The Functional Calculus for Sectorial Operators (Operator Theory: Advances and Applications, 169) 376437697X, 9783764376970

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Operator Theory: Advances and Applications Vol. 169

Editor: I. Gohberg Editorial Office: School of Mathematical Sciences Tel Aviv University Ramat Aviv, Israel Editorial Board: D. Alpay (Beer-Sheva) J. Arazy (Haifa) A. Atzmon (Tel Aviv) J. A. Ball (Blacksburg) A. Ben-Artzi (Tel Aviv) H. Bercovici (Bloomington) A. Bottcher (Chemnitz) K. Clancey (Athens, USA) L. A. Coburn (Buffalo) R. E. Curto (Iowa City) K. R. Davidson (Waterloo, Ontario) R. G. Douglas (College Station) A. Dijksma (Groningen) H. Dym (Rehovot) P. A. Fuhrmann (Beer Sheva) B. Gramsch (Mainz) J. A. Helton (La Jolla) M. A. Kaashoek (Amsterdam) H. G. Kaper (Argonne)

S. T. Kuroda (Tokyo) P. Lancaster (Calgary) L. E. Lerer (Haifa) B. Mityagin (Columbus) V. Olshevsky (Storrs) M. Putinar (Santa Barbara) L. Rodman (Williamsburg) J. Rovnyak (Charlottesville) D. E. Sarason (Berkeley) I. M. Spitkovsky (Williamsburg) S. Treil (Providence) H. Upmeier (Marburg) S. M. Verduyn Lunel (Leiden) D. Voiculescu (Berkeley) D. Xia (Nashville) D. Yafaev (Rennes) Honorary and Advisory Editorial Board: C. Foias (Bloomington) P. R. Halmos (Santa Clara) T. Kailath (Stanford) H. Langer (Vienna) P. D. Lax (New York) M. S. Livsic (Beer Sheva) H. Widom (Santa Cruz)

The Functional Calculus for Sectorial Operators

Markus Haase

Birkhauser Verlag Basel . Boston . Berlin

Author: Markus Haase Department of Pure Mathematics University of Leeds Leeds LS2 9JT UK e-mail: [email protected]

2000 Mathematics Subject Classification 47A60, 47D03, 47D06, 47D09, 47B44, 47A12, 47A55, 46B70, 46N20, 35K90,35J30,35J20,35JI5

A CIP catalogue record for this book is available from the Library of Congress, Washington D.C., USA Bibliographic information published by Die Deutsche Bibliothek Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data is available in the Internet at .

ISBN 3-7643-7697-XBirkhauser Verlag, Basel- Boston - Berlin 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. © 2006 Birkhauser Verlag, P.O. Box 133, CH-4010 Basel, Switzerland Part of Springer Science+Business Media Printed on acid-free paper produced from chlorine-free pulp. TCF 00 Cover design: Heinz Hiltbrunner, Basel Printed in Germany e-ISBN: 3-7643-7698-8 ISBN-lO: 3-7643-7697-X ISBN-13: 978-3-7643-7697-0 987654321

www.birkhauser.ch

Contents

Preface

xi

1 Axiomatics for Functional Calculi 1.1 The Concept of Functional Calculus 1.2 An Abstract Framework . . . . . . . 1.2.1 The Extension Procedure .. 1.2.2 Properties of the Extended Calculus 1.2.3 Generators and Morphisms 1.3 Meromorphic Functional Calculi ... 1.3.1 Rational Functions . . . . . . . 1.3.2 An Abstract Composition Rule 1.4 Multiplication Operators. 1.5 Concluding Remarks 1.6 Comments . . . . . . . . .

1

2 The Functional Calculus for Sectorial Operators

2.1

2.2 2.3

2.4 2.5

2.6

Sectorial Operators . . . . . . . . 2.1.1 Examples . . . . . . . . . 2.1.2 Sectorial Approximation . Spaces of Holomorphic Functions The Natural Functional Calculus 2.3.1 Primary Functional Calculus via Cauchy Integrals 2.3.2 The Natural Functional Calculus 2.3.3 Functions of Polynomial Growth 2.3.4 Injective Operators . . . . . . . . The Composition Rule . . . . . . . . . . Extensions According to Spectral Conditions 2.5.1 Invertible Operators ........ 2.5.2 Bounded Operators ........ 2.5.3 Bounded and Invertible Operators Miscellanies . . . . 2.6.1 Adjoints .. 2.6.2 Restrictions

1 3 4 5 7 9 10 12 13 15 16 19 19

24 25 26 30 30 34 37 39 41 45 45 46 47 48 48 50

vi

Contents

2.7

2.8 3

4

2.6.3 Sectorial Approximation . 2.6.4 Boundedness . . . . . . . The Spectral Mapping Theorem. 2.7.1 The Spectral Inclusion Theorem 2.7.2 The Spectral Mapping Theorem Comments . . . . . . . . . . .

Fractional Powers and Semigroups 3.1 Fractional Powers with Positive Real Part 3.2 Fractional Powers with Arbitrary Real Part . . . 3.3 The Phillips Calculus for Semigroup Generators. 3.4 Holomorphic Semigroups. . . . . . . . . . 3.5 The Logarithm and the Imaginary Powers 3.6 Comments . . . . . . . . . . . . . . . Strip-type Operators and the Logarithm 4.1 Strip-type Operators . . . . . . . . . 4.2 The Natural Functional Calculus .. 4.3 The Spectral Height of the Logarithm 4.4 Monniaux's Theorem and the Inversion Problem 4.5 A Counterexample 4.6 Comments . . . . . . . . . . . . .

5 The Boundedness of the HOO-calculus 5.1 Convergence Lemma . . . . . . . . . . . . . . . . . . . 5.1.1 Convergence Lemma for Sectorial Operators. . 5.1.2 Convergence Lemma for Strip-type Operators. 5.2 A Fundamental Approximation Technique 5.3 Equivalent Descriptions and Uniqueness 5.3.1 Subspaces. 5.3.2 Adjoints.............. 5.3.3 Logarithms . . . . . . . . . . . . 5.3.4 Boundedness on Sub algebras of Hoo 5.3.5 Uniqueness . 5.4 The Minimal Angle. . . . . . . . . . 5.5 Perturbation Results . . . . . . . . . 5.5.1 Resolvent Growth Conditions 5.5.2 A Theorem of Pruss and Sohr . 5.6 A Characterisation 5.7 Comments . . . . . . . . . . . . . . . .

50 52 53 53 55 57 61 61 70 73 76 81

88 91 91 93 98 100 101 104

105 105 105 107 108 111 112 113 113 114 116 117 119 119 125 127 127

Contents

vii

6 Interpolation Spaces 6.1 Real Interpolation Spaces 6.2 Characterisations..... 6.2.1 A First Characterisation. 6.2.2 A Second Characterisation 6.2.3 Examples....... 6.3 Extrapolation Spaces. . . . . . . . 6.3.1 An Abstract Method. . . . 6.3.2 Extrapolation for Injective Sectorial Operators 6.3.3 The Homogeneous Fractional Domain Spaces 6.4 Homogeneous Interpolation . . . . . . . . . . . . . 6.4.1 Some Intermediate Spaces. . . . . . . . . . 6.4.2 ... Are Actually Real Interpolation Spaces. 6.5 More Characterisations and Dore's Theorem. . . . 6.5.1 A Third Characterisation (Injective Operators) 6.5.2 A Fourth Characterisation (Invertible Operators) 6.5.3 Dore's Theorem Revisited . . . . . . . 6.6 Fractional Powers as Intermediate Spaces . . 6.6.1 Density of Fractional Domain Spaces. 6.6.2 The Moment Inequality . . . . . . . . 6.6.3 Reiteration and Komatsu's Theorem. 6.6.4 The Complex Interpolation Spaces and BIP 6.7 Characterising Growth Conditions 6.8 Comments................

131 131 134 134 139 140 142 142 144 146 149 149 152 153 153 155 156 157 157 158 160 162 164 168

7 The Functional Calculus on Hilbert Spaces 7.1 Numerical Range Conditions . . . . . 7.1.1 Accretive and w-accretive Operators 7.1.2 NormalOperators . . . . . . . . . . 7.1.3 Functional Calculus for m-accretive Operators 7.1.4 Mapping Theorems for the Numerical Range 7.1.5 The Crouzeix-Delyon Theorem. . . . 7.2 Group Generators on Hilbert Spaces . . . . . . . . . 7.2.1 Liapunov's Direct Method for Groups . . . . 7.2.2 A Decomposition Theorem for Group Generators. 7.2.3 A Characterisation of Group Generators . 7.3 Similarity Theorems for Sectorial Operators . . . . . . . . 7.3.1 The Theorem of McIntosh. . . . . . . . . . . . . . 7.3.2 Interlude: Operators Defined by Sesquilinear Forms 7.3.3 Similarity Theorems 7.3.4 A Counterexample . 7.4 Cosine Function Generators 7.5 Comments . . . . . . . . . .

171 173 173 175 177 180 181 185 185 188 190 194 195 197 202 205 208 212

Vlll

Contents

8 Differential Operators 8.1 Elliptic Operators: Ll_ Theory. 8.2 Elliptic Operators: LP- Theory. 8.3 The Laplace Operator . . . . . 8.4 The Derivative on the Line .. 8.5 The Derivative on a Finite Interval 8.6 Comments . . . . . . . . . . . . . .

219

9 Mixed Topics 9.1 Operators Without Bounded Hoo-Calculus . . . . . 9.1.1 Multiplication Operators for Schauder Bases 9.1.2 Interpolating Sequences 9.1.3 Two Examples . . . . . . 9.1.4 Comments . . . . . . . . . 9.2 Rational Approximation Schemes 9.2.1 Time-Discretisation of First-Order Equations 9.2.2 Convergence for Smooth Initial Data. 9.2.3 Stability . . . 9.2.4 Comments................ 9.3 Maximal Regularity . . . . . . . . . . . . . . 9.3.1 The Inhomogeneous Cauchy Problem 9.3.2 Sums of Sectorial Operators. 9.3.3 (Maximal) Regularity 9.3.4 Comments.

251

221 227 231 237 240 247 251 251 253 254 256 256 257 259 261 265 267 267 268 273 277

279

A Linear Operators A.1 The Algebra of Multi-valued Operators . . . . . . A.2 Resolvents. . . . . . . . . . . . . . . . . . . . . . . A.3 The Spectral Mapping Theorem for the Resolvent. A.4 Adjoints . . . . . . . . . . . . . . . . . . . . . . . . A.5 Convergence of Operators . . . . . . . . . . . . . . A.6 Polynomials and Rational Functions of an Operator A.7 Injective Operators . . . . . A.8 Semigroups and Generators

282 286 288 290 292 295 297

B Interpolation Spaces B.1 Interpolation Couples . . . . . . . . B.2 Real Interpolation by the K-Method B.3 Complex Interpolation . . . . .

303 303 305 310

C Operator Theory on Hilbert Spaces C.1 Sesquilinear Forms .. C.2 Adjoint Operators .. C.3 The Numerical Range

315 315

279

317 320

Contents

IX

C.4 C.5 C.6 C.7 C.8

Symmetric Operators. . . . . . . . . . . . . . . . . . . . . . Equivalent Scalar Products and the Lax-Milgram Theorem Weak Integration . . . . . . . . . . . . . . . Accretive Operators . . . . . . . . . . . . . The Theorems of Plancherel and Gearhart .

321 323 325 327 329

D The D.1 D.2 D.3 D.4 D.5 D.6

Spectral Theorem Multiplication Operators. . . . . . . . . . . . . Commutative C*-Algebras. The Cyclic Case. . Commutative C* -Algebras. The General Case. The Spectral Theorem: Bounded Normal Operators The Spectral Theorem: Unbounded Self-adjoint Operators. The Functional Calculus . . . . . . . . . . . . . . . . . . . .

331 331 333 335 337 338 339

E Fourier Multipliers E.1 The Fourier Transform on the Schwartz Space. E.2 Tempered Distributions . . . . . . . . E.3 Convolution. . . . . . . . . . . . . . . E.4 Bounded Fourier Multiplier Operators E.5 Some Pseudo-singular Multipliers . . . E.6 The Hilbert Transform and UMD Spaces. E.7 R-Boundedness and Weis' Theorem.

341 341 343 345 346 349 352 354

F Approximation by Rational Functions

357

Bibliography

361

Index

377

Notation

385

Preface

In 1928 the austrian author Egon Friedell wrote in the introduction to his opus magnum Kulturgesehiehte der Neuzeit [92]: Alle Dinge haben ihre Philosophie, ja noeh mehr: alle Dinge sind Philosophie. Alle Mensehen, Gegenstande und Ereignisse sind Verkorperungen eines bestimmten Naturgedankens, einer eigentiimliehen Weltabsieht. Der mensehliehe Geist hat naeh der Idee zu forsehen, die in jedem Faktum verborgen liegt, naeh dem Gedanken, dessen blofJe Form es ist. Die Dinge pfiegen oft erst spat, ihren wahren Sinn zu offenbaren. 1

And, a few lines after: DafJ die Dinge gesehehen, ist niehts. DafJ sie gewusst werden, ist alles. 2

Although spoken in the context of cultural history these words may also be applied towards the interpretation of mathematical thought. Friedell seems to say that nothing is just a 'brute fact' but the form of an idea which is hidden and has to be discovered in order to be shared by human beings. What really matters is not the mere fact (which in mathematics would be: the truth of a theorem) but is the form of our knowledge of it, the way (how) we know things. This means that in order to obtain substantial understanding ('revealing its true meaning') it is not enough to just state and prove theorems. This conviction is at the heart of my efforts in writing this book. It came out of my attempt to deepen (or to establish in the first place) my own understanding of its subject. But I hope of course that it will also prove useful to others, and eventually will have its share in the advance of our understanding in general of the mathematical world. 1 All things have their philosophy, even more: all things actually are philosophy. All men, objects, and events embody a certain thought of nature, a proper intention of the world. The human mind has to inquire the idea which is hidden in each fact, the thought its mere form it is. Things tend to reveal their true meaning only after a long time. (Translation by the author) 2That things happen, is nothing. That they are known, is everything. (Translation by the author)

Preface

xii

Topic of the book The main theme, as the title indicates, is functional calculus. Shortly phrased it is about 'inserting operators into functions', in order to render meaningful such expressions as An, e- tA , log A, where A is an (in general unbounded) operator on a Banach space. The basic objective is quite old, and in fact the Fourier transform provides an early example of a method to define J(A), where A = .6. is the Laplacian, X = L2 (lR) and J is an arbitrary measurable function on R A straightforward generalisation involving self-adjoint (or normal) operators on Hilbert spaces is provided by the Spectral Theorem, but to leave the Hilbert space setting requires a different approach. Suppose that a class of functions on some set n has a reproducing kernel, i.e.,

J(z) =

In

J(w)K(z, w) f-L(dw)

(z

E

n)

for some measure f-L, and - for whatever reason - one already 'knows' what operator the expression K(A, w) should yield; then one may try to define

J(A) :=

In

J(w)K(A, w) f-L(dw).

The simplest reproducing kernel is given by the Cauchy integral formula, so that

K(z,w) = (w - z)-l, and K(A,w) = R(w, A) is just the resolvent of A. This leads to the 'ansatz'

J(A) =

~

r J(w)R(w, A) dw,

21fZ } &ll

an idea which goes back already to RIESZ and DUNFORD, with a more recent extension towards functions which are singular at some points of the boundary of the spectrum. The latter extension is indeed needed, e.g. to treat fractional powers An, and is one of the reasons why functional calculus methods nowadays can be found in very different contexts, from abstract operator theory to evolution equations and numerical analysis of partial differential equations. We invite the reader to have a look at Chapter 9 in order to obtain some impressions of the possible applications of functional calculus.

Overview The Cauchy formula encompasses a great flexibility in that its application requires only a spectral condition on the operator A. Although we mainly treat sectorial operators the approach itself is generic, and since we shall need to use it also for so-called strip-type operators, it seemed reasonable to ask for a more axiomatic treatment. This is provided in Chapter 1. Sectorial operators are introduced in

Preface

xiii

Chapter 2 and we give a full account of the basic functional calculus theory of these operators. As an application of this theory and as evidence for its elegance, in Chapter 3 we treat fractional powers and holomorphic semigroups. Chapter 4 is devoted to the interplay between a sectorial operator A and its logarithm log A. One of the main aspects in the theory, subject to extensive research during the last two decades, is the boundedness of the Hoo-calculus. Chapter 5 provides the necessary background knowledge including perturbation theory, Chapter 6 investigates the relation to real interpolation spaces. Here we encounter the suprising fact that an operator improves its functional calculus properties in certain interpolation spaces; this is due to the 'flexible' descriptions of these spaces in terms of the functional calculus. Hilbert spaces play a special role in analysis in general and in functional calculus in particular. On the one hand, boundedness of the functional calculus can be deduced directly from numerical range conditions. On the other hand, there is an intimate connection with similarity problems. Both aspects are extensively studied in Chapter 7. Chapters 8 and 9 account for applications of the theory. We study elliptic operators with constant coefficients and the relation of the functional calculus to Fourier multiplier theory. Then we apply functional calculus methods to a problem from numerical analysis regarding time-discretisation schemes of parabolic equations. Finally, we discuss the so-called maximal regularity problem and the functional calculus approach to its solution. To make the book as self-contained as possible, we have provided an ample appendix, often also listing the more elementary results, since we thought the reader might be grateful for a comprehensive and nevertheless surveyable account. The appendix consists of six parts. Appendix A deals with operators, in particular their basic spectral theory. Our opinion is that a slight increase of generality, namely towards multi-valued operators, renders the whole account much easier. (Multivalued operators will appear in the main text occasionally, but not indispensably.) Appendix B provides basics on interpolation spaces. Two more appendices (Appendix C and D) deal with forms and operators on Hilbert spaces as well as the Spectral Theorem. Finally, Appendix F quotes two results from complex approximation theory, but giving proofs here would have gone far beyond the scope of this book. Instead of giving numbers to definitions I decided to incorporate the definitions into the usual text body, with the defined terms printed in boldface letters. All these definitions and some other key-words are collected in the index at the end of the book. There one will find also a list of symbols.

Acknowledgements This book has been accompanying me for more than three years now. Although I am the sole author, and therefore take responsibility for all mistakes which

XIV

Preface

might be found in it, I enjoyed substantial support and help, both professional and private, without which this project would never have been completed. On the institutional and financial side I am deeply indebted to the Abteilung Angewandte Analysis of the Universitiit Ulm, where I lived and worked from 1999 to 2004. The major progress with the manuscript was made at the Scuola Normale Superiore di Pisa, where I spent the academic year 2004/05 as a research fellow of the Marie Curie Training Network Evolution Equations for Deterministic and Stochastic Systems (HPRN-CT-2002-00281). I am very grateful to Professor Giuseppe Da Prato for inviting me to come to Pisa. H. Garth Dales (Leeds) has contributed substantially with his uncontestable competence regarding grammar and style, and to him go my warmest thanks. Imre Voros (Oxford) read parts of the manuscript and helped considerably in reducing the total number of misprints and other mistakes. lowe also thanks to Birkhiiuser editor Thomas Hempfling, especially for his patience and his calm, professional handling of the publishing procedure. Major support came and continues to come from many of my colleagues, friends, and family members. I am very much indebted to them, sometimes because of fruitful criticism or mathematical discussion, but more often for their encouragement and constant interest in my work. In particular I would like to mention Wolfgang Arendt, Andras Batkai, Charles Batty, Stefan Gabler, Bernhard Haak, Dietlinde Haase, Catherine Hahn, Annette and Peter Kaune, Hans Kiesl, Mate Matolcsi, Rainer Nagel, utf Schlotterbeck and Johannes Zimmer. Each of them has a special share in my work and in my life. I dedicate this book to my father, Hans Albert Haase (1929-1997).

February 2006,

Markus Haase

Chapter 1

Axiomatics for Functional Calculi We convey the fundamental intuition behind the concept "functional calculus" (Section 1.1). Then we present a formalisation of certain ideas common to many functional calculus constructions. In particular, we introduce a method of extending an elementary functional calculus to a larger algebra (Section 1.2). In Section 1.3 we introduce the notion of a meromophic functional calculus for a closed operator A on a Banach space and specialise the abstract results from Section 1.2. As an important example we treat multiplication operators (Section 1.4). In Section 1.3.2 we prove an abstract composition rule for a pair of meromorphic functional calculi.

1.1

The Concept of Functional Calculus

Consider the Banach space X := C [0, 1] of continuous functions on the unit interval with values in the complex numbers C. Each function a E X determines a bounded linear operator

Ma = (g

f----+

ag) : qa, 1]

---+

qa, 1]

on X called the multiplication operator associated with a. Its spectrum u(Ma) is simply the range a[O, 1] of a. Given any other continuous function f : u(Ma) ---+ rc one can consider the multiplication operator Mjoa associated with foa. This yields an algebra homomorphism

into the algebra of bounded linear maps on X. Since one has (z) = Mat and ((A - Z)-l) = R(A, Ma) for A E e(Ma), and since the operator (f) is simply multiplication by f(a(z)), one says that (f) is obtained by 'inserting' the operator Ma into the function f and writes f(Ma) := (f). Generalising this example to the Banach space X = CoOR) and continuous functions a E C(lR), one realises that boundedness of the operators is not an essential requirement. tWe simply write z := (z f---> z) for the coordinate function on C. Hence the symbols f(z) and f are used interchangeably.

Chapter 1. Axiomatics for Functional Calculi

2

The intuition of functional calculus now consists, roughly, in the idea that to every closed operator A on a Banach space X there corresponds an algebra of complex-valued functions on its spectrum in which the operator A can somehow be 'inserted' in a reasonable way. Here 'reasonable' means at least that J(A) should have the expected meaning iJ one expects something, e.g., if oX E g(A) then one expects (oX-z )-1 (A) = R(oX, A) or if A generates a semigroup T then etZ(A) = T(t). (This is just a minimal requirement. There may be other reasonable criteria.) In summary we think of a mapping J f------+ J (A) which we (informally) call a Junctional calculus for A. Unfortunately, up to now there is no overall formalisation of this idea. The best thing achieved so far is a case by case construction. We return to some examples. If one knows that the operator A is 'essentially' (i.e., is similar to) a multiplication operator, then it is straightforward to construct a functional calculus. By one version of the spectral theorem, this is the case if A is a normal operator on a Hilbert space (cf. Appendix D). As is well known, the Fourier transform on L2(JR) is an instance of this, and hence it provides one of the earliest examples of a non-trivial functional calculus. (The operator A in that case is just id/ dt.) In general, there is no canonical 'diagonalisation' of the normal operator A, and so the resulting functional calculus depends - at least a priori - on the chosen unitary equivalence. Hence an additional argument is needed to ensure independence of the construction (cf. Theorem D.6.1). Having this in mind as well as for the sake of generality, one looks for intrinsic definitions. It was the idea of RIESZ and DUNFORD to base the construction of J(A) on a Cauchy-type integral

J(A) =

~ ( 27TZ

lr J(z)R(z, A) dz.

(1.1)

The idea behind this is readily sketched. Let us assume that we are given a bounded operator A and an open superset U of the spectrum K := a(A). The so-called 'homology version' of the Cauchy Integral Formula says the following. One can choose a generalised contour (a 'cycle') f with the following properties: 1) f* C U \ K;

2) each point of rc \ f* has index either 0 or 1 with respect to f; 3) each point of K has index 1 with respect to f. (The set f* is the trace of the cycle.) With respect to such a contour f, one has

J(a) =

~ { 27TZ

lr

J(z)_1- dz z- a

(1.2)

for each a E K and for every holomorphic function defined on U. (See [48, Proposition VIII. 1. 1] for a proof.) Replacing a by A on both sides in the formula (1.2) and insisting that '1/(z - A)' should be the same as R(z, A), we obtain (1.1). The so-defined functional calculus is called the DunJord-Riesz calculus and it turns out that it is only a special case of a general construction in Banach

1.2. An Abstract Framework

3

algebras, cf. [79, VII.3 and VII.ll], [49, Chapter VII, §4]. Actually it yields an algebra homomorphism q,:= (f ~ J(A)) : A - - C(X),

where A is the algebra of germs of holomorphic functions on a(A). Let us continue with a second example. Consider the Banach space X := C[0,1] again, and thereon the Volterra operator V E C(X) defined by

(V f)(t) :=

fat J(8) ds

(t

E [0,1]).

As is well known, V is a compact, positive contraction with a(A) = {O}. Given a holomorphic function J defined in a neighbourhood of 0, simply choose c > 0 small enough and define 1. J(V) := -2

r

7rZ } 8Be(O)

J(z)R(z, V) dz.

Since IlVnll = lin! for all n E N, one can equally replace z by V in the power-series expansion of f. Now note that V is injective but not invertible, and that Z-I is not holomorphic at O. Nevertheless it seems reasonable to write V-I = (z-I)(V). This suggests an extension of the functional calculus to a larger algebra (of germs of holomorphic functions) which contains the function z-I. In the concrete example of the Volterra operator this extension is easy. (Consider functions holomorphic in a pointed neighbourhood of 0 and without essential singularity at O. This means that the principal part of the Laurent series is a polynomial in z-I. Now insert V as in Appendix A.7 into the principal part and as above into the remainder. Finally add the results.) In more general situations one cannot go back to power or Laurent series, but has to make use of an abstract extension procedure. This is the topic of the next section.

1.2

An Abstract Framework

In this section we describe abstractly how to extend a certain basic functional calculus to a wider class of functions. It is a little 'Bourbakistic' in spirit, and the reader who is not so fond of axiomatic treatment may skip it on first reading and come back to it when needed. To have a model in mind, assume that we are given an operator A on a Banach space X and a basic class of functions £ on the spectrum of A. Assume further that we have a 'method' q, = (f ~ J(A)) : £ - - £(X)

4

Chapter 1. Axiomatics for Functional Calculi

of 'inserting' the operator into functions from £, i.e., £ is an algebra and

(An). TUrning to rational functions we observe that by Theorem 1.3.2, for>. E e(A) one has (>. - z)-l E H(A) and (>. - z)-l(A) = R()., A). Hence if q E qz] is such that all zeros are contained in e(A), we obtain q-l E H(A) and q-l(A) = q(A)-l. Then r(A) = (p/q)(A) = p(A)q-l(A) = p(A)q(A)-l, again by Theorem 1.3.2. The last expression is exactly how r(A) is defined in Appendix A.6. D Next we turn to injective operators.

Proposition 1.3.4. Let A be an injective, closed operator on the Banach space X such that e( A) -=I- 0, and let (E (0), M (0), q» be a merom orphic functional calculus for A. Let p = LkEZ akzk E qz, Z-l] be a polynomial in z, z-l. Then p E M(O)A and p(A) = LkEZ akAk.

Proof. It follows from Theorem 1.3.2 and the assumptions that z, Z-l E M(O)A. Hence qz, z-l] C M(O)A. Now take p E qz, Z-l] as in the hypothesis of the statement. We may suppose that p ~ qz]. Hence we can write p(z) = L;=-n akz k where a-n -=I- 0 and n ~ 1. It follows from Theorem 1.3.2 that p(A) = T(A)-np(A)T(A)n = An(Tnp)(A), where T(Z) := z/(>' - z? and A := T(A)-l. By Theorem 1.3.2 c) we obtain

Chapter 1. Axiomatics for Functional Calculi

12

and this is contained in (.>. - A)2n (I:~~n ak_nA k ) A-n(.>. - A)-2n, by Lemma A.7.2. Applying Lemma A.7.1 and Lemma A.6.1 together with the previous Proposition 1.3.3, we see that this is equal to

(t

ak_nAk) (.>. - A)2n(.>. - A)-2n A- n =

k=O

(t

ak_nA k ) A- n

k=O

~ k~n ak Ak ~ k~n (ad)(A) C~n akz k) (A) ~ p(A). C

o

Taking into account also other points.>. E C such that .>. - A is injective leads us to the following. Theorem 1.3.5. Let A be a closed operator on the Banach space X with Q(A) i=- 0. Let r be any rational function on C with no pole of r being an eigenvalue of A.

Then r(A) is uniquely defined by any meromorphic functional calculus for A. Proof. Let (£(D),M(D), 7f /2 the growth behaviour of the resolvent of V is optimal, i.e. IIAR(A, V)II remains bounded. (One can show that the growth is like 1/ IAI2 when approaching 0 on the imaginary axis, and is more or less exponential when coming from the right.) Now suppose that one is given a function f, holomorphic on some open sector S

O. Then the Cauchy integral (1.1) still makes sense, r being a contour as in Figure 1 below. As we shall see in Chapter 2, in this way a primary calculus is set up which contains (germs of) functions holomorphic at 0 as well as (germs of) functions holomorphic on some sector S

0) and the logarithm log V can be defined (cf. Chapter 3) via this functional calculus.

va

It seems hard to judge if this is now the end of the story for the Volterra operator. Although there is a basic intuition of a kind of a 'maximal functional calculus' for each individual operator, this has not been made precise yet. The best achieved so far is a bundle of constructions of primary calculi for certain classes of operators, like bounded operators, operators described by certain resolvent growth conditions, or semigroup generators.

1.6

Comments

The contents of this chapter mainly rest on [107] where the notion of abstract functional calculus was introduced. As already said, this approach axiomatises the constructions prevalent in the literature. The Fundamental Theorem 1.3.2 thus contains the essence of what an unbounded functional calculus should satisfy. We remark that our treatment here is 'purely commutative' since we think of algebras of scalar functions. However, with some more-or-Iess obvious modifications one can also give an abstract setting for 'operator-valued' functional calculi, where non-commutativity must (and can) be allowed to a certain extent. Let us sketch the necessary abstract framework. Suppose again that . + A is sectorial (of angle w). Given a sectorial

20

Chapter 2. The Functional Calculus for Sectorial Operators

up II AR(A

A)II
. E C. In particular, M(A- 1,w') ::; 1 + M(A,w') for all

and x E X. Then one has x E 1>(A)

limt->oo tn(t + A)-nx = x limt---+oo An(t + A)-n x = 0, and limt->o An(t + A)-n x = x limt->o tn(t + A)-n x = 0.

-¢:::::} -¢:::::}

xE

~(A)

-¢:::::} -¢:::::}

d)

We have N(A)

n ~(A) = 0.

If ~(A) = X, then A is injective.

e) The identity N( An) = N( A) holds for all n EN.

f)

The family of operators (A + 5)6>0 is uniformly sectorial of angle w. Indeed, M(A + 5, w') ::; c(w')M(A, w')

(5) O,W' E (w,1f))

where c(w') = (sinw,)-l ifw' E (0,1f/2j andc(w') = 1 ifw' E [1f/2,1f). The family of operators (EA)s>o is uniformly sectorial of angle w. Indeed, M(EA, w') = M(A, w') for all w' E (w,1f) and all E > 0. The family of operators {(A + 5)(A + E + 5)-1 sectorial of angle w.

g) Let E > 0, n,m

h)

E

N, and x

E X.

IE> 0, 52: O}

is uniformly

Then we have

If the Banach space X is reflexive, one has 1>(A) = X and X

°

= N(A) EB ~(A).

i) If i= >. E C with larg>'1 + w < 1f, then >'A E Sect(w + larg>'l) and M(>'A,w')::; M(A,w'-larg>.1) for each w' E (larg>.1 +w,1fj. j) If 1>(A) = X then A' E Sect(w, X') with M(w', A') = M(w, A) for all w' E (w,1fj.

Proof. a) Set M := M(A). For each x E D(A) and>' >

°

we have

1 Ilxll ::; >.M IIAxl1 + M Ilxll·

°

This implies that Ilxll ::; M Ilxll for all x E 1>(A). If M this is impossible, since X i= and e(A) i= 0.

°

< 1, then 1>(A) = 0, but

Let >'0 < and M' > M. For each J-l E C with IJ-l- >'01 ::; I>'ol/M' we have that f-t E e(A) and R(f-t, A) = En (f-t - >.o)n R(>"o, A)n+1 (Neumann series, see

Chapter 2. The Functional Calculus for Sectorial Operators

22

Proposition A.2.3). Hence it follows that III.IIIR(II. A)II < M '" Ip, - >'oln Mn+l < MM '" (M)n ,.. ,.., - 1>'01 ~ I>'oln 1>'01 ~ M'

< -

(1

+

Ip,->'ol) ~ 1>'01 1-


7r - arcsin(I/M) and define M' := 1/ sin(7r - w'), then clearly M' > M. Choose p, E C such that 7r :2:largp,1 :2: w' and define >'0 := Rep,. Then >'0 < and Ip, - >'01 1>'01 /M', whence IIp,R(p,, A)II S (M' +1)M/(M'-M).

°

s

b) The identity (2.1) is true for all operators and all >. -1= 0, cf. Lemma A.2.1. The second statement follows readily. c) Consider the stated equivalence x E 'D(A) {==} limt--->oo tn(t+A)-nx = x. One implication is clear. For the other implication, let first x E 'D(A) and consider the identity x = t(t + A)-Ix + (l/t)[t(t + A)-I]Ax. After repeatedly inserting this identity into itself one arrives at 1

X

n

= [t(t + A)-I]nx + t ~[t(t + A)-I]k Ax. k=1

This shows limt->oo[t(t+A)-I]nx = x for x E 'D(A). By the uniform boundedness of the operators ([t(t + A)-I]n)t>o this is then true for all x E 'D(A). The other equivalences are treated similarly. d) is an immediate consequence of c). e) Evidently N(A) c N(An). But if x E N(An) and n :2: 2, then in particular x E 'D(An-l) and one has 0 = (t + A)-1 Anx = A(t + A)-1 An-Ix for all t > 0. Because An-Ix E 1{(A), one may apply c) to obtain An-Ix = o. By repeating this argument one finally arrives at Ax = o. f) Define (for the moment) A8 := A + '/>' + 0 . Because>. E and so

S7r-W I ,

we have >'(1

+ A)-IE + (A), but the weak and the strong closure of 1>(A) coincide, by the Hahn-Banach theorem. Hence it follows that x E 1>(A).

= Ax = limt->o(t+A)-l Ax = limt->o A(t+A)-lX = x by c). Therefore the sum is direct. For arbitrary x E X, by the reflexivity of X one can find numbers tn \. 0 and ayE X such that tn(tn + A)-Ix ----' y. But we have tnA(t n + A)-IX -+ o. The graph of A is weakly closed, hence y E N(A). This implies that A(tn + A)-Ix ----' X - y. Therefore x - y is in the weak closure of :R(A), but the weak and the strong closure of :R(A) coincide. It follows that x E N(A) + :R(A).

If x E N(A) n:R(A), then 0

i) If J-l ~ Sw', then J-lA- 1 ~ Sw'-larg>'l. Hence J-lR(J-l,AA) = (J-lA- 1 )R(J-lA-l,A) is uniformly bounded by M(A,W' -largAI) for such J-l. k) This follows since e(A) = e(A') and IIR(A, A)II = IIR(A, A),II = IIR(A, A')II for D all A E e(A), cf. Proposition A.4.2 and Corollary A.4.3. Remark 2.1.2. Let A E Sect(w) be a sectorial operator. Then

M(A, 'P) =

inf

w'E(w,cp)

M(A, Wi)

for each 'P E (w,7r]. This follows from the proof of Proposition 2.1.1 a). Actually, the function ('P f---+ M(A,'P)): (w,7r]---+ [1,00) is continuous, see [161, Proposition 1.2.2].

Chapter 2. The Functional Calculus for Sectorial Operators

24

Let A E Sect(w) on the Banach space X. We define Y := :R(A) and denote by B the part of A in Y, i.e.,

1J(B) := 1J(A) n :R(A)

with

By:= Ay

(y E 1J(B)).

It is easy to see that B E Sect(w) on Y with M(w', B) :s; M(w', A) for all w' E (w,7r]. Moreover, B is injective, so we call B the injective part of A. The identity

(n EN) is easily proved by induction. Proposition 2.1.1 yields the fact that N(A)n:R(A) = N(A) n Y = 0, and by a short argument one obtains

R(A, A)(x EEl y) =

1

>.X EEl R(A, B)y

for all x E N(A) and y E Y.

2.1.1

Examples

Let us browse through a list of examples. Multiplication Operators

Let (0, I;, /1) be a a-finite measure space, and let a E Mes(O, /1; q be as in Section 1.4. Define A := Ma to be the multiplication operator on LP(O, /1). Theorem 1.4.2 shows that K = essrana equals the spectrum a(A). We claim that A is sectorial of angle w E [0, 7r) if and only if K c Sw' This clearly is a necessary condition, but we also have 1 < 1 I R(A , A)II -_ dist(A, K) - dist(A, Sw) I

for all A E IC \ Sw by Theorem 1.4.2. So the pure location of the spectrum already implies the resolvent growth condition. (This is not true for general operators.) Generators of Semigroups

Let -A generate a bounded semigroup (T(t))t>o in the sense of Appendix A.8. Then with M := SUPt>o IIT(t)11 one has

(ReA>O)

(2.2)

by (the easy part of) the Hille-Yosida theorem (Theorem A.8.6). This shows that A is sectorial of angle 7r /2. An estimate of the form (2.2) does not suffice to ensure that -A generates a semigroup, neither if M = 1 (e.g. the Volterra operator on C [0, 1], cf. Section 8.5), nor if A is densely defined (see [233]). However, there is a perfect correspondence between generators of bounded analytic semigroups and sectorial operators of angle strictly less than 7r /2. See Section 3.4 for more information.

2.1. Sectorial Operators

25

Normal Operators and Numerical Range Conditions

Let X = H be a Hilbert space, and let A be a normal operator on H. By the Spectral Theorem D.5.1 the operator A is similar to a multiplication operator on some L2-space. Hence by our first example, A is sectorial of angle w if and only if

a(A) c Sw. If A is not normal but has its numerical range W(A) contained in Sw for some w E [0, 7r /2] then A is sectorial of angle w if and only if A has some resolvent point outside the sector. This follows from Appendix C.3. The Hilbert space theory is dealt with more deeply in Chapter 7. For more concrete examples see Chapter 8.

2.1.2

Sectorial Approximation

The present section formalises an important approximation tool in the theory of sectorial operators. A uniformly sectorial sequence (An)nEN of angle w is called a sectorial approximation on Sw for the operator A if

A E Q(A)

and

R(A, An) ---) R(A, A) in £(X)

(2.3)

for some A ~ Sw.1 From Proposition A.5.3 it follows that in this case (2.3) is true for all A ~ Sw. Moreover, A itself is sectorial of angle w. If (An)n is a sectorial approximation for A on Sw, we write An ---) A (Sw) and speak of sectorial convergence. Proposition 2.1.3. a) If An ---) A (Sw) and all An as well as A are injective, then A-I ---) A-I (Sw). b)

If An ---) A (Sw) and A An ---) A in norm.

E

£(X), then An

E

£(X) for large n

E N,

and

c) If An ---) A (Sw) and 0 E Q(A), then 0 E Q(An) for large n. d) If (An)n C £(X) is uniformly sectorial of angle w, and if An ---) A in norm, then An ---) A (Sw).

e) If A E Sect(Sw), then (A + E)e:>O is a sectorial approximation for A on Sw. f)

If A E Sect(Sw), then (Ae:)O(A) {:} limt-+oo t(t + A)-IX = x in the multi-valued case as well (cf. c) of Proposition 2.1.1). This shows AO n 1>(A) = O. Most statements of this section remain true in the multi-valued case, at least after adapting notation a little. As a rule, one has to replace expressions of the form B(B + >..)-1 by I - >..(B + >..)-1. For example we obtain that Ac = l/E - ((1 - E2)/E) (1 + EA)-1 is a sectorial approximation of A by bounded and invertible operators (cf. Proposition 2.1.3 f)). Note also that part k) of Proposition 2.1.1 holds even without the assumption that A is densely defined. In this case the adjoint A' is a multi-valued sectorial operator.

2.2

Spaces of Holomorphic Functions

In the next section we construct a functional calculus for sectorial operators. This is done by proceeding along the lines of Chapter 1 with the primary calculus constructed by means of a Cauchy integral. Since the spectrum of a sectorial

2.2. Spaces of Holomorphic Functions

27

operator is in general unbounded, one has to integrate along infinite lines (here: the boundary of a sector). As a matter offact, this is only possible for a restricted collection of functions. Dealing with these functions requires some notation, which we now introduce. As in Chapter 1 we write 0(0) for the space of all holomorphic functions and M(O) for the space of all meromorphic functions on the open set 0 C . z)

,

'=

J(>.) (g1e)(z)

. (>. - c)(>. - g(z))z

and prove its product integrability. The representation

F(>. z) = (J(>')) (

,

>.

>.

(>.-c)(>.-g(z))

) ((91 e)(Z)) z

shows that c -j. 0 is harmless since >'/(>' - g(z)) is uniformly bounded because of the conditions g(z) E S')) (

>.

1

>.-g(z)

)

((ge)(z)) z

= (J(>')) (A'l.Ha >. - g(z) z (Recall that 9 has no poles within S 0 such that sup Ilf(tA)11 :::; CfM(A, rp) t>O

for each sectorial operator A E Sect(w), Moreover, given B E [0, rp - w) one has

wE

(O,rp), on a Banach space X.

Ilf(AA)11 :::; CfM(A, rp - B)

for all A E te, larg AI :::; B. Proof. Write f = 'ljJ + a/(1 + z) + b with 'ljJ E H[f(S",), a, bE C. We can choose > 0 such that

C, S

By a change of variable in the defining Cauchy integral (or, if you wish, an application of the composition rule), we obtain 'ljJ(tA) = 'ljJ(tz)(A) for all t > O. Hence by Lemma 2.6.10 we have

2C 11'ljJ(tA) II :::; -M(A, rp). SJr

2.7. The Spectral Mapping Theorem

53

This implies that

Ilf(tA)11 : : :

2C -M(A, 'P) + S7r

lal

M(A) +

Ibl::::: [2C] -S7r + lal + Ibl

M(A, 'P)

since 1 ::::: M(A) ::::: M(A, 'P), cf. Proposition 2.1.1 a). To prove the second assertion, write ,\ = te-if' with 1111 : : : e. Note that eif' A E Sect(w + e). Since w + e < 'P, we may apply the (already proved) first part of the Proposition and obtain

Remark 2.6.12. Proposition 2.6.11 remains meaningful and true if A is a multivalued operator (cf. Remark 2.3.4). This will playa role in Section 3.4. For more results on boundedness and on approximation we refer to Section 5.2 and Theorem 9.2.4.

2.7

The Spectral Mapping Theorem

In this section we prove a spectral mapping theorem for the natural functional

calculus for sectorial operators. The first step is a spectral inclusion theorem.

2.7.1

The Spectral Inclusion Theorem

We begin with two auxiliary results. Lemma 2.7.1. Let A E Sect(w), and let ,\ E (f(B)) = 1>(f(A)) n Y and f(B)x = f(A)x E Y for all x E 1>(f(A)) n Y. Since f(A) is invertible by assumption, also f(B) is invertible with f(B)-l = f(A)-lly. Since f has polynomial limit 0 at 00, clearly f is in the domain of the primary functional calculus for B! Moreover, for large n E N the function 9 := zfn still has this property. This gives g(B) = Bf(B)n E C(X). But f(B) is invertible, whence B must be bounded. Since if(B) C {oo}, we conclude that if(B) = 0, hence Y = O. This shows that A is bounded, i.e., 00 tJ- if(A). We now consider the case f-L = 0, AD = O. Suppose again that f(A) is invertible and f has polynomial limit 0 at O. Let e E £ be a regulariser for f. Then ef E £, (ef)(A) is injective and (ef)(O) = O. Lemma 2.3.8 now shows that A must be injective. Let B := A-I, and define g(z) := f(Z-l). By Proposition 2.4.1, 9 E M[Sw]B, g(B) = f(A) is invertible and 9 has polynomial limit 0 at 00. By what we have shown above, B is bounded, whence A is invertible. Finally, we deal with the case f-L = 00. Suppose that 00 = f-L tJ- if(f(A)). By definition, f(A) must be bounded. Hence we can find A E C such that A - f(A) is invertible. By Theorem 1.3.2, 9 := l/(A - f) E M[Sw]A, g(A) is invertible and 9 has polynomial limit 0 at AD. By the results achieved so far we can conclude that AD tJ- if(A). 0 Cf. [108] for a slightly stronger result.

2.7.2

The Spectral Mapping Theorem

We are heading towards the final goal. As usual, we start with an auxiliary result.

Chapter 2. The Functional Calculus for Sectorial Operators

56

Lemma 2.7.6. Let A E Sect(w) and f E M[SwJ. Suppose that f has finite polynomiallimits at iJ(A) n {O,oo} and all poles of f are contained in e(A). Then the following assertions are true. a)

If {O, oo} c iJ (A), f (A) is defined by the nfc for sectorial operators.

b)

If A is invertible, f (A) is defined by the nfc for invertible sectorial operators.

c)

If A is bounded, f(A) is defined by the nfc for bounded sectorial operators.

In either case, f(A) E £(X). Proof. Let f be as required, and let >'0 be a pole of no EN, the function (>'0 - z)no II:= (1 + z)no f(z)

f.

