all ξ ∈ Rn and that the number of distinct eigenvalues of a(ξ) is constant, and the rank of a(ξ) is also constant, for ξ ∈ Rn \{0}. a) If A2 generates a C0 -semigroup on L2 (Rn )N , then Ap generates a k-times integrated semigroup on Lp (Rn )N provided k > n| 12 − p1 |. b) If A2 generates a C0 -semigroup on L2 (Rn )N and in addition σ(a(ξ)) = {iα1 |ξ|, . . . , iαN |ξ|} for all ξ ∈ Rn , where α1 , . . . , αN ∈ R, then Ap generates a k-times integrated semigroup on Lp (Rn )N provided k ≥ (n − 1) 12 − p1 . The proof of Theorem 8.4.7 is based on the following lemma. Lemma 8.4.8. Let the assumption of Theorem 8.4.7 a) be satisﬁed. Then the function ukt : Rn → L(CN ) deﬁned by t (t − s)k−1 sa(ξ) k ut (ξ) := e ds (t ≥ 0, ξ ∈ Rn ) (k − 1)! 0 1 1 n belongs to MN assumption b) is also satisﬁed, p (R ) provided k > n 2 − p . If the 1 N n then ut ∈ Mp (R ) provided k ≥ (n − 1) 2 − p1 . Proof. By assumption, a is homogeneous of order m and σ(a(ξ)) ⊂ iR for all ξ ∈ Rn . It follows from Lemma 8.4.1 and Proposition 8.4.2 that a(ξ) is diagonalisable for all ξ ∈ Rn . Therefore and by virtue of our assumptions, the minimal polynomial of the matrix −ia(ξ) has only (K + 1) simple roots λl (ξ) (l = 0, . . . , K) for some K satisfying 0 ≤ K ≤ N −1. Observe next that the eigenvalues λl (·) are homogeneous functions of degree m since a(·) is homogeneous of degree m. For t ≥ 0 and ξ ∈ Rn \ {0}, let q be a polynomial of degree K in one variable such that q(λl (ξ)) = eitλl (ξ) for l = 0, . . . , K. Then eta(ξ) = q(a(ξ)). We now examine the form of the coeﬃcients Cj (t, ξ) of q. Denote by L(ξ) the (K + 1) × (K + 1)-matrix whose l-th row is given by (λl (ξ)K , λl (ξ)K−1 , . . . , λl (ξ), 1) (l = 0, . . . , K). Since 9 det L(ξ) = (λl (ξ) − λj (ξ)) = 0 l<j≤K

for all ξ = 0, we have Cj (t, ξ) = (det L(ξ))−1

K l=0

(det Cjl (ξ))eitλl (ξ) ,

8. TRANSLATION INVARIANT OPERATORS ON LP (RN )

454

where Cjl (ξ) is the (K + 1) × (K + 1)-matrix deﬁned by replacing the element λl (ξ)K−j of L(ξ) by 1 and all other elements of L(ξ) of row l and column j by 0. Therefore, ta(ξ)

e

=

K

Cj (ξ)(a(ξ))K−j

j=0

=

K l=0

eitλl (ξ)

1 (det Cjl (ξ))a(ξ)K−j det L(ξ) j=0 K

(t ≥ 0, ξ = 0).

Moreover, since ξ → det L(ξ) is homogeneous of degree h := (K + 1)Km/2 and ξ → det Cjl (ξ) is homogeneous of degree h − (K − j)m for all l ∈ {0, . . . , K}, the functions Φl : Rn \ {0} → L(CN ) given by 1 (det Cjl (ξ))(a(ξ))K−j det L(ξ) j=0 K

Φl (ξ) :=

(l = 0, . . . , K)

are homogeneous of degree 0. Since by assumption the number of distinct eigenvalues of a(ξ) is constant for ξ ∈ Rn \{0}, it follows from [Kat82, Theorem II.5.13a] that the eigenvalues λl (·) are C ∞ -functions on Rn \{0}. Hence, eta(ξ) =

K

eitλl (ξ) Φl (ξ),

(8.22)

l=0

where Φl ∈ C ∞ (Rn \{0}, L(CN )) is homogeneous of degree 0 for l = 0, . . . , K. n It follows from Mikhlin’s theorem (see Theorem E.3) that Φl ∈ MN p (R ). By n assumption, either λl is identically zero on R \ {0} or it is homogeneous of degree m on Rn \ {0} and therefore satisﬁes (Hr ) with r = m. It follows from the proof of Theorem 8.3.9, Proposition 8.2.6 and Lemma 8.2.1 that t (t − s)k−1 isλl (ξ) ξ → e ds (k − 1)! 0 belongs to Mp (Rn ) provided k > n 12 − 1p . Since max1≤i,j≤N { aij Mp (Rn ) } is an k N n equivalent norm to a MN n , we conclude from (8.22) that u t ∈ Mp (R ) if p (R ) 1 1 k>n 2−p . If σ(a(ξ)) = {iα1 |ξ|, . . . , iαN |ξ|} for α1 , . . . , αN ∈ R and all ξ ∈ Rn , then the n representation together with Theorem 8.3.9 implies that ukt ∈ MN p (R ) if 1 (8.22) k ≥ (n − 1) 2 − p1 . Proof of Theorem 8.4.7. Thanks to Lemma 8.4.8 it is now no longer diﬃcult to extend the proof of Theorem 8.3.6 to the present situation of systems and to show

8.4. SYSTEMS OF DIFFERENTIAL OPERATORS ON LP -SPACES

455

that Ap is the generator of a k-times integrated semigroup S on Lp (Rn )N given by S(t)f := F −1 (ukt F f ) (t ≥ 0, f ∈ Lp (Rn )N ). We ﬁnish this section by applying Theorem 8.4.7 to certain systems arising in mathematical physics. We start with the wave equation on Rn . Example 8.4.9 (Wave equation on Rn ). Consider the classical wave equation wtt = !w

(t ∈ R, x ∈ Rn ).

Introducing the variable u := (∇w, wt )T , the wave equation can be written as a symmetric, hyperbolic system with ⎛ ⎞ 0 . . . . ξ1 ⎜ .. .. ⎟ ⎜ . ⎟ a(ξ) = i ⎜ . (ξ ∈ Rn ). ⎟ ⎝ . ⎠ ξn ξ1 ξ2 . . ξn 0 (n+1)×(n+1) It follows from Theorem 8.4.3 that the operator Ap in Lp (Rn )n+1 associated with a does not generate a C0 -semigroup on Lp (Rn )n+1 if p = 2 and n > 1. This property of the wave equation was observed ﬁrst by Littman [Lit63]. The eigenvalues of a(ξ) are λ0 (ξ) = 0 of multiplicity (n − 1) and λ1,2 (ξ) = ±i|ξ| each of multiplicity 1 (ξ ∈ Rn ). Hence, given p ∈ (1, ∞), by Theorem 8.4.7, the operator Ap on Lp (Rn )n+1 p n n+1 associated if k ≥ with a generates a k-times integrated semigroup on L (R ) (n − 1) 12 − p1 . Example 8.4.10 (Maxwell’s equations). We consider Maxwell’s equations in the case where current and charge densities are zero and units are chosen so that the speed of light is one. Then Maxwell’s equations can be written as ∂ u 0 − rot u u(0) u0 = , = , v rot 0 v v(0) v0 ∂t where u, v : R3 → C3 . Note that Maxwell’s equations symmetric, hyperbolic system satisfying the assumptions fact, a : R3 → L(C6 ) given by ⎛ 0 0 0 0 −ξ3 ⎜ 0 0 0 ξ 0 3 ⎜ ⎜ 0 0 0 −ξ ξ 2 1 a(ξ) = i ⎜ ⎜ 0 ξ −ξ 0 0 3 2 ⎜ ⎝ −ξ3 0 ξ1 0 0 ξ2 −ξ1 0 0 0

may be rewritten as a of Theorem 8.4.7 b). In ξ2 −ξ1 0 0 0 0

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

has eigenvalues λ0 (ξ) = 0, λ1,2 (ξ) = ±i|ξ| (ξ ∈ R3 ), each of multiplicity 2. Hence, by Theorem 8.4.7 b), the Maxwell operator Ap associated to a generates a once

8. TRANSLATION INVARIANT OPERATORS ON LP (RN )

456

integrated semigroup on Lp (R3 )6 . Note that, by Theorem 8.4.3, the Maxwell operator Ap does not generate a C0 -semigroup on Lp (R3 )6 if p = 2. Example 8.4.11 (Dirac’s equation). The relativistic description of the motion of a particle of mass m with spin 1/2 is provided by Dirac’s equation 3 ∂u ∂u mc2 (x, t) = c Mj (x, t) − M4 u(x, t) + V (x, t) ∂t ∂xj ih j=1

(x ∈ R3 , t ≥ 0).

Here, u is a function deﬁned on R3 × R+ which takes values in C4 , c is the speed of light, h is Planck’s constant, and M1 , M2 , M3 , M4 are 4 × 4 matrices given by ⎛ ⎞ ⎛ ⎞ 0 0 0 1 0 0 0 −i ⎜ 0 0 1 0 ⎟ ⎜ 0 0 i 0 ⎟ ⎟ ⎟ M1 := ⎜ M2 := ⎜ ⎝ 0 1 0 0 ⎠, ⎝ 0 −i 0 0 ⎠ , 1 0 0 0 i 0 0 0 ⎛ ⎞ ⎛ ⎞ 0 0 1 0 1 0 0 0 ⎜ 0 0 0 −1 ⎟ ⎜ 0 1 0 0 ⎟ ⎟ ⎟ M3 := ⎜ M4 := ⎜ ⎝ 1 0 0 0 ⎠, ⎝ 0 0 −1 0 ⎠ 0 −1 0 0 0 0 0 −1 . If V ≡ 0 and units are chosen so that all constants are equal to 1, then Dirac’s equation may be written as a symmetric, hyperbolic system on X := Lp (R3 )4 (1 < p < ∞) of the form d v 0 Ap v I 0 v = +i (t ≥ 0), w Ap 0 w 0 −I w dt v(0) v0 = , w(0) w0 where

⎛ ⎜ Ap := ⎝

∂ ∂x3

∂ ∂ +i ∂x1 ∂x2

⎞ ∂ ∂ −i ∂x1 ∂x2 ⎟ ⎠ ∂ − ∂x3

with domain D(Ap ) := {f ∈ Lp (R3 )2 : Ap f ∈ Lp (R3 )2 } in Lp (R3 )2 . It follows from Theorem 8.4.3 that the Dirac operator 0 Ap I 0 Dp := +i , Ap 0 0 −I with domain D(Dp ) := D(Ap ) × D(Ap ), generates a C0 -semigroup on Lp (R3 )4 if and only if p = 2. In the following, we show that the Dirac operator Dp generates a twice integrated semigroup on Lp (R3 )4 if 1 < p < ∞.

8.4. SYSTEMS OF DIFFERENTIAL OPERATORS ON LP -SPACES

457

In order to do so, let A be a linear operator on a Banach space Y , let A on Y × Y be given by 0 A A := , D(A) := D(A) × D(A), A 0 I 0 and let B be the bounded operator on Y × Y deﬁned by B := i . Then 0 −I the following holds true. Lemma 8.4.12. Let k ∈ N0 . Then, the operator A generates an (exponentially bounded) k-times integrated semigroup on Y × Y if and only if A and −A generate (exponentially bounded) k-times integrated semigroups on Y . I I Proof. Deﬁne U ∈ L(Y × Y ) by U := √12 . Then A = U DU −1 , where I −I A 0 D := . Since D generates a k-times integrated semigroup if and only 0 −A if A and −A both do so, the result follows from the remarks before Theorem 3.5.7. Lemma 8.4.13. Assume that A generates an exponentially bounded once integrated semigroup on Y × Y . Then A + B with domain D(A) generates an exponentially bounded twice integrated semigroup on Y × Y . Proof. It follows from Lemma 8.4.12 that A and −A generate once integrated semigroups on Y which are exponentially bounded. Moreover, Proposition 3.15.4 implies that A2 generates an exponentially bounded sine function (Sin(t))t≥0 on Y . Thus, ∞

R(λ2 , A2 ) =

e−λt Sin(t) dt

(λ > abs(Sin)).

0

We conclude that (λ − A − B) is invertible for λ > abs(Sin) and that λ+i A R(λ, A + B) = R(λ2 + 1, A2 ) (λ > abs(Sin)). A λ−i It follows from Theorem 3.15.6 that A2 − I generates a sine function (SinI (t))t≥0 on Y . Thus, ∞ 2 2 2 2 R(λ + 1, A ) = R(λ , A − I) = e−λt SinI (t) dt, 0

for λ suﬃciently large. For t ≥ 0, we set t t S11 (t) := SinI (s) ds + i (t − s) SinI (s) ds, 0 0 t t S22 (t) := SinI (s) ds − i (t − s) SinI (s) ds, 0 0 t S12 (t) := S21 (t) := A (t − s) SinI (s) ds. 0

458

8. TRANSLATION INVARIANT OPERATORS ON LP (RN )

We note that S12 (t) and S21 (t) are well deﬁned for t ≥ 0, since by Proposition t 3.15.2, 0 (t − s) SinI (s)x ds ∈ D(A2 ) for all x ∈ Y and all t ≥ 0. We now set S(t) :=

S11 (t) S21 (t)

S12 (t) S22 (t)

Then abs(S) < ∞ and we verify that ∞ R(λ, A + B) = λ2 e−λt S(t) dt

(t ≥ 0).

