Existence families, functional calculi and evolution equations

Lecture Notes in Mathematics Editors: A. Dold, Heidelberg B. Eckmann, Ztirich E Takens, Groningen

1570

Ralph deLaubenfels

Existence Families, Functional Calculi and Evolution Equations

Springer-Verlag Berlin Heidelberg NewYork London Paris Tokyo Hong Kong Barcelona Budapest

Author Ralph deLaubenfels Mathematics Department Ohio University 315 B Morton Hall Athens, Ohio 45701-2979, USA

Mathematics Subject Classification (1991): 47D05, 47D 10, 47A60, 34G 10, 35G 10, 47A10, 47A12, 47F05

ISBN'3-540-57703-3 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-57703-3 Springer-Verlag New York Berlin Heidelberg Library of Congress Cataloging-in-Publication Data. DeLaubenfels, Ralph, 1951- . Existence families, functional calculi, and evolution equations / Ralph deLaubenfels, p. cm. - (Lecture notes in mathematics; 1570) Includes bibliographical references and index. ISBN 0-387-57703-3 1. Evolution equations. 2. Linear operators. 3. Functional analysis. I. Title. II. Series: Lecture notes in mathematics (Springer-Verlag); 1570. QA3.L28 no. 1570 [QA377] 510 s-dc20 [515'.353] 93-47576 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 any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. 9 Springer-Verlag Berlin Heidelberg 1994 Printed in Germany SPIN: 10078796

46/3140-543210 - Printed on acid-free paper

To Karen, David, Michael and Evan

T A B L E OF C O N T E N T S

0.

Introduction

I.

Intuition and Elementary Examples

1

II.

Existence Families

7

III.

Regularized Semigroups

13

IV.

The Solution Space of an Operator and Automatic Well-Posedness

24

V.

Exponentially Bounded (Banach) Solution Space

38

VI.

Well-posedness on a Larger Space; Generalized Solutions

55

IX

VII. Entire Vectors and Entire Existence Families

60

VIII. Reversibility of Parabolic Problems

66

IX.

The Cauchy Problem for the Laplace Equation

71

X.

Boundary Values of Holomorphic Semigroups

73

XI.

The SchrSdinger Equation

76

XII. Functional Calculus for Commuting Generators of Bounded Strongly Continuous Groups

79

XIII. Petrovsky Correct Matrices of Generators of Bounded Strongly Continuous Groups

86

XIV. Arbitrary Matrices of Generators of Bounded Strongly Continuous Groups

92

XV. More Examples of Regularized Semigroups

94

XVI. Existence and Uniqueness Families

97

vii

XVII.

C-resolvents and Hille-Yosida Type Theorems

104

XVIII.

Relationship to Integrated Semigroups

110

XIX.

Perturbations

113

XX.

Type of an Operator

125

XXI.

Holomorphic C-existence Families

128

XXII.

Unbounded Holomorphic Functional Calculus for Operators with Polynomially Bounded Resolvents 133

XXIII.

Spectral Conditions Guaranteeing Solutions of the Abstract Cauchy Problem

154

XXIV.

Polynomials of Generators

158

XXV.

Iterated Abstract Cauchy Problems

164

XXVI.

Equipartition of Energy

175

XXVII. Simultaneous Solution Space

178

XXVIII. Exponentially Bounded Simultaneous Solution Space

183

XXIX.

Simultaneous Existence Families

187

XXX.

Simultaneous Existence Families for Matrices of Operators

191

Time Dependent Evolution Equations

196

Notes

201

Bibliography

218

Subject Index

228

XXXI.

VIII

0. I N T R O D U C T I O N

In this book, we wish to present the basic theory, and the more interesting developments, of existence families for the abstract Cauchy problem

d -~u(t,x) = A(u(t,x)) (t >_0), u(O,x)= x,

(0.1)

and regularized semigroups (these have been recently appearing in the literature as "C-semigroups"). These are generalizations of strongly continuous semigroups, that can be applied directly to the many differential and integral equations that may be modeled as an abstract Cauchy problem on a Banach space, where strongly continuous semigroups cannot be applied directly, that is, when the operator A in (0.1) does not generate a strongly continuous semigroup. Examples of this are the backwards heat equation, the SchrSdinger equation on LP,p ~ 2, the Cauchy problem for the Laplace equation and Petrowsky correct systems of constant coefficient partial differential equations. These families of operators unify well-posed and ill-posed problems, cover the entire spectrum of ill-posedness and unify and simplify results about ill-posed problems, just as strongly continuous semigroups do with well-posed problems. Of particular interest in this book will be the interplay between functional, or operational, calculus constructions, existence families and evolution equations. We do not attempt to give detailed physical applications. The examples of evolution equations are meant primarily to illustrate the great scope and broad, yet simple and intuitive, applicability of existence families and regularized semigroups. We shall see that regularized semigroups may also be used as a powerful tool for constructing functional calculi for unbounded operators. More complete descriptions of how physical problems may be modeled as an abstract Cauchy problem (0.1) may be found in any of the references for strongly continuous semigroups. Another central feature of this book is what is called the solution space, for an arbitrary closed operator. This is a unifying concept for all the generalizations of strongly continuous semigroups that have appeared, besides the existence families presented here. Even for the strongly continuous case, the introduction of the solution space greatly simplifies the proof of the fundamental relationship between the abstract Cauchy problem and semigroups. IX

In this book, we have tried to avoid those aspects of C-regularized semigroups that are obvious generalizations of strongly continuous semigroups, that is, where one takes a proof for strongly continuous semigroups and places a C everywhere. It is gratifying to find that many of these results may be immediately reduced to the corresponding result for strongly continuous semigroups with the solution space. In this book, we have included primarily results about existence families and regularized semigroups that would seem surprising to anyone familiar with strongly continuous semigroups. The existence of entire C-regularized groups with unbounded generators is an example. Another example is immense simplications and clarifications, both in the statement of results and their proofs, of old results, involving, for example, the SchrSdinger operator on LP,p ~ 2, and Petrowsky correct systems of differential equations. Many of the most fundamental results in this book, including Chapters II, V, VII and XII, are new, that is, have not appeared, at least in their present exposition, and are not intended to appear, in research journals. We have also organized and simplified a great deal of basic material that is currently scattered throughout many different papers. Some familiarity with the basic results of strongly continuous semigroups would be helpful in reading, or at least appreciating, this book, but is not necessary for most of the material, since these results will be a corollary of our results here. The theory of strongly continuous semigroups of operators has been extremely successful in dealing with the many situations that may be modeled as an abstract Cauchy problem (0.1). When A is closed, generating a strongly continuous semigroup corresponds to the abstract Cauchy problem having a unique mild solution, for all initial data x. What should be done when the abstract Cauchy problem (0.1) is not well-posed or does not have a mild solution for all initial data? It does not seem constructive to merely say it is not well-posed, throw up our hands with despair and refuse to deal with it. Ill-posed problems arise naturally (see [Pay]); the initial formulation of (0.1)is often made out of convenience or familiarity. The supremum norm and the L 1 norm, for example, are very natural choices for Banach space norms, but in many cases, such as the SchrSdinger equation, do not yield well-posedness. There are two approaches one can take. One can look for initial data in the original space that yield solutions. Or one can renorm a subspace in such a way that A, when restricted to that space, generates a strongly continuous semigroup. We shall see that existence families and regularized semigroups provide a simple yet powerful tool for either approach. For the first approach, we shall see that, for any bounded operator C, having X

a mild solution of the abstract Cauchy problem, for all initial data in the image of C, corresponds to having a mild C-existence family. We present in this book an intuitive method for choosing such C and constructing the desired existence family. For the second approach, we show that a mild C-existence family gives us a straightforward method for constructing and approximating a subspace of the original space, containing all mild solutions of the abstract Cauchy problem, on which A generates a strongly continuous semigroup. We call this the solution space for A (see Chapters IV and V). Besides existence and well-posedness, it is also important to have methods for representing or constructing solutions. We feel the essential concept here is an operator theoretic one of a good deal of independent interest, that of an JC-functional calculus for A. For bounded operators, this is a continuous algebra homomorphism, from the Banach algebra of functions ~- into the Banach algebra of bounded linear operators, mapping the constant function fo(s) = 1 to the identity operator, and mapping f l ( s ) = s to A. Some modifications are required to adapt this concept to unbounded operators, where the subject is relatively unformed (except for spectral operators--see [Du-S2]). Informally, one is replacing the complex variable z by A, and making sense out of f ( A ) . Again informally, the solution of the abstract Cauchy problem is u(t, x) = etAx, that is, f t ( A ) x , where ft(z) - e tz. Many of the most important results about semigroups of operators can be best understood with this perspective, and many new results, presented in this book, can be created and understood similarly. We may summarize the functional calculus approach to evolution equations, via the abstract Cauchy problem (0.1), as follows. We think of a strongly continuous semigroup generated by A as being etA. We think of a C-regularized semigroup generated by A as being etAg(A), for some g such that the complex-valued function z ~ etZg(z) decays sufficiently rapidly on the spectrum of A; here C is g(A). Finally, we think of a solution of (0.1) as being u(t,x) =_ etAx. Throughout this book, we will try to make literal sense out of these exponentials. To conclude our general introduction, we should make it clear that we do not consider regularized semigroups or existence families to be any sort of revolutionary replacement of strongly continuous semigroups. We view it more as a completion of the theory of strongly continuous semigroups, extending and finishing a beautiful and fundamental theory that strongly continuous semigroups began. For example, when there are solutions of the abstract Cauchy problem, it is natural to ask when they are reversible (see Chapter VIII). When {e~A}n~(z)>O is a strongly continuous holomorphic semigroup of angle y, r it is natural to ask in what X1

sense we can let Re(z) ~ 0; that is, can we make sense out of boundary values e isA (see Chapter X)? As with strongly continuous semigroups, an approach that allows us to deal with bounded operators has the best chance of being successful. Now we will give a rough chapter-by-chapter introduction. For an arbitrary bounded operator C, and an unbounded operator A, C-existence families for A, which we think of a s etAC, are introduced in Chapter II. We must quibble about whether we are obtaining mild solutions or strong solutions (Definition 2.1) of the abstract Cauchy problem (0.1), thus we have mild existence families, existence families and strong existence families. We introduce regularized semigroups in Chapter III. These are essentially existence families e~AC, where e~AC : Ce tA. This gives us an algebraic definition, W ( t ) W ( s ) = W i t § s)C, analogous to strongly continuous semigroups (/-regularized semigroups). To enable us to define the generator, A, we insist that C be injective (so that we may differentiate Ce tA at t -- 0, then apply C -1). In Chapter IV, for an arbitrary closed operator A on a Frechet space, we introduce its solution space, Z. This is the set of all initial data, x, for which the abstract Cauchy problem has a mild solution. When all solutions are unique, we may topologize Z in such a way that Z is a Frechet space, and A restricted to Z generates a strongly continuous, locally equicontinuous semigroup. This is the only chapter where X is a general Frechet space, rather than a Banach space. By restricting ourselves to exponentially bounded mild solutions of the abstract Cauchy problem, we obtain maximal Banach spaces, on which a closed operator A generates a strongly continuous semigroup, in Chapter V. We present numerous methods for testing whether a vector x is in one of these exponentially bounded solution spaces, and constructing the corresponding mild solution of the abstract Cauchy problem; some of these tests may be considered "pointwise Hille-Yosida theorems." In both Chapters IV and V, we see how regularized semigroups and existence families provide a way to approximate the solution spaces and their topologies. Chapter VI presents a strategy opposite to that of Chapters IV and V. Rather than shrink the original space, we enlarge it, to produce a space, W, in which the original space is continuously embedded, and an operator B, that generates a strongly continuous semigroup, such that the original operator A equals the restriction of B to X. In Chapter VII, a concept that has been used a great deal for strongly continuous semigroups, that of an entire vector, is used to give a simple XII

necessary and sufficient condition for an operator A to have an entire Cexistence family: the image of C must consist entirely of entire vectors. We may also construct this family with the power series for the exponential function. It should be mentioned that an entire strongly continuous group (or integrated group) occurs only in the essentially trivial case of a bounded generator. In Chapters VIII and IX, we apply Chapter VII to two famous ill-posed problems, the backwards heat equation and the Cauchy problem for the Laplace equation, giving simple proofs of the existence of unique entire solutions, for all initial data in a dense set. More generally, we show in Chapter VIII that any parabolic problem is reversible, for all initial data in a dense set. Chapter X considers the "boundary values" {eisA}seir of a strongly continuous bounded holomorphic semigroup {eZA)Re(z)>O , a n d gives a correspondence between the values of r for which {eisA(1 -- A)-r}seR is a polynomially bounded (1 - A)-~-regularized semigroup and the rate of growth of ]lezA]], as Re(z) goes to zero. In Chapter XI, Chapter X is applied to the SchrSdinger equation, with potential, on LP(R~)(1 _< p < co). In Chapter XII, we use regularized semigroups to define, for f E C~176 commuting generators iA1,..., iAn of bounded strongly continuous groups, a functional calculus f ~ f(A1,...,A,~). We explicitly define, for each such f, with the Fourier inversion theorem, a regularized semigroup, then define f ( A 1 , . . . , A,~) to be the generator. We use the construction of Chapter XII to consider, in Chapters XIII and XIV, matrices of polynomials of generators of bounded strongly continuous groups. In Chapter XIII, we consider Petrowsky correct matrices, and obtain generators of exponentially bounded (1 - IAI 2)-~-regularized semigroups, for appropriate r. In Chapter XIV, we show that any matrix of polynomials of generators of bounded strongly continuous groups generates an entire C-regularized group, for an explicit C with dense image. A few more examples of regularized semigroups appear in Chapter XV. Even when C does not commute with A, in Chapter XVI we show how we may give an algebraic definition of a pair of families of operators, an existence family and a uniqueness family, with a generator A. We think of these families as Wl(t) - etAC1 (for existence) and W2(t) =-- C2e tA (for uniqueness); a formal calculation leads to an algebraic definition that intertwines the two families: w2(OWl(

) = c2wl(t +

for all 8, t ~ O. XIII

= w2(t +

Chapter XVII gives analogues of Hille-Yosida type theorems, for Cexistence families, involving the natural analogue of the resolvent, the

C-resolvent. In Chapter XVIII, we show that an n-times integrated semigroup, another generalization of strongly continuous semigroups that has appeared recently, corresponds to an (r - A)-n-regularized semigroup. Some perturbation results, both additive and multiplicative, appear in Chapter XIX. We introduce the type of an operator in Chapter XX. This is an analogue of numerical range that satisfies mapping theorems similar to spectral mapping theorems. The results that one would expect for holomorphic C-existence families appear in Chapter XXI. We do not give any details of proofs here. In Chapter XXII we use regularized semigroups and unbounded versions of the Cauchy integral formula to define a holomorphic functional calculus, f ~ f(A), for large classes of unbounded operators A and functions f holomorphic on an open set containing the spectrum of A. We give numerous examples, including many powerful applications to the abstract Cauchy problem. This generalizes, to unbounded operators and unbounded (they may not even be polynomially bounded) functions, the Riesz-Dunford holomorphic functional calculus. The construction of Chapter XXII is applied, in Chapter XXIII, to characterize what we call spectral dense solution sets. These are subsets, V, of the complex plane, with the property that, whenever the spectrum of a densely defined operator A is contained in V, with the norm of the resolvent of A polynomially bounded outside V, then the abstract Cauchy problem (0.1) has a solution, for all initial data in a dense set. In Chapter XXIV we show that polynomials satisfy "type mapping theorems" (see Chapter XX). This leads to a large class of operators A and polynomials p such that p(A) generates an integrated semigroup, hence a regularized semigroup. In Chapter XXV we show how we may deal with many higher order abstract Cauchy problems by factoring them into an "iterated abstract Cauchy problem," with each factor generating a regularized semigroup. We use this to show that many second order problems, including the Cauchy problem for the Laplace equation, admit equipartition of energy, in Chapter XXVI. T e r m i n o l o g y 0.2. We will use the following terminology throughout the book. Except in Chapter IV, where X will be a Frechet space, X will be a Banach space. All operators are linear. We will write :D(A) for the domain of the operator A, a(A) for the spectrum, p(A) for the resolvent XIV

set. The space of b o u n d e d operators from X into itself will be denoted by B(X). We will write Ira(C) for the image of an operator C. We m a y make Ira(C)into a Banach space, [Ira(C)], with the norm

Ilyllt-,~(c)l ~ inf{llxll I c ~ = y}. W h e n A is closed, [Z)(A)] will mean the graph norm

79(A), made

into a Banach space with

IIxlIE~_O

I. I N T U I T I O N

AND ELEMENTARY

EXAMPLES

One of the first differential equations students see in their first class on the subject is

~'~u ( t ) =

a~,(t),

(1.1)

where a is a complex number. It does not take long to convince one that the solution is given by u(t) = etau(o). Soon after this, one learns to deal with a system of n constant coefficient linear differential equations by writing it as a single matrix differential equation = Aft(t),

d _

where A is an n x n matrix and ft is an n-vector whose components are the unknown functions, and again the solution is given by ft(t) = etAft(O). This simple example is a good demonstration of what we consider to be one of the major purposes of mathematics, to simplify and clarify. A complicated and chaotic problem is reduced to a simple and familiar problem. Even in finite dimensions, we see how a problem in differential equations is transformed into an interesting problem in operator theory, specifically, what is called a functional calculus (or operational calculus). What is meant by etA? This can be defined using the power series for the exponential function, 00 t k A k

e'A =-

h---Y-.'

(1.2)

k=O

but this still leaves the problem of how to compute it. A more useful representation may be constructed using the Jordan canonical form for a matrix. For any sufficiently differentiable complex-valued function f , define

f(

i )_ f(a)

a

0

f'(a)] f( f(a) J '

[ :][!a )-0

a

0

0

:a

f(oa) f'(a) f(a)

etc.; it is clear how to extend this definition to matrices formed by pasting together such Jordan blocks, as in the Jordan form. When A is an n • n matrix, this definition of f ( A ) (combined with the Jordan canonical form) defines a functional calculus for A, that is, ( f g)( A ) = f( A )g( A ), fo( A ) =

I , f l ( A ) = A, where fo(z) = 1 , f l ( z ) =- z. When gt(z) =_ e tz, then g,(A) may be shown to be the same as (1.2). A large class of partial differential initial-value problems and mixed problems may similarly be made to look like (1.1), as an abstract Cauchy problem (see (0.1)), where A is an operator on a locally convex space X and u(.,x) : [0, oc) ---, X. Again the natural thing to do would be to exponentiate A to get the solution u; u(t,x) - e t A x . But what does this mean? Let us consider another special case, but this time in infinite dimensions. Suppose iA is a self-adjoint operator on a Hilbert space, H. Then there exists a functional calculus, f ~ f ( A ) , defined for all bounded continuous functions on the real line, given by

f ( i A ) x - / R f ( s ) d E ( s ) x (x E H), where E is a projection-valued measure. This is the spectral theorem. It is now clear how to exponentiate A: e t A ---- gt(iA), where gt(s) = e -its. And in fact this turns out to define solutions of the abstract Cauchy problem (0.1), as a Fourier transform, given by

u(t,x) =_ etAx =_ / R e - i t S dE(s)x, for x in the domain of A~ In the operator-theoretic approach to the abstract Cauchy problem, one defines a strongly continuous semigroup generated by A, {T(t)}t>o =-{etA}t>_O , t o be a family of bounded operators with the properties of the family of functions s ~ ets; T(O) = I, T(s + t) = T(t)T(s), for all s, t > 0 and ~T(t)xlt=o = Ax, for all x in the domain of A. Thus this is a functional calculus, defined for exponential functions s ~ e t., for t > 0. The most well-known characterization of generators of strongly continuous semigroups, the Hille-Yosida theorem, is suggested by this functional calculus perspective: since

(s - a) -1 =

//

e-Ste ta dt

for any complex number a, for s sufficiently large, we expect the generator of a strongly continuous semigroup, A, to have nonempty resolvent set, with (s - .4) -1 =

e-Ste tA dr,

for s sufficiently large. (See Chapter XVII.)

Thus the Laplace transform is useful in studying semigroups of operators (see also Chapters X and XI). We shall see that the Fourier transform and (unbounded analogues of) the Cauchy integral formula play equally significant roles (see Chapters XII, XIII, XIV, XXII, and XXIII). The idea is to use classical analysis and the intuition suggested by the functional calculus perspective to write down operator-valued formulas, and then make sense out of them. Well-known examples include the construction of holomorphic semigroups, and fractional powers using the Cauchy integral formula and constructing e tA2, w h e n A generates a strongly continuous group, using the Fourier inversion theorem applied to e-Y2;

e~A2x =

(4~rs)- 89/R(etAx)e :~,2,2dr,

forxEX, s>0. The power series is also useful, for initial data x such that oo tk k=O

Then we may use (1.2) to define r

tk

_

k=O

(See Chapters VII, VIII and IX.) Informally, a functional calculus for A consists of taking a complexvalued formula, and replacing the complex variable with A; sometimes a great deal of imagination is required to make sense of the resulting formula. Thus functional calculi lead to strongly continuous semigroups, which lead to solutions of the abstract Cauchy problem. Are these implications reversible? It is well known that the existence of mild solutions of the abstract Cauchy problem, for all initial data, implies that A generates a strongly continuous semigroup (see Chapter IV for a new, greatly simplified proof of this fundamental result). What about the relationship between strongly continuous semigroups and functional calculi? The spectral theorem is an example where generating a certain type of semigroup implies the existence of a certain functional calculus. We shall see that a generalization of strongly continuous semigroups, known as regularized semigroups, besides their more obvious applications to the abstract Cauchy problem, may be used to construct functional calculi for unbounded operators.

Another approach to the abstract Cauchy problem (0.1) is the following. Suppose all solutions are unique, and let Z be the set of all initial data, z, for which the abstract Cauchy problem has a solution. In Chapter IV, we call this the solution space for a closed operator A. For any t > 0, define a map from Z into X , etAz -- u ( t , z ) . Thus, rather than consider the solutions one point at a time, we scoop them together, and consider the operator mapping a point to its corresponding solution at time t. This is the idea behind strongly continuous semigroups. When Z equals X, it can be shown that {etA}t>_O is a strongly continuous family contained in B ( X ) . In general, one could consider e tA as an unbounded operator; however, to get all the heavy artillery of operator theory, we really want bounded operators; this is where the power of strongly continuous semigroups arises. Thus our strategy is to choose an operator, C, such that {etAC}t>>_o is a strongly continuous family of operators in B ( X ) . This is what we call a C-ezistence family (Chapter II). When C commutes with A, so that etAC = Ce tA, then this family of operators may be characterized by algebraic properties analogous to strongly continuous semigroups, and is called a C-regularized semigroup. The idea is that C "smooths" or "regularizes" whatever bad or uncontrolled behaviour e tA may have; the more ill-posed the abstract Cauchy problem is, the more smoothing C must be, that is, the smaller its image must be. Our main interest in the abstract Cauchy problem in this book is the case where it has a mild solution for all initial data in a large set not equal to the entire space. This leads to a generalization of strongly continuous semigroups, known as an ezistence family. A mild C-existence family for A corresponds to the abstract Cauchy problem having a mild solution for all initial data in the image of C. Note that an /-existence family is a strongly continuous semigroup. Consider the following example: ( A f ) ( z ) - z f ( x ) , (z e R), on some standard Banach space of functions on the real line, say C0(R.). Clearly A does not generate a strongly continuous semigroup, for (etn f ) ( x ) ~. etXf(x) defines an unbounded operator, since e t* is an unbounded function. Itowever, there are solutions of the abstract Cauchy problem for all initial data in a dense set, namely for any initial data f such that llmlxl...oo etX.f(x) = O, for all t _> O. We shall see that these solutions are accessible through an existence family. Since e-::2e t:~ is a bounded function of x, for all l > O,

defines a mild C-existence family, where C - W(0). This construction, along with the Fourier transform, shows that, if A generates a bounded strongly continuous group, then iA generates a mild C-existence family, for C - - c - A ~ , producing solutions of the abstract Cauchy problem, for all initial data in a dense set (see Chapter XII). Much more generally, we show that, for any matrix of polynomials of commuting generators of bounded' strongly continuous groups, there exists a bounded operator C, with dense range, such that that matrix of operators generates a C-regularized semigroup (see Chapters XIII and XIV). Just as we think of a strongly continuous semigroup generated by A as etA, and the solution of the abstract Cauchy problem (0.1) a s etAx, we may think of a g(A)-existence family as etAg(A), with the solution of the abstract Cauchy problem given by etAg(A)y, when x = g(A)y is in the image of g(A). When C commutes with A, as one would expect with C = g(A), then W(t) = etAC will satisfy W ( t ) W ( s ) = C W ( t + s); thus we are led to an algebraic definition, of a C-regularized semigroup. Perhaps the simplest and most intuitive construction of a g(A)-regularized semigroup generated by A is with the Cauchy integral formula (or an unbounded analogue),

etAg(A) = / e ' ~ g ( w ) ( w - A)-I 2ri'dw

(1.3)

where we integrate over a cycle surrounding the spectrum of A. For example, when - A generates a holomorphic strongly continuous semigroup, then this construction, with appropriate g, gives us an existence family for A. This can be applied to produce solutions, for all initial data in a dense set, of numerous ill-posed abstract Cauchy problems such as the backwards heat equation (see Chapter XII). This also leads to a spectral intuition, that generalizes the spectral intuition of strongly continuous semigroups. When A generates a strongly continuous semigroup etA, then we expect the complex-valued function z ~-, e tz to be bounded on the spectrum of A; thus the spectrum of A must be contained in a left half-plane. In fact, this is necessary but not sufficient; there is another sense (the numerical range, after an equivalent renorming) in which A must be contained in a left half-plane. To generate a g(A)-regularized semigroup, we expect the function z ~ etZg(z) to be bounded on the spectrum of A. This simple spectral intuition can be used to generate sufficient conditions, involving only the location and rate of growth of the resolvent (z - A) -1, that guarantee that A will generate C-regularized semigroups, with the corresponding information about the abstract Cauchy problem (see Chapter XXIII).

Another popular integral formula is the multivariable Fourier inversion formula =

dx,

where .~ is the Fourier transform. If iA1,..., iAn generate commuting bounded strongly continuous groups, we define, for any polynomial p, in n variables, a g(A)-regularized semigroup generated by p(A1,..., Am) by

etP(A)g( A) = (2~v)-~ /R- ei~"4"T(etP(U')g(Y))(Z)dx

(1.4)

(see Chapters XII, XIII and XIV). We may also use (1.2) to define etAg(A), or more generally, etAC: -

k=O

whenever

oo

k

llAkCtl

k=O

(See Chapters VII, VIII and IX.) Thus functional calculus constructions give us existence families. We may turn this around: we show how we may use regularized semigroups to define functional calculi for unbounded operators, via f(A) defined to be the generator of et/(A)g(A). That is, we define f(A) by defining a regularized semigroup that it generates; use (1.3) with e tA replaced by e tl(A) and e t~ replaced by e t/(~),

etf(A)g(A ) _ / et](W)g(w)(w_ A)_ 1 dw 2~ri ' or (1.4) with p replaced by f,

etI(A)g(A ) = (2~r)-~" JR-ei~'A'T(e'l(U'~g(g))(Z)dx. Because regularized semigroups are so much more general than strongly continuous semigroups, this enables us to define functional calculi for large classes of (unbounded) operators and large classes of functions (see Chapters XII and XXII).

II. E X I S T E N C E

FAMILIES

In this chapter, we introduce a family of operators that corresponds to the abstract Cauchy problem (0.1) having a solution, for all initial data in the image of a specified bounded operator C (see Theorems 2.6 and

4.13). Throughout this chapter, C will be a bounded operator. First, we should make it clear what we mean by a solution of the abstract Cauchy problem. D e f i n i t i o n 2.1. we mean a map t fying (0.1). By a C([0,~z),X) such

By a solution of the abstract Cauchy problem (0.1) ~ u ( t , z ) e C([O, oc),[:D(A)])NCl([O, o o ) , X ) , satismild solution of (0.1) we mean a map t ~-~ u ( t , x ) e that v ( t , x ) =- fo u ( s , x ) d s e :D(A), for all t > 0, and d - ~ v ( t , z ) = d ( v ( t , x ) ) + x, (t _> 0).

(2.2)

Note that automatically v e C([0, ~ ) , [•(d)]). We want the solutions to be well-posed, in some sense; we will discuss this more in Chapters IV and V. In stating precisely what it means to be well-posed: a small change in the initial data x should produce a small change in the solution u(t,z); we are led naturally to the notion of a bounded operator; specifically, we want the map x ~ u(t, x), which we think of a s e tA , to be bounded. This is despite the fact that we distinctly do not expect the operator A to be bounded. In fact, the existence of a unique mild solution of the abstract Cauchy problem, for all initial data, in some Frechet space, corresponds precisely to the map T ( t ) z =_ u ( t , x ) defining a strongly continuous semigroup of operators, {T(t)}t>o, generated by A (see Corollary 4.11). But once we leave this idealized situation, then the map t ~ u ( t , x ) is no longer a bounded operator. It may still be possible to think of e tA as an unbounded operator (see the Notes; also see Definitions 22.6 and 7.5 and Terminology 27.1). However, to regain something analogous to well-posedness, we really need a family of bounded operators. We wish to deal with the case where a large (possibly dense) set of initial data, that does not happen to equal the entire space, produces unique solutions or mild solutions of the abstract Cauchy problem. Suppose Ira(C) is contained in this set of initial data. We cannot expect the map x ~ u ( t , x ) to be bounded; however, it can be shown that, if A is closed, then z ~ u(t, Cx) is bounded (see Theorem 4.13). Thus we make the following definitions. Note that we think of W ( t ) as being etnC.