Then, for suitably large

also has finite polynomial limits at iJ(A) n {a, oo}, but one pole less than f. Moreover, letting ro:= (>'0 - z)noj(1 + z)no, we see that ro(A) is bounded and invertible. Suppose that we are in the situation of a). Then f can have only finitely many poles. By induction we find a bounded rational function r such that r(A) is bounded and invertible and r fEE. Hence r regularises f to E and f(A) = r(A)-l(rf)(A) E £(X). (Cf. also Proposition 1.2.5.) A similar reasoning applies in the other cases. Namely, only finitely many poles of f are situated in the 'relevant' part of the domain of f. For example, if A is invertible, the poles of j may accumulate at 0, but for the nfc for A, behaviour D of f near is irrelevant (cf. Remark 2.5.1).

°

Proposition 2.7.7. Let A E Sect(w), and let f E M[SwJA have polynomial limits at the points iJ(A) n {O,oo}. Then iJ(f(A)) C f(iJ(A)).

Proof. Note that by assumption j(>.) is defined for each>' E iJ(A). Take f..1 E C such that f..1 tJ- j(iJ(A)). Then the function (f..1- f)-1 E M[SwJ has finite polynomial limits at {a, oo} n iJ(A) and all of its poles (namely, the points>. E Sw \ {a} where f(>.) = f..1) are contained in the resolvent set of A. Hence one may apply Lemma 2.7.6 to conclude that (f..1- f)-l(A) is defined and bounded. But this implies that f..1- f(A) is invertible, whence f..1 tJ- iJ(f(A)). Now suppose f..1 = 00 tJ- f(iJ(A)). This implies that the poles of f are contained in the resolvent of A. An application of Lemma 2.7.6 yields that f(A) is a bounded operator, whence 00 tJ- iJ(f(A)). D

Theorem 2.7.8 (Spectral Mapping Theorem). Let A E Sect(w), and let f E M[SwJA have polynomial limits at {a, oo} n iJ(A). Then

j(iJ(A)) = iJ(f(A)). Proof. Combine Proposition 2.7.7 and Theorem 2.7.5.

D

2.8. Comments

2.8

57

Comments

2.1 Sectorial Operators. Sectorial operators in our sense were introduced by KATO in [126] who however used the name 'sectorial' for something different (see below). At the same time BALAKRISHNAN in [24] considered operators A satisfying a resolvent estimate SUPt>o + < 00. These operators were later called non-negative operators by KOMATSU. On Banach spaces the concepts of non-negative and sectorial operators coincide (Theorem 2.1.1), but this is no longer true when one passes to more general locally convex spaces, cf. [161]. Many articles follow KATO'S definition of the name 'sectorial' to denote operators on Hilbert spaces associated with sectorial forms (cf. [130]). We have decided to call them Kato-sectorial instead, see Chapter 7. Other texts reserve the name 'sectorial operator' for generators of holomorphic semi groups , like [85] or [157]. Up to a minus sign these operators conform to our sectorial operators of angle strictly less than 7f /2. Finally, a sectorial operator is often required to have dense domain and range. However, we felt it more convenient to drop this additional density assumption. Most of the material of this section is adapted from [161, Section 1.2], where also the standard examples are presented. The notions 'uniform sectoriality' and 'sectorial approximation' unify methods well known in the literature. E.g., the family of operators Ac = (A + f)(l + fA)-1 is used in [192, Section 8.1] and [145, Section 2] and is called 'Nollau approximation' in [179]. Basics on multi-valued sectorial operators can be found in [160, Section 2].

Ilt(t A)-III

2.2 Spaces of Holomorphic Functions. The name 'Dunford-Riesz class' is taken from [216, Section 1.3.3.1 ] and [218, Section 2] (where the symbol V R is used). In [51] this class is denoted by w(Srp) but meanwhile the notation HOC is prevalent. Example 2.2.6 is from McINTOSH's seminal paper [167]. 2.3 The Natural Functional Calculus. As for the historical roots of the functional calculus developed here, we have already given a short account in Chapter 1. The 'mother of all functional calculi', so to speak, is provided by the Fourier transform or, more generally, by the spectral theorem for bounded normal operators on a Hilbert space (see Appendix D). The aim to obtain a similar tool for general bounded operators on a Banach space lead to the so called Dunford-Riesz calculus, see [79, Section VIL3] for the mathematics and [79, Section VIL11] for some historical remarks. This turned out to be only a special case of a general construction in Banach algebras; an account of it can be found in [49, Chapter VII, §4]' cf. also Chapter 1. However, in the Banach space situation only bounded operators were considered so far. The first approach to a (Banach space) calculus for unbounded operators was to reduce it to the bounded case by an application of a resolvent/elementary rational function, cf. [79, Section VII.9] or [3, Lecture 2]. This functional calculus

58

Chapter 2. The Functional Calculus for Sectorial Operators

is sometimes called Taylor calculus. Here as in the bounded case, only functions are used which are holomorphic in a neighbourhood of the spectrum, where 00 has to be considered a member of the spectrum if A is unbounded (see Section A.2). Transforming back to the original operator one obtains a formula of the form

J(A) = J(oo)

+~ r J(z) R(z, A) dz 21[21r

where r is a suitable finite cycle that avoids a(A), cf. Corollary 2.3.5. With the Hilbert space (i.e. multiplication operator) examples in mind it became clear that bounded ness of operators should not play an essential role in a general framework, neither as a requirement for the original operator A nor as a restriction for the result J(A). That J(A) is always bounded in the Taylor setting can be regarded as a mere coincidence due to the particular functions used. The decisive step to abandon bounded ness was to realise how to involve functions which are singular in certain boundary points of the spectrum. Despite later development this was first done for strip-type operators (see Section 4.1) by BADE [21]. He introduced the 'regularisation trick' which we have formalised in the extension procedure of Section 1.2. However, since 00 is somehow a 'hidden' spectral point, it was not obvious what was going on and in fact became fully clear only after McINTOSH [167] constructed the setup for sectorial operators. DE LAUBENFELS [65] constructs primary functional calculi for several classes of operators and uses regularisation by resolvents. More or less systematic accounts of the natural functional calculus for sectorial operators can be found, e.g., in [3], [216] or [222]. Our presentation differs from these approaches in two decisive points. First, they mostly consider only injective operators, with the obvious disadvantage that the usual fractional powers of a non-injective sectorial operator (which may even be bounded) is not covered by their methods. The second main difference between our and other treatments lies in that we do not make any density assumptions on either the domain or the range of the operator. This is due to the fact that there are important examples of operators which are not densely defined, like the Dirichlet Laplacian on C (0) where n is some open and bounded subset of ~n, cf. [10, Section 6.1]. Moreover, it seems to be advantageous to discard density assumptions in a systematic treatment because then one is not tempted to prove things by approximation and closure arguments (which usually are a tedious matter). The focus on injective operators (with dense range and dense domain) in the literature is of course not due to unawareness but a matter of convenience. As one can see, e.g., in the proof of Lemma 2.4.4, working only with injective operators often makes life easier. And McINTOSH [167] outlines how to treat the matter without the injectivity assumption. 2.4 The Composition Rule. Most of the main properties of the natural functional calculus as enumerated, e.g., in Theorem 1.3.2 are folklore (even if e) and f)

2.8. Comments

59

appeared quite late in printed form (in [107]). The composition rule (Theorem 2.4.2) however has been underestimated for a long time. Our account is based on [107], cf. [142]. We believe that only the composition rule opens the door for a fruitful use of the natural functional calculus beyond the mere definition. 2.5 & 2.6 Extensions According to Spectral Conditions and Miscellanies. These sections supplement the two previous ones. To sum up, we have defined four compatible 'natural' functional calculi for certain classes of (sectorial) operators. This is by far not the end of the story. For example, suppose that A is sectorial but there is E > 0 such that the vertical line {Re z = E} separates the spectrum of A. Then by using appropriate contours a new primary calculus can be defined, leading to a natural functional calculus for such operators. Obviously one can think of infinitely many modifications which all show the same pattern. The results on adjoints may also be found in [107]. Lemma 2.6.10 and as Proposition 2.6.11 are folklore results. The very Proposition 2.6.11 plays a decisive role in practically all non-trivial boundedness results in connection with functional calculus, cf. Theorem 9.2.4. 2.7 The Spectral Mapping Theorem. Spectral mapping theorems (SMTs) have a long tradition and are ultimately important. In semigroup theory for example a spectral mapping theorem a(e tA ) = eta-rAJ has a wealth of consequences regarding the asymptotics of the semigroup. (One can find a thorough discussion in [85, Chapter IV, Section 3].) It is well known that the SMT holds for the Dunford~ Riesz calculus, see [79, Section VII.3] or [49, Chapter VII, §4]. Also, there is a SMT for the Hirsch functional calculus, see below and [161, Chapter 4]. DORE [74] has obtained partial results for our functional calculus, with the key notion of polynomial limits, although implicit. Our presentation is based on [108] where even more general results are presented. The proofs are generic, whence similar results hold for other types of holomorphic functional calculi, cf. Remark 4.2.7. Other Functional Calculi. Although not particularly important for our purposes, we mention that there is a wealth of other functional calculi in the literature. The common pattern is this: Suppose that you are given an operator A and a function J for which you would like to define J(A). Take some (usually: integral) representation of J in terms of other functions 9 for which you already 'know' g(A). Then plug in A into the known parts and hope that the formulas still make sense. Obviously, our Cauchy integral-based natural calculus is of this type (the known parts are resolvents). Other examples are: 1) The Hirsch functional calculus, based on a representation

J(z) = a +

r _z_ J1(dt)

IIR+ 1 + tz

60

Chapter 2. The Functional Calculus for Sectorial Operators where J-L is a suitable complex measure on lR+. Details can be found in [161, Chapter 4]. In [107] it is shown that if A fails to be injective, there exist functions J which allow a representation as above but are not in the domain of the natural functional calculus. This can be remedied by extending the natural functional calculus by topological means, see [107, Section 5].

2) The Phillips calculus, based on the Laplace transform

J(z) =

r

lIR+

e- zt J-L(dt)

where J-L is a finite complex measure on lR+. (Here, the 'known' part is e-zt(A) = e- tA , i.e., -A is assumed to generate a bounded Co-semigroup.) We shall encounter this calculus in Section 3.3. 3) The Mellin transform calculus as developed in [193] and [218]. 4) A functional calculus based on the Poisson integral formula, see [36] and [64]. Of course, each of these calculi can be extended by the 'regularisation trick'. For the Phillips calculus, this is done in [216, Section 1.3.4]. Let us mention that there are first attempts to define a functional calculus for multi-valued sectorial operators, see [1] and [160].

Chapter 3

Fractional Powers and Semigroups In this chapter we present the basic theory of fractional powers A" of a sectorial operator A, making efficient use of the functional calculus developed in Chapter 2. In Section 3.1 we introduce fractional powers with positive real part and give proofs for the scaling property, the laws of exponents, the spectral mapping theorem, and the Balakrishnan representation. Furthermore, we examine for variable e > 0 the behaviour of (A + e)", and the behaviour of A"x for variable 0 Rea. Then zOO(l + z)-n E Ho(S 0, and let x E 1J(A'Y). Then the mapping (a

is holomorphic.

f-------*

Aax) : {a Eel 0< Rea < ReI'}

-----+

X

3.1. Fractional Powers with Positive Real Part

65

Proof. Without loss of generality we may suppose 'Y > O. Choose n E N, n > 'Y. Then A"'x = B",nh x and x E 'D(Bn) where B := AT/n. Now the claim follows from Proposition 3.1.1, b). D We now compare the operators A'" and (A + c)'" for c > Proposition 3.1.7. Let Re a E (0, 1) and c

o.

> O. Then

Tc := ((z + c)'" - z"')(A) E .c(X)

and A'" + Tc

=

(A + c)",.

Moreover, we have IITcl1 :S Cc Re "" where C = C(a, 'P, M(A, 'P)) for'P E (w,7I"). Proof. Let 'l/h(z) := (z + 1)'" - z'" - (z + 1)-1. A short computation shows that '1f;1 E Ho· Hence '1f;c(z) := c"''1f;l(Z/c) = (z + c)'" - z'" - c"'(c- 1z + 1)-1 is also contained in Ho. This shows that (z + c)'" - z'" E £(5'1'). By part c) of Theorem 1.3.2 and the composition rule we obtain A'" + Tc = (A + c)"'. Furthermore, we have

The claim now follows from sup 11(1 c>o

+ c- 1A)-III = M(A)

and

11'1f;1(c- 1A)II :S M(A, 'P)C('1f;I, 'P)

for 'P E (w,7I") (apply Proposition 2.6.11 with '1f; = 0).

D

Remark 3.1.8. With the help of the Balakrishnan representation (see below) the last proposition can be improved with respect to the constant C. Namely, one can explicitly determine a constant which depends only on M(A), Re a, and Isin a 71" I, see [161, Proposition 5.1.14].

The last proposition implies in particular that TI((A + c) 0 and c > o. Then the following assertions hold.

+ c)"').

b) A"'((A+c)-I)", = (A(A+c)-I)",.

c) limc->o(A + c)"'x = A"'x for each x E 'D(A"').

Proof. Apply the composition rule to the functions f(z) := (z + c)-1 and g(z) := z'" to obtain (z+c)-"'(A) = ((A + c)-I)'" E .c(X). Hence (z+c)-'" E H(A) and (A(A + c)-I)'" = Cz :"'c)"') (A) = z"'(A)(z + c)-"'(A) = A"'((A + c)-I)'" by the composition rule again, whence b) is proved. Furthermore, we have

66

Chapter 3. Fractional Powers and Semigroups

by the composition rule and Theorem 1.3.2 f). Together with b) this shows that

To prove the other inclusion of a), choose x E 'D(Aa) and n first law of exponents and b) we obtain

'D(An)

3

> Reo:. Applying the

(A + E)-n Aax = Aa(A + E)-n X = Aa((A + E)-1 )"((A + E)-1 t-ax

= (A (A + E)-1 )"((A + E)-1 t-ax. This yields ((A + E)-1)n-a x E 'D(An) by Proposition 3.1.1 g). Hence it follows that x = (A + E)n-a((A + E)-1 )n-a x E 'D((A + E)a). We are left to show c). The family of operators (A + E) (A + 1) -1 is a sectorial approximation of A(A + 1)-1 (apply Proposition 2.1.3 c) together with Proposition 2.1.1 f)). Lemma 2.6.7 together with b) implies that

in norm. In particular we have limc:""o(A + E)a X = Aa x for all x E 'D((A + l)a). However, 'D((A + l)a) = 'D(Aa). 0 Remark 3.1.10. The last result together with the rule (Aa)-1 = (A- 1)a (in the case where A is injective) reduces the definition of Aa for (general) sectorial operators A to the one for operators A E £(X) with 0 E e(A) where the usual Dunford calculus is at hand. Therefore, the last result may be viewed as an 'interface' to the literature where often the fractional powers are defined in a different way. Instead of proving a statement for fractional powers directly with recourse to the definition one can proceed in three steps: 1) The validity of the statement is proved for A E £(X) with 0 E e(A).

2) One shows that in the case where A is injective the statement for A follows from the statement for A -1. 3) One shows that the statement is true for A if it is true for all A + small E > 0).

E

(with

In fact, many proofs follow this scheme. Corollary 3.1.11. One has 'D(Aa) C 'D(A) and

~(Aa) C ~(A)

and each sectorial operator A.

for each Re 0: > 0

Proof. Suppose first that A is bounded, i.e., A E £(X). Then with appropriate finite path, we have

Aa =

~ 27fZ

rza R(z, A) dz = ~ Jr

Jr

2m

r

za-l AR(z,

A) dz

r

being an

3.1. Fractional Powers with Positive Real Part

67

by Cauchy's theorem. This gives 9«AO:) C 9«A) if A E £(X). For arbitrary A we apply this to the operator A(A + 1)-1 and obtain

9«AO:) = 9 0:7r. One can deduce M(Aa) :::; M(A) from this, see [161, (5.24) and (5.25)] and [210, (2.23)].

3.2

Fractional Powers with Arbitrary Real Part

To introduce fractional powers with arbitrary real part, i.e., in order to render the definition (0: E q meaningful we have to suppose that the sectorial operator A is injective. (Note that z'" E B(Scp) for all 0: E C and all rp E (0,7r).) Proposition 3.2.1. Let A E Sect(w) be injective, and let 0:,j3 E C. following assertions hold.

= A-a = (A-1)",. Aa+i3 with TI(Ai3) n TI(Aa+ i3 ) = TI(Aa Ai3).

a)

The operator Aa is injective with (Aa)-l

b)

We have A'" Ai3 C

c) IfTI(A) = X = :R(A), then Aa+i3 = A"'Ai3.

Then the

3.2. Fractional Powers with Arbitrary Real Part d) If 0 < Rea

< 1,

71

then

(x E 9«A)). e) If a E lR satisfies lal < 1r/w, then A'" E Sect(lal w) and one has

((3 f)

E

C).

Let Reao, Real> O. Then n(A"'l) n 9«A"'o) - Re ao < Re a < Re al. The mapping

c n(A"') for each a with

is holomorphic for each x E n(A"'l) n9«A"'o). Proof. a) follows from Theorem 1.3.2 f), cf. the proof of Proposition 3.1.1 e).

b) is immediate from c) of Theorem 1.3.2. c) For 1 < n E N we define

1 (1 Tn(A) := n(n + A)-l -:;: :;: + A

)-1

=

(n-l)

(n + A)-l -n- A (1:;: + A

)-1

Then it is easy to see that for kEN the following assertions hold. a) sUPn IITn(A)k II

< 00. b) limn---+oo Tn(A)k X = x for x E n(A) n 3f(A). c) 9«Tn(A)k) C n(Ak) n 9«Ak), and n(Ak) n 9«Ak) = n(A) n 9«A). We now choose kEN such that n(Ak) n9«Ak) c n(A!3). Let x E n(A"'+!3), and define Xn := Tn(A)k X. Then Xn E n(A"'+!3) n n(A!3) c n(A'" A!3) and Xn ----+ x. Furthermore, A'" A!3xn = Tn(A)k A"'+!3x ----+ A"'+!3x. This shows the claim. d) follows from Corollary 3.1.14 by replacing A by A-I. e) follows from a) and Proposition 3.1.2 together with the composition rule (Theorem 2.4.2). To prove f), let Reao, Re al > 0, x E n(A"'l) n9«A"'o), and - Reao < a < Real' If a t/: ilR, then it is clear from Proposition 3.1.1 that x E D(A"'). Let a = is E iR Choose n E N such that Reao,Real < n and define B := Al/n. Then x E n(B) n9«B), hence x E n(Bins) = n(Ais) = n(A"'). Since A"'x = A",+"'oA-"'ox the second statement follows from Corollary 3.1.6. D For actual computations the following integral representations are of great importance.

72

Chapter 3. Fractional Powers and Semigroups

Proposition 3.2.2 (Komatsu Representation)). Let A E Sect(w) be injective. The identities

Aax

sin no: 1 - [1 -x- _-A-IX + n 0: 1+0:

+

/00

sin no: - [1 -x + n

0:

11 0

11 1 1 1

t'3:+ (t+A)- A- xdt

0

(3.5)

t a - 1(t + A)-1 Ax dt]

1

r a (l + tA)- Axdt

-1

1 a t (l +tA-l)-IA-lxdt]

(3.6)

hold for IRe 0:1 < 1 and x E 1)(A) n :R(A). Note that the second formula is symmetric with respect to A and A-I, whence it can again be seen that A -ax = (A -1 )a x for x E 1)(A) n :R(A).

Proof. We suppose first that 0 representation (3.2) we obtain

< Re 0: < 1. Starting from the Balakrishnan

for x E 1)(A) n :R(A), where in the last step we have replaced t by rl in the second integral. This last formula makes sense even for -1 < Re 0: < 1. Hence by holomorphy we obtain (3.6) for IReo:l < 1. The representation (3.5) now follows from (*) with the help of the identity t- 1(t+A)-1 = r 1A-I - (t+A)-IA- 1. D A particularly important subclass of the injective sectorial operators are the invertible ones.

Proposition 3.2.3. Let A E Sect(w) such that 0 E Q(A). Then the mapping (0:

f------+

A-a) : {o: Eel Reo: > O} ----; £(X)

3.3. The Phillips Calculus for Semigroup Generators is holomorphic. For every

rp E (a, n: /2)

one has

sup{IIA-alll ReaE (a, 1), largal

x

and the equivalence

E

1>(A)

73

-¢=:::}

:srp}

< 00

A -ax

lim

=

(3.7)

x.

a---;O,largal~cp

Proof. The first assertion follows from A-a = (A- 1 )a and Proposition 3.1.1 a). To prove (3.7) we employ Proposition 3.2.1 d) and obtain

IIA-axll :S Klsinn:al n:

t io

XJ

CRea(t + 1)-1 dt Ilxll = K .Isinn:a l Ilxll, sm(n:Rea)

where K:= SUPt>o II(t + l)(t + A)-111 < 00. Hence

EL.

IIA-all :SKlsinn:al. n:Rea n:a sm( n: Re a) Re a

If A-ax ----t x, then x E 1>(A) by Corollary 3.1.11. The converse implication follows from Proposition 3.1.15 a). 0

3.3

The Phillips Calculus for Semigroup Generators

In this section we wish to give a formula for the fractional powers in the case that -A generates a bounded semigroup (T(t))t>o. This is actually part of a more general construction. For J-L E M[a, 00) we define its Laplace transform £(J-L) by

£(J-L)(Z) :=

1

(Rez 2: 0)

e- zt J-L(dt)

[0,00)

It is well known (or easy to see) that £(J-L) is a bounded holomorphic function on ((\ and continuous on C+. Moreover,

(s

E

IR)

is the Fourier transform of J-L (see Appendix E.1). From this follows that J-L is uniquely determined by £(J-L). A simple computation involving Fubini's theorem yields the product law

(J-L,V

E

M[a, 00)).

Now take rp E (n:/2, n:) and f E £(Scp). Obviously f E Hoo(C+) n C(C+), so we may ask whether f = £(J-L) for some J-L E M[a, 00). Clearly £(60) = 1, where 60 denotes the Dirac measure at a, and

£(e At dt) hence £(e- t dt)

=

(1

+ z)-l.

=

_1_

z-A

(ReA < 0),

The following lemma deals with the case

f E Ho.

74

Chapter 3. Fractional Powers and Semigroups

Lemma 3.3.1. Let rp E Crr/2,7r) and'IjJ E Ho(S 0, Re 0;

1 0

> O.

Proof. Replacing z by z + E, Lemma 3.3.4 shows that the function (z + E)-a is the Laplace transform of r(0;)-lta-1e-t: t dt. Hence the assertion follows from D Proposition 3.3.2. (We use the composition rule here.)

Corollary 3.3.6. Let - A be the generator of an exponentially stable semigroup T = (T(t))t>o. Then A-a

=-

in the strong sense, for all Re 0;

1

f( 0;)

1

00

ta-1T(t) dt

0

(3.10)

> O.

Proof. We find E > 0 such that the semigroup et:tT(t) is still bounded. If we apply Proposition 3.3.5 to this semigroup (which has generator -(A - E)) we obtain the assertion. D

Note that in no result of this section did we assume that the semigroup is strongly continuous at O.

3.4

Holomorphic Semigroups

In this section we survey the basic properties of (bounded) holomorphic semigroups. In contrast to general semigroups, the holomorphic ones are accessible via the natural functional calculus. This is one reason why the natural functional calculus is so important in the framework of parabolic problems (where the semigroups are holomorphic) but is of minor relevance in a hyperbolic setting. Let A be a sectorial operator of angle W E [0, 7f /2). In contrast to other sections we allow the operator A to be multi-valued. For 0 i=- A E C with larg AI < 7f/2 - w the function e- AZ is in £(Srp) for each

0 we compute

roo e-f..Lte-tA dt = roo e-f..Lt~ r e- tz R(z, A) dz io io 27rZ ir = -. 1

11

2m r

00

0

e-f..L t e- tz dt R(z, A) dz

by Lemma 2.3.2. (Here,

r

= -. 1

1

--R(z, A) dz 1

2m rfL+ z

=

(fL

+ A)-l

is an appropriate contour avoiding 0, see Section 2.3.)

e) Take n E N and A E S7r/2-w. Since both functions (1+z)-n and (1+z)ne- AZ are contained in [we obtain e- AA = (1 + A)-n((1 + z)ne-AZ)(A) by multiplicativity. f) Choose r.p E (0,7r/2 - w). Then (e- AZ (1

+ z)-l)(A)

->

(1

+ A)-l

in norm as

Chapter 3. Fractional Powers and Semigroups

78

S

'z(1 + Z)-l )(A)z ~ x for x E '.D(A) and z E (1 + A)x. By the uniform boundedness proved in c), we obtain e->'Ax ~ x even for all x E '.D(A). g) follows from the Spectral Mapping Theorem (Theorem 2.7.8).

D

Remark 3.4.2. We remark that by a)-d) of the previous proposition the operator family T := (e-tA)t>o is a bounded semigroup with generator -A (cf. Appendix A.8). Part g) impli;s in particular that r(e- tA ) = ets(-A) for each t ;:::: 0, where s(-A) := sup{ReA I A E a(-A)} is the spectral bound of the generator. Since always etwo(T) = r(e- tA ) (see [85, Proposition IV.2.2]), this in turn yields wo(T) = s(-A). Such an identity fails for general semigroups, see [85, IV.2.7]. Proposition 3.4.3. Let w E [0,7r/2), and let A E Sect(w) be single-valued. Then

f(A)e- AA E LeX) for all A E S7r/2-w and all f E A[Sw]. Let'P E (0,7r/2-w) and Rea> 0. Then

More precisely, for each choice of w' such that

E

(w, 7r /2 - 0) there is C = C(zae-Z, w' + 0) (Iarg AI :::; 0).

Proof. Take f E A[Sw]. Then f(z)e->'z is (super)polynomially decaying at 00 and f(z)e->'z - f(O) is polynomially decaying at 0, hence f(z)e->'z E E. By a standard functional calculus argument we obtain f(A)e- AA = (f(z)e->'Z)(A) E LeX). The last statement follows from Proposition 2.6.11 in noting that Aa Aae->'A = (zae-z)(AA) (composition rule). D Let 0 E (0,7r/2]. A mapping T: So ----+ LeX) is called a bounded holomorphic (degenerate) semigroup (of angle 0) if it has the following properties: 1) The semigroup law T(A)T(J-l) = T(A + J-l) holds for all A, J-l E So. 2) The mapping T : So

----+

3) The mapping T satisfies

LeX) is holomorphic. SUP>'ES",

IIT(A) II < 00 for each 'P E (0,0).

(By a), b) and c) of Proposition 3.4.1, (e->.A)>'ESo is a bounded holomorphic semigroup of angle 0 = 7r/2 - w, whenever A E Sect(w) with w E [0,7r/2).) By holomorphy, T is uniquely determined by its values on (0,00). Moreover, if the semigroup law holds for real values, then it holds for all A (see the proof of [10, Proposition 3.7.2]). Also by holomorphy and the semigroup law, the space NT := N(T(A)) is independent of A E So. Hence either each or none of the operators T(A) is injective. If we are given a bounded holomorphic semigroup T and restrict it to the positive real axis (0,00), we obtain a bounded semigroup as defined in Section A.8.

3.4. Holomorphic Semigroups

79

This semigroup has a generator B, defined via its resolvent by

R(>.., B) :=

1

00

e->"tT(t) dt

(Re A > 0).

Let A := -B. Then we have AO = NT by (A.l) on page 298. Hence A is singlevalued if and only if T(w) is injective for one/all W E So. The multi-valued operator A is sectorial (at least of angle 7r/2), since IIA(A+A)-lll

= IIAR(>..,B)II ::::: (supIIT(t)ID RIAl,

e /\

t>O

for all Re A > O. But even more is true: The multi-valued operator A is sectorial oj angle 7r/2 - 0, and we have T(A) = e->..A Jor all A E So.

Proof. Let

0 is a regulariser which compensates exponential growth at 00.

so

Chapter 3. Fractional Powers and Semigroups

e

Remark 3.4.5. Fix E (0, 1f /2]. An exponentially bounded holomorpbic semigroup of angle is a holomorphic mapping T : So ---+ C(X) satisfying the semigroup law and such that {T(>') I >. E Srp, 1>'1 :s:; I} is bounded for each 'P E (0, e). Given such a semigroup, for each 'P E (0, e) one can find numbers Mrp ;::: 1, wrp ;::: such that

e

°

In particular, TI (0,00) is an exponentially bounded semigroup in the sense of Section A.S. Thus it has a generator -A, which is characterised by

for sufficiently large Re >.. Given 'P E (0, e) and wrp as above, we have

Hence all statements on exponentially bounded holomorphic semigroups can be reduced to statements on bounded holomorphic semigroups. For example, given 'P E (-e, e), the mapping (t f------+ T(eirpt)) is an exponentially bounded semi group with generator _e irp A. (See the proof before Proposition 3.4.4.) From Corollary A.S.3 we know that the space of strong continuity of TI(o,oo) is exactly 1J(A). Employing Proposition 3.4.4 and part f) of Proposition 3.4.1 we see that even

for x E 1J(A) and each 'P E [0, e). Let us describe two situations where holomorphic semigroups appear in a natural way. Example 3.4.6 (Subordinated Semigroups). Let -A generate a bounded semigroup, and let a E (0,1). By scaling (Proposition 3.1.2) A'" E Sect(a1f/2) and since a < 1 the operator -A'" generates a bounded holomorphic semigroup of angle at least (1 - a)1f/2. For t > 0, the function e- tz '" belongs to &[S11"/2] and vanishes at 00, so there must be a function ft,,,, E L1 (0,00) such that

10

One then has e- tA '" x = 00 ft,,,,(s )T(s)x ds for all x E X, by the Phillips calculus (Proposition 3.3.2). Since ft,,,, is the inverse Fourier transform of the function r f---+ e-t(ir)"', one has

f t,'" (s) =

-1-1 2' 1f2

ilR

e sz -

tz '"

dz

(s, t > 0, a E (0,1)).

3.5. The Logarithm and the Imaginary Powers

81

Deforming the path yields the explicit formula

11

!t,G: (s) = ?f

00

esr cos IJ-tr '" cos G:IJ sin( sr sin 0 - trG: sin aO + 0) dr,

0

where 0 E [?f/2,?f] is arbitrary, see [230, p.263]. In the case a = 1/2 this reduces to the particularly nice form

°

see [230, p.268] or [10, Thm.3.8.3]. One can show that for all a E (0,1), t > the function ft,G: is positive [230, p.261]. The semigroups (etA"')t>o for a E (0,1) play an important role in the theory of Markov diffusion processes, and are called subordinated semigroups.

Example 3.4.7. Let A be a sectorial operator on a Banach space X, and let a E (0,1/2]. Then AG: E Sect(awA) (Proposition 3.1.2), hence (-AG:) generates a holomorphic semigroup. One can derive Balakrishnan-type representation formulae for these semigroups, namely

e- tA '" =

~ ?f

roo e-tr'" COS7rG: sin(trG: sin ?fa) (A + r)-l dr

io

(t > 0)

where the integral is absolutely convergent in the case where a E (0,1/2) and is convergent in the improper sense when a = 1/2. See [161, Section 5.5] for details.

°

Let A E Sect(w) on the Banach space X, and suppose that E e(A). We know from Proposition 3.2.3 that (A-Z)Rez>O is an exponentially bounded holomorphic semigroup of angle ?f /2, with 1>(A) as its space of strong continuity. In the next section we identify its generator.

3.5

The Logarithm and the Imaginary Powers

We return to our terminological agreement that 'operator' always is to be read as 'single-valued operator' (see p. 279). In fact, we work with an injective, singlevalued, sectorial operator A. The function log z has subpolynomial growth at and at 00, whence it belongs to the class B(Srp) for each t..p E (0, ?fl. Since A is assumed to be injective, the operator

°

log A := (logz)(A) is defined and is called the logarithm of the operator A. Because of the identity log(z-l) = -log z we have log(A -1) = -log A. The next result is fundamental.

82

Chapter 3. Fractional Powers and Semigroups

Lemma 3.5.1 (Nollau). Let A E Sect(w) be injective. If IImAI e(logA) and

R(A, log A) =

1

00

(A

D

-1 1 )2 ogt

-

>

+ 7r 2 (t + A)-l dt.

7r,

then A E

(3.12)

Hence IIR(A, log A) II ::; 7r M(A)(IIm AI - 7r)-1.

We call the formula (3.12) the Nollau representation of the resolvent of log A. Proof. Suppose first that A, A-I E £(X). We choose rp E (w,7r), a > 0 small and b > 0 large enough and denote by r the positively oriented boundary of the bounded sector Sp (a, b). Then

(A-log z)-l (A)

1 lb lb

1 = -. A 11 R(z,A)dz 2m r - og z

= -1

27ri

a

e-i'P . 1 j'P be is . R(te-"'P A) dt + R(be"S A) ds A - log t + irp , 27r -'P A + log b - is '

1

- 27ri (1)

=

Ib a

-1

1

=

1

00

D

1

(A_10gt)2+7r 2 (t+A)-ldt+ 27r

- 27r

(2)

a

ei'P . 1 j'P ae is . R(te"'P A) dt - R(ae"S A) ds A - log t - irp , 27r -'P A + log a - is \ '

(A

J-7r7r -

J7r be is . -7r A+logb_isR(be"S,A)ds

is

ae R ae is A) d s A + log a - is ( ,

-1 1 )2 ogt

+ 7r 2 (t + A)

-1

dt

=:

J(A),

where we have let rp ----; 7r in (1) and a ----; 0, b ----; 00 in (2). Hence we have proved the claim in the special case where A and A-I are both bounded. In the general case define AE := (A + c:)(1 + c:A)-l. Then (AE)E>D is a sectorial approximation of A (see Proposition 2.1.3). Let f(z) := (A -logZ)-l. Obviously f E HOO(S'P) for rp > w. We have already shown that f(AE) = J(AE) E £(X). It follows from the Dominated Convergence Theorem that

f(AE) ----; J(A) =

1

00

D

(A

-

-1 1 )2 ogt

+ 7r

2

(t + A)-l dt.

in norm. Applying Proposition 2.6.9 we obtain f(A) = J(A). From Theorem 1.3.2 f) we conclude that in fact f(A) = (A - log A)-I. Having shown this we compute

3.5. The Logarithm and the Imaginary Powers

roo

IIR(A, logA)11 ::S Jo

=rJffi.
' -logt)2

dt

83

r

M(A)

+ n 21t = Jffi. I(i 1m>. _ s)2 + n21 ds

MW

y'(s2 - ((1m >.)2 - n 2)2

- Jffi. (s2

M(A)

+ ((1m >.)2 -

n2)

+ 4s2(Im >.)2 ds = M(A)n

~ < M(A)n

y'(Im>.)2 - n 2 - IIm>'l- n'

where we have used the (easily verified) inequalities

(S2 _ ((1m >.)2 _ n 2))2

+ 4s 2(Im>.)2 :::: (s2 _ ((1m >.)2 _n 2))2 + 4s 2((Im>.)2 _n 2) = (s2 + ((1m >.)2 - n2))2,

y'(Im>.)2 -n 2

::::

IIm>.l- n.

D

Proposition 3.5.2. Let A E Sect(w) be injective. If 11m >'1 and for each cp > w there is a constant M", such that

M IIR(A,logA)II::S IIm>'I"'-cp In fact M",

and

(IIm>.1

> w, then>.

E

e(logA)

> cp).

= nM(A7f/",). Furthermore, 'D(logA) c 'D(A) n ~(A).

Proof. Let 0: := n/cp. Then A'" E Sect(o:w) is injective. The composition rule yields log(A"') = 0: log(A). Applying Nollau's Lemma 3.5.1 we obtain IIR(IL,o:logA)11 ::S M(A"')n/(lImlLl-n) Letting>.

= IL/O: and cp = n/o: we arrive at IIR(>.,logA)11 ::S M(A"')n/(IIm>'l- cp)

(11m >'1

> cp).

From the Nollau representation (3.12) for R(·, log A) it follows that 'D(log A) c C 'D(A-l) = ~(A). D

'D(A). But -logA = logA-l, hence also 'D(logA)

As promised, we can now identify the generator of the holomorphic semigroup

(A -Z)Re z>o, where A is a sectorial and invertible operator. Proposition 3.5.3. Let A E Sect(w) with 0 E e(A). Then -logA is the generator of the holomorphic semigroup (A -Z)Re z>o. In particular, 'D(log A) = 'D(A). Proof. By semigroup theory, there is c> 0 such that (e-ctA-t)t>o is a bounded semigroup. Clearly it suffices to show that

84

Cbapter 3. Fractional Powers and Semigroups

for some A E C with Re A Then

o < a < b < 00.

> c and 11m A> 7r1. So choose such a A and let

r

b t e->.t ~ z-t R(z, A) dz dt I a e->.t A- dt = Ib a 27rZ Jr 1 lIb 1 e->.az-a - e->.bz-b = -. e->.tz-t dtR(z, A) dz = - . A 1 R(z, A) dz 27rZ r

1

27rZ r = e->.a A -a(A + log A)-I - e->.b(A + log A)-I, a

+ og z

where r is an appropriate path avoiding 0 (see Section 2.5). The last equality is due to the fact that (A + log z)-I(A) = (A + log A)-I E £(X) by Nollau's Lemma 3.5.1. Since ReA> c we have A-bll----t 0 as b ----t 00. From 'D(logA) c 'D(A) and Proposition 3.2.3 we see that e->.a A -a(A + log A)-I ----t (A + log A)-I strongly. This concludes the proof. D

Ilc>.b

Remark 3.5.4. Suppose that A is a bounded sectorial operator. As a matter of fact, the fractional powers (AZ)Re z>o of A form a quasi-bounded holomorphic semigroup as well. What is its generator? If A is injective, the answer is of course log A. But if A is not injective, the generator is not single-valued and cannot be obtained by the natural functional calculus. Since also in this case the generator should be log A in a sense, we feel the necessity of a sophisticated functional calculus for multi-valued operators. However, we do not further pursue this matter here.

We conclude our investigation of the logarithm with the following nice observation. Consider the function f(z) := logz -log(z + c:) = log(z(z + c:)-I). A short computation reveals that f is holomorphic at 00 with f(oo) = O. Hence if A is sectorial and invertible, then f(A) E £(X), i.e., f E H(A). From the usual rules of functional calculus we see that 10g(A + c:) is a bounded perturbation of log A. In particular, 1> (log A) = 1> (log (A + c:)). Let us turn to the imaginary powers Ais of an injective operator A E Sect(w). Proposition 3.5.5. Let A following assertions hold.

E

Sect(w) be injective, and let 0

=1=

s

a) If Ais E £(X), then 'D(A O. Moreover, sUPo::;e::SlII(A + f)isll < 00 and lime:-> (A + f)is x = Ais x for all x E :R(A).

°

e) Let 0 E Q(A). If Ais E £(X), then sUPO

£(X) is a homo-

b)

For)., p tf- Hep the identity ((A - z)(p- Z))-l (B) = R(A, B)R(p, B) holds.

c)

We have (J(Z)(A - Z)-l )(B) = R(A, B)f(B) = f(B)R()., B) for A tf- Hep.

d)

If C is a closed operator commuting with the resolvents of B, then C also commutes with f(B). In particular, f(B) commutes with B and with R()., B) for all A E (!(B).

Proof. a), c) and d) are proved analogously to the corresponding statements in Lemma 2.3.1. The only difficulty is in b). We give a sketch of the proof but leave the details to the reader. Let J(z) := (A - Z)-l(p - Z)-l with)., p tf- Hep. Let us suppose that)., p lie on the same side of Hep, say above. f(B) is given by a Cauchy integral on the contour (~ - i6I) e (~ + i(2) where 61,62 E ('P, 00) are close to 'P. Since the integrand has no singularity within the lower half-plane, we shift the lower path (as 61 -- 00) towards 1m z = -00 without changing its value. This value must therefore be equal to o. As 62 increases, nothing happens in the beginning, but when crossing A we obtain a residue, which is (p - A)-l R(A, B),

94

Chapter 4. Strip-type Operators and the Logarithm

____________________~--------------------~7w '

a(B )

7 w' --------------------~----------------------

Figure 8: Cauchy integral for strip-type operators.

and when crossing f.1 we obtain in addition (..\ - f.1)-1 R(f.1, B). After this the path wanders off to 1m Z = +00 without changing the value any more. So this value is equal to O. Summing up, we obtain

f(B) = R(..\, B) f.1-..\

+ R(f.1, B) = R("\, B)R(f.1, B) ..\-f.1

by the resolvent identity. (When..\ = f.1 a similar reasoning applies, also in the case that ..\, f.1 are situated on different sides of the strip.) D We define F[Hwl := U

w F(H"l, larg/-LI E (w,7rj. Keeping /-L fixed we see that the right-hand side of this equation is defined even for Iarg >"1 E (w', 7r j and is in fact holomorphic as a function of >.. (Lemma 4.3.2). Since the norm of the resolvent blows up if one approaches a spectral value (Proposition A.2.3), no >.. with larg>..1 E (w',7rj can belong to u(A). Furthermore, if we choose

q such that 2/p < a/q. We define the weight w : JR. ---) [0,00) by

w(x)

:=

2°

x< x 0:

{ eax 1

Now we let X := LP(JR., e2X dx) n Lq(JR., w(x)dx) with its natural (sum) norm. It can be shown that X has in fact the UMD property. The space Cc(JR.) of compactly supported continuous functions is dense in X.

Proof. Let f E X. Then 1[-n,n]f ---) f in X as n ---) 00. Hence we may suppose f is compactly supported. Now we approximate f in Lq(JR.) by functions fn E Cc(JR.) such that supp(fn) C [a, b], where a < b and a, b do not depend on n.

that

Since

Lq((a, b), w(x)dx) ~ Lq(a, b) the sequence

fn

tends to

f

'----*

LP(a, b) ~ LP((a, b), e2X dx)

o

in the norm of X.

On X we consider the left shift group (T(t))tEIR defined by

[T(t)f](x)

:=

f(x

+ t)

(x E JR., t E JR.).

Then we have the norm inequalities

Ilfllp + Ilfllq ::; Ilfllx, e~t Ilfllp + e~t Ilfllq

IIT(t)fllx ::; e-~t

IIT( -t)fllx ::;

1:

Proof. Obviously, IIT(t)fllp = C(2/p)t

IIT(t)fll~ =

= e- at

If(xW eax dx + e- at

-00

::; e- at [00 If(xW eax dx + The computation for

IIT( -t)fll~

E

X, t 2': 0).

Ilfllp for t E RIft 2': 0, we have

If(xW w(x - t) dt

J

(f

100

is similar.

t If(xW eax dx + 100 If(xW dx

k

If(xW dx ::;

t

Ilfll~· o

4.5. A Counterexample

103

In particular, it follows that IIT(t)11 :::; 1 for all t 2: 0 and that T is a group. Since Cc(lR) is dense in X and (t t------t T(t)f) : lR -----t X is continuous for each f E Cc(lR), we conclude that T is in fact a Co-group. Claim. We have IIT(t)11 = 1 for all t 2: l.

Proof. Let to 2: O. Choose tl > to arbitrary. Since LP((to, h), e2X dx) ~ LP(to, tI) and this is not embedded into Lq(to, h), there is no inequality of the form Ilfllq :::; C Ilfllp with f E Cc(to, tl)' Hence there is a sequence gn E Cc(to, tI) such that Ilgnllq 2: n Ilgnllp for all n. Letting fn := gn/ Ilgnllq we have Ilfnllq = 1, Ilfnll :::; l/n, and fn == 0 on (-00, to]. Thus, IIT(to)fnllx = e-(2/p)to Ilfnllp + Ilfnllq 2: 1 and Ilfnllx :::; l/n+ 1. Hence 1 2: IIT(to)ll2: 1/(1 + l/n) for all n E N. D Claim. The operator A given by

1>(A) = {f (where

I'

E

X I I' E X}

and Af = f'

denotes the distributional derivative of f) is the generator A of T.

Proof. Let us denote the derivative operator on distributions by 8. If f E 1>(A) there is agE X such that r1(T(t)f - f) ~ 9 in X as t '\. O. Since X '--+ V(lR)' we see that 9 = 1'. Hence, 1>(A) C 1> := {f E X I f' E X} and Af = f' for f E 1>(A). Now, 1 - A is bijective (since T is a contraction semigroup) and 1 - 8 : 1> -----t X is injective (since I' = f E X implies that f is more or less the exponential function). This implies the claim. D Claim. We have s(A)

:=

sup{Re A I A E a(A)} :::; -2/p.

Proof. Let Xp := LP(lR, e2X dx) with the norm Ilfllp defined above, and denote by Ap the generator of the left shift group on Xp. Since IIT(t)fllp = e-(2/p)t Ilfllp for all t E lR and all f E X p, we have so(Ap) :::; -2/p. By the same reasoning as above one can show that 1>(Ap) = {f E Xp I f' E Xp} with Apf = f' for f E Ap. Next, we claim that 1>(Ap) '--+ X. Actually, by the Closed Graph Theorem, only inclusion has to be shown. If f E 1>(Ap) we let 9 := e(2/ p)'f and note that 9 E LP(lR, dx) and g' = (2/p)g + e(2/ p)"j' E LP(JR., dx). Hence 9 E W1,P(lR), and since W1,P(lR) '--+ Co(JR.), there is a constant C > 0 such that If(x)1 :::; Ce-(2/p)x for x E R This immediately implies that f E Lq((O,oo),dx). Moreover, we have If(x)1 e(a/q)x :::; Ce(a/q-2/p)x for all x E JR., and since a/q > 2/p we conclude that f E Lq(( -00,0), eaXdx). Altogether we obtain f E X. Now take A E C with -2/p < Re A. Since (A - A)f = (A - Ap)f for f E 1>(A) c 1>(Ap) we see that A - A is injective. If f E X, then f E Xp and 9 := R(,\, Ap)f E 1>(Ap) C X. But this implies that 9 E 1>(A) since g' = Ag - f EX. Hence A - A is also surjective. This proves that A E e(A). D We take the last step. Obviously, the space X is not only a Banach space but even a Banach lattice and the semigroup T is positive, i.e., T(t)f 2: 0 for all o :::; f E X and all t ::::: O. By [10, Theorem 5.3.1] we conclude that so(A) = s(A) :::; -2/p < 0 = wo(T), whence we are done.