(λ > abs(S)).

0

Thus, A + B generates a twice integrated semigroup on Y × Y . Finally, consider again the situation of the Dirac equation. The eigenvalues of the symbol of Ap may be computed to be λ1,2 (ξ) := ±i|ξ|. Hence, by Theorem 8.4.7 it follows that Ap and −Ap generate exponentially bounded once integrated 0 A p semigroups on Lp (R3 )2 . By Lemma 8.4.12, Ap := generates an expoAp 0 nentially bounded once integrated semigroup on Lp (R3 )2 × Lp (R3 )2 . Furthermore by Lemma 8.4.13, Dp = Ap +B generates a twice integrated semigroup on Lp (R3 )4 . We have thus proved the following result. Theorem 8.4.14. Let 1 < p < ∞. Then the Dirac operator Dp = Ap + B on Lp (R3 )4 , with domain D(Ap ) × D(Ap ), generates a twice integrated semigroup on Lp (R3 )4 .

8.5

Notes

Section 8.1 Most of the content of this section is more or less standard. The results given in Proposition 8.1.3 and Proposition 8.1.6 are based on the well known properties of Fourier multipliers listed in Appendix E. Section 8.2 An excellent reference for more information on Fourier multipliers is [Ste93]. A proof of Bernstein’s result (Lemma 8.2.1) can be found for instance in [H¨ or83]. Proposition 8.2.3 is due to Hieber [Hie91a]. The remaining part of this section follows the lines of [Hie95]. Section 8.3 The result described in Theorem 8.3.3 on Lp -spectral independence for pseudo-diﬀerential operators is due to Hieber [Hie95]. Corollary 8.3.4 was ﬁrst shown by Iha and Schubert [IS70]. For further results on invariance of the Lp -spectrum of certain classes of pseudo-diﬀerential operators, see [Sch71] and [LS97]. Theorem 8.3.6 and Theorem 8.3.9 m on integrated semigroups generated by operators associated to symbols a ∈ Sρ,0 or to m homogeneous symbols a of the form a(ξ) = |ξ| are due to Hieber [Hie91a], [Hie95]. For related results see also [Lan68], [Sjo70], [BE85], [deL94]. It seems that Corollary 8.3.11 on the boundary group of the Poisson semigroup does not exist in the literature. It is

8.5. NOTES

459

however strongly related to Lp estimates of the wave equation; see [Per80]. The result described in Theorem 8.3.12 on the cosine function generated by the Laplacian in Lp (Rn ) was ﬁrst proved by Littman [Lit63] by direct calculations (not using the theory of cosine functions). Section 8.4 The Cauchy problem for systems of diﬀerential operators with constant coeﬃcients of the form described in Theorem 8.4.3 and Theorem 8.4.6 was investigated in detail by Brenner (see [Bre66] and [Bre73]). Theorem 8.4.3 and Theorem 8.4.6 are due to him. Our proof follows essentially the lines of [Bre66]. For the rest of the section we follow closely [Hie91c]. Example 8.4.11 and Theorem 8.4.14 can be found in [Hie91d]. For further information on the systems discussed in Section 8.4, see also Chapter 1 of [Fat83].

Appendix A

Vector-valued Holomorphic Functions Let X be a Banach space and let Ω ⊂ C be an open set. A function f : Ω → X is holomorphic if f (z0 + h) − f (z0 ) f (z0 ) := lim (A.1) h→0 h h∈C\{0}

exists for all z0 ∈ Ω. If f is holomorphic, then f is continuous and weakly holomorphic (i.e. x∗ ◦ f is holomorphic for all x∗ ∈ X ∗ ). If Γ := {γ(t) : t ∈ [a, b]} is a ﬁnite, piecewise smooth contour in Ω, we can form the contour integral Γ f (z) dz. This coincides b with the Bochner integral a f (γ(t))γ (t) dt (see Section 1.1). Similarly we can deﬁne integrals over inﬁnite contours when the corresponding Bochner integral is absolutely convergent. Since 1 2 ∗ f (z) dz, x = f (z), x∗ dz, Γ

Γ

many properties of holomorphic functions and contour integrals may be extended from the scalar to the vector-valued case, by applying the Hahn-Banach theorem. For example, Cauchy’s theorem is valid, and also Cauchy’s integral formula: 1 f (z) f (w) = dz (A.2) 2πi |z−z0 |=r z − w whenever f is holomorphic in Ω, the closed ball B(z0 , r) is contained in Ω and w ∈ B(z0 , r). As in the scalar case one deduces Taylor’s theorem from this. Proposition A.1. Let f : Ω → X be holomorphic, where Ω ⊂ C is open. Let z0 ∈ Ω, r > 0 such that B(z0 , r) ⊂ Ω. Then

W. Arendt et al., Vector-valued Laplace Transforms and Cauchy Problems: Second Edition, Monographs in Mathematics 96, DOI 10.1007/978-3-0348-0087-7, © Springer Basel AG 2011

461

462

A. VECTOR-VALUED HOLOMORPHIC FUNCTIONS

f (z) =

∞

an (z − z0 )n

n=0

converges absolutely for |z − z0 | < r, where f (z) 1 an := dz. 2πi |z−z0 |=r (z − z0 )n+1 We also mention a special form of the identity theorem. Proposition A.2 (Identity theorem for holomorphic functions). Let Y be a closed subspace of a Banach space X. Let Ω be a connected open set in C and f : Ω → X be holomorphic. Assume that there exists a convergent sequence (zn )n∈N ⊂ Ω such that limn→∞ zn ∈ Ω and f (zn ) ∈ Y for all n ∈ N. Then f (z) ∈ Y for all z ∈ Ω. Note that for Y = {0}, we obtain the usual form of the identity theorem. Proof. Let x∗ ∈ Y 0 := {y ∗ ∈ X ∗ : y, y ∗ = 0 (y ∈ Y )}. Then x∗ ◦ f (zn ) = 0 for all n ∈ N. It follows from the scalar identity theorem that x∗ ◦ f (z) = 0 for all z ∈ Ω. Hence, f (z) ∈ Y 00 = Y for all z ∈ Ω. In the following we show that every weakly holomorphic function is holomorphic. Actually, we will prove a slightly more general assertion which turns out to be useful. A subset N of X ∗ is called norming if

x 1 := sup |x, x∗ | x∗ ∈N

deﬁnes an equivalent norm on X. A function f : Ω → X is called locally bounded if supK f (z) < ∞ for all compact subsets K of Ω. Proposition A.3. Let Ω ⊂ C be open and let f : Ω → X be locally bounded such that x∗ ◦ f is holomorphic for all x∗ ∈ N , where N is a norming subset of X ∗ . Then f is holomorphic. In particular, if X = L(Y, Z), where Y, Z are Banach spaces, and if f : Ω → X is locally bounded, then the following are equivalent: (i) f is holomorphic. (ii) f (·)y is holomorphic for all y ∈ Y . (iii) f (·)y, z ∗ is holomorphic for all y ∈ Y, z ∗ ∈ Z ∗ . Proof. We can assume that x 1 = x for all x ∈ X. In order to show holomorphy at z0 ∈ Ω we can assume that z0 = 0, replacing Ω by Ω − z0 otherwise. For small h, k ∈ C\{0}, let f (h) − f (0) f (k) − f (0) u(h, k) := − . h k

A. VECTOR-VALUED HOLOMORPHIC FUNCTIONS

463

We have to show that for ε > 0 there exists δ > 0 such that u(h, k) ≤ ε whenever |h| ≤ δ and |k| ≤ δ. Let r > 0 such that B(0, 2r) ⊂ Ω and M :=

f (z) < ∞.

sup z∈B(0,2r)

Then by Cauchy’s integral formula, for |z| < r, |h| ≤ r, |k| ≤ r, h, k = 0, 1 1 1 1 1 1 ∗ ∗ u(h, k), x = f (z), x − − − 2πi |z|=2r h z−h z k z−k h−k f (z), x∗ dz. = 2πi |z|=2r z(z − h)(z − k)

x∗ ∈ N , 1 dz z

Hence, |u(h, k), x∗ | ≤ |h − k|M/r2 . Since N is norming, we deduce that

u(h, k) ≤ |h − k|

M . r2

This proves the claim. Corollary A.4. Let Ω ⊂ C be a connected open set and Ω0 ⊂ Ω be open. Let h : Ω0 → X be holomorphic. Assume that there exists a norming subset N of X ∗ such that for all x∗ ∈ N there exists a holomorphic extension Hx∗ : Ω → C of x∗ ◦ h. If supx∗ ∈N |Hx∗ (z)| < ∞, then h has a unique holomorphic extension H : Ω → X.

z∈Ω

Proof. Again we assume that · 1 = · . Let Y := y = (yx∗ )x∗ ∈N ⊂ C : y ∞ := sup |yx∗ | < ∞ , x∗ ∈N

and let H : Ω → Y be given by H(z) := (Hx∗ (z))x∗ ∈N . It follows from Proposition A.3 that H is holomorphic. By x ∈ X → (x, x∗ )x∗ ∈N , one deﬁnes an isometric injection from X into Y . Since H(z) ∈ X for z ∈ Ω0 , it follows from the identity theorem (Proposition A.2) that H(z) ∈ X for all z ∈ Ω. We will extend Proposition A.3 considerably in Theorem A.7. Before that we prove Vitali’s theorem. Theorem A.5 (Vitali). Let Ω ⊂ C be open and connected. Let fn : Ω → X be holomorphic (n ∈ N) such that sup n∈N z∈B(z0 ,r)

fn (z) < ∞

whenever B(z0 , r) ⊂ Ω. Assume that the set ' ( Ω0 := z ∈ Ω : lim fn (z) exists n→∞

464

A. VECTOR-VALUED HOLOMORPHIC FUNCTIONS

has a limit point in Ω. Then there exists a holomorphic function f : Ω → X such that f (k) (z) = lim fn(k) (z) n→∞

uniformly on all compact subsets of Ω for all k ∈ N0 . Proof. Let l∞ (X) := {x = (xn )n∈N ⊂ X : x ∞ := sup xn < ∞}. Then l∞ (X) is a Banach space for the norm · ∞ and the space c(X) of all convergent sequences is a closed subspace of l∞ (X). Consider the function F : Ω → l∞ (X) given by F (z) = (fn (z))n∈N . It follows from Proposition A.3 that F is holomorphic. (One may take N to be the space of all functionals on l∞ (X) of the form (xn )n∈N → xk , x∗ where k ∈ N, x∗ ∈ X ∗ , x∗ ≤ 1). Since F (z) ∈ c(X) for all z ∈ Ω0 , it follows from the identity theorem (Proposition A.2) that F (z) ∈ c(X) for all z ∈ Ω. Consider the mapping φ ∈ L(c(X), X) given by φ((xn )n∈N ) = limn→∞ xn . Then f = limn→∞ fn = φ ◦ F : Ω → X is holomorphic. Finally, we prove uniform convergence on compact sets. Let B(z0 , r) ⊂ Ω and k ∈ N0 . It follows from (A.2) that 1 (k) 1 fn (w) fn (z) = dw. k! 2πi |w−z0 |=r (w − z)k+1 (k)

Now the dominated convergence theorem implies that fn (z) converges uniformly on B(z0 , r/2) to f (k) (z). Since every compact subset of Ω can be covered by a ﬁnite number of discs, the claim follows. If in Vitali’s theorem (fn ) is a net instead of a sequence, the proof shows that f (z) = lim fn (z) exists for all z ∈ Ω and deﬁnes a holomorphic function f : Ω → X. Next we recall a well known theorem from functional analysis. Theorem A.6 (Krein-Smulyan). Let X be a Banach space and W be a subspace of the dual space X ∗ . Denote by B ∗ the closed unit ball of X ∗ . Then W is weak* closed if and only if W ∩ B ∗ is weak* closed. For a proof, see [Meg98, Theorem 2.7.11]. Now we obtain the following convenient criterion for holomorphy. Theorem A.7. Let Ω ⊂ C be open and connected, and let f : Ω → X be a locally bounded function. Assume that W ⊂ X ∗ is a separating subspace such that x∗ ◦ f is holomorphic for all x∗ ∈ W . Then f is holomorphic. Here, W is called separating if x, x∗ = 0 for all x∗ ∈ W implies x = 0 (x ∈ X). Proof. Let Y := {x∗ ∈ X ∗ : x∗ ◦ f is holomorphic}. Since W ⊂ Y , the subspace Y is weak* dense. It follows from Vitali’s theorem (applied to nets if X is not

A. VECTOR-VALUED HOLOMORPHIC FUNCTIONS

465

separable) that Y ∩ B ∗ is weak* closed. Now it follows from the Krein-Smulyan theorem that Y = X ∗ . Hence, f is holomorphic by Proposition A.3.