D e f i n i t i o n 2.3. The family of operators {W(t)}t>>.o C_ B(X) is a mild C-existence family for A if (1) the map t ~ W(t)z, from [ 0 , ~ ) into X, is continuous, for all x E X; and (2) for all x E X,t > 0,So W(s)xds E :P(A), with

A (~tW(s)xds) = W(t)x-Cx. In particular, u(t, Cx) = W(t)x is a mild solution of the abstract Cauchy problem, for any x E X. D e f i n i t i o n 2.4. The family of operators {W(t)}t>o C_ B([V(A)]) is a C-existence family for A if (1) the map t ~ W(t)x, from [ 0 , ~ ) i n t o [:P(A)], is continuous, for all x E :P(A); and (2) for all x E ~)(A),t >_O,

j~ot AW(s)x ds = W(t)x - Cx. In particular, u(t, Cx) =- W(t)x is a solution of the abstract Cauchy problem, for any x E X. D e f i n i t i o n 2.5. We will say that the mild C-existence family for A, {W(t)}t>_o, is a strong C-existence family for A if {W(t)lI~(A)l}t>_o is a C-existence family for A. It is obvious that these families are producing solutions of the abstract Cauchy problem (0.1), for certain initial data. We also gain uniform control over the solutions, an analogue of well-posedaess: small changes in the initial data produce small changes in the solutions; however, a different yardstick is used to measure the initial data, than is used to measure the solutions (see Theorem 2.6). Clearly the goal, both in producing as many solutions as possible and in obtaining more desirable analogues of wellposedness, is to choose C whose image is as large as possible. The choice of C tells us how far from well-posedness we are. In Theorem 4.13, we shall see that, when the solutions of the abstract Cauchy problem are unique, having a mild C-existence family is equivalent to the abstract Cauchy problem being well-posed in the strongest sense of strongly continuous semigroups, on a Frechet subspace, Z, continuously embedded in X, that contains the image of C (when the existence family

is exponentially bounded, Z will be a Banach space). In terms of wellposedness, this is saying that there exists a method of measurement (the topology of this subspace) with respect to which small changes in initial data produce small changes in the solution, with the same method of measurement being used. The choice of C also gives an approximation of the topology of Z, because the natural topology on the image of C, HYI[[Im(C)] -inf{[[x[[[Cx = y}, is stronger than the topology on Z. T h e o r e m 2.6. (1) Suppose there exists a mild C-existence family, {W(t)}t>_0, for A. Then there exists a mild solution of the abstract Cauchy problem (0.1), for all x e I m ( C ) . The sequence of solutions u(t, C z , ) W(t)xn -+ O, uniformly on compact subsets of [0, oo), whenever Xn --* O. (2) Suppose there exists a C-existence family, {W(t)}t>o, for A and A is closed. Then there exists a solution of (O.1), for all z 6 C CD(A)). Both u(t, Cxn) =- W(t)xn and Au(t, Cxn) converge to O, uniformly on compact subsets of[0, c~), whenever Az,,~ and z,~ converge to O. See Theorem 4.13 for a converse to Theorem 2.6(a); in fact, the analogue of well-posedness will follow automatically, when A is closed and (0.1) has a mild solution, for all x 6 Ira(C). It is natural to want to know when a mild existence family is a strong existence family. The following proposition states a correlation between W ( t ) x being differentiable and x being in the domain of A. Coronary 2.8 then provides sufficient conditions for a mild existence family to be a strong existence family. P r o p o s i t i o n 2.7. Suppose A is closed, {W(t)}t>o is a mild C-existence family and the map t ~ W(t)x, from [0,co) into X , is differentiable at t = to. Then W(to)x 6 I)(A) and AW(to)x = d(w(t)x)lt=to. C o r o l l a r y 2.8. Suppose A is closed, {W(t)}t>0 is a mild C-existence family for A and either

(1) the map t W(t)x e for all x e V ( 3 ) ; (2) the map t ~ W ( t ) x 6 C([0,oo),[V(A)]), for d l x 6 V(A); or (3) W ( t ) A C_ A W ( t ) , for all t >_ O. Then {W(t)}t>o is a strong C-existence family for A. A strongly continuous semigroup is defined algebraically, but it can be shown to be an/-existence family and a mild/-existence family. In the

next section, we will show that, when A and C commute, then an algebraic definition (regularized semigroup; see Definition 3.1), generalizing the algebraic definition of a strongly continuous semigroup, is equivalent to the definition of a (mild) C-existence family (see Theorems 3.5(1), 3.7 and 3.8). In general, we are not guaranteed uniqueness of the solutions. However, when A commutes with its existence family, we are (see Theorems 3.5(2) and 3.7(a)). If we focus on exponentially bounded solutions, then we are always guaranteed uniqueness, that is, at most one exponentially bounded solution. P r o p o s i t i o n 2.9. Suppose A is closed and there exists w 6 R such that A has no eigenvalues in (~o,~). Then alt exponentially bounded solutions and mild solutions of the abstract Cauchy problem (0.1) are unique. L e m m a 2.10. Suppose A is closed, u(t) is a O(e "t) mild solution of(O.1) and Re(z) > w. Then f ~ e-Ztu(t)dt 6 7?(A) and

(z - A) Proof:

e-Ztu(t) dt = x.

~0(X)

Fix z such that Re(z) > w and let L =_ f ~ e - Z t u ( t ) d t .

An

integration by parts shows that L equals z f ~ e -zt fg u(s)dsdt; thus, since A is closed, L is in D(A) and AL = z f ~ e-ZtA(fo u(s)ds)dt = z I0~ e - z ' ( u ( t ) - x ) ~ t = z L - ~ . I E x a m p l e 2.11. On X1 x X2, define

G2 '

V(A) --- v ( a , ) x v ( a ~ ) ,

where D(G2) C_ D(B), B is a closed operator from (a subspace of) X2 into XI, and Gi generates a strongly-continuous semigroup, for i = 1,2. Then there exists a strong C-existence family for A, for s sufficiently large, where C -

(s -

0]

a2)

-1

"

R e m a r k 2.12. The operator A, of Example 2.11, generates a (A A)-l-regularized semigroup (see Definition 3.1), for A sufficiently large, and an integrated semigroup (see Chapter XVIII). Either of these facts yields unique solutions of the abstract Cauchy problem, for initial data 10

in D(A 2) = :D(G~) x T~(G~). Example 2.11 yields unique solutions of the abstract Cauchy problem, for initial data in C(V(A)) = V(G1) x V(G~). P r o o f o f T h e o r e m 2.6: It is obvious that u(t, Cy) = W(t)y is a (mild) solution of (0.1), with z = Cy, if {W(t)}t>.o is a (mild) C-existence family for A. By the uniform boundedness principle, IIW(t)ll is bounded on compact subsets of [0, cr if {W(t)}t>o is a mild C-existence family, and IlW(t)lltv(A)l is bounded on compact subsets of [0,oo), if {W(t)}t>o is a C-existence family and A is closed. This implies the analogues of well-posedness in (a) and (b). II P r o o f of P r o p o s i t i o n 2.7: For any k 6 N, let xk =- kft~ +~ W(s)xd8. Then zk converges to W(to)z, while axk = k(W(to + ~)z-W(to)z) converges to as k ~ oo. Thus this result follows from the fact that A is closed. II

-~W(t)zlt=to,

P r o o f of C o r o l l a r y 2.8: For (1), this follows from Proposition 2.7, since we may then integrate mW(s)z = dW(s)z, from 0 to t. For (2), this follows from the fact that A is closed and s ~ AW(s)z, from [0, ~ ) ~ X, is continuous, so that W(s)zds) fo AW(s)zd8, for x 6 :D(A). Note finally that hypothesis (3) implies hypothesis (2). |

a(f~

=

P r o o f of P r o p o s i t i o n 2.9: Suppose u is an exponentially bounded solution or mild solution of (0.1), with u(0) = 0. Then by Lemma 2.10, its Laplace transform is trivial, for Re(z) sufficiently large, which implies that u = 0, as desired. It P r o o f o f E x a m p l e 2.11:

Formally,

e tG2

0

"

To make this bounded, we need an (8 - G2) -1 to the right of B. This explains our choice of C. For t > 0, let

W(t)= [et: 1 fo e(t-'~ e C2(8 - C2)-1

.

For any continuously differentiable f : [0, oo) ~ Xi (i = 1, 2), t _> 0, it is well known that fte~G'f(r)dr 6 V(Gi), with

Gi (~terG'f(r) dr) -- etG'f(t)-- f(O)- ~oterG'f'(r) dr 11

(2.12)

(see Theorem 3.4(c)). This implies that, for any z 6 X, x X2,t >_ 0, ft W(s)zds 6 D(A), and a calculation, using (2.12), implies that W(t) satisfies Definition 2.3. Thus {W(t)}t>o is a mild C-existence family for A. Also by (2.12), if x 6 ~D(A), and t > 0, then W(t)x 6 D(A), and a calculation shows that

AW(t)=

= [e'a'(Clzl+ B(s-G2)-"z2) + foe(t-'~)G*B(s-a2)-*ewG'G2z2dw ] (s - G2)-'(eta'a2z2)

J

This is a continuous function of t. Thus t ~ W(t)z is a continuous map from [0, oo) into [D(A)], for all z in D(A). It is straightforward to show that A is closed. Thus, by Corollary 2.8(2), {W(t)}t_0 is a strong C-existence family for A. I

12

III. R E G U L A R I Z E D S E M I G R O U P S

Throughout this chapter, C will be a bounded, injective operator. We think of a C-existence family as being W(t) = etAC. When etAC = Ce tA, a very short calculation shows (at least if A is a complex number) that CW(t + s) = W(t)W(s). Thus we are led to the following algebraic definition, which is obviously analogous to the definition of a strongly conthiuous semigroup (an/-regularized semigroup). Definition 3.1. The strongly continuous family {W(t)}t_>0 C_ B(X) is a C-regularized semigroup if (1) W(O)= C, and (2) W(t)W(s) = CW(t + s), for all s,t > O. A generates {W(t)}t>>.o if Ax = C -1

1

(W(t)x

- Cx)

]

,

with ~D(A) = {z I the limit exists and is in ha(C)}. When we do not wish to specify the C, we will say merely regularized

semigroup. We will show that, if A generates a strongly continuous semigroup

{etA}t>_O that commutes with C, then A is the generator of the C-regularized semigroup {CetA}t>_O. More generally, if .4 generates a Cl-regularized semigroup {W(t)}t>_o that commutes with C~, then A generates the C1C2regularized semigroup {C2W(t)}t>_o (Proposition 3.10). E x a m p l e 3.2. The following simple example demonstrates how broad a class of operators and abstract Cauchy problems may be dealt with using regularized semigroups. Define Af(z) = zf(z), on C0(R). Then

W(t)f(x) =_e-~2et~f(:r) is a W(0)-regularized semigroup generated by .4. A similar construction, aided by the Fourier transform or the Cauchy integral formula, shows that, if L4 generates a strongly continuous group, then .4 generates an e-A2-regularized semigroup (see Chapter XIV and Example 22.24). 13

E x a m p l e a n d R e m a r k s 3.3. Example 3.2 addresses the first objection to regularized semigroups: "Does anything generate a regularized semigroup that does not generate a strongly continuous (or integrated; see Chapter XVIII) semigroup?" The subsequent objection is usually: "Everything generates a regularized semigroup, therefore it's a meaningless concept; to say that something generates a regularized semigroup says nothing." Here is a simple example of an operator that does not generate a regularized semigroup. Define A f =_ fl, on X - {f E C[0,1]If(0 ) = 0}, with maximal domain. Then it is not hard to see that the abstract Cauchy problem (0.1) has no nontrivial solutions; thus by Theorem 3.5, A cannot generate a Cregularized semigroup, for any C. So not every closed operator generates a regularized semigroup. If the abstract Cauchy problem has mild solutions, for all initial data in the image of C, and C commutes with A, then an extension of A generates a C-regularized semigroup (see Theorem 4.15). Thus information about generating a regularized semigroup is the most fundamental information, telling us whether or not to expect nontrivial solutions of the abstract Cauchy problem. Much more than this, the choice of C tells us how many solutions to expect (see Theorems 3.5, 3.13 and 3.14); the larger the image of C, the more solutions we are guaranteed. Thus, if we say, instead of, "A generates a C-regularized semigroup, for some C," rather, "A generates a C-regularized semigroup, for this specific choice of C," then we have gained very specific and worthwhile information about A and the corresponding abstract Cauchy problem. Rather than merely saying, for a given A, that the abstract Cauchy problem is or is not well posed (does or does not generate a strongly continuous semigroup), it is more constructive to identify C such that A generates a C-regularized semigroup; the choice of C provides a continuous scale of measurements of how far from being well-posed the abstract Cauchy problem is. As we shall see later (Chapter IV), a C-regularized semigroup generated by A may be used to construct a Frechet space (a Banach space if the regularized semigroup is exponentially bounded), Z, continuously embedded in X, on which A generates a strongly continuous semigroup, so that the usual form of weU-posedness occurs. Again the choice of C is informative, because the image of C will be contained in Z. T h e o r e m 3.4. Suppose {W(t)}t>_o is a C-regularized semigroup gener-

ated by A. Then (a) A is closed; (b) Im(C) C_:D(A); 14

(c) if f : [O, oo) --. X is continuously differentiable and t >_O, then

f ~ W ( s ) f ( s ) d s e :P(A), with A [fotw(s)f(s)ds]=W(t)f(t)-Cf(O)-fotW(s)f'(s)ds; and (d) for all t >_O, W(t)A C_ AW(t), with

w ( t ) z = Cz +

W(s)Ax ds,

for all ~ E ~( A ). It is clear from Theorem 3.4(c) and (d) that {W(t)}t>o is a strong Cexistence family for A. Thus Theorem 2.6 guarantees the existence of solutions and mild solutions of the abstract Cauchy problem. With regularized semigroups, we are also guaranteed uniqueness of the solutions, even if merely an extension of A generates a C-regularized semigroup. T h e o r e m 3.5. (1) If {W(t)}t>_o is a C-regularized semigroup generated by A, then {W(t)}t>o is a strong C-existence family for A. (2) If an extension of A generates a regularized semigroup, then all solutions and mild solutions of the abstract Cauchy problem (0.1)

are unique. In Theorem 3.7 we give a partial converse to 3.4(c) and (d). First we must introduce the C-resolvent set, which will play the same role with C-regularized semigroups that the resolvent set plays with strongly continuous semigroups (see Chapter XVII). D e f i n i t i o n 3.6. The complex number A is in pc(A), the C.resolvent set of A, if (A - A) is injective and Im(C) C Im(A - A). It will often be the case that a regularized semigroup will be generated by a proper extension of the operator of interest. In the next two theorems, we investigate the relationship between having an existence family and having an extension that generates a regularized semigroup. In (b) of the following, note that, whenever {W(t)}t>0 is an exponentially bounded C-regularized semigroup, then pc(A) is nonempty (Proposition 17.2). 15

T h e o r e m 3.7. Suppose A is closed and {W(t)}t>0 is a strongly continuous family of bounded operators, with W(t)A C_AW(t), for a/l t > 0. (a) If {W(t)}t>_o is a mild C-existence family for A, then {W(t)}t>_o is a C-regularized semigroup generated by an extension of A. (b) If {W(t)}t>o is a C-existence family for A, then {W(t)}t_>0 is a C-regularized semigroup generated by an extension of A, if either (1) ~)(A) is dense; or (2) pc(A) is nonempty.

Theorem 3.8. Suppose {W(t)}t_>0 is a C-regularized semigroup, generated by an extension of A. Then (1) if f ~ W ( s ) x d s e V(A), for all t >__O,z e X, then {W(t)},>_0 is a mild C-existence family for A; (2) i f W ( t ) leaves 7)(A) invariant, for all t > O, then {W(t)}t>o is a C-existence family for A; and (3) if, in addition to (2), A is closed and densely defined, then {W(t)}t>_o is a strong C-existence family for A. The solution or mild solution is given by u(t, Cz) = W(t)z. For the abstract Cauchy problem, Theorems 3.8 and 2.6 indicate how the distinction between an extension of A being the generator and A itself being the generator does not seem crucial; this is why the concept of an existence family (Chapter II) is in many ways more natural for dealing with the abstract Cauchy problem. However, it is of interest to know when A itself is the generator. For example, in future chapters we will define functional calculi, f ~ f(A), for A, by constructing, for each function f, a regularized semigroup and then defining f ( A ) to be the generator. Note that the hypotheses of Proposition 3.9 are satisfied when p(A) is nonempty and W(t)A C.C_AW(t), for all t > 0.

Proposition 3.9. Suppose an extension of A generates the regularized semigroup {W(t)}t>0 and there exists bounded, injective G such that 79(A) = Im(G),I)(A 2) = 1re(G2), and GW(t) = W(t)G, for a/l t > 0. Then A is the generator. The following proposition shows that there is no harm in applying additional smoothing, that is, the generator remains the same.

Proposition 3.10. Suppose {W(t)}t>o is a Cl-regularized semigroup generated by A and C~W(t) = W(t)C2, for o21 t > O. Then {CzW(t)}t>_o is a C1C2-regularized semigroup generated by A. 16

Proposition 3.11. Suppose an extension of A, A, generates a C-regularized semigroup. Then the following are equivalent.

(1) c (z,(3)) c_ Z~(A). (2) C -1AC = ~t. The hypothesis on D(A) in the preceding proposition does not seem like a natural one, since it requires extensive knowledge about the generator, .4. In practice, all that we may have information about may be A itself. The following Corollary provides a sufficient condition that is easier to verify; in particular, if the regularized semigroup is exponentially bounded, then pc(ft) automatically contains a left half-plane (see Proposition 17.2). C o r o l l a r y 3.12. Suppose A is dosed, A C_A, fl generates a C-regularized semigroup and pc(A) n pc(A) is nonempty. Then fl = C-1AC. It is interesting that we may use a regularized semigroup generated by A to characterize all solutions of the abstract Cauchy problem. We also obtain uniform control over the "smoothed" or regularized solutions Cu, for all initial data x for which the abstract Cauchy problem has a solution. See Chapters IV and V for bona fide well-posedness of the solutions, at least on a subspace, when A generates a regularized semigroup. In the following two theorems, any C such that A generates a Cregularized semigroup may be used. This is useful, because, if one allows the image of C to be small, it is easier to obtain a C-regularized semigroup; but regardless of how restrictive C is, we still may use the C-regularized semigroup to obtain all solutions of the abstract Cauchy problem; that is, it is not necessary to try to choose an optimal C, in any sense. T h e o r e m 3.13. Suppose A generates a C-regularized semigroup {W(t)},>o. Then the following are equivalent: (1) there exists a mild solution of the abstract Cauchy problem (0.1); (2) W(t)x 9 Im(C), for a/l t ~ 0, and t -- C-1W(t)x, from [0,ec)

into X, is continuous. The solution is given by

u(t,x)=c-lw(t)x. Whenever {x~}~=l is a sequence of initial data, for which (0.1) has a solution, converging to zero, then Cu(t, xn) converges to zero, uniformly on compact subsets of [0, co).

17

T h e o r e m 3.14. Suppose A generates a C-regularized semigroup {W(t)}t_>0. Then the following are equivalent: (1) there exists a solution of the abstract Cauchy problem (0.1); (2) W(t)x e C (:D(A)), for all t >_0 and t --* C-1AW(t)x, from [0,oo) into X, is continuous; and (3) W(t)x 9 Ira(C), for all t >_0 and t ~ C - ' W ( t ) x , from [0, cr X , is differentiable. The solution is given by u(t,x) = C-1W(t)x. Whenever {xn} is a sequence of initial data, for which (0.1) has a solution, converging to zero, then Cu(t, xn) converges to zero, uniformly on compact subsets of [0, ec).

P r o o f of T h e o r e m 3.4:

!

(d) Fix t >_ 0, z e P(A). For s _> 0,

(w(t)z)]= w(t) [~( w ( s ) x -

cx)],

so that, since W(t) is bounded, lim 1 [W(s)(W(t)x) - C (W(t)x)] s---* 0 S

exists and equals W(t)CAx = CW(t)Ax. This implies that W(t)z 9 •(A), with AW(t)x = W(t)Ax. Note also that ! [ c w ( s + t)~ - c w ( t ) ~ ] = ! [W(s)W(t)x - c w ( t ) ~ ] , S

S

so that dCW(s)x[s=t exists, and equals CW(t)Ax. Thus, for t > 0,x 9 /)(A), CW(t)x = C2x +

/0

CW(s)Axds = C

I'

W(s)Axds,

since C is bounded. Since C is injective, this implies (d). oo (a) Suppose {X ~},~=1 C_ 7)(A), with x,~ ~ z and Az,, ~ y. Fix t > 0. By the uniform boundedness principle, [IW(s)[[ is bounded on [0,t]. Thus, {W(s)xn) converges to W(s)x, and { d W ( s ) x n } = {W(s)Axn} (by (d)) 18

converges to W(s)y, as n --* oo, both uniformly on [0,t]. This implies that

W ( t ) z - Cz = lim (W(t)z,, - Czn) n -..~ oo

= lim

=

/o'

W ( s ) A x n ds

W(s)yds,

so t h a t limt-.,0 1 ( W ( t ) x - Cx) exists and equals Cy. This implies that x 9 79(A), with Ax = y, as desired. (c) For fixed t, let z - f o W ( s ) f ( s ) d s .

:[/0

lh ( W ( h ) x - Cx) = -~

( C W ( s + h) - C W ( s ) ) f ( s ) d s

l[Z+

= -~

= -s

1/'+'

limh..-,o

Jo

CW(s)f(s)ds

]

W(w)f(w - h)dw-

:foh

-~

CW(s)f(s)ds,

exists, with

lim 1 ( W ( h ) z - C x ) =

h---*O -h

CW(w)f(w - h)dw-

]

CW(s)(f(s-h)-f(s))ds

+ -h at so t h a t

For h > 0 ,

j0 t - C W ( s ) f ' ( s ) d s + C W ( t ) f ( t ) - C 2 f ( O ) .

Thus z E D(A), with Ax = - f t W ( s ) / ' ( s ) d s + W ( t ) f ( t ) - C f(O). I (b) For all x 9 X , limt__,0 ~ f o W ( s ) x d s = W(O)x = Cx. Thus, (b) follows from (c). I P r o o f o f T h e o r e m 3.5: (a) is clear from Theorem 3.4. For (b), suppose ft, is the generator of the C-regularized semigroup {W(t)}t>o and A C_ ill. Suppose v'(t) = A(v(t)), for all t > 0 and v(0) = 0. For 0 < s < t,

---~W(t - s)v(s) = W ( t - s)A(v(s)) - fi~W(t- s)v(s) = O, 19

by Theorem 3.4. Thus Cv(t) = W(t)v(O) = O, so that, since C is injective, v(t) =_O, proving uniqueness. | P r o o f o f T h e o r e m 3.T: (a) Since A is closed and W(t)A C_AW(t), by Corollary 2.8, {W(t)}t>0 is a strong C-existence family for A, with

w(t)~ = c~ +

W(~)Axe~,

(3.15)

for all t >_O, x E "17(.4), as in (b). To see that {W(t)}~>_0 is a C-regularized semigroup, suppose 8, t >_ 0 and x E X. We calculate as follows:

/s+t

W(r)Cx dr =

~L

W(s + t - w)Cx dw

.18

=

~

w ( s + t - w)

(/0

W(r)~dr

)]

dw

= W(8) ( f o t W ( r ) x d r ) , so that differentiating with respect to s implies that

W(s+t)Cx-W(s)Cx=

W(s)d

(/0

W(r)xdr

)

= W(8)(W(t)x-Cx).

Thus {W(t)}t>o is a C-regularized semigroup. It is clear from (3.15) that its generator is an extension of A.

(h) As in (a), W(t)W(,)~ = CW(t + ~)~, W(0)~ = C~, W e ~(A), , , t > 0. If :D(A) is dense, then the same is true for all x e X, since W(t) and C are bounded. If there exists A e pc(A), then, since W(t)A CC.AW(t), W(t) commutes with (~ - A)-IC; thus, for any x e X,

()~- A)-IC[W(t)W(s)x] = W(t)W(s)()~- A ) - l c x = CW(t + s ) ( A - A)-ICx = (~- A)-Ic[cw(t

+

,)~],

so, since ( A - A)-IC is injective, {W(t))f>0 satisfies (2) of Definition 3.1. An identical argument shows that W(0) = C. Assertion (3.15) implies that the generator is an extension of A. II 2O

P r o o f of T h e o r e m 3.8: Assertions (1) and (2) follow immediately from Theorem 3.4. For (3), all that needs to be shown is that

o~W(s)xds 6 :D(A),

(3.16)

for all x 6 X , t ~_ 0. Since A is closed and W(t)A C_ AW(t), (3.16) is OO true for x 6 7)(A),t >_ O. For arbitrary x 6 X , t >_ O, let < xn >n=0 be a sequence in :D(A) converging to x. Then f~ W(s)xn ds converges to f~ W(s)x ds, while

A (1tW(s)x~ds)

= W(t)xn-Cxn

(Theorem 3.4) converges. Thus, since A is closed, (3.16) follows. I P r o o f o f P r o p o s i t i o n 3.9: Suppose x 6 :D(A). Then CG~x = GC~

= G

Let ,4 be the generator of {W(t)}t>0.

W(t)~l~=o

= ~W(t)e~l,=o = C ~ G ~

= CAGx.

Since C is injective, Gflx = AGx, thus AGx 6 ~(A), so that Gx 6 7)(A2). This means there exists y 6 X such that Gx = G2y, so that, since G is injective, x = Gy 6 7)(A), as desired. I P r o o f o f P r o p o s i t i o n 3.10: It is clear that {C2W(t)}~>o is a C~C2regularized semigroup. Let A1 be its generator. Since C2 is bounded, it is easy to show that A C_ A1. Suppose x 6 T)(A1). Then, for all t >_ 0, by Theorem 3.4(d), c2 ( N ( t ) ~ - C1~) =

i'

C2W(~)A~e~

= C~ ( ~ ' W ( s ) A l x d s )

;

since C~ is injective, this implies that (w(t)x - cx) =

W ( s ) A l ~ es,

which implies that x E 77(A), with .4z = Alx, as desired. I 21

P r o o f of P r o p o s i t i o n 3.11: (1) ~ (2). Note that by definition of the generator, C-I~iC is the generator of the C2-regularized semigroup {CW(t)}t>o; thus, by Proposition 3.10,

C - 1 A C = ft.

(3.17)

Let V ( B ) - C ( V ( A ) ) , B x = Ax, forallx E V(B). Suppose x E :/)(A). Then Cx e D(B) and B(Cx) = 7iCx = CTix, by Theorem 3.4(d), thus x E D ( C - a B C ) and C - 1 B C x = Ax. This is saying that

~i C_ C - 1 B C C_ C - 1 A C C_ C - I ~iC = A, by (3.17). This proves (2). (2) - . (1) is obvious from the definition of I)(C-1AC). 1 P r o o f of C o r o l l a r y 3.12: By Proposition 3.11, it is sufficient to show that C (:D(.4)) C_ T)(A). There exists r e pc(A) npc(~i). If x e :/)(A), then c x = c ( ~ - ~ ) - 1 ( ~ _ ~ ) ~ = (T - A ) - I c ( T

- A ) x = (~ - A ) - ' C ( r

- ~)~,

which is in ~(A). | P r o o f of T h e o r e m 3.13: Suppose (0.1) has a mild solution, u. Let v(t) =_ f~ u(s)ds. Then, since C is bounded and CA C AC,

d (cv)(t) = A(Cv(t)) + Cx Vt > O. By Theorem3.5(2),

Cu(t) = C v(t)=

=

which implies that t ~ C - 1 W ( t ) x is continuous and u(t) = C - l W ( t ) x . Conversely, suppose W(t)x E Im(C), for all t _> 0, and t ~ C -x W(t)x is continuous. Let v(t) =_ fd C - 1 W ( s ) x ds. Then by Wheorem 3.4, Cv(t) e D(A), with ACv(t) = W(t)x - Cx; thus v(t) E •(C-1AC), with c-'

ACv(t)=

C-1W(t)x-

9 = v'(t)-

~.