104

Chapter 4. Strip-type Operators and the Logarithm

Corollary 4.5.2. There is a Banach space X with the UMD property and an injective, sectorial operator A on X such that A E BIP(X) and ()A > Jr.

4.6

Comments

4.1 and 4.2. Strip-type Operators and their Natural Functional Calculus. The natural functional calculus for strip-type operators appears first in [21]. It is discussed in [65] in a general setting and used in [34] and companion papers. Our presentation here follows the lines of [103]. Assume that one is given an operator B with spectrum in a horizontal strip Hw and such that the resolvent is bounded on some horizontal lines lR ± i'P with 'P > w. Then one can construct the 'functional calculus' as we did in Section 4.2, since the basic Cauchy integrals converge. However, it may happen that this 'functional calculus' is the zero mapping. If one requires that the calculus behaves well for rational functions, i.e., is really a functional calculus for B, then B is forced to be strip-type. This was shown in [101]. 4.3 The Spectral Height of the Logarithm. Theorem 4.3.1 is due to the author [101]. The Priiss-Sohr result (Corollary 4.3.4) is part of a celebrated theorem of PRUSS and SOHR [193, Theorem 3.3]. (See Corollary 5.5.12 for the second part.) Its original proof rests on the Mellin transform calculus for Co-groups, cf. also [216, Proposition 3.19], [218], and [161, Chapter 9]. Corollary 4.3.5 is originally due to McINTOSH [167]. 4.4 Monniaux's Theorem and the Inversion Problem. For the proof of Theorem 4.4.3 MONNIAUX [172] utilises the theory of analytic generators of Co-groups. In the same paper she constructs Example 4.4.1 to show that the conclusion may fail when one discards the UMD assumption. This example also shows that (strong) strip-type operators are more general than sectorial operators. Up to now, Monniaux's theorem seems to be the only positive result on the inversion problem. 4.5 A Counterexample. The example of a group with differing growth bound and abszissa of uniform boundedness of the resolvent is due to the author [103] and is an adaptation of an example given by WOLFF [225]. We are indebted to BATTY for bringing this result to our attention. Although we do not know it for sure, we expect that in our example the operator does not have a bounded natural Hoo_ calculus on some strip (cf. Section 5.3.3 for terminology). The derivative on LP(lR), p '" 2 does not have bounded Hoo-calculus on strips, see Section 8.4. KALTON [123] has given an example of a sectorial operator with bounded Hoo-calculus whose sectoriality angle differs from the HOO-angle, cf. Section 5.4.

Chapter 5

The Boundedness of the Hoo-Calculus This chapter mainly provides technical background information. We start with the so-called Convergence Lemma (Section 5.1) and some fundamental boundedness and approximation results (Section 5.2). Then we prove equivalence of boundedness of Ffunctional calculi for different subalgebras F of Hoo (Section 5.3). We also introduce and study the HOO-angle (Section 5.4) and present permanence results with respect to additive perturbations (Section 5.5). Finally, we prove a technical lemma which indicates the connections with Harmonic Analysis (Section 5.6).

One of the main issues in the theory of functional calculus of sectorial operators is the question for which operators A and functions f E HOC! the resulting operator f(A) is bounded. Without exaggerating one may say that Chapters 6-8 are devoted to this question or at least have it as a leitmotif The present chapter collects the general and more technical aspects of the matter, serving as a main source for later reference.

5.1

Convergence Lemma

In this section we prove a basic approximation result, called the Convergence Lemma. Whereas in Section 2.6.3 we treated only approximation of the operators, we now consider the approximation of functions. More precisely, our hypothesis is pointwise convergence of holomorphic functions, thereby bringing us in need of the following fact. Proposition 5.1.1 (Vitali). Let Dee be open and connected, and let (fa)a be a locally bounded net of holomorphic functions on D. If the set {z E D I (fa(z))a converges} has a limit point in D, then fa converges to a holomorphic function uniformly on compact subsets of D. Proof See [14, Theorem 2.1] for an elegant proof. (For sequences instead of nets D this proof can also be found in [10, Theorem A.5].)

5.1.1

Convergence Lemma for Sectorial Operators.

The next lemma, although very elementary, is fundamental.

106

Chapter 5. The Boundedness of the H OO -calculus

c Ho(Scp) be a net of functions converging pointwise to a function f. Suppose that there are C, s > 0 such that

Lemma 5.1.2. Let A E Sect(w), and let 'P E (w,7r). Let (fol"

(5.1) independent of a. Then f E Ho(Scp) and Ilfa(A) - f(A)11

~

O.

Proof. Because of (5.1) the family (fo,}a is locally bounded. Vitali's theorem implies that f is holomorphic, hence f E HO'(Scp). Moreover, fa ~ f uniformly on compact sets. Now the claim follows from a version of the Dominated Convergence Theorem. 0 Example 5.1.3. Let A E Sect(w). Then Aa ~ A (Sw) as a / I (sectorial approximation). Indeed, by Proposition 3.1.2 the family (Aa)c 0 such that

IF(t, z)1

::; C min {Izls ,Izl-s}

for all t E [a, b], z E S'P. Define f(z) :=

lb

F(t, z) dt. Then the following asser-

tions hold. a) f E Ho(S'P); b) (t

f-----t

F(t,A)): [a,bj---+ £(X) is continuous.

c) f(A) =

lb

F(t, A) dt.

Proof. Assertion a) is clear and b) is an immediate consequence of Lemma 5.1.2. To prove c) we compute (21fi)

lb

lb 1r ",g. i (l

F(t, A) =

F(t,z)R(z,A)dzdt

F( t, z) dt) R(z, A) dF

i

fez )R(z, A) dz

= (21fi)f(A).

D

Let us introduce the following notation. For 'ljJ E £(S'P) and t > 0 we write

'ljJt := (z f-----t'ljJ(tz)) It is then clear that 'ljJt E £(S'P) again, and 'ljJ(tA)

(5.2)

= 'ljJt(A) for all t > o.

5.2. A Fundamental Approximation Tecbnique Theorem 5.2.2. Let A E Sect(w),

ip E

109

(w, n), and 'ljJ, B

E

Ho(Scp). Then the

following statements hold. a) For each f E Hoo(Scp) the mapping (t I---' (f'ljJt)(A)) : (0,00) continuous. Moreover, there is a constant C = C('ljJ) such that

b)

One has sup

t>o

°

1

----t

C(X) is

dr IIB(tA)'ljJ(r A)II - < 00.

00

r

0

Proof. a) Choose C,s > with 1'ljJ(z) I S; Cmin{lzI S , Izl-T Let [a,b] and define F(t, z) := f(z)'ljJ(tz) on [a, b] x Scp. Then we have IF(t, z) I S; Ilflloo C min { Izl s , Izl- s

}

max {t S , C

S

}

C (0,00),

•

This shows that F satisfies the hypothesis of Lemma 5.2.1, whence continuity is established. To prove the boundedness simply apply Lemma 2.6.10. This is possible since we have I(f'ljJt)(z) I S; C Ilflloo,cp min { Itzl S ,Itzl- S for all t

}

> 0, z E Scp.

b) Choose w' E (w, ip) and define

r

:=

11

1

8Sw '. Then

00 IdzlI -dr IIB(tA)'ljJ(rA) II -dr ;S IB(tz)'ljJ(rz)I-1 oro r z r 00

= roo r IB(z)'ljJ(rC1 z)1

io ir s; IBI~)I

lr

Idzl

s; r roo 1'ljJ(reiargz) I dr IB(z)1 Idzl

~ dr

~~1 (1

r

Izl

00

ir io

1'ljJ(re ciW') I ~)

r

Izl

. o

The constant hidden in the symbol ';S' is of course M(A,w')/2n. Let A E Sect(w),

ip E

l

(w, n], 'ljJ b

'ljJa,b(Z):= a 'ljJ(tz)

E

Ho(Scp), and define

tdt

(0 < a < b < 00, z E Scp).

One knows from Example 2.2.6 that 'ljJa,b(Z) where c is the constant

1

----+

c pointwise as (a, b)

dt 'ljJ(t)-. o t The interesting question now is, for which x E X one has e:=

'ljJa,b(A)x

----+

ex

00

as

(a, b)

----+

(0,00).

The first step towards a general result is the following.

(5.3) ----+

(0,00),

(5.4)

Chapter 5. The Boundedness of the H oo -calculus

110

Lemma 5.2.3. Let A E Sect(w), 'P E (w, nJ, and 'IjJ E HO' (S'P)' Define 'ljJa,b and c

as in (5.3) and (5.4). Then the following assertions hold.

a) 'ljJa,b

HO'(S'P)'

E

b) sUPa,b

II'ljJa,bll oo < 00.

c) 'ljJa,b(Z)

~

C as (a, b)

l

b

d) 'ljJa,b(A) = a 'IjJ(tA)

~

(0,00) uniformly on compact sets of S'P'

dt

T'

e) sUPa,b II'ljJa,b(A) I < 00.

Proof. We write 'ljJa,b(Z) = =

b'IjJ(tz) dt r t io

_

1'IjJ(tbz) dt r t io

_

r 'IjJ(tz) dtt

io

1'IjJ(taz) dt = h(bz) _ h(az), r io t

where h(z) = fo1 'IjJ(tz) dtlt as in Example 2.2.6. Using the results shown there, a)c) follow immediately. To prove assertion d) we apply Lemma 5.2.1 to the function F(t, z) := 'IjJ(tz). Assertion e) is immediate from 'ljJa,b(A) = h(bA) - h(aA) and Proposition 2.6.11. D Proposition 5.2.4. Let A E Sect(w), 'P E (w, nJ, and'IjJ E HO'(S'P)' Define

h(z)

:=

1

1,00

1'IjJ(tz) -, dt dt g(z):= 'IjJ(tz) -, o t I t

l

b

dt

T

as in Example 2.2.6. Let also 'ljJa,b(Z):= a 'IjJ(tz) the following assertions hold. a)

If

b)

If

c:=

1

00

0

'IjJ(t) -dt t

as above. Then for x E X

1 r1 11 'IjJ(tA)xll dtt < 00, then h(A)x = a'-,.O lim 'ljJa,l(A)x = r 'IjJ(tA)x dt. io t

io

1,00 11'IjJ(tA)xll 1

dt - < 00 and A is injective, then t g(A)x = lim 'ljJ1,b(A)x = b/oo

dt 1,00 'IjJ(tA)x-.

c) If x E :R(O(A)) for some 0 E HO'(S'P)' then lim

a-+O,b-+oo

'ljJa,b(A)x =

l

t

1

100

dt 11'IjJ(tA)xll - < 00 and o t

1

b 00 'IjJ(tA)x -dt = 'IjJ(tA)x -dt = cx. a-+O,b-+oo a tot lim

5.3. Equivalent Descriptions and Uniqueness

111

Proof. Find C, s > 0 such that 1~(z)1 :::; Cmin{lzl S ,Izl- S }. a) It is clear that ~a,l(Z) -+ h(z) pointwise, with

t J

XJ

l~a,l(Z)1 :::; o

1~(tz)1

Also,

l~a,l(Z)1 :::;

(Xl

dt

t : :; C Jo Jor

1

1~(tz)1

min{t S , C

dt

dt

S

t

}

2C

= ----;-.

C

t :::; ---; Izl s .

Hence one can apply Lemma 5.1.2 to the functions ~a,d(1 + z) and obtains the convergence (1 + A)-1~a,1(A) -+ (1 + A)-lh(A) in norm as a -+ O. On the other hand, by assumption and Lemma 5.2.3 d) one has

(1

+ A)-l~a,l(A)x =

(1

+ A)-l J1 ~(tA)x dt a

t

-+

(1

+ A)-l r1 ~(tA) dt

Jo

as a '\, O. Since (1 + A)-l is injective, it follows that h(A) =

t

fo1 ~(tA) dtlt.

b) is proved in the same way as a), where one has to regularise at 0 with the function z(1 + Z)-l. Note that one needs A(1 + A)-l to be injective in the end of the argument. c) The first assertion is straightforward from Theorem 5.2.2 b). For the second, let x = B(A)y and consider the functions ~a,b(z)B(z), which converge to the function cB(z). Since sUPa,b II~a,blloo < 00 by Lemma 5.2.3, one can apply Lemma 5.1.2 D and obtains ~a,b(A)x = ~a,b(A)B(A)y -+ cB(A)y = cx. Remark 5.2.5. Part b) of Proposition 5.2.4 is false in general if A is not injective. More precisely, if c = fooo ~(t) dtlt !- 0 and 0 !- x E N(A), then ~(tA)x = 0 for all t > 0 but g(A)x = cx !- O.

The final result of this section is just a corollary of the above considerations. Theorem 5.2.6 (McIntosh Approximation). Let A E Sect(w), and let

such that fooo ~(t) dtlt = 1. Then

b dt (Jb dt) (A)x a ~(tz)t J a ~(tA)xt = for all x

5.3

E

a-+O,b----+oo

--+

1

00

o

~(tA)x

~ E

dt t

HO'[Sw]

=

x

1>(A) n J«A).

Equivalent Descriptions and Uniqueness

Let A E Sect(w) be a sectorial operator on the Banach space X and let 'P E (w, Jr). Suppose we are given a sub algebra F c HOO (S O. From Lemma 2.6.7 we see that rn(Ae) ---t rn(A) as c "" 0 for each n. A standard argument from functional analysis now implies that there is T E C(X) with f(Ae) ---t T. An application of Proposition 2.6.9 yields T = f(A). The same proof applies in the case where A is injective and f E HOO(S'P) n C(K). D Proposition 5.3.7. Let A E Sect(w) ,

be a bounded HOO(Sw)-calculus for A with bound C 2': O. Then A is sectorial with wA :::; w. Take a sector S

for A. If such a 11> exists, the operator A is sectorial with W A :::; wand has dense domain and dense range. Moreover, coincides with the natural functional calculus on HOO(S WA with cp 2: w.

Proof. Uniqueness follows from Proposition F.4. Define fn(z) := n(n + Z)-l. Then fn ---t 1 pointwise on Sw as n ---t 00 and sUPn Ilfnt < 00. Since 11> is

5.4. Tbe Minimal Angle

117

b.p.-continuous, n(n + A)-l =

x E 1>(A)

and y = Ax E DA(B,p)

for all x, y EX. Then the following assertions hold. a) p(A) c p(Ao,p) with

R(A, Ao,p) = R(A, A)IDA(O,P) b) Ao,p E Sect(w) in the Banach space DA(B,p).

(A

E

p(A)).

133

6.1. Real Interpolation Spaces

c) If A is injective or invertible, so is Ao,p. d) If f

E

M[Sw]A, then f

E

D A ((), p), i. e.,

(x, y)

E

f(Ao,p)

M[Sw]Ae,p with f(Ao,p) being the part of f(A) in

~

2)(f(A)) n DA((),p),

X

E

y

= f(A)x

{

E

DA((),p)

for all x, y EX. Now we can state Dore's theorem. Note that that there are sectorial operators without a bounded Hoo-calculus (Remark 5.4.4).

Theorem 6.1.3 (Dore). Let A be an invertible sectorial operator on the Banach space X, and let () E (0,1) and p E [1,00]. Then for each cp E (WA, n) the natural Hoo(Scp)-calculus for Ao,p is bounded. In particular, DA((),p) c 2)(f(A)) for all f E Hoo[Sw]. Proof. We first treat the case p = 00. Employing the resolvent identity and the second part of Lemma 6.1.1, one sees that for each Wi E (W A, n) there is a constant c(w' ) such that sup llzo AR(z, A)xll :::; c(w' ) Ilxllo,oo zE8S w '

for all x E DA((),OO). Take cp E (wA,n), f E Hoo(Scp) and x E DA((),p). Then

r

f(z) R(z, A)xdz ( f(z) ) (A)x = ~ 1+z 2n2 Jr 1 + z

1

° °

( f(z)) [z AR(z, A)x] dz, 2nz r 1 + z z

= A -1 -1.

where r = asw' for some Wi E (w, cp). This shows that (f /(1 whence x E 2)(f(A)). Moreover, for each t > 0 one has

litO A(t + A)-l f(A)xllx

=

Jr

It + zllzlo

r

E

2)(A),

~ II r (f(Z)~O °zO AR(z, A)x dzll x 2n Jr t + z z

: :; ~ r If(z)1 to Idzl c(w' ) Ilxll 2n

+ z))(A)x

= 0,00

~ 2n

r If(tz)lldzl c(w' ) Ilxll

Jr 11 + zllzlo

0,00

1 Idzl I :::; Ilfllcp 2n Jr 11 + zllzlo c(w ) Ilxllo,oo' Hence f(A)x E DA((), 00) with Ilf(A)xllo,oo :::; independent of x.

c Ilfllcp Ilxllo,oo for some constant C

Let us turn to the case 1 :::; p < 00. By the Reiteration Theorem B.2.9 one can view D A((), p) as a real interpolation space between D A((), 00) and D A ((3, 00 ), where () < (3 < 1. Combining this with Proposition 6.1.2 concludes the proof. 0

Chapter 6. Interpolation Spaces

134

Example 6.1.4. The hypothesis of invertibility cannot be dropped from Theorem 6.1.3. Let A be an unbounded, invertible, sectorial operator on a Hilbert space H such that the natural Hoo-calculus for A is not bounded (see Corollary 9.1.8). From the composition rule it is clear that A-1 cannot have a bounded Hoo-calculus. Form the diagonal operator

on the space H EB H. Then it is clear that DA(B,p) = D A(B, A) EB H with the induced operator Ao,p being the diagonal operator diag(Ao,p, A -1). Since obviously f(A) = diag(f(A), f(A- 1 )), A cannot have a bounded Hoo-calculus on DA(B,p), for any pair (B,p). Corollary 6.1.5. Let A E Sect(w) such that 1J(A"') = DA(Rea,p) (with equivalent norms) for some Re a E (0,1) and some p E [1,00]. Then for each E > 0 and each 'P E (w, n] the natural HOO(Srp)-calculus for A + E is bounded.

Proof. Since (A + E)-'" : X from Theorem 6.1.3.

---+

1J(A"') is an isomorphism, the assertion follows D

We shall prove below that the identity 1J(A"') = DA(a,p) for one a implies already that it holds for all a. Also we shall prove a much more general form of Dore's theorem without employing the Reiteration Theorem.

6.2

Characterisations

In the present section we use the functional calculus to give several descriptions of the real interpolation spaces associated with a sectorial operator A on a Banach space X. Given Rea> 0 the space 1J(A"') will be considered endowed with the norm Ilxllx + IIA"'xllx, which turns it into a Banach space continuously embedded in X. Hence the pair (1J(A"'), 1J(Af3)) is an interpolation couple whenever a,(3 E {z I Rez > O} U {O}. Note that, since A"'(A + 1)-'" = (A(A + 1)-1)", is bounded, one has 1J(A"') = 1J((A + 1)"') not only as sets but also with equivalent norms. (Apply the Closed Graph Theorem.) Therefore, in considerations involving only the space 1J(A"') one may always suppose that A is invertible. In that case the homogeneous norm IlxllAQ := IIA"'xllx is an equivalent norm on 1J(A"').

6.2.1

A First Characterisation

In this section we consider a description of the real interpolation spaces as follows. Theorem 6.2.1. Let A E Sect(w), 'P E (w, n), and Rea> 'Ij; E O(Srp) be a function with the following properties:

a) 'Ij;, 'lj;z-'" E E(Srp);

o.

Furthermore, let

6.2. Characterisations

135

b) limz-+o~(z)z-Q -::f 0;

c)

~(z) -::f 0 for all z E S'P;

d)

I ~(sz) I < 00.

sup

(6.3)

zES 0),

where C = max(C"" C,,), cf. Proposition 2.6.11. Consequently, there is a continuous inclusion (X,

1)(A~))o,p

{x

C

E X

I cORea'ljJ(tA)x E L~((O,oo);X)}

for each (B,p) E A. Proof. Let x = a + b, where a E X and b 'ljJ(tA)a + ta"((tA)Aab, and this yields

11'ljJ(tA)xll : : : c'" I all x + tReac" IIAabli x

E

1>(Aa). Then one has 7J;(tA)x =

:::::

C( Ilall x + tRea Il bl 1l (A 0, and let 'ljJ E E(S 0, one has

[s 1--4 (f1jJ)(stA)x]

E L!((O, 1); X)

for each t > O. An application of Proposition 5.2.4 a) yields

Ilh1(tA)xll

=

1111 (f1jJ)(stA)x ~s II S

lt

II (f1jJ)(sA)xll d:

=:

(J(t).

Employing the Hardy-Young inequality (Lemma 6.2.6), we obtain rORea(J(t) E L~(O,oo) with

IlcORea(J(t)llu: S

B~ea Cf IICORea1jJ(tA)xIlLl:(O,oo)'

From (6.7) we can now infer that

CORea K(t Rea , X,X, 1J(Aa))

E L~(O, (0)

and estimate the L~-norm of this function in terms of Ilxll and IlrORea1jJ(tA)xIILl:'

o

Combining Lemma 6.2.7 and Proposition 6.2.8 completes the proof of Theorem 6.2.1.

6.2. Characterisations

6.2.2

139

A Second Characterisation

In this section we formulate and prove a result which is more general than Theorem 6.2.1, at least when the interpolation parameter is restricted to the open interval (0,1).

e

Theorem 6.2.9. Let A E Sect(w), t.p E (w, n) and Re a > O. Take any function

oi- 'ljJ E 0(8'1')

such that 'ljJ, 'ljJz-n

E

£(8'1')' Then one has

I CORen'ljJ(tA)x E L~((O,oo);X)}

(X, 1)(An ))o,p = {x E X with the equivalence of norms

Ilxllx + IlcORen'ljJ(tA)xIIL!:((o,oo);X)

Ilxll (X,n(A"'))o,p for all e E (0, 1),p E [1,00].

(We shall give an even more general result for the case of an injective operator

A in Section 6.5.1 below.) As before, it follows from Lemma 6.2.4 that we only have to take care of the inclusion

{x

I CORen 11'ljJ(tA)XII

L~(O, oo)} C (X,1)(A n ))o,p'

E

The main ingredients of the proof are already known. There are just some further technicalities. Lemma 6.2.10. Let A E Sect(w),t.p E (w,n), f E HO'(8'P) and Rea> O. Define

h(z) Then h

E

HO'(8'P) and h(A) =

:=

1

1

00

1

00

ds s-n f(sz) -. S

1

s-n f(sA) ds. s

Proof. If one chooses 0 < E < Re a small enough such that z±c f E Hoo, one also has z±ch E Hoo. This shows that h E HO'. The rest is simply Fubini's theorem and the definition the HO'-functional calculus by the Cauchy integral. Note that the function s f------t s-n f(sA) is in L!((I, 00); .c(X)). D We return to our main objective.

Proof of Theorem 6.2.9. As in the proof of Theorem 6.2.1 we try to write the constant function 1 as a sum of appropriate functions. Hence we start as before and choose a function f E HO'(8'P) such that J := zn f E HO'(8'P) and

rOO (f'ljJ)(s) ds =

io

s

1

140

Chapter 6. Interpolation Spaces

(Lemma 6.2.5). Then we define

hI (z):=

r (f1jJ)(SZ) ds io l

and

h2(Z):=

1

00

(f1jJ)(sz) ds.

S I S

By Example 2.2.6, hI, h2 E [(Sip) and hI

z ah2(z) = is -

1

by Lemma 6.2.10 -

for all t

> O. In

Now fix

00

1

+ h2 = 1.

za J(sz)1jJ(sz) -ds =

1

Moreover,

00

s-a(f1jJ)(sz) -ds

S I S

a function in HOO(S J«T) is an isometric isomorphism. Now we consider the commuting diagram

and rename some of its components:

The map t is bounded and injective, i.e., it is an embedding, and after some settheoretical work we may view X-I as a proper superspace of X. We arrive at the

6.3. Extrapolation Spaces

143

commuting diagram

which shows that T-Ilx = T. Hence no confusion can arise in renaming T-I by T. When we define Xl := :R(T) , X 2 := :R(T2) ... we obtain T

X-I-X

I

T

I

X--XI

I

I

T XI~X2

with T being an isometric isomorphism at every stage. Iterating this procedure yields an upwards directed series of new spaces (X-n)nEN, i.e., X=XO~X-I~X-2~···

~X-n~···

where as before we may always view the embeddings as proper inclusions. Moreover, we have (compatible) isometric isomorphisms T : X-n ----t X- n - l . As a last extension we finally construct the algebraic inductive limit of the directed family (X-n)nEN,

U X-no

U := X_= := ~nEN X-n =

nEN

The space U may be called the universal extrapolation space corresponding to T. On U we define the following notion of convergence. Let (Xoo)aEP CUbe a net and x E U. Then Xa ---->X

in U

:~

3n E N, ao E P:

x, Xa E X-n (a::::: ao)

and

Ilxa - xllx_

n

---->

o.

(This does not give a proper topology on U but is well adapted to our purposes, cf. the comments in Section 6.8 below.) One easily sees that the limit of a net in U is unique, and that sum and scalar multiplication are 'continuous' with respect

144

Chapter 6. Interpolation Spaces

to the so-defined notion of convergence. Finally, since the operator T is defined on each space X- n , n E N, it is a fortiori defined on the whole of U. The sodefined mapping T : U ----t U is obviously surjective, whence it is an algebraic isomorphism, continuous with respect to the notion of convergence defined above.

6.3.2

Extrapolation for Injective Sectorial Operators

We now apply this abstract procedure to the case of an injective sectorial operator A on X. The operator T - which is constitutive for the extrapolation method is defined by T:= A(l + A)-2 = (1 + A)-I(l + A-I)-I. Then T is injective since A is. Moreover, ~(T) = 'D(A) n ~(A). By the abstract method described above, we obtain a sequence of nested spaces X = Xo C X-I C X- 2 C ... C X-n C ... U with U being the 'universal' space. Recall that T extends to a 'topological' isomorphism on U. Let us now extend the operator A to the whole of U. Using the isometric isomorphism T : X-I ----t X one can just transfer the operator A to X-I by defining By construction, A-I is an injective sectorial operator on X-I, isometrically similar to A. Clearly, X c'D(A-d. Moreover, A is the part of A-I in X, i.e., 'D(A)

= {x E X 1 A_IX E X} and A_IX = Ax (x E'D(A)).

(This is due to the fact that the operator T - considered as an operator on X commutes with A.) The inverse of A-I is given by [A_Itl = T- I A-IT with the appropriate domain 'D(A=D = ~(A-I) = T-I~(A). Iterating this procedure we obtain a sequence of isometrically similar sectorial operators A-n on X- n , with each A-n being the part of A_(n+l) in X-no Since X-n C 'D(A_(n+1)), we obtain an extension of A to the whole of U. This extension, by abuse of notation, is again denoted by A. It has the pleasant feature that it is invertible, i.e., A : U ----t U is an isomorphism, continuous with respect to the notion of convergence we introduced above. Within the space U we can define an array of spaces as follows. Already within X we have the following natural spaces:

:= DI := 'D(A) with norm IlxiiD := 11(1 + A)xll. R:= RI := ~(A) with norm IlxIIR:= 11(1 + A-I)xll·

• D

•

• D

n R := 'D(A) n ~(A) with the norm

IlxllDnR = 11(2 + A + A-I)xll = IIT-Ixll·

6.3. Extrapolation Spaces

145

IIxllD2 = 11(1 + A)2xll. 1J(A-2) with the norm IIxllR2 = 11(1 + A- 1)2xll.

• D2

:=

1J(A2) with the norm

• R2

:=

9«A2)

=

These can be arranged into a diagram.

x

/~ R D /~/~ DnR

A

~

Here a downward meeting of two lines means intersection and an upward meeting of two lines means sum of the spaces. E.g., X = D + Rand D = D2 + (D n R). The operator A + 1 acts as an isometric isomorphism in the /-direction, i.e. A+1 A+1 D2~D~X

A+1

or D n R~R. (One may replace A + 1 by A + >. for each>' > 0, but then the isomorphisms cease to be isometric.) Furthermore, the operator A acts as an isometric isomorphism in the ~-direction, e.g., D~R A

or D2 ---->-- D n R. Via the isometric ismorphism T : X -1 to X- 1 . That is, we form the spaces

----+

X this diagram can be transported

• D-1

:=

T- 1(D) with the norm

Ilu11D_1

• R-1

:=

T- 1(R) with the norm

Ilu11 R_1= IITullR

=

IITullD

and

and obtain a situation as shown in Figure 9. Note that we already know D-1 = 1J(A_ 1), R-1 = 9«A-d. By applying T- 1 again and again, we generate the spaces D-2' D_ 3 , ... and R-2' R_ 3 , .... Since A is an isomorphism on U, we define

We frequently write D(n) := x(n)

and

R(n):= x(-n)

for n E N, and iJ := D(1) and R := R(1) in the case n = 1. These spaces are called the homogeneous spaces associated with the injective sectorial operator A. Figure 10 illustrates the situation. As before, T, A, (A + 1) and (A- 1 + 1) act as isometric isomorphisms in the directions 1,~, /, """ respectively. If 1J(A) is dense in X, then all inclusions in the /-direction are dense. If 9«A) is dense then all inclusions in the "",-direction are dense.

146

Cbapter 6. Interpolation Spaces

DnR Figure 9: The first extrapolation space.

6.3.3

The Homogeneous Fractional Domain Spaces

Having constructed the universal space U we look at the behaviour of the functional calculus. Let A E Sect(w) and'P E (w,n]. Recall the definition of the function algebra

B(S(A 2a). But A2" = Aa Aa even as operators in X, and so x = Aa y E X. This shows D(a) n R(a) C X and together with a) and b) this implies the stated set equality. From a) we obtain the norm equivalence

(which holds for x E 1>(A2a)). Now we replace x by A-ax, and we are done. d) Take n

X

> Rea and expand I = [A(l + A)-1 + (1 + A)-Ij2n.

E

1>(Aa) n 9{(Aa)

o

In many concrete situations extrapolation spaces can be explicitly identified with spaces of well-known objects. In Section 8.3 we shall exemplify this for the negative Laplace operator -~ on LP(JRd),p E [1,00), identifying the universal extrapolation space with a space of distributions.

149

6.4. Homogeneous Interpolation

6.4

Homogeneous Interpolation

We make the overall assumption that A E Sect(w) is an injective sectorial operator on the Banach space X. In this section we characterise the real interpolation spaces (D(a), R(f3))O,p between the homogeneous fractional domain spaces by means of functional calculus. To achieve this we freely use the extrapolation space U constructed in Section 6.3.

6.4.1

Some Intermediate Spaces

Fix p E [1,00],

e E JR, and 0 -=I- 'IjJ E O[Swl with z-°'IjJ E HO'[Swl. XO,,p,p:= {x

For x E Xo,,p,p we let

E U I

r°'IjJ(tA)x

Ilxllo,,p,p :=

Then we define

E L~((O,oo);X)}.

Ilr°'IjJ(tA)xII L !:·

(Note that it may well be that x E Xo,,p,p but x tJ. X.) Proposition 6.4.1. Let p E [1,00],

statements hold. a)

eE

JR, and 'IjJ as above. Then the following

The operator A ° (defined on U) induces an isometric isomorphism A O: Xo,,p,p -----; Xo,z-e,p,p'

b)

The space Xo,,p,p is continuously included in A-O X-I.

c) The space Xo,,p,p is a Banach space. Proof. a) is clear from the identity

rIJ'IjJ(tA)x = (tA)-°'IjJ(tA)Aox = (z-IJ'IjJ)(tA)AIJ x for each x E U. To prove b) it suffices to consider the case that e = 0 (by a)). Therefore 'IjJ E HO', by assumption. Take x E Xo,,p,p, i.e., x E U such that 'IjJ(tA)x E L~(X). Apply Lemma 6.2.5 with a = 0 to find a function f E HO' with ft(f'IjJ)(t) dt/t = 1. By Theorem 5.2.2 the function t f---+ T(A)f(tA) is bounded and absolutely integrable in £(X). (We use the abbreviation T(Z) := z(1+z)-2 as usual.) In particular, by Holder's inequality it is contained in L~' (£(X)), where p' is the conjugated exponent to p. Hence we obtain T(A)(~'IjJ)(tA)x E L!(X) with

10()() IIT(A)(~'IjJ)(tA)xll x

dt:s t

Ilxll x

'1IT(A)~(tA)11 L!: (C(X)) I

O,,",p

•

Applying Proposition 5.2.4 we obtain fooo(f'IjJ)(tA)xdt/t = x in U (or, if you wish, in some space X-m). Consequently T(A)x E X, i.e., x E X-I, with

Ilxll x

-1

=

IIT(A)xllx

:s Ilxllx

O,,",p

'1IT(A)~(tA)11 L!: (C(X)) I

•

Chapter 6. Interpolation Spaces

150

The proof of c) is now easy. Again, it suffices to consider the case that () = O. Let (xn)n C XO,'I/J,p be a Cauchy sequence. By b) there is x E X-I with Xn ~ x in X-I. Then 'ljJ(tA)xn ~ 'ljJ(tA)x in X-I uniformly in t. On the other hand there is f E L~(X) with 'ljJ(tA)x n ~ f(t) in the L~(X)-norm, hence a fortiori in the L~(X_I)-norm. This shows that f(t) = 'ljJ(tA)x almost everywhere, whence we are done. 0 We can now prove the main result. Theorem 6.4.2. The spaces (Xe,'I/J,p,

11.lle,'I/J,p)

are independent of the chosen 'ljJ.

Proof. By a) of Proposition 6.4.1 it suffices to prove the theorem in the case that () = O. We choose VJ E (w, 7r), 0 -=I 'ljJ, "( E H(f(Sp), and x E Xo",P" Then we apply Lemma 6.2.5 (with ex = 0) to find f E H(f(S 0 and some constants M 1 , M 2 . Taking the infimum with respect to the representations x = a + b yields

(t > 0) for some constant M. This proves the continuous inclusion (D(n), R((3))e,p C XO,p'

Jo

For the converse inclusion choose a function 'IjJ E Ho(5cp) with oo 'IjJ(t) dtlt = 1 and define 1 dt 00 dt g(z) := / , 'IjJ(tz)h(z) := 'IjJ(tz) -, 1 t o t as in Example 2.2.6 and Proposition 5.2.4. Without loss of generality one may suppose that the decay of 'IjJ at 0 and at 00 is rapid enough to ensure that both functions 'ljJl := zng and 'ljJ2 := z~(3h are contained in Ho(5cp). Take x E XO,p and write x = g(tA)x + h(tA)x (t > 0).

1

Now observe that g(tA)x E D(n), because Ang(tA)x

= rn(tA)ng(tA)x = rn'IjJl(tA)x

E

X.

Analogously, h(tA)x E R((3) with A~(3h(tA)x

Therefore, for each s

= t(3(tA)~(3h(tA)x = t(3'IjJ2(tA)x.

> 0 one obtains

K(s, x, D(n), R((3)) ::; r Ren II'ljJl(tA)xllx

+ st Re (3II'IjJ2(tA)xllx'

Letting s := r(Ren+Re(3) yields

(t > 0). The right-hand side (as a function of t) is in L~. This concludes the proof.

D

6.5. More Characterisations and Dore's Theorem

153

We remark that in the proof we made essential use of Theorem 6.4.2. Indeed, it was important that one can choose every HOO-function to describe the space XO,p. With the help of Proposition 6.4.4 we are now able to show that each real interpolation space between homogeneous spaces is in fact a space XO,p' Theorem 6.4.5 (Homogeneous Interpolation). Let A be an injective, sectorial operator on the Banach space X, and let a, (3 E C with Re a i= Re (3. Then, for all () E (0, l),p E [1,00]' ( X(Q) , X(3))

o,p

-- X (1-0) ReQ+ORef3,p

with equivalent norms. Proof. Without loss of generality we may suppose that Re a > Re (3 (replace () by 1 - () if necessary). Define 8 := (1 - ())a + ()(3. Then a' := a - 8 = ()(a - (3) and (3' := 8 - (3 = (l-())(a- (3). Hence Rea', Re(3' > and () = Rea' /(Re a' + Re(3'). By Proposition 6.4.4 we have

°

( X(Q-O) , X(f3-0))

O,p

= (X(QI) , X(-f3' ))

O,p

= (D(QI) , R(f3I))

O,p

= X O,p·

Applying A -0 = A - Re 0A -iIm 0 to this identity concludes the proof. (Note that by Corollary 6.4.3 the operator A-i1mo is an isomorphism on XO,p') D Corollary 6.4.6. Let () E (0, l),p E [1,00]. Then

6.5

(D, R)o,p = X l - 20 ,p'

More Characterisations and Dore's Theorem

In this section we revisit Dore's theorem and give evidence to the heuristic statement that the functional calculus properties of an injective sectorial operator A improve in its interpolation spaces. But first, we complement the characterisations of the interpolation spaces obtained in Section 6.2.

6.5.1

A Third Characterisation (Injective Operators)

With Theorem 6.4.5 we have established a powerful description of real interpolation spaces in terms of functional calculus. However, the spaces (X(Q), X(f3))o,p are quite uncommon and we look for spaces which are naturally included in the original space X. Lemma 6.5.1. Let A be an injective sectorial operator on the Banach space X. Then

n 9«AQ))o,p = (X, D(Q))o,p n (X, R(Q))o,p, (X, 1)(AQ))o,p = (X, D(Q))o,p n X, (1)(AQ), 9«AQ))o,p = AQ(1 + A)-Q(D(Q), X)o,p

(X, 1)(AQ)

Chapter 6. Interpolation Spaces

154 for all Reo: > 0, () E [0, l],p E [1,00].

Proof The last assertion follows from the fact that the isomorphism A-"'(1 + A)'" sends 'D(A"') to D("') and ~(A"') to X, see Lemma 6.3.2. For the first two assertions we use Proposition B.2.7. Indeed, the second assertion follows directly from (B.4) and D("') n X = 'D(A"') (see Lemma 6.3.2).

To prove the first, let us denote for the moment Y := D("') , Z := R("'). Then we have Y n Z = 'D("') n ~(",) = 'D(A"') n ~(A"') c X and

(X n Y) + (X n Z) = ('D("') n X) + (~(",) n X) = 'D(A"') + ~(A"') = X, by Lemma 6.3.2. Hence we obtain (X, Y)o,pn(X, Z)o,p c (X +Y)n(X +Z) This yields

(X, Y)o,p n (X, Z)o,p = [(X, Y)o,p n X] n [(X, Z)o,p n X] = (X,X n Y)o,p n (X,X n Z)o,p = (X,X n Y n X n Z)o,p = (X, Y n Z)o,p.

= X.

0

Remark 6.5.2. In the first statement of the last lemma one writes 'vertical' interpolation spaces as intersections of 'horizontal' ones. As a matter of fact, one can also go in the reverse direction. E.g., one has the equality

(D, R)o,p = (D-l' Dh-o,p n (R-l' R)o,p, which can be deduced from an interpolation identity that is in the spirit of Proposition B.2.7 (see [111, Theorem 1]). Combining Lemma 6.5.1 with Theorem 6.4.5 we arrive at a third representation theorem. Theorem 6.5.3. Let A be an injective sectorial operator on a Banach space X, let cpE (wA,n), Reo:>O, (}E (0,1), andpE [1,00].

a) One has

(X,'D(A"'))o,p = {x

E

X I rORe"'~(tA)x

E L~((O,oo);X)}

with equivalence of norms

whenever ~ E O(S 0 in Theorem 6.4.5. Thus we obtain

(X, x(a))o,p = Xo Re a,p' Now we intersect both sides of this identity with X and employing Lemma 6.5.1 we arrive at (x,1>(Aa))o,p = XORea,p n X. The rest follows from Theorem 6.4.2.

o

The proof of b) is similar.

When A is invertible, some simplifications apply. This is the content of the next section.

6.5.2

A Fourth Characterisation (Invertible Operators)

Let us specialise the given characterisations to the case of an invertible sectorial operator A. If A is invertible, the array of extrapolation spaces becomes 'halftrivial' since we obtain

D(a) = 1>(Aa), Moreover, D -1

=X

and R-1

~(Aa)

= X-I.

=X

(Rea> 0).

Consequently, one has

(X, 1>(A a ))o,p = XORea,p by Theorem 6.4.5. Comparing this with the Characterisation Theorem 6.5.3 a) this means that in the case where A is invertible one may leave out the 1I'llx-part and simply has a norm equivalence

Il x ll(x,1)(A o,e E (O,l],p E [1,00] there is a constant c such

o i=- 'l/J that

II xlix ::::: c Ilt-ORea~(tA)xIIL!.'((O,oo);X)

(x EX).

Note that ~(tA)x E U is always defined but the right-hand side might be infinite, in which case the inequality holds trivially.

ea

e

Proof. Without harm we can replace a by and suppose that = 1. Choose f E H[)(S'P) such that II := za+1 f, ~f E H[)(S'P) and Jooo(f~)(t) dtlt = 1 (Lemma 6.2.5). Define as usual

h(z) :=

1 1

o

dt (f'l/J)(tz)-, t

and

g(z):=

1

00

1

dt (f'l/J)(tz)-. t

Chapter 6. Interpolation Spaces

156 Then h+g Then

= 1 (Example 2.2.6).

Take x E X with rORen'ljl(tA)x E L~((O, (0); X).

(f'ljl)(tA)x = A-(n+1) [C 1 h(tA)] [Cn'ljl(tA)x] E L!((l, (0); X) since by assumption the last factor is in L~, the first factor is bounded, and the second factor is in L~/. Hence by Proposition 5.2.4 we obtain Ilg(A)xll with C1

=

=

111

00

(f'ljl)(tA)x

~t II :::; C1 Ilc Ren'ljl(tA)xII L!.'((l,oo);X) ,

IIr 1 11 L.p/( 1,00 ) IIA-(n+1)11 SUPt>lllh(tA)II. On the other hand we have

(f'ljl)(tA)x = [t n f(tA)][Cn'ljl(tA)x] E L!((O, 1); X) since the first factor is in L~' ((0, l),X). Again by Proposition 5.2.4 we obtain Ilh(A)xll with C2

=

=

1111 (f'ljl)(tA)

~t II:::; c21ICRen'ljl(tA)xIIL!.'((0,l);X)'

IltRenl1 L.p/( 0,1 ) SUPt 0, e E (0, 1),p E [1,00]). Proof. By Lemma 6.5.1,

is an isomorphism which commutes with A. Hence the assertion follows from Theorem 6.5.6. 0 We may apply Lemma 6.5.1 to pass from the 'horizontal' interpolation spaces to the 'vertical' ones. Corollary 6.5.8 (Dore). Let A be an injective sectorial operator on the Banach space X, and let cp E (WA, ]f). Then A has a bounded HOO(S'P)-calculus in each of the spaces

(X,1>(A") n J«A"))IJ ,p

(Re a> 0,

e E (0, 1),p E [1,00]).

(For the special case a = 1 and under density assumptions, this corollary was proved by DORE [73].) Assuming A to be invertible and taking a = 1 gives back Dore's Theorem 6.1.3. Note that this new proof does not use the Reiteration Theorem.

6.6

Fractional Powers as Intermediate Spaces

In this section we examine the domains 1>(A") as intermediate spaces. Here A is any sectorial operator on a Banach space X, not necessarily injective.

6.6.1

Density of Fractional Domain Spaces

°
(Ai3) dense in 1> (A"). However, within the real interpolation spaces we have a quite different behaviour.

Theorem 6.6.1 (Density). Let A be a sectorial operator on the Banach space X and let 0< Rea < Re(3, e E (0, 1),p E [1,00). Then the space 1>(Ai3) is dense in (X,1>(A"))IJ,p'

°

°

Proof. Without loss of generality one may suppose that A is invertible. Choose cp E (WA,]f) and -=1= 7/J E O(S'P) such that z-IJ"7/J E Ho(S'P)' Choose -=1= T E Ho(S'P) such that (n/J) E Ho(S'P)' Then also Z-IJ"(T7/J) E Ho(S'P) and either function 'IjJ, T'IjJ may be used to describe (X, 1>(A"))Ii,p (Corollary 6.5.5). We choose

158 f

Chapter 6. Interpolation Spaces

E Ho(S'P) in such a way that

f'lj;,z f3 f'lj;

(Lemma 6.2.5). Define

gn(Z)

:=

1

00

1.