Notes: Usually, Vitali’s theorem is proved with the help of Montel’s theorem which is only valid in ﬁnite dimensions. A vector-valued version is proved in the book of Hille and Phillips [HP57] by a quite complicated power-series argument going back to Liouville. The very simple proof given here is due to Arendt and Nikolski [AN00] who also proved Theorem A.7 (see also [AN06]).

Appendix B

Closed Operators Let X be a complex Banach space. An operator on X is a linear map A : D(A) → X, where D(A) is a linear subspace of X, known as the domain of A. The range Ran A, and the kernel Ker A, of A are deﬁned by: Ran A := Ker A :=

{Ax : x ∈ D(A)}, {x ∈ D(A) : Ax = 0}.

The operator A is densely deﬁned if D(A) is dense in X. An operator A is closed if its graph G(A) is closed in X × X, where G(A) := {(x, Ax) : x ∈ D(A)}. Thus, A is closed if and only if Whenever (xn ) is a sequence in D(A), x, y ∈ X,

xn − x → 0 and Axn − y → 0, then x ∈ D(A) and Ax = y. It is immediate from this that if A is closed and α, β ∈ C with α = 0, then the operator αA + β with D(αA + β) = D(A) is closed. An operator A is said to be closable if there is an operator A (known as the closure of A) such that G(A) is the closure of G(A) in X × X. Thus A is closable if and only if Whenever (xn ) is a sequence in D(A), y ∈ X,

xn → 0 and Axn − y → 0, then y = 0. When A is closable, D(A) = x ∈ X : there exist xn ∈ D(A) and y ∈ X

such that xn − x → 0 and Axn − y → 0 , Ax = y.

468

B. CLOSED OPERATORS For an operator A, D(A) becomes a normed space with the graph norm

x D(A) := x + Ax .

The operator A : D(A) → X is always bounded with respect to the graph norm, and A is closed if and only if D(A) is a Banach space in the graph norm. Note that if A is replaced by αA + β where α = 0, then the space D(A) is unchanged and the graph norm is replaced by an equivalent norm. Let A be a closed operator on X. A subspace D of D(A) is said to be a core of A if D is dense in D(A) with respect to the graph norm. Thus, D is a core of A if and only if A is the closure of A|D , or equivalently for each x ∈ D(A) there is a sequence (xn ) in D such that xn − x → 0 and Axn − Ax → 0. An operator A on X is said to be invertible if there is a bounded operator A−1 on X such that A−1 Ax = x for all x ∈ D(A) and A−1 y ∈ D(A) and AA−1 y = y for all y ∈ X. Proposition B.1. Let A be an operator on X. The following assertions are equivalent: (i) A is invertible. (ii) Ran A = X and there exists δ > 0 such that Ax ≥ δ x for all x ∈ D(A). (iii) A is closed, Ran A is dense in X, and there exists δ > 0 such that Ax ≥ δ x for all x ∈ D(A). (iv) A is closed, Ran A = X and Ker A = {0}. Proof. The equivalence of (i) and (ii) is an easy consequence of the deﬁnition. Since any bounded operator has closed graph, and since G(A−1 ) = {(y, x) : (x, y) ∈ G(A)}, any invertible operator is closed. Thus, (i) and (ii) imply (iii) and (iv). When (iii) holds, G(A) is complete, and the map (x, Ax) → Ax is an isomorphism of G(A) onto Ran A, so Ran A is complete and (ii) follows. When (iv) holds, the inverse mapping theorem can be applied to the map A from D(A) (with the graph norm) to X, showing that A−1 exists as a bounded map from X to D(A) and hence to X. Let λ ∈ C. Then λ is said to be in the resolvent set ρ(A) of A if λ − A is invertible, and we write R(λ, A) := (λ − A)−1 . The remarks in the previous paragraphs show that if ρ(A) is non-empty, then A is closed. The function R(·, A) : ρ(A) → L(X) is the resolvent of A. The spectrum of A is deﬁned to be: σ(A) := C \ ρ(A),

B. CLOSED OPERATORS

469

and the spectral bound is: s(A) := sup{Re λ : λ ∈ σ(A)} if the supremum exists (s(A) := −∞ if σ(A) is empty). The point spectrum σp (A), and approximate point spectrum σap (A), of A are deﬁned by: σp (A) σap (A)

:= {λ ∈ C : Ker(λ − A) = {0}} , := λ ∈ C : there exist xn ∈ D(A) such that

xn = 1 and lim (λ − A)xn = 0 . n→∞

Thus, σp (A) and σap (A) consist of the eigenvalues and approximate eigenvalues of A, respectively. It is clear that σp (A) ⊂ σap (A) ⊂ σ(A). Proposition B.2. Suppose that A has non-empty resolvent set, and let μ ∈ ρ(A). Let λ ∈ C, λ = μ. Then a) λ ∈ ρ(A) if and only if (μ − λ)−1 ∈ ρ(R(μ, A)). In that case, ! "−1 R(λ, A) = (μ − λ)−1 (μ − λ)−1 − R(μ, A) R(μ, A).

(B.1)

b) λ ∈ σp (A) if and only if (μ − λ)−1 ∈ σp (R(μ, A)). c) λ ∈ σap (A) if and only if (μ − λ)−1 ∈ σap (R(μ, A)). d) The topological boundary of σ(A) is contained in σap (A). Proof. Parts a), b) and c) follow immediately from the identity ! " λ − A = (μ − λ) (μ − λ)−1 − R(μ, A) (μ − A). Part d) follows from a), c) and the corresponding result for bounded operators. Alternatively, d) may be proved directly in exactly the same way as for bounded operators. Corollary B.3. For any operator A, ρ(A) is open and σ(A) is closed in C. Moreover, if μ ∈ ρ(A), λ ∈ C and |λ − μ| < R(μ, A) −1 , then λ ∈ ρ(A), and R(λ, A) =

∞

(μ − λ)n R(μ, A)n+1 ,

n=0

where the series is norm-convergent. Hence,

R(λ, A) ≤

R(μ, A)

. 1 − |λ − μ| R(μ, A)

Moreover, R(·, A) is holomorphic on ρ(A) with values in L(X) and R(μ, A)(n) = (−1)n R(μ, A)n+1 n!

(n ∈ N).

470

B. CLOSED OPERATORS

Proof. is immediate from (B.1) and the Neumann expansion, (I − T )−1 = ∞ This n n=0 T , when T is a bounded operator with T < 1. Proposition B.4. Let A be an operator on X, and let λ, μ ∈ ρ(A). Then R(λ, A) − R(μ, A) = (μ − λ)R(λ, A)R(μ, A).

(B.2)

Proof. The identity (B.2) follows by rearranging (B.1). Proposition B.5. Let A be an operator on X, and U be a connected open subset of C. Suppose that U ∩ ρ(A) is nonempty and that there is a holomorphic function F : U → L(X) such that {λ ∈ U ∩ ρ(A) : F (λ) = R(λ, A)} has a limit point in U . Then U ⊂ ρ(A) and F (λ) = R(λ, A) for all λ ∈ U . Proof. Let V = {λ ∈ U ∩ ρ(A) : F (λ) = R(λ, A)}, μ ∈ ρ(A), x ∈ D(A), y ∈ X. For λ ∈ V , F (λ)(λ − A)x = x,

(B.3)

F (λ)y = R(μ, A)y − (λ − μ)R(μ, A)F (λ)y,

(B.4)

using (B.2). By uniqueness of holomorphic extensions (Proposition A.2), (B.3) and (B.4) are valid for all λ ∈ U . Now, (B.4) implies that F (λ)y ∈ D(A) and R(μ, A)(λ − A)F (λ)y

= F (λ)y + (λ − μ)R(μ, A)F (λ)y = R(μ, A)y.

Since R(μ, A) is injective, (λ − A)F (λ)y = y for all λ ∈ U . This and (B.3) imply that λ ∈ ρ(A) and F (λ) = R(λ, A). The equation (B.2) is known as the resolvent equation or resolvent identity. A function R : U → L(X), deﬁned on a subset U of C, is said to be a pseudo-resolvent if it satisﬁes the resolvent equation; i.e., if R(λ) − R(μ) = (μ − λ)R(λ)R(μ)

(λ, μ ∈ U ).

The following proposition is easy to prove. Proposition B.6. Let R : U → L(X) be a pseudo-resolvent. Then a) Ker R(λ) and Ran R(λ) are independent of λ ∈ U . b) There is an operator A on X such that R(λ) = R(λ, A) for all λ ∈ U if and only if Ker R(λ) = {0}. An operator A is said to have compact resolvent if ρ(A) = ∅ and R(λ, A) is a compact operator on X. Since the compact operators form an ideal of L(X), it is immediate from (B.2) that this property is independent of λ ∈ ρ(A). When A has compact resolvent, then σ(A) is a discrete subset of C. This follows from (B.1) and the fact that the spectrum of a compact operator has 0 as its only limit point. The following is easy to prove.

B. CLOSED OPERATORS

471

Proposition B.7. Let A be an operator on X with non-empty resolvent set, and let T ∈ L(X). The following are equivalent: (i) R(λ, A)T = T R(λ, A) for all λ ∈ ρ(A). (ii) R(λ, A)T = T R(λ, A) for some λ ∈ ρ(A). (iii) For all x ∈ D(A), T x ∈ D(A) and AT x = T Ax. For an operator A, the powers An (n ≥ 2) are deﬁned recursively: ' ( D(An ) := x ∈ D(An−1 ) : An−1 x ∈ D(A) , An x

:=

A(An−1 x).

Note that D((λ − A)n ) = D(An ) for all λ ∈ C, n ∈ N. It is easy to see that An is invertible if and only if A is invertible, and then (An )−1 = (A−1 )n . If A is densely deﬁned and ρ(A) = ∅, then D(An ) is a core for A, for each n ∈ N. To see this, let λ ∈ ρ(A). Then R(λ, A) has dense range D(A). It follows that the range D(An−1 ) of R(λ, A)n−1 is dense in X. Let x ∈ D(A). There is a sequence (ym )m∈N in D(An−1 ) converging to (λ − A)x. Let xm := R(λ, A)ym . Then xm ∈ D(An ), xm − x → 0 and Axm − Ax → 0. Let A be an operator on X, and let Y be a closed subspace of X. The part of A in Y is the operator AY on Y deﬁned by D(AY ) AY y

:= {y ∈ D(A) ∩ Y : Ay ∈ Y }, :=

Ay.

The following results are easy to prove. Proposition B.8. Let A be an operator on X, and let Y be a closed subspace of X. a) If D(A) ⊂ Y , then ρ(A) ⊂ ρ(AY ) and R(λ, AY ) = R(λ, A)|Y for all λ ∈ ρ(A). b) Suppose that ρ(A) = ∅ and there is a projection P of X onto Y such that P R(λ, A) = R(λ, A)P for some λ ∈ ρ(A). Then A maps D(A)∩Y into Y , AY is the restriction of A to D(A) ∩ Y , λ ∈ ρ(AY ) and R(λ, AY ) = R(λ, A)|Y . One situation where the conditions of Proposition B.8 b) are satisﬁed is described in the following. Proposition B.9. Let A be a closed operator on X with ρ(A) = ∅, and suppose that there are a compact subset E1 and a closed subset E2 of C such that E1 ∩E2 = ∅ and E1 ∪ E2 = σ(A). Then there is a bounded projection P on X such that R(λ, A)P = P R(λ, A) for all λ ∈ ρ(A), P (X) ⊂ D(A), σ(AY ) = E1 and σ(AZ ) = E2 , where Y := P (X), Z := (I − P )(X). Moreover, P is unique, and A|Y ∈ L(Y ).

472

B. CLOSED OPERATORS

The projection P is known as the spectral projection of A associated with E1 . Proof. Take μ ∈ ρ(A) and consider R(μ, A) ∈ L(X). Then σ(R(μ, A)) = E1 ∪ E2 , where E1 := {(μ − λ)−1 : λ ∈ E1 } and

{(μ − λ)−1 : λ ∈ E2 } E2 := {(μ − λ)−1 : λ ∈ E2 } ∪ {0}

if D(A) = X, otherwise.