Since, by Proposition 3.11, A = C-1AC, this concludes the proof. | 22

P r o o f o f T h e o r e m 3.14: Suppose (0.1) has a solution, u. Then by Theorem 3.5(2), Cu(t) = W(t)x, so that W ( t ) x ~ C(:D(A)), for all t >_ 0, with = d C_IAW(t)x, so that t ~ C - 1 A W ( t ) x is continuous and t ~ C - 1 W ( t ) x = u(t) is differentiable. Conversely, if t ~ C - 1 A W ( t ) z is defined and continuous, let u(t) C - 1 W ( t ) x . As in the proof of Theorem 3.13, u(t) E :D(A) = :D(C-1AC), with Au(t) = C - ~ A W ( t ) x , so that, since C -1 is closed,

A u ( s ) d s = C -1

]0

AW(s)xds=C-l(W(t)x-Cx)=u(t)-x,

thus u is a solution of (0.1). If t ~ C - 1 W ( t ) z is differentiable, then by Proposition 2.7, Cu(t) e :D(A), with ACu(t) = Cu'(t); thus u(t) e :D(C-1AC) = :D(A), with

Au(t) = C-1ACu(t) = u'(t); thus again u is a solution of (0.1). |

23

IV. T H E S O L U T I O N S P A C E OF A N O P E R A T O R AND AUTOMATIC WELL-POSEDNESS

Many physical problems may be modeled as an abstract Cauchy problem (0.1), where A is an operator on a locally convex space. It is well known that, in order for this model to have any practical value, it is not enough to have plenty of solutions; (0.1) should also be well-posed; informally, this means that small changes in x, the initial data (corresponding to small errors in measurement) should yield small changes in u(t, x). Although there is sometimes uncertainty about what precisely constitutes well-posedness, one condition that most would agree implies wellposedness is having A generate a strongly continuous semigroup. This implies that a sequence of solutions u(t, zn) converges to u(t, x), uniformly on compact subsets of [0, c~), whenever the initial data zn converge to x. It is often the case that A does not generate a strongly continuous semigroup. Consider the following list, which includes what are perhaps the most well-known partial differential equations: heat equation, SchrSdinger equation, wave equation, Cauchy problem for the Laplace equation, backwards heat equation, all on LP(g/), for appropriate ~ C_ It'*, with appropriate boundary conditions. For each equation, when written as an abstract Canchy problem (0.1), let us ask whether the operator A that appears generates a strongly continuous semigroup, for 1 _< p < c~. The answers are, respectively, "yes," "sometimes," "sometimes," "no" and "no." Yet in all these cases, a unique solution exists, for all initial data in a dense set. The class of operators that generate C-regularized semigroups is much larger than the class of operators that generate strongly continuous semigroups. In all the examples in the previous paragraph, we shall see in later chapters that A generates a C-regularized semigroup. The choice of C measures how ill-posed the problem is. When C is chosen to be ()~-A) -n, for some n E N, then i/k, on LV(I:tk), for 1 _< p _< c~,p i~ 2, may be shown to generate a C-regularized semigroup, but not a strongly continuous semigroup (see Chapter XI). This yields solutions of the SchrSdinger equation, for all initial data in the domain of A n+l. Much "worse" operators, corresponding to what are traditionally referred to as ill posed or improperly posed problems, generate C-regularized semigroups. For example, if A = - / k , so that (0.1) becomes the backwards heat equation, then A generates a C-regularized semigroup (see Chapter VIII). If A - [_~A 0/],so that (0.1) becomes the Cauchy problem for the Laplace equation, then A generates a C-regularized semigroup (see Chapter IX). h

J

24

When Im(C) is dense, as is the case in the examples above, then (0.1) has a unique solution for all x in a dense set. Thus, C-regularized semigroups may often be used to produce unique solutions for all initial data in a dense set. In this chapter and the next, we will address the question of whether the solutions are well-posed, in some sense. In this chapter, we show that, if A is closed and there exists a nonempty set of initial data for which the abstract Cauchy problem (0.1) has a unique mild solution, then there exists a Frechet space, Z, that contains all such initial data, on which A generates a strongly continuous semigroup. This is saying that we can always make (0.1) well-posed, regardless of how ill-posed our original formulation was. And if one adopts the working hypothesis that all physically correct models are well-posed, this construction may be considered a way of automatically correcting one's first guess; the space on which A acts is often chosen out of convenience. As a corollary, we obtain a much shorter, easier proof, than currently exists, of the well-known relationship between the abstract Cauchy problem and strongly continuous semigroups (Corollaries 4.11 and 4.12). This is a fundamental result that has seen many proofs, all of them somewhat involved, even when X is a Banach space. More generally, we use the solution space to show that, for any bounded operator C, the existence of a unique solution of (0.1), for all initial data in the image of C, corresponds to A having a mild C-existence family (see Chapter II). When A and C commute and C is injective, this is a C-regularized semigroup (see Chapter III.) The only objection to the solution space is that it is not practical, in general, to try to construct it explicitly. C-existence families and Cregularized semigroups provide a simple method of approximating the solution space and its topology, as follows. We will write Y ~ X to mean that Y is continuously embedded in X, that is, Y C_ X and the identity map from Y to X is continuous. We show that, when A generates a C-regularized semigroup, then Jim(C)]--~ Z ~ X, and AIz , the restriction of A to Z, generates a strongly continuous semigroup. If the regularized semigroup is exponentially bounded, then we shall see in the next chapter that we may choose a Banach space; in general, Z is a Frechet space. If p(A) is nonempty, the converse is also true. The norm on the solution space Z can then be expressed in terms of the regularized semigroup: Ilxllz --" sup{llC-1W(t)xlllt > 0}, where {W(t)}t>0 is the C-regularized semigroup generated by A. This chapter shows that, in a technical sense, at least if one is willing to make renormings, the concepts of existence family and strongly continuous 25

semigroup are the same. But there are important practical differences. An existence family is often very easy to produce and construct, and one does not have to leave the original norm, which may be very simple or physically meaningful. The solution space, on which the restriction of A generates a strongly continuous semigroup, may be very difficult to construct, with a norm that is unpleasant or impossible to work with. Throughout this book, there are numerous examples of regularized semigroups that are constructed in the most simple-minded and intuitive manner. Even if one's only goal is to find subspaces, Y, on which the restriction of A, Air , generates a strongly continuous semigroup, we submit the following algorithm. First find a regularized semigroup generated by A. Then use the construction of this chapter (Proposition 4.14) to produce Y. This chapter makes it clear how regularized semigroups may be used, first, to characterize all initial data that yield a solution of the abstract Cauchy problem (0.1) (see Proposition 4.14), then, to find a norm with respect to which those solutions are well-posed. Perhaps more importantly, for practical purposes, is the fact that the choice of C, when A generates a C-regularized semigroup, tells one how to measure how far from well-posedness one is, and to approximate the space on which A is well-posed. In this chapter only, the space X will always be a Frechet space, topologized by the seminorms {H Hj}jEN. We shall see that this is the natural generality for these results (see Remark 4.9 and Example 4.10). In the next chapter, we will discuss how to obtain well-posedness on a continuously embedded Banach subspace, when the original space is a Banach space. Throughout this chapter, A will always be a closed operator, C E B(X). D e f i n i t i o n 4.1. We will write Z ~ X to mean that Z C X and Z i s continuously embedded in X, that is, the identity map, from Z into X, is continuous. D e f i n i t i o n 4.2. By AIz we will mean the part of A in Z, that is, :D(AIz ) equals {x e / ) ( A ) N Z IAx e Z} and AIzx = Ax, for all x e I)(AIZ). D e f i n i t i o n 4.3. We will write [/)(A)] for the Frechet space 7)(A) with the graph topology, that is, topologized by the seminorms

IIxlltv(A)],5 =-IIxllJ + IIAzll . We will write [Im(C)] for the Frechet space topologized by the seminorms

Ilx[l[zmtC)l,j - i n f { l l Y l l j l C y 26

- x}.

D e f i n i t i o n 4.4. By C([0, or ogized by the seminorms llr

X) we will mean the Frechet space topol-

a, b ~ Q + , j E N.

sup I]r tell,b]

D e f i n i t i o n 4.5. The strongly continuous family of bounded operators {T(t)}t>o, on a locally convex space Y, is a locally equicontinuous semigroup if T(O) = I, T(t)T(s) = T(s q- t), for all s,t >_ 0 and for all s < co, {T(t)[ 0 < t < s} is equicontinuous. The operator A generates {T(t)}t>o if

Ax = lim -1 ( T ( t ) x - x), t-+O t

with maximal domain, that is,/)(A) equals the set of all x for which the limit exists. D e f i n i t i o n 4.6. Suppose all solutions of the abstract Cauchy problem (0.1) are unique; that is, there are no nontrivial solutions of (0.1), when x = 0. We will denote by Z the solution space for A, which we define to be the set of all x for which the abstract Cauchy problem (0.1) has a mild solution. We topologize Z with the family of seminorms ll llo,bj -- sup

tel~,b]

llu(t, )llJ,a,b e

Q+,j e N,

where t ~-. u(t,x) is the unique mild solution of (0.1). It is convenient to note that Z is topologically equivalent to a subspace of C([0,oo),X), via the embedding

(Az)(t) -

u(t,z).

(4.7)

T h e o r e m 4.8. The space Z is a Frechet space and T(t)x = u(t,x) is a locally equicontinuous semigroup on Z generated by A[z. R e m a r k 4.9. When X is a Banach space, then there will, in general, be no Banach solution space, that is, no Banach space that contains all solutions of the abstract Cauchy problem, on which A generates a strongly continuous semigroup, because a strongly continuous semigroup on a Banach space is automatically exponentially bounded (see Example 5.8). We shall see later (Theorem 5.5) that, for fixed w E It, there exists a Banach space on which A generates a strongly continuous semigroup, which 27

will contain all x for which the abstract Cauchy problem has a unique O(e t('~-~)) solution, for any e > 0. To get all solutions of the abstract Cauchy problem, even when X is a Banach space, we must pass to a Frechet space. Since our solution space is still a Frechet space when X is, this seems to be the natural setting for our results in this chapter. Here is a simple example of an operator, A, for which Z contains :D(A), but the abstract Cauchy problem has no nontrivial exponentially bounded solutions. Let f~ _ {x + i y t x >_ O,e x2 o. Then z = {.l w ( t ) .

9 Im(C),Vt > O, and t ~ C-1W(t)x is continuouous },

and

IIxllo,b,~= sup IIC-1W(t)xll~, for all j 9 N,a,b 9 Q.

T h e o r e m 4.15. Suppose X is a Banach space, p(A) is n o n e m p t ~ C is injective and CA C_ AC. Then the following are equivalent. (a) The abstract Cauchy problem (0.1) has a unique solution, for all

9 C(D(A)). (b) The abstract Cauchy problem has a unique mild solution, for all x 9 Ira(C). x

29

(c)

All solutions of the abstract Cauchy problem are unique and [Ira(C)] ~ Z.

(d) The operator A generates a C-regularized semigroup. (e) All solutions of the abstract Cauchy problem are unique and there exists a mild C-existence family for A. (f) There exists a mild C-existence family for A, {W(t)}t>.o, such that W(t)A C_ AW(t), for all t >_O. The C-regularized semigroup generated by A is then given by

W(t)=T(t)C, where T(t) is as in Theorem 4.8. Examples of operators that generate C-regularized semigroups are i/k, on LP(It~), 1 < p _< ec (with C -= (1 + i/k) -k, where k depends on p and n ) ; - / k , on LP(fl)(with C = e -(zx)2) and [ ~ Lp(f~)• LP(f~)(with C =- e -(zx)2) (see Chapters XI, VIII and IX). Note that the corresponding abstract Cauchy problems are the Schr6dinger equation, the backwards heat equation and the Cauchy problem for the Laplace equation. Because these are so ill-posed, Theorem 4.15 might be surprising. Note that in the three examples just given, the image of C is dense in X. E x a m p l e 4.16. Let A be as in Example 3.2,

(Af)(x) =_xf(x), with maximal domain, on C0(It)). Then Z equals the set of all continuous f : It -* C that decay more rapidly on [O,c~) than any exponential, that is, for all t _> O, there exists a constant Mt such that If(x)[ _< Mte -tx, for all x > O. The space Z is topologized by the family of seminorms

Ilflla,b

sup I:(x)eb I + sup If(x)ea l, x~O

~0

for all a, b E Q+ (compare with Example 5.6). Note that, for any continuous g, not vanishing on any interval, such that x ~ et~g(x) is bounded on It, for all t > 0, A generates a Cg-regularized semigroup W,(t)f(x) =_etZg(x)f(x), 30

where Cg = Wg(0). Thus C-1W(t)f(x) = etXf(z), so that the construction of Proposition 4.14 yields our Z, regardless of which g is chosen. It is clear how to extend this construction to arbitrary multiplication operators, A. E x a m p l e 4.17. Let Z = C0(It) NC01([0,oo)), where C01([0,c~)) ~ {f e Cl([0,oo))l lim ft(x) = 0 = lim f(x)}, with the norm on X given by

Ilfll

Ilftlco(R) + Ilf'llcoct0,oo>>.

Let A ~ ddx ' with maximal domain, 79(A) =_{f e X lf' G X}. We call X a bumpy translation space; it is good for counterexamples and other novel behaviour (see Examples 4.18, 5.8, 5.20, 19.11 and 28.6 and the Notes to Chapter IV). Behaviour that is acceptable in the negative real axis may not be acceptable in the positive real axis (e.g., nondifferentiable continuity), thus right-translation, the semigroup that A wants to generate, cannot map X into itself.

^ A V

A ~

^ ~

A A A Ib~' V

V

~ V

A/_

A ~

W

V

If the abstract Cauchy problem has a mild solution, it is given by right translation, u(t, f ) ( x ) = f ( x - t), (x G It, t >_ 0); thus it is not hard to see that Z = X ~ C l ( I t ) , with topology generated by the seminorms

Ilfllb

Ilfllco(R> + sup lf'(x)l, x>_-b

f o r b E Q+. E x a m p l e 4.18. Let X be the bumpy translation space of the previous example. Let A be a bounded perturbation of d

( A f ) ( x ) = - f ' +qf, 7)(A)=_ {f G X I f ' GX}, 31

where q E X, and q is not differentiable at any point in ( - c o , 0 ) . Note that all right translates of q, z ~ q(x - t) (t > 0), fail to be in X. A calculation shows that, if u is a mild solution of the abstract Cauchy problem, with u(0, f) = f, then

u(t,f)(x) = exp

[/o'

]

q(x - s)ds / ( x - t) (x 9 R , t > 0).

Note that

d [u(t,f)(x)] = (q(x) - q(x - t))exp [fotq(z - s)ds] f ( z +exp

[/o

q(z-s)ds

]

f'(z-t)(xER,

t)

t>O);

for x,t > O , x - t < 0, this will be defined if and only if f ( x - t ) = 0. Thus, since u(t,f) must be in C01([0,oo)), it follows that f = 0 in ( - c o , 0 ) . We conclude that Z = {f e Cl([0,co))lf(0) = 0}, or, to be more precise, {f E X lf(x) = 0 Vx < 0}. Notice how destructive the bounded perturbation of the operator in Example 4.17 is (see also Example 19.11). The operator of Example 4.17 is densely defined, has spectrum contained in the imaginary axis and generates a bounded (1 -k A)-l-regularized group

(14:(t)f)(x)=((1-l-A)-If)(x-t) =

e-af(x-Fs-t)d8

~0~176

(x,t E P..)

and an exponentially bounded integrated semigroup (see Chapter XVIII) (S(t)f)(z) =

~0t f ( x

- s)ds (x E It, t >_ 0).

Example 4.1 9. Suppose (-cr 01 c.C_p( B ), with { ( 1 + r)ll(r + B )-1 Ill r _> 0} bounded. This includes, but is not limited to, all operators, B, such that - B generates an exponentially decaying strongly continuous holomorphic semigroup (in fact, it may be shown to equal all squares of such operators). It is well known that one may construct fractional powers, {Bt}teR, for such operators, and that {B-t}t>0 is a strongly continuous semigroup of bounded operators. Let - A be the generator (we think of A as log B). 32

Then it may be shown that Z is an extension of the locally convex space

Coo(B), with seminorms

II~ll~ = liB ~11, in the sense that C~176 C_ Z, and the topology of Z, restricted to Coo(B), is equivalent. In Chapter XV, we shall see that the family of unbounded operators {Bt}t>o may be "regularized," that is, there exists an injective bounded operator C such that {BtC}t>o is a C-regularized semigroup. P r o o f of T h e o r e m 4.8: We will show that A(Z) (see (4.7))is a dosed subspace of C([0, oo),X). Suppose < z,~ >C Z and Az,~ --. r in C([0,cr Then T(t)z,~ ---* r uniformly on compact subsets of [0,oo). For any n,

T(t)zn = A

(/0tT(s)zn ds ) + zn;

thus, letting n ---. cr and using the fact that A is closed and the convere :D(A) and gence is uniform, we find that f0t r r

= A

(//

ds

r

)+

r

Vt >_ 0,

thus r e Z and r = T(t)r so that r = A(r as desired. Since A is a homeomorphism from Z onto A(Z), this shows that Z is a Frechet space. To show that T(t) maps Z into itself and is a semigroup, we must show that, if z E Z and s > 0, then u(s, x) E Z, and

,(t + ~,~) = ~,(t,~,(~,~:)). Let

v(t) =_

j~Ot u(r + s,x)dr = /t+s u(r,x)dr. ,,IS

Then

v'(t) = u(t + s,x) = A =A

( // v(t)+

(/0

u(r,z)dr

u(r,x)dr

)

) +z

+z=A(v(t))+u(s,z)-z+z.

33

(4.20)

Thus v(t) = fot u ( r , u ( s , x ) ) d r , proving (4.20). For a,b 9 Q+, j 9 N, s > 0, IlT(8)xll-,b,j = sup{llu(t,x)llsI t 9 [a q- s,b + 8]) < Ilmll~,d,j, when c,d 9 Q+, c < a + 8 and d > b + s . Thus T(s) 9 B ( Z ) . The strong continuity of {T(s)}8>0 follows from the fact that s ~-+ u(s,z), from [a,b] into X, is continuous, hence uniformly continuous, for any z 9 Z, a,b 9 Q+. Since Z is a Frechet space, the strong continuity implies that {T(t)}t>0 is locally equicontinuous (see [Komu]). Let A be the generator of {T(s))s>0. If x 9 Z)(.4), then for t > 0, define

1/0'

xt =- -[

u(s, x) ds.

Then xt 9 7)(A) M 7)(A) and Azt = f~xt, for all t > 0. Since x 9 the maps s ~ u(s,x) and s ~-~ A ( u ( s , x ) ) = AT(t)x, from [0,oo) into Z, are both continuous. This implies that A z t ~ Ax and zt --} z, in Z, as t~0. Thus, since Z ~ X, and A is closed, z 9 Z)(A) and Ax = / i x 9 Z. This is saying that A C Alz. To see that Alz C_ A, suppose x E 7)(Aiz). Then both z and A z are in Z. We claim that

u(t, x) = f0 t u(s, A x ) d s + z, Vt >_ O.

(4.21)

To see this, let w(t) =. f~(t - s)u(s, Ax) ds + tx. We need to show that w is a solution of (2.2). Since A is closed,

dw(t) =

/o' (/o I' A

u(r, A x ) d r

)

ds + tAx =

fo

t(u(s, Ax)

Ax) ds + t A x

as desired, proving (4.21). By (4.21), u ( t , z ) is a continuously differentiable function of t, from [ 0 , ~ ) into X and u'(t,z) = u(t, Az). Ifa, b E Q + , j e N and t > 0, then

] II (T(t)x

] - x) - Axlla,b,j = sup IIt (u(s + t,x) - u(s,x)) - u'(s,z)[Ip se[~,bl

34

This converges to 0 as t ~ 0, since the map t ~ u(t,z) is continuously differentiable. This means that x E 7)(,4), thus .4 = AIz , as desired. II P r o o f o f C o r o l l a r y 4.11: (a) ~ (b). This is well known and straightforward by letting u(t, x) - etAx; see Chapter III or any of the references on locally equicontinuous semigroups. (b) --. (a). It is clear that Z ~-~ X. Since both X and Z are Frechet spaces, and X = Z (as sets), this implies that their topologies are equivalent. Thus, by Theorem 4.8, T(t)x = u(t, x) is a locally equicontinuous semigroup on X, generated by A. 1 When p(A) is nonempty, the following lemma shows the relationship between strong solutions and mild solutions. L e m m a 4.22. Suppose r E p(A) and x E I)(A). Then the following are equivalent. (a) The function w(t) is a solution of (0.1). (b) The function u(t) - (r - A)w(t) is a mild solution of(0.1), with z replaced by (r - A)x. Proof: (a) -+ (b). For any t > 0, since A is closed, we may integrate both sides of (0.1), to obtain, for any t > 0, = d

(/0)

w(s)ds

+ x,

thus f : w(r)dr e :D(A2), so that we may apply (r - A) to both sides, as follows: ( r - A)w(t) = A

(/o'

( r - A)w(s)ds

)

+ ( r - A)x.

(4.23)

(b) --* (a). We apply ( r - A) -1 to both sides of (4.23). Since w(s) e ~ ( A ) and s ~-* Aw(s) is continuous and A is closed, we may take A inside the integral to obtain =

Aw(s)es +

as desired. I P r o o f of C o r o l l a r y 4.12: (a) --* (b) is straightforward, as in Corollary 4.11. (b) --~ (a). Suppose r E p(A). For any x E X, by Lemma 4.22, u(t, x) ( r - A ) u ( t , ( r - A ) - l x ) defines a unique mild solution of (0.1), thus (a) follows from Corollary 4.11 1 35

P r o o f of T h e o r e m 4.13:

(a) --, (c). Define W : X -* C([0,c~),X) by

(Wz)(t) = W(t)x = T(t)Cz = (ACz)(t),

where A is defined by (4.7). To see that W is closed, suppose x,~ ~ x and W z n ~ r Since A(Z) is closed in C([O, o o ) , X ) (this was shown in the proof of Theorem 4.8), r = T ( t ) z , for some z E Z. Thus = . -lira .~(wxn)(~

z = r

= .-.oolimc , .

= Cx,

since C E B ( X ) . Thus r = W z , as desired. Thus by the closed graph theorem, W is bounded, which clearly implies that W ( t ) E B ( X ) , for all t>0. The other properties of a C-existence family for A all follow from the properties of T(t). (c) ~ (b). Fix a,b E Q + , j E N. By the uniform boundedness principle, {W(t)}te[a,b] is equicontinuous, thus there exists Ma,bj and j l , j 2 , . . . j k such that k

sup IIW(t):~llj _< Ma,b,j ~ l l ~ l h , , W te[a,b]

e x.

i=l

Suppose z E Ira(C) and C y = z. Then Ilzllo,b,j = IlCyllo,b,j = supte[~,b] IIw(t)yNi, thus, taking infima over all y such that C y = x, we conclude that k

Ilxllo,b,s _< M~,,b,j Z II~llf,,,,r i=l

| P r o o f of P r o p o s i t i o n 4.14: This follows from Theorem 3.13. I P r o o f of T h e o r e m 4.15: The equivalence of (f) and (d) follows from Theorems 3.4(d), 3.5(1) and 3.7(a) and Proposition 3.9. By Theorem 4.13, (b), (c) and (e) are equivalent. The equivalence of (a) and (b) follows from Lemma 4.22, since, if r E p(AJ, then by Lemma 4.22, u ( t , z ) =_" (r - A ) u ( t , ( r - A ) - l z ) is a mild solution of (0.1) if and only if u ( t , ( r - A ) - I z ) is a solution of (0.1), with x replaced by (r - A ) - l z , and, since C A C A C , C(I)(A)) equals Im((r - A)-lC). The fact that (a), (bJ and (e) imply (f) follows by noting that, by uniqueness and Lemma 4.22, W ( t ) z = u(t, C x ) = (r - A J u ( t , C ( r - A ) - l z ) = (r - A ) W ( t ) ( r -

36

A)-lx.

To see t h a t (f) implies (e), s u p p o s e u is a mild solution of (0.1), with x = 0. T h e n v(t) - fo u(s)ds is a strong solution of (0.1). For any s,t >_ 0,0 = ~ W ( t - s)v(s), so t h a t Cv(t) = W(t)v(O) = 0; since C is injective, this implies t h a t v = 0, so t h a t u --- 0, as desired. |

37

V. E X P O N E N T I A L L Y B O U N D E D SOLUTION SPACES

(BANACH)

We have seen that, in general, given an operator A, on a Banach space X, there is no Banach space embedded in X , that contains all solutions of the abstract Cauchy problem (0.1), on which A generates a strongly continuous semigroup (see Remarks 4.9, 5.20 and Example 4.10). However, there is such a Banach space that contains all bounded solutions of the abstract Cauchy problem; more generally, for any w E R, there exists a Banach space that contains all O(e ~t) solutions of the abstract Cauchy problem, on which A generates a strongly continuous semigroup. We prove "pointwise Hille-Yosida"-type theorems (Theorems 5.10, 5.13, 5.14 and 5.15), that is, conditions on the resolvent evaluated at a fixed point x, that guarantee that the abstract Cauchy problem has an exponentially bounded solution for that particular choice of initial data. These conditions can be easily translated into global HiUe-Yosida-type theorems, including the well-known Hille-Yosida theorem (see Chapter XVII). Of particular interest is a pointwise condition that involves only (z A)-lx, for z in a right half-plane (Theorem 5.15), rather than all powers of the resolvent, since this seems like a particularly simple sufficient condition to verify. This chapter, along with the previous chapter, also shows the relationship between general C-existence families and exponentially bounded C-existence families. It is the same as the relationship between Frechet spaces and Banach spaces; that is, a C-existence family for A corresponds to a restriction of A to a Frechet space generating a strongly continuous semigroup, while an exponentially bounded C-existence family corresponds to a restriction to a Banach space generating a strongly continuous semigroup. More specifically, an existence family corresponds to the topology of uniform convergence on compact subsets of [0, oo), while a bounded uniformly continuous existence family corresponds to the topology of uniform convergence on [0, oo) (see the definition of Z, in Chapter IV, the definition of Zw, in this chapter, and Propositions 4.14 and 5.18). Throughout this chapter, C will be in B(X). We will write [Im(C)] for the Banach space Im(C) with norm {{x{{[rm(c)l

=

inf{{ly}l l Cy = x}.

Also, A will be a fixed closed operator, on a Banach space X. D e f i n i t i o n 5.1. We will denote by Z~ the O(e ~t) solution space, the set of all x for which the abstract Cauchy problem (0.1) has a mild solution, u, such that t ~-~ e-~tu(t) is bounded and uniformly continuous. 38

Note that Z~ contains all x for which the abstract Cauchy problem has an O(e (w-e)t) mild solution, for some e > 0. When A has no eigenvalues in (a, oo), for some real a, then by Proposition 2.9, exponentially bounded mild solutions of the abstract Cauchy problem are unique. When x is in Zw, we will then denote by u(t,x) the unique mild solution such that t ~ e-~tu(t, x) is bounded and uniformly continuous. We then define the following norm on Z~: II llz

--- sup e-

'llu(t,

)ll 9

t>0

We will call Z0 the Hille-Yosida space for A, as in [K]. Since Z~ is merely the Hille-Yosida space for (A - w ) , it is sufficient to state our results for w = 0; we will leave it to the reader to make the translation to arbitrary w. D e f i n i t i o n 5.2. We will denote by Y the weak bounded solution space or the weak Hille- Yosida space for A, the set of all x for which d

-~w(t,x)= A(w(t,x))+tx(t>O),w(O,x)=O,

(5.3)

has a solution such that w' is Lipschitz continuous. Note that Z0 C_ Y. If z E Z0, then w(t, z) is obtained from u(t,z) by integrating twice, w(t,x) = fo(t - s)u(s,z)ds. Here is the justification for our terminology (see also Theorem 5.5(3), (4) and (6)). Let v(t,x) = w'(t,z). When 79(A)is dense, so that A* is defined, then for any z* E 79(A*), the map t ~ < z*,v(t,z) > is Lipschitz continuous, thus is differentiable almost everywhere, with d

d~ < z*,v(t,z) > = < A*z*,v(t,z) > + < z*,z > a.e.(t > 0), thus we are getting a "weak mild" solution of the abstract Cauchy problem. As with Proposition 2.9, when there exists w E 1~ such that A has no eigenvalues in (w,c~), then the Lipschitz continuous solution of (5.3) is unique, so that we may define a norm on Y: Ilzll " -

sup

{1--LIIw'(r,z

r>s_>0

)-

ll ll};

r -- S

this may also be expressed as Ilzllv = ess

sup

d

ii~.ll0{l~ < z', 39

w'(t,x) > I , I x ' ( z ) l } .

Note that II llY = Ilxllzo, for all x e Zo, since ~ < x*,w'(t,x) > = < x*,u(t,z) > almost everywhere, for all z E Zo,z* E X*. D e f i n i t i o n 5.4. We will say that an operator, B, satisfies the HilleYosida conditions if (O, oo) C_p(B) and IIs(s - B)-Xll _< 1, for an s > 0. T h e o r e m 5.5. Suppose A has no eigenvalues in (0, e~). Then (1) Y ~-* X; (2) Zo and Y are Banach spaces;

(3) Zo is the closure, i. Y, of~)(AIr); (4) AIy satisties the Hille-Yosida conditions; (5) T( t )x =- u( t, x) is a strongly continuous semigroap on Zo, generated by alzo, such that IIT(t)l I _< 1, for all t > O; (6) :P(AIr) equals the set of all z in 7P(A) for which the abstract Caachy problem (0.1) has a bounded Lipschitz continuous mild solution;

(7) D(AIz0) equals the set of all x for which the abstract Cauchy problem has a bounded (strong) solution with a bounded uniformly continuous derivative; (8) Z0 is maximal-unique, that is, i f W satis~es (1), (2) and (5), then

W'--'Zo. E x a m p l e 5.6.