E Ho(8'P) and

Jooo(f'lj;)(s)ds/s = 1

ds (f'lj;)(sz)-. S

n

Then gn E £(8'1') as was shown in Example 2.2.6. We claim that (i) gn(A)(X) c 1)(Af3) and (ii) gn(A)x -+ x within (X, 1)(A"'))o,p for x E (X,1)(A"'))o,p. Claim (i) follows from the identity

which is a function in Ho(8'P). To prove Claim (ii) we take x E (X,1)(A"'))o,p. We have to show

CO Ren( T'Ij; )(tA)gn(A)x

-+

C ORe ",( T'Ij;)(tA)x

in L~((O,oo);X). The Convergence Lemma (Proposition 5.1.4, see also c) of Proposition 5.2.4) shows that the convergence is at least pointwise. Employing the Dominated Convergence Theorem we have to prove domination. Now

and, by Proposition 5.2.4 and Theorem 5.2.2,

~~~ Ilhoo (f'lj;)(SZ)T(tZ) ~s (A) I = ~~~ Ilhoo (f'lj;)(SA)T(tA) ~s I ::; sup roo II (f'lj;)(SA)T(tA)11 ds =: c < t>o Jo s This yields IlrORe"'(T'Ij;)(tA)gn(A)Xllx by assumption.

00.

::; c IlrORe"''Ij;(tA)xllx' which is in 0

L~(O, 00)

6.6.2

The Moment Inequality

We first establish that - in the language of interpolation theory - the space 1)(Af3) is always of class JonKo between X and 1)(A"'), provided 0 < Ref) < Reo: and is defined as := Re f) / Re 0:. The following result is of auxiliary nature.

e

e

Proposition 6.6.2. Let A E Sect( w),

Rea and J E cf. Lemma 6.2.5. Define

h(z)

:=

Ho such that z8 J E Ho

1 1

o

dt (f'¢)(tz) -

g(z):=

and

Ho.

r

1

/,00 (f'¢)(tz)-dt

t I t

as in Example 2.2.6. Then 1 = h + g, and zag E Hence g(A)X C 1>(Aa). Take x E X such that J; j := za J E Then clearly

io IIJ(tA)'¢(tA)xll

and Jooo(f'¢)(t) dtlt = 1,

dt

t < 00

Ho since gz8 is still bounded. rRea

11,¢(tA)xll dtlt < 00. Let

r

1

and

io IIj(tA)Ca'¢(tA)xll

dt

t < 00.

Proposition 5.2.4 implies that in fact

h(A)x =

1 1

o

dt J(tA),¢(tA)x-, t

and since J(tA)'¢(tA)x E 1>(Aa) for all t E (0,1) and the operator Aa is closed, we have h(A)x E 1>(Aa) with

Aah(A)x =

1 1

o

C

a

dt J(tA),¢(tA)x-.

t

o

Thus we arrive at x = h(A)x + g(A)x E 1>(Aa).

Specialising '¢ to a function which describes the real interpolation spaces as in Theorem 6.2.9 yields the following. Corollary 6.6.3. Let Re'Y

< Re f3 < Re a with Re'Y >

(1)(A'),1>(A a ))O,l where ()

E (0,1) is

C

1>(A,8)

C

°or

'Y = 0. Then

(1)(A'),1>(A a ))o,oo

(6.11)

such that Re f3 = (1 - ()) Re'Y + () Re a.

Proof. Since (1 + A)-' : X -----; 1>(A') is an isomorphism which carries 1>(A,8-,) to 1>(A,8) and 1>(Aa-,) to 1>(Aa), we may suppose that 'Y = 0. Then the statement follows from the characterisation in Theorem 6.2.9 by taking an appropriate 0 function '¢.

Cbapter 6. Interpolation Spaces

160

It is well known that the left-hand inclusion in (6.11) implies (and is in fact equivalent to) an inequality of the form IIA,6xll ::; C IIA'xI1 1- OIIAaxIIO. The usual interpolation-theoretic proof requires a description of the interpolation spaces different from the K-method, cf. [29, Theorem 3.5.2] or [157, Theorem 1.2.15]. However, our methods allow also a direct proof. Proposition 6.6.4 (Moment Inequality). Let A be a sectorial operator on the Banach space X. Let 0.,(3,,,,( E C such that Re"'( < Re(3 < Rea and Re"'( > 0 or "'( = O. Then there is a constant C such that

where B is defined as B := (Re (3 - Re "'() j (Re a - Rwy). Proof. Choose any 'ljJ E Hf) such that 'ljJza, 'ljJz-a are still bounded functions. 'ljJ(sz)dsjs, g(z) := 'ljJ(sz) dsjs as in Example 2.2.6. Then Define h(z) := z-ah and zag are bounded functions, whence it := z-(a-,6)h and 9 := z,6-'g are both in Hf). For x E 'D(Aa) we have

I;

This yields

It"

IIA,6xll::; tRea(l-OCh IIAaxl1 +rReaOCg IIA'xll

(see Proposition 2.6.11). Taking the infimum with respect to t

> 0 we arrive at

where C:= max{Cg,q,J. The term in brackets is less than (B(l- B))-l.

D

Remark 6.6.S. A more detailed analysis would reveal the kind of dependence of the constant C on A and on 0.,(3,,,,(. The classical estimate is [161, Lemma 3.1.7]. Corollary 6.6.6. Let A E Sect (w) and 0 ::; Re"'( or Re"'( > O. Then

where B E (0,1) is such that Re(3

6.6.3

< Re (3 < Re a,

where either "'(

=0

= (1- B) Re"'( + BRea.

Reiteration and Komatsu's Theorem

The next result could be obtained from Corollary 6.6.6 by means of the Reiteration Theorem B.2.9. However we can give a direct proof.

6.6. Fractional Powers as Intermediate Spaces

161

Proposition 6.6.7 (Reiteration). Let A be a sectorial operator on a Banach space X. Then the following assertions hold. a)

If 0

< Re,8:::;

Rea, then

(X,1>(A"))oRe{3 = (X, 1>(A~))o,p Rea'P for all () E (0, 1),p E [1,00]. b)

If 0

< Rwy < Re,8:::; Rea, a

x E (X,1>(A"))o,p

E (0,1), p E [1,00], and x E X, then

-¢=:::}

where () := (1 - a) (Rwy / Re a)

x E 1>(A')

and

A'x E (X, 1>(A~-'))a,p,

+ a(Re,8/ Re a).

c) Ifa,,8,,,!,p,a,() are as in b), then

Proof. a) Choose a function 'Ij; E E such that z-"'Ij; E and one can apply Theorem 6.2.9 twice.

Ho.

Then also 'lj;z-~ E

Ho

b) Choose ;j; E E such that z-";j; E E and use it to describe (X,1>(A"))o,p. Observe that also 'Ij; := z-,;j; E E and z-(~-')'Ij; = z-~;j; E E. Hence we may use 'Ij; to describe (X, 1>(A~-'))a,p. (Note that (), a E (0,1), whence Theorem 6.2.9 is applicable.) Now, since () Re a - Re,,! > 0, we have

if x E (X,1>(A"))o,p. Applying b) of Proposition 6.6.2 yields (X,1>(A"))o,p 1>(A'). Moreover,

C

for all t > O. This concludes the proof of the stated equivalence. c) follows immediately from b) since here one may suppose without loss of generality that A is invertible. 0

Theorem 6.6.8 (Komatsu). Let A be a sectorial operator on a Banach space X, and let () E (0, 1), p E [1, 00]. Suppose that the identity (6.12) holds for some a E C with Rea> O. Then it holds for all such a. Moreover, if A is invertible, its natural HOO(S",)-calculus is bounded on X for each 'P E (WA, 7f).

Proof. Without loss of generality we may suppose that A is invertible (replace A by A+l). Then by Dore's theorem, A has bounded HOO-calculus on 1>(A lI ,,), hence

Chapter 6. Interpolation Spaces

162

on X since A-On maps X isomorphic ally onto D(AOn). In particular, A E BIP and the spaces D(A,8) depend only on Re,8 (see Proposition 3.5.5). We show first that for fixed 0 we have D(A,80) = (X, D(A,8))o,p for arbitrary Re,8 > O. To obtain this we may suppose that Re a -I Re,8. Suppose that Re,8 > Rea. Then we let "( := ,8 - a and 01 := O(Re,8 - Re"()/(Re,8 - oRe"() and find by Proposition 6.6.7 the identities

Now A-or: (X, D(A,8-o,))Ol,P -----+ (D(Ao,), D(A,8))Ol,P is an isomorphism. Employing the assumption we obtain

The case Re,8 < Re a is proved analogously. To complete the proof, let Re"( > 0 and 0 < 01 < 1 with 0 -I 01 . Find Re,8 > 0 such that 010 1 = Re"(IRe,8. Then by Proposition 6.6.7 we have

6.6.4

The Complex Interpolation Spaces and BIP

So far we have encountered real interpolation spaces only. We have seen that these spaces allow characterisations in terms of the functional calculus and that the operator improves when restricted to these spaces. However, Komatsu's Theorem 6.6.8 shows that an identity of the form (X, D(An))o,p = D(AOn) can hold only if the operator A (or a translate of it) has bounded Hoo-calculus in the orginal space X. On the other hand, using complex interpolation spaces we have the following intriguing result. Theorem 6.6.9. Let A be sectorial and densely defined such that A Then, for 0 E (0,1), we have

for all 0 ::; Re a

+ 1 E BIP(X).

< Re,8 with either Re a > 0 or a = O.

(The reader may consult Appendix B.3 for background information on the complex interpolation spaces.) Proof. Replacing A by A + 1 we may suppose without loss of generality that A is invertible. Moreover, applying the isomorphism An on both sides of the identity, reduces the whole statement to the identity

(Rea>

o,e E

(0,1)).

6.6. Fractional Powers as Intermediate Spaces

163

From A E BIP also follows that 2J(k~) depends only on the real part Re a, whence we may suppose that a > O. By assumption, 2J(A) is dense in X and (A-iS)sEiR is a Co-group on X, whence one can find M 2: 1,1] 2: 0 such that

(s

E IR).

Now, two inclusions have to be shown. To prove the first, we take x E 2J(AaO) and consider the function

(z

E

S).

Clearly 1(8) = x and 1 : S -----+ X is holomorphic. Since the norms IIAiSl1 of the imaginary powers grow at most exponentially as s ----; 00, sUPzES 111(z)llx < 00. From Proposition 3.5.5 it follows that 1 : S -----+ X is continuous and 111(1 + is) 1I'D(Aal

= IIA a 1(1 + is) Ilx ::; M e(1-0l2 e- s2 e17alsIIIAOaxllx.

So in fact 1 E F(X, 2J(Aa)), and this implies that x E [X,2J(Aa)]o. The converse inclusion [X, 2J(Aa)]o C 2J(Aa) is shown by using the fact that 2J(Aa) is dense in [X,2J(Aa)]o (see Theorem B.3.3). Take x E 2J(Aa), and let 1 E V(X,2J(Aa)) be such that 1(8) = x (cf. Proposition B.3.4). Then 1 is of the form I(z) = 2:,7=1 'Pj(z)aj for certain 'Pj E Fo(C,q and aj E 2J(Aa). Define

(z Then 9 E Fo(X, X) with g(8) B.3.1: Ilxll'D(AOal

E

S).

= AOa x . Now we apply the Three Lines Lemma

= IIAooxllx = 111(8)llx :S sup {llg(is)llx, Ilg(l + is)ll x }, sEiR

but for each s E IR we have Ilg(is)llx ::; eo2 e-s21lAias l(is)llx ::; Me o2 e- s2 e17alsllll(is)llx, Ilg(1

+ is) Ilx ::; e(1-ol2 e- s2 1lA aA ias 1(1 + is) Ilx ::; Me(1- 0 l 2e- s2 e17alsllll(1 + is)II'D(Aal·

Hence we can find a constant c > 0 (not depending on 1 and x of course) such that Ilxll'D(AOal ::; C IIIII F . Taking the infimum with respect to 1 (and applying Proposition B.3.4) we find Ilxll'D(AOal ::; c Ilxll[x,'D(AalJo. Now density accounts for the rest. D

Remark 6.6.10. It is unknown in general whether the equality [X, 2J(A)]o = 2J(AO) for some 8 E (0,1) is sufficient to have A+1 E BIP. It is true on Hilbert spaces due to the fact that (X,2J(A))O,2 = [X,2J(A)]o for all 8 E (0,1) (see [158, Corollary 4.3.12]).

164

Chapter 6. Interpolation Spaces

Example 6.6.11. Consider the negative Laplacian -.6 p on X := LP(lR d ) , where p E (1,00) and p -1= 2. It has domain 1)(-.6p ) = W 2 ,p(lRd ) and is an injective, sectorial operator with spectral angle and bounded Hoo-calculus. However,

°

for all () E (0, l),p E [1,00]. See Section 8.3, in particular Remark 8.3.6. It should be clear from these considerations that an identity of the form

(X, 1)(A))o,p = 1)(Ao) is -

apart from the Hilbert space case -

6.7

extremely rare.

Characterising Growth Conditions

When we examined perturbations B of a sectorial operator A in Section 5.5, an important condition proved to be an estimate of the form sup litO B(t + A)-III < 00 t>O

(cf. Lemma 5.5.1 and Proposition 5.5.3). We remarked (in Remark 5.5.2) that one can describe such a condition in terms of interpolation spaces. The present section is to give evidence to this assertion. Proposition 6.7.1. Let A be an arbitrary sectorial operator on the Banach space X, let B : Y --- X be a bounded operator, where Y is another Banach space, and let () E [0, 1]. Then the equivalences

1«B) c (X, 1)(A))o,oo ~ sup IltO(t + A)-I BIIY---;1)(A) < 00 t>1

~ sup Ilto(t + A)-I BIIY---;D < 00 t>1

hold, with the latter applying only in the case where A is injective. Furthermore, if A is injective, then also the equivalences

1«B) c (X, 1«A)h-o,oo ~

~

sup Ilto(t + A)-I Blly---;x < 00

O 0. Then we have x = c- 1 1000 7(s-1 A)x ds/ s for all x E 'D(A) (Proposition 5.2.4). So, for x E 'D(A2) this integral converges in 'D(A), whence

roo IIC7(s-1 A)xll dss = c- 1 roo 11C(1 + s-l A)-l s-l A(l + S-l A)-lxll

IICxl1 :S C 1

io io

= c- 1

roo Ils1-0C(s + A)-IsO A(s + A)-lxll

io

:S c'c- 1

1 00

IlsO A(s + A)-lxll

~s

ds s

ds

s

:S c'c-11Ixll(x,1)(A))o,1

'D(A2). Since 'D(A2) is dense in (X, 'D(A))O,l, we obtain the desired result. We are left to deal with the cases = 0, 1. It is easily seen that condition (6.13) with = 1 is equivalent to IIClI1)(A)->Y < 00. So let = O. If C is bounded

for x

E

e

e

e

for the norm 11·ll x , then clearly (6.13) holds, by sectoriality of A. For the converse suppose that IltC(t + A)-lXlly :S 'Y Ilxll for each x E X and t > 1. Let x E 'D(A). Then

IICxl1 = IltC(t + A)- l XII for all t > 1. As t --)

00

+ IIC(t + A)-l Axil

we obtain IICxl1

:s 'Y Ilxll + c 1 IIAxl1

:s 'Y Ilxll for all x E'D(A).

D

Proposition 6.7.4. Let A be an injective sectorial operator on the Banach space X, let C : 'D(A) --+ Y a bounded operator, where Y is a second Banach space, and let e E [0,1]. Then the condition

(6.14) is trivially satisfied for the boundedness of

e=

--+

Y.

e E (0,1),

(6.14) is equivalent to

CA- 1 : (X,9((A)h-o,l --+ Y

on 9((A). In the case where

c: b

O. In the case where

e=

1 it is equivalent to the boundedness on'D(A) of

6.7. Characterising Growth Conditions

167

Proof. Suppose first that () E (0,1). Writing s = lit yields

Now one can apply Proposition 6.7.3 with A replaced by A-I and () replaced by 1 - (). The case () = bounded, one has

°

being easy, suppose now that ()

°

hence (6.14). Conversely, suppose that all < t < 1. Then

=

1. If C :

IIC(t + A)-lxll ::; "( Ilxll

D

--+

Y is

for all x E X and

IICxl1 ::; t IIC(t + A)-lxll + IIC(t + A)-l Axil::; t"( Ilxll + "( IIAxl1 for x E 1J(A). Hence if we let t

-+

0, we obtain

IICxl1 ::; "( Ilxlli>'

o

Again, we combine the last two propositions. Corollary 6.7.5. Let A be an injective sectorial operator on the Banach space X, let C : 1J(A) --+ Y be a bounded operator, where Y is another Banach space, and let () E [0, 1]. Then the condition

is equivalent to the boundedness on 1J(A) of C: X --+ Y, { C : (~, D)O,1 C:D--+Y,

--+

Y,

if () = 0, if ()E(O,l), if () = 1.

Proof. In the cases () = 0,1 this is just a combination of Propositions 6.7.3 and 6.7.4. Suppose that () E (0,1). Since A is injective, it induces a topological isomorphism A: (D,1J(A)) --+ (X, :R(A))

of Banach couples. Hence the boundedness of CA- I : (X,:R(A)h-o,1 --+ Y is then equivalent to the boundedness of C : (D,1J(A)h-o,1 --+ Y. This yields the boundedness of C : (X,1J(A))O,1 + (D,1J(A)h-o,1 --+ Y as a characterisation. However, (B.6) of Proposition B.2.7 applies and we obtain the identity

(X,1J(A))O,1

+ (D,1J(A)h-o,1 = (X, D)O,I.

o

168

6.8

Chapter 6. Interpolation Spaces

Comments

6.2 and 6.5. Characterisations of Real Interpolation Spaces. KOMATSU was probably the first who studied thoroughly the possible descriptions of real (i.e., LionsPeetre) interpolation spaces for sectorial operators. The description by holomorphic semigroups (6.9) and (for ex E N) the proper 'Komatsu description' (6.8) can be found already in [132]. At least for the case ex = 1 (Lemma 6.1.1) they are now folklore and reproduced in many texts on parabolic equations, e.g. in [157]. Theorem 6.5.3 and the general characterisations in Section 6.2 are due to the author [106, 110]. DORE [73] proves a special case of Theorem 6.5.3 b). 6.1, 6.4, and 6.5. Dore's Theorem and Homogeneous Interpolation. DORE'S results Theorem 6.1.3 from [72] and Corollary 6.5.8 (in the special case ex = 1) from [73] are by far not the first accounts of the fact that a sectorial operator improves some properties when restricted to its real interpolation spaces. Indeed, in 1975 DA PRATO and GRISVARD [58] discovered that an invertible sectorial operator always has maximal regularity in its interpolation spaces (cf. Section 9.3 below). And AUSCHER, McINTOSH and NAHMOD [19] pointed out that McIntosh's theorem on Hilbert spaces (Theorem 7.3.1) may also be read in this way (cf. Remark 7.3.2). Around the same time it turned out that an operator-valued version of Dore's theorem yields back the original results of DA PRATO and GRISVARD (cf. Section 9.3). The crucial point lies in the fact that one has at hand a functional calculus description for the interpolation space where one is allowed to vary the auxiliary function (cf. the proof of Theorem 6.5.6). This idea is from [167] and [19], and in fact the whole Section 6.4 is a generalisation of Hilbert space results from (parts of) [19] to general Banach spaces. The central Theorem 6.1.3 is due to the author [110] and is itself only a special case of a far more general result which allows the functional calculus to be operator-valued and asserts even R-boundedness of the functional calculus in the case where p E (1,00). See [110] for more details. 6.3 Extrapolation Spaces. The concept of extrapolation space has proved quite useful in semigroup theory, in particular in the context of perturbations (see [85, Section 111.3]). In the 'classical' approach one considers the norm of the (desired) extrapolation space X_Ion the original space X and takes the (abstract) completion. This embeds X densely into the new space X-I. For example the homogeneous spaces x(a) would be the completion of X with respect to the norm IIAaxll. In the case where A is densely defined this works well, but unfortunately breaks down otherwise. (The completion is then simply too small.) A one-step extrapolation without density was first given by HAAK, KUNSTMANN and the author [100] in the context of perturbation theory. Section 6.3 with its 'universal' extrapolation space goes back to the author's article [110, Appendix]. The construction can be considerably generalised. Let

6.S. Comments

169

X be a Banach space and T c £(X) a set of pairwise commuting, injective, bounded linear operators on X, closed under multiplication. To each T E Tone can trivially construct a one-step extrapolation space X T :) X together with an extension of T to an isometric isomorphism T : X T ----+ T. For two elements S, T E T the space XST may be regarded as a common superspace of X T and Xs since ST = TS. Hence the set of spaces (XT)TE'T is upward directed, and the inductive limit U := limT X T is an extrapolation space on which all operators -----7 T E T become isomorphisms. To identify extrapolation spaces in concrete situations with known spaces it is highly desirable to have a categorical definition of extrapolation spaces by means of a universal property. A first attempt can be found in [109]. Our notion of convergence on the universal space U is adapted to our purposes but is very unlikely to be induced by a proper vector space topology on U. Of course one thinks of the inductive limit topology as an alternative, but apart from very special cases (e.g., where one has weakly compact embeddings X-n "-+ X-(n+l)) one cannot guarantee that this topology is Hausdorff. This is so unpleasant a feature that we wanted to avoid it at any cost. 6.6 Intermediate Spaces and 6.7 Growth Conditions. The moment inequality (Proposition 6.6.4) and the Reiteration Theorem (Proposition 6.6.7) are well known, see e.g. [210], [211], [158]' or [161, Chapter 11] and the references therein. We treat the matter for the sake of completeness, but also since it shows how the functional calculus descriptions of the interpolation spaces can be used. Theorem 6.6.9 on BIP and the complex interpolation spaces seems to go back to the paper [200] of SEELEY for differential operators. Formulated and proved in full generality it may be found in [215, Theorem 1.15.3] or [161, Theorem 11.6.1]. Growth conditions as treated in Section 6.7 are common in perturbation theory. The characterisations given here in their full generality go back to the paper [100] by HAAK, KUNSTMANN and the author.

Chapter 7

The Functional Calculus on Hilbert Spaces We start with providing necessary background information on the functional calculus on Hilbert spaces. In Section 7.1 we show how numerical range conditions account for the boundedness of the Hoc-calculus. In the sector case this is essentially von Neumann's inequality (Section 7.1.3), in the strip case it is result by CROUZEIX and DELYON (Section 7.1.5). From von Neumann's inequality we obtain certain 'mapping theorems for the numerical range' (Section 7.1.4). Section 7.2 is devoted to CO-groups on Hilbert spaces. We discuss Liapunov's direct method (Section 7.2.1) from Linear Systems Theory and apply it to obtain a remarkable decomposition and similarity result for group generators (Section 7.2.2). This allows us to prove a theorem of DE LAUBENFELS and BOYADZHIEV on the boundedness of the Hoc-calculus on strips for such operators. This result can be approached also in a different way, yielding in addition a characterisation of group generators (Section 7.2.3). Section 7.3 is devoted mainly to the connection of the functional calculus with similarity theorems. The main tool is McINTOSH'S fundamental result on the boundedness of the Hoc-calculus (Section 7.3.1). The similarity questions we are interested in deal with operators defined by sesquilinear forms. We introduce these operators in Section 7.3.2 and obtain in Section 7.3.3 a characterisation of such operators up to similarity. Afterwards, several theorems on similarity are proved, related also to the so-called square root problem. We give an example of a Co-semigroup which is not similar even to a quasi-contractive semigroup (Section 7.3.4). Finally, we present applications to generators of cosine functions, showing in particular that after a similarity transformation those operators always have numerical range in a horizontal parabola (Section 7.4).

a.

Preliminaries on Functional Calculus

Before we can come to more substantial things we have to do some preliminary work. Let U c C be open and invariant under complex conjugation, e.g., U = Sw or U = Hw for some w. Let f : U --+ Coo be meromorphic. Then the function 1*, defined by

1*:= IS

(z

f----+

f(z)) : U --+ Coo

called the conjugate of the function

f. Obviously, 1* is meromorphic again.

Chapter 7. The Functional Calculus on Hilbert Spaces

172

Moreover, the spaces HO'(Sw), HOO(Sw),'" and F(Hw), HOO(Hw), ... are invariant under conjugation. The following proposition gives some background information on the functional calculus for sectorial operators on Hilbert spaces. Proposition 7.0.1. Let A E Sect(w) be a sectorial operator on a Hilbert space H. Then the following assertions hold.

a) The operator A is densely defined and H = N(A) EB 9«A). In particular, A is injective if and only if 9« A) is dense in H. b)

The operator A * is also sectorial of angle w with M (A, w') = M (A * ,w') for all w' E (w,n]. In particular, we have WA = WA*.

c) The operator A is injective (invertible, bounded) if and only if A* is injective (invertible, bounded).

d) The identity f(A*) = [f*(A)]* holds whenever f it holds for all f E B[Sw].

e) The identity

E

A[Sw]. If A is injective,

(A")* = (A*)Q

(7.1)

is true for all Re a > 0, and even for all a E C in the case that A is injective.

f) If A E BIP(H) then also A* E BIP(H) with eA = eA *. g) Let'P E (w, n]. If the natural HO'(S",)-calculus for A is bounded, the same is true for A* and the bounds are the same. Proof. a) follows from Proposition 2.1.1 since a Hilbert space is reflexive. b) follows from Corollary C.2.2. c) If A is injective, then 9«A) is dense. By Proposition C.2.1 e) this implies that N(A*) = O. The converse implication follows from A** = A. d) Let f E HO'(S",). Then . ( f*(Z)R(z,A)dZ)* = -1. { f(z)R(z,A)*dz (-21mh 2mh 1 . ( f(z)R(z, A*) dz = f(A*), = 2- 1. ( f(z)R(z, A*) dz ~ -2 mh mh

[f*(A)]* =

since f is such that (z t----+ z) just reverses the orientation of f. Since g* = 9 if g(z) = (1 + z)-l we thus obtain f*(A)* = f(A*) for all f E £(S",). Now, if f E A such that F(z) := (1 + z)-n f(z) E £ we have

f(A*) = (1 + A*t F(A*) Q) [(1 + A)n]*[F*(A)]* (;) [F*(A)(l

+ At]* ~

[f*(A)]*.

Here we have used Proposition C.2.3 in (1) and Proposition C.2.1 in (2). Equation (3) is justified by Proposition C.2.1 and the fact that 'D(An) is a core for f*(A) by Proposition 2.3.11 c).

7.1. Numerical Range Conditions

173

The proof of the statement in the case that A is injective and fEB is similar. One has to use the identity [A'A]* = AA*' which holds for each n E N. e), f) and g) are consequences of d).

D

Of course, there is an analogous result for strip-type operators, but we refrain from stating it explicitly.

7.1

Numerical Range Conditions

In this section we examine sectorial and strip-type operators A on a Hilbert space H, with the additional property that the numerical range W(A) is contained in the relevant set (sector or strip). The ultimate goal is to obtain boundedness of the natural Hoo-calculus for such operators.

7.1.1

Accretive and w-accretive Operators

Let us recall that an operator A on a Hilbert space is called accretive if its numerical range W(A):= ((Axlx) I x E 1)(A), Ilxll = I} (see Appendix C.3) is contained in the closed right half-plane Sn:/2. Since any restriction of an accretive operator is likewise accretive, an accretive operator need not be closed. However, if we assume in addition that -1 E [J( A) we obtain a so-called m-accretive operator, cf. Appendix C.7. By Proposition C.7.2 and Theorem C.7.3 the following assertions are equivalent for a closed operator A on H. (i) A is m-accretive, i.e., W(A) C Sn:/2 and :R(A + 1) is dense in H. (ii) -A generates a Co-semigroup (T(t))t>o with IIT(t)11 :::; 1 for all t ~ O.

II

(iii) {Re z < O} C [J(A) and ('x + A)-III:::; (Re ,X)-I for all Re'x > O. In particular, A E Sect (7r / 2) . As a matter of fact, one can consider also operators whose numerical range is contained in a smaller sector. Let w E [0, 7r /2]. An operator A on a Hilbert space H is called w-accretive if W(A) c Sw. Note that this means IIm(Aulu)l:::; (tanw) Re(Aulu)

(u

E

1)(A)).

The operator A is called m-w-accretive if it is w-accretive and :R(A + 1) is dense in H. Hence an operator is (m-) accretive if and only if it is (m-)7r /2-accretive. Furthermore, each m-w-accretive operator is m-accretive. A O-accretive operator is symmetric (since our Hilbert spaces are complex). An operator is positive if and only if it is m-O-accretive. The following proposition gives a useful characterisation.

Chapter 7. The Functional Calculus on Hilbert Spaces

174

Proposition 7.1.1. Let A be an operator on the Hilbert space H, let wE [0,7r/2), and define {} := 7r /2 - w. The following assertions are equivalent.

(i) The operator A is m-w-accretive. (ii) The operators e±iOA are m-accretive. (iii) The operator -A generates a holomorphic Co-semigroup (T(Z))zESe on So such that IIT(z) II ::; 1 for all Z E So. If one of these equivalent conditions is satisfied then A E Sect(w).

Note that (ii) does not imply (i) if w = O. For the meaning of (iii) cf. Section 3.4. Proof. (i){::}(ii). This is clear from Corollary C.3.2.

(i):::}(iii). From (i) and Proposition C.3.1 we infer that A E Sect(w). Since w < 7r /2 we conclude that -A generates a bounded holomorphic semigroup T : So ------> .c( H). For each 'P E (-{), (}) the operator _ei

w deal WIth the first summand. We define S(z):= T(z) - . Then Imz ::; -w T(z) dz = S(z) Idzl on 8Hw. Hence

S:=

r

j8Hw

T(z) dz =

r

j8Hw

S(z) Idzl

is a positive operator. Moreover, by Corollary C.6.4 we can estimate (7.4)

7.1. Numerical Range Conditions

183

so we have to specify 11811. To this end we write

~ (R(z,A) -!) + ~ (! -~) + ~ (~- R(z,A*)) 27l'z z 27l'z Z Z 27l'z Z

T(z) =

1 R(z,A)A 27l'i z

=-

-Imz R(z,A*)A* +- - - 1 ----'---'---'-7l'Izl2

z

27l'i

When we form 8 = J8Hw T(z) dz the first and the last summand vanish by Cauchy's theorem (deform the path according to w ----> 00). Hence we arrive at

l

-Imz - 2 dz = 8Hw 7l'Izi

8 =

1

l-+

w 2 dr = 2. IR 7l' Ir iwl

w 2 dr IR 7l' Ir - iwl

(7.5)

We now treat the second summand, namely

~

r

2m J8Hw

J(z) R(z, A*) dz

=~

rJ(r-iw)R(r+iw,A*)dz-~ rJ(r+iw)R(r-iw,A*)dz 27l'z JIR

27l'z JIR

r J(z)R(z + 2iw, A*) dz - ~ 1 J(z)R(z - 2iw, A*) dz 27l'Z 1 . r J(z)R(z - 2iw, A*) dz (j ~ r J(z)R(z + 2iw, A*) dz - -2 27l'zJIR 7l'zJIR = ~ r J(z) [R(z + 2iw, A*) - R(z - 2iw, A*)] dz 27l'z JIR

=

~

27l'ZJlrnz=-w

= -2w

7l'

Irnz=w

rJ(t)R(t + 2iw, A*) R(t - 2iw, A*) dt,

JIR

where in (*) we have used Cauchy's theorem. Write

R(t+2iw, A*)R(t - 2iw, A*) = ((t - A*)2 + 4w2r1 = ((t - B + iC)2 + 4w2r1

= (( t

-

B)2

+ 4w 2 -

C 2 + i [2tC - C B _ BC]) -1

= (Mt + iNt )-l = M t-2 (1 + iRt)-l M t-2 1

1

with M t := (t_B)2+4w 2-C 2, Nt := 2tC-BC-CB and R t := M t- 1 / 2N t M t- 1 / 2. Note that all operators M t , Nt, R t are self-adjoint and one has M t ~ 3w 2. Now II (I + iRt)-lll :::; 1 for all t since Rt is self-adjoint. Employing Proposition C.6.3 we obtain

112~i faHw J(z) R(z, A*) dzll = 2;

III

J(t)Mt-! (I + iRt )-l Mt-! dtll

:::; IIJllHw 2;

III

Mt- 1 dtll·

(7.6)

184

Chapter 7. The Functional Calculus on Hilbert Spaces

Now, M t ~ (t - B)2 C.4.7. This gives

+ 3w 2 whence Mt- 1 ~ ((t - B)2 + 3w 2) -1

by Proposition

(7.7)

f ((t - B)2 + 3w2r1 dt = ; . J~ y3w Proof of Claim. The assertion is readily seen to be true by using the Spectral Theorem. Without this powerful tool one can argue as follows (with 7] := v'3w): Claim:

f ((t-B)2+7] 2r 1 dt= ~ f (R(t-i7],B) -R(t+i7],B)) dt ~ ~7]~

= ~ f(R(t _ i7],B) _ _ 1_.) + (_1_. _ _ 1_.) _ (R(t + i7],B) _ _ 1_.) dt 2Z7] J~ t - Z7] t - Z7] t + Z7] t + Z7] = _1 f R(t - i7], B)B _ R(t + i7], B)B dt + f ~ 2i7] J~ t - i7] t + i7] J~ t 2 + 7]2 = _1_ f R(z,B)B dz+ ~ = ~ 2i7]

JaH

'1

z

7]

7]'

by Cauchy's theorem (make a change of variable w := that w- 1 R(w- 1 , B) is holomorphic at 0).

Z-l

in the integral and note

The Claim together with (7.7) shows that IIJ~Mt-1dtll ~ 1f/(v'3w). Combining this with (7.3), (7.4), (7.5), and (7.6) yields

Ilf(A)11 ~ 1lfliH

W

2 + IlfliHw1fy3w 2w ; = (2 + 2/J3)

IlfliH . W

D

As a corollary we obtain a statement for w-accretive operators. Corollary 7.1.17. Let A be m-w-accretive for some w E [0, 1f /2]. Then

Ilf(A)11

~

(2 + ~) Ilfllsw

(7.8)

for all f E R(Sw). If A is injective, then (7.8) holds for all f E HOO[Sw]. Proof. Let B := logA and c := 2 + 2/v'3. Then B is Hw-accretive by Corollary 7.1.14. Hence

by the composition rule.

D

Remark 7.1.18. A different (and easier) proof can be given for the following, considerably weaker, statement: If A is Hw-accretive and rp E (w, 1f), then the natural HOO(H 0 such that

c

{z

en.

I Re z > a}.

Then

(t 2': 0). The theorem has two components. First it states that the spectral condition s(-A) := sup{Re'\ I ,\ E O"(-A)}

0 such that IIT(t) II ::::: M ew1tl for all t E R Thus,

Ilxll~ = (Qx I x) = roo IIT(t)xI12 dt ~ roo M- 2e- 2wt dt IIxl1 2 =

io

io

2w

1M2

IIxl1 2

for all x E H. Hence in this case the new scalar product is equivalent to the old one and Q is invertible (Proposition C.5.1). For the proof of the Liapunov inclusion note that since T is exponentially stable, the resolvent of its generator A is given by

_A- 1

= R(O, A) =

1

00

T(s) ds

(see Proposition A.8.1). Therefore,

Q(_A- 1 )

= =

1 1 1 (1 00

T(t)*T(t)

00

8

00

T(s)dsdt

=

1 1 00

T*(t)

00

T(s)dsdt

T(t)* dt) T(s) ds.

Similarly,

Adding the two identities we obtain

_(QA- 1 + (A*)-lQ)

=

11 00

00

T(t)* dtT(s) ds

= (A*)-l A- 1 .

This yields QA- 1 = _(A*)-l(A-l + Q), whence (7.10) holds. Suppose T is a group. Then Q is invertible and AO = Q-l A*Q is the adjoint with respect to the new scalar product (see Lemma C.5.2). Multiplying the Liapunov inclusion from the left by Q-l yields A c -AO - Q-l. But -Ao - Q-l is a bounded perturbation of a Co-semigroup generator, whence it is also a generator of a C osemigroup. This implies readily that A = -A ° - Q-l. Multiplying by Q from the left yields (7.11). D Corollary 7.2.5 (Liapunov's theorem for groups). Let T be an exponentially stable Co-semigroup on the Hilbert space H. Suppose that T is a group. Then for each E E (0, -wo(T)) there is an equivalent Hilbert norm 11·110 on H such that

(t

~

0).

Chapter 7. The Functional Calculus on Hilbert Spaces

188

Remarks 7.2.6. 1) Since the 'Liapunov inclusion' QA C -J - A*Q is not an equation in general, we do not want to call it a 'Liapunov equation' (which is the name in the finite-dimensional setting). However, one can reformulate the inclusion as a system of equations ( Ax Qy) 1

+ (Qx

1

Ay)

=-

(x y) 1

(x, y E 1>(A)).

In [57, p.160 and p.217] this system is called 'Liapunov equation'. 2) We shall prove an infinite-dimensional Liapunov theorem for holomorphic semigroups in Corollary 7.3.8. The next proposition shows in which cases the 'direct method' works. Consult [105] for a proof. Proposition 7.2.7. Let A be the generator of an exponentially stable Co-semigroup T on H, and define Q := Jo= T(t)*T(t) dt. Then the following assertions are equivalent.

(i) The semigroup T is a group. (ii) The operator Q is invertible and T(t) has dense range for some t (iii) The operator Q is invertible and T*(t) is injective for some t (iv) Both operators Q and

Q := Jo= T(t)T*(t) dt

> O.

> O.

are invertible.

(v) The operator Q is invertible and no left half-plane is contained in the residual spectrum of A.

As a consequence we obtain that the direct method works in the case of holomorphic semigroups if and only if the generator is bounded.

7.2.2

A Decomposition Theorem for Group Generators

We come back to our original goal. Let A be the generator of a Co-group T on the Hilbert space H. Recall the definition of the group type e(T) in Appendix A on page 302. We fix w > e(T) and define (xly)o:=

=

l

1=

(T(t)xIT(t)y)e-2wltldt (T(t)x T(t)y) e- 2wt dt 1

+

1=

(T(-t)xIT(-t)y)e- 2wt dt

(7.12)

for x, y E H, i.e., we apply the Liapunov method simultaneously to the rescaled 'forward' and 'backward' semi groups obtained from the group T. From Proposition 7.2.4 it is immediate that (·1')0 is an equivalent scalar product on H. The following theorem summarises its properties.

189

7.2. Group Generators on Hilbert Spaces

Theorem 7.2.8 (Decomposition Theorem). Let iA be the generator of a Co-group T on a Hilbert space H and let w > (}(T). With respect to the (equivalent) scalar product ( ·1·)0 defined by (7.12) the operator A is m-Hw-accretive.

If one writes A = B + iC with B, C self-adjoint and -w ::; C ::; w with respect to (.1. )0' then 1J(A) = 1J(AO) is C-invariant, and [B, CJ = BC - CB has an extension to a bounded operator which is skew-adjoint with respect to (·1· )0. Note that by Proposition 7.1.2 the decomposition A = B

+ iC exists.

Proof. For s E JR, x E H one has

IIT(s)xll~ = ~ IIT(t)T(s)xI12 e-2wltl dt = ~ IIT(t + s)x11 2 e-2wltl dt = ~ IIT(t)xI1 2 e-2wlt-sl dt = ~ IIT(t)xI1 2 e-2wltle2w(ltl-lt-sl) dt ::; e2wlsl

Ilxll~

since It I - It - sl ::; lsi for all s, t E JR by the triangle inequality. This gives IIT(t)llo ::; ew1tl for all t E JR, whence iA is m-Hw-accretive by Proposition 7.1.2. Now, let

QIJ) :=

1

00

T(t)*T(t)C 2wt dt,

Qe:=

1

00

T( -t)*T( _t)e- 2wt dt,

and

Q := QIJ) + Qe =

~ T(t)*T(t)e-2wltl dt.

Then (x I Y)o = (Qx I y) for all x, y E H. The Liapunov equations for QIJ) and Qe read

QIJ)(iA - w) = -I - (-iA* - w)QIJ) Qe( -iA - w) = -I - (iA* - w)Qe·

(7.13) (7.14)

Subtracting the second from the first yields

QA c A*Q + 2iw(QIJ) - Qe). If we multiply by Q-l we arrive at A c AO + 2iwQ-l(QIJ) - Qe). Since 1J(A) = 1J(AO) by Proposition 7.1.2, we actually have equality and

The C-invariance of 1J(A) is clear from the formulae (7.13) and (7.14) and the identity 1J(A) = 1J(AO) = Q-l1J(A*). Furthermore, employing again (7.13) and

190

Chapter 7. The Functional Calculus on Hilbert Spaces

(7.14) as well as AO

iCA

= A - 2iC, we compute

= wQ-l(QEBiA - QeiA) = wQ-l( -1 + 2wQEB + iA*QEB - 1 + 2wQe - iA*Qe) = wQ-l( -2 + 2wQ + iA*(QEB - Qe)) = -2WQ-l + 2w 2 + iwAoQ-l(QEB - Qe) = -2WQ-l + 2w 2 + iAoC = -2WQ-l + 2W2 1+ i(A - 2iC)C = -2wQ-l + 2w 2 + 2C2 + iAC.

This shows that [B, Cj = [A, Cj has an extension to a bounded operator which is skew-adjoint with respect to ( ·1· )0' D A simple renaming of terms and forgetting the new scalar product yields the following stunning corollary. Corollary 7.2.9. Let A generate a Co-group T on a Hilbert space H. Then there exists a bounded operator C such that B := A - C generates a bounded Co-group. Moreover, the operator C can be chosen in such a way that 1)(A) is C-invariant and the commutator [A, Cj = AC - CA has an extension to a bounded operator

onH. Remark 7.2.10. Let A be an Hw-accretive operator on H. By Proposition 7.1.2, A = B + iC with B, C self-adjoint and -w ::::: C ::::: w. In general, however, 'D(A) is not C-invariant. In fact, let H := L2(JR) and B = d/dt be the generator of the shift group. Furthermore, let C := (f f--) wmf) where m(x) = sgnx is the sign function. Then C is bounded and self-adjoint, and A := B + C generates a Co-group T with IIT(t)11 : : : ew1tl . Obviously, 'D(A) = 'D(B) = W 1,2(JR) is not invariant with respect to multiplication by m. This example shows that the additional statement in Theorem 7.2.8 is not a matter of course and is due to the particular way of renorming.

Theorem 7.2.8 together with the Crouzeix-Delyon Theorem 7.1.16 immediately imply Theorem 7.2.1.

7.2.3

A Characterisation of Group Generators

We now turn to a second proof of Theorem 7.2.1. In fact, we prove a more general result, characterising generators of Co-groups on Hilbert spaces. In this section we consider a strip-type operator A space H. Let 'Pl, 'P2 E JR \ [-w, wj. Then we have

E

Strip(w) on the Hilbert

7.2. Group Generators on Hilbert Spaces

191

for each x E H. (Use the resolvent identity and the fact, that the resolvent is uniformly bounded on the horizontal lines IR + iCPl and IR + icp2). We say that A admits quadratic estimates if for every cp E IR \ [-w,w] there exists c = c(A, cp) such that lllR(t

+ icp, A)x112 dt :::; c(A, cp) IIxl1 2

(x

E

H).

From the remarks above and the Closed Graph Theorem it follows that A admits quadratic estimates if and only if there exists cp E IR \ [-w, w1 such that (tI------+R(t+icp,A)x) E L2(IR,H)

for all x E H. If A admits quadratic estimates then also -A and A each A E C.

+ A do so, for

Example 7.2.11. Let A E Strip(w), and suppose that iA is the generator of a C osemigroup T on H. We claim that A admits quadratic estimates. In fact, we can find constants M,wo such that IIT(t)11 :::; Me wot for all t 2: O. Then

R(A, A)x

= lDO e-AtT(t)x dt

for all Re A > wo0 An application of the Plancherel Theorem C.B.1 now yields lllR(t - icp, A)x11 2 dt

= lIIR(it, A - cp)x112 dt = 27r lDO Ile-'P T(s)xI1 2 ds S

: : : 7rM2 IIxl1 2 cP - Wo

for all cp > w, Wo and all x E H. Since also iA * = - (iA) * generates a Co-semigroup, the operator A* admits quadratic estimates as well. We can now state the main result. Theorem 7.2.12 (Characterisation Theorem). Let H be a Hilbert space, let w 2: 0, and let A E Strip(w) on H such that A is densely defined. Then the following assertions are equivalent.

(i) The natural HDO(Hc,J-calculus for A is bounded, for one / all a> w. (ii) The operator iA generates a Co-group. (iii) The operator iA generates a Co-semigroup. (iv) The operators A and A* both admit quadratic estimates. IfiA generates the Co-group T, then w(T) :::; w.

Chapter 7. The Functional Calculus on Hilbert Spaces

192

Proof. The implication (ii)=}(iii) is obvious, and (iii)=}(iv) is in Example 7.2.1l. To prove (i)=}(ii) one only has to note that the boundedness ofthe Hoo-functional calculus for A on some horizontal strip Ha implies the Hille-Yosida conditions for iA and -iA (see Theorem A.8.6). Hence iA generates a Co-group T. Futhermore, we see that in this case the group type of T is at most as large as Q. To establish the implication (iv)=}(i) needs a little more effort. We fix

Q

E

(w, M) and define the auxiliary function 'ljJ by (z

E H a ),

2 C 2)2

dt = l.

where c is chosen such that

( 'ljJ( t) dt = { (

J~

J~

= 4M3 rTf). For a given 1 E HOO(Ha) we now define the

(One can easily compute c approximants In by

In(z) :=

i:

+t

M

(f'ljJt)(z) dt = I(z)

i:

'ljJ(z + t) dt

(7.15)

where here and in the following for a function 9 on the strip Ha we denote by gt the function gt := (z f---7 g(t + z)) : Ha ----t 0 such that

d

c

1'ljJ(z) I = 1M2

+ z 2 12

For fixed n E N one can find dn 1

--------~2

1 + IRe z + tl


0 such that dn 1 + IRe zl

2

(z E H a ,

It I :::; n).

7.2. Group Generators on Hilbert Spaces

193

With the help of this we can compute

Ifn(z)l:S;

Ilflloo

j nI'l/J(z + it)1 dt:s; Ilflloo jn -n

-n

:s; Ilflloo 1 +

d

(1 + IRe z + tl

2 2dt )

~e Zl2 (k1: t2 dt) = Ilflloo 1 +d~~:zl2

for z E Ha. This proves a). To establish b), let n E Nand z E Ha. Then

Ifn(z)1

< Ilflloo kl'l/J(z + t)1 dt < Ilfll oo

k

(1 + IRedz + t12)2 dt =

Ilflloo

k

(1 +dt2 )2 dt.

Part c) follows easily, since by b) and Vitali's theorem it suffices to show that fn(z) ----) f(z) for all z E R But this is obvious from (7.15). The statement d) is immediate from the Convergence Lemma (Proposition 5.1.7). To prove e) we let 'T)(z) := 1/(p,2 + z2). Hence'l/J = C'T)2. Choose WI E (W, a) and let 'Y := 'YWI = 8Hwl ' Now we fix t E lR and compute

2

f

JlR

ds