Then E1 and E2 are compact and disjoint. By the functional calculus for bounded operators (see [DS59, p.573]), there is a unique bounded projection P on X such that R(λ, A)P = P R(λ, A) for all λ ∈ ρ(A), σ(R(μ, A)|Y ) = E1 and σ(R(μ, A)|Z ) = E2 . Since 0 ∈ σ(R(μ, A)|Y ), Y ⊂ D(A) and A|Y is bounded by the closed graph theorem. The remaining properties follow easily from Proposition B.2 a). Suppose that A has compact resolvent, let λ ∈ ρ(A) and μ ∈ σ(A). Let P be the spectral projection of A associated with {μ}. Then there exists m ∈ N such that (R(λ, A)P − (λ − μ)−1 P )m = 0 (see [DS59, Theorem VII.4.5]). Hence, (A − μ)m P = 0. Given an operator A on X, let G(A∗ ) := {(x∗ , y∗ ) ∈ X ∗ × X ∗ : Ax, x∗ = x, y ∗ for all x ∈ D(A)} , which is a weak* closed subspace of X ∗ × X ∗ . If (and only if) A is densely deﬁned, then G(A∗ ) is the graph of an operator A∗ in X ∗ , known as the adjoint of A. For the remainder of this appendix, we shall assume that A is densely deﬁned, and we shall consider properties of A∗ . When A is closed, the operator A can be recovered from A∗ in the following way. Proposition B.10. Let A be a closed, densely deﬁned operator on X, and let x, y ∈ X. The following assertions are equivalent: (i) x ∈ D(A) and Ax = y. (ii) x, A∗ x∗ = y, x∗ for all x∗ ∈ D(A∗ ). Hence, D(A∗ ) is weak* dense in X ∗ . Proof. The implication (i) ⇒ (ii) is immediate from the deﬁnition of A∗ . For the converse, suppose that (x, y) ∈ / G(A). By the Hahn-Banach theorem, there exists (x∗ , y ∗ ) ∈ X ∗ × X ∗ such that x, x∗ + y, y ∗ = 0 but u, x∗ + Au, y ∗ = 0 for all u ∈ D(A). The latter condition implies that y ∗ ∈ D(A∗ ) and A∗ y ∗ = −x∗ . Thus, x, A∗ y ∗ = −x, x∗ = y, y ∗ , so (ii) is violated. If D(A∗ ) is not weak* dense in X ∗ , then by the Hahn-Banach theorem there exists y ∈ X such that y = 0 and y, x∗ = 0 for all x∗ ∈ D(A∗ ). By the previous part, A0 = y, which is absurd.

B. CLOSED OPERATORS

473

If A is closable (and densely deﬁned), it is easy to see that (A)∗ = A∗ , so D(A ) is weak* dense by Proposition B.10. Conversely, it is easy to see that if D(A∗ ) is weak* dense, then A is closable. ∗

Proposition B.11. Let A be a closed, densely deﬁned operator on X. Then a) A∗ is invertible if and only if A is invertible, and then (A∗ )−1 = (A−1 )∗ . b) σ(A∗ ) = σ(A), and R(λ, A∗ ) = R(λ, A)∗ for all λ ∈ ρ(A). c) σ(A) = σap (A) ∪ σp (A∗ ). Proof. a) If A is invertible, it is easy to verify that (A−1 )∗ A∗ x∗ = x∗ for all x∗ ∈ D(A∗ ) and A∗ (A−1 )∗ y ∗ = y ∗ for all y ∗ ∈ X ∗ . Thus, A∗ is invertible and (A∗ )−1 = (A−1 )∗ .

−1 Now suppose that A∗ is invertible, and let δ = (A∗ )−1 . Since Ker A∗ = {0}, Ran A is dense in X, by a simple application of the Hahn-Banach theorem. For x ∈ X, there exists x∗ ∈ X ∗ such that x∗ = 1 and x, x∗ = x . Let y ∗ = (A∗ )−1 x∗ ∈ D(A∗ ), so that y ∗ ≤ δ −1 and A∗ y ∗ = x∗ . Hence,

Ax ≥ δ |Ax, y ∗ | = δ |x, A∗ y ∗ | = δ x . It follows from Proposition B.1 that A is invertible. b) This follows from a) by replacing A by λ − A. c) This follows from applying Proposition B.1 and the fact that Ran A is dense in X if and only if Ker A∗ = {0} (by the Hahn-Banach theorem), with A replaced by λ − A. Now, let H be a Hilbert space with inner product (·|·)H . Identifying H ∗ with H by means of the Riesz-Fr´echet lemma, we obtain the following. If A is a densely deﬁned operator on H, the adjoint A∗ of A is deﬁned by D(A∗ ) := x ∈ H : there exists y ∈ H such that (Au|x)H = (u|y)H for all u ∈ D(A) , A∗ x

=

y.

We say that A is selfadjoint if A = A∗ . Example B.12 (Multiplication operators). Let (Ω, μ) be a measure space, H := L2 (Ω, μ), m : Ω → R a measurable function. Deﬁne the operator Mm on H by D(Mm ) := {f ∈ H : mf ∈ H}, Mm f := mf. It is easy to see that Mm is selfadjoint.

474

B. CLOSED OPERATORS

˜ be Hilbert spaces. Two operators A on H and A˜ on H ˜ are called Let H, H ˜ unitarily equivalent if there exists a unitary operator U : H → H such that ˜ D(A) ˜ Ax

= U −1 D(A), = U −1 AU x.

It is easy to see that, in that case, A˜ is selfadjoint whenever A is. Now we can formulate the spectral theorem as follows; we refer to [RS72, Theorem VIII.4] for a proof. Theorem B.13 (Spectral Theorem). Each selfadjoint operator is unitarily equivalent to a real multiplication operator. Thus, selfadjoint and real multiplication operators are eﬀectively the same thing. In proofs, we frequently regard an arbitrary selfadjoint operator as being a real multiplication operator. A selfadjoint operator A is always symmetric; i.e., (Ax|y)H = (x|Ay)H for all x, y ∈ D(A). In particular, (Ax|x)H ∈ R for all x ∈ D(A). We say that A is bounded above if there exists ω ∈ R such that (Ax|x)H ≤ ω(x|x)H

(x ∈ D(A)).

In that case, ω is called an upper bound of A. If A is a multiplication operator Mm , then this is equivalent to saying that m(y) ≤ ω

for almost all y ∈ Ω.

It is easy to see (for example, from the spectral theorem) that for any selfadjoint operator A, we have σ(A) ⊂ R and ω is an upper bound for A if and only if σ(A) ⊂ (−∞, ω]; i.e., ω ≥ s(A). Similarly, we say that A is bounded below by ω if (Ax|x)H ≥ ω(x|x)H

(x ∈ D(A)).

The deﬁnition of selfadjointness is not easy to verify in practice. Here is a handy criterion, for a proof of which we refer to [RS72, Theorem X.1]. Theorem B.14. Let A be an operator on H and let ω ∈ R. The following are equivalent: (i) A is selfadjoint with upper bound ω. (ii) a) (Ax|y)H = (x|Ay)H (x, y ∈ D(A)), b) (Ax|x)H ≤ ω(x|x)H (x ∈ D(A)), and c) there exists λ > ω such that Ran(λ − A) = X. Finally, we mention one or two topics concerning bounded operators. By the closed graph theorem, an operator T on a Banach space X is bounded if T is closed and D(T ) = X. Conversely, a densely deﬁned, closed, bounded operator is

B. CLOSED OPERATORS

475

everywhere deﬁned. By convention, a bounded operator T on a Banach space X will be assumed to be deﬁned on the whole of X. The spectral radius of T will be denoted by r(T ), so r(T ) = sup{|λ| : λ ∈ σ(T )} = inf T n 1/n : n ∈ N . In order to allow a convenient citation in the book, we state the following standard fact whose proof is straightforward. Note that a family of bounded linear operators is equicontinuous if and only if it is bounded. Proposition B.15. Let X, Y be Banach spaces, Tn ∈ L(X, Y ) (n ∈ N) such that supn∈N Tn < ∞. The following are equivalent: (i) (Tn x)n∈N converges for all x in a dense subspace of X. (ii) (Tn x)n∈N converges for all x ∈ X. (iii) (Tn x)n∈N converges uniformly in x ∈ K for all compact subsets K of X.

Notes: The material of this appendix is standard, and can be found in various books, for example [Kat66, Chapter 3].

Appendix C

Ordered Banach Spaces Let X be a real Banach space. By a positive cone in X we understand a closed subset X+ of X such that X+ + X+ R+ · X + X+ ∩ (−X+ ) X+ − X+

⊂ X+ ; ⊂ X+ ;

(C.1) (C.2)

= {0}; and = X.

(C.3) (C.4)

Then an ordering on X is introduced by setting x ≤ y ⇐⇒ y − x ∈ X+ . The space X together with the positive cone is called a real ordered Banach space. The elements of X+ are called positive. Remark C.1. Property (C.3) is frequently expressed by saying that X+ is a proper cone, and (C.4) says that X+ is generating. We assume these properties throughout without further notice. If x∗ ∈ X ∗ , then we say that x∗ is positive and write x∗ ≥ 0 if x, x∗ ≥ 0 for all x ∈ X+ . ∗ The set X+ := {x∗ ∈ X ∗ : x∗ ≥ 0} is closed and satisﬁes (C.1), (C.2) and (C.3). For x, y ∈ X such that x ≤ y we denote by

[x, y] := {z ∈ X : x ≤ z ≤ y} the order interval deﬁned by x and y. One says that the cone X+ is normal if all order intervals are bounded.

478

C. ORDERED BANACH SPACES

∗ Proposition C.2. The cone X+ is normal. The cone X+ is normal if and only if ∗ ∗ ∗ X+ − X+ = X . ∗ Thus, if X+ is normal then (X ∗ , X+ ) is also an ordered Banach space with ∗ normal cone. We call X+ the dual cone of X+ . If the cone X+ is normal then there is a constant c ≥ 0 such that

y ≤ x ≤ z =⇒ x ≤ c max( y , z ).

(C.5)

Indeed, passing to an equivalent norm one can even arrange that c = 1. If X is a real ordered Banach space we tacitly consider the complexiﬁcation of X. So in this book an ordered Banach space is always the complexiﬁcation of a real ordered Banach space. Thus, any C ∗ -algebra is an ordered Banach space with normal cone. Let X be an ordered Banach space. A linear mapping T : X → X is called positive if T x ∈ X+ for all x ∈ X+ . Then we write T ≥ 0. If S, T : X → X are linear, we write S ≤ T if T − S ≥ 0. If X+ is normal, every positive linear mapping T : X → X is continuous. Moreover, there is a constant k ≥ 0 such that ±S ≤ T =⇒ S ≤ k T

(C.6)

if S, T : X → X are linear. A real ordered Banach space X is a lattice if for all x, y ∈ X there exists a least upper bound x∨y of x and y (i.e., x∨y ∈ X, x∨y ≥ x, x∨y ≥ y and w ≥ x, y implies w ≥ x ∨ y). In that case, there also exists a largest lower bound x ∧ y = −((−x) ∨ (−y)). One sets x+ = x ∨ 0, x− = (−x)+ , |x| = x ∨ (−x) = x+ + x− . Then X is called a real Banach lattice if in addition the following compatibility condition is satisﬁed: |x| ≤ |y| =⇒ x ≤ y

(C.7) for all x, y ∈ X. Thus, the cone of a Banach lattice is always normal. In this book, a Banach lattice is the complexiﬁcation of a real Banach lattice. Important examples of Banach lattices are the spaces Lp (Ω, μ) (1 ≤ p ≤ ∞), where (Ω, μ) is a measure space, and C(K) := {f : K → C : f continuous}, where K is a compact space. Let X be a real Banach lattice. A subspace Y of X is called a sublattice if x∈Y

implies |x| ∈ Y.

The space Y is called an ideal if x ∈ Y, y ∈ X, |y| ≤ |x| implies y ∈ Y.

C. ORDERED BANACH SPACES

479

Let (Ω, μ) be a σ-ﬁnite measure space and X = Lp (Ω, μ), where 1 ≤ p < ∞. Then Y is a closed ideal of X if and only if Y = {f ∈ X : f |S = 0 a.e.} for some measurable subset S of Ω. If M ⊂ X is a subset, then M d := {x ∈ X : |x| ∨ |y| = 0 for all y ∈ M } is a closed ideal of X. One says that M is a band if M = M dd . In that case, M ⊕ M d = X. If X is a complex Banach lattice, then a subspace Y of X is called a sublattice (ideal, band) if a) x ∈ Y =⇒ Re x ∈ X, and b) Y ∩ XR is a sublattice (ideal, band) of XR , where XR denotes the underlying real Banach lattice. An ordered Banach space has order continuous norm if each decreasing positive sequence (xn )n∈N converges; i.e., If xn ≥ xn+1 ≥ 0 (n ∈ N), then

lim xn exists.

n→∞

The spaces Lp (Ω, μ) (1 ≤ p < ∞) have order continuous norm, but L∞ (Ω, μ) and C(K) do not if they have inﬁnite dimension. Also, the dual of a C ∗ -algebra has order continuous norm. Let X be a Banach lattice. Then the following assertions are equivalent: (i) If 0 ≤ xn ≤ xn+1 and supn∈N xn < ∞, then (xn )n∈N converges. (ii) X is a band in X ∗∗ . (iii) c0 is not isomorphic to a closed subspace of X. In assertion (ii), we identify X with a closed subspace of X ∗∗ via the canonical evaluation mapping. A Banach lattice X satisfying the equivalent conditions (i), (ii), and (iii) is called a KB-space. Every reﬂexive Banach lattice and every space of the form L1 (Ω, μ) are KB-spaces. Moreover, if X is a KB-space then X has order continuous norm. The space c0 does have order continuous norm but is not a KB-space. Each closed ideal of a KB-space is a band.