Let A be as in Example 3.2,

(Af)(x)=xf(x), with maximal domain, on Co(It)). Then Z~ equals Co((-cr the set of all f E C0(tt) supported on (-oo,r

that is,

with the same norm

Ilfllz. = Ilfllco(O,xER

sup

t>o,~O,nENand

{snl[(8- A)-nx[I Is > 0,n e N } is bounded. Then

II~llY =

sup{s~l]( 8 - A)-~xll 8 > o,n + 1 E N } .

When A has no eigenvalues in a right half-plane, we may represent solutions of the abstract Cauchy problem as complex integrals of the (possibly unbounded) resolvent, and also obtain characterizations of Z~ or Y.

D e f i n i t i o n 5.11. Suppose a E P~ and A has no eigenvalues in Re(z) > a. Let ~ a be the set of all x E NRe(z)>aIm(z - A) such that the map z ~ ( z - A ) - l x is holomorphic and bounded on Re(z) > a. 42

Then we define, for any t >__0, x E :D~,w > a,

K(t)z - ~

eZt( z - A ) _ l z +iR

dz 2~iz2 "

Note that, by the residue theorem, this is independent of w > a. Even when A itself generates a bounded strongly continuous semigroup, we only expect II(z - A)-lll to be bounded in left half-planes, rather than O(M -1), unless the semigroup is holomorphic. For y E :Da, we obtain solutions of the three-times integrated abstract Cauchy problem.

Proposition 5.12. Suppose y E :Da, for some a E It. Then, for all t > O, f2 g ( s ) y d s e V ( A ) and A (~tK(s)yds)

= K ( t ) y - lt2y.

T h e o r e m 5.13. Suppose A has no eigenvalues in the open right halfplane Re(z) > O. Then the following are equivalent: (a) x e Y; and (b) x e v o , for all a > O, and the map t ~ K ( t ) x ,

from [0,oo) into X,

has a Lipschitz continuous derivative. Then K ( t ) x = w(t, x), from Definition 5.2.

Theorem 5.14. Suppose A has no eigenvalues in the open right hMfplane Re(z) > 0. Then the following are equivalent: (a) x E Zo; and (b) x E ~a, for all a > 0, and the map t ~-* K(t)x, from [0, oo) into X , has a bounded uniformly continuous second derivative.

Then K"(t)x = u(t,x), from Definition 5.1. The following is a simpler sufficient condition for being in the HilleYosida space, than appears in Theorem 5.10, because it involves only the resolvents, ( z - A ) -1, rather than arbitrarily high powers of the resolvents; the only possible increase in difficulty is that we need the resolvents in a half-plane Re(z) > a, rather than a half-line (w, oo). We also should point out that this is not a necessary condition, as was the case in Theorem 5.10. 43

Theorem 5.15. Suppose k G N, t~ > O, A has no eigenvalues in Re(z) > i~, X E ])(Ak+l),Ak+lz E Nn~(~)>6Im(z - A) and the map z ~ (z A ) - l A k + l z is holomorphic and O(Izl k-l) in Re(z) > 5. Then z E Zb, for all b > 6. The following three results are proven exactly as are Theorems 4.13 and 4.15 and Proposition 4.14. T h e o r e m 5.16. Suppose A has no eigenvalues in (0, eo). Then the following are equivalent. (a) The abstract Cauchy problem (0.1) has a mild bounded uniformly continuous solution, for all z G Ira(C).

(b) Jim(C)] ~ Z0. (c) There exists a bounded, strongly uniformly continuous mild Cexistence family for A. The mild C-existence family for A is then given by W ( t ) = et'tl~o C. T h e o r e m 5.17. Suppose CA C_ AC and p(A) is nonempty. Then the following are equivalent. (a) The abstract Cauchy problem (0.1) has a unique solution, for all x E C(I)(A)), with u and u t bounded and uniformly continuous functions from [0,c~) into X. (b) The abstract Cauchy problem has a mild bounded uniformly continuous solution, for all x G Ira(C).

(c) [xm(c)] ~ z0. (d) All solutions of the abstract Cauchy problem are unique and there exists a bounded, strongly uniformly continuous mild C-existence family for A. (d) The operator A generates a bounded, strongly uniformly continuous C-regularized semigroup.

Proposition 5.18. Suppose A generates a bounded strongly uniformly continuous C-regularized semigroup {W(t)}t>0. Then Zo = {x [t ~-* C -1 W(t)z is uniformly continuous and bounded}

and

ll~JJZo = sup IIc-' w(t)~ll. t>0

44

R e m a r k 5.19. If A is densely defined and generates a bounded Cregularized semigroup {W(t)}t>o, then '{W(t)}t>0 is strongly uniformly continuous. The same is true if {W(t)}t>0 is stable. It is not clear if all bounded regularized semigroups are strongly uniformly continuous. R e m a r k a n d E x a m p l e 5.20. The following example shows that, even when A generates a bounded strongly uniformly continuous regularized semigroup, as in Theorem 5.17 (and an exponentially bounded onceintegrated semigroup; see Chapter XVIII and Example 4.18), U,,eR Z,, may not contain all initial data for which the abstract Cauchy problem has a solution, that is, we may have

zeUz . ~ER

Let X be the bumpy translation space X - C 0 ( R ) n C01([0,oo)), as in Example 4.17, A dx d ,:D(A) - { f E XI f' E X } (see also Example 5.8). Then, as shown in Example 4.18, A generates a bounded strongly uniformly continuous (1 + A)-l-regularized semigroup. if x E U~el~ Z~, then u(t, x) would be exponentially bounded. However, there are solutions of the abstract Cauchy problem that are not exponentially bounded. Choose

f(x) =_sm(e )1(-oo,01+ g(x)l[0.oo), 9

~7 2

where g is chosen so that f E X and Ilfll = 1. Then u(t, f ) ( z ) = f ( x - t) $2 is the solution of the abstract Cauchy problem and Ilu(t, f)ll >- e , for all t>_0. We have already seen that, to obtain a space on which A generates a strongly continuous semigroup, containing all initial data for which the abstract Cauchy problem has a mild solution, we must allow Frechet spaces. This example shows that this is true even when A generates a bounded strongly uniformly continuous C-regularized semigroup, with the image of C as large as the domain of A. In this example, Z0 equals {f E C~ ([0, c~)) N C1 (R) I f, is bound ed and uniformly continuous ouR}, with Ilfllzo = supxeR If(x)[ + supxeR If'(x)l 9 n More generally, if X - Nk=0 Cok ([k, oo)) with

IIfll

sup Iftk>(x)l

~'~ ~>k 45

d D(A) - { f E X I f' E X}, then A generates a bounded and A dz, strongly uniformly continuous (1 -- A)-n-regularized semigroup, and Z0 equals C~' ([0,oo)), Ilftlzo = E2_-0 sup~>_o If(k)(m)l 9 Proof of Theorem

5.5: Assertion (1) is obvious from the definition of

II~llr-

The fact t h a t Z0 is a Banach space and {T(t)}t>_o is a strongly continuous semigroup generated by Alz o follows as in Chapter IV, T h e o r e m 4.8; note t h a t the uniform continuity of the solutions implies the strong continuity of {T(t)}t>_o. For any t > 0, z E Z0,

IIT(t)~llzo --sup Ilu(t + s, ~)11 = sup Ilu(r,x)ll ~ Ilxllzo. r>t

s>0

To see that Y is a Banach space, suppose < xn > is a Cauchy sequence in Y. T h e n there exists Lipschitz continuous v : [0,cr --* X such that the maps t ~ w'(t,x,) converge to v in the Lipsckitz norm, as n --+ c~. Let w(t) = ft v(s)ds, for t > 0. There exists z E X such that z,, ~ z, in X , a s n --+ oo. Since A is closed and, for all n E N , t > 0,

w'(t, zn) = A(w(t,z,~)) + tz,, it follows t h a t w(t) z e Y, with w(t, z)

e D(A) and w'(t) = A(w(t)) + tz, for all t > 0. Thus = w(t), so that x,~ ~ z in Y, as n ~ cr

This proves (2) and (5). For (4), fix s > 0. Note that, for x E Y, the same proof as in L e m m a 2.10 shows that

(s - A) (S fo~176

= ( s - A) (S2 /oCCe-'tw(t,z)dt) = z,

(5.21) for all s > 0; note that the quantities in parentheses are in D(A) because A is dosed. Thus Y C_ I m ( s - A) = ~((s - A)-I). To show t h a t ( s - A) -1 maps Y into itself, for a fixed x 6 Y, we must construct a map r ~ w ( r , ( s A ) - l z ) , satisfying (5.3). A candidate is suggested by the formal identity erA( s -- A) -1 : f o e-~te(~+t)A dt, which, after integrating both sides with respect to r twice, and thinking of w " ( r ) as being erA, gives us the following definition.

v(r)-

e-'t[w(r+t,z)-w(t,z)-rw'(t,x)]dt.

/0 ~176

46

Then e - ' t [w'(r + t , z ) - w'(t,z)] dr,

r

=/o ~~

and

e-St[w'(r+t,z)-(r+t)z-(w'(t,z)-tz)ldt

Av(r)=

j~0~176 -

rA [ o o e-Stw'( t, z) dt, Jo

so that, by (5.21), r

7"

vt(r) - Av(r) = s(X + A(s - A ) - l x ) = s (Z + ( ( A - s) + s)(s - A ) - l x )

= r(s - A)-lx. This proves that ( s - A) -1 maps Y into itself, with w ( r , ( s - A ) - l x ) for x E Y. This implies that, for all s > 0, (s - AIr) is a bijection, with

= v(r),

/ o o -st d drd < z * , w ' ( r , ( s - A I r ) - l x ) > = J0 e ~ < z*,w'(r + t , z ) > d t , (5.22) for all r,s > O,z q Y,z* q X*. To prove the desired estimates on II(s- AIr)-lll, we win use the last expression for ]lzlJy, in Definition 5.2. By (5.22), for s > 0, ess

sup

I ~d < x * ,

t_>o,llz"1151

w'(t,(s

AIr)-ix) >1

d

1 < -ess sup I~/< ~*,w'(t,~) > 1 < s t~0,11z'l151

II~llr.

Also, by (5.21),

I< z * , ( s - AIr)-I~ >1 = J

/o

e-" -~ d < x * , w ' ( t , x ) > dtl
O, as desired, proving (4). To prove (3), suppose x 6 :D(A[y). By (4), there exists y 6 Y such that z - ( 1 - A ) - l y . Thus w'(t,x) = ( 1 - A ) - l w ' ( t , y ) 9 D(A); letting v =. w', we may now rewrite (5.3) as v(t,x) = A

(/0')

v(s) ds

-{- tx =

/0

Av(s) ds + tz,

so that v is a mild solution of (0.1), that is, z 9 Z0. Since Z0 is a closed subspace of Y, this implies that the closure of ~(AIy) is contained in Z0. Conversely, since the generator of a strongly continuous semigroup is densely defined (see Theorem 3.4(b)), and we've already shown (5), it follows that Z0 equals the closure of :D(A[zo), which is contained in the closure of D(A[r). This proves (3). For (6), suppose z e T}(AIY). Then both z and Az are in Y and, by (3), z 9 Z0. Thus u(t, z) is a bounded mild solution of (0.1). We will show that w'(t, A x ) = u(t,z) - z, which, by the definition of Y, will imply that u(t, z) is Lipschitz continuous. Let w(t) =_ St u(s, z) ds - tz. Then by the definition of a mild solution, w(t) 9 D(A), for all t > 0, with Aw(t) = u(t,z) - z - t a x , so that w'(t) - Aw(t) = t a x . This implies that w(t) = w(t, A z ) , as desired. Conversely, suppose (0.1) has a bounded Lipschitz continuous mild solution, u(t, z), and z E ~ ( A ) . Since we already know that z 9 Z0 C_ Y, all that we need show is that A z q Y; this follows by showing that w(t, Az) = fo u ( s , z ) d s - tz is the desired solution of (5.3). For (7), suppose x 9 :D(Alz0). Since z and A z are in Z0, both u(t,z) and u(t, A z ) are bounded and uniformly continuous. As in the proof of (6), u(t,z) = f o u ( s , A z ) d s + z, so that t ~ u(t,z) is a bounded strong solution of (0.1), with a bounded uniformly continuous derivative u(t, Az). Conversely, suppose (0.1) has a bounded (strong) solution with a bounded uniformly continuous derivative. Then z 9 Zo rl D(A), so all that remains is to show that A z fi Zo. Let u(t) =__ u'(t,z). Then it is simple to verify that u is a mild solution of (0.1), with initial data A z , that is, A ( f o u(s)ds) = u ( t ) - Az, as desired. For (8), suppose W satisfies (1), (2) and (5). Since W ~ X , u ( t , z ) etAIwz is the unique bounded mild solution of (0.1) and is uniformly continuous. Thus W C_ Z0. There exists K such that Ilxll ___ KIIxllw, Vz 9 W. For any t >_ 0, Ilu(t,z)ll -- IIe'Atw xll ___ Klle 'Atw zllw _< KIIxllw, thus Ilxllz0 _< KIl~llw, for all 9 9 w , which is saying that W ,--, Z0. I 48

L e r n m a 5.23. Suppose A has no eigenv~lues in (0,oo), x E Im(s - A) n, for all s > O, n E N and there exists M < oo such that I l s n ( s - a)-nxll _< MIIxll,Vn E N , s > O.

Then the map s ~ (s - A)-lz, from (0, oo) into X, is infinitely differentiable, with d n( ( s - z ) - l x ) = ( - 1 ) n n ! ( s - A)-(n+l)x. (~'~s) Proof." Fix r > 0. For complex s such that Ir - s I < r, define CO

R(s) = ~ ( r -

s ) k ( r - A)-(k+Dz.

(5.24)

k=O

We will show that R(s) = (s - A)-lx, when s is real. The hypotheses on A imply that the series in (5.24) converges uniformly, in the norm of X, on {s I Is] < s0}, for any So < r. Thus R(s) is in X. Note that each partial sum in (5.24) is in 79(A), with A

(r - s ) k ( r - A)-(k+l)z

[ A ( r - A) -1] ( r - s ) k ( r - A)-kz.

=

k=0 The hypotheses on A guarantee that the latter sum converges, as n ~ co, thus, since A is closed, R(s) E 7:)(A), with OO

AR(s) = ~ ( r k=0

s ) k A ( r - A)-(k+Dx.

Thus we may calculate as follows. CO

( s - A)R(s)= ~ ( r k=0

s) ~ [ ( r - A ) - ( r - s)] (r - A ) - ( k + l ) x

CO

= ~ ( r - s) k [ I - ( r - sl(r- A)-'] ( r - A)-k* k=O --X.

Thus z E I m ( s - A) and (s - A ) - l x equals R(s), when s is real and { r - s{ < r. This implies that r ~ ( r - A ) - l x , from (0,oo) into X, is analytic, with d n x] = ~rr [ ( r - A) -1

R(s)ls=r

=

-

49

A)-("+')=. m

L e m m a 5.25 ( f r o m [A2]). Suppose W is a Banach space, M < oo,g : (0, ~ ) --, W. Then the following are equivalent.

(a) The function g is the once-integrated Laplace transform of a Lipschitz continuous function of Lipschitz constant M, that is, there exists G: [ 0 , ~ ) --. W such that a(o) = O, I l a ( t ) - a ( s ) l I 0 an d g(s) = s

e-StG(t)dt, (s > 0).

~0~176

(b) The function g is infinitely differentiable and

$n+l

11---7-g(n)(s)ll _< M,

P r o o f of T h e o r e m 5.10:

Vs > 0, n + 1 E N .

(a) ~ (b). By Theorem 5.5(4),

s " l l ( s - A)-"IIY _< 1,

for all s > 0, n E N, thus (b) follows from the fact that Y ~ X. (b) --* (a). By Lemmas 5.23 and 5.25, there exists Lipschitz continuous G(t, x) such that

(s- A)-lz = s

e-StG(t,z)dt,

(5.26)

for all s > 0. Since A ( s - A ) - l x = s ( s - A ) - l z - x, we may apply these same lemmas to g(s) =- ] ( s - A ) - l x , on the Banach space (since A is closed) [D(A)], to obtain Lipschitz continuous t ~ GA(t,z), from [0,c~) to [D(A)], such that e-stGA(t,x) dt, - A/-lx = s for all s > O. However, applying integration by parts to (5.26) gives us

l (s - A)-Xx = s

fo fo' e-'t

G(r,x)drdt,

for all s > 0. By the uniqueness of the Laplace transform, GA(t,x) fo G(r, x) dr, for all t _> 0. 50

This implies that fo a ( r , x ) d r E D(A), for all t > 0 and the map t ~-* A fo G(r, x) dr, from [0, oo) into X, is continuous. We will show that w(t) = ]o G(r, x) dr is the desired solution of (5.3), by showing that the Laplace transforms of Aw and wt(t) - tz are equal. For s > 0, since A is closed,

e-~tAw(t)dt = A

e-~tG(t,z)dt

=(A-s+s)l(s-A)-lz = (s- A)-lz-

Ix 8

= ~

e-"(,o'(t)-

tx) dr.

This implies that w(t) = w(t, x), the desired solution of (5.3), so that

xEY. For the assertion about Ilxllr, first note that we have shown that

w'(t, z) = G(t, z) (t > O, z e Y), so that

IlzllY =

sup r>s>0

{ - - L I I G ( r , x ) - G(8,x)ll,llxll}; r -- 8

by Lemmas 5.25 and 5.23 (in that order),

sup {---L-1 lle(r,x ) - a(~,x)ll}

r>8>O

r -- 8

8 n'l-1 d n -- sup{ll---~-., ( ~ ) ( ( s - a)-~x) III s > 0 , n +

1 ~ N}

= sup{s"ll(s - a)-"zll I s > 0,n ~ N}, concluding the proof. | P r o o f o f P r o p o s i t i o n 5.12:

fot K ( s ) y d s

By Fubini's theorem,

=

dz

(e z t - 1 ) ( z - A ) - l y 2~riz3 ; +JR

51

thus, since A is closed, the growth conditions on II(zthis is in :D(A), with

A(

/0

K(s)yds) =

L

A)-ayll imply that

(e z t - 1 ) ( A - z + z ) ( z - A)-ly 2riz a

+iR

+iR

2--~5z3 Y

dz 2 + f~+ia(eZt)(z - A)-ly 2~riz = g(Oy-

12 ~t y,

by a calculus of residues argument. | L e m m a 5.27. Suppose x E 7)~, for all a > O. Then for a/l s > 0,

fo~e-StK(t)xdt = ~ ( s - a)-lx. P r o o f : For fixed s > 0, choose w between s and 0. Then, by Fubini's theorem,

/o

e-'tK(t)z dt =

L/o

et(~-*)(z - A) - lx dt dz

2riz 2

+iR

= f~

1

+iR s - Z (z -

A)_lx dz

2~rz 2'

which a calculus of residues argument shows to be equal to ~ ( s - A ) - l x .

II

P r o o f o f T h e o r e m 5.13: (b) -~ (a). By Proposition 5.12, K(t)x is a solution of (5.3), if we can show that g(t)z e 7)(A). This follows by using the fact that A is dosed: { 1 Jt rt+h g(s)x ds) g ( t ) z = limh-~0 ~ f / + h Z ( s ) z d s and limb-.0 n k~ / Z ' ( t ) x - tx, so K(t)x e D(A) and

Ag(t)x = g ' ( t ) x - tx; this shows that x e Y, with K(t)x = w(t, x). (a) ---* (b). As in the proof of Lemma 2.10, x e Im(z - A) and (z - A ) - X z = z

e-"~w'(r,z)dr,

52

when Re(z) > 0. Since w ~ is Lipschitz, this implies that x E :Da, for all a>0. Also, an integration by parts and Lemma 5.27 show that K(t)x and w(t, x) have the same Laplace transform, thus are equal, so that K(t)x =

w(t,.), l P r o o f o f T h e o r e m 5.14: (b) ~ (a). As in the proof of Theorem 5.13 (b) ~ (a), K(t)z E I)(A), with

AK(t)x = K'(t)x - tx, for all t _> 0. Repeating the argument given there implies that Kt(t)x E D(A), with

dK'(t)x = g " ( t ) x - x, so that x e Z0, with u(t, x) = KH(t)x. (a) ~ (b). By Theorem 5.13, x E T)0 and

K(t)z =

]0 t w'(r,x) dr =

]0

(t- r)u(r,x)dr,

so that K(t)x is twice differentiable and K"(t)x = u(t,x). | P r o o f o f T h e o r e m 5.15:

First note that, by writing

( z - A) -1 = 1(1 § A ( z - A)-I), it can be shown that (z - A)-IArnx is O(Izl k-l) in Re(z) > 6, whenever 0 _< m < k § 1. Thus (z - A)-I(A - a)k+lx is O(Mk-1), for any a > 6. For a > 6, the resolvent identity implies that n-1

( z - A ) - I ( A - a ) -n = ~

1 ( A - a) i-n + (z_a)i+l

1

i:O

a)"

( z - A) -1

for all n E N, Re(z) > a; thus, for j = 0,1,2,

~-j 1 ( z - A ) - l ( A - a ) J x = Z (z_a)i+l ( A - a ) i + j x i=O 1 + (z - a) k+'-j(z - A)-*(A

53

-

a)k+lx"

Thus, for a > ~, x, Ax and A2x are in :Da (see Definition 5.11), so that we may apply Proposition 5.12, with y = x,Ax,A2x, as follows.

Z ( t ) x - 5t x = d

K(s)xds

=

g(s)Axds,

(5.28)

so that K(t)x is differentiable, with

K'(t)x-tx=

K(t)dx= = .zt2Ax +

t2nx + A

f

(/o'

K ( s)Az ds) (5.29)

K(s)A2z ds,

so that both K'(t)x and K(t)Ax are differentiable. Now suppose b > ~. Let v(t) - K"(t)x. Differentiating both sides of (5.28), and using the fact that A is closed, now implies that v is a mild solution of (0.1); this implies that u(t) =- e-btv(t) is a mild solution of u'(t) = (A - b)~(t) + ~, ~( 0) = 0.

All that remains is to show that u is bounded and uniformly continuous. By (5.29),

ebtu(t) = x q- tAx + g(t)A2x (t ~_ 0). By choosing w such that b > w > a > ~, we see that, since [[K(t)A2x[[ is O(e ~t) (see Definition 5.11), u e C0([0,c~),X), thus is bounded and uniformly continuous, as desired. |

54

V I . W E L L - P O S E D N E S S O N A LAI:tGEI:t S P A C E ; GENERALIZED SOLUTIONS

Another way to approach ill-posed problems is to expand one's definition of solution, so that all initial data in the original space yield a solution, if one allows "generalized" solutions. A well-known example of this is the theory of distributions, generalizing the idea of a function. This is in contrast to the approach in Chapters IV and V, where we restricted our initial data (to the solution spaces) so as to have wellposedness, that is, the restriction of A generated a strongly continuous semigroup. When A generates a C-regularized semigroup, then we may construct a Frechet space, W, such that X is continuously embedded between C ( W ) and W, on which an extension of A generates a strongly continuous semigroup. When the regularized semigroup is exponentially bounded, then this enlarged space may be chosen to be a Banach space. This leads to characterizations of generators of regularized semigroups, in terms of generators of strongly continuous semigroups, on spaces larger than X and smaller than X (Theorems 6.6 and 6.7). Throughout this chapter, C will be a bounded, injective operator with dense range such that C A C_ AC. T h e o r e m 6.1. Suppose A generates a strongly uniformly continuous bounded C-regularized semigroup {W(t)}t>o. Then there exists a Banach space W and an operator B such that (1) C extends to a bounded operator, C, on W , (2) B generates a strongly continuous contraction semigroup on W , such that etBC = Ce tB, for MI t >_ O, (3) CetBx = W ( t ) z , for all t >_ O,x E X , (4) A = B[x and

(5)

x

w.

W may be chosen so that

II llw : sup{[lw(t)xlllt ~ 0), Vx ~ X. T h e o r e m 6.2. Suppose A generates a C-regularized semigroup {W(t)}t>_o. Then there exists a Frechet space W and an operator B such that (1) C extends to a bounded operator, C, on W , 55

(2) B generates a locally equicontinuous semigroup etnC = (~e tB, for all t > O,

on

W , such

that

(3) (~etBx = W(t)x, for all t > O,z e X , (4) A = Blx and

(s) [ ~ ( w ) ] ~ x ~ w . W may be chosen to have topology generated by the seminorms II~ll~,b =

suP(llW(t)~lllt e [a,blL W e X, a,b e Q+.

E x a m p l e 6.3. Let A be as in Example 3.2, (Af)(x)

-

zf(~:),

with maximal domain, on C0(R). We showed that A generates a Cregularized semigroup, where ( C f ) ( x ) =- e-X2f(x). Then Y, from the proof of Theorem 6.2, the completion of C ( X ) , equals the set of all continuous f : R ~ C such that

llfllr~ sup lf(x)e-:l
0, then we similarly choose Wg as the set of all continuous f such that llflla,b,g=

If(x)etXg(x)l < 00,

sup

xER,tE[a,bl

for all a, b E Q+, topologized by that family of seminorms. Thus our enlarged spaces are not sharp, as were the solution spaces of Chapters IV and V; compare this example with Example 4.16. 56

P r o p o s i t i o n 6.4. Suppose there exists a Banach space W, a bounded extension, C, of C, on W and an operator B such that

x

w,

B generates a strongly continuous contraction semigroup on W, etBC = Ce tB, for all t >_ 0 and A = B[x. Then an extension of A, C - s A C , generates a bounded strongly uniformly continuous C-regularized semigroup on X .

Proposition

6.5. Suppose there exists a Frechet space W , a bounded extension, C, of C, to W and an operator B such that

[r

x ,-. w,

B generates a locally equicontinuous semigroup on W, etBC = Ce tB and A = B[x. Then an extension of A, C - 1 A C , generates a C-regularized semigroup on X . T h e o r e m 6.6. The following are equivalent, if p(A) is nonempty.

(a) The operator A generates a bounded strongly uniformly continuous C-regularized semigroup. (b) There exists a Banach space Y such that [C(X)] ~ Y ---, X, and AIY generates a strongly continuous contraction semigroup. (c) There ex/sts a Banach space W and an operator B such that B

generates a strongly continuous contraction semigroup on W , C extends to a bounded operator, C, on W , etBC = Ce tB, for all

t>_O, x

w,

and A = B I x . T h e o r e m 6.7. The following are equivalent, if p(A) is nonempty.

(a) The operator A generates a C-regularized semigroup. (b) There exists a Frechet space Y such that [C(X)] ~ 57

Y ~-* X,

and A i r generates a locally equicontinuous semigroup. (c) There exists a Frechet space W and an operator B such that B generates a locally equicontinuous semigroup on W , C extends to a bounded operator C, on W, etBC = Ce tB, for all t > O, [O(w)] ~ x ~ w,

and A = B l x . P r o o f o f T h e o r e m 6.1: Let Y be the completion of X, with respect to the norm Ilxlly =_ IICxll. Extend C to Y by defining Cy = limn--.oo Cxn, co is a sequence in X converging with the limit taken in X, whenever {Xn}n=t to y, in Y. It is not hard to see that 6' is bounded and injective on Y, and C(Y) equals the closure, in X, of C(X). For any t > 0, extend W(t) to Y by 17V(t)y =_ C-1W(t)Cy; note that, since W(t) is bounded and commutes with C, W(t)Cy 9 C(Y), for all y 9 Y. Let A be the generator of the C-regularized semigroup {l~(t)}t>0. Let W be the Hille-Yosida space for A. Note that, by Proposition 5.18, II~llw - sup I1~ -1 w ( t ) ~ l l Y t>o

Thus X C_ W and (5) is clear. Let B -~ A. To see that A = t>O,

-- sup t t w ( t ) x l l . t>o

suppose x 9 79(BIx). Then, for all

BIx,

which converges to C B x , as t ~ 0, in X, because [C(W)] ~ X. Thus z 9 79(A) and Ax = Bx, so that B]x C_ A. Conversely, suppose x 9 :D(A). Then, for all t >__0,

CetBx - Cx = W(t)x - Cx =

/0

W ( s ) A x ds = C

(f)

e'BAx ds

so that, since C is injective, etB x

-- X =

~0 t eSBAx ds,

which implies that x E ~ ( B ) and Bz = Az E X, so that x E I)(BIx), as desired. | 58

P r o o f of T h e o r e m 6.2: This follows from Chapter IV exactly as Theorem 6.1 followed from Chapter V. | P r o o f of P r o p o s i t i o n s 6.4 a n d 6.5: For z E X, let W(t)z = CetBz. Boundedness and strong uniform continuity follow from the fact that [C(W)] ~-* X. Since Ce tB = etBC, {W(t)}t>o is a C-regularized semigroup. Let A be the generator of {W(t)}t>o and suppose x e :D(A). Since x e :D(B) and [C(W)] '--+ X, W(t)z is differentiable, with ~W(t)xlt=o = CBx = CAz. Thus z E 7)(A), with ~ix = Az, so that an extension of A generates {W(t)}t>o. To apply Proposition 3.11, we need to consider z e ~D(A). Then W(t)Cx E Im(C), with C - l W ( t ) C x = W(t)x, for all t _> 0, thus dc-1w(t)Cx[t=o exists, in the norm of X, and equals CAx; since [C(W)] ~ X, this implies that dw(t)Cx[t=o exists, in the norm of W, and equals C2Ax. Since B is the generator of the (C)2-regularized semigroup {etB(c)2}t>o = {W(t)Cz}t>o (see Proposition 3.10), this implies that Cz e :D(B), with Bx = ~lz, thus Cz e D(B[x) = D(A). Thus, C(:D(~I)) C_:D(A), so that, by Proposition 3.11, .4 = C-lAG. | P r o o f o f T h e o r e m 6.6: This follows from Theorems 5.17 and 6.1 and Propositions 6.4 and 3.11. | P r o o f o f T h e o r e m 6.7: This follows from Theorems 4.15 and 6.2 and Propositions 6.5 and 3.11. ]

59

VII. ENTIRE VECTORS AND ENTIRE EXISTENCE FAMILIES

This chapter treats the possibility of having entire solutions of the abstract Cauchy problem. We shall see that surprisingly many operators have such solutions, for a dense set of initial data, accessible through an entire C-regularized group, with the image of C dense. One of the functional calculus constructions of e t A mentioned in Chapter I is the power series representation for the exponential function. For certain vectors x, known as entire vectors for A (Definition 7.5), this will define an entire vector-valued function, which will be an entire solution of the abstract Cauchy problem (0.1). The existence of an entire C-existence family will be seen to correspond to the image of C being contained in the set of entire vectors. This will provide a simple way to determine for what C an operator A has an entire C-existence family. We should remark here that a strongly continuous semigroup is entire only in the essentially trivial case where the generator is bounded (see Remark 7.11). D e f i n i t i o n 7.1. If {W(z)}zeC is an entire family of bounded operators, that is, the map z ~ W(z), from C into B(X), is entire, such that, for all 8 E R, the family {W(teie)}t>>_o is a mild C-existence family, we will say that { W ( z ) ) z e c is an entire C-existence family. Note that Proposition 2.7 implies that for all x E X , W(z)x E I)(A) and d W ( z ) x = AW(z)x, for all z e C. In particular, {W(tei~ o is a strong C-existence family, for all 0 C R. D e f i n i t i o n 7.2.