1p,2 + (s + iWI)21

2kl(s + iWI + iP,~~S + iWI -

ip,)1

I

I

2

2

(J.IS+i:'S+i~12) (J.IS+~;'-i~12) 2(f ds )!(f ds )! JlR (p, + wd + S2 JlR (p, - wd + S2

< 2

2

21T Vp,2 -

wf

Using this we can estimate I (f'T)t)( A) I for each t E lR by

2

194

Chapter 7. The Functional Calculus on Hilbert Spaces

Thus, for arbitrary x, y E H one has

l(fn(A)xly)1

=

Ii: Ii: Ii:

((f1jJt)(A)x y) dtl 1

c

(lIt (A) (fllt)(A)x 1 y) dtl

c

((fllt)(A)R(iJ-L - t, A)x 1 R(iJ-L - t, A*)y) dtl

< cL(A, WI) IlflLXl
(A"2)

2'

2'

2'

2'

~,2

.

This yields H = (D(1/2),R(1/2))1/2,2' and by Theorem 6.4.5 this is exactly what we intended to prove.

7.3.2

Interlude: Operators Defined by Sesquilinear Forms

This section has a preparatory character. We briefly review the construction of operators by means of elliptic forms. This construction provides us with two interesting questions concerning similarity that will be answered in the subsequent section with the help of McIntosh's theorem. Let dEN and 0 C ~d an open set, and let H := L2(0). Suppose that we are given a set of coefficients aij E L'X> (0) (i, j = 1 ... , d) satisfying a uniform strict ellipticity condition, i.e., d

Re

L

ai,j(x)~i~j 2: fJ 1~12

i,j=l

for some fJ > O. Given bi , Ci E L oo (0), i = 1, ... ,d, and m E LOO (0) one considers the operator d

d

A:= - "~i,j=l D·(a· J ~,J. D) ~ +" ~i=l (b·D· ~ ~ - D·c·) ~ ~

+

m,

which acts (in the distributional sense) from W 1 ,2(0) to D(O). Taking the part of this operator in L2 (0) yields the so called maximal L2-realisation of the differential operator. In general, only a further restriction of this operator (which amounts to incorporating boundary conditions) leads to an operator with reasonable properties. Such restrictions are furnished by considering a closed subspace V c W 1 ,2(O) containing 1)(0) and thereon the sesquilinear form

The operator A is then defined by

(u, Y)

E

A

: (A) and Au = A u for all u E V, whence A is a restriction of A. The choice V = W~,2(0) yields Dirichlet and the choice

198

Chapter 7. The Functional Calculus on Hilbert Spaces

v = W 1 ,2 yields Neumann boundary conditions.

Other choices of V lead to socalled Robin or mixed boundary conditions, d. [16,17]. By a standard argument (see [210, Sec.2.2, Ex.l] or [87, Chapter 6]) one establishes the so-called GArding inequality, i.e., Re a( u, u) + Ao Ilull~2 ~ 81Iull~Tl,2

(u E V)

for some constants Ao E IR and 8 > O. As we shall see below, this inequality implies that the operator A + Ao is m-w-accretive for some w E [0, 7r /2). We now turn to the abstract description. Let H be a Hilbert space, V c H a dense subspace and a E Ses(V) a sesquilinear form on V. For A E C we define the form a.>, E Ses(V) by a.>,(u, v) := a(u, v)

+ A (u

1

v)H

for u, v E V. The form a is called elliptic if there is Ao two conditions hold.

~

0 such that the following

1) The form Rea.>,o = (Re a) + Ao (.I')H is a scalar product on V; this turns V into a Hilbert space such that the inclusion mapping V cHis continuous. 2) The form a is continuous with respect to this scalar product on V. (A moment's reflection shows that in our example above indeed the two conditions are satisfied, see Remark 7.3.3 below.) With an elliptic form a on V c H we associate an operator A in the following way. We define a norm on V by

Ilull~

:= Rea(u)

+ Ao Ilull~

for u E V. Condition 1) then implies that V is a Hilbert space. Since V is continuously embedded in H (also by condition 1)), there is a continuous embedding of H* into V* (injectivity follows since V is dense in H). If we identify H with its antidual by means of the scalar product (,1·) H of H (Riesz-Frechet theorem), we obtain a sequence of continuous embeddings V

c

H(~

H*)

c

V*.

In doing this, H becomes a dense subspace of V*. Now we define the mapping A : V -----+ V* by A:= (u f------+ a(u, .)) : V -----+ V*. By means ofthe identifications above we obtain (A + Ao) (u) = a( u, .) + Ao ( u The operator B := A +Ao : V -----+ V* is an isomorphism.

1 .)

H"

Proof. Consider the form a.>,o. Then Rea.>,o is the scalar product of V, hence a is coercive (see (C.5) on page 324). By condition 1), a is also continuous, hence D satisfies the hypotheses of the Lax-Milgram theorem (Theorem C.5.3).

7.3. Similarity Tbeorems for Sectorial Operators

199

Employing the embedding H c V* we define

D(A)

.-

{u E V 1 there is y E H s.t. a(u,·) = (yl·)H} {u

EV

1

Au

E H}

and Au := A u regarded as an element of H. Then the two fundamental identities

a(u,v) = (Au,v) a(u,v) = (Aulv)H

(u E V) (u E D(A))

(7.19)

(7.20)

hold for v E V. The operator A is called associated with the form a (notation: A rv a). Note that the definition of A is actually independent of the chosen AO. An operator A on H is called Kato-sectorial if there is a dense subspace V cHand an elliptic form a E Ses(V) on V such that a rv A. Clearly, if A is Kato-sectorial, then A + A is Kato-sectorial for each A E C. Furthermore, also A * is Kato-sectorial (if A rv a then A* rv a). Remarks 7.3.3. 1) We use the name 'Kato-sectorial' because these operators were treated extensively by KATO. In [130] he called them just 'sectorial' but nowadays this very name is used as we have defined it in Chapter l. 2) In our introductory example of an elliptic differential operator with boundary conditions the starting point is a little different in that the space V already carries a (natural) Hilbert space structure such that V is densely and continuously embedded in H. We denote this given scalar product on V by (·1· )v. Instead of the conditions 1) and 2) one rather has 3) Rea>Ju) 2 611ull~ for some 4) la(u, v)1

: : : M Iluliv Ilvllv

6> 0 and all u E V.

for some M ~ 0 and all u, v

E V.

Here, condition 4) says that a is continuous on V, and this together with condition 3) implies that Re a), is an equivalent scalar product on V. So indeed conditions 1) and 2) are satisfied. The following characterisation of Kato-sectorial operators is well known. Proposition 7.3.4. Let A rv a where a : V x V ~ H is an elliptic form. Then there is w E [0, 7f /2) such that A + AO is an invertible m-w-accretive operator on H. (Here, AO is such that the conditions 1) and 2) above hold.) Conversely, let A be an m-w-accretive operator on H for some w E [0, 7f /2). Then A is Kato-sectorial. More precisely, there is a dense subspace V cHand an elliptic form a E Ses(V) such that A rv a and Re a is positive. Proof. Suppose that A rv a and a: V x V ~ H is elliptic, and let AO be as above. Then the operator B := A + AO : D(A) ~ H is bijective and B- 1 is a bounded operator on H since B is the restriction of!3 : V ~ V* to the range H. Because H is dense in V*, D(A) = 'D(B) is dense in V, hence a fortiori in H. This means

Chapter 7. The Functional Calculus on Hilbert Spaces

200

that B is a densely defined closed operator in H with 0 E e(B).

Claim: The form a.xo is continuous with respect to the scalar product Re a.x o. Proof of Claim. Since the embedding V cHis continuous, there is a constant Ml such that Ilull~ ::; Ml Ilull~ for all U E V. The continuity of the form a (condition 2) yields the existence of a constant M2 such that la(u, v)1 ::; M 2 11 u ll v Ilvllv. Putting the two inequalities together we obtain la.xo(u,v)1 ::; (M2 + AoMl ) Ilull~. A short glance at Proposition C.1.3 shows that the form a.xo is sectorial of some angle w which depends on the continuity constants of a.xo with respect to Re a.x o. Since (Bu Iu )H = a.xo (u, u) for u E 1) (B) the operator B is m-w-accretive. Conversely, let A be m-w-accretive. On 1)(A) we define the sesquilinear form a by a( u, v) := (Au I v) H and a scalar product (!) by

(ulv)v:= (Rea)(u,v)+(ulv)w The form a is continuous with respect to this scalar product. In fact, since a is sectorial of angle w, by Proposition C.1.3 we have

la(u, v)1 ::; (1 + tanwh/Re a (uh/Re a(v) ::; (1 + tanw) Iluli v Ilvllv for all u,v E 1)(A). Obviously, the embedding (1)(A),II.ll v ) cHis continuous. Hence it has a continuous extension L : V ----+ H, where V is the (abstract) completion of 1)(A) with respect to 11.ll v Claim: The mapping L is injective. Proof of Claim. Let x E V and LX = o. This means that there is (un)n C 1)(A) such that Ilu n - umll v ---) 0 and Ilunll H ---) O. Hence for all n, mEN we have

Rea(un)

Re (Aun Iun)H = Re (Aun IUn - um)H Re[a(un , Un - um)] + Re (Aun I um)H

< la(un,un - um)1 + I(Aun Ium)HI < M Ilunll v Ilun - umll v + IIAunll H Ilumll H Since (Un) is V-Cauchy, C := sUPn Ilunll v

+ Re (Aun Ium)H

.

< 00. Hence

Rea(un ) ::; MClimsup Ilu n - umll v ---) 0 as n ---)

00.

m

This shows that Ilunll v ---) 0, whence x = O. We therefore can regard V as being continuously and densely embedded into H. The form a has a unique extension to a continuous sesquilinear form on V (again denoted by a). Clearly, the form a is elliptic (with AO = 1). Hence it remains to show that A is associated with a.

7.3. Similarity Theorems for Sectorial Operators

201

We denote by B the operator which is associated with a and choose u E 1)(A). The equation a(u, v) = (Au Iv)H holds for all v E V since it holds for all v E 1)(A), 1)(A) is dense in V, a is continuous on V, and V is continuously embedded in H. This shows u E 1)(B) and Bu = Au, whence A c B. On the other hand, we have -1 E e( B) by construction and -1 E e( A) since A is m-w-accretive. This implies that A = B. 0 Combining Proposition 7.3.4 with Proposition 7.1.1 yields the following corollary. Corollary 7.3.5. An operator A on H is Kato-sectorial if and only if there is A E IR such that A + A is m-w-accretive for some w < n:/2 if and only if -A generates a holomorphic semigroup T on some sector So such that

IIT(z)11 :::; eARez

(z E So)

for some A E IR.

Note that the corollary says that A is Kato-sectorial if and only if A is 'quasim-w-accretive' for some w E [0, n: /2). The construction of operators by means of sesquilinear forms, though quite powerful, raises two highly non-trivial questions. The first is due to the fact that, in contrast to many other concepts, the notion of 'Kato-sectoriality' relies heavily on the particular chosen scalar product. That is, an operator may be Katosectorial with respect to the original but not Kato-sectorial with respect to some new (although equivalent) scalar product on H. (A concrete example is given in [12].) In fact, a result of MATOLCSI says that if A is not a bounded operator, one always can find a scalar product on H such that A is not Kato-sectorial (not even quasi-m-accretive, see [162]). It is therefore reasonable to ask for a characterisation of Kato-sectorial operators modulo similarity. Such a characterisation can indeed be given with the help of the functional calculus (see Corollary 7.3.10 below.) To understand the second problem, we have to cite a stunning theorem from

[127]. Theorem 7.3.6 (Kato). Let A be m-accretive and 0: E [0,1/2). Then the following assertions hold: 1) 1)(A") = 1)(A*") =: 1)".

2) 3)

IIA*"ull:::; tan (n:(1 + 20:)/4) IIA"ull for all u E 1)". Re (A"u IA*"u) 2: (cosn:o:) IIA"uIIIIA*"ull for all u E 1)".

Proof. See [127, Theorem 1.1]. The proof proceeds in two steps. First the statements are proved under the additional hypothesis of A being bounded with Re (Au Iu) 2: /j IIul12 for all u E H and some /j > O. (This is the difficult part of the proof.). The second (easy) step uses 'sectorial approximation' (in our language) and Proposition 3.1.9. 0

202

Chapter 7. The Functional Calculus on Hilbert Spaces

Kato's theorem is remarkable also in that it fails for a = 1/2. KATO was not aware of this when he wrote the article [127] but only shortly afterwards, LIONS in [154] produced a counterexample. For some time it had been an open question whether at least for m-w-accretive operators A it is true that

(7.21 ) If A is associated with the form a E Ses(V) and, say, we have AO = 0, then (7.21) is equivalent to V = 'D(A1/2), as LIONS in [154] and KATO in [128] have shown. Finally, McINTOSH gave a counterexample in [163]. However, this was not the end of the story. In fact, even if the statement is false in general it might be true for particular operators such as second order elliptic operators on L2 (lR.n) in divergence form. For these operators the problem became famous under the name Kato's Square Root Problem and has stimulated a considerable amount of research which led to the discovery of deep connections between Operator Theory and Harmonic Analysis (see also the comments in Section 7.5). Let us call an operator A on the Hilbert space H square root regular if A + A is sectorial and 'D((A + A)1/2) = 'D((A + A)l/h) for large A E lR.. Whether an operator is square root regular does depend on the particular chosen scalar product. (If the scalar product changes, A stays the same while the adjoint A * changes.) Hence it is reasonable to ask the following: Suppose that A is Kato-sectorial with respect to some scalar product. Is there an equivalent scalar product on H such that A is Kato-sectorial and square root regular with respect to the new scalar product? This is indeed the case (see Corollary 7.3.10 below).

7.3.3

Similarity Theorems

After so much propedeutical work we can now bring in the harvest. The first similarity result has an interesting historical background (see the comments in Section 7.5 of this chapter). Theorem 7.3.7 (Callier-Grabowski-Le Merdy). Let -A be the generator of a bounded holomorphic Co-semigroup T on the Hilbert space H. Let B be the injective part of A on K := :R(A). Then T is similar to a contraction semigroup if and only if BE BIP(K). Recall that the injective part of A is the part of A in :R(A). Clearly, A is m-w-accretive if and only if B is. Proof. Suppose that T is similar to a contraction semigroup. Changing the scalar product we may suppose that A is m-accretive. Therefore, B E BIP(K) by Theorem 7.1.7.

To prove the converse, let B E BIP(K). Since -A generates a bounded holomorphic semigroup, A is sectorial with WA < 1f/2. By Theorem 7.3.1 we can

203

7.3. Similarity Theorems for Sectorial Operators

change the norm on K to Uooo Ilf(tA)xI1 2 dt/t)1/2 where f E HO'[SWA] is arbitrary. If we choose f(z) := zl/2e- z we obtain the new norm

on K. Employing the semigroup property it is easy to see that each T(t) is contractive on K with respect to this new norm. However, on N(A) each operator T(t) acts as the identity. Since H = N(A) EEl K we can choose the new scalar product on H in such a way that it is the old one on N(A), the one just constructed on K and N(A) ~ K. With respect to this new 0 scalar product, the semigroup T is contractive. For a different proof of Theorem 7.3.7 using interpolation theory, see the comments in Section 7.5. As an immediate corollary we obtain the announced Liapunov theorem for holomorphic semigroups. Corollary 7.3.8 (Liapunov's theorem for holomorphic semigroups). Let A be an

operator on a Hilbert space H such that - A generates a holomorphic Co -semigroup o. If A + A E BIP(H) for some A > 0 then, for every E E (0, -s( -A)) there is an equivalent Hilbert norm 11·110 on H such that

T and s( -A)
o. We choose w' E (wA,7r/2) and let R:= VE2(1 + tan 2 w'). Now AR(A, B) = A(A + E)-l[(A + E)R(A + E, A)] and the factor A/(A + E) is uniformly bounded for IAI ~ R. The second factor is uniformly bounded for Re A ::; -E hence we have to check that sup{II(A+E)R(A+E,A)11 I -E::; ReA::; 0, IAI ~ R}
R together with -E ::; ReA::; 0 implies that IIm(A+E)1 = IImAI > VR2 - E2 = ctanw' , whence larg(A + E)I ~ w'. Since w' > WA, the claim is proved. So -B generates a bounded holomorphic Co-semigroup. Moreover, B + A E BIP(H) for some A > O. By b) of Proposition 3.5.5, Bis E £(H) for all s E R Applying Corollary 3.5.7 we obtain B E BIP(H). We can now apply the CallierGrabowski-Le Merdy Theorem 7.3.7. This yields an equivalent scalar product such that (ectT(t)k:::o becomes contractive. 0 Recall that one can explicitly write down the new scalar product in Proposition 7.3.8. In fact, it follows from the argument in the proof of Theorem 7.3.7

Chapter 7. The Functional Calculus on Hilbert Spaces

204 that

(x

E

H).

The function II·II~ is a Liapunov function for the dynamical system given by

U+(A-E)U=O. Now we can state the main result of this section. Theorem 7.3.9 (Similarity Theorem). Let A be a sectorial operator on the Hilbert space H with WA < n:/2. Suppose that A satisfies the equivalent properties (i)(iv) of Theorem 7.3.1. Then for each cp E (wA,n:/2) there is an equivalent scalar product ( ·1·)0 on H with the following properties.

1) N(A) -.l :R(A) with respect to (·1· )0. 2)

The operator A is m-cp-accretive with respect to

(·1· )0.

3) One has 'D(An) = 'D(Aon) for 0::::: 0:::::: n:/(4cp). Here, AO denotes the adjoint of A with respect to (·1· )0· 4)

One has

Ilf(A)llo : : : Ilfll'l'

Note that n:/(4cp)

for all f E £(S'I').

> 1/2. Hence in particular 'D(AI/2)

=

'D(Ao(1/2l).

Proof. We choose WA =: W < cp' < cp, and define (3 := n:/(2cp') and B := Ai3. Then B E Sect(w'), where w' = {3w = (w/cp')(n:/2) < n:/2. Hence -B generates a bounded holomorphic Co-semigroup. By hypothesis, A satisfies condition (i) of Theorem 7.3.1. Applying Proposition 3.1.4 we see that B has the same property. By Theorem 7.3.7 there is an equivalent scalar product (·1·)0 that makes B maccretive and such that N(A) -.l :R(A). Now, A = B 1 /i3, whence by Corollary 7.1.13, A is m-n:/2{3-accretive with respect to the new scalar product. Moreover, Kato's Theorem 7.3.6 says that 'D(A ni3) = 'D(Bn) = 'D(Bon) = 'D(Aoni3) for all o : : : 0: < 1/2. Thus, assertion 3) follows since

{3

0< 0:{3 < -

2

n:

=-

4cp'

and

n: 4cp'

n:

->-. 4cp

If we take f E £(S'I') then

Ilf(A)llo = Ilf(zl/i3)(B)llo : : : Ilf(zl/i3)II~ by Proposition 3.1.4 and Theorem 7.1.7.

=

Ilfll2"rJ

=

Ilfll'l" : : : Ilfll'l' D

As a corollary, we obtain a simultaneous solution to the similarity problems posed in Section 7.3.2 (see page 201). Corollary 7.3.10. Let A be a closed operator on a Hilbert space. Then A is Katosectorial with respect to some equivalent scalar product if and only if - A generates a holomorphic Co-semigroup and A+A E BIP(H) for large A ERIn this case the scalar product can be chosen such that A is K ato-sectorial and square root regular.

7.3. Similarity Tbeorems for Sectorial Operators

205

Corollary 7.3.11 (Franks-Le Merdy). Let A be an injective sectorial operator on the Hilbert space H such that A E BIP(H). Then for each

0) to conclude that

ei'P[A2 ei'P[A2

+ e 2 ) + A2] = (~)2e-i'P + e 2 ei 'P + ei'P A2 cos 'P

is m-accretive. Letting e "" 0 we see that (wi cos'P)2e-i'P +ei'P A2 and finally that

ei'P (( ~)2 + A2) = (~)2ei'P + ei'P A2 cos'P cos 'P is m-accretive. (Note that ei'P differs from e-i'P only by a purely imaginary number and this does not affect m-accretivity by (ii) of Proposition C.7.2.) This finishes the proof of the first part of the theorem. The second part follows from Proposition 7.1.1, cf. also [10, Chapter 3.4 and Chapter 3.9]. To prove the last assertion, note first that the operator w2 +A 2 is m-accretive, hence sectorial. Thus the square root is well defined. Since iA generates a group, _A2 generates a cosine function with phase space n(A) x H. By general cosine function theory (see the remarks at the beginning of this section), _(A2 +w 2) also generates a cosine function with the same phase space. Fattorini's Theorem 7.4.2 implies that B := i(w 2 + A 2)1/2 generates a group and B2 = _(A2 + w2). Then n(B) = n(A) follows from the uniqueness of the phase space (see Proposition 7.4.1). 0 Combining this result with the Similarity Theorem 7.2.8 for groups yields the following corollary: Corollary 7.4.4. Let - B be the generator of a cosine function on a Hilbert space H. Then B is Kato-sectorial and square root regular with respect to some equivalent scalar product. In particular, >. + B E BIP(H) for large>. E R

Proof. First, one can find (3 such that B+(3 is sectorial. Since -(B+(3) generates a cosine function as well, we can apply Fattorini's Theorem 7.4.2. Thus, the operator iA := i((3 + B)1/2 generates a strongly continuous group T on H. Choose w > 8(T). By Theorem 7.2.8 we can find a new scalar product (·1')0 that renders A m-Hw-accretive and is such that n(A) = n(AO). Apply now Theorem 7.4.3 together with Corollary 7.3.5 to conclude that A2 = B + (3 is Kato-sectorial. This implies that B is Kato-sectorial. Finally, we apply Theorem 7.4.3 to the operators A and AO and obtain n((w 2 + (3 + B)~) = 'D((w 2 + A2)~) = 'D(A) = 'D(AO)

= 'D((w 2 + A02)~) = n((w 2 + (3 + Bt~). This completes the proof.

o

7.4. Cosine Function Generators

211

Let us turn to a second corollary. Namely, Theorem 7.4.3 can be seen as a mapping theorem for the numerical range (cf. also Section 7.1.4). We define the horizontal parabola lIw = {z I (Imz)2 < 4w 2 Rez} for w > 0, and lIo := [0,00).

Corollary 7.4.5. Let A be an Hw-accretive operator on the Hilbert space H for some w E [0, n/2]. Then W(A2) C {z2 I z E Hw}, i.e., the numerical range of B := w 2 + A2 is contained in the horizontal parabola lIw

= {z Eel

Rez?: 0 and IImzl ::; 2wVRez}.

This is equivalent to saying that B is m-accretive and 1

(7.23)

IIm(Bulu)l::; 2wRe(BuluPllull for all u E 'D(B)

= 'D(A2).

Proof. By Theorem 7.4.3 we obtain that ei'P((w/ COStp)2 + A2) is m-acccretive for every 0 ::; Itpl < n /2. Specialising tp = 0 yields that B = w 2 + A2 is m-accretive. Now we write ei'P

(~ + A2) cos tp 2

= ei'P

(B

+

(~ cos tp 2

Since this operator is m-accretive and tan 2 tp obtain W(B + w 2 tan 2 tp) C S7r/2-'P' i.e.,

w 2))

=

ei'P (B

+ w 2 tan 2 tp)

.

= tan 2 ( -tp), by Proposition 7.1.1 we (u E 'D(B))

for 0 <

o. Does it follow that B already generates a cosine function? This is in fact true since CROUZEIX has proved in [54] the following result, which we quote without proof.

Chapter 7. The Functional Calculus on Hilbert Spaces

212

Theorem 7.4.7 (Crouzeix). There is an absolute constant c

> 0 such that

Ilf(A)11 ::; c IlfllIIw for all f E ROO(IIw) and every m-IIw-accretive operator on a Hilbert space H. The proof follows in principle the lines of the proof of Theorem 7.1.16. However, it is much harder in detail. Corollary 7.4.8. Let B be an operator on a Hilbert space H. Then -B generates a cosine function if and only if there is A E lR and an equivalent scalar product on H with respect to which A + B is m-IIw-accretive. Proof. One direction is Corollary 7.4.6. For the converse, suppose that A+ B is mIIw-accretive. Without loss of generality we may suppose that B is m-IIw-accretive and invertible. Then we can set up a holomorphic functional calculus for B on horizontal parabolas IIa for 0: > w. Using Crouzeix' Theorem 7.4.7 together with similar arguments as in Section 5.3 we conclude that the natural Hoo(IIa)-calculus is bounded for every 0: > w. Inserting into this calculus the functions etiz (1/2) for t E lR we obtain the Co-group generated by _Bl/2. (The density of 'D(B) in H 0 follows from sectoriality.) Hence - B = (iB 1 / 2 )2 generates a cosine function. At the end of this section we would like to illustrate CROUZEIX's result with an example. In Section 7.3.2 we considered operators A defined by means of an elliptic form a: V X V -----+ H. Suppose that condition 1) of page 198 is satisfied, i.e., we find AD E lR such that Re a + AD ( ·1· )H is a scalar product on V which turns V into a Hilbert space, continuously embedded into H. Instead of condition 2) (which, by some trivial equivalences, merely postulates continuity of 1m a with respect to this scalar product) we require the stronger condition IIma(u, u)1

::; M Iluli H Iluliv

(u E V).

(7.24)

Then the numerical range of A + AD is contained in IIw for some w ~ O. Using approximation by rational functions, one can conlcude that -(A + AD) and hence - A generates a cosine function. In the concrete case of an elliptic form with Loo-coefficients a(u, v)

=

In {~ ai,jDiuDjv + l::)iDi uv + CiuDi 1,,)

V

+ dUV}

dx

(u E V)

2

on H = L2(O) and with V being some closed subspace of Wl,2(O) containing 'D(O), condition (7.24) is satisfied if ai,j = aj,i for all i,j.

7.5

Comments

7.1 Numerical Range Conditions. The material on normal or self-adjoint operators is standard. The proof is usually given via a spectral measure respresentation, cf. [158, Section 4.3.1].

7.5. Comments

213

7.1.3 Functional Calculus for m-accretive Operators. The fundamental Theorem 7.1. 7 has been folklore for a long time, due to its connection with von Neumann's inequality (cf. Remark 7.1.9 and below). The classical reference for the dilation theorem is the book [209] by Sz.-NAGY and FOIAS. Therein one can find also the construction of a functional calculus for so-called completely non-unitary contractions. The proof of Theorem 7.1.7 via the dilation theorem can also be found in [145, Theorem 4.5]. Our second proof is taken from [3], where it is attributed to FRANKS. This proof is in the spirit of Bernard and Fran«ois DELYON'S proof of the von Neumann inequality in [68]. Other proofs of von Neumann's inequality can be found in [185, Corollary 2.7] and [190, Chapter 1]. 7.1.4 Numerical Range Mapping Theorems. To establish 'mapping theorems for the numerical range' Theorem 7.1.7 seems to be at the core of the theory since it ties closely contractivity and m-accretivity to the functional calculus. KATO [129] uses this connection to prove more general results. Based on that, CROUZEIX and DELYON [55] essentially prove Corollary 7.1.14. KATO [127] gives a proof of Corollary 7.1.13 different from ours, see also [210, Lemma 2.3.6]. In general, 'mapping theorems for the numerical range' are a delicate matter. For example, SIMARD in [203] constructed an example of an m-H7r / 2 -accretive operator on the space H = ([2 such that e A is not m-accretive. 7.1.5 The Crouzeix-Delyon Theorem. The Crouzeix-Delyon Theorem 7.1.16 from [55] can be considered a major breakthrough in deriving boundedness of functional calculi from numerical range conditions. For a long time, the case of m-accretive operators was more or less the only example of this connection. The new insights came when a new proof of von Neumann's inequality was found by directly manipulating the corresponding Cauchy integrals (see above). This has been developed further by CROUZEIX [53, 54] to obtain similar results for operators with numerical range in a parabola (see Theorem 7.4.7) and more general convex domains. 7.2 Group Generators on Hilbert Spaces. Theorem 7.2.1 is due to BOYADZHIEV and DE LAUBENFELS [34, Theorem 3.2], although in a little different form. The original proof proceeds in two steps. First, assuming w < 7r /2 without loss of generality, the authors construct the operator e B (see Chapter 4; their construction however relies on the theory of regularised semigroups). Then they show that this operator is sectorial and has a bounded HCXl-functional calculus on a sector. The statement then follows by means of some special case of the composition rule. Our first proof for Theorem 7.2.1 is just a combination of the Decomposition Theorem 7.2.8 and the Crouzeix-Delyon Theorem 7.1.16. Since the proof of the latter is quite involved, this proof cannot be regarded as simple. However, there is no need of the full power of the Crouzeix-Delyon theorem. In fact, the weaker version mentioned in Remark 7.1.18 suffices. Regarding the Spectral Theorem as

214

Chapter 7. The Functional Calculus on Hilbert Spaces

known, this can be proved by a simple perturbation argument, as was done by the author [105]. 7.2.1 Liapunov's Direct Method. In the case where A is bounded the Liapunov method is used in Chapter I of the book [59]. There the operator equation QA + A *Q = -lis directly linked to the problem of finding a Liapunov function for the semigroup. (This is sometimes called 'Liapunov's direct method'.) For the unbounded case the relevant facts are included in [57, Theorem 5.1.3], where a characterisation of exponential stability of the semigroup is given in terms of the existence of an operator Q satisfying the Liapunov equation(s). (Extensions of this result can be found in [98], [15] and [2].) It is shown in [232] that this method in fact gives an equivalent scalar product if the semigroup is a group. In the paper [156] on exact controllability LID and RUSSELL establish almost the same results (and others). The Liapunov theorem for holomorphic semigroups (Proposition 7.3.8) is due to ARENDT, Bu, and the author [12, Theorem 4.1]. In that paper, Liapunov type theorems are established also for hyperbolic holomorphic semigroups and quasi-compact holomorphic semigroups. Moreover, Proposition 7.3.8 is applied to semilinear equations. 7.2.2 The Decomposition Theorem. Theorem 7.2.8 is due to the author [105]. The following well-known theorem by SZ.-NAGY [207] can be regarded as the 'limit case': Every generator of a bounded group is similar to a skew-adjoint operator. This result cannot be deduced directly from Theorem 7.2.8. However, ZWART [232] gives a proof using the Liapunov renorming and some approximation argument. DE LAUBENFELS [67, Theorem 2.4] proves that, given a Co-group T on a Hilbert space, one has IIT(t)llo :::; ewltl for some equivalent scalar product (·1·)0 and some w strictly larger than the group type ()(T). (This is covered by Theorem 7.2.8 a).) However, this proof is based on the boundedness of the HOO-calculus and on Paulsen's theorem (see below), so the new scalar product cannot easily be made explicit. One can wonder whether, given a group T such that IIT(8)11 :::; Ke llisl (8 E lR), one may always take w = () in Theorem 7.2.8. However, SIMARD [203] has shown that this is not possible in general. 7.2.3 A Characterisation of Group Generators. Theorem 7.2.12 is due to the author [101]. The rough idea behind the proof of the crucial implication (iv)=}(i) consists in adapting McINTOSH's methods from [167] (in particular his use of quadratic estimates) to strip-type operators. Corollary 7.2.14 is a result by LID [155, Theorem 1]. One may consult [101] for other characterisations of groups on Hilbert spaces, including results of LID [155, Theorem 2] and ZWART [232, Theorem 2.2]. Further results on quadratic estimates in connection with semigroups

7.5. Comments

215

can be found in [96, 202, 122, 27]. 7.3.1 McIntosh's Theorem. Theorem 7.3.1 is due to McINTOSH [167], based on ideas of YAGI [226], see also [169]. Norm inequalities such as (i) of McIntosh's theorem are called quadratic estimates or square function estimates in the terminology of these authors. The importance of quadratic estimates pertains when studying bounded ness of HOO-functional calculus on LP-spaces, see [51], [151] and [141, Section 11]. AUSCHER, NAHMOD and McINTOSH [19] have made explicit the connection with interpolation theory (d. also Chapter 6 and the comments there). The proof given in Remark 7.3.2 is taken from the author's paper [110]. The proof of the implication (ii)::::}(i) in Theorem 7.3.1 via the Unconditionality Lemma 5.6.1 is taken from [150, Theorem 4.2]. Let us point out that in Theorem 7.3.1 the equivalence of (iii) and the other statements is proved without making use of interpolation theory (cf. Remark 7.3.2). One can add further equivalences to McIntosh's Theorem 7.3.1, e.g.

(iv) There is equivalence of norms

Ilxll~ where 0

=1=

1 Sa

'¢ E Ho(Sp) and 'P E (WA

2

dz

11,¢(zA)xII H 2

Z

+ a, 7r).

This is done by KUNSTMANN and WEIS [141, Theorem 11.13]. One can easily deduce this from the fact that in any Banach space the homogeneous interpolation space (D(a), R(a)h/2,p can be described as

which in turn follows from the results of Section 6.4. (One has to note that the embedding constants associated with the family of sectorial operators (e ili A)IIiI::;a are uniformly bounded. We do not go into details.) 7.3.2 Sesquilinear Forms. The classical reference for operators constructed via sesquilinear forms is KATO'S early paper [127] as well as his book [130]. This is the reason why we called this operator 'Kato-sectorial'. Proposition 7.1.1 essentially is KATO'S 'First Representation Theorem' [130, Chapter VI, Theorem 2.1]. It is also included as Theorem 1.2 in [12]. The applications to PDE in the literature are numerous, see the recent book [180] by OUHABAZ and the references therein. In [12, Example 3.2] an example of two similar operators on a Hilbert space is given, one variational but not the other. As mentioned already, by MATOLCSI'S result [162] this is not a pathological case. It would be interesting if there is a proof for Kato's Theorem 7.3.6 that differs essentially from KATO'S original one. Apart from [210, Lemma 2.3.8] and [158, Theorem 4.3.4]' which more or less copy KATO'S arguments, we do not know of

216

Chapter 7. The Functional Calculus on Hilbert Spaces

any other account. The Square Root Problem (cf. Remark 7.3.2) has been solved by AUSCHER, HOFFMAN, LACEY, LEWIS, McINTOSH, and TCHAMITCHIAN [18]. Surveys and a deeper introduction to these matters as well as the connection with Calderon's conjecture for Cauchy integrals on Lipschitz curves are in [165, 166, 168] and [20]. The first counterexample to KATO's original question was given by McINTOSH [163]. A different one can be obtained from [20, Section 0, Theorem 6] employing an additional direct sum argument.

7.3.3 Similarity Theorems and 7.3.4 A Counterexample. Similarity problems have a long tradition in operator theory. In 1947 SZ.-NAGY [207] had observed that a bounded and invertible operator T on a Hilbert space H is similar to a unitary operator if and only if the discrete group (Tn)nEZ is uniformly bounded. His question was whether the same is true if one discards the invertibility of T. In [208] he showed that the answer is yes if T is compact, but FOGUEL [90] disproved the general conjecture by giving a counterexample. By von Neumann's inequality, if T is a contraction, then T is not only power-bounded but even polynomially bounded, i.e., sup{[[p(T)[[ [p E qz], [[p[[oo :::; I} < 00, where [[p[[oo denotes the uniform norm on the unit disc. So HALMOS [113] asked whether polynomial boundedness in fact characterises the bounded operators on H that are similar to a contraction. This question remained open only until recently when PISIER [188] found a counterexample, see also [60]. Meanwhile, PAULSEN [184] had shown that T is similar to a contraction if and only if T is completely polynomially bounded. His characterisation is a special case of a general similarity result [185, Theorem 8.1] for completely bounded homomorphisms of operator algebras (see [185] for definitions and further results). We address this result as Paulsen's theorem in the following. One can set up a semigroup analogue of SZ.-NAGY'S question (see above) namely whether every bounded Co-semigroup on a Hilbert space is similar to a contraction semigroup. The corresponding result for groups is true as was also proved by SZ.-NAGY in the very same article [207]. (The question is a little more special than the original one since not every power-bounded operator is the Cayley transform of a Co-semigroup generator.) It was answered in the negative by PACKEL [181]. By using this result, CHERNOFF [43] provided an example of a bounded operator with the generated Co-semigroup being bounded but not similar to a contraction one. Furthermore, via Lemma 7.3.15 he constructed a Co-semigroup which is not even similar to a quasi-contractive semigroup. (Our setting up this historical panorama is mainly based on [146, Introduction]. More on similarity problems and their connection with the theory of operator algebras can be found in [189].) Being probably unaware of the history of this problem, CALLIER and GRABOWSKI in an unpublished research report [97] proved Theorem 7.3.7 in the case where the semigroup is exponentially stable. Their arguments are based on two

7.5. Comments

217

facts from interpolation theory: first, both the (complex) interpolation space [H, 1)(A)h/2 and the (real) interpolation space (H, 1)(A)h/2,2 are equal to 1)(Al/2) if A E BIP(H) (see Theorem 6.6.9) and, second, this real interpolation space is given by

(cf. (6.9) in Section 6.2.3). It was LE MERDY [150, Theorem 4.2] who observed that the result follows from McIntosh's theorem. Theorem 7.3.7 as it stands is also a consequence of Corollary 7.3.11 and is stated as such in [146, Theorem 4.3]. LE MERDY proves Corollary 7.3.11 with the help of Paulsen's theorem (see above). The same is done by FRANKS in [91, Section 4]. Note that - by using the scaling technique and the results of KATO on accretive operators - the Franks-Le Merdy theorem and the CallierGrabowski-Le Merdy theorem are actually equivalent, and that Paulsen's theorem is not necessary to prove the Similarity Theorem 7.3.9. However, LE MERDY [148] proved with the help of Paulsen's theorem that the exponential e A of an m-H,,/2accretive operator A is always similar to an m-accretive operator. (As mentioned above, SIMARD has shown that eA need not be m-accretive itself.) Up to now it is unknown whether one can avoid Paulsen's theorem here. It has been noted in [12, Theorem 3.3] that the Franks-Le Merdy result solved the first similarity problem posed (see page 201). However, from their proof it was not clear whether also the second problem concerning the square roots can be solved. That it actually can is an observation due to the author [102, Theorem 4.26] and sheds an interesting light on the orginal square root problem. It also complements a result of YAGI [226, Theorem B] which says that a sectorial invertible operator A on a Hilbert space has bounded imaginary powers if 1)(A"') = 1)(A*"') for all a contained in a small interval [0, E). It is a consequence of Theorem 7.3.9 combined with the scaling technique that the converse holds modulo similarity. The following problem seems to be open until now. Problem: Is there an m-accretive operator A on a Hilbert space H such that

for every equivalent scalar product ( ·j·)o? (Note that, by Theorem 7.3.9, such an operator necessarily has to satisfy the relation WA = 7r/2.) Returning to the semigroup version of SZ.-NAGY'S question (see above), by Theorem 7.1.7 (which is essentially von Neumann's inequality) one obtains that a bounded Hoo-calculus (or equivalently BIP) is as necessary a condition for similarity to a contraction semigroup as it was polynomial bounded ness in the discrete case. Moreover, the Callier-Grabowski-Le Merdy Theorem 7.3.7 implies that for bounded holomorphic semigroups in fact the condition BIP already suffices. Using

218

Chapter 7. The Functional Calculus on Hilbert Spaces

PISIER'S counterexample to HALMOS'S problem LE MERDY [145, Proposition 4.8] succeeded in showing that in general BIP is not sufficient. Theorem 7.3.13, based on CHERNOFF'S ideas from [43] and due to the author [102, Theorem 4.31]' states that one can even exclude similarity to a quasi-contraction semigroup. 7.4 Cosine Function Generators. The mapping theorem for the numerical range established in Theorem 7.4.3 and Corollary 7.4.5 is from [102, Corollary 5.17]. Its proof is based on Theorem 7.4.3, also due to the author [105]. McINTOSH [164] shows that an operator B satisfying the conclusion of Corollary 7.4.5 also has the square root property 'D(BI/2) = 'D(B*1/2). Employing this one can eliminate the cosine function theory from the proof of Corollary 7.4.4. CROUZEIX' result Theorem 7.4.7 from [54] is a milestone. This author is about to prove a general theorem on bounded Hoo-calculus for operators with numerical range in a convex subset of the plane, cf. [53]. For more information on cosine functions in general consult [10] and the references therein.

Chapter 8

Differential Operators We treat constant coefficient elliptic operators on the euclidean space ~d. The main focus lies on the connection of functional calculus with Fourier multiplier theory. The Ll-theory is the subject of Section 8.1 while the LP-case is presented in Section 8.2. Then we apply the obtained results to the negative Laplace operator (Section 8.3). The universal extrapolation space for the Laplace operator is identified with a space of certain (equivalence classes) of tempered distributions. In Sections 8.4 and 8.5 we treat the derivative operator on the line, the half-line and finite intervals. The UMD property of a Banach space X is characterised by functional calculus properties of the derivative operator on X-valued functions.

Preliminaries