Notes: We refer to the monograph [Sch74] by Schaefer and to the survey article [BR84] for all this and for further information.

Appendix D

Banach Spaces which Contain c0 We let c0 be the Banach space of all complex sequences a = (ar )r≥1 such that limr→∞ ar = 0, with a = supr |ar |. For n ≥ 1, let en := (δnr )r≥1 , so en = 1 and

m

αn en = max |αn |

n

n=1

for all m ∈ N and α1 , . . . , αm ∈ C. A complex Banach space X is said to contain c0 if there is a closed linear subspace Y of X which is isomorphic (linearly homeomorphic) to c0 . This is equivalent to the existence of a sequence (xn )n≥1 in X and strictly positive constants c1 and c2 such that

m

c1 max |αn | ≤ αn xn ≤ c2 max |αn | (D.1) n n

n=1

m m for all m ∈ N and α1 , . . . , αm ∈ C. Then the map n=1 αn xn → n=1 αn en extends to an isomorphism of the closed linear span of {xn } onto c0 . Since c0 is not reﬂexive, a reﬂexive Banach space cannot contain c0 . Moreover, for any measure space(Ω, μ), the space L1 (Ω, μ) does not contain c0 . ∞ A formal series n=1 xn in X is said to be unconditionally bounded if there is a constant M such that

m

xnj (D.2)

≤M j=1

whenever m ∈ N and 1 ≤ n1 < n2 < · · · < nm . The series n en in c0 is unconditionally bounded (with M = 1), but it is divergent. It follows that any Banach space which contains c0 has a divergent, unconditionally bounded series. In this appendix, we shall give a converse result showing that if X contains a divergent, unconditionally bounded series, then X contains c0 .

482

D. BANACH SPACES WHICH CONTAIN C0

Lemma D.1. Suppose that n xn is a divergent, unconditionally bounded series in a complex Banach space X, and let M be as in (D.2). Then

m

αn xn ≤ 4M max |αn |

1≤n≤m n=1

for all m ∈ N and α1 , . . . , αm ∈ C. Proof. First suppose that αn ≥ 0 for n = 1, 2, . . . , m. By rearranging x1 , x2 , . . . , xm , we may suppose that 0 ≤ α1 ≤ α2 ≤ · · · ≤ αm . Then m n=1

α n xn = α 1

m

xn + (α2 − α1 )

n=1

m

xn + · · · + (αm − αm−1 )xm .

n=2

Hence,

m

αn xn ≤ α1 M + (α2 − α1 )M + · · · + (αm − αm−1 )M = αm M.

n=1

The general case follows by decomposing each complex number αn as where αnj ≥ 0 and |αnj | ≤ |αn |.

3 j=0

αnj ij

Lemma D.2. Suppose that X contains a divergent, unconditionally bounded series. Then there is a sequence (yj )j≥1 in X such that yj = 1 for all j and

m

3

β j yj max |βj |

≤ 2 1≤j≤m

j=1

for all m ∈ N and β1 , . . . , βm ∈ C. Proof. Let n xn be a divergent, unconditionally bounded series, and let

m

γk := sup αn xn : m > k, αn ∈ C, |αn | ≤ 1 .

n=k+1

By Lemma D.1, γk is ﬁnite, and clearly (γk ) is a decreasing sequence. Let γ := limk→∞ γk . Then γ > 0, since

m

γk ≥ sup xn : m > k

n=k+1

and

n

xn is divergent. Replacing xn by (5/4γ)xn , we may assume that γ = 5/4.

D. BANACH SPACES WHICH CONTAIN C0

483

Choose k1 ≥ 1 such that γk1 < 3/2. Since γk1 > 1, there exist k2 > k1 and αn ∈ C (k1 < n ≤ k2 ) such that |αn | ≤ 1 and

k

2

ν1 := αn xn > 1.

n=k1 +1

Iterating this, we may choose k1 < k2 < . . . and αn ∈ C (n > k1 ) such that |αn | ≤ 1 and

k

j+1

νj := α x n n > 1.

n=kj +1

Let yj :=

νj−1

kj+1

α n xn .

n=kj +1

Then yj = 1. Moreover, if m ∈ N and βj ∈ C (j = 1, . . . , m) and jn is chosen so that kjn < n ≤ kjn +1 (n > k1 ), then

m

km+1

3 3 −1

βj yj = βjn νjn αn xn max βjn νj−1 αn ≤ max |βj |.

≤ 2 k1
j=1

n=k1 +1

Theorem D.3. Suppose that X contains a divergent, unconditionally bounded series x . Then X contains c0 . n n Proof.

Let (y

j ) be as in Lemma D.2. Let m ∈ N and β

j∈ C (j = 1, . . . , m). Then

m

m

3

j=1 βj yj ≤ 2 maxj |βj |. We shall establish that j=1 βj yj ≥ 12 maxj |βj |, so that (yj ) satisﬁes the condition (D.1), and therefore X contains c0 . Choose k such that |βk | = maxj |βj |, and choose x∗ ∈ X ∗ such that x∗ = 1 and βk yk , x∗ = |βk |. Let βj (j = k), βj := −βk (j = k). Then

m

m m

∗

β y ≥ Re β y , x = 2|β | + Re βj yj , x∗ j j j j k

j=1

j=1 j=1

m

3 1

≥ 2|βk | − β y max |βj | = max |βj |. j j ≥ 2|βk | −

j 2 2 j

j=1

484

D. BANACH SPACES WHICH CONTAIN C0

This completes the proof.

Notes: Theorem D.3 is due to Bessaga and Pelczynski [BP58]. They also showed that X c0 if (and only if) there is a sequence of unit vectors (yj ) in X such that contains ∗ ∗ ∗ j |yj , x | < ∞ for all x ∈ X . Our proof, which is adapted from [LZ82], establishes such a property but in a speciﬁc way which eliminates some of the cases considered in [BP58]. Moreover, this proof shows (when constants 5/4 and 3/2 are replaced by constants arbitrarily close to 1) that X contains c0 “almost isometrically”, thereby establishing a positive solution to the “distortion problem” in c0 . This was ﬁrst proved by James [Jam64]. Another direct proof of Theorem D.3 is given in a paper of Eberhardt and Greiner [EG92]. There are numerous other characterizations of Banach spaces which contain c0 , some of which may be found in the books of Guerre-Delabri`ere [Gue92], Lindenstrauss and Tzafriri [LT77] and Megginson [Meg98]. Note in particular that a Banach lattice X does not contain c0 (as a subspace, or equivalently as a sublattice) if and only if X is a KB-space, that is, every bounded increasing sequence in X converges [LT77, Theorem II.1.c.4], [Mey91, Theorem 2.4.12].

Appendix E

Distributions and Fourier Multipliers In this appendix we collect basic facts on distributions and Fourier multipliers. They are needed at various places in the book; those which are essential to understanding Parts I and II are also explained at the appropriate point in the text, while other results from this appendix are needed only for examples in Chapter 3 or for the applications in Part III. n is an element α = First, we consider distributions non R . A multi-index ∂ α (α1 , . . . , αn ) ∈ Nn0 . We write |α| for j=1 αj , Dj for ∂x and D for D1α1 · · · Dnαn . j We denote by D(Rn ) (or by Cc∞ (Rn ) in other contexts) the space of all complexvalued C ∞ -functions on Rn with compact supports (the test functions), and by S(Rn ) the Schwartz space of all smooth, rapidly decreasing functions on Rn , i.e. S(Rn ) := {ϕ ∈ C ∞ (Rn ) : ϕ m,α < ∞ for all m ∈ N0 , α ∈ Nn0 } , where

ϕ m,α := sup (1 + |x|)m |Dα ϕ(x)|. x∈Rn

When equipped with the topology deﬁned by the family of all norms · m,α , S(Rn ) is a Fr´echet space, and D(Rn ) is a dense subspace of S(Rn ). We denote by D(Rn ) the space of all distributions, i.e., linear maps f : ϕ → ϕ, f of D(Rn ) into C such that for each compact K ⊂ Rn there exist m ∈ N and C > 0 such that |ϕ, f | ≤ C sup sup |Dα ϕ(x)| |α|≤m x∈Rn

for all ϕ ∈ D(Rn ) with supp ϕ ⊂ K. Let S(Rn ) be the space of all temperate distributions, i.e., continuous linear maps from S(Rn ) into C. Then S(Rn ) is embedded in D(Rn ) in a natural way.

486

E. DISTRIBUTIONS AND FOURIER MULTIPLIERS

We consider D(Rn ) to have the topology arising from the duality with D(Rn ), so a net (fα ) of distributions converges to 0 in D(Rn ) if and only if ϕ, fα → 0 for all ϕ ∈ D(Rn ). Any locally integrable f : Rn → C can be identiﬁed with a distribution by ϕ, f := ϕ(x)f (x) dx (ϕ ∈ S(Rn )). (E.1) Rn

We shall make such identiﬁcations freely. A function f : Rn → C is said to be absolutely regular if there exists k ∈ N0 such that x → (1 + |x|)−k f (x) is Lebesgue integrable on Rn . For an absolutely regular function f , the corresponding distribution is temperate. Any continuous linear map T : S(Rn ) → S(Rn ) induces an adjoint. We now describe how this enables operators of multiplication, diﬀerentiation, Fourier transform and convolution to be extended from functions to distributions. Let g : Rn → C be a C ∞ -function. Then ϕ · g ∈ D(Rn ) for all ϕ ∈ D(Rn ). Given a distribution f ∈ D(Rn ) , we can deﬁne g · f by ϕ, g · f := ϕ · g, f

(ϕ ∈ D(Rn )).

(E.2)

If, for each multi-index α, there exists mα ∈ N and cα > 0 such that |(Dα g)(x)| ≤ cα (1 + |x|)mα

(x ∈ Rn ),

(E.3)

then the map ϕ → ϕ · g is continuous from S(Rn ) into S(Rn ), and therefore g · f ∈ S(Rn ) whenever f ∈ S(Rn ) . Given a distribution f ∈ D(Rn ) , the derivatives Dj f (j = 1, . . . , n) are deﬁned in D(Rn ) by ϕ, Dj f := −Dj ϕ, f

(ϕ ∈ D(Rn )).

(E.4)

Then Dj maps S(Rn ) into itself. Integration by parts shows that this notation is consistent when diﬀerentiable functions are identiﬁed with distributions, and the product law extends to derivatives of products of diﬀerentiable functions and distributions as discussed above. For higher order derivatives, (E.4) becomes ϕ, Dα f = (−1)|α| Dα ϕ, f

(ϕ ∈ D(Rn )).

(E.5)

Next, we consider convolutions. For functions f and g, the convolution f ∗ g is deﬁned by (f ∗ g)(x) := f (x − y)g(y) dy Rn

whenever the integral exists. For ϕ, ψ ∈ S(Rn ), ψ ∗ ϕ ∈ S(Rn ) and the map ψ → ψ ∗ϕ is continuous. Hence, the convolution ϕ∗f of ϕ ∈ S(Rn ) and f ∈ S(Rn ) can be deﬁned by ψ, ϕ ∗ f := ψ ∗ ϕ, ˇ f

(ψ ∈ S(Rn )),

(E.6)

E. DISTRIBUTIONS AND FOURIER MULTIPLIERS

487

where ϕ(x) ˇ := ϕ(−x), and then ϕ ∗ f ∈ S(Rn ) . An easy calculation shows that this is consistent when functions are identiﬁed with distributions. An alternative way to deﬁne ϕ ∗ f is as follows. For x ∈ Rn and ψ ∈ S(Rn ), let τx ψ(y) := ψ(y − x). For ϕ ∈ S(Rn ), τx ϕˇ ∈ S(Rn ) and the map x → τx ϕˇ is continuous on S(Rn ). For f ∈ S(Rn ) , let (ϕ ∗ f )(x) := τx ϕ, ˇ f

(x ∈ Rn ).

(E.7)

Then ϕ ∗ f is a continuous, bounded function. These two deﬁnitions of ϕ ∗ f are consistent when functions are identiﬁed with distributions. Moreover, Dj (ϕ ∗ f ) = (Dj ϕ) ∗ f. For f ∈ L1 (Rn ), the Fourier transform Ff of f is deﬁned by: (F f )(ξ) = e−ix·ξ f (x) dx (ξ ∈ Rn ),

(E.8)

(E.9)

Rn

n where x · ξ := j=1 xj ξj . The Fourier inversion theorem [H¨ or83, Theorem 7.1.5] shows that F is a linear and topological isomorphism of S(Rn ), and (F −1 ϕ)(ξ) = (2π)−n (Fϕ)(−ξ)

(ϕ ∈ S(Rn ), ξ ∈ Rn ).