A strongly continuous family of bounded operators

{W(t)}teR is a C-regularized group if W(0) = C and W(t)W(s) = CW(t+ s), for all real s, t. The generator is defined exactly as with C-regularized semigroups, except that in "limt--.o," t may be positive or negative. Essentially the same argument as with strongly continuous groups (see [Paz, Section 1.6]) shows that A generates a C-regularized group if and only if both A and ( - A ) generate C-regularized semigroups. The following is not a natural definition to make, when dealing with strongly continuous semigroups (see Remark 7.11). D e f i n i t i o n 7.3. An entire C-regularized group is an entire family of bounded operators {W(z)}zeC such that w(o) = c,

CW(z +

=

60

(z,

e

c).

E x a m p l e 7.4.

Let A be as in Example 3.2,

(Af)(x) = xf(x), with maximal domain, on C0(R). Then it is clear how to an entire C-regularized group

(W(z)f)(x) =_e-~2e~:f(x), D e f i n i t i o n 7.5.

Let

{W(t)},>_o extends

for x real, z complex.

C+(A) =_ A~__oI)(A'~). The C + vector x is an

entire vector for A if oo

8 n

,IIA xll < n=0

for all s > 0. We will write 8(A) for the set of all entire vectors for A. For x E E ( A ) , z E C, we define o~ zk eZAx -- ~ -~. Akx. k.=O

It is clear that, for x E $(A), this defines an entire family of vectors R e m a r k 7.6. It is well known that a generator of a bounded, strongly continuous group has a core of entire vectors. This may be shown in the following manner. If iA is the generator, then - A 2 generates a bounded strongly continuous holomorphic semigroup given by

e- A2x = (4

z)-t

(e"Ax)e dr,

for x E X, Re(z) > 0 (this is just the Fourier inversion formula for the function s ~ e-ZS2; see Chapter XII). Then it is not hard to verify that Im(e -zA2) C C(A), whenever Re(z) > 0; thus, since x=

lime

1

2

-,,A x (xE X), and A x = lim e-~A2Ax (xEI)(A),

n"--~ O 0

n---~ ( : ~

C(A) is a core. The proof of the following consists of straightforward series arguments. 61

P r o p o s i t i o n 7.7. (1) x 9 g(A) if and only i f x 9 C ~ 1 7 6 and N zk

}2 Akx k=O

converges uniformly in z on compact subsets of C, as N ~ co.

(2) The operator A maps g(A) into itself, with eZAAx = AeZAx = d ezAx ~ for all x 9 g(A), z 9 C -~z (3) For all complex z, e zA maps g(A) into itself, with ezAe~Ax = e(Z+W)Ax, for all x 9 E ( A ) , w 9 C .

(4) For any 8 >__ O,s

is contained in the solution space for e~eA,

with

u(t, x) = e' 'eAx, for all x 9 s

Perhaps the most natural way to think of g(A) is as a subspace of Z, the solution space (see Chapter IV), consisting of the solutions of the abstract Cauchy problem with entire extensions. T h e o r e m 7.8. Suppose A is closed. Then the following are equivalent. (a) I m ( C ) _Cg(A). (b) There exists an entire C-existence family for A. The entire C-existence family for A is then # y e n by W ( z ) =_-ezAC, where e zA is as in Definition 7.5.

Corollary 7.9. Suppose {W(z)}zeC is an entire C-existence family for A. Then for all z E C, (1) for all k E N , (Tz) d } W ( z ) = A k W ( z ) E B ( X ) ; and (2) W ( z ) m a p s X into g(A). Corollary 7.10. Suppose C A C_ A C and A is closed. Then the following are eq uivalen t.

sin(c) c Z(A). (b) A n extension of A generates an entire C-regularized group that m a p s X into g(A). 62

R e m a r k 7.11. The definition analogous to Definition 7.3, for strongly continuous semigroups (or, more generally, for exponentially bounded ntimes integrated semigroups--see Chapter XVIII), reduces to the trivial case of the generator being bounded. In particular, if +A and +iA all generate strongly continuous semigroups (or exponentially bounded ntimes integrated semigroups), then A must be bounded. This may be easily shown, using the following fact. P r o p o s i t i o n . Suppose A is a closed operator whose spectrum is contained in a bounded set, with I](w- A)-I[[ O(Iwl ), for some integer k, for w large. Then A E B(X). P r o o f of T h e o r e m 7.8: and Theorem 4.13 that

(a) --+ (b). It is clear from Proposition 7.7

w ( z ) x - ezAcx (x e x ) is the desired entire C-existence family for A. (b) ~ (a). We will show the following by induction on n. For a l l n E N , z E C , x E X ,

W(z)x E :D(A~), and ( d ) n w ( z ) x = A~W(z)x.

(7.12)

By Proposition 2.7, (7.12) is true for n = 1. Suppose (7.12) is true for arbitrary n E N, z E C, x E X. Let xk - k(( ~z) d n-1 W( z'-~ l'~x [ d "~n-lW{z~x~ (~z)nW(z) x as k --+ eo, and by the induction hypothesis,

Axk = k(A~W(Z+-k)X-A W(z)x)= k ( ( d ) ~ w ( z + 1-~)x-( d ) ~ w ( z ) x ) , n

which converges

to

(d)n+lW(z)x,

a s ]g ~

oo, so

that, since A is closed,

AnW(z)x = (d)nW(z)x E :D(A), with An+IW(z)x = (~z)n+lW(z)x, completing the induction and proving (7.12). The assertion (7.12) now implies that x E Ca(A), and the MacLaurin series for W(z) is given by

W(z) = ~ k=0

ZkAk k! '

with the convergence of the sum uniform in z, on compact subsets of N. By Proposition 7.7(1), Ira(C) C_$(A), as desired. I 63

P r o o f o f C o r o l l a r y 7.10: (a) ~ (b). A calculation shows that W ( z ) ~ ezAC, as in Theorem 7.8, is an entire C-regularized group. Since CA C_AC, C leaves E(A) invariant, with ezACx = CeZAx, for all z E C. Proposition 7.7 now implies that W(z) maps X into s for all z. By Theorem 3.7(a), an extension of A generates {W(z)}zec. (b) ~ (a). Since A is closed, Theorem 3.8(1)implies that {W(z)}~ec is an entire C-existence family for A. Thus (a) follows from Theorem 7.8. | P r o o f o f R e m a r k 7.11: The Proposition follows by the construction of the Riesz-Dunford functional calculus, except that we do not yet know that A is bounded, so we cannot draw any conclusions from the properties of the P~iesz-Dunford functional calculus. Define T E B(X) by

T =_flz [=M( w - A) -1 27ri dw for M sufficiently large. We want to show that T = I, the identity operator. For this, note that, by the residue theorem,

27ri(T- I) = fM=M ( W - A)-I = fl~ _

1 ) dw

( ( w - A + A ) ( w - A) -1 - 1) ~ i= M

: ~zl=U

Fix co E

p(A).

(7.13)

w

A ( w - A) -1 dWw

We will show by induction that, for any n E N,

2~ri(T- I ) ( A - w ) -n = flz

I=M

An(A-w)-n(w -

dw A) -1 wn"

(7.14)

For n = 1, (7.14) follows by applying (A - co)-1 to both sides of (7.13). Suppose (7.14) holds for n = m _> 1. Then

2 ~ i ( T - I ) ( A - w ) -(re+l) = flz =

I=M

A) -1

wm dw

( W - - A -t- A ) A m ( A - a J ) - ( m + l ) ( w

I=M

= - flz T

Am(A-w)-(m+D(w_

I=M

I=M

- A) -1 wm+l

Am(A_w)_(m+D dw

W mq-1

A'~+I(A -- ~)-(m-{'l)(w -64

A) -1

wm+l;

by the Cauchy integral formula, the first integral is zero, thus this completes the induction, proving (7.14). Choosing n > k + 1 and letting M ~ oc in (7.14) now implies t h a t

( A - o o ) - ~ ( T - I) = ( T - I ) ( A - o v ) -~ = O, so that, by applying (A - w) n to b o t h sides, we conclude that T = I. Since A is closed, X = .lm(T) C I)(A), with

Ax = A T x = flz I=M A(w - A ) - l x ~dw / This implies t h a t A E B ( X ) . I

65

VxEX.

VIII. REVERSIBILITY

OF PARABOLIC

PROBLEMS

Imagine a world where diffusion is reversible. A consequence of the results in this chapter is the existence of such a Frechet space, that is, a space on which the Laplacian generates a strongly continuous group (see Example 8.6). More generally, we use regularized semigroups to show that any parabolic problem is reversible on a dense set, with weU-posedness on a dense Frechet subspace, as guaranteed by Theorem 4.8 (see Corollary 8.3). More precisely, we show that, if - A generates a strongly continuous holomorphic semigroup, then A generates an entire C-existence family, with Ira(C) dense (Theorem 8.2). We will need fractional powers. Since the complex-valued function z ,-* z ~ is analytic on C - ( - o c , 0 ] , to define fractional powers of an operator B, we need to have B contained in C - (-oo,0], in some sense. D e f i n i t i o n 8.1. Suppose B is a closed, densely defined operator such that ( - o o , 0 ) is contained in p(B), with {r]](r + B)-llllr > 0} bounded. It can be shown (see [Balak] or [F1]) that there exists 0 < r such that the spectrum of B is contained in S--o= {reir > 0, Ir < 0} and {[[z(z- B ) - l [ [ [ z ~ S0} is bounded. Suppose first that zero is in p(B). For 0 < a0 < ~, we then define the fractional power (-BO), as the generator of the exponentially decaying strongly continuous holomorphic semigroup {Ta(t)}t>o, defined by

To(t) = fr e - t ~ ( w - A) -1 dw 2~ri ' where ~r > r > 0, ~r < ~, t > 0, Fr = boundary of Sr (see [F1]). When zero is not in p(B), another formula (also in [FI]) defines the fractional powers, with the same properties. We will not need this formula, merely the existence of fractional powers of operators as in the above definition. Suppose ( - B ) generates a strongly continuous holomorphic semigroup. Then there exists real k, and 0 < ~, such that the spectrum of (B - k) is contained in S0, with the growth condition on the resolvent of Definition 8.1. If (x > 1 is chosen so that ~0 < ~, then the function w ~ e-The ztv is bounded on Se, for any complex z. Thus, in the following theorem, we expect (B - k), and hence B, to generate an entire e-(S-k)a-regulaxized 66

group, which we think of as e - ( B - k)~ e zB. We could construct this regularized group with the Cauchy integral formula, with a construction similar to that which was used in Definition 8.1 (see Remark 8.5); however, it is simpler to use entire vectors, as in the previous Chapter, to show that such a group exists. T h e o r e m 8.2. Suppose - B generates a strongly continuous holomorphic semigroup of angle O. Then there exists k E R such that B generates an ir entire e-(B-k) ~ -regularized group, whenever 1 < a < L _ O 9 2

Theorem 4.15 then gives Us the following. C o r o l l a r y 8.3. Suppose - B generates a strongly continuous holomorphic semigroup. Then (1) the abstract Cauchy problem (0.1) has a unique entire solution, for all initial data z in a dense subspace; and (2) there exists Z, dense in X , such that Z is a Frechet space, [Im(e -(B-k)" )] '--* Z "--* X, where k and a are as in Theorem 8.2, and strongly con tin uous group.

BIz generates

an entire

E x a m p l e 8.4. Unlike the generator of a strongly continuous holomorphic semigroup, the generator of a strongly continuous semigroup may not generate a C-regularized group, for any choice of C. Let X -= { f E C[0,1]l f(0 ) = 0}, and B equal the generator of the strongly continuous semigroup {etB}t>o on X, defined by right translation, (etBf)(x)=

f(z-t) 0

x>t z 0. Note that, for all real t, W(t) must be injective; for, if W(t)x = 0, then C2z = W ( - t ) W ( t ) x = 0, so that, since C is injective, x = 0. For t > 1, e ts = 0. Thus Ce tB is not injective, for any choice of C, so that B cannot generate a C-group. More generally, this argument shows that, if B generates a strongly continuous semigroup {etB}t>_0 , and B generates a C-regularized group, then e t/3 is injective, for all t > 0. 67

R e m a r k 8.5. It may be shown that the regularized group of Theorem 8.2 is given by

W(z) = ~r eZ~e-W~ 4~

k - B) -~ dw 2ri'

which we think of a s ezAe -A~ , as with the Riesz-Dunford functional calculus (see Chapter XXII). E x a m p l e 8.6: B a c k w a r d H e a t E q u a t i o n . The backward heat equation has the form

Ou -~(x,t)+Au(s

(s

t_>0)

u(~,t) = o (r e OD, t > O) u(s

= f(s

(s e D),

where D is a bounded open set in R n with smooth boundary OD. Theorem 8.2 yields unique global solutions in X = LP(D) (1 _< p < oo); in fact, the map t ~ u(.,t), from [0,oo) into X, extends to an entire function, for f in a dense subspace of X, because A - A, :D(A) -W2'P(D) gl W~'P(D), generates a strongly continuous holomorphic semigroup on those spaces. This is saying that the heat equation is reversible on a dense set. What is most surprising is that this produces a Frechet space, Z, dense and continuously embedded in LP(D), on which this reversibility is well-posed, in the strongest sense; that is, - A generates a strongly continuous semigroup (see Theorems 4.8 and 4.13). This is hard to imagine, a space where diffusion is continuously reversible. L e m m a 8.7. Suppose a( A ) C_~, O~ is a finite union of smooth (possibly

unbounded) curves, g : O~ --~ C,

(1) ~

g(z) dz=O, D

and (2) A n Izg(z)lll(z -

A)-lxll

A

dlzl,

n Ig(z)llla(z -

and

A~ Ig(z)lll(z - a)-X xll dlzl are finite. 68

A)-lxll

dlzl,

Then y =_ fo~g(z)(z- A ) - l x d z e 7)(A), with Ay =

[

zg(z)(z - A)-lx dz.

JO

P r o o f : This follows from the fact that A is closed, and, by writing A = z - ( z - A),

L

A ( g ( z ) ( z - A)-lx) dz= fo zg(z)(z- A ) - l x d z - ( f 0

g(z) dz)

= foa zg(z)(z - A)-lx dz. | L e m m a 8.8. Suppose {T(z)}zcSe is a strongly continuous holomorphic semigroup. Then T(z) is injective and has dense range, for a/l z E So. Proof." Suppose zo E $6 and T(zo)x = 0. Then for all n E N,

( d )nT(zo )x = A"T(zo)x = O, thus, since T(z) is holomorphic in So, T(z)x = O, for all z E S0, so that x = limt-..0 T(t)x = O. This shows that T(zo) is injective. To show that it has dense range, suppose x* is in the annihilator of the image of T(zo). Then

0 =< x*,T(zo)x > = < T(zo)*X*,x >, for all x E X, thus 0 = (A*)nT(zo) *x* = ( d ) " T ( z 0 ) * x * , for all n E N. As in the previous paragraph, this implies that

0 =< x*,T(z)x >,Vz E So, so that < x*,x >= O, for all x E X. Thus T(zo) has dense range. | P r o o f o f T h e o r e m 8.2: There exist positive 0 and real k, such that ( k B) generates an exponentially decaying strongly continuous holomorphic semigroup of angle 9. This means that we may define fractional powers of (B - k), as in Definition 8.1. 69

Let

C =- e -(B-k)" - fr e-~~ (w + k - B) -1 dw 2ri' where ~ > r > (~ - 0), r < a r < ~ and r , is as in Definition 8.1. We will show that Im(C) C_g(A) (see Definition 7.5). By Lemma 8.7 and the residue theorem, with g(w) - wn-le -w~ Im(C) C_C~176 with

A'~Cz = fr w'~e-~" (w + k - B ) - l z dw

2rri'

for any x E X , n + 1 E N; thus, for any t >_ 0,

oo 7., t,~ IIA~CxII _< fp etlWl[e-~'~ I I[(w + k - B)-*xl[ dlwl2~r n=O

"

r

which is finite. This shows that the image of C is contained in the set of entire vectors for B. By Corollary 7.10, an extension of B generates an entire C-regularized group. Since p(B) is nonempty, it is generated by B (see Proposition 3.9). |

70

IX. THE CAUCHY

PROBLEM

FOR THE

LAPLACE EQUATION

We have already seen how regularized semigroups cover extreme cases of ill-posedness, such as the backwards heat equation. Another famous example of an ill-posed problem is the Cauchy problem for the Laplace

equation in an infinite cylinder Au(x, ~') = 0 u(x,~') = 0

((x, ~7) e [0, oc) x D) ((x,g) e [ 0 , ~ ) • OD)

u(0, g ) = f ( f )

Ou Ox (0, ~') = g(~)

(g 9

(9.1)

(~ 9 D),

where D is a bounded open set in R n, with smooth boundary. This may, as with the backward heat equation (see Chapter VIII), be shown to have unique entire solutions, for all f,g in a dense subspace of X =_ LP(D) (1 _< p < ~ ) . As with Corollary 8.3, we will also, by Theorems 4.13 and 3.8(3), have well-posedness on a Frechet space densely and continuously embedded in LP(D). This will follow by writing (9.1) as

(d)2u(x) : Bu(x), (x > 0), u(O)= f, ( d ) u ( 0 ) : g, where B - - / k , on LP(D), with :D(B) - W2,P(D) n WI'B(D). The relevant property of B is that - B generates a bounded strongly continuous semigroup. With the usual matrix reduction, this becomes a first-order abstract Cauchy problem, d

~xg(X) = Ag(x), (x > 0) if(0) = (f,g),

(9.2)

where

A-

[B

In this chapter, we will show that there exists an operator C, with dense range, such that an extension of A generates an entire C-regularized group. More generally, we will show the following. 71

T h e o r e m 9.3. Suppose there exists complex A such that (-oo,O) C p(AB) and {r[[(r + AB)-I[[ [r > 0} is bounded. Then there exists a > 1 such that an extension of A, on X x X , l)(.4) - 1)(B) x X , generates an entire e - ( ~ ) " -regularized group that leaves 1)( A ) invariant. C o r o l l a r y 9.4. Suppose B is as in Theorem 9.3. Then (1) (9.2) has a unique entire solution, for all initial data (f,g) in a dense subspace; and (2) there exists Z, dense in X x X , such that Z is a Frechet space,

[Im(e -('xB)" )] ~ Z ~ X x X, where a is as in Theorem 9.3, and AIz generates an entire strongly continuous group. P r o o f o f T h e o r e m 9.3: There exists a square root, G -= (AB) 89 such that - G generates a b o u n d e d holomorphic strongly continuous semigroup. By Corollary 7.10 and T h e o r e m 8.2, it is sufficient to show that, whenever xl and x~ are entire vectors for G, then ~ = (x],x2) is an entire vector for .4. But this is not hard to see, when one notes t h a t ( A ) 2n = B n l ,

so that, for t >_ 0, so

k

k=0

oo =

r

~2n

t(2,~+1) /~

X1

(2n + 1)[

n=0

[[(Bnx2,Bn+]xl)[I] ,

which is finite, since B maps g(G) into g(G) (see Proposition 7.7). |

72

X. B O U N D A R Y

VALUES OF HOLOMORPHIC SEMIGttOUPS

The heat semigroup e zzx, where /k is the Laplacian on LP(R~), constitutes a holomorphic function on the right half-plane (RtIP) which is bounded on every sector Se = {zl larg(z)[ < 8}, for 8 less than ~. Its boundary values, e iszx, however, constitute a strongly continuous group (the Schr6dinger group) only when p = 2. It is important to "regularize" e i~zx for p ~ 2 and study the analogous SchrSdinger "group" in this case. In general, when e ~A is a bounded (in sectors So, for 6 < ~) holomorphic strongly continuous semigroup of angle ~, it is natural to ask when boundary values exist, in some sense. As indicated in the first paragraph, to include many interesting examples, this "sense" needs to be weaker than the sense of a strongly continuous group. In this chapter, we show that, when A generates a bounded strongly continuous holomorphic semigroup of angle ~, then the amount of regularizing, C, required to make e isA into a C-regularized group eisAc depends on how rapidly IlezAII g r o w s as z approaches the imaginary axis. This is useful for applications, because we may restrict our attention to bounded operators e zA, for Re(z) strictly greater than zero, which are much easier to work with than the unbounded operators e isA, whose definition, in general, may be somewhat mysterious. More specifically, we may look at the behaviour of IlezAII in sectors Se, as 0 approaches ~, or in half-planes Re(z) > a > 0, as a approaches 0. By definition of a bounded holomorphic semigroup, [lezAII is bounded in Se, for any 0 < ~; we show that iA generating a (1 - A)-~-regularized 1 e) r ), where O = group that is 0(1 +lsl r) corresponds to IlezAII being 0 ((e-gTXs arg(z). Generating a bounded (1 -A)-~-regularized group corresponds to IlezAI] being O((n~--~)~). We assume throughout this chapter that A generates a bounded strongly continuous holomorphic semigroup of angle ~ {eZA}zeRgp. T h e o r e m 10.1. Suppose that 7 -> 0 and there exists M < oc such that

IlezAII < M( -

Izl )" Re(z)

Then for all r > 7, there exists Mr,.r < oo such that iA generates a A)-r-regularized group {W~(s)}s,R such that

(1 -

IIWr(s)l I '7, there exists C~ such that iA generates a (1 - A) -~regularized group {Wr(s)}seR such that IIw~(s)ll _< c r ( 1 + [sir).

(c) II(1+

[z[r)-lezA(x

--

A)-~ll is uniformly

bounded in the right half-

plane, for all r > "7. It is theoretically interesting that a bounded ( 1 - A)-r-regularized semigroup corresponds to the same growth conditions on right hMf-planes Re(z) > a rather t h a n sectors. The proof of the following theorem is essentially the same as the proof of Theorem 10.2. Theorem

10.3. Suppose "7 ~_ O. Then the following are equivalent.

(a) For all r > "7, there exists Mr < oo such that HezA H 7. (c) ]]ezA(1 -- d)-r[] is uniformly bounded in the right half-plane, for all r > % Proof of Theorem

10.1: Fix r > 7. We have

(1 - A ) - r x -

1

e-Uu r - l e u A x du

r(r)

([gomu, Proposition 11.1]); thus, for z = t + is, (1 - A ) - ~ e z A x - I'(-r)

e-~ur-le((t+~)+is)Ax du,

so that we m a y estimate as follows: 1 Mfo~ e--Uur-l( u - ~ y ( v / ( u

I1(1 - A)-reZAII ___ r - ~

+ 0 2 + s2)~ du

e-~u~-l-~(~/(u + t)~ + s~)~ du, (10.4) 74

which is convergent since r - 1 - 7 > - 1 . As with strongly continuous holomorphic semigroups, since ( 1 - A ) - r e zA is holomorphic and bounded in every rectangle {t + is[O < t < 1, Is] < a},a > 0, its boundary values exist when t --* 0 and define a (1 - A) -~regularized group, {Wr(s))seR. It is straightforward to verify that the generator of the (1 - A)-~-regularized semigroup {(1 - A)-reZA)zeRHp is A, thus iA generates {W~(s)}seR. All that remains is the growth estimate on [[Wr(s)[[ = limt--+0 [ [ ( 1 A)-~e(t§ which, by (10.4), is less than or equal to

M

e_~u~_l_~(X/~+s2)~du"

jr0(X)

The integral may be shown to be less than or equal to

(1§

e-Uur-l-~(V~-~-~2)'Ydu ,

jr0~176

by considering separately Is[ 1, for if Is[ < 1, then u 2 § s 2 _< u 2 + l, while if ls[ ___ 1, then u 2 + s 2 = s 2 ( ( 7 ~) 2 § 2+1). | P r o o f o f T h e o r e m 10.2: The implications (a) --+ (b) and (c) --* (b) are essentially the same as the proof of Theorem 10.1. (b) ~ (c) is a consequence of the maximum principle for analytic functions. (b) --* (a). Fix r > 7. Since {ezA) is a bounded strongly continuous holomorphic semigroup, there exists K~ < oc such that [](1 - A)retA H 0 (see [Paz, Chapter 2]). Forz=x+iy, withx >0,

[[eZAl[ = [[(1- A) re~AWr(y)[I _< [1(1 - A) ~e~A[[[[W~(y)l I

_< g~x-~Cr(1 + [y]~) _< ( g ~ ) ( C ~ ) ( ~ ) nr

75

~.

XI. THE SCHRI~DINGEIt EQUATION

A much less extreme form of ill-posedness, than appears in Chapters VIII and IX, is the Schrhdinger equation, on LP(IU~),p ~t 2, or Co(I:tn),

(t,x)au

= i(

u(t,x) - V ( x ) u ( t , x ) )

(t e R ) , u(O,x) =

We apply the previous chapter to the Schrhdinger operator with potential, i(A - V), for V real valued, with V+ a Kato perturbation, V_ E L~176 and show that -

(n

-

v))-'Lea

is a polynomially bounded regularized semigroup on LP(tt~), for r > 1 2n I~ - ~[, w sufficiently large (Theorem 11.4). This produces polynomially bounded mild solutions of the Schrhdinger equation for all initial data f in : D ( ( / k - Y)r), on L(Rn). It is well known that, even for V - 0, the Schrhdinger operator does not generate a strongly continuous semigroup on L P ( R n ) , p ~ 2, or C0(R'~). For V - 0, we may cut r in half. 1 1 T h e o r e m 11.1. Suppose 1 ~ p < oc. Then for all r > n]-~ - ~],i/k, on LP(l:tn), generates a (1 - / k ) - r - r e g u l a r i z e d group, {Wr(s))seR, that is

0((1 + On C0(R n) or BUC(R'~), i A generates a (1 - A)-'-regularized group, {Wr(s)}seR that is O(1 + [s[~), for all r > ~. D e f i n i t i o n 11.2. We will denote by K n the Kato class of measurable functions on t t n, as defined in [Sim, p. 453]. This includes, but is not limited to, L~ D e f i n i t i o n 11.3.

For V E K '~, it is shown in [Sim, Theorem A.2.7] that H =_/X-V,

defined as a quadratic form, is a self-adjoint operator on L2(Rn). The following theorem states that H, with appropriate domain, generates an exponentially hounded (w - H)-r-regularized group, for r twice as big as in Theorem 11.1. 76

T h e o r e m 11.4. Suppose 1 < p < 0% V+ E K = , V - E L~176 Let H be as in Det~nition 11.3. Then there exists w E t t such that for a11 r > 2nl I - lj, {eisH(w_ H ) - r } s e R is an ( w - H)-"-regularized group on z,(it")

that

i~ 0((1 + I~l~li-+l)).