In this chapter we discuss a fairly concrete class of operators, namely (elliptic) differential operators with constant coefficients. We shall work on the whole space JRd and only sketch generalisations to differential operators on domains, cf. Section 8.6. In our discussion we will make extensive use of Fourier theory, and we refer the reader to Appendix E for notation and background information. For a scalar function oftempered growth a E 1'(JRd) we define the associated operator A := (u f-----7 F-1a(Fu)) : TD(JRd; X) -----+ TD(JRd; X). Given any Banach space X C TD(JRd; X) the operator Ax is defined as the part of A in X, i.e.,

1)(Ax) := {u E TD(JR d; X)

I Au EX}.

As the embedding X ~ TD(JRd; X) is continuous, Ax is a closed operator on X. For the special choices Xp := LP(JRd; X), p E [1,00] we use the abbreviation Ap. This setting covers in particular differential operators with constant coefficients since they are induced by proper polynomials. E.g., the polynomial

a(s) =

L

lal:Sm

aa(is)a

(8.1)

220

Chapter 8. Differential Operators

yields the operator

Au:=

L

aa Dau

(8.2)

lal:. - A) is injective, and

(>. - A) ~ 1 = F~ 1 (>. - a) ~ 1F. Let X be a Banach space continuously embedded into TD(JRd; X). Then>. E Q(Ax) if and only if X is invariant under>' - A. In this case R()., Ax) = [(>' - A)~1l:x:. Proof. Suppose that >. ~ a(JRd). Then (>. - A)~1 E 1'(JRd) follows from a general result about derivatives of quotients. Indeed, one proves by induction that, given f, 9 functions of d variables, for each multiindex a E Nd one has

Da(L) =~ 9 gla l+1' where fa is a polynomial in the derivatives Di3g, Di3 j, (0 ~ f3 ~ a) and with integer coefficients. Applying this fact to the case f = 1 and 9 = >. - a, one sees easily that each derivative Da(>. - a)~1 must be polynomially bounded. The remaining statements are then straightforward to prove. D In general, very little may be said about such differential operators. This changes drastically when we restrict to the class of homogeneous elliptic operators. These arise from homogeneous polynomials

a(s) =

L

aa(is)a

(8.3)

lal=m satisfying the condition s~O

a(s)~O

===}

(8.4)

and a non-triviality postulate

(8.5) If one requires the even stronger condition O~s

===}

Rea(s) >0,

(8.6)

the polynomial a and the corresponding operator A are called strongly elliptic. For our further investigations we are in need of some auxiliary information. Lemma 8.0.2. Let a: JRd

hold.

----*

C satisfy (8.3)~(8.5). Then the following assertions

221

8.1. Elliptic Operators: L1-Theory

> 0 such that

a)

There are constants

b)

There is W E [0,7r) such that a(JR d) = a(JRd) c Sw. Let Wa be the least of these w. If a is strongly elliptic, i.e., if (8.6) holds, then Wa < 7r/2.

c) With

C3

:=

C1, C2

c2/2 and L :=

(C3)-m

one has

(lsi

Da(A

-a

)-1

~ L, IAI

= 1).

Pa

= (A-a)lal+l

where Pa is a polynomial of degree degpa :::; (m -1) lexl. Proof. a) The existence of C2 follows from the inequality Isal :::; Isl 1al . By (8.4) and the compactness of the unit sphere C2 := inflsl=l la(s)1 > O. The rest follows from the homogeneity.

b) Since by a) one has limlsl--->oo la(s)1 = 00, a(JRd) = a(JRd) is obvious. Condition (8.5) then insures that a small neighbourhood U of -1 does not intersect the image of a, whence by the homogeneity the whole cone generated by U does not intersect the image of a. If a is strongly elliptic, then {z I Izl = 1, largzl E [7r/2,7r]} does not intersect a(JR d). Hence this is true also for {z I Izl = 1, larg zl E [w, 7r]}, with some W E (0,7r/2). c) is a simple computation and d) is proved by induction on lexl.

D

In the following we look at the part Ap in LP(JRd; X) of a constant-coefficient elliptic operator A. By Lemma 8.0.1, the attempt to determine the resolvent leads to the question whether a certain function - here: (A - a)-l - is an LP(JRd; X)Fourier multiplier. Moreover, general functional calculus philosophy lets us expect the formula f(A)u = F- 1 (1 0 a)u, at least for certain u. Boundedness of f(A) is therefore somehow linked to f 0 a being an LP(JRd; X)-Fourier multiplier. The next two sections will render more precise this intuitive reasoning.

8.1

Elliptic Operators: L1-Theory

rc

Let a : JRd ------t satisfy (8.3)-(8.5), let A be the corresponding operator with symbol a, and let C1,C2,C3,L,w a as in Lemma 8.0.2.

Chapter 8. Differential Operators

222 Proposition 8.1.1. For each A

is.. = (A - a)-I. The mapping (A

is holomorphic, and

f------+

tt.

a(JRd) there is a unique r>.. E V(JRd) such that

r>..) : C \ a(JR d)

-----+

Ll(JR d)

IIAr>..lll = Ih/l>"llll.

Proof. First of all, the homogeneity of a implies that A tt. a(JRd) if and only if IL := We claim that there is an r", E L 1 (JRd) such that ~ = (IL - a)-I. To establish this, let v", := (IL - a)-I. Then DO:v", = Po:/(IL - a)lo:l+1 for some polynomial Po: of degree degpo: ::; (m - 1) 1001 (Lemma 8.0.2). Hence

AI IAI tt. a(JRd).

for lsi ~ L. This gives v", E Mm (defined in Theorem E.4.2), hence v", = is.. for some r>.. E L 1 (JRd). Now let E := IAI- l/m and use the formulae from Appendix E.2 and the homogeneity of a to compute

(A - a)-l = Fr>.. with r>.. = IAI- l U(>"1-1/",r W Since U~ is isometric on L1, we have liAr>.. 111 = Ilr", Ill. It is now clear that R(A, Ad is convolution with r>.., and

Hence

so the holomorphy follows from Lemma E.3.1.

0

With this result at hand we are in a position to prove the main theorem on homogeneous elliptic constant-coefficient operators. Theorem 8.1.2. Let a : JRd -----+ C satisfy (8.3)-(8.5), and let A = L:lo:l=m ao:DO: be the corresponding operator on TD(JR; X), with Al being its part in Ll(JRd; X). Then the following assertions hold.

a) R(A, Adu = r>.. * u for all u E Ll(JR d; X) and A tt. a(JRd). b)

IIR(\ Al)11 = Ilr>..lll

for

Att. a(JRd).

c) Al is an injective, densely defined, sectorial operator of angle d) a(Ad

WA 1

= Wa.

= a(JRd).

Proof. a) is clear from Lemma 8.0.1 and Proposition 8.1.1. b) follows from a) and Lemma E.3.1. c) Proposition 8.1.1 shows that IlAr>..lll is constant on rays originating at O. Since IIAR(A, Adll ::; IIAr>..lll and (A f------+ r>..) is continuous, the sectoriality of Al is immediate. Since II(A - a)-liloo ::; Ihlll = IIR(\Al)ll, the sectoriality angle of Al must in fact be equal to Wa. Clearly S(JR d ; X) c 1)(Ad, so Al is also densely

8.1. Elliptic Operators: L1-Theory

223

defined. Let us prove that Al is injective. If u E Ll (JR d; X) such that Al u = 0, then 0 = :FAl U = au. But 1£ E Co and a(s) i- 0 whenever s i- O. So u(s) = 0 for all s i- 0, whence also 1£(0) = 0 by continuity. This implies that u = 0 since the Fourier transform is injective. d) Fix A E Q(A 1 ). For each u E Ll(JR d; X) we have u = (A - A)R(A, Adu. Taking Fourier transforms yields

1£ = (A - a):FR(A, Adu, Since for given s E JRd we can find a function u such that 1£( s) a(s) i- A for all s E JRd, i.e., A ~ a(JRd).

i- 0, it follows that 0

We also can say something about the domain 2)(Ad. Proposition 8.1.3. Let a and A be as above, and let 1 ::; k ::; m - 1. Then W'm,l(JRd; X)

c

2)(Ad C (L1(JRd; X), 2)(Adh

='

1

C Wk,l(JRd; X).

Proof. It is clear from the definition that W1,'m(JRd; X) C 2)(Ad. We claim that for 13 ENd, 1131 ::; m -1 the function sP(1 + a(s))-1 is an L1(JRd;X)-multiplier. This implies in particular that the operator DP(1 + A)-1 restricts to a bounded operator on the space Ll(JRd; X). In order to prove the claim, we use Theorem E.4.2. By induction on 10:1 one proves that for any 0: ENd there exists a polynomial Pa,p such that Da (

and degpa,p ::; (m - 1) 10:1

sP

1 + a(s)

) _ -

(1

Pa,p

+ a(s))la l+l

+ 1131. Hence one can estimate

C IsIIPI-m-lal IDa ( 1 +sPa(s) ) I < - a,p

(lsi

2: L)

(see Lemma 8.0.2), with constants ca,p > 0 not depending on s. This implies that C ,FLl by Theorem E.4.2, as long as 1131 ::; m - 1.

sP(1 + a)-1 E Mm-IPI

Now, fix 13 E N d with 1131 ::; m - 1, and choose v E Ll such that (is)P(1 + a)-I. Let t > 0, and define E:= C 1/ m. Then

(is)P(t + a)-1 = Em(is) PUc:(1 + a)-1 = Em-IPIUc::Fv

= Em-IPI:FU~v = t 1;;'1_1 :FU~v. Hence tl-IPI/m DP(t + A 1 )-1 is induced by convolution with U~v, whence

is constant. By Corollary 6.7.5 this means

DP: (Ll(JRd; X), 2)(A 1 t) 1£1

1 ---->

=,

Ll(JRd; X),

v=

Chapter 8. DiHerential Operators

224 where 'D(Al)·

= iJ is the homogeneous domain as defined in Chapter 6. Hence

D

for all 1 :::; k :::; m - 1. Let us now turn to the functional calculus. For may define

f

E

H[)(S(f(Ad) if and only if there is v E Ll(JR.d; X) such that (ef)(Adu = e(Adv. By taking Fourier transforms, this identity is equivalent to [(ef) 0 a]u = 11. This already proves the implication '-¢='. To prove the converse, suppose that there is such a v. Then (f 0 a)u = 11 except possibly on the set {O} U {e 0 a = O}. However, e has at most countably many zeroes, so by Lemma 8.1.5 a) and the fact that a is a polynomial, {e 0 a} is an JR.d-null set. Since both U,11 are continuous functions and from 0, one has 11 = (f 0 a)u on JR.d \ {O}.

f

0

a is continuous apart

Within the stated equivalence the implication (iv)=?(i) is obvious, and so is the equivalence (ii){::}(iii) since f 0 a is a scalar function. The implication (i)=?(iii) follows easily from Theorem E.5.4. Suppose that (iii) holds, i.e., that f 0 a is an

Chapter 8. DiHerential Operators

226

L1-Fourier multiplier. This means that there is a bounded measure f.L E M(JRd) with J1 = f oa. Now, with e E Ho(S.)>. is as in Proposition 8.1.1. For 1 ::::; k ::::; m - 1 one has wrn,p(~d;

X)

C

2)(Ap)

C (LP(~d;

X), 2)(Ap)h

m'

1

C Wk,p(~d;

X).

If X is a UMD space, then 2)(Ap) = wrn,p(~d; X). Proof. As in the case p = 1 the assertions about the resolvent set, the representation of the resolvent as convolution with r>. and hence the sectoriality of Ap follow from Proposition 8.1.1 and Young's inequality. It is also clear that Ap is densely defined since D(~d; X) is contained in 2)(Ap). Let us prove injectivity. If Apu = 0, then U E LP and au = O. Since a#-O away from the origin, supp C {O}, whence by Lemma E.2.1 the distribution is a linear combination of derivatives of 50' Hence its inverse Fourier transform v is a polynomial which, by Lemma 8.1.5, must then be equal to O. Thus u is weakly zero, hence O.

u

u

The proof of the embedding 2)(Ap) C (LP(~d; X), 2)(Ap))k/m,l C X) works exactly as in the L1-case. In fact, in the proof of Proposition 8.1.3 we have shown that the operator t 1 - ,6I/ mD,6(t + A)-l is induced by convolution with a function U~v, for c = C 1 / m and a fixed function v E L1. So the rest is just an application of Young's inequality. Wk,p(~d;

1

Now, let X be a UMD Banach space. Then the vector-valued Mikhlin theorem (Theorem E.6.2) is at hand to show that the functions s,6(1 + a)-l are in fact LP(~d; X)-Fourier multipliers for all 1,81 ::::; m. Cf. the computation in the proof of Proposition 8.1.3. D

228

Chapter 8. Differential Operators

Remark 8.2.2. One can show that actually a(Ap) = a(~d), see [10, Lemma 8.1.1] and its use in the proof of [10, Lemma 8.3.2].

As to the functional calculus, if

f

E

£(S",),

O. Hence -.6. is strongly elliptic. So for each Banach space X and each P E [1,00] the operator -.6. p is a sectorial operator on LP(JRd; X) of angle O. It is injective and densely defined in the case where p I=- 00. If p = 00, its kernel consists of the constant functions. Its domain satisfies

W 2 ,p(JRd; X) C 1)(.6. p) C (LP(JRd; X), 1) (.6. p)) 1

1

C

W1,p(JR d; X).

2 '

If p E (1, (0) and X is a UMD space, -.6. p has a bounded HOO(S 0), where

(Re>. > 0). Moreover, (A

f------+

G A ): {ReA> O}

---+

COORd) nLl(JR d) is holomorphic.

Proof. Essentially one has to show that

for all Re>. > O. This is done in two steps. If >. > 0, one uses the formula for the inverse Fourier transform and the well-known identity

employing some simple change of variables. Next, one shows that the function

is holomorphic. For this it suffices to show that the derivative _1,1 2 e- A1 ' 12 is majorised by an integrable function, locally uniformly in A. (This is obviously the case.) The inverse Fourier transform is a bounded linear operator from Ll to Co, whence it follows that in fact GA(t) is holomorphic in A for each t E JRd. So we obtain the final result by uniqueness of analytic continuation. D

8.3. The Laplace Operator

233

Corollary 8.3.2. One has (Re>.>O). From the concrete representation of the semigroup we can derive a formula for the operators (E - D.)-a, Re a > 0.

Proposition 8.3.3. Let

E

> 0, Rea> 0, and p E [1,00]. Then

(E - D.p)-a u = Be,a * u where Be,a E L1 (JR d ) is given by

(x

=f 0).

Proof. We apply Proposition 3.3.5 to the operator -D.l. Here, kernel representations and operators are two sides of the same coin (isometrically). Hence the integral 1

Be,a := r(a)

['XJ

10

ta-le-ctC t dt

converges in L1 (JR d) yielding the convolution kernel for (E - D.l) -a. It is clear that this carries over to the LP-setting. The only remaining thing to check is the given pointwise respresentation of Be,a on JRd \ {O}. But if K c JRd \ {O} is compact, the integral above is absolutely convergent within L1 (K) n C (K), whence the pointwise representation follows.

0

The convolution kernel Be,a is called a Bessel potential of degree a.

Proposition 8.3.4. Let X be a UMD space, and let p E (1,00) and'P E (O,n). Then the operator -D.p has a bounded HOO(S' (JRd) with

cI>(JR d) := {f

E

S(JR d) I (P,1)

= 0 V pEP}.

The Fourier transform restricts to an isomorphism

By adjoint action, one defines the Fourier transform as an isomorphism

and it is clear that this corresponds to the induced operator

The space cI>' (JR d) allows for a much greater variety of Fourier multiplier operators as S' (JR d), namely with singularities at O. In particular, for each 0: E C not only the operator

is an isomorphism (it is already on S'), but also

(Use Taylor's formula.) The operator

is an isomorphism on cI>'(JRd) and restricts to A(l universal space U may then be identified with

U=

U T-nX = U(2 nEW

+ A)-2

on X

= LP(JRd). The

~ - ~-ltLP(JRd).

nEW

Within this space one finds the Bessel potential spaces

(which may be identified with subspaces of S' (JRd)). The homogeneous fractional domain spaces

are the well-known Riesz potential spaces.

235

8.3. The Laplace Operator

Remark 8.3.5. Let a be any homogeneous elliptic polynomial, and let A be the associated operator on LP(JR.d). Then as for the Laplace operator one can find the universal space of A within 1]1' (JR.d). The homogeneous fractional domain spaces are x(a) = (-A)-aLP(JR.d) = {u E 0, let u E S(lR) such that u( s) = e- as2 , and set v := eirt:..pu. Then 0(s) = e-(ir+a)s2. Note that u and v can be determined explicitly, namely u = G a and v = Ga +ir (see Proposition 8.3.1). Computing LP-norms yields as a "'" O.

o

From Proposition 8.3.8 one derives easily that in the case where p I- 2 the operator i~p does not generate an exponentially bounded semigroup. Theorem 8.3.9. If p E (1,00) and X is a UMD space, then Ap := et:..p is a bounded, injective, sectorial operator of angle 0 on LP(lR; X). One has Ap E BIP(LP(lR)) if and only if p = 2.

Proof. We have Ap = T(l) where T('>') = eAt:..p is the holomorphic semigroup generated by ~p. So Ap is bounded and injective. To prove sectoriality, one must bound the operator family [t(t + et:..p)-lL>o' The operator t(t + et:..p)-l is a Fourier-multiplier, with symbol

mt(s)

:=

t t

+ e-

S

2'

which is a function of bounded variation Var'::'oo(mdx = 2(t + 1)-1 ::; 2. We may apply Bourgain's version of the Marcinciewicz theorem (Theorem E.6.2 a)) - or rather a simpler version sometimes called Steckin's theorem - and conclude that mt is an LP-multiplier with IlmtllM p ::; 2cp for some constant cp > 0 independent of t > O. The second assertion follows from Proposition 8.3.8. 0

237

8.4. The Derivative on the Line

Theorem 8.3.9 shows in particular that the familiar one-dimensional heat semigroup on LP (lR), p E (1, 00 ), is the semigroup of fractional powers of a sectorial operator.

8.4

The Derivative on the Line

We fix any Banach space X and consider the operator A = d/ dt on TD(lR; X) and its restrictions Ap to the spaces LP(lR; X) for p E [1,00]. The symbol of this operator is (s E lR), a(s) = is which is a homogeneous elliptic polynomial of degree 1. Hence we may in principle employ the results of the previous sections. However, since we can write down the resolvent explicitly, we prefer to give direct arguments wherever possible.

Proposition 8.4.1. Let p E [1,00]. The derivative operator Ap = d/dt on the space LP(lR; X) is sectorial of angle 7r /2 with domain 1)(Ap) = W1,P(lR; X) and spectrum a-(Ap) = ilR. It is injective and densely defined if p E [1,00). It has dense range if p E (1, 00 ). Its resolvent is given by

R()" A)u =

r)..

* u,

where

_ { -e )"tl (0,00) )..t e l( -00,0)

r)..

-

(ReA < 0), (ReA> 0).

One has 9«Ad = {u E Ll(lR;X) I u(O) = O}, and Aoo restricts to a densely defined sectorial operator with dense range Ao on Co(lR; X), with domain CMlR; X). Proof. The operator Ap is defined by Apu

= u'

with

by definition of the latter space. Given Re A < 0 it is easily proved that 1

A - is

(sElR).

Hence A E g(Ap) and R(A, Ap) is just convolution with the kernel -e)..tl(o,oo)' Young's inequality shows that IIR()"Ap)11 ::::; II-e)"tl(o,oo) 111 = IReAI- 1. Analogous arguments apply in the case that Re A > O. The estimate for the resolvent implies readily the sectoriality of the operator Ap- Injectivity (in the case p < 00 and of the operator Ao) is clear since a distribution u satisfying u' = 0 must be a constant function. We determine the spectrum of Ap. Let cp be any test function with cp(O) and consider the sequence

fn(t)

:=

cp

(~)

.

(ni IlcplILP)

-1

= 1,

Chapter 8. Differential Operators

238

Then it is easy to see that IlfnilLP = 1 but f~ --+ O. This shows that 0 is an approximate eigenvalue of Ap for any p E [1,00] and of Ao. (If p = 00, the constants are in the kernel of AX) , whence here 0 is in fact an eigenvalue.) To deal with the other purely imaginary numbers, just observe that the operators A and A + ir are similar via the isometric isomorphism (Uf)(s) := e- irs f(s). As to the dense range, observe that if'P E 'D(IR; X) with J'P = 0 the function 'lj;(t) := J~oo 'P(r) dr is a primitive of 'P and also contained in 'D(IR; X). Hence the inclusion

{ 'P E 'D(IR; X) I

J

'P =

O}

c

:R( Ap)

holds. The space on the left is dense in LP(IR; X) if p E (1,00), and in Co(lR; X). Indeed, let 'lj; be any test function. Then 'lj;n(t) := 'lj;(t) - (l/n)'lj;(t/n) defines a test function 'lj;n with J'lj;n = 0 and 'lj;n --+ 'lj; in LP if 1 < p :::; 00. In the Ll_ case we note that J 'P' = 0 for each 'P E 'D(IR; X). Since test functions are dense within W1,1(1R; X), we have l' = 0 for all f E 2)(Ad = W1,1(1R; X), whence :R(Ad = {f E Ll(lR; X)) I f = O}. 0

J

J

From the explicit formula for the resolvent we derive the norm estimate

Hence by the Hille-Yosida theorem (Theorem A.8.6) the operator -Ap generates a strongly continuous isometric group on LP(IR; X) if p E [1,00). However, we do not have to employ this theorem. Proposition 8.4.2. Let p E [1, 00). Then - Ap = -d/ dt generates the right shift group on LP(IR; X) defined by

[S(t)u] (s)

(s, t E IR, u E LP(IR; X)).

u(s - t)

The same is true for -Ao on Co(lR; X). Proof Clearly (S(t))tEIR is a strongly continuous group of isometries on the mentioned spaces. Let B denote its generator as defined in Appendix A.8. Then for Re>. > 0 and u E 'D(IR) we have R(>', B)u =

1

00

e-)"t S(t)u dt =

1

00

e-)"tu (· - t) dt = [e->.tl(o,oo)]

*u

= -R( ->., A)u = R(>', -A)u. Hence R(>", B)

o

= R(>., -A), i.e. B = -A.

Let us turn to the functional calculus. The general results of Sections 8.1 and 8.2 show that for 'P E (7r/2,7r) and f E HO'(S 0),

which is exactly (3.8). (The contour of course is r = asw ' for some Wi E (7f/2, cp).) For further results on the functional calculus we refer to the general approach in Propositions 8.1.6 and 8.2.3. More interesting is the following theorem. Theorem8.4.3. Let X be any Banach space, andletpE (1,00). Then the Jollowing assertions are equivalent.

(i) X is a UMD space. (ii) For one/any r > 0 the operators (d/dt)±ir are bounded on LP(lR.;X). (iii) The operator d / dt on LP (lR.; X) is in BIP. (iv) The natural Hoo(S.. * u for a certain function s>.. E L1(IR). Since the mapping

is isometric, one sees that for every f E F(H",),

0, the operator f(B) is given by convolution with some L 1 -function gf. Clearly F9f = IR • From this it is easy to derive the following lemma.

fl

> 0, and let f

HOO(H",). Then f(B) is bounded on U(IR) if and only if there is J-l E M(IR) such that FJ-l = IR •

Lemma 8.4.5. Let 0 and consider the spaces

YT := {LP((O,T);X) Co((O, T]; X)

~fPE [1,(0), If P = 00.

243

8.5. The Derivative on a Finite Interval

One could describe YT as a factor space of Y, but we refrain from doing so. By

we denote the restriction mapping. For u E YT we denote by v = u~ any extension of u to the half-line, i.e., v E Y with RTv = u. The right shift semigroup on YT is defined by

(t 2: 0)

(which of course does not depend on the actual choice of u~). It is easy to see that ST is a strongly continuous semigroup on YT and ST(t) = 0 whenever t 2: T. The negative generator of this semigroup is denoted by AT. In order to describe its domain we define W~'P((o, T); X) := {u E W1,P((0, T); X) I u(O) = O}

(8.9)

for p E [1,00] and also C~((O,T];X):= {u E C1([0,T];X)

I u(O) = u'(O) = O}.

Then we can assert the following facts.

= dj dt (in the distributional sense) with

Lemma 8.5.4. One has AT

ijpE[I,oo), ijp = 00. For Re.>. < 0 one has

or - equivalently - R('>',AT)RT = RTR(.>.,Ay). Moreover, e(AT) = C and R(.>.,AT)U(S) = _e A8 for all .>.

E

1 8

e-Atu(t)dt

(s E [0, T])

c, u E Y

T •

Proof. Suppose that Re.>.

R(\AT)U =

< 0, and let u

-1

= RT

00

E YT

eAtST(t)udt =

(-1

00

•

Then

-1

eAtS(t)u~ dt)

00

eAtRT[ST(t)u~]dt

=

RT[R(\Ay)u~]

by the definition of a generator. Since RT is surjective, it follows that 1) (AT ) = RT 1)(A y ), and this amounts to the characterisation given in the theorem. It also shows that AT = d/dt. Now observe that the shift on YT is nilpotent, i.e., ST(t) = 0

244

Chapter 8. Differential Operators

°

J;

for all t 2: T. Hence actually R(A,Ar)u = eAtSr(t)udt for all ReA < and this has an obvious holomorphic extension to the entire complex plane. Thus in fact e(Ar) = 0. The operator family (va)Re a>O forms a strongly continuous, exponentially bounded, holomorphic semigroup of angle 7r /2 on Yr. Proof. As - Ar generates the nilpotent semigroup Sn we may apply Proposition 3.3.5. This yields

1 Vau(s) = f(a) 1

= f(a)

roo

Jo

1 ta-1Sr(t)udt(s) = f(a)

Jor ta-1u(s -

1 t) dt = f(a)

roo

Jo

Jor (s -

ta-1u(· -t)dt(s) t)a-lu(t) dt.

The second assertion follows from the general theory in Chapter 3.

o

The operators va are called Riemann-Liouville fractional integrals and the semigroup (Va)Rea>O is called the Riemann-Liouville semigroup on (0, T). We now prove an analogue of Theorem 8.4.3 and Corollary 8.5.3. This requires several steps. Lemma 8.5.6. Let T, Y r , A r , V be as above, and let

0.

Proof. Given A ¢:. [aI, (0), we form the sequence r).. := (l/(A - an))n and compute

By Lemma 9.1.1 the operator associated to r).. is bounded (even compact) and it is easily seen that this operator equals R(A, A). In particular, O"(A) C [aI, (0). For A = IAI ei' - tl- 2 dt, one sees easily that A is also a strong strip-type operator with Wst(A) = 0 and R(>', A) ::; 7r Mo/ 11m >'1 for >. ~ R The functional calculus for a so-obtained sectorial/strip-type operator A is straightforward. Proposition 9.1.4. Let a = (an)n C (0,00) be a strictly increasing sequence with limn an = 00, and let A be the associated multiplication operator on X. Let W E (0,7r) and f E Hoo(Sw). Then f(A) is the multiplication operator associated with the sequence (f(an))n. The same conclusion holds true if one regards A as a strip-type operator and takes a bounded function on a strip f E Hoo(Hw), W > O.

Proof. Take e E HfJ(Sw) first. Then e(A) is defined by the Cauchy integral, and a straightforward application of Cauchy's theorem yields [e(A)x]n = e(an)xn for all n. Hence e(A) is the multiplication operator associated with (e(an))n. The operator e(A) is injective if and only if e(a n ) i=- 0 for all n. For general f E H OO take a regulariser e for f. Then x E 1>(f(A))

¢=::}

::Jy EX: (ef)(A)x = e(A)y

¢=::}

::Jy E X \:In : e(an)f(an)xn = (ef)(an)x = e(an)Yn

¢=::}

::Jy E X \:In : f(an)xn = Yn,

since e(a n ) i=- 0 for all n. This proves the claim. In the strip case the arguments 0 are analogous.

If the basis is unconditional, every bounded sequence is a multiplier. Hence in this case a sectorial multiplication operator A always has a bounded Hoo-calculus. If the basis is conditional, one always finds a bounded sequence b such that the corresponding operator is not bounded, cf. [224, Theorem II.D.2]. The whole problem is now to ensure that this sequence is of the form b = (f(an))n. This is the subject of the next section.

9.1.2

Interpolating Sequences

In this section we deal with the problem to find a bounded holomorphic function f which coincides on certain predefined points Zn with certain previously fixed numbers an. Denote for the moment by Z the space of all entire functions f E O(iC) such that Ilfllz := sup e-llmzI2If(z)1 < 00. zEIC

Then Ilfllz is a norm on Z which turns it into a Banach space. Moreover, Z embeds canonically in to each space Hoo(Hw), W > O. Let us denote by

Chapter 9. Mixed Topics

254

the restriction mapping.

Theorem 9.1.5 (Interpolating Sequences). There exists a bounded linear operator T : £00 (71.) -----7 Z with RoT = I on £00 (71.). Proof. For a given sequence a E £00(71.) we define Sa(z) := LnEZ a(n)e-(z-n)2 for E o is a Schauder basis of HP('ll'). If p -=J 2, this basis is not unconditional, as follows from [224, Proposition II.D.9].

Chapter 9. Mixed Topics

256

Let us now consider the multiplication operator A on X = HP(1l') corresponding to the sequence (2 n )n>o. It follows from Lemma 9.1.2 that A is invertible and sectorial of angle 0, and from Theorem 9.1.7 that the natural HOO(S7r)-calculus for A is not bounded. However, A E BIP(X,O), since Ais is the multiplication operator associated with the sequence (2 ins )n>O and

where (U(t))tETIf. is the (isometric) rotation group on LP(1l'). Write Xp := ker R for the kernel of the Riesz projection. Then LP(1l') = Xp EB HP(1l') and the operator I EB A is an invertible sectorial operator of angle 0 on LP(1l') with bounded (even isometric) imaginary powers but without bounded H OO -calculus.

9.1.4

Comments

The first example of a sectorial operator without bounded imaginary powers was given by KOMATSU [131, Section 14, Examples 6 & 8] (reproduced in [161, Example 7.3.3]), modelled on the space Co. McINTOSH and YAGI [169, Theorem 4] gave an example of an invertible operator A on a Hilbert space without bounded H OO _ calculus. The method is to construct A as a direct sum of a sequence of finitedimensional operators with certain special properties. The idea of using conditional bases for constructing counterexamples was introduced by BAILLON and CLEMENT [23]. VENNI [219] subsequently used it to give an example of an operator A with Ais being bounded for some but not all s E 1Ft The method was elaborated further by LANCIEN, LE MERDY and SIMARD, cf. [143, 145, 203, 147]. In those papers it is usually referred to Carleson's theorem to obtain that the sequence of powers (2n)n>1 is an interpolating sequence for the halfplane. Our proof of this fact, i.e., essentially Theorem 9.1.5, was abstracted from [51, Lemma 5.3], where an example is given of an operator A on LP(lR) (p #- 1,2,00) that has bounded imaginary powers but not a bounded Hoo-calculus. Actually, this example is more or less the analogue of our second example in Section 9.1.3, due to LANCIEN [143], cf. [9, Section 4.5.3]. Although the construction is elegant, the examples are artificial in a sense and in fact the natural differential operators on the natural reflexive spaces all do have a bounded Hoo-calculus, cf. Section 8.6. However, if one passes to pseud~ differential operators, things are different, as HIEBER [117] has pointed out.

9.2

Rational Approximation Schemes

In this section we turn to an application of functional calculus to questions of numerical analysis.

257

9.2. Rational Approximation Schemes

9.2.1

Time-Discretisation of First-Order Equations

Assume that one is given an autonomous first-order initial-value problem X'

= F(x),

x(O) = Xo

in a state Banach space X. An exact solution is a differentiable curve (u(t))t>o starting at Xo and satisfying -

d dt u(t) = F(u(t))

(t > 0),

i.e., at any time t > 0 the velocity of u at t coincides with the value of the vector field F at the point u(t). In simulating the dynamical system numerically one tends to work with piecewise affine curves instead of exact solutions. So given a fixed t > 0 one might think of dividing the interval [0, t] into n E N equal parts 0 = to < tl < ... < tn = t, each of length h := tk+1 - tk = tin, and consider a continuous curve u : [0, t] ----+ X starting at u(O) = Xo and being affine on each interval [tj- 1 , tj]. Using the notation Uk = U(tk), the vector r5uk := h- 1 (Uk+1 - Uk) is the constant velocity vector of u on [tk, tk+l]. In order that u can count as an approximate solution to our original problem, this velocity vector should have something to do with the vector field F. Onecertainly natural - choice is to require

that means, at each tk the curve u starts off with velocity F(U(tk)). This method is commonly called the forward (or explicit) Euler method. When instead we require r5uk = F( uk+d, i.e., we want u to arrive at tk+1 with the right velocity, it is called the backward (or implicit) Euler method. Let us now look at the case that F(x) = -Ax, where A is a sectorial operator of angle W A < 7r 12. The forward Euler method leads to the equation

Uk+l = Uk - hAuk = (1 - hA)Uk for all k :::; n - 1. Hence Un = (1 - hA)nxo. Obviously, if we want to apply this method in an infinite-dimensional setting where the operator A is unbounded, this has some drawbacks since the approximation u will only be defined if the initial value is sufficiently smooth, i.e., is contained in TI(AN) for some (high) power N E N. Using instead the backward Euler method yields

Uk+l - Uk = -hAuk+l, which amounts to Uk+l = (1 + hA)-lUk and there are no restrictions for the initial value Xo. (Note that our assumptions on A ensure that (1 + hA)-l E £(X).) So we are lead to consider

Un =

(1 + ~t A) - nXo

= r (t)n ~ A Xo

258

Chapter 9. Mixed Topics

(with r(z) = rbe(Z) := (1 + z)-l) as an approximation of e-tAxo. Of course, apart from the backward Euler scheme there are other methods which work well in the infinite-dimensional setting. We might, for example, require 1 6Uk = h -1 (Uk+! - Uk) = F (Uk+! 2+ Uk) = -2"A(Uk+!

+ Uk),

which amounts to the fact that the straight curve UI[tk,tk+ll has its velocity coinciding with the vector field F exactly at the midpoint between its beginning Uk and its end Uk+1. A short computation then yields

with

r(z)

=

rcn(Z)

:=

1- lz

--i-. 1 +"2z

This method is called the Crank-Nicolson scheme. We cast these examples into a proper definition. A rational approximation scheme for the operator A is given by some rational function r with poles outside the spectral sector SWA' It consists in computing

nh= t for n

E

N, t > O. We call the scheme accurate of order p (p as

z

----t

E

N) if

O.

While the backward Euler scheme is accurate of order 1, the Crank-Nicolson scheme is accurate of order 2. In the following we consider rational functions r and examine the properties of the corresponding approximation schemes, i.e., of the operator family

r(hAt

(h > 0, n EN).

Of course one may ask for convergence r(tA/n)nx ----t e-tAx in general or, more precisely, for convergence rates depending on the smoothness of x. Apart from that, one may ask for the stability of the approximation scheme determined by r with respect to the operator A. By this we mean simply the uniform boundedness

K:=

sup

h>O,nEN

Ilr(hAtll
p. Suppose that Ir(O)1 = 1. Then for (p, q, a) to be adapted it is necessary that r(z) = r(O) -r(O)az P +O(zp+l) near z = O. This determines p, a, and finally q. The inequality largal + ptp ::; 7r /2 is now easily deduced from the fact that Ir(O)1 = 1 and Ir(z)1 ::; 1 for z E Scp. 0

The next lemma describes the asymptotic behaviour of the rational function

r(z)n at O. Lemma 9.2.2. Let tp E (0,7r/2], let r be an A(tp)-stable rational function, and let (p, q, a) be adapted to r. Then the following assertions hold.

a) If Ir(O)1 < 1, then for each () E (0, tp) there are constants C 2': 0, c E [0,1) such that (z E Se, Izl ::; 1, n EN).

b) If Ir(O)1 = 1, then for each () E (O,tp) there are constants C 2': O,c that

>0

such

(z E Se, Izl ::; 1, n EN).

260

Chapter 9. Mixed Topics

Proof. Fix a constant C f ;:::: 0 with Ir(z) - r(O)e-azPI :S C f Izl q for z E S'P' Izl :S 1, and let (J E (0, i.p) a) Suppose that Ir(O)1 < 1. By the Maximum Principle one has Irl < 1 in the interior of S'P' Hence 0 < d := sup{lr(z)1 I z E Se, Izl :S I} < 1. Choose c E (d,l). Then

I

Ir(zt - r(Ote- nazP = Ir(z) - r(O)e- azp

II L;~: r(z)jr(O)n-j-le-a(n-j-l)zP I

n-l

:S C f Izl q Lj=o dn-1e-(n-j-l) Re(az P) :S Cfndn- 1 Izl q = Ccn Izl q for all n E N, Izl :S 1 and some other constant C ;:::: O. This proves a). In the case Ir(O)1 = 1 we define Cl := inf{Reaz P I z ESe, Izl = I} > 0 and obtain le- azP I :S e-cllzlP for all z ESe. Next, we claim that there exists C2 > 0 such that

(z ESe, Izl :S 1).

(9.4)

Assume the contrary. Then for some sequence (zn)n we have Ir(zn)1 > e-lznIP/n. After passing to a subsequence we may suppose that (zn)n converges to some Zo E Se with Izol :S 1. Hence Ir(zo)1 ;:::: 1. The Maximum Principle and (9.2) now imply that Zo = O. Writing tn := IZnl and using that (p, q, a) is adapted it follows that

(n EN), but this is impossible, since tn --. 0 and q define c := min{ Cl, C2} and estimate

Ir(z)n - r(Ote-nazpl = Ir(z) - e-azPII

> p. Having thus established

(9.4), we

L;~: r(z)jr(o)n-j-le-a(n-j-l)Zpl

:S C f Izl q L;~: e-jc2lzlP e-c1(n-j-l)lzI P :S C f Izl q ne-(n-l)clzI P :S (C feC ) Izl q ne-nclzIP.

D

We now use Lemma 9.2.2 in the special case of accurate functions r to obtain optimal approximation results for smooth initial data.

Theorem 9.2.3 (Convergence Theorem). Let r be a rational function accurate of order pEN and satisfying (9.2). Then for every a E (0, p] and (J E (0, i.p) there is a constant C = C(r, (J, a) such that for every sectorial operator A on a Banach space X such that WA E [0, (J) one has an estimate

for all x E 1>(AQ;) and all h > 0, n E N.

9.2. Rational Approximation Schemes

261

Proof. Fix 0 E (0, cp). The triple (1,p + 1,1) is adapted and r(O) Lemma 9.2.2 we find constants C, c > such that

°

(z E So, Izl

~

= 1, hence by

1,n EN).

Also, by using the A(cp)-stability, we find Ir(z)n - e-nzl ~ 2 for all z E S(AO:), we have

Ilr(hAtx - e-nhAxll = II(hAt fn(hA)xll ~ C f M(A, O)hO:

o

This concludes the proof.

9.2.3

IIAO:xll.

Stability

°

We recall that in Theorem 9.2.3 the case that a = is excluded. We shall see in a moment that in fact a norm convergence statement like

in general is only true if Ir( 00) I < 1. However, we are going to show the stability of the appproximation scheme, requiring nothing else than A( cp )-stability of r.

°

Theorem 9.2.4 (Stability Theorem). Let cp E (0,1T/2], and let r be an A(cp)-stable rational function. Then for every 0 E (0, cp) there is a constant C = C(O, r) ~

such that

sup

nEN,h>O

Ilr(hA)n I

~

CM(A, B)

for every sectorial operator A with W A E [0, B). As in the previous section, the proof of this theorem rests on a careful asymptotic analysis of the rational function r. The crucial fact is the following.

262

Cbapter 9. Mixed Topics

Proposition 9.2.5. Let rp E (0, 1f /2]' let r be an A( rp) -stable rational function, and let (p, q, a) be adapted to r. Then for 9n defined by

9n(Z) := r(zt - r(ote- anzP and () E (0, rp) there is c E [0,1) such that

{ (where s =

if Ir(O)1 < 1, if Ir(O)1 = 1.

O(c n ) O(n- S )

= (q - p)/p is as defined in (9.3)) and

Sr

r

ias o,IzI'?1

Ir(O)ne-naZ PI ~ = O(c n ).

Izl

Proof. Given an A(rp)-stable rational function r and an angle () E (0, rp) according to Lemma 9.2.2 one can write

{c Izl Izl

p Ir(z)n _ r(O)ne-aZ I ~

C

cn

q

ne-cnlzl

(where c E [0,1) in the first and p < q, C >

r

iaso,lzl~l

19n(z)1

~ Izl

=

r

iaso,lzl9

r

in the second case). If Ir(O)1 < 1,

Ir(z)n - r(Ote- anzP I ~

~

Ir(z)n _ r(Ote- anzP Ildz l

Izi

2C

~ 1)

°

Izl

If however Ir(O)1 = 1, one has

i aso,lzl9

Izl

(n E N, z E So,

P

~c

r

iaso,lzl9

r tqne-cntP dt ~ 2CP nl-~ iot io 1

XJ

~ 2Ccn .

Izl q ne-cnlzlP

t~-le-ct dt.

t

Idzl

Izl

°

This proves the first statement. To prove the second, choose c > in such a manner that le- azP I ~ e-clzlP for all z E So (cf. the proof of Lemma 9.2.2). Then

r

Ir(Ote-anzP Ildzl ~ 2

tX! e- cntP dt il t

~ 2e- cn .

o iaso,lzl?l Izl cnp The strategy for the proof of Theorem 9.2.4 is to apply Proposition 9.2.5 not only to the function r(z) but as well to r(z-l) (which is also an A(rp) stable rational function). We choose (p, q, a) adapted to r and (p', q', b) adapted to r(z-l) and consider the functions fn(z) := r(zt - r(Ote-nazP - r(oote-nbz-pl. (Note that fn

E

Ho(Se) for every

r(hAt = fn(A)

+ r(o)n

e E (0, '1').)

(e- anzP ) (hA)

We thus can write

+ r(oot

(e-nbz-pl) (hA).

The two last summands are unproblematic, as the following lemma shows.

(9.5)

9.2. Rational Approximation Schemes

263

Lemma 9.2.6. Let 'P E (0, n /2]' and let p, p' E N and a, bEe \ {o} such that largal + P'P, largbl + P''P :S n/2. Then for any B E (0, 'P) there is a constant C

such that

sup

h>O,nEN

Ile- anzp (hA)11

+

sup

h>O,nEN

for every sectorial operator A with W A

E

Ile-bnZ-pl (hA)11 :S C M(A, B)

[0, B).

Proof. This follows immediately from Proposition 2.6.11, since by hypothesis we have e- azP , e- bz - pf E E(S W > max{wA,wB} and 'P E (WA,7r),'ljJ E (WB,7r) with max{'P,'ljJ} < W and 'P+'ljJ < 7r. Then Scp +S'ljJ = Smax{cp,'ljJ}' For A E IC such that largAI E [w,7r] we write

A A2 A - (z + w) = (A - z) (A - w)

+

AZW A---Z---w-')--:(A---W-')--:(A--:--Z--:-) ,

7:"(

and all functions are in O(Scp x S'ljJ). The second summand -let us call it f>..(z, w) - is even in HfJ(Scp x S'ljJ), as a moment's reflection shows. This yields directly

AR(\ C)

=

[A _

(~+ w)] (A, B) =

AR(A, A),AR(A, B) + f>..(A, B) E £(X).

Moreover, for J-l = AI IAI we have

c) Let a E (0,1). Then

(Z + w)(1

+ z)(1 + w)

(1

+ z)(1 + w)

(z + w)(1

+ z)(1 + w)'

and the second summand is in HfJ(Scp x S'ljJ) since ZC:Wl-C: I(z + w) is bounded for each E E [0,1]. Note further that

((z + w)(::wZ)(1 + w)) (A, B)

11

1( =')2 ( ) (z"w )( )R(w,B)dwR(z,A)dz 27rZ r1 r2 z +w 1 + z 1+w

-11

z" ) [BR( -z, B)]R(z, A) dz. -(27rZ r 1 1 + z

= (1 + B)-l_.

9.3. Maximal Regularity

271

It follows that fa(A,B) E £(X), where fa(z,w) := z1+ a (z Therefore,

A1+ a (1

+ A)-(1+ a) = C fa(A, B)(l + A)-a:::)

+ w)-l(l + z)-l.

(1 + A)-a fa(A, B)C,

(9.10)

hence N(C) C N(A1+a(1 + A)-(1+ a)) = N(A), cf. Proposition 3.1.1 d). By symmetry, N(C) c N(B) as well. But N(A) n N(B) c N(C) is clear by a), so c) is proved. d) For x E 1>(C), formula (9.10) shows that A1+ a (l+A)-(1+ a)x E 1>(Aa), whence necessarily x E 1>(Aa) (cf. Proposition 3.1.1 g)). By symmetry and a), this proves d). e) Another look on (9.10) yields ::R(A1+a) = ::R(A1+a(l + A)-(1+ a)) C ::R(C). Employing symmetry again establishes the left-hand inclusion in e). To complete the proof take x E 1>(C). We have to prove y := Cx E ::R(A) + ::R(B). By definition we have

and this is obviously contained in ::R(A) + ::R(B). Hence

y = (1 + B)-l(l + A)-ly + B(l + B)-l(l + A)-ly + A(l + A)-Iy E

::R(A) + ::R(B)

+ ::R(B) + ::R(A) = ::R(A) + ::R(B).

0

Let us state some immediate but important consequences. Corollary 9.3.2. Let A, B, C as in Theorem 9.3.1. Then the following assertions hold.

a) If A or B is injective/invertible, so is C. b)

If A or B has dense range, so has C.

c) C is densely defined if and only if both A and B are densely defined. If this is the case, then C = A + B. Proof. Employing some standard facts about fractional powers (Proposition 3.1.1, Corollary 3.1.11) the statements are more or less direct consequences of Theorem 9.3.1. Suppose that A, B, C are densely defined, and let (x, y) E C. Then