The Fourier transform therefore induces an isomorphism of S(Rn ) , also denoted by F : ϕ, Ff := Fϕ, f (ϕ ∈ S(Rn ), f ∈ S(Rn ) ). (E.10) A simple application of Fubini’s theorem shows that this notation is consistent when f ∈ L1 (Rn ) and f is identiﬁed with a distribution in S(Rn ) . The following relations, which are elementary for functions, extend to distributions f : F −1 f = (2π)−n (F f )ˇ= (2π)−n F fˇ, where ϕ, fˇ := ϕ, ˇ f , FDj f = iξj · Ff, F (ϕ ∗ f ) = (Fϕ) · (Ff )

(E.11) (E.12)

(ϕ ∈ S(Rn )).

(E.13)

(ϕ, ψ ∈ S(Rn )),

(E.14)

Plancherel’s theorem states that ¯ ¯ = (2π)n ϕ, ψ F ϕ, F ψ

where ψ¯ is the complex conjugate of ψ, and hence F extends by continuity to a linear operator F on the Hilbert space L2 (Rn ) such that (2π)−n/2 F is unitary. This also says that, for each f ∈ L2 (Rn ), the distribution Ff belongs to L2 (Rn ). Many of the concepts above can be extended to the case of distributions on an open subset Ω of Rn . Let D(Ω) be the space of test functions on Ω, i.e., C ∞ functions of compact support in Ω, and D(Ω) be the space of distributions on Ω,

488

E. DISTRIBUTIONS AND FOURIER MULTIPLIERS

i.e., linear functionals f on D(Ω) such that for each compact K ⊂ Ω there exist m ∈ N and C > 0 such that |ϕ, f | ≤ C sup sup |Dα ϕ(x)| |α|≤m x∈Ω

for all ϕ ∈ D(Ω) with supp ϕ ⊂ K. Locally integrable functions on Ω can be identiﬁed with distributions, and the derivatives Dj of a distribution f are deﬁned by ϕ, Dj f := −Dj ϕ, f (ϕ ∈ D(Ω)). For m ∈ N and 1 ≤ p ≤ ∞, the Sobolev space W m,p (Ω) is deﬁned by W m,p (Ω) := {f ∈ Lp (Ω) : Dα f ∈ Lp (Ω) for all α ∈ Nn0 with |α| ≤ m}, where Dα f is understood in the sense of distributions. Thus, f ∈ W m,p (Ω) if and only if for each α ∈ Nn0 with |α| ≤ m there exists fα ∈ Lp (Ω) such that |α| ϕfα dx = (−1) (Dα ϕ)f dx (ϕ ∈ D(Ω)). Ω

Ω

In the special case when n = 1, f ∈ W m,p (Ω) if and only if f ∈ C m−1 (Ω), f (m−1) is absolutely continuous, and f (j) ∈ Lp (Ω) for j = 0, 1, . . . , m. Equipped with the norm

Dα f p ,

f W m,p (Ω) := |α|≤m

W (Ω) becomes a Banach space. The closure of D(Ω) in W m,p (Ω) is denoted m,p by W0 (Ω). For p = 2, we also use the notation m,p

H m (Ω) := W m,2 (Ω) and H0m (Ω) := W0m,2 (Ω). Equipped with the equivalent norm ⎛

f H m (Ω) := ⎝

⎞1/2

Dα f 22 ⎠

,

|α|≤m

H m (Ω) is a Hilbert space with the inner product (f |g)H m (Ω) = D α f Dα g dx. |α|≤m

Ω

Note that Plancherel’s theorem and (E.12) show that H m (Rn ) = {f ∈ L2 (Rn ) : ξ α · Ff ∈ L2 (Rn ) for all α ∈ Nn0 with |α| ≤ m}, where ξ α is the function ξ → ξ1α1 ξ2α2 · · · ξnαn . Hence, f ∈ H m (Rn ) if and only if ξ → (1 + |ξ|2 )m/2 (F f )(ξ) belongs to L2 (Rn ).

E. DISTRIBUTIONS AND FOURIER MULTIPLIERS

489

Now we consider Fourier multipliers. If g is a C ∞ -function satisfying the estimates (E.3), then the map ϕ → F −1 gFϕ := F −1 (g · (Fϕ)) is a continuous linear map on S(Rn ). It is a classical problem to seek conditions on a function such that such a map becomes continuous on a function space X = Lp (Rn ) (1 ≤ p ≤ ∞) or C0 (Rn ). Let m : Rn → C be an absolutely regular function. For ϕ ∈ S(Rn ), we deﬁne m · (Fϕ) ∈ S(Rn ) by ψ, m · (F ϕ) := ψm · (F ϕ) dx (ψ ∈ S(Rn )). Rn

Then we consider the distribution F −1 (m · (F ϕ)) ∈ S(Rn ) . We call m a Fourier multiplier for X if F −1 (m·(F ϕ)) ∈ X for all ϕ ∈ S(Rn ) and there exists a constant C such that

F −1 (m · (Fϕ)) X ≤ C ϕ X (ϕ ∈ S(Rn )). Then the map ϕ → F −1 (m · (F ϕ)) extends to a bounded linear operator Tm : f → F −1 mF f on X (in the case when X = L∞ (Rn ), the extension is weak* continuous). When m is a C ∞ -function, Tm f agrees with the distribution F −1 (m · Ff ) deﬁned earlier. We denote the space of all Fourier multipliers for X by MX (Rn ), or by Mp (Rn ) when X = Lp (Rn ), with the usual identiﬁcation of functions which coincide a.e. We put

m MX (Rn ) := Tm L(X) . Fourier multipliers are bounded functions, and m MX (Rn ) ≥ m ∞ (see Proposition E.2). It follows easily that MX (Rn ) is a Banach space. Note also that MC0 (Rn ) ⊂ M∞ (Rn ). For a ∈ Rn , deﬁne τa ∈ L(X) by τa f (x) := f (x − a). If m ∈ MX (Rn ), it is easy to see that Tm τa = τa Tm (a ∈ Rn ). (E.15) Conversely, we have the following result. Proposition E.1. Let X = Lp (Rn ) (1 ≤ p ≤ ∞) or C0 (Rn ), and assume that T ∈ L(X) satisﬁes (E.15). Then there exists m ∈ MX (Rn ) such that T f = F −1 mFf

(f ∈ X).

For a proof of Proposition E.1, see [H¨ or60]. n For N ∈ N, we let MN (R ) be the space of all matrices m = (mij )1≤i,j≤N , p where mij ∈ Mp (Rn ). Each such matrix m deﬁnes a bounded operator F −1 mF on Lp (Rn )N , where F : Lp (Rn )N → Lp (Rn )N acts on each coordinate function, n and matrix multiplication operates as usual. The norm on MN p (R ) is taken to be −1 p n N the norm of the operator F mF when L (R ) is given the norm of Lp (Rn × {1, . . . , N }). Note that M1p (Rn ) = Mp (Rn ).

490

E. DISTRIBUTIONS AND FOURIER MULTIPLIERS

Proposition E.2. Let 1 ≤ p ≤ ∞, N ∈ N. Then the following hold true: n a) MN p (R ) is a Banach algebra. n ∞ n N ∞ n N b) MN 2 (R ) = L (R , L(C )) := {(mij )1≤i,j≤N : mij ∈ L (R ) }. n N n c) MN p (R ) = Mp (R ), where 1/p + 1/p = 1. n N n N n N n d) MN 1 (R ) ⊂ Mp (R ) ⊂ M2 (R ). Moreover, for m ∈ M1 (R ), 1−θ θ

m MN n ≤ m

n m

n , MN p (R ) MN 1 (R ) 2 (R ) 1 1 where θ := 2 p − 2 .

(E.16)

n n e) Given a ∈ MN p (R ) deﬁne at by at (ξ) := a(tξ) for t > 0, ξ ∈ R . Then N n at ∈ Mp (R ) for all t > 0 and

at MN n = a MN n p (R ) p (R )

(t > 0).

n f) Let (aj )j∈N ⊂ MN p (R ). Assume that there exists a constant C > 0 such that aj MN n ≤ C for j ∈ N. Let a ∈ L∞ (Rn ) such that aj (x) → a(x) p (R ) n for almost all x ∈ Rn as j → ∞. Then a ∈ MN n ≤ C. p (R ) and a MN p (R )

Proof. We give only sketches of the proofs; details may be found in [H¨or60] or [Ste93]. a) follows from the formal identity F −1 (m1 m2 )F = (F −1 m1 F)(F −1 m2 F ), b) is an easy consequence of Plancherel’s theorem and c) is easily proved by duality, showing even that the equalities are isometric. n For d), we can assume by c) that 1 ≤ p ≤ 2. Let m ∈ MN p (R ). By c), −1 p n N p n N F mF is bounded on L (R ) and on L (R ) . Moreover the two versions of the map agree on Lp (Rn )N ∩ Lp (Rn )N . By the Riesz-Thorin theorem [H¨or83, Theorem 7.1.12], F −1 mF extends to a bounded linear operator on L2 (Rn )N . This n N n N n N n shows that MN p (R ) ⊂ M2 (R ). The inclusion M1 (R ) ⊂ Mp (R ) and the inequality (E.16) also follows from the Riesz-Thorin theorem. e) follows from the fact that F −1 at F = Jt−1 F −1 aFJt , where Jt is the isometry, (Jt f )(ξ) := t−n/p f (tξ), on Lp (Rn ), and f) is proved by taking limits through the deﬁnitions of Fourier multipliers. An extremely useful suﬃcient condition for a function m to belong to M1p (Rn ) for 1 n2 }. Deﬁne the Banach space MM by ' ( MM := m : Rn → K : m ∈ C j (Rn \ {0}), |m|M < ∞ , (E.17) where the norm | · |M is deﬁned by |m|M := max

sup

|α|≤j ξ∈Rn \{0}

We then have the following result.

|ξ||α| |D α m(ξ)|.

(E.18)

E. DISTRIBUTIONS AND FOURIER MULTIPLIERS

491

Theorem E.3 (Mikhlin). Let 1 < p < ∞. Then MM → Mp (Rn ). For a proof of Mikhlin’s theorem, we refer to [Ste93, Theorem VI.4.4]. The following results on Fourier multipliers will be useful in Chapter 8. Theorem E.4. Let 1 ≤ p ≤ ∞. Then the following hold true: a) Let a be a real homogeneous polynomial on Rn of degree m > 1. Then eia ∈ Mp (Rn ) if and only if p = 2. b) Let a ∈ C ∞ (Rn ) satisfy

α |ξ|−β e−i|ξ| a(ξ) := 0

(|ξ| ≥ 2), (|ξ| ≤ 1),

where α > 0 and β ≥ 0. (i) If α = 1 and 1 < p < ∞ (respectively, p = 1), then a ∈ Mp (Rn ) if and only if n 12 − p1 ≤ αβ (respectively, n2 < αβ ). (ii) If α = 1 and 1 < p < ∞ (respectively, p = 1) then a ∈ Mp (Rn ) if and only if (n − 1) 12 − 1p ≤ β (respectively, n−1 2 < β). c) Deﬁne a1 : R3 → C and a2 : R3 → C by a1 (ξ) a2 (ξ)

:= (−i)(ξ1 + ξ22 + ξ32 − i), := ξ1 + ξ22 + ξ32 + i.

3 (i) If p = 2, then a−1 1 ∈ Mp (R ). −1 (ii) Let a := a1 a2 . Deﬁne the operator Ap on Lp (R3 ) by Ap f := 1 F 1 (aF f3 ) p 3 −1 p 3 with D(Ap ) := {f ∈ L (R ) : F (aFf ) ∈ L (R )}. If 2 − p > 8 , then σ(Ap ) = C.

For a proof of the assertions of Theorem E.4 we refer to [H¨ or60] (assertion a)), [FS72], [Miy81] and [Per80] (assertion b)), [KT80] (assertion c)i)) and [IS70] (assertion c)ii)). Finally, we note one instance of Mikhlin’s Theorem. For x ∈ R, deﬁne ⎧ ⎪ (x > 0), ⎨1 sign x := 0 (x = 0), ⎪ ⎩ −1 (x < 0). Then sign ∈ MM . By Mikhlin’s theorem, sign ∈ Mp (R) for 1 < p < ∞. For ϕ ∈ S(R), one ﬁnds that F −1 (−i sign)Fϕ is a function given by 1 ϕ(y) −1 (F (−i sign)Fϕ)(x) = lim dy. (E.19) ε↓0 π |x−y|≥ε x − y This is known as the Hilbert transform of ϕ. Thus, we have the following.