On C o ( R ~) or B U C ( I t " ) , {ei'H(w - H ) - ~ } s e R is an ( w - H ) - ~ - r e g u l a r ized group that is 0 ( ( ! + 1,1~)), ~or a l l r > n Theorem 11.1 follows immediately from Theorem 10.1 and the following lemms L e m m a 11.5. Suppose l O,f E LV(R'~)(1 _< p _< oo) and

for Re(z) > O,x,y E R. n. P r o o f o f T h e o r e m 11.4: Let w _-- max{llV_lloo,#} + 1. We argue as in T h e o r e m 11.1 w i t h / k replaced by H d- 1 - w. Since V + w - 1 is a real-valued nonnegative Kato perturbation, so t h a t - z ( V -t- w - 1) is a dissipative Kato perturbation, Ilez(H+l-w)ll2 _< 1, for all Re(z) > 0 (see [Go2, Corollary 6.8]). By L e m m a 11.6, Ilez(H+l-~lll < II(Re(z))-~exp ( - R e ( ~al~a-~)zzz) lll _< c~(nez-~(~))'~, where c ~ is independent of z, so that we may argue exactly as in the proof of T h e o r e m 11.1, to conclude that, for all r > 2hi I !1, {e~sH(w-- H ) - r } = {eiS(~-l)e~(H+l-")(1 -- ( g + 1 - w ) ) - r } is a O((1 + 2 Isl2nl~- 89

(w - H)-~-regularized semigroup. |

78

XII. FUNCTIONAL CALCULUS FOR COMMUTING GENERATORS OF BOUNDED STRONGLY CONTINUOUS GROUPS

In this chapter, we give an example of how regularized semigroups may be used to construct functional calculi for unbounded operators, or families of commuting unbounded operators. The idea is to construct a regularized semigroup, e t f ( A ) g ( A ) , for appropriate g, then define f ( A ) as the generator. By "appropriate" g, intuitively, we want the map w ~ e t f ( w ) g ( w ) to be bounded on the spectrum of A; depending on the structure of A, we may impose stronger growth conditions on this map. In this chapter, we will use the Fourier transform for our functional calculus construction. In Chapter XXII, we will use unbounded analogues of the Cauchy integral formula in a similar way. Even for multivariable polynomials, p, choosing an appropriate domain for p ( B 1 , . . . , Bn), where B 1 , . . . Bn a r e commuting operators, can be difficult. We shall see that regularized semigroups provide a very simple way of specifying a domain, namely, the domain of the generator of a regularized semigroup. We will need some standard multivariable terminology. We will write x = ( x l , . . . x n ) , for vectors in t t ~, a = ( a l , . . . a ~ ) for vectors in (N U {0}) ~. We will write x ~ = x~ 1 ~ , Ixl n Iz l , n ak. Ek=l Ek=l =

=

T e r m i n o l o g y 12.1. Throughout this chapter, and the next two, iB1,..., i B n will be commuting generators of bounded strongly continuous groups of operators. We will write B - ( B I , . . . , Bn),

B `~ - B ~ 1 . ' ' B ~ ' .

Let 9r be the Fourier transform. We define A to be the set of all inverse Fourier transforms of L 1 functions, that is, A = { f E C0(Rn) l~-f E LI(Rn)}. We will write ei(x'B) for 1-I~=1 eixk B~, where {e itBk }tER is the strongly continuous group generated by i B k . We will use the following well-known functional calculus. Define a bounded operator f ( B ) by f ( B ) - (27r)-~ JR. e i(x's) . T f ( x ) d x ,

(12.2)

whenever f C A. Note that this is essentially the Fourier inversion theorem. 79

The map f ~ f ( B ) has the following properties. (1) It is an algebra homomorphism. (2) There exists M < ec such that

llf(B)H -< M]IZ'f]]~, for all f E A.

(3)

I~(~5- B,) -~, =

( ~ j - ~,)-",

(B),

j=l

for any a E N n, Im(Aj) ~ O. Let - I B ] 2 be the generator of the strongly continuous semigroup

e -tlBI2 - f,(B), ft(x) - e -tIll2. Then in the usual way for generators of strongly continuous semigroups (see any of the references for strongly continuous semigroups), we have fractional powers (1 + IBI2) - r , for r > 0; in this case, they are given by (1 + [BI2) - r = ((1 + [x[2)-r)(B), defined by (12.2). For k E N, we will denote by B C k ( R n) the algebra of complex-valued functions on R ~, with bounded continuous derivatives, up to order k, with norm

IIfIIBC~(R-)-~

llD~fllo~ 9

l ~, there exists M ( s ) < o0 such that, for any f e ~ C k ( R " ) , (f(x)(1 + Ix12)-~) e A, and H (f(x)(1 + [x12)-~) (B)]] _< M(s)HfllBCk(R. ). R e m a r k 12.4. The map A f - (f(x)(1 + [x[2)-S)(B) defines a multivariable example of a ((1 + [B[2) -s) regularized functional calculus (see Definition 22.11). D e f i n i t i o n 12.5. For f E C~176 let A ( f ) be the set of all real, positive bounded functions g E C ~ ( R n ) , such that z ~-+ eZlg defines an entire map from the complex plane into B C k ( R n ) ) . Before proceeding further, we should verify that this set is nonempty. 80

L e m m a 12.6. For any f E C~176

A ( f ) is nonempty.

Definition 12.7. For any f E C~176176 g E r -

complex z, let

+

where s - 1 + [~]. For any z E C, define

WsAz) = T h e o r e m 12.8. The family {Ws,g( Z) }zeC is an entire WLg(O)-regularized

group. Propositon 12.9. If f E A N C~176

then f ( B ) is the generator of

{WLg(Z)}zeC , for any g E A(f). Definition 12.10. For f E C~176 f ( B ) is defined to be the generator of {Wl,g(z)}zec, for any g E A ( f ) . The operator is well-defined, that is, the definition is independent of which g E A ( f ) is chosen, for the following reason. Temporarily write f(j, B) (j = 1,2) for the generator of {Ws,g i (Z)}zeC- Then by Proposition 3.10, both f(1, B) and f(2, B) generate the regularized group {Wl,glg2(z)}~ec , thus are equal. Proposition 12.9 shows that this definition is consistent with (12.2). In the following theorem, note that any N th degree polynomial will satisfy (1). T h e o r e m 12.11. Suppose f E C~176 N, M > 0 such that for all x E t t n,

~- 1 + [~],w • I~,N E

(1) I(D"f)(x)l < M(1 + I 1) N, whenever [a[ < k; and (2) Re(f(x)) < w. Then, for all r > ( ~ ( N - 1) + ~), f ( B ) generates a norm continuous (1 + [B[2)-~-regularized semigroup that is O(e~t). The following states two ways in which our map f ~-~ f(B) is an algebra homomorphism. The consistency of Definitions (12.2) and 12.10 81

has already been established. When p is a polynomial of degree N in n variables, p(x) =_ ~l~I0 is uniformly bounded. The same argument applied to [[Gt(B) - Gs(B)[] implies that {Gt(B)}t>o is norm continuous. Since f ~ f ( B ) is an algebra homomorphism, {Gt(B)}t>0 is a C0(B)regularized semigroup. By (3) of (12.2), C0(B) = (1 + ]B]2) -r. Thus {Gt(B)}t>0 is a (1 + ]S[2)-r-regularized semigroup. Choose g E A ( f ) . By (12.2)(1), g(B)(1 + [B[2)-s~(B) = (1 + [B[2)-rWl,g(t) (see Definitions 12.7 and 12.10). By Proposition 3.10, f ( B ) is the generator of

{~(B)},>0. I

P r o o f of T h e o r e m 12.12: For (1), we use (1) of (12.2) and Proposition 12.3, as follows. Suppose g E A ( f ) . Then

d --d~zWI,g(Z)lz=oh(B) = (Gf,g,of) ( B ) h ( B ) = (Gf,g,ofh) (B) = Wl,g(O)(fh)(B ). The definition of f ( B ) as the generator of a Wf,g(O)- regularized group now concludes the proof of (1). 84

For (2), we apply (1) with h(x) - (1 + Ix[2) -t. By Proposition 12.3, h and ph are in ,4. Thus, by (1), 7)([B[ 2~) = Im((1 + [B[2) -~) C_ 7)(p(B)), with

p(B)(1 + JB]2) -t = (p(x)(1 + ]x]~) -t) (B) = ~ c~,BC'(1+ IB[2) -~, I~l__ 0), g ( O ) = ~,

(13.1)

where (pi,j) is a m a t r i x of polynomials in n variables and iB1,..., iBn is a family of commuting generators of bounded strongly continuous groups of operators, intuitively, one wants to take the exponential of the m a t r i x of operators (pi,j(B)). Since the spectrum of Bk is real, for all k, this means we want the complex matrix (pi,j(x)) to be contained in a left half-plane, "in some sense," for all x in R n. The strongest sense is to have the numerical range of the complex matrix contained in a half-plane Re(z) < w, since this implies that the matrix-valued exponential et(p"i(x)) has norm less t h a n or equal to e TM, for all t >_ 0, x E R n. A sense t h a t is easier to calculate, or at least estimate, is to have the spectrum of (pi,j(x)) contained in a left half-plane, for all x E R ~. The system (13.1) is then said to be Petrovsky correct (see [G-S]). The intuition we just described is not quite true, in general, without i0 on Lq(Rn)(1 _< q < oo), then some modification. W h e n Bk =- 0xk, (pi,j(B1,...,Bn)) generates a strongly continuous semigroup, whenever the numerical range of (pi,j(x)) is contained in the left half-plane, if and only if q = 2. This is true even for a one-by-one matrix: take p(x) =_i]x]2, so t h a t p(B) is the Schrhdinger operator iA; then it is well known that p(B) generates a strongly continuous semigroup if and only if q = 2. Higher order abstract Cauchy problems are a special case of (13.1), via the usual matrix reduction to a first order problem. The obvious choice io on some space of functions on t t ~, so that (13.1) becomes for Bk is b-~-x~, an arbitrary system of constant coefficient differential equations. But certainly other choices are possible, such as ~ on a space of functions on some bounded domain, with appropriate boundary conditions. In this chapter, we use regularized semigroups to describe simply and explicitly how a perturbation of the intuition described above is true, when the m a t r i x (pLj(x)) is Petrovsky correct. The operator (Pi,j(B1,..., Bn)) then generates a (1 + IBI2) -~- regularized semigroup, where n

IBI

IBkl k=l

86

and r is a constant depending on n, the order of the matrix and the maximum value of the degree of Pi,j (Theorem 13.9). When (pi,j(x)) has numerical range contained in a left half-plane, independently of x, then the dependence on the order of the matrix disappears (Theorem 13.4). Our goal in this chapter has been to obtain results involving nothing more than the most simple-minded spectral intuition, that is, the behaviour of the complex-valued matrices (pi,j(x)), for x in R n. We will use the same terminology as in the previous chapter. Throughout this chapter, m and n are fixed natural numbers. M a t r i x O p e r a t o r T e r m i n o l o g y 13.2 We give X TM the Banach space norm m

li(xl,...xm)lJ =

iixkll. k=l

Throughout this chapter and the next, M -- (Pi,j)i,'~=l will be an m x m matrix of polynomials. The number N will be ma:Q,j{ r of Pi,j }. We define the operator M ( B ) , on X TM, by

.A4(B) =_ (pi,j(B)), D(Jt4(B)) --= O(IB[2~)m,/_-__ 1 q - [ I ( N q- 2 ) 1. (See Theorem 12.2(2).) If Y:fi,j E L~(Rn), and G -= (fi,j), we define the operator G(B) -(fi,j(B)) e B ( X TM) similarly. Note that l ~i,j Ix e R m, Ixl = 1), m where < 9> is the inner product in C m, < x , y >=_- Y~i=~ xi~.

T h e o r e m 13.4. Suppose w C R, k = ([~] q- 1) and for all x C R n,

87

Then, for all r > (~ (N - 1) + ~), an extension of M (B) generates a norm continuous O(e ~t) (1 + [B[2)-~-regularized semigroup, that leaves D(.M ( B ) ) invariant.

R e m a r k 13.5. It is interesting that this result is independent of m, the size of the matrix (compare with Theorem 12.11). E x a m p l e 13.6. In Theorem 13.4, let Bk -- iDk, D - i ( D 1 , . . . , D n ) , X - the Sobolev space w~,q(PJ~)(1 _< q < oo,l + 1 E N). Then since wJ,q(I:t n) C_ :D(/k~) = Ira((1 + A)- 89 Theorems 13.4, 3.8(3) and 5.16 give us the following result about systems of constant coefficient Cauchy problems, ~tff(t,/)(x) = (p~,j(D))ff(t,/)(x)

ff(0, f ) ( x ) = f(x) (t ___0,x e Rn). (13.7)

C o r o l l a r y . Suppose w E R and for all x E R n, n.r.(pi,j(x)) C {z e C [ R e ( z ) ~_ w} and j - ~ + lV + ( g - 1)[~].

Then for all nonneg~tive integers e, 1 < q
( ~ ( N - 1 ) + ~ + N ( m - - 1)). Then an extension of M ( B ) generates an exponentially bounded norm continuous ( i + [B[2)-r-regularized semigroup, that leaves O ( M ( B ) ) invaria~t. Specifically, if w is as in Definition 13.8, then this regularized semigroup is O(e(W+e)t), for all E > 0 E x a m p l e 13.11. As in Example 13.6, the obvious application is to Petrovsky-correct systems of constant coefficient Cauchy problems. 88

Corollary. Suppose (Pi,j) is Petrovsky-correct and j = ~ + N m + (N 1)[~]. Then for all nonnegative integers ~, 1 _< q < co, there exists Zt,q, a Banach space, such that (wJ+t,q(p~n)) m t.~ Zt,q r

(W~"q(l~n)) m,

and (13.7) is we//-posed on Zt,q, that is, for all f ~ Zt,q, there exists a unique mild solution of(13.7) in Zt,q and g(t, f=) --* 0 in Zt,q, uniformly in t on compact subsets, whenever fn -~ 0 in Zt,q. Of course, similar results hold for X = C0(tt'~),C~(R '~) or C0~(R.'~), since translation is uniformly bounded and strongly continuous on these spaces. E x a m p l e 13.12. The equation describing sound propagation in a viscous gas is (see [G-S, Example 3, page 134]) 02 u

03 u -

02 u +

After the usual matrix reduction, this becomes d . = .M(D)ff, 0 where D - z9d , 3 d ( x ) - - [ (_x2)

1 ].SincethespectrumofAd(x ) (_2x2)

is - x 2(1 + v / ~ - 1), which has real part bounded above, for x real, this is a Petrovsky-correct system, so that Theorem 13.9 implies that an extension of 3/I(D) generates an exponentially bounded (1 + A)-r-regularized 7 semigroup, for all r > ~. The following lemma is a straightforward calculation, using the product rule and the fact that ]x~[ < Ix[% For matrix-valued functions ~ - ( f i j ) , with fi,j E BCk(Itn), we define

IIGIIBc (R.,B(C-))--

l O,pij fixed, there exists M < ec such that

II (et(P"J(~))Y'i=~(1 + Ix12)

(1 + [ZI)k(N-1)-2"llet(V'"())"J=~]IB(C",)

_< M sup xER

IlBc~m-,.(c~))

~

89

9

P r o o f o f T h e o r e m 13.4: Let ~t(x) --- et(P',i(x))(1 + [xl2) -r, for t _> 0, x C R,n. As in the proof of Theorem 12.11, with Lemma 13.13 replacing Lemma 12.17, {Gt(B)}t_>0 is a (1 + ]BI2)-r-regularized semigroup. Using dominated convergence and Proposition 12.3, a calculation shows that G~(B)(1 + IB[2) - t is (norm) differentiahle at t = 0, and

d 6 t ( B ) ( 1 + ]B[2)-e[t=o = (1 + [B[2)-~M(B)(1 + IB[2) -t.

Since I ) ( M ( B ) ) = Ira((1 + IB[2)-t), this implies that an extension of M ( B ) generates the regularized semigroup. The invariance of • ( M ( B ) ) under the regularized semigroup is clear from the fact that ~t(B) and (1 + IB]2) -~ commute. II Lemma 13.13 tells us that the proof of Theorem 13.9 should be the same as the proof of Theorem 13.4, if we can estimate liet(p',j(~))',~=~]t. Thus we need some lemmas about matrices. Although the following may be well known, we include their proofs for completeness. Lemma 13.14. For any m E N, there exists Kl(m) such that

det(M)lIM-111 _< KI(m)IIM]I m-l, for MI m x m invertibte matrices M. P r o o f : Look at the cofactor expansion for M -1. | L e m m a 13.15. For any m E N, there exists K2(m) with the following property. Whenever e > 0 and M is an m • m matrix with

a(M) C_ {z e C I R e ( z ) < -e}, then for all t > O,

IletMH K2(m) m--I IIMHm-1. P r o o f : Let {bk}km=l be the (not necessarily distinct) eigenvalues of M. It is elementary to verify that there exists F, a contour of length less than 3~rm2E, contained in the left half-plane Re(z) < 0, that surrounds 90

a(M),

bk] > ~, for

such that ]z -

2~iie~Mii = II~r e'z(z

all z E r . Then, using L e m m a 13.14,

- M)-I dzii

_.0, the family of fractional powers of a fixed operator B. The standard hypothesis on B is that of Definition 8.1. In Example 15.1, we show that there exists bounded, injective C, with dense range, such that {BtC}t>0 is a C-regularized semigroup. We also consider sums of commuting generators of integrated semigroups (Example 15.2; see Definition 18.1) and the linearized form of the Ricatti operator (Example 15.3). Example 15.1. Suppose B is dosed, ~ ( B ) is dense and (-co,0) C p(B), with {[[r(r + B)-I[[ [r > 0} bounded. For t > 0, let B t be the usual fractional power of B (see Definition 8.1). Then, by the properties of strongly continuous holomorphic semigroups, {Bte -B 89}t>_0 is an e -B 89 regularized semigroup (note that -B 89is constructed in such a way that it generates a strongly continuous holomorphic semigroup {e -sB 89}s_>0). We remark that this regularized semigroup is far from being exponentially bounded, in general. If we choose

(Bf)(x) =- z2f(z), on C0(R), with maximal domain, then a calculus calculation shows that for any t > 0,

lIB, e-B 89ii = E x a m p l e 15.2. See Definition 18.1 for the definition of an integrated semigroup. When B1 and B2 generate commuting exponentially bounded integrated semigroups, even if p(B1 + B2) is nonempty, it is difficult to write down an integrated semigroup generated by B1 + B2. The simplest way to deal with sums of generators of integrated semigroups is with regularized semigroups. T h e o r e m . Suppose B1 and B2 generate exponentially bounded n-times integrated semigroups that commute and Az - BlZ + B2z, T~(A) :D(B1B2). Then an extension of A generates an (r - B1)-n(r - B2) -nregularized semigroup that leaves ~)(A ) invariant. Proof: By Theorem 18.3, there exists r, in p(B1)VIp(B2), such that, for i = 1,2, Bi generates an exponentially bounded ( B i - r)-n-regularized semigroup, Wi(t). Let W(t) =. Wl(t)W2(t). This is a (B1 - r)-'*(B2 94

r)-n-regularized semigroup. The following argument shows that an extension of A generates W(t). Suppose x is in 7:)(A). For t > 0,

~(w(t)~-

w(0)~)

Since IlWl(t)ll is bounded, for t small, and x is in/)(B1) fl V(B2), we have lim 1 ( W ( t ) x - W(O)x)= WI(O)(W2(O)B2x) + W2(O)(WI(O)BlX)

t---~O t

= W(OlAx. Thus an extension of A generates W(t). Since Wl(t) and W2(t) commute, W(t) commutes with all resolvents of B1 and B2. This implies that W(t) leaves :D(A) invariant. | E x a m p l e 15.3. We define r on B(X), by

r

- A1B % BA~.

When A1 and A2 generate strongly continuous semigroups, then, formaily, etr = etA1Be tA2. Since etA1 and e tA2 are only strongly continuous and not norm continuous, etr will not be strongly continuous. However, (r - A1)-le tA1 and etA2(r -- A2) -1 are norm continuous, thus

W(t)(B) =_ (r - A1) -1 (e'|

A2)-1

is strongly continuous. Thus, when A1 and A2 generate strongly continuous semigroups, we expect r to generate a A-regularized semigroup, where A(B) = ( r A 1 ) - I B ( r - A2) -1. By a happy coincidence, the range of A is a comfortable domain for r Theorem. Suppose A1 and A2 generate exponentially bounded n-times integrated semigroups, and r is in p(A1) CIp( A2 ). Define A, ~, on B(X), by A(B) = ( r - A 1 ) - I B ( r - A2) -1, ~(B) A1B + BA2,/)(~) = range of A.

95

Then an extension o f r generates a A(n+l)-regularized semigroup, that leaves 2)(r inwariant. Proof: For i = 1, 2, let Si(t) be the (r - Ai)-n-regularized semigroup generated by Ai (as guaranteed by Theorem 18.3). Define W(t), on B(X), by (W(t))(B) -= ( r - A1)-lSl(t)BS2(t)(r A2) -1. Using the mean value theorem and the fact that ~ S i ( t ) ( r - A i ) - l z exists and equals Si(t)Ai(r - Ai)-lx, for all z in X, i = 1,2, it is not difficult to show that (r - Ai)-lSi(t) is a continuous function of t, in the operator norm of B(X). Thus W(t) is a strongly continuous function of t. It is clear that W(0) equals A n+x. To see that W(t) is a An+X-regularized semigroup, we calculate

W(t)W(s)(B) = ( r - A1) -1Sl(t)[(r- A1)-xSx(s)BS2(s)(r - A2)-l] 9 9s 2 ( t ) ( r - A2) -x

= ( r - Ax) -2 [(r - A1)-nsx(t + s)BS2(t + s ) ( r - A2)-n] 9 9( r -- A 2 ) - 2 ,

= [ A n + l w ( s + t)] ( B ) ,

as desired. Suppose now that B is in 2)(r Then W ( t ) ( B ) = (7' - Ax)-2Sl(t)(A-X(B)) S 2 ( t ) ( r - A2) -2. Again using the mean value theorem, one may show that ( r - A i ) - 2 S i ( t ) is a differentiable function of t, in the operator norm of B(X), for i = 1, 2. Thus W(t)(B) is differentiable, with d W ( t ) ( B ) = [ d ( r - A x ) - 2 S x ( t ) ] A-I(B)S2(t)(r-A2) -2

+ ( r - A1)-2Sl(t)A-l(B) [ d s2(t)(r- A2) -2] = (r-- A1)-lSl(t)Al(r

- A1)-IA-I(B)(r

- A 2 ) - l s 2 ( t ) ( r - A2) -1

+ ( r - A 1 ) - l S l ( t ) ( r - A 1 ) - I A - I ( B ) ( r - A2)-IA2S2(t)(r - A2) -1 -- ( r - - A 1 ) - I s I ( t ) ( A 1 B ) S 2 ( t ) ( r

- A2) -1

+ (r - A1) -1SI(t)(BA2)S2(t)(r - A2) -1 = W(0r

This implies that an extension of r generates W(t). Since W(t) commutes with A, W(t) leaves 2)(r invariant. | 96

XVI. EXISTENCE

AND UNIQUENESS

FAMILIES

Let us recall the difference between a C-existence family, or mild Cexistence family, and a C-regularized semigroup. We think of all these families as being etAc; for a C-regularized semigroup, etAC = Ce tA, and this leads to two advantages: an algebraic definition of a C-regularized family, and not only existence, but uniqueness, of solutions of the abstract Cauchy problem (0.1). A close look at the proofs of uniqueness, when A generates a C-regularized semigroup, reveals that what is being used is Ce tA. If we drop the hypothesis that C commutes with A, then we could have two families of operators, etAC (for existence) and Ce tA (for uniqueness). More generally, we could have bounded operators C1, C2, and consider families of operators etAc1, C2 etA. It is only C2 that need be injective, since etAc 1 is merely providing existence of solutions, for all initial data in the image of C1. The family of operators C2e tA, besides providing uniqueness, also suggests the possibility of defining a generator, as with C-regularized semigroups: A - C~ a(~C2e d tA )]t=o. Finally, since

(C2e tA)(esAC1) = C2 e(t+s)Ac1 ----C2 (e(t+s)Ac1) = (C2e (t+s)A)C1, we obtain an algebraic definition of the pair of families of operators ( etAC1, C2e tA ) (Definition 16.1). To summarize: existence and uniqueness have been separated into two families of operators, that are equal when C1 and C2 are equal and commute with A. To define the algebraic properties of the (C1, C2) existence and uniqueness family {(Wl(t),W2(t))}t>o, it is necessary to intertwine W1 and W2, as in Definition 16.1 (3) (see Remark 16.2). The basic properties of the generator are in Theorem 16.5. The relationships with the abstract Cauchy problem are in Theorem 16.5 and Propositions 16.6, 16.7 and 16.8. D e f i n i t i o n 16.1. The pair {(Wa(t),W2(t)}t>_0 of strongly-continuous families of bounded operators is a mild (C1, C2) existence and uniqueness family if (1) Wi(O) = Ci, for i = 1, 2; (2) C2 is injective; and

(3) W2(t)WI(s ) -~ C2Wl(t -~- s) : W2(t ~v s)C1,

97

for all s,t >_ O.

The operator A generates (W1,W2) if

Ax = C~a ( dw2(t)x,t=o) , :D(A) =_{x ] -~W2(t)x exists, and equals W2(t)C~ 1 -~W2(t)xlt=o , for all t _> 0}. Intuitively, Wl(t) = etAC1, W2(t)

=

C2e tA.

R e m a r k 16.2. When Ca equals C2, and commutes with Wl(t) and W2(t), for all t > 0, then Wl(t) equals W2(t), and Wl(t) is a Cl-regularized semigroup. A consequence of Theorem 3.4(d) is that A generates Wl(t). First, we would like to give a simple example of an operator that generates a (C1, C2) existence and uniqueness family, but does not generate a regularized semigroup. E x a m p l e 16.3. Let X -= {continuous f : l:t ~ C I liml::l--*oof(x) e~2 = 0}, and let [[f[[ = supxe~[f(x)eX2[, A =_d/dx, with maximal domain. Let (Clf)(x) =- e-~2 f(z), C2 =- e l ,

(Wa(t)f) (x) = e-(X+t)2f(x q- t) (W2(t)f) (x) = e-X2 f(x + t). (Intuitively, Wl(t) = etAC1, W2(t) = C2etA, where etA is translation,

(etA f)(z) = f ( z + t).) Then it is straightforward to show that (W1,W2) is a (C1, C2) existence and uniqueness family, generated by A. In fact, the abstract Cauchy problem has a unique solution, for all initial data in a dense set(see Theorem 16.5(b)). However, the following demonstrates that regularized semigroups, unlike existence families, are inadequate to produce these solutions. P r o p o s i t i o n 16.4. /fA is as in Example 16.3, then the abstract Cauchy problem (0.1) has a unique solution, for all initial data in a dense set, but A does not generate a C-regularized semigroup, for any C. T h e o r e m 16.5. Suppose A generates the mild (C1,C2) existence and uniqueness family (W1, W2). Then

(a) A is dosed; (b) {Wl(t)}t>o is a mild Ca-existence family for A; and (c) ~ ( a ) _DIm(Ca). 98

P r o p o s i t i o n 16.6. Suppose A generates a mild (C1,C2) existence and uniqueness family. Then all solutions and mild solutions of the abstract Cauchy problem (0.1) are unique. As with regularized semigroups (see Theorem 3.7), we may obtain partial converses of Theorem 16.5(b). As long as 9(A) is sufficiently large, it is sufficient that an extension of A be such a generator. P r o p o s i t i o n 16.7. Suppose {Wl(t)}t>_o and {W2(t)}t>>_o are strongly continuous families of bounded operators, and fo Ws(s)z ds E D(A), for all z G E. Then the following are equivalent. (a) (W1, W2) is a mild (C1, C2) existence and uniqueness family, generated by an extension of A. (b) {Wl(t)}t>_o is a mild Cl-existence family for A, W2(O) = C2, and a w 2 ( t ) z exists and equals W2(t)Az, for M1 z E :D(A), t _> 0. Similar results for strong solutions are in the following.