Xn

:=

n 2 (n + A)-I(n + B)-Ix

----7

x and Yn:= n 2 (n + A)-I(n + B)-Iy

But (xn' Yn) E C and Xn, Yn E 1>(A) n 1>(B), hence (xn, Yn) E A + B.

----7

y. 0

Remark 9.3.3. Let A, B, C be as in Theorem 9.3.1. An elementary but fairly tedious discussion yields that for W E (max{W A, W B}, 7r) and f E £ (8w ) one has f(z + w) E £(8

(A)

n 1>(B)

: Au + Bu = x

and

Au E E.

Note that one does not require u E E. The pair (A, B) is said to have (abstract) maximal regularity if E = X is a space of maximal regularity. Lemma 9.3.4. Let A, B, C be as above, and suppose that A is injective. Define

f(A, B) := For a Banach space E

(_Z_) (A, B). Z+W

c X the following assertions are equivalent:

(i) E c X is a space of maximal regularity for (A, B). (ii) E c

~(C)

n 1>(f(A, B)) and f(A, B)E c E.

c

~(C)

and f(A,B) restricts to a bounded operator on E.

(iii) E

In particular, (A, B) has maximal regularity ifC is invertible and f(A, B) E £(X). Proof. The equivalence of (ii){:}(iii) is an easy consequence of the continuity of the embedding E C X and the Closed Graph Theorem. By general functional calculus and Theorem 9.3.1 a) one has

AC- l c f(A, B)

and

~(C)

n 1> (f(A, B)) = 1>(AC- l ) = ~(A + B).

From these identities the equivalence (i){:}(ii) readily follows.

D

Theorem 9.3.5 (Da Prato-Grisvard). Let A, B be resolvent-commuting, sectorial

operators on the Banach space X satisfying the parabolicity condition WA +w B < 1f. If A or B is invertible, then each space E := (X,1>(AO))o,p, () E (0, 1),p E [1,00]' is a space of maximal regularity for (A, B).

Proof. By hypothesis and Corollary 9.3.2 the operator C is invertible, hence the inclusion E := (X, 1>(AO))o,p C ~(C) is trivial. So one is left to show that E C 1>(f(A, B)) and f(A,B)E C E. By Theorem 9.3.1 d) we have ~(C-l) = 1>(C) C 1>(Al -c) for each E E (0,1). On the other hand, E = (X,1>(AO))o,p c 1>(Ac) for some (small) E E (0, I), see Corollary 6.6.3. Hence

f(A, B)

=:l

[Al-cC- l ] Ac

and Al-cC- l E £(X). This shows 1>(A")

c 1>(f(A, B)),

hence E C 1> (f(A, B)).

We now prove the invariance of E under f(A, B). Choose a function 'ljJl E £(S(A))o,p) for all

n

W~'P((O, T); X)

e E (0,1).

Here W~'P((O, T); X) = (LP((O, T); X), W~'P((O, T); X))o,p is a fractional Sobolev space. Proof. This follows from Theorem 9.3.1 d) in noting that for any sectorial operator A on a Banach space one has n"E(O,l) 1>(A") = nOE(O,l)(X, 1>(A))o,p. 0

Remark 9.3.8. Proposition 9.3.7 is designed to be just an example of what can be done with Theorem 9.3.1. In particular, one can choose a different interpolation parameter q instead of p, and one obtains a similar result on the space of continuous functions. In the case that T = 00, the operator C is not surjective in general. In fact it can be shown that if we work on X = LP((O, (0); X), and C is invertible, then already A must be invertible, i.e., the semigroup (e-tA)t>o is exponentially stable. This is known as Datko's theorem, see [10, Theorem 5.1.2]. Let us turn to the so-called maximal regularity problem. We call a Banach space E c L!c([O, T); X) (continuous inclusion) a space of maximal regularity for A if for every fEE the corresponding mild solution u of (9.7) satisfies:

u E WI!) [0, T); X) n L!)[O, T); 1>(A))

and

Au, u' E E.

(Note that we do not require u E E, although in many cases this is automatically satisfied.) This means that the solution should have the best regularity properties one can expect, with respect to both operators A and d/ dt, and the space E is invariant under the mappings (J f------t Au), (J f------t u'). (It is this invariance property that renders the whole concept ultimately important for applications, see the comments in Section 9.3.4.) Theorem 9.3.9 (Da Prato-Grisvard). Let T < OO,p E [1, (0), q E [1,00]' e E (0,1), and Re 0: > 0. Let A be a sectorial operator on the Banach space X with W A < 7r /2. Then the spaces

and are spaces of maximal regularity for A.

275

9.3. Maximal Regularity

Proof. Fix p E [1, (0) and consider the usual operators A and B = d/dt on X = LP((O, r); X). By Theorem 9.3.5, E = LP((O, r); (X, 2)(A"))e,q) = (X,2)(A"))e,q is a space of abstract maximal regularity for the pair (A, B); this means that E is in fact a space of maximal regularity for A. Interchanging the roles of A and B yields that E = wg,P((O, r); X) is a space of maximal regularity for A. The remaining statements are proved analogously by working on X = CoCCO, r]; X). 0

Let r be finite, for the time being. A theorem of BAILLON [22] states that E = Co((O, r]; X) is a space of maximal regularity for A if and only if either A is bounded or X contains a copy of Co, cf. [81]. In reasonable concrete situations, e.g. on reflexive spaces, this is never the case, whence continuous maximal regularity is not a promising property to study. Things change when we pass to the so-called LP -maximal regularity for p E (1, 00 ). More precisely, we say that the operator has the MRp,r-property, where p E (1, (0), r E (0,00], if E = LP((O, r); X) is a space of maximal regularity for A. It turns out that many common operators on common Banach spaces do indeed possess this property, but this fact is far from being trivial. We digress a little on that property. From the Variation of Constants formula (9.8) it follows readily that for finite r and A E IR one has A E MRr,p if and only if A + A E MRr,p. Hence as far as maximal regularity is concerned, one can always suppose that A is invertible. One can further show that the MRr,p-property is independent ofr E (0,00) [71, Theorem 2.5]. If A is invertible, i.e., if the semigroup is even exponentially stable, one can include r = 00 in this statement. Therefore, we can omit the reference to r and write just A E MRp- (It turns out that it is even independent of p, see below.) The problem of maximal regularity is essentially a question of operator-valued Fourier multipliers, i.e., vector-valued singular integrals. To see this, let for simplicity A be invertible and T = 00. Loosely speaking, A E MRp is equivalent to fELP((O,oo);X) ===}- Ae-sA*fELP((O,oo);X), i.e., (J 1--+ Ae- sA * 1) (originally defined on the dense set 'D((O, 00); 2)(A))) is a bounded operator on the space LP((O, (0); X). Since the norm ofthe kernel Ae- sA grows like S-1 as s ~ 0, this is a singular integral operator. Lemma 9.3.10. Let A be a densely defined, invertible, sectorial operator on the Banach space X such that WA < 7f /2. Then for p E (1,00) the following assertions are equivalent:

(i) A

E MRp.

(ii) The function A(is multiplier.

+

A)-1 E LOO(IR;£(X)) is a bounded LP(IR;X)-Fourier

Proof. Denote by S = (J 1--+ define Y := LP((O, (0); 2)(A))

(i)*(ii). Take any f

E

U

= e-· A

* 1) the solution operator as in (9.8), and

n w~'P((O, (0); X).

Set m(s) := A(is + A)-I.

'D((O, 00); 2)(A)). Then ASf

= SAf and F(AS1)(s) =

Chapter 9. Mixed Topics

276

m(s)f(s). By hypothesis, IIASfllp ::::: c Ilfllp for some constant c independent of f. Hence (9.11) By shifting functions this inequality extends to all f E 1>(lR.; 'D(A)), and then even to all f E S(lR.; X), because 'D(A) is dense in X. Thus (ii) follows, by definition (see Appendix E.4).

°

(i)=}(ii). Fix a constant c > such that (9.11) is satisfied for all f E S(lR.; X), and take f E 1>((0, (0); 'D(A)). Then (SJ)' = f - ASf, whence Sf E Y and

IISfll y

'"

II(SJ)'llp+ IIASfll p + IISfllp;S

IIASfll p +

IISfllp;S

IIASfll p +

Ilfllp

since the semigroup is exponentially stable. However, ASf = F(mj) as above, whence IIASfllp ::::: c Ilfll p. Since 1>((0, (0); 'D(A)) is dense in LP((O, (0); X), we see that in fact S maps LP((O,oo);X) boundedly into Y, hence A E MRp. D Using an important result of BENEDEK, CALDERON and PANZONE from [28] one can prove that if A E MRp for some p E (1, (0) this actually holds for all p E (1, (0), see [71, Theorem 4.2.]. Thus we can omit even the reference to p and write A E MR instead of A E MRp. From Lemma 9.3.10 and Plancherel's theorem it follows readily that on a Hilbert space X = H every sectorial operator A with WA < 7r /2 satisfies A E MR 2 • It was proved by COULHON and LAMBERTON in [50] that the Poisson semigroup on X = L2(lR.; Y) has maximal regularity if and only if Y is a UMD space. (See Appendix E.6 for the definition of UMD spaces.) This not only showed that there exist operators without LP-maximal regularity, but at the same time stressed the importance of UMD spaces for possible positive results. In 1987 DORE and VENNI [75] found the following abstract result which was a milestone at the time. Theorem 9.3.11 (Dore-Venni). Let X be a UMD space, and let A, B be resolventcommuting, sectorial operators on X such that A, B E BIP(X) and BA + BB < 7r. Then C:= (z+w)(A,B) = A+B, C E BIP(X) with Be::::: max{BA,BB}, and f(A, B) E .c(X), where f(z, w) = z/(z + w). In particular, if C is surjective then

the pair (A, B) has abstract maximal regularity. The Dore-Venni theorem can be used to prove maximal LP-regularity of operators with bounded imaginary powers on UMD spaces. Corollary 9.3.12. Let X be a UMD space, and let A E BIP(X). Then A E MRp for each p E (1, (0).

Proof. We define as usual the operators A, B on X := LP((O, r); X). Since X is a UMD space, the space X is also a UMD space, and the derivative operator B has bounded imaginary powers on X, see Theorem 8.5.8. Hence we can apply the Dore-Venni theorem to conclude that C = A+B, i.e., 'D(C) c 'D(A), whence the assertion follows. D

9.3. Maximal Regularity

277

So far, one still could believe that on UMD spaces every sectorial operator A with WA < 7r/2 has maximal LP-regularity. However, this hope is in vain, as KALTON and LANCIEN [124] showed that among a very large class of Banach spaces, a space X with the property that every negative generator of a holomorphic semigroup on X has maximal regularity, must be isomorphic to a Hilbert space. Around the same time WEIS finally succeeded to give a characterisation which more or less settled the question. Theorem 9.3.13 (Weis). Let X be a UMD Banach space, and let A be a densely defined, sectorial operator on X with W A < 7r /2. Then A E MR if and only if the set {A(is + A)-l I s -1= O} is R-bounded.

(See Appendix E.7 for the definition of R-boundedness.) The proof is a combination of Lemma 9.3.10 and Weis' operator-valued Mikhlin multiplier theorem (Theorem E.7.4) together with some simple facts about R-boundedness, cf. [141].

9.3.4

Comments

Basic facts on the inhomogeneous Cauchy problem can be found in practically every book on evolution equations. In [85, Section VI.7.a] one can find a rigorous derivation of Duhamel's principle, including a discussion of regularity. LUNARDI [157] extensively treats the Holder-regularity theory. The seminal paper for regularity of solutions is [58] by DA PRATO and GRISVARD who introduced the sum-of-operators point of view. Theorems 9.3.1, 9.3.5, and 9.3.9 are already in their paper. They do not talk explicitly about functional calculus but the basic Cauchy-type integrals are already there. A functional calculus for more than one sectorial operator was introduced for the first time by ALBRECHT in his 1994 thesis and subsequently studied by LANCIEN, LANCIEN and LE MERDY [142] and ALBRECHT, FRANKS and McINTOSH [4]. (However, injectivity of at least one of the operators was a crucial assumption.) Sums of commuting operators can also be dealt with by a functional calculus for a single operator, but this calculus then has to involve also operator-valued functions, cf. [142], [46]. The material of Chapter 1 can be easily modified to cover this, cf. the comments in Section 1.6. However, the sum operator C is not as easily obtained as by means of the joint functional calculus. Our proof of the Da Prato-Grisvard Theorem 9.3.5 is inspired by [110], cf. also [46]. Maximal LP-regularity has been an issue so important in the last twenty years that there are numerous texts treating the topic. As an outstanding instruction we recommend DORE'S survey [71]. The importance of the concept of maximal regularity lies in the fact that its validity allows to obtain solutions to non-linear equations via the Banach fixed point theorem, see [9, 6.2.10] for the rough structure and [45,47, 86] for 'real' applications. Theorem 9.3.11 and its corollary is basically due to DORE and VENNI [75], with slight generalisations due to PRUSS and SOHR [193J. Its original proof is in

Chapter 9. Mixed Topics

278

the spirit of functional calculus, and in fact MONNIAUX [172] and finally UITERDIJK [218] brought to light the underlying functional calculus ideas, see also [161, Chapter 9]. After important work of CLEMENT, DE PAGTER, SUKOCHEV and WITVILIET [44] (see Proposition E. 7.3) WEIS [223] finally gave the long desired characterisation, a detailed account of which is now in [141]. A discrete approach to Weis' theorem was given by ARENDT and Bu [11]. There is also a functional calculus approach to Weis' theorem. Indeed, KALTON and WEIS [125] proved the following result, see also [141, Theorem 12.13].

Theorem 9.3.14 (Kalton-Weis). Let A, B be two resolvent-commuting, sectorial operators with dense domain and range on a Banach space X. Let cP E (W A, 7r) and 'if; E (w B, 7r) such that cp + 'if; < 7r, and suppose that the following two conditions are satisfied. 1) A has a bounded HOO(Scp) calculus. 2)

The set {AR(>.,B) I

Then C

>.

~

S,p} is R-bounded.

= (z + w)(A) = A + Band f(A, B)

E £(X), where f(z, w)

= z/(z + w).

One way to prove this theorem is to show that an operator with bounded scalarvalued Hoo-calculus always has a bounded operator-valued Hoo-calculus when one restricts to functions with R-bounded range (see [77] for a proof). Theorem 9.3.14 is not a generalisation of the Dore-Venni theorem, but it can replace it as long as only maximal regularity is concerned. Finally, we remark that there are maximal regularity results for non-commuting operators [173, 221, 159]. In view of the prospective applications this seems a promising field for further research. A functional calculus approach in the non-commutative setting - if there is any - is still to be developed.

Appendix A

Linear Operators This chapter is supposed to be a 'reminder' of some operator theory, including elementary spectral theory and approximation results, rational functional calculus and semigroup theory. There is a slight deviation from the standard literature on operator theory in that we deal with multi-valued operators right from the start.

A.I

The Algebra of Multi-valued Operators

Let X, Y, Z be Banach spaces. A linear operator from X to Y is a linear subspace of the direct sum space X EEl Y. A linear operator may fail to be the graph of a mapping. The subspace

AO

:=

{x

E

X

I (0, x)

E

A} c X

is a measure for this failure. If AO = 0, the relation A c X EEl Y is functional, i.e., the operator A is the graph of a mapping, and it is called single-valued. Since in the main text we deal with single-valued linear operators almost exclusively (not without significant exceptions, of course), we make the following Agreement: Unless otherwise stated, the term 'operator' always is to be understood as 'single-valued linear operator'. We call an operator multi-valued if we wish to stress that it is not necessarily single-valued (but it may be).

The image of a point x under the multi-valued operator A is the set

Ax:= {y

EY

I (x,y)

E

A}.

This set is either empty (this means, the multi-valued operator A is 'undefined' at x) or it is an affine subspace of Y in the 'direction' of the space AO. With a multi-valued operator A C X EEl Y we associate the spaces kernel domain range

N(A)

.-

1) (A)

.- {x E X I there is y E Y such that (x,y) E A}, .- {y E Y I there is x E X such that (x,y) E A}.

::R(A)

{xEXI(x,O)EA},

280

Appendix A. Linear Operators

The multi-valued operator A is called injective if N(A) = 0, and surjective if

::R( A) = Y. If 1J (A) = X, then A is called fully defined.

Let A, B c X E9 Y and C C Y E9 Z be multi-valued operators, and let A E C be a scalar. We define the sum A + B, the scalar multiple AA, the inverse A-I, and the composite C A by

A + B := {(x, Y + z) EX E9 Y I (x, y) E A, (x, z) AA:= {(X,AY) EX E9 Y I (x,y) E A},

A- 1 := {(y,x) CA:= {(x,z)

E YE9X E

I (x,y)

X E9 Z 13y

E

A}, Y: (x,y)

B},

E

E

E

A

A

(y,z)

E

C}.

Then we have the following identities:

1J(A + B) = 1J(A) n 1J(B), 1J(A-l) 1J(CA) = {x

E

=

::R(A),

1J(AA) = 1J(A), 1J(A) 13y E 1J(C): (x,y)

E

A}.

The zero operator is 0 := {(x,O) I x E X} and I := {(x, x) I x E X} is the identity operator. (This means that in general we have only OA C 0 but not OA = 0.) Scalar multiples of the identity operator AI are abbreviated as A, in particular we write A + A instead of AI + A. Proposition A.I.I. Let X be a Banach space, and let A, B, C valued linear operators on X.

c X E9 X be multi-

a)

The set of multi-valued operators on X is a semigroup with respect to composition, i.e., the associative law A(BC) = (AB)C holds. The identity operator I is the neutral element in this semigroup. Moreover, the inversion law (AB)-1 = B- 1A- 1 holds.

b)

The set of multi-valued operators on X is an abelian semigroup with respect to sum, with the zero operator 0 as its neutral element.

c) For A -::J 0 one has AA = (AI)A = A(AI). d)

The multi-valued operators A, B, C satisfy the following monotonicity laws:

Ac B Ac B

===} ===}

AC c BG, GA c CB, A + G c B + G, AA c AB.

281

A.l. The Algebra of Multi-valued Operators

e) The following distributivity inclusions hold:

+ B)C c AC + BC, CA+CB c C(A+B), (A

with equality if C is single-valued; with equality if~(A) C 1l(C).

In particular, there is equality in both cases if C E £(X) (see below). A multi-valued operator A C X EB Y is called closed if it is closed in the natural topology on X EB Y. If A is closed, then the multi-valued operators ,\A (for ,\ i- 0) and A-l are closed as well. Furthermore, the spaces N(A) and AO are closed. Sum and composition of closed multi-valued operators are not necessarily closed. For each multi-valued operator A one can consider its closure A in X EB Y. A single-valued operator is called closable if A is again single-valued. If a multi-valued operator A is closed, then every subspace V C 1l(A) such that {(x, y) E A I x E V} = A is called a core for A. A single-valued operator A is called continuous if there is c

IIAxl1 ::; cIlxll

(x E 1l(A)).

Every continuous operator is closable. continuous and fully defined. We let £(X, Y) := {A C X EB Y

2': 0 such that

An operator is called bounded if it is

I A is a bounded operator}

be the set of all bounded operators from X to Y. If X = Y we write just £(X) in place of £(X, X). For a single-valued operator A there is a natural norm on 1l(A), namely the graph norm (x E 1l(A)). Ilxii A := Ilxll + IIAxl1 Then A is closed if and only if (1l(A),

II.II A )

is complete.

Lemma A.1.2. Let X and Y be Banach spaces. A single-valued operator A C X EBY is continuous if and only if 1l(A) is a closed subspace of X.

Proof. The continuity of the operator is equivalent to the fact that the graph norm is equivalent to the original norm. Closedness of the operator is equivalent to the fact that 1l(A) is complete with respect to the graph norm. Therefore the assertion follows from the Open Mapping Theorem. 0 Lemma A.1.3. Let A be a closed multi-valued operator on the Banach space X, and let T E £(X). Then AT is closed.

282

Appendix A. Linear Operators

Proof. Let (Xk' Yk) E AT with Xk ----t x and Yk ----t y. Then there is Zk such that TXk = Zk and (Zk' Yk) E A. Since T E .c(X), one has Zk = TXk ----t Tx. The closedness of A implies that Z E 'D(A) and (z, y) E A, hence (x, z) EAT. 0 A multi-valued operator A C X EB Y is called invertible if A-I E .c(Y, X). We denote by .c(X)X := {T I T, T- 1 E .c(X)} the set of bounded invertible operators on X. Lemma A.1.4. Let A be a closed multi-valued operator on the Banach space X, and let T be an invertible multi-valued operator. Then T A is closed. Proof. Suppose that (xn, Yn) ETA with Xn ----t x and Yn ----t y. Then there are Zn such that (xn, zn) E A and (zn, Yn) E T. Since T- 1 E .c(X), Zn = T- 1Yn ----t T- 1 y. The closed ness of A implies that (x, T- 1y) E A, whence (x, y) ETA. 0 Note that the last result in general is false without the assumption that T is invertible.

A.2

Resolvents

In this section A denotes a multi-valued linear operator on the Banach space X. The starting point for the spectral theory is the following lemma. Lemma A.2.1. The identity

holds for all A E C. Proof. For A = 0 the assertion is (almost) trivial. Therefore, let A -I- O. If x, Y E X and Z := (1/ A)y, one obtains

(x, y)

E

A(A + A)-1 {:} (x, z)

{:} (z, x - AZ)

E

E

(A + A)-1 {:} (z, x)

A {:} (x - Az, z)

E

E

(A + A)

A-I {:} (x - Az, AZ)

r

E

AA- 1

+ AA -1 {:} (X, X - AZ) E [I + AA -1 1 {:} (x, AZ) E I - [I + AA -1 r 1 {:} (x, y) E I - [I + AA -1 r 1 . {:} (X - Az, x)

E

I

We call the mapping

(A

f------+

R(A,A)

:=

(A - A)-I) : C

----+

{multi-valued operators on X}

the resolvent of A. The set

Q(A)

:=

pEe I R(A, A) E .c(X)}

is called the resolvent set, and a(A) := C \ Q(A) is called the spectrum of A.

0

283

A.2. Resolvents

>., J-l

E

rc

1 - [1 + (.\

-

J-l)R(J-l, A)t 1 = (.\ - J-l)R(>., A)

Corollary A.2.2. For all

the identity

holds true. Proof. Just replace .\ by (.\ - J-l) and A by J-l- A in Lemma A.2.1.

D

Proposition A.2.3. Let A be a closed multi-valued linear operator on the Banach space X. The resolvent set e(A) is an open subset of rc. More precisely, for J-l E e(A) one has dist(J-l, O"(A)) 2: IIR(J-l, A)II- 1 and 00

R('\, A) = :~:)J-l- .\)k R(J-l, A)k+l

for

1.\ - J-li

< IIR(J-l, A)II- 1 .

k=O

The resolvent mapping R(., A) : e(A) ---) £(X) is holomorphic and the resolvent identity

R(>., A) - R(J-l, A) = (J-l - .\)R(J-l, A)R('\, A) holds for all

>., J-l

E

e( A).

If A E £(X) one has 0 i= O"(A) C {z E

rc

1 Izl::; IIAII} and

00

R(>., A) = L.\ -(k+l) Ak

(1.\1

> IIAII)·

k=O

Proof. As an abbreviation we write R(.\) instead of R(>., A). Take J-l E e(A) and rc such that 1.\ - J-li < IIR(J-l)11- 1 . Then a well-known result from the theory of bounded operators states that 1 + (.\ - J-l)R(J-l) is invertible with .\ E

(1 + (.\ - J-l)R(J-l))-l = L(J-l- .\)k R(J-l)k k2:0

E £(X)

being its inverse. Combined with Corollary A.2.2 this gives R(.\) E £(X) and

R(.\) =

.\~

J-l

(1 - L(J-l- .\)k R(J-l)k) = k2:0 J-l

~.\ L(J-l- .\)k R(J-l)k k2:1

= L(J-l- .\)k R(J-l)k+l. k2:0 Let .\, J-l E e(A), and let a E X. Set x := R(.\)a. Then (x, a) E (.\ - A) and hence (x, a + (J-l- .\)x) E (J-l- A). This implies that

R(.\)a = x = R(J-l) (a + (J-l- .\)x) = (R(J-l)

+ (J-l- .\)R(J-l)R(.\))a.

The proofs of the remaining statements are well known.

o

284

Appendix A. Linear Operators If A E £(X) we call

r A := r(A) := inf{r > 0 I o-(A)

c Br(O)}

the spectral radius of A. Each mapping R : 0 identity

---->

£(X) with

0 -=I-

0

c

C such that the resolvent (>', J-l E 0)

holds, is called a pseudo-resolvent. Since we allow multi-valued operators, we obtain the following proposition, which fails to be true for single-valued operators. ----> £(X) be a pseudo-resolvent. Then there is one and only one multi-valued operator A on X such that 0 C Q(A) and R(>') = R(>', A) for all >. E O.

Proposition A.2.4. Let R : 0

Proof. If R(>.) = R(>', A), then A = >. - R(>.)-l. This shows that the operator A is uniquely determined by each single R(>.). Thus we define A,\ := >. - R(>.)-l. What we have to show is that all the A,\ are equal, i.e., that

(>., J-l EO). By interchanging the roles of J-l and>' it is clear that we are done as soon as we know one inclusion. Let (x, y) E J-l + R(>.)-l. This means that x = R(>.)a where a := Y - J-lX. Then

R(J-l)(Y - >.x) = R(J-l) (a + (J-l- >.)x) = R(J-l) (I + (J-l- >.)R(>.))a = R(>.)a = x. But this gives (x, y - >.x) E R(J-l)-l, whence (x, y) E

>. + R(J-l)-l.

o

----> £(X) and R2 : O2 ----> £(X) be pseudoresolvents. If R1(Z) = R 2(z) for some z E 0 1 n O2 , then R 1(z) = R2(Z) for all such z.

Corollary A.2.5. Let R1 : 0 1

Let T E £(X). We say that the operator T commutes with the multi-valued operator A if

(x, y) E A

(Tx, Ty) E A

===}

for all x, y E X. This is equivalent to the condition T A c AT. Obviously, T commutes with A if and only if T commutes with A -1. An immediate consequence of this fact is the following proposition.

Proposition A.2.6. Let T

E

£(X), and suppose that Q(A) -=I- 0. Then the following

assertions are equivalent.

(i) [T, R(>', A)]

:=

TR(>., A) - R(>., A)T

(ii) [T,R(>.,A)] = 0 for all

>. E Q(A).

=

0 for some>. E Q(A).

A.2. Resolvents

285

(iii) T A cAT. Let A, B be closed multi-valued operators, and suppose that e(A) #- 0. We say that B commutes with the resolvents of A if B commutes with R().., A) for each A E e(A). If e(B) #- 0, it is sufficient that this is the case for a single A. Moreover, it follows that A commutes with the resolvents of B. One should note that each single-valued operator A with e(A) #- 0 commutes with its own resolvents. (This is due to the identity B- 1Bx = x + N(B) for x E 'D(B) which is true for all operators B.) Proposition A.2.7. Let A be a multi-valued operator such that e(A) T E £(X) be injective. 1fT commutes with A, then T- 1AT = A.

#- 0,

and let

Proof. Because of AT ~ T A we have T- 1AT ~ T- 1T A = A. The converse inclusion T- 1AT c A is equivalent to the following statement: If (Tx, Ty) E A, then (x, y) E A, for all x, y E X. Let A E e(A). From (Tx, Ty) E A it follows that (Tx, T(y - AX)) = (Tx, Ty - ATx) E (A - A). This gives R(A, A)T(AX - y) = Tx. Now, T commutes with R(A, A), hence TR(A, A)(AX-Y) = Tx. By injectivity ofT we obtain R(A, A)(AX - y) = x, whence (x, y) E A. This completes the proof. D

Let A be a multi-valued operator on the Banach space X, and let Y be another Banach space, continuously embedded in X. We denote by Ay := Aly the part of A in Y, i.e., Ay := An (Y EB Y). In the case where A is single-valued this means 'D(Ay)

= {x

EY

n 'D(A)

I Ax E Y}

and

Ayy

= Ay

(y E'D(Ay)).

Proposition A.2.8. Let A be any operator on X, and let Y c X be another Banach space with continuous inclusion. Then the following assertions hold.

a) If A is closed, then also Ay is closed.

= [Ayj-1. c) A - Ay = (A - A)y holds for all A E C-

b)

[A-1jy

d)

If A E e(A), then the two assertions

(i) A E e(Ay); (ii) Y is R()..,A)-invariant; are equivalent. In this case R().., Ay) = R().., A)y. Proof. a) Suppose that A is closed. Let (Yn,zn) E AY,(Yn,zn) --+ (y,z) within Y EB Y. Since the inclusion Y c X is continuous, (Yn, zn) --+ (y, z) within X EB X, and since A is closed, it follows that (y, z) E AnY EB Y = A y . b) and c) are D trivial, and d) follows from a), b) and c).

Corollary A.2.9. Let X, Y, A be as above. If'D(A)

c

Y, then e(A)

c

e(Ay).

286

A.3

Appendix A. Linear Operators

The Spectral Mapping Theorem for the Resolvent

We define Coo := C U {oo} to be the one-point compactification of C (Riemann sphere) with the usual conventions for computation 1. For a multi-valued operator A we call the set

&(A) := { a(A) a(A)U{oo}

if A E .c(X), if A ~ .c(X),

the extended spectrum of A. Proposition A.3.1 (Spectral Mapping Theorem). Let A be a closed multi-valued operator on the Banach space X. We have

&(AA) = A&(A),

&(A + A) = &(A)

for all A E C. In particular, &(R(A, A))

+ A,

&(A- 1 ) = [&(A)r 1

and

= [A - &(A)]-l for all A E C.

This follows from Corollary A.3.3 below. Let A E Coo. The eigenspace of A at A is defined as

N(A A) := { N(A - A) , Aa

for for

A E C, A = 00.

Similarly, we define the range space of A at A as

9«A A) .= , .

{

9«A -.\) 1>(A)

for .\ for.\

E

'·00

= 00

for 0

# >. E C,

and 0

= 0·00 =

1/00.

A.3. Tbe Spectral Mapping Tbeorem for tbe Resolvent

287

Lemma A.3.2. Let p, E C and A E CcX). Then

N(oX, p, - A) = N(p, - A, A)

= N(A -1, A), ~(A, p, - A) = ~(p, - oX, A) and ~(A, A-I) = ~(A -1, A). and N(oX, A-I)

as well as

Proof. We only show the first and the last equality. The other two are proved similarly. Let y E X. In the case where A = 00, one has

Y E N(oX, p, - A) {::} Y E ~(oX,A-l) {::} since p, - A =

00

and

(0, y) E

P, -

A

{::}

3x: (y,x) E A-I {::}

II A = O.

(0, y) E A

{::}

Y E N(p, - oX, A),

3x: (x,y) E A {::}

y E ~(11 oX, A),

In the case where A -I-

00,

we have

Y E N(oX, p, - A) {::} (y,O) E (p, - A) - A {::} (y,O) E A - (p, - A) {::} Y E N(p, - oX, A). Note further that ~(O,A-l) = ~(A-l) = 1J(A) = ~(1/0,A). Hence to finish the proof of R(oX, A-I) = R(A-l, A) we may suppose that A -I- O. Then YE~(A,A-l) {::} 3x:(X,y)EA- I -A {::} 3x: (y+AX,X) EA

{::} 3x: (y

+ Ax, _A- l y)

E A - A-I {::} Y E ~(A-l, A).

0

Corollary A.3.3. Let p, E Co Then

xa-(p, - A) = p, - Xa-(A)

and Xa-(A- l )

= [Xa-(A)r l

for X E {P, A, R, S}.

The following is a characterisation of the approximate point spectrum in terms of approximate eigenvectors. Proposition A.3.4. Let A be a closed multi-valued operator on X, and let A E Co The following assertions are equivalent.

(i) A E AO"(A). (ii) There is (xn' Yn) E A-A such that

Ilxnll = 1 for all nand Yn

---; 0 as n ---;

00.

Proof. Without loss of generality we may suppose that B := A - A is injective. Since B is closed, one can consider the induced (single-valued) operator B : 1J(B) ----) XI BO. Clearly B has a closed range if and only if B has closed range. Now, 0 EEl BO c B, whence B is a closed operator B c X EEl (XI BO). The inverse B- 1 : XI BO ----) X is therefore a single-valued closed operator. By Lemma A.1.2 this operator is continuous if and only if its domain - which is ~(B) - is closed. This means that B has a closed range if and only if there is a constant c such that IIBxllx/BO 2: c Ilxll for all x E 1J(B). From this readily follows the stated equivalence. 0

288

Appendix A. Linear Operators

Corollary A.3.5. Let A be a closed multi-valued operator on the Banach space X. Then aa(A) C Aa(A).

The boundary aa(A) has to be regarded as a boundary in Coo. Proof. Let A E aa(A). Suppose first that A E C. Then there is a convergent sequence An ----+ A with An E [J(A). Since a(A) is closed, A E o-(A). By Proposition A.2.3 one has IIR(An' A)II ----+ 00. From this follows that there is a sequence (zn)n such that Ilznll = 1 and IIR(An, A)znll ----+ 00. Let Xn := IIR(An, A)znll- 1 R(An, A)zn and Wn := IIR(An' A)znll- 1 Zn· Then Ilxnll = 1, Wn ----+ 0, and (xn' w n) E An -A. Defining Yn := Wn +(A-An)Xn we obtain (xn' Yn) E A-A and Yn ----+ O. Hence A E Aa(A), by Proposition A.3.4. If A =

o E aa(A- 1 ). A.4

we use the Spectral Mapping Theorem to apply the above to Then we use Corollary A.3.3. 0

00,

Adj oint s

We denote the dual space of a Banach space X by X'. Note that there is a canonical identification of (X EB X)' and X' EB X' that we often employ without explicitly mentioning it. The canonical duality between X and X' is denoted by (.,.) : X x X' Given A

c

-+

rc.

X and B C X' we define

M.l:= {x' E Xii (X, x') = 0 "Ix EM}, NT

:=

{x E X I (x, x')

=

0 V x' EN}.

Identifying X with a closed subspace of X" we therefore have N.l n X = NT. Let A C X EB X be a multi-valued operator. The adjoint of A - usually denoted by A' - is defined by

(X',y') E A' If we let J := [(u, v)

f-----)

:{o}

(V,X') = (U,y') for all (u,v) EA.

(-v, u)] : X EB X

-+

X EB X, we may write

A' = [JA].l. It is clear from the definition that A' is always a closed operator. The following lemma is an easy consequence of the Hahn-Banach theorem. Lemma A.4.1. Let A be a multi-valued operator on the Banach space X. Then

'D(A') = {x' E X' I x' A is single-valued and continuous}, A'X = {y' E X'I y'::) x'A} (x E 'D(A')).

289

A.4. Adjoints We collect the basic properties of adjoints.

Proposition A.4.2. Let A, B be multi-valued linear operators on X. following statements hold.

Then the

a) A' = (A)' is w* -closed. b)

(A-I)' = (A')-l.

c) (AA)' = AA', for 0 i- A E Cd) A"n(XEBX) = A, where we identify XEBX canonically with a closed subspace of X" EB X".

e) :N(A')=~(A)~ and:N(A)=~(A')T.

f) A'O = 1>(A)~ and AO = 1>(A') T. g) 1>(A') c h)

and ~(A')

(AO)~

C :N(A)~.

If A E C(X), then A' E C(X') and IIA'II = IIAII.

i) A c B j) A' + B'

=}

B' c A'.

c (A + B)' with equality if A E C(X).

k) A' B' c (BA)' with equality if B E C(X). If A E C(X) and B is closed, one has A' B'w* = (BA)'.

Proof. a) Since J is a topological isomorphism, A' = (JA)~ = JA~ = (JA)~. b) This follows from (JA)-l = J(A-l) and (A-l)~ = (A~)-l. c)Wehave(x',y')E(AA), 0, ReA> wo(T), and all x E X.

and

(~x,

Dcx) E A

299

A.S. Semigroups and Generators

Proof. We compute (AR(A) - I)

1€

T(s) ds =

= = =

1 1 1

1€ 1€

00

Ae->'tT(t)

00

(Ae->.t

00

Ae->.t

1

00

1;G·1

T(s) dsdt

-1€

T(s) ds

T(t+s)ds-Ae->.t

(I t+€ ... -1€ ... )dt

1€

T(S)dS) dt

Ae->.t (1€+t ... -1t ... ) dt

OO

e->'t(T(t + c:) - T(t)) dt = (T(c:) - I)R(A).

Dividing by c: completes the proof of the first statement. Using this we obtain Vc = R(A)(AVc - D€). This shows that (Vcx, AVcX - D€x) E (A - A) for every xEX. D Corollary A.8.3. Let T be an exponentially bounded semigroup on the Banach space

X, and let x EX. The following assertions are equivalent.

(i) x

E 1>(A).

(ii) T(t)x

-+

x as t '\. O. x as c: '\. O.

(iii) ~ J;T(s)xds

-+

(iv) AR().., A)x

x as A -+

-+

00.

In particular, 1>(A) is the space of strong continuity of T. One has the inclusion T(t)X c 1>(A) for each t > o. Proof. (iv)=}(i) and (ii)=}(iii) are obvious. (iii)=}(i) follows, since ~(Vc) c 1>(A) by Proposition A.S.2. (i)=}(ii). Let x E 1>(A), and pick y E (A - A)x, where A > wo(T). Then T(c:)x - x = (T(c:) - I)R(A)y = c:V€(AR(A) - I)y as c:

-+

O. Now supt:s;1IIT(t)11
w where w > wo(T). This follows from Proposition A.S.l. Given x E 1>(A) we choose f-L E e(A) and y E (f-L - A)x to obtain

A

AR(A)X = AR(A)R(f-L)Y = -\-(R(f-L)y - R(A)y) /\-f-L as A -+

00.

-+

R(f-L)y = x D

Appendix A. Linear Operators

300 Corollary A.8.4. Let x EX. Then we have x E 1)(A), Ax n 1)(A) =I-

0

{=}

.

1

hm -(T(t)x - x) =: y

t'"O t

and in this case {y} = Axn1)(A). In particular, AOn1)(A) X T := AO EB 1)(A) is a closed subspace of x.

= 0,

exists, whence the space

It follows from Corollary A.8.4 that the part B := An (1)(A) EB 1)(A)) of A in 1)(A) is single-valued.

Proposition A.8.5. Let T be an exponentially bounded semigroup with generator A on the Banach space X. Let Y := 1)(A). The space Y is invariant under the semigroup T. The semigroup T restricts to a strongly continuous semigroup on Y, with generator B = A n (Y EB Y) . The following result is one of the cornerstones of the theory of Co-semigroups. (See [85, Section 11.3] or [10, Theorem 3.3.4] for proofs.) Theorem A.8.6 (Hille-Yosida). Let A be a (single-valued) linear operator on the Banach space X. Suppose that A has dense domain and there are M 2: 1, w E lR such that (w, A) C Q(A) and IIR(>', Atll

::; (>. ~w)n

(A.4)

for all n E N and all >. > w. Then A generates a Co -semigroup satisfying the estimate IIT(t)11 ::; Me wt for t 2: 0.

Remark A.8.7. Unfortunately there is no similar characterisation for generators of general exponentially bounded semigroups. The resolvent condition (A.3) guarantees that A generates a so-called integrated semigroup SO, see [10, Theorem 3.3.1]. For x E 1)(A) one has S(t)x = J~ T(s)xds, where T is the semigroup generated by the part B of A in 1)(A). Then A generates an exponentially bounded semigroup if and only if SOx E C 1 ((0, (0), X) for each x E X. Employing [7, Theorem 6.2] one can show that this is always true if the Banach space X has the Radon-Nikodym property. Let us note an important result on bounded perturbations. For a proof see [85, Theorem II1.1.3]. Proposition A.8.8. Let A be the generator of a Co-semigroup on the Banach space X, and let BE £(X) be a bounded operator. Then A+B generates a Co-semigroup onX. Finally we deal with the important case of groups. Proposition A.8.9. Let T be an exponentially bounded semigroup with generator A. The following assertions are equivalent:

A.S. Semigroups and Generators

301

(i) There exists to > 0 such that T(to) is invertible. (ii) Each T(t) is invertible and the mapping T : IR --+ £(xy, defined by T(t)

T(t) := { I T( _t)-l

(t > 0), (t = 0), (t < 0)

is a strongly continuous group homomorphism.

(iii) The operator -A generates an exponentially bounded semigroup and A is single-valued. Proof. (i)=>(ii)=>(iii). Let T(t o) be invertible. Then the semigroup property readily implies that each T(t) is invertible. Moreover, since J((T(t)) c 2:l(A) for each t > 0, we have that A is densely defined. Hence A is single-valued and we are in the standard (Co-) case. So we can refer to [85, Subsection 3.11], or [186, Section 1.6] for the remaining arguments. (iii)=>(i). We denote by S the semigroup generated by -A. Choose a number w > max(wo(T), wo(S)). Let Y := 2:l(A), and let B := An (Y EEl Y) the part of A in Y. From Proposition A.8.5 we know that B generates the Co-semigroup obtained by restricting T to Y. Analogously, -B generates the Co-semigroup obtained by restricting S to Y. (Note that 2:l(-A) = 2:l(A).) The theorem from [85, Section 3.11] now yields that T(t)S(t)y = S(t)T(t)y = y for all y E Y. Let to > 0, and suppose that T(to)x = 0 for some x EX. Then

to

R().., A)x = io e-AtT(t)x dt

=:

f(>.)

for Re>. > w. Obviously f has a holomorphic continuation to an entire function which is bounded on every right half-plane. Claim: f(>.)

= R(>', A) for all Re>. < -w.

Proof of Claim. Consider the function F : C --+ X EB X/A defined by F(>') .(f(>.) , >.f(>.)-x)+A. Then F is entire and F(>.) = 0 for Re>. > w, since (f(>.) , x) E >. - A for these >.. By the uniqueness theorem for holomorphic functions, F(>') = 0 for all >. E C. Hence (f(>.) , x) E >. - A even for Re>. < -w. However, these>. are contained in the resolvent set of A, whence the claim is proved. From the claim we conclude that f is in fact bounded also on some left halfplane, hence it is constant. However, f(>.) --+ 0 as Re>. --+ 00. Thus, f(>.) = 0 for all >. E c. This implies that R(>', A)x = 0 for all >. in some left halfplane. Since A is single-valued, x = o. So we have shown that T(to) is injective. We obtain that T(to) : X ~ Y is an isomorphism. But since T(to) : Y ~ Y is also an isomorphism, we must have X=Y. 0

Appendix A. Linear Operators

302

One usually writes T again instead of T and calls it a Co-group. In general one cannot omit the assumption 'A is single-valued' from (iii). Indeed, let S be a Co-group on a Banach space Y and let X := Y EB C with T(t)(y, A) := (S(t),O) for all t E R If B generates S, then

A = {((y, 0), (By, A)) lyE 'D(B), A E C} generates (T(t))t?o and -A generates (T( -t))t?o. Given a Co-group (T(t))tElR we call

B(T):= inf{B > 0 I 3M 2:: 1: IIT(t)ll::::; Me Oltl

(t E JR.)}

the group type of T. Let us call the semigroups Tffi and Te , defined by

Tffi(t)

:=

T(t)

and Te(t):= T( -t)

(t 2:: 0),

the forward semigroup and the backward semigroup corresponding to the group T. Then we obviously have B(T) = max{wo(Tffi), wo(Te)}.

References The basic results on multi-valued operators, covered by Section A.l and Section A.2, are contained, e.g., in [89, Chapter I] or the monograph [52]. Proposition A.2.4 and its corollary on pseudo-resolvents is not included in these books and may be new. The same is true for Propositions A.2.7 and parts of Section A.3. An exhaustive treatment of adjoints of multi-valued operators can be found in [52, Chapter III]. The results on convergence in Section A.5 are generalisations of wellknown facts for single-valued operators that can be found, e.g., in [130, Theorem IV.2.23 and Chapter VIII,§I]. Polynomials of operators (Section A.6) are studied in [79, Chapter VII.9] including the spectral mapping theorem (Proposition A.6.2). By [52, Theorem VI.5.4], the spectral mapping theorem for polynomials remains valid for multi-valued operators. We provided the results for rational functions and on injective operators (Section A.7) without direct reference, but it is quite likely that these facts have been published somewhere. The results of [52, Chapter 6] let us expect that there is a generalisation of Sections A.6 and A.7 to multivalued operators. Section A.8 is an adaptation from the standard monographs in semigroup theory, like [186], [85], and [10]. However, if multi-valued operators are involved (as in Proposition A.8.2 and Proposition A.8.9), we do not know of a direct reference. Semigroups in connection with multi-valued operators are studied in [26] and [25].