492

E. DISTRIBUTIONS AND FOURIER MULTIPLIERS

Proposition E.5. Let 1 < p < ∞. Then the Hilbert transform is a bounded linear operator on Lp (R).

Notes: The material on distributions is very standard and can be found in many books, for example [H¨ or83]. The basic material on Fourier multipliers can be found in [Ste93].

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[You92] P. H. You. Characteristic conditions for a C0 -semigroup with continuity in the uniform operator topology for t > 0. Proc. Amer. Math. Soc. 116 (1992), 991–997. [Zab75] A. Zabczyk. A note on C0 -semigroups. Bull. Acad. Polon. Sci. 23 (1975), 895–898. [Zab79]

A. Zabczyk. Stabilization of boundary control systems. Systems Optimization and Analysis, Lecture Notes in Control Information Sci. 14, Springer-Verlag, Berlin (1979), 321–333.

[Zai60]

S. Zaidman. Sur un th´eor`eme de I. Miyadera concernant la repr´esentation des functions vectorielles par des int´egrales de Laplace. Tˆ ohoku Math. J. (2) 12 (1960), 47–51.

[Zar93]

M. Zarrabi. Contractions ` a spectre d´enombrable et propri´et´es d’unicit´e des ferm´es d´enombrables du cercle. Ann. Inst. Fourier (Grenoble) 43 (1993), 251–263.

[Zem94] J. Zem´ anek. On the Gelfand-Hille theorem. Functional Analysis and Operator Theory. Banach Center Publ. 30, Polish Acad. Sci., Warsaw (1994), 369–385.

Notation Function and Distribution Spaces AAP(R+ , X)

space of asymptotically almost periodic functions . . . . . 307

AP(I, X), AP(I)

space of almost periodic functions . . . . . . . . . . . 292, 297, 307

BSV([a, b], X)

space of functions of bounded semivariation . . . . . . . . . . . . 48

BSVloc (R+ , X)

space of functions of locally bounded semivariation . . . . . 48

BUC(I, X), BUC(I) space of bounded, uniformly continuous functions . . . . . 15 Lipω (R+ , X)

space of Lipschitz continuous functions . . . . . . . . . . . . . 63, 77

D(Ω)

space of distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485, 488

D(Ω)

space of test functions . . . . . . . . . . . . . . . . . . . . . . . . 15, 485, 487

E, E(R+ , X)

space of totally ergodic functions . . . . . . . . . . . . . . . . 301, 308

FL1 (Rn )

Fourier algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 436

MM

Mikhlin space of Fourier multipliers . . . . . . . . . . . . . . 436, 490

n MN p (R )

space of matrices of Fourier multipliers . . . . . . . . . . . . . . . 489

MX (Rn ), Mp (Rn ) space of Fourier multipliers on X or Lp (Rn ) . . . . . . . . . . 489 Mε

strong Mikhlin space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 436

S(Rn )

space of temperate distributions . . . . . . . . . . . . . . . . . . . . . . 486

S(Rn )

Schwartz space of rapidly decreasing functions . . . 319, 485

E

quotient of space of totally ergodic functions . . . . . 301, 308

C(I, X), C(Ω)

space of continuous functions . . . . . . . . . . . . . . . . . . . . . . . . . . 15

c(X)

space of convergent sequences . . . . . . . . . . . . . . . . . . . . . . . . . . 41

526

NOTATION

C k (I, X), C k (Ω)

space of k-times continuously diﬀerentiable functions . . . 15

C ∞ (I, X), C ∞ (Ω)

space of inﬁnitely diﬀerentiable functions . . . . . . . . . . . . . . 15

C0 (I, X), C0 (Ω)

space of continuous functions vanishing at inﬁnity . . . . . . 15

c0

space of null sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10, 481

Cc (I, X), Cc (Ω)

space of continuous functions with compact support . . . 15

Cc∞ (I, X), Cc∞ (Ω)

space of inﬁnitely diﬀerentiable functions with compact support . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

∞ CW ((ω, ∞), X)

Widder space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64, 78

Cω1 (R+ , X)

space of functions with continuous, exponentially bounded derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

H 2 (C+ , X), H 2 (C+ ) Hardy spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 H m (Ω), H0m (Ω)

Sobolev space of order (m, 2) . . . . . . . . . . . . . . . . . . . . . . . . . 488

Lp (Ω, μ)

space of p-integrable functions on a measure space . . . . 175

Lp (I)

space of Lebesgue p-integrable functions . . . . . . . . . . . . . . . 14

Lp (I, X)

space of Bochner p-integrable functions . . . . . . . . . . . . 13, 14

lp

space of p-summable sequences . . . . . . . . . . . . . . . . . . . . 10, 132

L∞ (I, X), L∞ (I)

space of bounded measurable functions . . . . . . . . . . . . . . . . 14

l∞ (X)

space of bounded sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

L∞ ω (I, X)

space of exponentially bounded functions . . . . . . . . . 77, 226

L1loc (R+ , X)

space of locally Bochner integrable functions . . . . . . . . . . . 13

m Sρ,0

space of symbols of pseudo-diﬀerential operators . . . . . . 430

W m,p (Ω), W0m,p (Ω) Sobolev space of order (m, p) . . . . . . . . . . . . . . . . . . . . . . . . . 488

Dual Spaces and Subspaces V

antidual of V . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 420

X∗

dual space of X . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

X

sun-dual of X . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

X0

space of vectors converging to 0 . . . . . . . . . . . . . . . . . . . . . . . 360

NOTATION

527

Xe

space of totally ergodic vector in X . . . . . . . . . . . . . . . . . . . . 267

Xap

space of almost periodic vectors in X . . . . . . . . . . . . 290, 361

Xe0

space of totally ergodic vectors with means 0 . . . . . . . . . 267

Norms and Dualities (·|·)H

inner product on a Hilbert space H . . . . . . . . . . . . . . . . . . . . 45

(·|·)

duality between a space and its antidual . . . . . . . . . . . . . . 421

·, ·

duality between a space and its dual . . . . . . . . . . . . . . . 7, 485

· D(A)

graph norm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 468

· W

Widder norm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64, 78

|α|

norm of multi-index α . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485

| · |M

Mikhlin norm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 436, 490

· p

Lebesgue-Bochner norm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13, 14

| · |Mε

strong Mikhlin norm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 436

· ω,∞

exponentially bounded norm . . . . . . . . . . . . . . . . . . . . . . 77, 226

|π|

norm of partition π . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

Functions, Integrals and Abscissas abs(f ), abs(dF )

abscissa of convergence . . . . . . . . . . . . . . . . . . . . . . 27, 30, 56, 57

χΩ

characteristic function of Ω . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

Cos

cosine function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203

hol(fˆ), hol(Tˆ)

abscissa of holomorphy of fˆ or T . . . . . . . . . . . . . . . . . . . 33, 35

hol0 (fˆ) b g(t) dF (t) a b g(t) dt a f (t) dt I

abscissa of boundedness of fˆ . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

ω(f ), ω(T )

exponential growth bound of f or T . . . . . . . . . . . . . . . . 29, 30

ω1 (T )

exponential growth bound of classical solutions . . . . . . . 343

Riemann-Stieltjes integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 Riemann integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 Bochner integral of f over I . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

528

NOTATION

sign

signum function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138, 491

Sin

sine function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206, 218

En

Newtonian potential on Rn . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404

eλ

exponential function t → eλt . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

eλ ⊗ x

the function t → eλt x . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 292

f ∗ g, T ∗ f

convolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21, 24, 487

kt , kz

Gaussian kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

S(g, π)

Riemann sum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

S(g, F, π)

Riemann-Stieltjes sum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

ux

orbit of T through x . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30, 337

V (π, F )

variation of F over π . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

V (F ), V[a,b] (F )

total variation of F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

Operators Δ

distributional Laplacian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

Δ0

Laplacian on C0 (Ω) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 406

Δp

Laplacian on Lp (Rn ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

ΔX

Laplacian on X . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

Δmax

Laplacian on C(Ω) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403

ΔL2 (Ω)

Dirichlet-Laplacian on L2 (Ω) . . . . . . . . . . . . . . . . . . . . . . . . . 140

Ap

system of diﬀerential operators on Lp (Rn ) . . . . . . . . . . . . 449

AX

pseudo-diﬀerential operator on X . . . . . . . . . . . . . . . . . . . . . 431

K(X)

space of compact operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

L(X, Y ), L(X)

space of bounded linear operators . . . . . . . . . . . . . . . . . . . . . . 24

OpX (a), Opp (a)

pseudo-diﬀerential operator on X or Lp (Rn ) . . . . . . . . . . 430

Ker A

kernel of A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261, 467

Ran A

range of A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261, 467

NOTATION

529

A

closure of an operator A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 467

Φ

Riesz operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

ΦS

Riesz-Stieltjes operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

A∗

adjoint of an operator A . . . . . . . . . . . . . . . . . . . . . . . . . . 472, 473

AH

operator associated with quadratic form . . . . . . . . . . . . . . . 419

AY

part of an operator A in Y . . . . . . . . . . . . . . . . . . . . . . . 136, 471

B −z

fractional power of B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

B 1/2

square root of B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

D(A)

domain of an operator A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 467

Dα

higher order partial derivative . . . . . . . . . . . . . . . . . . . 485, 486

Dj

partial derivative ∂/∂xj . . . . . . . . . . . . . . . . . . . . . . . . . . 485, 486

R(λ, A)

resolvent of an operator A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 468

Spectrum and Resolvent Set spB (f )

Beurling spectrum of f . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322

spC (f )

Carleman spectrum of f . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295

sp(f )

half-line spectrum of f . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272

spw (f )

weak half-line spectrum of f . . . . . . . . . . . . . . . . . . . . . . . . . . 325

ρ(A)

resolvent set of an operator A . . . . . . . . . . . . . . . . . . . . . . . . . 468

ρu (A, x)

imaginary local resolvent set . . . . . . . . . . . . . . . . . . . . . . . . . . 371

σ(A, x)

local spectrum of A at x . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299

σ(A)

spectrum of A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 468

σp (A)

point spectrum of A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 469

σu (A, x)

imaginary local spectrum of A at x . . . . . . . . . . . . . . . . . . . 371

σap (A)

approximate point spectrum of A . . . . . . . . . . . . . . . . . . . . . 469

r(T )

spectral radius of an operator T . . . . . . . . . . . . . . . . . . . . . . . 475

s(A)

spectral bound of A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188, 469

530 s0 (A)

NOTATION pseudo-spectral bound of A . . . . . . . . . . . . . . . . . . . . . . . . . . . 345

Subsets of Rn or C C+

open right half-plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

C−

open left half-plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362

T

unit circle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316

N

set of positive integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

N0

set of non-negative integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

∂Ω

topological boundary of Ω . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 401

R+

set of non-negative real numbers . . . . . . . . . . . . . . . . . . . . . . . . 13

Σα

sector of angle α . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

Transformations fˇ

reﬂection of f . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

fˆ, T

Laplace or Carleman transform of f or T . . . . . . 27, 32, 295

F

Fourier transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44, 487

L

Laplace transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

LS

Laplace-Stieltjes transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

F

conjugate Fourier transform . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

dF

Laplace-Stieltjes transform of F . . . . . . . . . . . . . . . . . . . . . . . 55

Cauchy Problems (ACP0 )

homogeneous abstract Cauchy problem . . . . . . . . . . . . . . . . 108

(ACPf )

inhomogeneous abstract Cauchy problem . . . . . . . . . . . . . 117

(ACPk+1 )

(k + 1)-times integrated abstract Cauchy problem . . . . 129

ACP0 (R)

abstract Cauchy problem on the line . . . . . . . . . . . . . . . . . . 118

D(ϕ)

Dirichlet problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 401

P∞ (u0 , ϕ, f )

inhomogeneous heat equation . . . . . . . . . . . . . . . . . . . . . . . . . 415

Pτ (u0 , ϕ)

heat equation with inhomogeneous boundary conditions 408, 412

NOTATION

531

Other Notation (Hr )

growth hypothesis for symbols . . . . . . . . . . . . . . . . . . . . . . . . . 439

Freq(x), Freq(f )

set of frequencies of vector x or function f . . 267, 293, 315

dN(x)

subdiﬀerential of the norm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

supp

support of a function or distribution . . . . . . . . . . . . . . . . . . 318

B(x, ε)

closed ball, centre x, radius ε . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

D

closure of a set D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

B(x, ε)

open ball, centre x, radius ε . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

m(Ω)

Lebesgue measure of Ω . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

Mη x, Mη f

mean of vector x or function f at η . . . . 266, 293, 308, 315

x·ξ

scalar product of x and ξ in Rn . . . . . . . . . . . . . . . . . . 430, 487

x≤y

ordering in a Banach space . . . . . . . . . . . . . . . . . . . . . . . . . . . . 477