Proposition 16.8. Suppose {Wl(t)}t>o and {W2(t)}t>o are strongly continuous famih'es of bounded operators, ~)(A) is dense, and t ~-+Wl(t)x is a continuous map from [0, oe) into [:D(A)], for all z e D(A). Then the following are equivalent. (a) (W1, W2) is a (C1, C2) existence and uniqueness family, generated by an extension of A. (b) {Wx(t)[[V(A)l}t>o is a Cl-existence family for A, W2(0) = C2, and d w 2 ( t ) z exists and equals W2(t)Az, for all z e D(A), t >_O. E x a m p l e 16.9. For n G N, define, on X1 x X2, A = (G01

G~B) '

I)(A)=2)(G1)x[D(B)ND(G2)],

where :D(G~) C_ :/)(B), B is closed, and Gi generates a strongly-continuous semigroup, for i = 1,2 and there exist rn E N, w E p(Gx) such that (w - G1)-mB is bounded. Then an extension of A generates a (C1, C2) existence and uniqueness family, where

cl [0'

0 c2 n] c2

cx m 0]

Proposition 16.4 is shown by the following two lemmas. 99

L e m m a 16.10. Suppose W(t) is a C-regularized semigroup generated by A, from Example 16.3. Then there exists I~, a complex-valued measure of bounded variation on hounded intervals, such that, for all f in C ~ ( R ) ,

(cf)(~) = J~ I(~ + r)d~,Cr) (W(t)f)(z) = frt'f(z + t + r) d~(r). P r o o f : Fix f E C ~ ( R ) . Since the generator of a C-regularized semigroup commutes with C (see Theorem 3.4), C f e 79(A), and -~z(Cf) = C(-~1~x). Let gt(x) -- g(x + t), for x real. Since C is bounded and f e C ~ ( t t ) , limb--.0 ~ (ft+h -- ft) exists, in X, and equals f[, for all t > 0. Thus, since C is bounded, d c ( f t ) = d c ( f t ) . Since d , on C0(tt), with maximal domain, generates a strongly continuous semigroup, and ~ ( C f ) t = "~z(Cf)t (in C0(it)), it follows that c(f~) = (el),, for all t > O. For all real x, there exists #~, a complex-valued measure of bounded variation on bounded intervals, such that

(Cg)(z) = JR g(r)d~,~(r), for all g e X. For f e C ~ ( R ) , x e It,

f~

:(~) d.~(~ + x) =

f~

: ( w - ~) d.~(w) = [C(:_.)] (x) =

(C:)_.(~)

= (ci)(o)= JR I(r)d~,o(~). Thus d#x(r + x) = d#o(r), for all real r, z, so that

(Cf)(x) = JR f(w)d#o(W- x) = /R f(r + x)dlzo(r), the desired representation of C. Since d fit f(x + t -F r) d#(r) = d~ fR f(x q- t + r) dtt(r), for any f in C ~ ( i t ) , W(t) is as stated. II The following lemma shows that we cannot have both C and W(t) in B(X), for any t > O. i00

L e m m a 16.11. Suppose t >_ O, and W(t), asia Lemma 16.10, is in B ( X ) . Then the support o f # is contained in {-t}. P r o o f : For any real x, the map f ~ ez2(W(t)f)(x) is a linear functional on X, of norm less than or equal to IIW(t)ll. Thus there exists a complexvalued measure, Cz, of total variation Ir < IIW(t)ll, such that

e x2

f ( x + t + r)d#(r) =

e

f(w)dCx(w),

for all f e C ~ ( I t ) , which implies that

IIW(t)l/~> Ir

= / R e~:~-~ dl~l(w - x - t)

= frt e-2~C'+~)e-C~+~)~dl~'l(~) =

~ ' e - ~ / 4 d l # l ( - t - ~-).

A short argument with Liouville's theorem shows that, because this is a bounded function of x E It, the support of I/zl must be contained in

{-t}. i P r o o f o f T h e o r e m 16.5: (a) It's clear from the definition that d w 2 ( t ) x

= W2(t)Az, for all x E :D(A), t _> 0. Suppose {xn}n=l ~ C_ V ( A ) , x . - ~ x and Axn --, y, as n -+ ~ . Then W2(t)zn converges to W2(t)x, and d w 2 ( t ) z n = W2(t)Azn converges to W2(t)y, both uniformly on compact subsets of [0, or). The strong continuity of W2(t) now implies that ~tW2(t)x exists, and equals W2(t)y, for all t > 0. This implies that x E D(A), and Ax = y, as desired. (b)

ForxeX,

t>0,1et

y ~

~0tW l ( s ) z d s . i01

For any r, r + h > 0,

= n~ w 2 ( ~ )

-

(W~(8)xd.)

which converges to W 2 ( r ) ( W l ( t ) x - Clx), as h ~ O. Thus y E :D(A), with Ay = Wl(t)x - ClX, as desired. Assertion (c) follows from (b), since Clx = limh-~0 ~ f:Wl(s)xds, for any x E E. ] P r o o f o f P r o p o s i t i o n 16.6: Suppose u is a solution or mild solution of (0.1), with x = 0. Then, for all t > s > 0, d [ W 2 ( t - s)u(s)] = 0, so that C2u(t) = W2(t)u(O) = 0. Since C2 is injective, this implies that u(t) = O, for all t _> 0, as desired. ] P r o o f of P r o p o s i t i o n 16.7: (a) --* (b) is immediate from Theorem 16.5(b) and the definition of the generator. ( b ) ~ (a). For t _> w >_ 0, x E X ,

J;

W2(r)Clxdr =

io'-

=L

W2(t - s)Clxds

t-~ d [W2(t- 8) ( L ' W x ( r ) x dr)] ds d---~

= W~(w)

(io

Wl(r)xdr

w2(~)clx = w2(w)wl(t

-

)

,

so that

~)~.

|

P r o o f o f P r o p o s i t i o n 16.8: (a) ~ (b). (W1, W2), x e :/)(A). By Theorem 16.5(b),

(/o') L'

Wl(t)x - ClX -b fil = Clx +

Let .4 be the generator of

Wl(s)xd8

AWl(8)xds,

102

since A is closed (by Theorem 16.5(a)), and by hypothesis, W1 ( s )z G T~(A) and s ~ AWx(s)z is continuous. The derivative formula for W2 follows immediately from the definition of the generator. (b) ~ (a). For x e D(A), 0 = ~W2(t - s)Wl(s)z, for all t > s > 0, thus W2(O)W~(t)x = W 2 ( t - s)Wl(s)x = W2(t)WI(O)x. Since D(A) is dense and Wi(t) E B(X), for i = 1, 2, t _> 0, the same is true for all z E X. Since d w 2 ( t ) z = W~(t)Az, for all z E D(A), the generator of (Wx, W~) is an extension of A. | P r o o f o f E x a m p l e 16.9: As in Example 2.11, formally

0

e tG2

'

and we want Wl(t) = etAC1,W2(t) = C2etA; thus our choice of C1 and C2 become clear, as we make the following definitions. For t > 0, let

W~(t)= (et: 1 fo e(t-~')aaB(s-G2)-"e"a2dw). eta~ (s - G~)-"

For the uniqueness family, define, for x2 in Zl(G~), t > 0, x

W2(t)x = ( ( w - G 1 ) - m e t a l

=

(xl,x2),

fte(t-~)al(w-Gx)-mBe~a2dy)

0

etG2

X.

Since G2 generates a strongly continuous semigroup, :D(G'~) is dense. Our hypotheses imply that W2(t) is bounded. Thus W2(t) may be extended to a strongly continuous family of bounded operators on X1 x X~. A calculation shows that W2(t) and Wl(s) satisfy (3) of Definition 16.1, for ms 6 D(G~), and hence for all x, since all relevant operators are bounded. Thus {Wl(t), W2(t)}t>0 is a (C1, C2) existence and uniqueness family. To see that A is the generator, note that, for x G 79(A), the map t ~-~ W2(t)x is differentiable, with

dw2(t)zl,=o =

j z,

which is in the image of C2. A calculation shows that, for x G D(A), dw2(t)x exists and equals W2(t)Az. Thus an extension of A is the generator. | 103

XVII. C-RESOLVENTS AND HILLE-YOSIDA TYPE THEOREMS Theorems about semigroups that involve resolvents are desirable, because resolvents are often relatively easy to calculate or estimate. For C-regularized semigroups or C-existence families, we need a generalization of resolvent that we introduced in Definition 3.6, the C-resolvent. E x a m p l e 1%1. Except when C is the identity, there is no reason to expect the generator of a C-regularized semigroup, even a bounded regularized semigroup, to have nonempty resolvent set. Consider A = [ 0G

G k ] , 7 9 ( A ) = : D ( G ) x 7)(Gk),

where k 6 N and G is unbounded and generates a strongly continuous exponentially decaying semigroup (so that 0 6 p(G)). Then an extension of A generates the bounded G-k-regularized semigroup

W(t)=_ [etaGo-k

te ta etGG-k ] "

However, for k > 2, it is not hard to show that A, and any extension of A, has empty resolvent set. If (A - G) fails to be injective (surjective), then the same is true of (A - A), while if A 6 p(G), then a calculation shows that, at least on a dense set, (A_A)-I

[ ( A - G ) -1 =

o

G k ( A - G ) -2]

(A-G)

/"

For k > 2, this implies that (A - A)-* is an unbounded operator, thus all extensions of A, including the generator of {W(t)}t>o, have nonempty resolvent set. The following proposition shows that the natural analogue of resolvent, for generators of C-regularized semigroups, is the C-resolvent, introduced in Definition 3.6. P r o p o s i t i o n 17.2. Supposew 6 R and A generates a O(e ~t) C-regularized

semigroup. Then {z IRe(z) > w} C_pc(A). The map z ~-~ (z - A)-IC is a holomorphic map from Re(z) > w into B(X). Hille-Yosida type theorems are a consequence of our "pointwise tIilleYosida theorems" in Chapter V. 104

T h e o r e m 17.3. Supposew E R, A is closed, C e B ( X ) , (o~,oo) C_pc(A) and I v ( C ) C_Im(s - A) n, for all s > w, n E N, with { l l ( s - ~)~(s - A)-~CII I , > ~o,n ~ N} bounded.

Then for all r > w, there exists a mild (r - A)-lC-existence family for A, {W(t)}t>o, such that e-~tW(t) is Lipschitz continuous. One surprising consequence of the following is that, when A is closed and densely defined, (w, ~ ) C_pc(A), and {W(t)}t>0 is a C-regularized semigroup generated by an extension of A, that leaves :D(A) invariant, then f~ W(s)x ds e 19(3), for all x e X, t >_ 0. We know that this is true when A itself is the generator (Theorem 3.4), but it seems surprising that the domain of A in this situation is sufficiently large; apparently having nonempty C-resolvent forces the domain of A to be large. Note also that, in the following, if Im(C) is dense, then :D(A) is automatically dense (Theorem 3.4). T h e o r e m 17.4. Suppose C E B ( X ) is injective, w E R, A is dosed, 79(A) is dense, (w, co) C_ pc(A) and CA C_ AC. Then the following are equivalent. (a) An extension of A generates a O(e ~t) C-regularized semigroup that leaves 7)( A ) invariant. (b) There exists an O(e wt) strong C-existence family, {W(t)}t>o, for A, such that W(t)A C_ AW(t), for all t > O. (c) Ira(C) C_I v ( s - A) ~, for all s > w, n E N, with { l l ( s - ~)~(s - A)-~CII Is > ~ , n e N} bounded.

(d) Ira(C) C_Im(s - A) ~, for all s > w, n e N, with {11(~-~)"(s-

A)-~zll I s > ~ , n e N} bounded, Vz e Ira(C).

Proposition 3.9 then immediately gives us the following. C o r o l l a r y 17.5. Suppose C e B ( X ) is injective, w E R, I)(A) is dense, (w, cr C_p(A) and CA C_ AC. Then the following are equivalent. (a) A generates an O(e ~ C-regularized semigroup. (b) There exists an O(e '~t) strong C-existence family, {W(t)}t>o, for A, such that W(t)A C_AW(t), for all t > O. (c) {ll(s- ~)"(~- A)-nCII Is > ~,n e N} is bounded. And as an immediate corollary, using Theorem 3.4(a) and (b) and Proposition 17.2, we obtain the famous Hille-Yosida-PhiUips-Miyadera theorem for strongly continuous semigroups.

105

C o r o l l a r y 17.6 ( H U l e - Y o s i d a - P h i l l i p s - M i y a d e r a ) . Suppose w E R. Then the following are eq uivalen t. (a) A generates a O(e ~t) strongly continuous semigroup. (b) :D(A) is dense, (w, oo) C_p(A) and

{ll(s- ~)~(s- A)-~ll I~ > o~,~ e N}

is bounded.

The following is a particularly simple sufficient condition for the existence of a mild existence family, since it involves only polynomial growth of the resolvent (not its powers, as in the previous theorems) in a halfplane. T h e o r e m 17.7. Suppose A is dosed, 5 > O, k E N and whenever Re(z) > 5 , ( z - A) is injective, Ira(C) C_ I m ( z - A) k+l and the map z ~-. ( z - A ) - I C is holomorphic and O(Izl k-') on Re(z) > 5. Then, for all r,b > 5, there exists a mild uniformly continuous O(e br (A - r)-(k+l)C-existence family for A.

C o r o l l a r y 17.8. Suppose w, k e N, { z IRe(z) > ~ } c_ p( A ) and

{[zl'-k[I(Z -- A)-'C[[ l Re(z) > ~a} is bounded. Then there exists a mild uniformly continuous exponentially bounded (A - r)-(k+l)C-existence family for A, for all r > w. E x a m p l e 17.9. Theorem 17.7 may be applied to the operator considered in Example 16.9, to conclude the following. P r o p o s i t i o n 17.10. IrA and C1 are as in Example 16.9, then there exists a mild uniformly contin uous exponentiaUy bounded ( A - r ) -2 Cl-existence family for A, for all r > a;. This is in most ways a weaker result than appears in Example 16.9. The advantage of this technique is that resolvents are sometimes easier to calculate than semigroups, especially for larger matrices. Let us write ...

A11 A12 9

AI~)

Ooo

9

9

o

"

\ A~I ...... A~ 106

'

71

acting on x X1, to mean that Aij maps a subspace of Xj into Xi for i=1

l0 is Lipschitz continuous. | P r o o f o f T h e o r e m 17.4: We may assume that w = 0. Let Y be the weak solution solution space for A and let Z0 be the Hille-Yosida space for A (see Definitions 5.1 and 5.2).

108

(d) --, (b). By Theorem 5.10, Im(C) C__ Y, thus, as in the proof of Theorem 5.16, Jim(C)] ,--* Y. We claim that Ira(C) is contained in Z0. Suppose x E X. There exists < xn >C :D(A) such that xn --* x, in X, thus Cxn --* Cx, in [Im(C)] and hence in Y. Since CA C AC, Czn E Z)(AIy ) C_Z0, for all n, thus Cx E Z0, proving the claim. Theorem 5.16 now implies that there exists a mild C-existence family for A, W(t)x =_ u(t, Cx). Since CA C_ AC, the uniqueness of the solutions of (0.1) implies that W(t)A C_ AW(t), for all t _> 0. By Corollary 2.8, {W(t)}t>o is a strong C-existence family for A. (b) --* (c). By Theorem 5.16, [Im(C)] "--* Y (note that, since :D(A) is dense, the C-existence family is strongly uniformly continuous). The expression for [[xlly in Theorem 5.10 now implies (c). (b) ~ (a) follows from Theorem 3.7(a) and Theorem 3.8(3). II P r o o f o f T h e o r e m 17.7: By Theorem 5.15, Im((A - r)-(k+I)C) C_g b . Thus this follows from Theorem 5.16. | L e m m a 17.12. Suppose B is an injective operator, from a subspace of X1 into X2, and for i = 1,2, there exists injective Di E B(Ei) such that DzB and D1B -1 are bounded. Then B is closable and -B is injective. Suppose xn ~ 0 and Bx~ --+ y. Then (D2B)x~ ~ 0, and D2(Bx~) ~ D2y, so that D2y = 0, thus, since D~ is injective, y = 0.

Proof:

This means that B is closable. Suppose B-z = 0. Then there exists {zn) C_ 7)(B) such that Bz~ ~ 0 and z,~ --* z, that is, B - I ( B x n ) --+ z. The same argument, as given above, implies that x = 0, so that B is injective. | P r o o f of P r o p o s i t i o n 17.10: For r in p(G1)Np(G2), ( r - A) is injective, with (r_A)-I

( ( r - G 1 ) -1 =

(r-G1)-IB(r-G2)

0

(r -

-1)

a2)

" I

0

Since :D(G~) C :D(B), and B is closed, ( r - A) -1 (0 ( s - G , ) - " ) is in

B(X1 x X2). By Lemma 17.12, ( r - A) is closable, and ( r - A ) is injective, for r e p(G1) fl p(G2). Thus H(z - A)-ICH is bounded on right half-planes. The image of (z - A) 2 contains X1 x :D(G~-I), since :D(G~') _ /)(B), thus Ira(C) C_ I m ( z - A) 2, for Re(z) large. Theorem 17.7 thus implies that there exists a mild exponentially bounded (A - r)-2C-existence family for A. II

109

XVIII. RELATIONSHIP

TO INTEGRATED

SEMIGROUPS

Another way to smooth an unbounded semigroup is by integrating. The following definition presents the algebraic properties that one would obtain by integrating a semigroup n times, and requires only (strong) continuity, rather than n times (strong) continuous differentiability, as one would obtain by integrating a strongly continuous semigroup n times. D e f i n i t i o n 18.1. If n E N, an n-times integrated semigroup is a strongly continuous family of operators {S(t)}t>o such that S(0) = 0 and

S(t)S(,) - (n-i)!

[C

I"

(s+t-r)n-'S(r)dr-

(s+t-r)n-lS(r)dr

]

'

for all s, t > 0. {S(t)}t>_.o is nondegenerate if, whenever S(t)x = 0, for all t > 0, then x must equal 0. The generator is defined by

tn :t Z)(A) = {xl3y such that S(t)x = --X+n! /o S(r)ydrVt _> 0}, with Ax = y. We will use Chapter IV to show that generating an n-times integrated semigroup (not necessarily exponentially bounded) corresponds to generating an (r - A)-%regularized semigroup, whenever r e p(A). Here is a simple example of a once-integrated semigroup that is not exponentially bounded.

Example 18.2. Let Af(z) =_ zf(z), on C0(7), where 7 = {x + ie x2 Ix > 0}, with maximal domain. The operator A generates the once-integrated semigroup

S(O/(z)

=

//

e ~" d r =

1

(e'" - 1)/(z).

" a calculation s h o w s that Then I I S ( t ) l l 2 = sups_>0 ((e'Z--1)a :+::), is bounded below as t ~ co. Note that A generates the A-l-regularized semigroup

W ( t ) f ( z ) =_

Ile-l'~S(t)ll

letzf(z). Z

With a slight modification, we obtain a space, X, and an operator, A, such that the abstract Cauchy problem (0.I) has a unique mild solution, for all z E Z~(A), but no nontrivial solutions of (0.I) are exponentially bounded (see Example 4.10). 110

T h e o r e m 18.3. Suppose X is a Banach space, r E p(A) and n E N. Let Z be as in Definition 4.6. Then the following are equivalent. (a) The abstract Cauchy problem (0.1) has a unique solution, for all x e ~(A~+'). (b) The abstract Cauchy problem has a unique mild solution, for M1

e ~(A~). (c) A11 solutions of the abstract Cauchy problem are unique and

[~(A~)] ~ z. (d) The operator A generates an (r - A)-~-regularized semigroup,

{w(t))L>0.

(e) The operator A generates an n-times integrated nondegenerate semigroup, {S(t))L>0. We then have W(t)x : ( ~ ) ~ S ( t ) ( r - A)-nx and S(t) = (r - A ) n g ~ w ( t ) x , for M1 z E X , where g f(t) -= f~ f ( s ) d s . Here is an example of how we may use Theorem 18.3 to produce HilleYosida type theorems for integrated semigroups. C o r o l l a r y 18.4. Suppose :D(A) is dense and n + 1 E N. [ollowing are equivalent.

Then the

(a) The operator A generates an exponentially bounded n-times integrated semigroup. (b) There exists w E tt such that (w, oo) C_ p(A) and

{(~- w)k+l ( d )k

\ A ) - I } Is > w,k + 1 E N} /

is bounded. P r o o f of T h e o r e m 18.3: The equivalence of (a) through (d) is in Theorem 4.15, with C - ( r - A) -n, since :D(A n) = I m ( ( r - A)-n). It is straightforward to show that (e) implies (a) (see [A2] or [Wh]), since an induction argument, using the definition of D(A), shows that k-1

S(t)x = ~

t(n_j) (n-_-f)!AJx + Jk(S(t)Akx),

j=O

for x E :D(Ak),0 < k < n. 111

(18.5)

Given (a) through (d), let {T(t)}t>_o be as in Theorem 4.8. By Theorem 4.15, W(t) = T ( t ) ( r - A) -n. For any x E X , J ' ~ W ( t ) x = J n T ( t ) ( r A ) - n x , thus is in D((AIz)'~), since T(t) is a strongly continuous semigroup generated by Aiz. Let S(t)z - (r - A)'~J=W(t)x, for all x E X , t > O. By translating A, we may assume that r = 0. A tedious calculation shows that {S(t)}t___0 is an n-times integrated semigroup. Note that, for all x E X,

~

t S ( r ) x d r = A n J n + l T ( t ) A - n x e D(A),

since, by the properties of strongly continuous semigroups, J=+lT(t)A-~x is in •((AIz)n+l); similarly,

A( ~0t S(r)xdr) = a ' ~ J n ( T ( t ) -

I)a-"x

= S(t)z-

t" n~.X.

(18.6)

Let -d be the generator of {S(t)}t>o. If x E D(A), then by (18.6), ~.x = f : S ( r ) A x d r , thus x E 7)(4) and .4x = As, that is, A C_ A. Conversely, if x E D(.A), then by (18.6) and the definition of the generator,

I

S(r)A-

x er =

/0'

S(r)A-(n+l)

4x er,

for all r >__O, so that we may differentiate n + 1 times to obtain

A-'~x = A-(n+x) fi~x, since, by (18.5), (~7) d n S(t)lt=oX = z, for all x E D((fi,)n). Thus A = A, which proves (e). I P r o o f o f C o r o l l a r y 18.4: Since, for any r E C, (A - r) generates an exponentially bounded n-times integrated semigroup if and only if A does (this is a tedious calculation), we may assume, by translating A if necessary, that w = 0 and 0 E p(A). By Theorem 18.3, A generates an ntimes integrated semigroup if and only if A generates an A-'~-regularized semigroup. The resolvent identity shows that n--1

(s - A ) - I A -n = E

1.._l__Ai_n 1 si+1 "~ ~-~(S -- A) -1, Vn 9 N, s > 0.

i----0

Thus, since ( - 1 ) k k ! ( s - A ) - ( k + l ) A - ~ = ( ~d) k ( s - A ) -1A -n, this corollary follows from Corollary 17.5. I

112

XIX. P E R T U R B A T I O N S

In this chapter we consider both additive and multipUcative perturbations. We see how regularized semigroups arise naturally as bounded, commuting multiplicative perturbations of the generator of a bounded, strongly continuous group (Theorem 19.6). This theorem is best possible (see Examples 19.7-19.9). It is interesting that, even when the multipUcative perturbation is bounded and has real spectrum and numerical range, the perturbed operator may not be the generator of a strongly continuous group (see Example 19.8). Since BA = A + (B - 1)A, a bounded multiplicative perturbation is, in general, an unbounded additive perturbation. We give other examples of additive unbounded perturbations, that transform the generator of a Cl-regularized semigroup into the generator of a C2-regularized semigroup, where C2 may have smaller range than C1. All the arguments for perturbations of generators of strongly continuous semigroups may be mimicked to obtain predictable analogous results for C-existence families, by placing a C everywhere. These are not the results of interest, and, as we commented in the introduction, we would like to avoid such things in this book. In fact, most such arguments may be avoided by using the solution space to reduce such results to the case C = I. An example is the following, which we leave as an exercise, using Theorem 5.16 and the fact that a bounded perturbation of the generator of a strongly continuous semigroup on a Banach space is also the generator of a strongly continuous semigroup. E x e r c i s e 19.1. Suppose there exists an exponentially bounded mild Cexistence family for A and B E B([Im(C)],X). Then there exists an exponentially bounded mild C-existence family for (A + B). Similar results may be obtained for existence and uniqueness families, with the mimicry we described above, but we do not feel these are worth mentioning. The hypothesis on the perturbing operator, B, in Exercise 19.1, is too limiting to be of much interest. Of much more interest is to perturb A with an operator, B, that is unbounded, even relative to A. When there exists a C 1-existence family for A, we would like a C2-existence family for (A + B), where, in general, C2 is more smoothing than Ci, that is, CiiC2 E B ( X ) , but C~IC1 may not be. Two examples of this are Theorems 19.2 and 19.4. T h e o r e m 19.2. Suppose (1) A generates a bounded uniformly continuous Cl-regularized semigroup {W(t)}t>o that commutes with C2; 113

(2) c?~c. e B(X); (3) B is closed in X and (4) W - {x[t ~ c ~ l w ( t ) z is a bounded uniformly continuous map from [0, ~ ) into X } C 7)(B), with t ~ C g l W ( t ) B z bounded and uniformly continuous, for all z E W. Then there exists an exponentially bounded mild C2-existence family f o r ( A + B)lw. Here is an example where B is unbounded relative to A. E x a m p l e 19.3. Suppose G generates a bounded strongly continuous semigroup, {T(t))t>o, on X, B is closed in X , r E p(G) a~d D(G n) C_ D(B). Define A, on X x X, by

~[0" ~] .(~ . ~ . ~ Let

62 ~-

[;

0 ]

( r - G) -n

e B(X x X),

C1 ~ I, W ( t ) ~

b,~ 0] T(t)

"

Then Theorem 19.2 may be used to show that there exists a bounded mild C2-existence family for

A= [G~ B]G , V(A)---X• since W, from (4), equals X x D(G n) and

, , ~ , ~ , [; ~] 0 ~,,:,,

,,

, ~, ~,(, ~,,,,

whenever x2 e D(Gn). T h e o r e m 19.4 ( N i l p o t e n t p e r t u r b a t i o n s ) . Suppose there exist bounded, injective C1, C2 such that (1) A generates an exponentially bounded Ct-regularized semigroup;

(2) BC2 ~ B(X); (3) [(r - A)-IC1BC2] g = O, for all r e pc,(A); and (4) C2 and C1 commute with BC2,(r - A)-1C1, C2 and C1, for all

~ pc, (A). 114

Then, for all s E pcl(A), there exists a mild exponentially bounded ( s - ( A + B))-lCl(ClC2)g-existence family for(A+ B), where :D(A + B) = 7)(A)AI)(B). E x a m p l e 19.5 Let G,A and A be as in Example 19.3, except that 7)(.4) - :D(G) x [7)(G) x / ) ( B ) ] , there exists A e p(B) such that ( A - B ) -1 commutes with T(t), for all t _ 0, and we remove the hypothesis about the domain of G n being contained in the domain of B. Let C - (1 - A ) - I ( A - B) -2. Then Theorem 19.4 may be used to show that there exists a mild exponentially bounded C-existence family for .4, by letting C1 = I, C2 -= ( A - B ) - I , N _= 2, so that ~

,0 ] C 2 =

[00 ( 1 - G ) - I B ( A 0

B)-I]"

This example may be extended to upper triangular matrices of arbitrary size, with commuting entries. This example is weaker than 19.3, in that commuting is required, but is stronger in that there is no relationship between :D(G) and :D(B). Our next theorem involves a multiplicative perturbation by a bounded operator. Note that, in general, this may be the same as an unbounded additive perturbation, since AB = A + A(B - I). T h e o r e m 19.6. Suppose n E N U {0}, B is bounded and A generates a bounded strongly continuous group that commutes with B. Then (a) There exists bounded injective C( -= e-A~), with dense range, such that BA generates an entire C-regularized group; (b) If {lle"Sllh is for t arge, then B A generates an exponentially bounded (1 + A) -(~+1)- regularized semigroup; (c) If, in addition to (b) 0 e p(B), then BA generates an exponentially bounded (n + 1)-times integrated semigroup.

Example 19.7. This is a bounded operator B, and an operator A, as in Theorem 19.6, such that sp(B) = {0}, but p(BA) = 0. This shows that the invertibility of B is necessary in Lemma 19.12 and Theorem 19.6 (c) (at least for n > 0). Let G be any unbounded generator of a bounded, strongly continuous group, on X, and, on X x X, define

115

Then A generates a bounded strongly continuous group and IletBII = ~0 0 [~ 0 ] hasemptyresolvent, sinceIm(r-BA) C_ X X D(G), for all complex r. E x a m p l e 19.8. Here we present B, A as in Theorem 19.6 (c), with n = 0, such that BA does not generate a bounded strongly continuous semigroup. Let X be the set of all functions, f, from t t 2 into the complex plane, such that the map y ~ f(x,y) is in LI(R), for all z E t t and the map x ~ JR If(z,Y)l dy is uniformly continuous and bounded, on It, with norm

Let h(y) = :~-~kr162(1 + (1/2k))l[k_l,k)(y), and define the bounded operator B by

(B/)(x, y) = h(y)f(x, y). Let A be cg/Ox, the generator o f (etA f)(x, y) = f(x "-]-t, y). Note that B and e tA commute, and the spectrum of B is contained in the closure of the range of h, which is contained in the interval [1,2]. If e tBA existed, it would be given by

(etBA f)(x,y) = f ( x "11-th(y),y), since

(d/dt)f(z + th(y),y) = h(y)(Of /Oz)(x + th(y),y) = (BAf)(z + th(y),y). We will show that e BA is unbounded. For any N, define fg as follows. Let AN = uN=x[1 + (1/2k + 1),1 + (1/2k)] • [ k - 1,k), BN = uN=I[1 + (1/2k),l + ( 1 / 2 k - 1)] x [ k - 1,k). Then

fN(*,y) =

1

for x in

0

for x outside AN t.J BN

linear in x

on AN and BN

A N 17 B N

116

(see graphs below).

Z i

n

m

1 +2 N~ + l

l+2N

1+2N_1

"'"

~

2

Figure 1. Projection of z = f/v(x, y) onto x-z plane.

Y

f

%,

w

I I

I

I

6

L_ -J Ii 1 + 2k §

Figure 2.

il I § 2k +---~

11 I § 2--k

{(z,y)lfN(r,,y ) ~ 117

0}

Ii

I-{-~

2k- 1

z

Note that, for all x, fl~ IfN(x,y)ldY -< 1. Thus, IIfNII = 1, for all N. However,

=/N(h(y),y)

(~B*/N)(O,y)

00

= ~/N(1

+

(1/2~),y)lt~_l,k)(y)

k=l N

= ~] it~-~,~(y) k=l = I[O,N) (y).