Appendix B

Interpolation Spaces In this appendix we present a short survey of basic definitions and results from the interpolation theory of Banach spaces.

B.I

Interpolation Couples

Let X, Y be two normed spaces. A closed linear (single-valued) and injective operator ~ C X EEl Y is called an (interpolation) coupling between X and Y. A pair of normed spaces (X, Y) together with an interpolation coupling ~ is called an interpolation couple. If (X, Y) is an interpolation couple and both X, Yare Banach spaces, then (X, Y) is called a Banach couple. Lemma B.l.l. Let X, Y be normed spaces. If there is a Hausdorff topological vector space Z and continuous linear injections ~x : X ---+ Z and ~y : Y ---+ Z, then (X, Y) is an interpolation couple with respect to the coupling ~ defined by

(x, y) E

~

{==}

~x(x)

=

~y(y)

(B.l)

for x E X, Y E Y. Conversely, if ~ is some coupling between X and Y, then there is a normed space Z and continuous injections ~x : X ---+ Z and ~y : Y ---+ Z such that (B.l) holds. Proof The first assertion is straightforward to prove. Suppose that ~ is some coupling between X and Y. Then the operator -~ is a closed subspace of the normed space X EEl Y. Define the normed space Z as the quotient space Z := X EEl Y / - ~ and denote by [x, y] the image of (x, y) under the canonical projection of X EEl Y onto Z. Then let ~x := (x f----+ [x, 0]) : X ---+ Z and ~y := (y f----+ [0, y]) : Y ---+ Z. Clearly, ~x and ~y are linear and continuous. Moreover, they are injective. To see this, consider x E X such that ~x(x) = 0. This means that (x,O) E (-~). Since ~ is injective, x = 0. The injectivity of ~y follows similarly from the single-valuedness of L Now, for x E X, Y E Y, we have ~x(x) = ~y(y) {==} [x,O] = [O,y] {==} (x,-y) E (-~) {==} (x,y) E L This proves (B.l). D

Appendix B. Interpolation Spaces

304

Given an interpolation couple (X, Y) (with coupling L) the normed space X

+ Y := (X EB Y) /( -L)

defined in the proof of Lemma B.l.l is called the sum of the couple (X, Y). Furthermore, the space xnY:= 'D(L) with the norm Ilxllx + IIL(X) Ily is called the intersection of the couple (X, Y). If the coupling L is obtained by (B.l) based on imbeddings LX, Ly into some Hausdorff topological vector space Z, then X n Y and X + Y embed canonically in Z. In fact, the most common situation is when X and Yare already subspaces of some space Z with continuous inclusion mappings. Then the coupling is natural and reference to it is often omitted. Also, the spaces X n Y and X + Y appear naturally as subspaces of Z. Viewing X and Y as subspaces of X + Y, one has Ilxllx+y = inf{llall x + Ilbll y I a E X, bEY, a + b = x},

(B.2)

Ilzllxny = Ilzllx + Ilzlly

(B.3)

for the norms on X + Y and X nY, respectively. If both X and Yare complete, i.e., if (X, Y) is a Banach couple, then X and X n Yare also complete.

+Y

The interpolation couples form the objects of a category whose morphisms are the pairs (TI' T2) of continuous linear mappings

such that LoTI = T2 0 an operator

L.

This condition ensures that one can unambiguously define

T:X+Y----+X+Y with Tlx = Tl and Tly = T2· One then has T E ,c(X + Y, X + Y) and Tlxny E 'c(X n Y,X n Y). Hence sum and intersection are (covariant) functors from the category of interpolation couples into the category of normed spaces. Usually notation is abused a little and one simply writes T instead of T 1 , T2. Lemma B.1.2. Let Y

c X, Y c X, and let T

E

'c(X, X). IfT(Y) c Y, then

T E 'c(Y, Y).

Proof. This is an easy application of the Closed Graph Theorem. By Lemma B.l.2, a morphism T : (X, Y) by the following properties. a) T E 'c(X

+ Y,X' + Y')

b) T(X) eX', T(Y)

c y'.

and

----+

D

(X, Y) is already characterised

305

B.2. Real Interpolation by the K-Method

A functor F from the category of interpolation couples into the category of normed spaces is called an interpolation functor if the following assertions hold. 1) X nYc F(X, Y) eX

+Y

for all interpolation couples (X, Y).

2) F(T) = TIF(x,y) for every morphism T : (X, Y) ----) couples.

(X, Y) of interpolation

3) If (X, Y) is a Banach couple, then also F(X, Y) is a Banach space.

B.2

Real Interpolation by the K-Method

Let (X, Y) be an interpolation couple. We define

K(t, x) := K(t, x, X, Y) := inf{llall x for x E X

+ t llbll y

I x = a + b, a E X, bEY}

+ Y and t > O.

Lemma B.2.1. The following assertions hold true. For every t > 0 the mapping x More precisely,

a)

min(t, 1) llxllx+y

f----+

K(t, x) is an equivalent norm on X

+ Y.

:s: K(t, x) :s: max(t, 1) Ilxllx+Y

+ Y. For fixed x E X +Y

for all x E X

b)

the mapping t in particular, it is continuous.

f----+

K (t, x) is non-decreasing and concave;

c) K(t, x, X, Y) = tK(l/t, x, Y, X) for all x

X

+ Y, t > O.

d)

K(t,x):S: (t/s)K(s,x) for 0 < s (l/t)K(t, x) is non-increasing.

e)

For x E X

f)

IfY C X, then K(t,x):S: Ilxllx for aUt> 0, x E X.

nY

one has K(t, x)


o.

= tK(I/t, x, Y, X), which holds

b) By a) it suffices to consider the case () = o. So suppose that K(.,x) E L~. Because Ilxllx+Y :::; K(t, x) for t 2: 1, we have Ilxll~+Y dtlt < 00. This implies that Ilxllx+Y = O. c) In the case where p = 00 one can choose c = 1, by definition. Now suppose 1 :::; p < 00. If () E {O, I}, then by b) every c does the job. Hence we can suppose E (0,1) and estimate

It

e

COK(t,x) = (()p)'P1

(1 t

00

dS) ~ K(t,x):S (()p)'P (l

s-OP---;

1

t

DO

dS) ~

s-OPK(s,x)P---;

1

:::; (()p)'P Ilxllo,p for x E (X, Y)o,p. d) We let t = 1 in c) and obtain Ilxllx+Y :S c((),p) Ilxllo,p. For x E X n Y we have K(t, x) :S min(l, t) Ilxllxny. Now the assertion follows from the implication p

= 00 V () E (0,1)

===}

CO min(l, t) E L~(O, 00).

e) Suppose first that x E (X, Y)o,oo, i.e., that there is c > 0 such that K(t,x) < c for all t > o. For each n E N choose x = xn+Yn EX +Y with Ilxnllx+n IIYnlly < c. Then SUPn Ilxnllx < c and Ilx - xnllx+Y :::; Ilylly :::; Cn ----+ o. On the other hand, if (ii) holds, then K(t,x)n:::; Ilxnllx and K(t,x n ) ----+ K(t,x), since each K(t,·) is a

307

B.2. Real Interpolation by the K-Method norm on X + Y. Hence (i) follows. f) is similar to the proof of part e).

D

The next result shows in particular that the mapping

(X, Y)

(X, Y)e,p

f-----+

is an interpolation functor whenever p E [1,00]' () E [0,1]' except for the case where p < 00 and () E {O, I}.

Theorem B.2.3. Let (X, Y) and assertions hold.

(X, Y)

be interpolation couples. The following

a) If (X, Y) is a Banach couple, then (X, Y)e,p is a Banach space for all p E

[1,00], ()

E

[0,1].

b) One has always (X, Y)e,p

C

(X, Y)e,q

for 1'5: p '5: q '5: 00 and () E [0,1].

c) Let T : (X, Y)

(X, Y)

be a morphism of interpolation couples. Then it restricts to a bounded linear mapping T : (X, Y)e,p ---+ (X, Y)e,p such that ---+

IITII(X,Y)e,p-+(X,Y)e,p

'5: IITII~-~x IITII~-+Y

for all p E [1,00] and () E [0,1]. Proof. a) is [157, p. 9/lD].

b) By Proposition B.2.2 c) we have (X, Y)e,p C (X, Y)e,oo, whence Ilxlle,oo c Ilxlle,p for all x E (X, Y)e,p' If p < q < 00, then Ilxlle,q=

dt) i ( JorOO reqK(t,x)qT

=

'5:

(rOO dt) i J rePK(t,x)p(reK(t,x))(q-P)T o

1

'5: ( roo rep K(t, x)p dt) q sup(re K(t, x)) q~p = Ilxlllp Ilxll;~ '5: c Ilxll e p' Jo t t>O " , c) is [157, p. 10].

D

Corollary B.2.4. Let (X, Y) be an interpolation couple, p E [1,00] and () E (0,1). Then there is c = c(p, (}) such that

(x E X nY). Proof. Take X =

X, Y = Y and T

= I in Theorem B.2.3 c).

D

Appendix B. Interpolation Spaces

308

Remark B.2.5. Let us briefly consider the special case Y c X. Then clearly (X, Y) is an interpolation couple. Since in this case K (t, x) is bounded, we certainly have

for all () E (0,1] and all p E [1,00] (and also for () = only has to check the condition at 0, and

°

and p = 00). Hence one

lire K(t, x) IIL~(o,l) is an equivalent norm on (X, Y)e,p (in the relevant cases). A similar remark holds in the case that X c Y, where only the behaviour of K (t, x) at 00 is relevant, and so IlreK(t,x)IIL~(l,oo) is an equivalent norm on (X, Y)e,p (in the relevant cases). Proposition B.2.6. Let Y eX, and let

Y c (X, Yh,oo c (X, Y)".,oo

°< () < C

(J

< 1. Then

(X, Y)e,l c (X, Y)o,oo =

x.

Proof. This is [157, p. 9].

D

The following result is often useful. Proposition B.2.7. Let (X, Y) be a Banach couple, and let () E [0, l],p E [1,00]. Then the following identities hold.

(X

+ Y, Y)e,p n X = (X, Y)e,p n X = (X + Y, X)e,p n (X + Y, Y)e,p = (X, X n Y)e,p + (Y, X n Y)e,p = (X + Y, Xh-e,p n (X + Y, Y)e,p = (X, X n Y)e,p + (Y, X n Yh-e,p =

(X, X

n Y)e,p, (X + Y, X n Y)e,p, (X + Y, X n Y)e,p,

(B.4) (B.5) (B.6)

(X, Y)e,p, (X, Y)e,p.

(B.7) (B.8)

Proof. In (B.4) the chain of inclusions ':J' holds trivially. Let x E (X +Y, Y)e,pnx. Then K(t, x, X, XnY) S; Ilxllx, and we only have to account for the case 0 < t S; l. Let x = a + b + c, where a E X and b, c E Y. Then b + c = x - a E X nY, whence

Ilallx + t lib + cll xny = Ilall x + t lib + cll y + t Ilx - all x S; Ilall x + Ilbll y + t Ilxll y + t Ilxll x + Ilallx S; 2 Ulallx + Ilblly + t Ilcll y] + t Ilxllx·

K(t, x, X, X n Y) S;

Taking the infimum first with respect to a, b (with c fixed) and afterwards with respect to c yields K(t, x, X, X n Y) S; 2K(t, x, X + Y, Y) + t Ilxllx. For the remaining arguments we are in need of the so-called 'modular law': B c C

==?

(A

+ B) n C = (A n C) + B,

309

B.2. Real Interpolation by tbe K-Metbod

which holds for all subspaces A, B, C of a common superspace. The two identities (B.5) and (B.6) are proved by using (B.4) and the modular law (several times):

(X

+ Y,X n Y)e,p C (X + Y, X)e,p n (X + Y, Y)e,p = [(X + Y, X)e,p n (X + Y)] n [(X + Y, Y)e,p n (X + Y)] = {X + [(X + Y, X)e,p n Yl} n {Y + [(X + Y, Y)e,p n Xl} = {X + (Y,X n Y)e,p} n {Y + (X,X n Y)e,p} = {X n [Y + (X, X n Y)e,pl} + (Y, X n Y)e,p = (X n Y) + (X,X n Y)e,p + (Y,X n Y)e,p = (X, X n Y)e,p + (Y, X n Y)e,p C (X + Y, X n Y)e,p.

This kind of proof also works for the last two identities (B.7) and (B.8):

n (X + Y, Y)e,p = [(X, X + Y)e,p n (X + Y)] n [(X + Y, Y)e,p n (X + Y)] = [X + ((X, X + Y)e,p n Y)] n [Y + ((X + Y, Y)e,p n X)] = [X + (X nY, Y)e,p] n [Y + (X, X n Y)e,p]

(X,Y)e,p C (X, X + Y)e,p

= (X n Y, Y)e,p + [X n (Y + (X, X n Y)e,p)] = (X n Y, Y)e,p + [(X n Y)e,p + (X, X n Y)e,p] C

(X

n Y, Y)e,p + (X, X n Y)e,p

C

(X, Y)e,p'

D

The Reiteration Theorem Reiteration is one of the most important techniques, making interpolation theory the powerful tool that it is. Let (X, Y) be a Banach couple, and let E be an intermediate space, i.e., a space for which the continuous embeddings X nYc E c X + Y hold. Fix () E (0,1). We say that E is of class Jo in (X, Y) if there is c ~ 0 such that

(x E X n Y). We write E E Je(X, Y) if this holds. Lemma B.2.8. Let X nYc E C X + Y, and let () E [0,1]. Then the following assertions are equivalent.

(i) E E Je(X, Y). (ii) (X, Y)e,l C E. Proof. See [158, p. 28].

D

Appendix B. Interpolation Spaces

310

Note that for () E {O, I} the characterisation given in Lemma B.2.8 must fail, since in this case (X, Y)e,l = 0. Fix () E [0,1]. We say that E is of class Ke in (X, Y) if there is a k ::::: Osuch that

K(t,x) ~ kt e IlxilE

(x

E E,

t> 0).

°

One writes E E Ke(X, Y) if this holds. In the case where < () < 1 one has E E Ke(X, Y) if, and only if E c (X, Y)e,oo. Let (X, Y) be a Banach couple, and let X nYc E l , E2 c X + Y be two intermediate spaces. Then (Eo, Ed is another interpolation couple. The reiteration theorem deals with this situation.

Theorem B.2.9 (Reiteration Theorem). Let (X, Y) be a Banach couple, ()l ~ 1, and () E [0,1].

°

~ ()o


0, A E JR, a E X

n Y}

is total in Fo(X, Y). Proof. This in fact needs an intricate argument a more detailed account of which can be found in [138, Theorem IV.l.1] or [29, Lemma 4.2.3]. We only sketch the proof. Note that leoz2+AZ I ::; ce- ol1m zI2 for c = e H1A1 . Hence the considered set is in fact contained in Fo(X, Y) and consists of very rapidly decreasing functions. Let f E Fo be arbitrary. One has to approximate f by finite sums of elements of M. The proof of this fact consists of several steps.

1) One has eEz2 f ----* f in Fo as c\.O for each f E Fo· 2) By Step 1) one may suppose without loss of generality that f decays rapidly. For such f consider fn(z) := I:kEZ f(z+i(27rkn)). (The sum converges absolutely, uniformly for Z in compacts of S.) This yields a 27rni-periodic function fn E F. 3) Let bnk(s) be the k-th Fourier coefficient of the 27rn-periodic function fn(s+i· ). An intricate argument using Cauchy's theorem shows that ank(s) := e-sk/nbnk(s) is independent of s E (0,1). By continuity, ank(O) = ank(1) EX n Y. 4) Fejer's theorem yields that fn(z) can be (uniformly in z) approximated by sums ofterms of the form anke kz / n . Multiplying by eoz2 shows that each function eoz2 fn is in the closed span of M. 5) Finally, fn ----* f uniformly on compacts, hence with t5 > 0, n E N to be chosen appropriately, IIe oz2 fn - filF can be made arbitrarly small. 0

Appendix B. Interpolation Spaces

312

We are now in a position to define the complex interpolation spaces. Let

(X, Y) be an interpolation couple, and let () E [0,1]. We define [X, Y]o := {f(())

If

E F(X, Y)}

with the norm

Ilxll o := inf{llfll F If

f(())

E F(X, Y),

=

x}.

In passing to functions of the form eo(z-O)2 f one can see that in the definition of [X, Y]o one can replace the space F by the space Fo. We obtain the following theorem.

Theorem B.3.3. Let (X, Y) and assertions hold. a)

(X, f)

be interpolation couples. The following

The space [X, Y]o is an intermediate space, i.e., X nYc [X, Y]o

c

X

+ Y.

n Y is dense in [X, Y]o for each ()

b)

The space X

c)

If(X,Y) is a Banach couple, then [X,Y]o is aBanachspaceforall() E [0,1].

E (0,1).

----> (X, f) be a morphism of interpolation couples. Then it restricts to a bounded linear mapping T : [X, Y]o ----> [X, flo such that

d) Let T : (X, Y)

for all () E (0,1).

Proof. a) The first inclusion is clear by considering constant functions. The second follows from the Maximum Modulus Principle. b) is immediate from Lemma B.3.2. c) follows from abstract nonsense, since [X, Y]o can be written as the factor space F(X, Y)/ N, where N is the kernel of the evaluation mapping f f----) f(()).

o

d) is straightforward.

Proposition B.3.4. Let (X, Y) be a Banach couple. Define V(X, Y) := span{if ® x I if E Fo(C,q, x E X Then

Ilall[X,Yle = inf{llfIIF(x,y) for any a E X n Y, () E (0,1).

I f E V(X, Y)}

n Y}

313

B.3. Complex Interpolation

Proof. We reproduce the proof from [158, Remark 2.1.5]. Take any g E Fo(X, Y) such that f((}) = a. Then define r(z)

z-(}

:= --()'

and

z+

h(z):=

g(z) - e(z-IJ)2 a r(z) .

Then Ir(z)1 ::; 1 for z E S with Ir(z)1 -+ 1 as Izl -+ 00. Hence h E Fo(X, Y). By Lemma B.3.2 one can find a function k E M such that Ilk - hIIF(x,y) is arbitrarily small. Now define

f(z)

:= e(z-IJ)2 a +

r(z)k(z),

which is obviously contained in V(X, Y) and satisfies f((}) = a. Moreover,

Ilg -

fIIF(x,y)

= Ilrh + e(z-IJ)2 a - fIIF(x,y) = Ilr(h - k)IIF(X,y) :s;

Ilh -

kIIF(x,y) . D

Now the statement follows.

An important feature of the complex interpolation spaces is their connection with the real interpolation spaces. Proposition B.3.5. Let (X, Y) be an interpolation couple, and let () E [0,1]. Then

[X, Y]IJ is of the class K((}) and of the class J((}), i.e., the inclusion (X, Y)IJ,1

C

[X, Y]IJ

C

(X, Y)IJ,oo

holds true. Proof. Choose X = X, Y = Y and T = I in c) of Theorem B.3.3. By definition, this gives [X, Y]IJ E JIJ(X, Y). For the proof of the second inclusion we refer to [158, Proposition 2.1.10]. The key step is, by applying the conformal mapping e7riz and the Poisson kernel for the half-plane, to write each f E F(X, Y) can be written on the open strip S as a sum f = fo + h, where fo, h are given by integrals of f on the axes Re z = 0 and Re z = 1, respectively (see [158, Lemma D 2.1.9] or [138, Ch. IV.1.2]). Let us remark that there is also a reiteration theorem for the complex method, see [29] or [138].

References Our main references are [157] and [158]. Other expositions are [29], [138] and [215]. We included proofs for the convenience of the reader, omitting those which seem simple or at least straightforward. The Reiteration Theorem B.2.9 is used in the proof of Dore's Theorem 6.1.3, but later we give an alternative proof without employing the Reiteration Theorem (see Corollary 6.5.8). Proposition B.2.7 may be new for some readers, and in fact it is not contained in the mentioned textbooks. However, one can find more results of this kind in [111].

Appendix C

Operator Theory on Hilbert Spaces This chapter provides some facts on linear operators on Hilbert spaces, including adjoints (of multi-valued operators), numerical range, symmetric and accretive operators, and the Lax-Milgram theorem. The main difference from standard texts lies in the fact that we have avoided employing the Spectral Theorem in dealing with spectral theory of self-adjoint operators (Proposition C.4.3 - Corollary C.4.6). We take for granted the basic Hilbert space theory as can be found in [49, Chapter I], [194, Chapter II], or [196, Chapter 4]. During the whole chapter the letter H denotes some complex Hilbert space. The scalar product on H is denoted by ('1')'

C.l

Sesquilinear Forms

Let V be a vector space over the field of complex numbers. We denote by Ses(V) := {a

1

a :V x V

-----t

C sesquilinear}

the space of sesquilinear forms on V. Given a E Ses(V), the adjoint form a is defined by a(u,v) := a(v,u) (u,v E V). The real part and the imaginary part of the form a E Ses(V) are defined by 1

Rea := 2(a + a) respectively. Hence a notation

and

1

Ima := -(a - a) 2i '

= (Rea)+i(Ima) for every a E Ses(V). Using the shorthand a(u) := a(u,u)

for a E Ses(V) and u E V, we obtain

(Rea)(u) = Re(a(u))

(u

E

V).

Appendix C. Operator Theory on Hilbert Spaces

316

Given a E Ses(V) one has the equations 1

2(a(u + v) + a(u - v)) = a(u) + a(v), a(u, v) =

~(a(u + v) -

(C.1)

a(u - v) + i(a(u + iv) - a(u - iv)));

(C.2)

these are called parallelogram law and polarisation identity, respectively. (The importance of the polarisation identity lies in the consequence that each sesquilinear form a is already determined by the associated quadratic form a(·).) A form a is called real if a( u) E ~ for all u E V. It is called symmetric if a = a. Note that Re a and 1m a are always symmetric forms. A sesquilinear form a E Ses(V) is called positive (or monotone) if Re a( u) 2: 0 for all u E V. Lemma C.l.l. A form a E Ses(V) is real if and only if it is symmetric. Proof. Obviously, a symmetric form is real. Let a be a real form. We have 1

a(u, v) + a(v, u) = 2(a(u + v) - a(u - v))

E ~

for all u, v E V. Replacing u and v by iu and iv, respectively, we obtain also

ia(u, v) - ia(v, u)

E ~

for all u, v E V. Combining these facts yields Im(a(u, v)) = - Im(a(v, u)) and Re(a(u, v)) = Re(a(v,u)). But this is nothing else than a(u,v) = a(v,u). D Proposition C.1.2 (Generalised Cauchy-Schwarz Inequality). Let a, bE Ses(V) be symmetric, and let c 2: 0 such that

la(u)1 ::::; cb(u)

(u

E

V).

(Note that this implies that b is positive.) Then la(u,v)1 ::::; cy'b(u) y'b(v) for all u,v E V. Proof. The simple proof can be found in [199, Chapter XII, Lemma 3.1].

D

A positive form a is sometimes called a semi-scalar product. It is called a scalar product if it is even positive definite, i.e., if it is positive and if a( u) = 0 implies that u = 0 for each u E V. If a E Ses(V) is a semi-scalar product on V, then by

Ilxll a

:=

y'a(u)

a seminorm on V is defined. The form a is continuous with respect to this seminorm. (This follows from the Cauchy-Schwarz inequality.)

317

0.2. Adjoint Operators

Let w E [0, 7r /2). A form a E Ses(V) is called sectorial of angle w if a(u)

-I- 0

::::} larga(u)1 5. w

(u

E V).

The form a is called sectorial if it is sectorial of some angle w < if a is sectorial, then Re a is positive.

7r

/2.

Obviously,

Proposition C.1.3. Let a E Ses(V) such that Re a ?: O. The following assertions are equivalent.

(i) The form a is sectorial. (ii) The form a is continuous with respect to the seminorm induced by the semiscalar product Re a. More precisely: If la(u, v)1 5. M JRea(u) JRea(v) for all u, v E V, then a is sectorial of angle arccos M- 1 . Conversely, if a is sectorial of angle w, then la(u, v)1 5. (1

+ tanw) JRea(u) JRea(v)

for all u,v E V. Proof. Suppose that la(u,v)1 5. M JRea(u) JRea(v) for all u,v E V. Then, letting u = v, we have la(u)1 5. MRea(u), whence M ?: 1 and a(u) -I- 0 ::::} larga(u)1 5. arccosM- 1 . for all u E V. Conversely, if a is sectorial of angle w < 7r /2, then 11m a( u) I 5. (tan w) Re a( u) for all u E V. The generalised Cauchy-Schwarz inequality, applied to Rea and Ima, yields la(u,v)1 5. I(Rea)(u,v)1

+ I(Ima)(u,v)l5. (1 +tanw)JRea(u)JRea(v)

for all u,v E V.

C.2

D

Adjoint Operators

In this section we provide the theory of Hilbert space adjoints of multi-valued

linear operators. The results are more or less the same as for Banach space adjoints (cf. Section A.4). We use different notations to distinguish between Banach space adjoints (A') and Hilbert space adjoints (A*). Let A c H EB H be a multi-valued linear operator on H. The Hilbert space adjoint of A, usually denoted by A *, is defined by (x,Y)EA*

:~

(vlx)=(uIY) for all (u,v) EA.

If for the moment we define J := ((u,v) we may write

f-----t

(-v,u)) : H EB H

---+

(C.3) H EB H, then

A* = [JAl~

where the orthogonal complement is taken in the Hilbert space H EB H. Hence A* is always a closed operator. We list the basic properties.

Appendix C. Operator Theory on Hilbert Spaces

318

Proposition C.2.1. Let A, B be multi-valued linear operators on H. following statements hold.

Then the

a) A* = (:4)*.

b)

(A-l)* = (A*)-l.

c) (AA) * = ~A *, for 0 # A E Cd) A** = A.

e) N(A*) = ~(A)~ and N(A) = ~(A*)~.

f) A*O =

1)(A)~

g) 1)(A*) c h) If A

E

and AO =

(AO)~

and

1)(A*)~.

~(A*)

C(H), then A*

E

c

N(A)~.

C(H) and

IIA*II = IIAII = JiTA*Ajf.

i) ACB

:::}

j) A* + B*

c (A + B)* with equality if A E C(H).

B*cA*.

k) A* B* c (BA)* with equality if BE C(H). If A E C(H) and B is closed, one has A* B* = (BA)*.

Proof. a) Since J is a topological isomorphism, A* = (JA)~ b) follows from (JA)-l = J(A-l) and (A-l)~ = (A~)-l.

= JA~ = (JA)~.

c) We have

(x,y)E(AA)*

d) A**

=

(J(JA)~)~

{o}

(-Avlx)+(uly)=O\i(u,V)EA

{o}

(-v I ~x)

{o}

(~x,

=

y)

E

(JJA)~~

+ (u I y) = 0 \i (u, v) A*

{o}

E A

(~x, ~y) E ~A*

{o}

(x, y)

E

A*.

= A~~ = A.

e) We have

xEN(A*)

{o}

(x,O)EA*

{o}

(-vIX)=O\ivE~(A)

{o}

xE~(A)~.

The second satement follows from the first together with d). f) We have y E A*O {o} (0, y) E A* {o} (u I y) = 0 \iu E 1)(A) The second statement follows from the first together with d). g) If (x,y) E A*, v E AO, and u E N(A), then (-vlx)

{o}

Y

E 1)(A)~.

= 0 and (uIY) = 0 by

(C.3). h) Let A E C(H). Then A* is closed and single-valued by f). We show that 1)(A*) = H. In fact, let x E H. Then (u f-----+ (Aulx)) is a continuous linear functional on H. By the Riesz-Fn§chet theorem [196, Theorem 4.12] there is y E H such that (Au Ix) = (u I y) for all u E H. But this means exactly that (x,y) E A*. The equation I All = IIA*II is easily proved by using the identity liT I = sup{I(Tulv)1 I Ilull = Ilvll = I}, which holds for every bounded operator

C.2. Adjoint Operators

319

on H. This implies that IIA* All ~ IIA* IIIIAII = IIAf But if x E H is arbitrary, we have IIAxl12 = (AxIAx) = (A*Axlx) ~ IIA*Allllxl1 2by the Cauchy-Schwarz inequality. Hence IIAI12 ~ IIA* All· i). We have A c B =} JA c JB =} (JB).l.. c (JA).l...

j) Let (x, y) E A*, (x, z) E B*. The generic element of J(A + B) is (-v - w, u), where (u,v) E A and (u,w) E B. So (x,y).l (-v,u) and (x,z).l (-w,u), hence (x,y+z).l (-v-w,u). If A E C(H), we write B = (A + B) - A and note that A* E C(H) by h). k) Let (x, y) E A* B*. Then there is z such that (x, z) E B* and (z, y) E A*. If (u, v) E BA, one has (u, w) E A and (w, v) E B for some w. Hence (v I x) = (wlz) = (uly). Since (u,v) E BA was arbitrary, (x,y) E (BA)*. Suppose now that B E C(H) and (x, y) E (BA)*. Define z := B*x. It suffices to show that (z,y) E A*. Take (u,w) E A and define v:= Bw. Hence (u,v) E BA. Therefore

( u Iy) = (v Ix) = (Bw Ix) = (w IB* x) = (w Iz) , whence (z, y) E A* by (C.3). Finally, suppose that B is closed and A E C(H). The assertions already proved yield (A* B*)* = B** A** = (B)(JI) = BA. Hence A*B* = (BA)*. 0 Corollary C.2.2. Let A be a multi-valued linear operator on H. Then

(,\ - A)* = (~- A*) for every ,\

E C.

and R(,\, A)* = R(5., A*)

In particular, we have e(A*)

= {~

I ,\ E e(A)} =

e(A).

Let A be a single-valued operator on H. From Proposition C.2.1 f) we see that A* is densely defined if and only if A is closable, i.e., A is still single-valued; and A* is single-valued if and only if A is densely defined. Proposition C.2.3. Let A be a densely defined (single-valued) operator on H with e(A) -=I- 0. For a polynomial p E qz] define p*(z) := p(z). (Hence p* is obtained

from p by conjugating all coefficients.) Then we have [P*(A)]* = p(A*). The same statement holds if p is a rational function with poles inside e( A *). Proof. Let r = p/q be a rational function with poles inside the set e(A*). Suppose first that degp ~ degq. Hence r(A*) and r*(A) are bounded operators. The function r may be written as a product r = TI j rj where each rj is either of the form a('\ - z)-l or of the form a('\ - Z)-l + (3. Now the claimed formula [r*(A)]* = r(A*) follows from Proposition C.2.1 k) and Corollary C.2.2. From this and Proposition C.2.1 b) we can infer that p*(A)* = p(A*) holds for all polynomials p having their roots inside Q( A *). Now suppose that deg p > deg q.

Appendix C. Operator Tbeory on Hilbert Spaces

320 Since [J(A) i=-

degq + degq1

0, =

we can find a polynomial q1 having its roots inside [J(A*) with f:= p/(q1q). Then

degp. Define

r(A*) = q1(A*)f(A*) = [q~(A)]*[r*(A)]* Q) [r*(A)q~(A)]*

\;! [q~(A)f*(A)]* = [r*(A)]*, where we have used Proposition C.2.1 k) in (1) and Proposition A.6.3 in (2).

C.3

0

The Numerical Range

From now on, all considered operators are single-valued (cf. the terminological agreement on page 279.) Given an operator A on H we call

W(A):= {(Aulu) I u

E

1>(A), Ilull = 1}

C

(A), Ilull = 1, such that Au = Au. This yields (Aulu) = (>'ulu) = Allul1 2 = A. Hence A E W(A). Suppose that A t/:- W(A) and define 15 := dist(A, W(A)). By definition, one has I(Ax I x) - AI;:::: 15 for all x E 1>(A) with Ilxll = 1. Hence 1((A-A)xlx)1 = I(Axlx)-AllxI121 ;::::r5llxl1 2 for all x E 1>(A). But I((A - A)X I x) I :::; II (A - A)xllllxll, whence II(A - A) xii;:::: r5llxll for all x E 1>(A). Since A is closed, this implies that (A - A) is injective and has closed range, i.e., A t/:- Au(A). Moreover, it shows that IIR(A, A)II :::; 15- 1 if A E [J(A). If A E £(X), then W(A*) = {X I A E W(A)}. Now, if A E Ru(A), then clearly X E Pu(A*), whence A E W(A). 0

Corollary C.3.2. Let A be a single-valued, closed operator on H, and let the set U C O. In this case, {ReA < O} c [J(A) and 1

II(A + A)-lll ~ Re A for all Re A > O. Moreover, 0 ~ t(t > O.

t

+ A)-l

~

1 and 0 ~ A(t + A)-l ~ 1 for all

322

Appendix C. Operator Theory on Hilbert Spaces

Proof. Let A be a single-valued, closed, and densely defined operator such that W(A) C [0, (0). Since C \ [0, (0) is open and connected, the stated equivalence is a consequence of Proposition C.3.1 and its corollary. Moreover it follows that in this case {ReA < O} C e(A), and the stated norm inequality holds. Since t + A is self-adjoint, we know that (t + A)-l also is. Furthermore 0 :s; A(t + A)-l :s; 1 {:} o :s; t(t + A)-l :s; 1 {:} (tu I (t + A)u) :s; ((t + A)u I (t + A)u) \/u E 'D(A). However, this is true if and only if IIAul1 2 + t (Au I u) ~ 0, which is always the case. D

Lemma C.4.4. Let A be a closed and densely defined operator on H. a)

If -0: :s; A :s; 0: for some 0:

b)

If A E £(H) is self-adjoint, then IIAII = SUp{IAI I A E W(A)}. In particular, W(A) is a bounded subset of R

c)

If 0 :s; A :s; 1, then 0 :s; A 2 :s; A.

~ 0,

then A

E

£(H) and IIAII :s; 0:.

Proof. a) Define a(u,v) := (Aulv) and b(u, v) := (ulv) on V:= 'D(A). The hypothesis implies that la(u)1 :s; o:b(u) for all u E H. Moreover, a is symmetric. An application of the generalised Cauchy-Schwarz inequality (Proposition C.1.2) yields I(Aulv)l:s; o:llullllvll for all u,v E 'D(A). Since A is densely defined, this inequality holds for all v E H, whence we have IIAul1 :s; 0: Ilull for all u E 'D(A). Since A is closed, this implies that 'D(A) is closed. Hence A E £(H) and IIAII :s; 0:. b) Let A E £(H) be self-adjoint. Then I(Au I u)1 :s; IIAlillul1 2 for all u E H. Hence supIW(A)I:s; IIAII· IfsupIW(A)I:s; 0:, then I(Aulu)l:S; 0:1Iul1 2 for all u. Since W(A) c JR., this is equivalent to -0: ::; A ::; 0:. Now a) implies that IIAII ::; 0:.

c) We have A - A2 = A2(1 - A) since A and 1 - A are.

+ A(l -

A)2, and both summands are positive D

The following result is usually proved with the help of the spectral theorem.

Proposition C.4.S. Let A ~ 0: for some 0: E lR, and define 0:0 := inf W (A). Then 0:0 E a(A). If 0:0 E W(A), then even 0:0 E Pa(A).

Proof. Without loss of generality we may suppose that 0:0 = O. Hence A ~ O. Let Q := A(l + A)-I. Then 0 :s; Q :s; 1. Moreover, Q :s; A in the obvious sense since for x E 'D(A) we have ( A( A + 1) -1 X I x) - (Ax Ix)

=

(A( (A

+ 1) -1 - I)x Ix)

= - (A(A + 1)-1 Ax I x) = - ((A + 1)-1 Ax I Ax)

:s; O.

Applying c) of Lemma C.4.4 we obtain O:S; IIQxI12:s; (Qxlx):S; (Ax Ix)

for all x E 'D(A). Now if there is x E 'D(A) such that Ilxll = 1 and (Ax I x) = 0 it follows that 0 = Qx = A(A+1)-lX. But this implies that 0 =I- (A+1)-lx E N(A),

c.s.

Equivalent Scalar Products and tbe Lax-Milgram Tbeorem

323

whence 0 E Pa(A). Similarly, if Xn E 1J(A) such that Ilxnll = 1 and (Ax n I x n ) ----; 0, then QX n ----; O. Hence (xn)n is a generalised eigenvector for Q and 0 E a( Q). But this immediately implies that 0 E a(A). D Corollary C.4.6. Let A be a bounded and self-adjoint operator on A. Then

sup W(A), infW(A) E a(A).

In particular, IIAII = r(A), where r(A) denotes the spectral radius of A. Proof. Let 0: := inf W(A) and (3 := sup W(A). Then 0 = inf W(A - 0:) infW((3 - A), whence 0 E a(A - 0:) n a((3 - A) by Proposition C.4.5. From D Lemma C.4.4 b) we know that r(A) ~ IIAII = max{lo:l, 1(31} ~ r(A). Note that positive operators are special cases of m-accretive operators (see Section C.7 below). Proposition C.4.7. Let S, T E £(H) be self-adjoint, and let 0 ~ S ~ T in the sense that (Sx I x) ~ (Tx I x) for all x E H. If S is invertible, then T is invertible and T- 1 ~ S-l.

Proof. By Corollary C.4.6, if S is invertible, there is 0 < 0: E ~ such that 0: ~ S. Hence 0: ~ T, whence T is invertible. Let x E H, and define y := T- 1 x, z := S-lx. Then (T- 1 x I x) 2 = (y I Sz)2 ~ (y I Sy)( z I Sz) ~ (y ITy)( z I Sz)

= (T- 1 x I x) (S-l X I x) by Cauchy-Schwarz. Consequently, (T- 1 x I x) ~ (S-l x I x).

C.5

D

Equivalent Scalar Products and the Lax-Milgram Theorem

Let H be a Hilbert space. We denote by H* the antidual of H, i.e., the space of continuous conjugate-linear functionals on H, endowed with the norm II'PIIH* := sup{I'P(x)I I x E H, Ilxll = I}

('P E H*).

One sometimes writes ('P, x) instead of 'P( x), where x E H, 'P E H*. Let a be a continuous sesquilinear form on H. Then we have an induced linear mapping La := (u f------+ a(u, .)) : H ------t H*, (C.4) which is continuous. The Riesz-Frechet theorem [196, Theorem 4.12] implies that for each u E H there is a unique Qu E H such that

a(u,·) = La(u) = (Qui·).

324

Appendix C. Operator Theory on Hilbert Spaces

Obviously, Q is a linear and bounded operator (IIQII = IILall) and a is uniquely determined by Q. On the other hand, given Q E C(H), the form aQ defined by aQ(u, v) := (Qu Iv) is sesquilinear and continuous. Hence the mapping

(Q

t-------+

aQ) : C( H) ------) {continuous, sesquilinear forms on H}

is an isomorphism. We have aQ = aQ*, whence the form a is symmetric if and only if Q is self-adjoint, and the form is positive if and only if Q 2 o. Proposition C.5.1. Let Q E C(H). Then the form aQ is a scalar product on H if and only if Q is positive and injective. The norm induced by this scalar product is equivalent to the original one if and only if Q is invertible. Proof. The form aQ is positive semi-definite if and only if Q 2 o. This is clear from the definitions. If Q is not injective, then obviously aQ is not definite. Let Q be injective and positive. By Proposition C.4.5 we conclude that 0 ~ W(Q). But this means that aQ is definite. Since aQ is continuous, equivalence of the induced norm is the same as the existence of a 8 > 0 such that 8 ::; Q. But this is equivalent to Q being invertible by Proposition C.4.5. 0

A scalar product on H that induces a norm equivalent to the original one is simply called an equivalent scalar product.

Lemma C.5.2. Let A be a multi-valued linear operator on H, and let (·1·)0 := aQ be an equivalent scalar product on H. Denote by AO the adjoint of A with respect to the new scalar product. Then (x, y) E AO for all x, y E H. In particular, AO

0 we have II(A + A)u11 2 - A211ul1 2 = IIAul1 2 + 2A (Au I u) for all u E 'D(A). This shows (i)=}(iv). Dividing by A and letting A ----* 00 yields the reverse implication. The implication (iii)=}(iv) is obvious. Suppose that (i) holds, and let Re A '2 Define a := 1m A. Then (i) holds with A replaced by A + ia. Since we have already established the implication (i)=}(iv), we know that II((A + ia) + ReA)ull '2 (Re A) Ilull for all u E 'D(A). But this is (iii). D

o.

Note that an operator A is symmetric if and only if ±iA both are accretive. An operator A is called m-accretive if A is accretive and closed and 9« A + 1) is dense in H. Proposition C.7.2. Let A be an operator on H, a E IR and A >

assertions are equivalent.

(i) A is m-accretive.

o. The following

Appendix C. Operator Theory on Hilbert Spaces

328 (ii) A

+ ia

is m-accretive.

> o. A)(A + A)-III

(iii) A + E is m-accretive for all (iv) -A E e(A) (v) {Re A < O}

and

c

I (A -

E

e(A) and

1

IIR(A, A) II S IRe AI (vi) (-00,0)

c

s 1.

e(A) and SUPt>o

(Re A < 0).

Ilt(t + A)-III s 1.

(vii) A is closed and densely defined, and A* is m-accretive. The operator (A - l)(A + 1)-1 is called the Cayley transform of A. Proof. Let B be a closed accretive operator on H. By Proposition C.3.1 and its corollary, if 'R( B + J-l) is dense for some J-l with Re J-l > 0, then {Re J-l < O} c e( B). This consideration is of fundamental importance and is used several times in the sequel. In particular, it shows (i){:}(ii){:}(iii). The equivalence (iv){:}(i) holds by (ii) of Lemma C.7.1. Similarly, (v){:}(i) and (vi){:}(i) hold by (iii) and (iv) of Lemma C.7.1. From (vi) it follows that an m-accretive operator is sectorial (see Section 2.1), and as such is densely defined since H is reflexive (see Proposition 2.1.1 h)). Moreover, (vi) implies (vi) with A replaced by A*, whence (vi)*(vii) follows. Finally, suppose that (vii) holds. Since we already have established the implication (i)*(vii), we conclude that A = A = A** is m-accretive. D Theorem C.7.3 (Lurner-Phillips). An operator A is m-accretive if and only if-A generates a strongly continuous contraction semigroup. Proof. If A is m-accretive, parts (vi) and (vii) of Proposition C.7.2 show that the Hille-Yosida theorem (Theorem A.8.6) (with w = 0 and M = 1) is applicable to the operator -A. Conversely, suppose that -A generates the Co-semigroup T such that IIT(t)11 S 1 for all t 2 o. Take x E n(A). Then the function t f------t T(t)x is differentiable with derivative t f------t -AT(t)x. Thus, t ----+ IIT(t)xI1 2 is differentiable with

dl t=o IIT(t)xll 2 dt

= (T(O)x I - AT(O)x) + ( -AT(O)x IT(O)x) = -2 Re (Ax I x) .

Since T is a contraction semigroup, the mapping t f------t IIT(t)xI1 2 is decreasing. This implies that djdtlt=o IIT(t)xI1 2 O. Hence, Re (Ax I x) 2 0, i.e., A is accretive. Since 1 E e( -A) we conclude that A is in fact m-accretive. D

s

Theorem C.7.4 (Stone). An operator B on the Hilbert space H generates a Co group (U(t))tElR of unitary operators if and only if B = iA for some self-adjoint operator A.

329

G.8. The Theorems of Plancherel and Gearhart

Proof. If B = iA, then Band -Bare both m-accretive. By the Lumer-Phillips theorem both operators generate Co-contraction semigroups. Hence B generates D a unitary Co-group. This proof also works in the reverse direction.

c.s

The Theorems of Plancherel and Gearhart

We state without proof two theorems which are 'responsible' for the fact that life is so much more comfortable in Hilbert spaces. Theorem C.8.l (Plancherel). Let f E Ll(JR, H) n L2(JR, H), and define F(f)

JR -----; H by

F(f)(t) Then F(f)

E

:=

l

f(s)e- ist ds

Co(JR, H) nL2(JR,H) with

(t E JR).

IIF(f)liP = V27fllfliP·

A proof can be found in [10, page 46]. More about the Fourier transform is contained in Appendix E. Let X be a Banach space, and let T be a Co-semigroup on X with generator A. From Proposition A.8.1 we know that

{Rez;::::w}cg(A)

sup{IIR(z,A)111 Rez;::::w}

and

(C.6)

for every w > wo(T). (Recall that wo(T) is the growth bound of T, see Appendix A, Section A.S.) Let us define the abszissa of uniform boundedness of A as

so(A)

:= inf{w E

Hence in general we obtain that so(A) can say more.

JR I (C.6) holds}.

(C.7)

:s; wo(T). However, on Hilbert spaces one

Theorem C.8.2 (Gearhart). Let A be the generator of a Co-semigroup on a Hilbert

space H. Then so(A)

=

wo(T).

One of the many proofs of this theorem uses the Plancherel theorem and Datko's theorem, which states that wo(T) < 0 if and only if T(·)x E L2(JR+, H) for every x E H. It can be found in [10, page 347].

References The following books underly our presentation: [130] and [199, Chapter XII] for sesquilinear forms and numerical range, [52, Chapter III] for adjoints, [49, Chapter X, §2] and [194, Chapter VIII] for symmetric and self-adjoint operators, [210, Chapter 2] for accretive operators and the Lax-Milgram theorem, and [85, Chapter II.3.b] for the Lumer-Phillips theorem and Stone's theorem.

Appendix C. Operator Theory on Hilbert Spaces

330

The idea how to establish part c) of Lemma C.4.4 is from [144, Chapter XVIII, §4].

Theorem C.8.2 goes back to GEARHART [95], but its present form is due to [191] and GREINER [174, A-III.7].

PRUSS

Appendix D

The Spectral Theorem The Spectral Theorem for normal operators exists in many versions. Basically one can distinguish the 'spectral measure approach' and the 'multiplicator approach'. Following HALMOS'S article [112] we give a consequent 'multiplicator' account of the subject matter. In this form, the Spectral Theorem essentially says that a given self-adjoint operator on a Hilbert space acts 'like' the multiplication of a real function on an L2_ space. In contrast to usual expositions we stress that the underlying measure space can be chosen locally compact and the real function can be chosen continuous.

D.I

Multiplication Operators

A Radon measure space is a pair (0',J.l) where 0, is a locally compact Hausdorff space and J.l is a positive functional on Cc(o'). By the Riesz Representation Theorem [196, Theorem 2.14] we can identify J.l with a cr-regular Borel measure on O. If the measure J.l has the property (D.l) then the Radon measure space is called a standard measure space. Property (D.l) is equivalent to the fact that a non-empty open subset of 0, has positive J.l-measure. If this is the case, the natural mapping C(o') ----) L1!JO" J.l) is injective. Let f E C(o') be a continuous function on O. The multiplication operator Mf on L2(0" J.l) is defined by

The following proposition summarises the properties of this operator. Proposition D.l.l. Let f E C(o'), where (0" J.l) is a standard measure space. Then the following assertions hold.

332

Appendix D. The Spectral Theorem

a)

The operator (Mf' 'D(Mf )) is closed.

b)

The space Cc(O) is a core for M f .

c) One has (Mf)* = My. d) One has Mf E £(L2) if and only if f is bounded. If this is the case, one has

IIMfllc(p) = 1111100·

e) The operator Mf is injective if and only if f-l( {f = O} compact K cO, i.e., {f = O} is locally f-l-null. f)

n K) =

0 for every

One has u(Mf) = f(O). Furthermore, R(>..,Mf) = M(>. -f)-' for all..\ E (}(Mf)·

g) If Mf c 0, then f = o. h) Let also 9

E C(O). Then MfMg C Mfg. One has equality if 9 is bounded or if M f is invertible.

Proof. a) Suppose that gn E 'D(Mf) such that gn -+ 9 and fgn -+ h in L2. For