X+

positive cone in X . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 477

d

Z → X

continuous dense embedding . . . . . . . . . . . . . . . . . . . . . 184, 418

Z → X

continuous embedding of Z in X . . . . . . . . . . . . . . . . . . . . . 184

Index A Abel-convergence, 244, 257 Abel-ergodic, 263 Abelian theorem, 243, 245 abscissa of boundedness, 33, 285 of convergence, 27, 31, 56, 58 of holomorphy, 33 absolutely continuous, 15, 18 convergent, 14 regular, 486 adjoint, 472, 473 almost periodic function on the half-line, 307 function on the line, 292 orbits, 294 vector, 290, 361 almost separably valued, 7 analytic Radon-Nikodym property, 61 representation, 84 antiderivative, 15 antidual, 420, 422 antilinear, 420 approximate eigenvalue, 469 point spectrum, 469 unit, 23 approximation theorem, 41, 67 asymptotically almost periodic, 307, 365 norm-continuous, 389

B B-convergence, 244 Banach lattice, 478 band, 479 Bernstein, 90, 100, 435 Beurling spectrum, 322

Bochner, 9 integrable, 9 integral, 9 boundary group, 172 semigroup, 171 bounded above, 214, 474 holomorphic semigroup, 150 semivariation, 48 variation, 15, 48 Brenner, 450, 452

C Carleman spectrum, 295 spectrum and C0 -groups, 295 transform, 295 Cauchy problem abstract, 108 inhomogeneous, 117 on the line, 118 Ces` aro-convergence, 244 Ces` aro-ergodic, 262 character, 289 classical solution, 108, 117, 203 closable operator, 467 closed operator, 467 closure, 467 Coifman-Weiss, 175 compact resolvent, 470 complete orbit, 120 completely monotonic, 90, 106 complex inversion, 75, 259 representation, 81 Tauberian condition, 247 cone, 477 converges in the sense of Abel, 244 in the sense of Ces`aro, 244

534 convex, 91 convolution, 21, 24, 26, 486 core, 468 cosine function, 203 countable spectrum, 374, 385 countably valued, 6

D Da Prato-Sinestrari, 142 Datko, 340 densely deﬁned, 467 Desch-Schappacher, 161 Dirac’s equation, 456 Dirichlet boundary conditions, 140, 423 kernel, 257 Laplacian, 424 problem, 401 regular, 402 dissipative, 137 distribution, 485, 487 semigroup, 231 dominated convergence, 11 dual cone, 478 Dunford-Pettis property, 270 theorem, 19

E eigenvalue, 469 elliptic equation, 170 maximum principle, 402 operator, 425, 431 polynomial, 431 ergodic vector, 266 eventually diﬀerentiable, 284 exponential growth bound, 29, 30, 338

F Fattorini, 227 feebly oscillating, 249 Fej´er, 257 kernel, 258

INDEX ﬁrst order perturbation, 160 form domain, 420 Fourier coeﬃcients, 257 inversion theorem, 45, 487 multiplier, 489 sums, 257 transform, 44, 487 type, 61, 387 fractional powers, 163 frequencies, 267, 293, 311 Fubini, 12 function absolutely continuous, 15, 18 absolutely regular, 486 almost separably valued, 7 asymptotically almost periodic, 307 Bochner integrable, 9 completely monotonic, 90 convex, 91 countably valued, 6 feebly oscillating, 249 holomorphic, 461 Laplace transformable, 28 Lipschitz continuous, 15 locally bounded, 462 measurable, 6 normalized, 100 of bounded semivariation, 48 of bounded variation, 15, 48 of weak bounded variation, 48 Riemann integrable, 50 Riemann-Stieltjes integrable, 49 simple, 6 slowly oscillating, 247 step, 6 strongly continuous, 24 test, 15, 485, 487 totally ergodic, 296, 308, 315 uniformly ergodic, 296, 308, 328 weakly measurable, 7 fundamental theorem of calculus, 18

INDEX

G Gaussian semigroup, 150, 153, 156, 170, 172, 183 Gelfand, 280 generating cone, 477 generator inﬁnitesimal, 114 of C0 -group, 119 of C0 -semigroup, 112 of cosine function, 205 of integrated semigroup, 122 of semigroup, 126 of sine function, 218 Glicksberg-deLeeuw, 389 graph norm, 468 Grothendieck space, 270 group C0 , 119, 295 boundary, 172 integrated, 179

H H¨ ormander, 173 half-line spectrum, 272, 310, 315 Hardy, 256 Hardy-Littlewood, 253 Hilbert transform, 198, 491 Hille-Yosida operator, 141 theorem, 134 holomorphic function, 461 semigroup, 148 hyperbolic equation, 427 semigroup, 388 system, 449 hypoelliptic, 431

I ideal, 192, 478 identity theorem, 462

535 imaginary local resolvent set, 371 spectrum, 371 improper integral, 13 inﬁnitesimal generator, 114 Ingham, 327 integral absolutely convergent, 14 Bochner, 9 improper, 13 Laplace, 27 Laplace-Stieltjes, 55 Riemann, 50 Riemann-Stieljes, 49 integrated semigroup, 122 integration by parts, 50 intermediate points, 49 interpolation property, 90 inversion complex, 75, 259 Phragm´en-Doetsch, 73 Post-Widder, 42, 73 invertible, 468 irreducible, 394

J jump, 99

K Kadets, 300 Karamata, 251 Katznelson-Tzafriri, 317, 391 KB-space, 479 kernel, 261, 394, 467 Krein-Smulyan, 464

L L-space, 359 Laplace integral, 27, 32 transform, 63, 110 transformable, 28

536

INDEX

Laplace-Stieltjes integral, 55 transform, 63 Laplacian and boundary group, 173 and boundary integrated group, 183 and cosine functions, 448 ﬁrst order perturbation, 160 generates Gaussian semigroup, 150 on continuous functions, 403 square root, 170 with Dirichlet boundary conditions, 140, 424 with inhomogeneous boundary conditions, 408 largest lower bound, 478 lattice, 478 least upper bound, 478 Lebesgue point, 16 Lipschitz continuous, 15 local integrated semigroup, 232 spectrum, 299, 371 locally bounded, 462 Loomis, 297 Lotz, 272 Lumer-Phillips, 139

M Maxwell’s equations, 455 mean-ergodic, 262 measurable, 6 Mikhlin, 491 mild solution, 108, 117, 119, 203, 368, 408, 413, 415 molliﬁer, 23, 319 multi-index, 485 multiplication operator, 419, 473

N Newtonian potential, 404 non-resonance, 380 normal cone, 477

normalization, 100 normalized antiderivative, 28 function, 100 norming, 462

O operator adjoint, 472, 473 associated with form, 418 closable, 467 closed, 467 elliptic, 425, 431 Hille-Yosida, 141 invertible, 468 kernel, 394 multiplication, 419, 473 Poisson, 404 positive, 478 pseudo-diﬀerential, 430 resolvent positive, 188 Riesz, 72 Riesz-Stieltjes, 66 sectorial, 162 selfadjoint, 150, 473 symmetric, 474 order continuous norm, 479 interval, 477 ordered Banach space, 477

P Paley-Wiener, 46 parabolic domain, 412 equation, 427 maximum principle, 410 problem, 408 part, 471 partitioning points, 49 period, 292 periodic vector, 290

INDEX perturbation compact, 161 ﬁrst order, 160 of C0 -semigroup, 144 of cosine function, 210, 213 of Hille-Yosida operator, 143, 144 of integrated semigroup, 187, 232 of resolvent positive operator, 195 of selfadjoint operator, 420, 423 of sine function, 220 relatively bounded, 159 Petrovskii correct systems, 232 Pettis, 7 phase space, 210 Phragm´en-Doetsch, 73 Phragm´en-Lindel¨ of, 176 Plancherel, 45 point spectrum, 469 Poisson equation, 404 operator, 404 semigroup, 152, 170, 447 positive cone, 477 element, 477 functional, 477 operator, 478 Post-Widder, 42, 73 Pr¨ uss, 82 primitive, 15 principal part, 431, 449 value, 13 proper cone, 477 pseudo-diﬀerential operator, 430 pseudo-resolvent, 470 pseudo-spectral bound, 345

R Radon-Nikodym property, 19, 72 range, 261, 467 real Banach lattice, 478 ordered Banach space, 477

537 representation, 69, 78 Tauberian condition, 247 realization, 430 regular point, 295, 310 regularized semigroup, 232 relatively compact orbit, 288 relatively dense, 288, 310 representation analytic, 84 complex, 81 Paley-Wiener, 46 real, 69, 78 Riesz-Stieltjes, 66 resolvent, 335, 468 compact, 470 equation, 470 identity, 470 positive, 188 set, 468 Riemann integrable, 50 integral, 50 sum, 50 Riemann-Lebesgue, 45 Riemann-Liouville semigroup, 175 Riemann-Stieltjes integrable, 49 integral, 49 sum, 49 Riesz operator, 72 Riesz-Stieltjes operator, 66 representation, 66

S sandwich theorem, 185 Schwartz space, 318, 485 sectorial operator, 162 selfadjoint operator, 150, 473 semigroup, 126 C-, 232 C0 , 111 Abel-ergodic, 263

538 asymptotically almost periodic, 365 asymptotically norm-continuous, 388 boundary, 171 bounded holomorphic, 150 Ces´aro-ergodic, 262 distribution, 231 eventually diﬀerentiable, 284 Gaussian, 150, 153, 156, 170, 172, 183 holomorphic, 148 hyperbolic, 388 irreducible, 394 k-times integrated, 122 local integrated, 232 norm-continuous, 201 once integrated, 122 Poisson, 152, 170, 447 regularized, 232 Riemann-Liouville, 175 smooth distribution, 232 totally ergodic, 266, 373 separating, 8, 262, 464 sesquilinear form, 420 similar operators, 144 simple function, 6 pole, 269 sine function, 206, 218 slowly oscillating, 247 smooth distribution semigroup, 232 smoothing eﬀect, 158 Sobolev space, 488 spectral projection, 472 bound, 188, 343, 469 radius, 475 synthesis, 291, 293, 391 theorem, 474 spectrum, 468 approximate point, 469 Beurling, 322 Carleman, 295

INDEX half-line, 272, 310, 315 imaginary local, 371 local, 299 point, 469 weak half-line, 325 square root, 164 step function, 6 strong convergence, 31 strong splitting theorem, 364 strongly continuous, 24 subdiﬀerential, 137 sublattice, 478 sun-dual, 137 support, 318 symbol, 430 symmetric, 474

T Tauberian condition, 243, 247 theorem, 88, 243, 247 temperate distribution, 485 tempered integrated semigroup, 232 test function, 15, 485, 487 Theorem Abel, 247 Analytic Representation, 84 Approximation, 41, 67 Bernstein, 100 Bochner, 9 Brenner, 450, 452 Coifman-Weiss, 175 Complex Inversion, 75 Complex Representation, 81 Countable spectrum, 374 Da Prato-Sinestrari, 142 Datko, 340 Desch-Schappacher, 161 Dominated Convergence, 11 Dunford-Pettis, 19 Fattorini, 227 Fej´er, 257 Fubini, 12 Gelfand, 280

INDEX Glicksberg-deLeeuw, 389 H¨ormander, 173 Hardy, 256 Hardy-Littlewood, 253 Hille-Yosida, 134 Identity, 462 Ingham, 327 Kadets, 300 Karamata, 251 Katznelson-Tzafriri, 317, 391 Krein-Smulyan, 464 Loomis, 297 Lotz, 272 Lumer-Phillips, 139 Mikhlin, 491 Non-resonance, 380 Paley-Wiener, 46 Pettis, 7 Phragm´en-Doetsch Inversion, 73 Phragm´en-Lindel¨ of, 176 Plancherel, 45 Post-Widder Inversion, 42, 73 Real Representation, 69, 78 Riesz-Stieltjes Representation, 66 Sandwich, 185 Spectral, 474 Splitting, 364, 368, 389 Tauberian, 88 Trotter-Kato, 146 Uniqueness, 40, 294 Vitali, 463 totally ergodic function, 296, 308, 315 semigroup, 266, 373 vector, 266, 290, 361 transference principle, 175 trigonometric polynomial, 292, 365 Trotter-Kato, 146

U UMD-space, 198 unconditionally bounded, 304, 481 uniform ellipticity, 425

539 uniformly convex, 303 ergodic, 296, 308, 328 unimodular eigenvector, 361 uniqueness sequence, 40 theorem, 40, 294 unitarily equivalent, 474

V variation of constants formula, 118, 158 Vitali, 463

W wave equation, 170, 332, 425, 455 weak bounded variation, 48 half-line spectrum, 325 splitting theorem, 368 weakly almost periodic, 294 almost periodic in the sense of Eberlein, 294 asymptotically almost periodic, 334 holomorphic, 461 measurable, 7 regular point, 325 Weierstrass formula, 216 well-posed, 115

Y Young’s inequality, 22, 24