Thus HeBAfNI] >- N. Since IIfNII = 1, and N was arbitrary, this shows that e BA is unbounded. E x a m p l e 19.9. Let n be arbitrary. This is an example o f / ) , A as in Theorem 19.6 (b) and (c), such t h a t / ) A does not generate an (1 + .4)-'~regularized semigroup or an n-times integrated semigroup. This shows that Theorem 19.6 (b) and (c) are best possible. Let B, A be as in Example 19.8 9 On X n+l, define

"o

0

?___

".~

~176 ~176176

A calculation shows that ever,

neit[3 n is O(t'~), *~

/)ti=

since

Ile'tBII is bounded.

How-

~176

"

"

"'.

BA BA

generates an n-times integrated semigroup if and only if B A generates a strongly continuous semigroup. Hence, by Example 19.8, B A does not generate an n-times integrated semigroup. This implies that /)A does not generate a (1 + BA)-n-regularized semigroup (see Theorem 18.3); 118

since (1 + / ~ A ) - I ( 1 + A) is bounded and injective,/~J cannot generate a (1 + J)-'*-regularized semigroup. E x a m p l e 19.10. For 1 0, on W. This is saying that W is contained in Z~(A+B), for some w E R, so that, by Theorem 5.16, since Ira(C2) C W, the conclusion follows. | P r o o f of T h e o r e m 19.4: For s E Pc1 (A), define

N-1 R(s) = E

[ ( s - A)-'C, BC~] k ((s - A)-1Cl)(C1C2) N-~.

k----O

By (2) and (4), R(s)x E D(A+B), for all x E X. Let C = C1(CIC2) N. We will show that (s-(A+ B))-IC exists and equals R(s). For z e D ( A + B ) , the calculation follows:

R(s)(8-(A + B))x N-1 = c, ~ [(s - A)-'C, BC2]~(C, C2)N-~x k=0 N-1

[(s- A)-~C~BC~]~( ( . - A)-~C, BC~)(C,C~)~-"-~

-c, ~ k=O

"-C 1 [(ClC2)Nz-

[(S--

A)-ICIBC2]Nx]

= Cx. Thus, (s - (A + B)) is injective, and, for all x e X,

R(s)(s- (A -{- B))R(s)x = CR(s)x = R(s)Cx, so that, since R(s) is injective,

(s - (A + B))R(s)x = Cz. This implies that pc~(A) C pc(A + B), with, for s E pcl(A),

R(s) = (s - (A + B))-IC,

N II(s - (A + B ) ) - l C l l _< ~ l l ( s

- A)-'C, IIk+'IlBC211klIC,C21IN-k.

k=O

121

Theorem

17.7 and Proposition 17.2 now conclude the proof. |

Our technique in proving Theorem 19.6 is to use the Fourier inversion formula to construct the C-regularized semigroup generated by BA, which may be thought of as

esBAc -- JR eitA~(A)(t) dr, where

fs(r) - g(r)e "rB, C = / eltAjr(g)(t) dr,

and g is a suitable function; the choice ofg will depend on the behaviour of {eSB}s>0. (In this formula, (iA) generates a bounded strongly continuous group.) To prove Theorem 19.6 (c), we will need the following. L e m m a 19.12.

Suppose A and B are as in Theorem 19.6 (c). Then

(O, oo) C_p(BA). P r o o f : Since {lleitBll}tea is O(t'~), for t large, the spectrum of B is real (see [Davl]). Thus, the hypotheses on B imply that sp(B) C_ t t - {0}. For s > 0, define the bounded operator

R(s) = ~ r ( s - wA)-l(w - B) -1 dw 2~ri ' where F is a cycle that bounds sp(B) and does not intersect the imaginary axis. Note that ( s - wA) -1 = 1/w((s~)/Iwl 2 - A) -1 exists and w ~-* ( s wA) -1 is a holomorphic map, from ( C - i R ) i n t o B(X),sincesp(A) CiR. Since F is bounded, a n d ( s - w A ) - l x e 7)(A), for all w 9 F, R(s)x 9 for all x 9 X. Hence, we may calculate as follows, for x 9 X:

27ri(s- BA)R(s)x = ~ r ( s - B A ) ( s - wA)-l(w - B ) - l x d w = ~r (s - wA + ( w - B ) A ) ( s - wA)-l(w - B ) - l x d w = ~ r ( w - B)-ixdW+ ~ r A ( S - w A ) - l x d w = 2~rix, 122

by Canchy's theorem, and the usual Gelfand functional calculus for bounded operators. Thus ( s - B A ) R ( s ) x = x, for all x E X. It is clear that, for x E 7)(A), R ( s ) ( s - BA)x = ( s - BA)R(s)x = x. Thus s e p(BA), with

( s - BA) -1 = R(s). P r o o f o f T h e o r e m 19.6: It is sufficient to assume that iA generates a bounded strongly continuous group, and, in (b) and (c), {[let~H}teR is

o(t"). Let .4 - {strongly continuous f : R. ---, B(X)IVx e X, the maps

r ~-, f(r)x and r ~ f'(r)x are in L2(R,X)}. For f E .4, define the operator f(A) by, for z in X,

(19.13) where the limit is taken in L2(K,X). When f is scalar-valued, this functional calculus appears in [Davl, Chapter 8]. The same arguments that appear there may be used to show that (19.14) f(A)g(A) = (fg)(A), Vf, g E .4; 3M < oo such that lif(A)zl] < M(Hf(r)xll2 + Ilf(r)xH2),

(i + A) -1 = g(d),

VxEX;

where g(r) - (i + r) -1.

(19.15) (19.16)

For (a), define

Wl(z)-=gz(A),

wheregz(r)=ezrBe -r2 ( z E C , r E R ) .

For (b), define

esrB W2(s) - f~(d),

where fs(r) - (i + r) n+l"

Assertion (19.15)implies that {W2(s)},>0 is an exponentially bounded strongly continuous family of bounded operators and {Wl(z)}zeC is an entire family of bounded operators. Assertions (19.14) and (19.16) imply that {W2(s)}8>0 is an (i -{- A)-(n+l)-regularized semigroup, while (19.14) 123

implies that {1411(z)}zeo is an entire W1 (0)-group; it was shown in Lemma 8.8 that C -- WI(O) e -A2, is injective and has dense range. For (a) and (b), all that remains is to show that {Wl(z)}zec and {W2(s)},>0 are generated by BA. (19.14) and (19.16) allow us to make the following calculations, for z E X: =

W2(s)(i + A)-lx =

lim

N--* oo

esrB X

dr

N

lliml;

"+r-i dr e-irt eitA BeSrB x ( i~ + r)n_~2) ~-~rdt

= B[W2(s)x - iW2(s)(i + A)-lx] = BAW2(s)(i + A)-lx, (19.17)

d W l ( Z ) ( i + A)-'x = BAWI(Z)(i + A)-lx,

(19.18)

exactly as with (19.17). Assertion (19.17) implies that an extension of BA generates {W2(s)}s>0 and (19.18) implies that an extension of BA generates {Wl(z)}zec. (a) and (b) now follow from Proposition 3.9, with G = (i + A) -1. Since (i + A)(i + BA) -1 is bounded and injective, (b) implies that BA generates an exponentially bounded (i + BA)-(n+D-regularized semigroup. By Theorem 18.3, BA generates an exponentially bounded ( n + 1)times integrated semigroup, proving (c). |

124

XX. TYPE

OF AN

OPERATOR

In considering the spectrum of an operator, A, the goal is to think of A as a complex number, or a set of complex numbers. For example, if one wants A to generate a bounded, strongly continuous semigroup, which we think of as etA, we want the spectrum of A to be contained in the left half-plane Re(z) < 0, since this is the set of all z for which t ~-~ e tz, from [0, oo) into the complex plane, is bounded. Throughout this book, we have constructed g(A)-regularized semigroups W(t), generated by A, by thinking of W(t) as etAg(A), with g chosen so that the map t ~ etZg(z) is bounded, for all z in the spectrum of A. But it is well known that specifying the location of the spectrum is not enough. For generation theorems, it is also necessary to have conditions on the rate of growth of the resolvents. For example, the ttille-Yosida theorem states that a densely defined operator A generates a bounded strongly continuous semigroup if and only if the spectrum is contained in the closed left half-plane and

{llRe(z)(z- A)-llllRe(z)

< 0}

is bounded. One very successful approach is to specify both the spectrum and the numerical range (see any of the references on strongly continuous semigroups). The Lumer-Phillips theorem states that a densely defined operator generates a strongly continuous semigroup of contractions if and only if the spectrum and numerical range are contained in the closed left half-plane. There are two disadvantages to using the numerical range. First, it does not, in general, satisfy the sort of mapping theorems that the spectrum does; that is, it is not true in general that n.r.(f(A)) = f(n.r.(A)), even when f is chosen to be a polynomial. Consider, on (3 2,

16 + 4i

"

Then a calculation shows that the numerical range of A is contained in S~ ~- {re ~r [r > 0, [r < ~}, but the numerical range of A 2 is not contained in S~ Thus, if f ( z ) =__ z 2, then n.r.(f(A)) is not contained in f(n.r.(A)). Second, there does not appear to be a reasonable analogue of the numerical range for the more general classes of semigroups that we are considering, such as regularized semigroups. 125

We would like, associated with any operator, a set of complex numbers, with the following properties: (1) it satisfies mapping theorems; (2) its location tells us something about generation of regularized semigroups. Throughout this chapter, V and O will be open subsets of the complex plane whose complement contains a half-line and whose boundaries, OV, O0, are positively oriented countable systems of piecewise smooth, mutually nonintersecting (possibly unbounded) arcs. We will write R H P for the open right half-plane {z e CI Re(z) > 0}, L H P for the open left half-plane, 5'r -- {rei~[r > 0, [r < r < r < ~'), H, = (z e Cl IZ (z)l < D e f i n i t i o n 20.1. Suppose a _> - 1 . We will say that the operator B is ofa-type V if sp(B) C_ V and I I ( w - B)-lll is 0((1 § Iwl)"), for ~ ~t v . E x a m p l e s 20.2. (1) If B E B ( X ) , then B is of (-1)-type V, whenever V is an open set conta.ining sp(B). (2) Let (Bf)(z) -- zf(z), on L2([1,co)). Then, for all 0 > 0, B is of (-1)-type Se. for all e > 0, B is of 0-type H~, but is not of (-1)-type H~. This illustrates the fact, that, although many operators are of (-1)type V, for some V, many natural choices of V require that we consider operators of a-type V, for a > - 1 . Another example of this is B _= - A , the Laplacian, on LP(ttn), 1 < p _< cr with 7)(B) = { f E LP(R.n)IAf E LP(R'~)}, where A is taken in the sense of distribution. In particular, if 1 < p < or, then :/:)(B) = W2'P(R n) (the Sobolev space)(see [H-V]). For all 0 > 0, B is of (-1)-type (S0 - 1), while for all e > 0, there exists a > - 1 , depending on both p and n, such that B is of s-type H~, but B is not of (-1)-type H~ (this may be seen by using the functional calculus for commuting generators of bounded strongly continuous groups; see Chapter XII). Consider also an operator, A, that generates an exponentially decaying strongly continuous semigroup, {etA}t>_O. The operator A is of 0-type L H P and is of (-1)-type L H P if and only if {etA} extends to a bounded holomorphic strongly continuous semigroup. The following example provides a prototype operator of a-type RHP. (3) For any (~ > - 1 , k > 0, let Oa,k ------{x-t-iylx > RHP, (B,,kf)(z) =- zf(z), on L2(Oa,k). Then, for all k > 0, Ba,k is of a-type V and is not of #-type V, for any fl > a. (4) Let F be the unit circle in the complex plane and let B - ~ , the generator of rotation on LP(F), 1 < p _< cr V defined to be a union of 126

open balls of radius less than one, centered at the integers. Then iB is of 0-type V. (5) If D(B) is dense, then it is well known that B is of (-1)-type Sr for all r > 8, if and only if ( - B ) generates an exponentially decaying holomorphic strongly continuous semigroup of angle (~ - 8 ) ( 0 < 8 < ~). (6) More generally, if ( m + 2) E N, A E p(B) and :D(B)is dense, then B is of m-type Sr for all r > t~, if and only if ( - B ) generates an exponentially decaying holomorphic (A - B)-(m+l)-regularized semigroup of angle (~ t?) (see the next chapter). Hence, being of m-type V is analogous to having the numerical range contained in V; the desired property here is generating a ()~ - B) -(re+Dregularized semigroup rather than a strongly continuous contraction semigroup. In terms of the abstract Cauchy problem, being of m-type - S e (9 < ~) corresponds to the abstract Cauchy problem having a unique exponentially decaying solution, for all x E/9(Bm+l), while having the numerical range contained in -Se corresponds to the abstract Cauchy problem having a unique nonincreasing solution, for all x E D(B). (7) In many places ([B-C], [Bo-dL3], [Do-V1, 2], [Duol, 2], [M], [M-Y], [Pr-Soh], [Ril, 2], IV] and [Y1, 2], for example), operators of type w are considered. These are densely defined operators, A, such that sp(A) C S~ and for all r > w, there exists Me < oo such that ]]w(w - A)-II] _< Me, for all w ~ Sr Note that a densely defined operator with 0 in its resolvent set is of type ~; if and only if it is of (-1)-type Sr for all r > w. Even on a Hilbert space, when f E H~176162and B is of type r f ( B ) may not be bounded (see [M-Y] and [B-C]). (8) In [Bel, 2] operators of n-type V~, where n E N is arbitrary, 0 < a < 1 and V~ - {x + iy I x < lyl ~) are considered. A simple example of such an operator (from [Bell)is

on X x X, D(B) -- ~P(G) x/P(G), where G generates a strongly continuous semigroup.

127

XXI. HOLOMORPHIC

C-EXISTENCE

FAMILIES

In this chapter, we define exponentially bounded holomorphic existence families (Definition 21.5) so as to generalize holomorphic strongly continuous semigroups, a class of semigroups that has found wide applicability. In keeping with our policy, in this book, of not presenting proofs that are obvious modifications of proofs for strongly continuous semigroups, we have included no more of the proofs than the basic construction of the desired family of operators. Our main results may be summarized as follows. For an operator, A, to generate an exponentially bounded holomorphic k-times integrated semigroup, it is sufficient that it be of ( k - 1 -e)-type V, for an appropriate sector V, for some e > 0 (see Definition 21.1). If A is densely defined, it is necessary and sufficient that it be of (k - 1)-type V. In order that there exist an exponentially bounded holomorphic C-existence family for A, it is sufficient that [[A(w- A)-ICH be O([w[-~), for some positive E (Theorems 21.8 and 21.9). If A is densely defined and CA C_ AC (this is automatically true when A generates a C-regularized semigroup), it is necessary and sufficient that II(A(w - A)-ICII be bounded (Theorems 21.15 and 21.16). D e f i n i t i o n 21.1. So =- {reir

> 0,]r

< 0}, Vo - {rei#[r > 0,]r


0 > 0. The n-times integrated semigroup {S(t)}t>0 is a holomorphic n-times integrated semigroup of angle 0 if it extends to a family of bounded operators {S(z)}zeso satisfying: (1) the map z ~-* S(z), from So into B(X), is holomorphic; (2) { ( d ) " S(z)}zeS~ is a semigroup; and (3) for all r < O, {S(z)} is strongly continuous on S-~. D e f i n i t i o n 21.3. Suppose ~ >_ 0 > 0. The C-regularized semigroup {W(t)}~>0 is a holomorphic C-regularized semigroup of angle 0 if it extends to a family of bounded operators {W(z)}zese satisfying: (1) the map z ~ W(z), from So into B(X), is holomorphic; (2) W(z)W(w) = C W ( z + w), for all z,w e So; and (3) for all r < O, {W(z)} is strongly continuous on Sr D e f i n i t i o n 21.4.

The family of operators in Definition 21.2 (21.3) is exponentially bounded if, for all r < | there exist finite M~, w~, such that [[S(z)il(llW(z)l[) _ 0 > 0. The exponentially bounded mild C-existence family {W(t) }t>o is an exponentially bounded holomorphic C-existence family of angle 0 for A if it extends to a family of bounded operators {W(z))zeso satisfying: (1) the map z ~ W(z), from So into B(X), is holomorphic; (2) whenever 1r < O, {W(teir o is an exponentially bounded mild C-existence family for ei*A; ~nd (3) for all ~b < | {W(z)} is strongly continuous on S-~. If [[W(z)[[ is bounded on S--~, for all r < O, then { W ( z ) } , e s o is a bounded holomorphic mild C-existence family. T h e o r e m 21.6. Suppose A is closed, ~ > 0 > 0, S(o+~) C pc(A), C is injective and commutes with (w - A)-IC, for all w e pc(A) and {W(z)}zeso C_ B ( X ) . Then the following are equivalent. ( a ) { W ( z ) } ze so is an exponentially bounded holomorphic C-regularized semigroup of angle 0 generated by an extension of A. (b) {W(z)}zr is an exponentially bounded holomorphic mild Cexistence family of angle 0 for A. T h e o r e m 21.7. Suppose there exists reM r such that [r, oo) _ p(A), and 2 > -- 0 > O. Then the following are equivalent. (a) The operator A generates an exponentially bounded holomorphic n-times integrated semigroup { S( z ) } ~eso , of angle O. (b) The operator A generates an exponentially bounded holomorphic ( A - r) -n- semigroup, {W(z)}zeSo, of angle O. (c) There exists a holomorphic semigroup {T(z)}zeso satisfying the following. (1) If z E So and x E V(A), then T(z)x E D(A), with ~ T ( z ) x AT(z)z = T(z)Ax. (2) If r < 0 and x e D(An), then T(z)x converges to x, as z --* 0 in So. (3) For all r < O, there exists anite M,, such that IIT(z)xll < M~,etU*lzl[[(A - r)nx[[, for all z e D(An), z e So. We then have (d-~) n S ( z ) = T ( z ) , W ( z ) = ( A -

r ) - n T ( z ) , for z e S o .

T h e o r e m 21.8. Suppose A is closed and there exists 7r > !b > ~ such that V~ C_pc(A), and w ~-. ( w - A ) -1 , from Vr into B ( X ) , is holomorphic. 129

Then there exists a bounded holomorphic mild C-existence family of angle (r ~) for A if either

(a) {z e ~ ( g C ) l A C z ~ Ira(C)} is dense and IIA(~- A)-ICII is bounded in V~, or (b) there exists e > 0 such that I [ A ( w - A)-ICI[ is bounded and

o (Iwl -r in v,. T h e o r e m 21.9. Suppose A is closed and there exist 7r > r > ~ , k > 0 such that (k § V~) C_ pc(A), and w ~-* ( w - A ) - I c , from (k + Vg,) into B ( X ) , is holomorphic. Then there exists an exponentially bounded holomorphic mild C-existence family of angle (r - i ) for n if either (a) {x e D ( A C ) [ A C x e Im(C)} is dense and [[A(w- A)-IC[[ is bounded in (k + Vvz), or (b) thereexistse > 0 such that [IA(w-A)-~Cll isO (Iwt -*) in (k+Vr C o r o l l a r y 21.10. Suppose A is closed and there exists z > r > such that V~ C_ pc(A), the map w ~ (w - A ) - I C is holomorphic and C is injective and commutes with (w - A ) - I C , for all w EVr Then an extension of A generates a bounded holomorphic C-regularized semigroup of angle ( r - i ) that leaves I)( A ) invariant if either (a) D(A) is dense and [[A(w - A)-IC[[ is bounded in V,), or (b) there exists E > 0 such that

IIA(w - A)-'Cll is bounded and

O (Iwl -~) in V~. C o r o l l a r y 21.11. Suppose A is closed and there exist r > r > i , k > 0 , such that (k + V,p) C_ pc(A), the map w ~ (w - A ) - I C is holomorphic and C is injective and commutes with (w - A ) - I C , for all w E V,p. Then an extension of A generates an exponentially bounded holomorphic Cregularized semigroup of angle ( r ~ ) that leaves D( A) invariant if either (a) D(A) is dense and [[A(w - A)-IC[[ is bounded in (k + V,~), or (b) there exists e > 0 such that [[A(w-A)-ICI[ isO ([w[ -~) in (k+ Vr R e m a r k 21.12. In the preceding results, in order that the map w (w - A ) - I C , from V~ into B ( X ) , be holomorphic, it is sufficient to have A closed and I m ( C ) C_ I m ( ( w - A)3), for w e Vr with II(w- A ) - l ( r A) -1 (s - A) -~ II locally bounded. This may be shown with the identity (r- A)-Ic - (s- A)-Ic = (s- r)(r- A)-l(s-

Note that 21.13(b) is similar to Theorem 17.7. 130

A)-Ic.

T h e o r e m 21.13. Suppose there exist 7r > ~b > ~,k > O, such that ( k+ Sr ) C p( A ). Then A generates an exponentially bounded holomorphic n- times in tegrated semigroup of angle (r - ~) if either

(a) D(A) is dense and Z is o f ( n - 1)-type (k + S---~), or (b) there exists e > 0 such that A is of (n - 1 - e)-type (k + -~). P r o o f of T h e o r e m 21.8:

For r > 0, let

rr =- {se +i• Is >_ r} u (re i~ I - r < 0 < r oriented counterclockwise. Define, for z E S(r

W(z) - fr

,

eZW(w-A ) - I c dw

27ri"

By Cauchy's theorem, this definition is independent of r > 0. The construction of T(z) relies on the following. L e m m a 21.14. Suppose k and r are nonnegative, r > ~,( k + Sr C p(A), larg(z)l < ( r ~), and n is a nonnegative integer. Let Fr,k ~- k + r r , where l~r is defined in the proof of Theorem 21.8. Then, for x E D(An),

/ eZW(w-A)-lxdw:~r eZW(w-A)-lAnxdwwd +--2ri Z

n-1

r,k

r,k

-fi.zJAS.x.

.~----0

P r o o f o f T h e o r e m 21.13: With rr,k as in Lemma 21.14, x E X , define

T(z)x - fr for larg(z)l < ( r

r.k

eZ~( w _ A)_lx dw 2ri'

~),r > O.

T h e o r e m 21.15. Suppose A is closed, {x E D ( A C ) I A C z E Ira(C)} is dense, ~ >_ 0 > 0 and S(~+o) C_ pc(A). Then the following are

equivalent. (a) There exists a bounded holomorphic mild C-existence family of angle 0 for A. (b) For a/l r < (~ + 0), I I a ( w - A)-ICI[ is bounded in S~,.

131

T h e o r e m 21.16. Suppose A is closed, {x E ~)(AC)[ ACx e Ira(C)} is dense, 3 >- 0 > 0 and for a11 r < ( 3 + | there exists kr E It such that (kr + Sr _Cpc(A). Then the following are equivalent. (a) There exists an exponentially bounded holomorphic mild C-existence family of angle 0 for A. (b) For all r < (3 + 0), IlA(w- A)-ICll is bounded in (kr + S,p). When A generates a bounded holomorphic C-regulaxized semigroup of angle O, then S~+o C_ pc(A); this is a consequence of Proposition 17.2. When there exists a bounded holomorphic mild C-existence family of angle O for A, then it may be shown that Im(C) C_ Im(w - A), for all w E S{.+o. T h e o r e m 21.17. Suppose 7)( A) is dense. Then the following are equivMen t. (a) The operator A generates an exponentially bounded n-times integrated semigroup of angle O. (b) For all r < (3 + O), there exists kr > 0 such that ( k,h + Sr ) C_p( A ) and [[(A- r ) - ~ A ( w - A)-I[I is bounded in (kr + SO) , for some r e p(A). (c) For all tb < (3 + 0), there exists kr > 0 such that A is of(n - 1)type (kv~ + Sr As in Examples 16.9 and 17.9, we have the following, where we use the same terminology as in 17.9. E x a m p l e 21.18. Suppose n,m E N U O,s E p(G~) rip(G2),

A =- [Go1 G2B] ' 7)(A) =_I)(G1) x [~)(B)NI)(G2)], where (1) Gi generates a strongly continuous holomorphic semigroup, for i = 1,2; (2) (s - G1)-mB is bounded; (3) 7)(a~) C__7)(B); and (4) B is dosed. Then A is closable and there exists an exponentially bounded holomorphic mild C-existence family for A, where n

132

XXII. UNBOUNDED HOLOMORPHIC FUNCTIONAL CALCULUS FOR OPERATORS WITH POLYNOMIALLY BOUNDED RESOLVENTS

In studying linear operators, it is desirable to gain as much information as possible by looking at the spectrum. Perhaps the most well-known functional calculus for arbitrary bounded operators on a Banach space is the Riesz-Dunford functional calculus,

f(A) -

dw f(w)(w - A) -1 27ri'

(22.1)

where f is holomorphic in an open neighborhood containing the spectrum of A and F surrounds the spectrum of A. Of more interest in applications are unbounded operators, such as differential operators. Two problems immediately arise when one tries to extend (22.1) to unbounded operators. The spectrum of A may be unbounded and functions holomorphic on the spectrum of A may be unbounded. Since fr [[(w - A)-I[[ d[w[ is no longer finite, it is not surprising that, even for bounded holomorphic f , f(A) may not be bounded, even when 1 [[(w- A)-I[] is O (1-T~), as with a bounded operator. For example, let D be the open unit disc in the complex plane, let - i A be d , on LI(OD), the generator of the rotation group on 0D and let f - 1[0,~). Then it is well known that f ( d ) , the projection of LI(OD) onto H i ( D ) , is unbounded. Even on a Hilbert space, f(A) may be unbounded, although f is bounded and holomorphic in an open neighborhood containing the spectrum of A (see [M] and [B-C]). For polynomially bounded f , one could modify (22.1) by replacing ( w A ) - l x d w w i t h ( w _ d ) - l ( r _ d ) n x (r-w)d~ , for n sufficiently large, x in the domain of A n, r in the resolvent set o f d (see [M], [Bo-dL1]). However, this does not include functions of the most interest (particularly in considering the abstract Cauchy problem (0.t) or (22.2) below), such as exponentials and cosines. When A is unbounded, there are several ways to get bounded operators into t h e picture and use their functional calculi to define a functional calculus for A. One could apply (22.1) to (r - A) -~, for some r in the resolvent set of A; this defines f ( A ) only for f holomorphic in a neighborhood of infinity (see [Du-S1]), hence is even more restrictive than the requirement of polynomial growth.

133

One could also attempt to define f ( A ) by defining a strongly continuous semigroup that it will generate. Unfortunately, even when g maps the ctg(A) spectrum of d into the left half-plane, and I ] ( w - A ) - ' II is O ( ~ )1, may not be bounded. However, it may be shown that e t g ( A ) ( r - A) -1 is bounded; more generally, if II(w - d ) - l l l is O(IwlC~), we will show that etg(A)(r - A) -m, where m = [c~]+ 2, is bounded, for all nonnegative t. We will show that this defines a regularized semigroup, so that g(A) may be defined as its generator. In previous chapters, we have used functional calculus techniques to construct regularized semigroups. In this chapter, we use regularized semigroups to construct a holomorphic functional calculus for operators with polynomially bounded resolvent (see also Chapter XII). The functional calculus of this chapter may then be used to construct Cregularized semigroups generated by operators with spectrum contained in arbitrary strips and sectors of angle less than ~. We may also write down solutions of the abstract Cauchy problem, u(t, x) = e t A x , explicitly, using our functional calculus construction of e t A . By constructing cos(itB), for an appropriate square root, B, of A, we may similarly write down solutions of the second order abstract Cauchy problem (d)2u(t)

d (u(t)) (t E It), u ( 0 ) = x, u'(0) = y.

=

(22.2)

Throughout this chapter, the sets V and O will be as in Chapter XX. D e f i n i t i o n 22.3. Let 7-IL(V) be the set of complex valued functions, f, hotomorphic on V such that s u p { R e ( f ( z ) ) l z E V} < ec. ("L" stands for "left.") For 3' E R, let ~.y(V) be the set of functions, f, holomorphic on V, such that sup{ f--lL12R-(l+I z )~

z

E V} < oo. Let ~L,../(V) ~ ~'~L(V) N ~-[~(V).

L e m m a 22.4. Suppose A is of a-type V. Then there exists 0 such that 0 C_ V and A is of a-type O. T h e o r e m 22.5. Suppose B is of a-type V, f E ~-[L(V) and m - [a] + 2. Then, for ~ ~_ V, t >_ O,

ws(t)-

1

o e ' s ( " ) ( w - D)

-

A)m'

where O is as in Lemma 22.4, de/ines a norm continuous (B - A) -mregularized semigroup. Note that, by the residue theorem, the definition of W/(t) is independent of O. 134

D e f i n i t i o n 22.6. For f , B as in Theorem 22.5, f ( B ) is defined to be the

generator of {Ws(t)},>0. This definition is independent of A for the following reasons. Temporarily writing f~(B) for the generator of the (B - A)-m-regularized semigroup {Wf,~(t)}~>0 in Theorem 22.5, note that, by Proposition 3.10, for A1,A2 ~ V, both f ~ ( B ) and f~2(B) generate the ( B - A 1 ) - m ( B - A 2 ) -mregularized semigroup {Ws,~I(t)(B

-

,x~)-m}~_>o :

{(B

-

~,~)-mws,~(t)},_>o

(see Lemma 22.35), thus are equal. For 22.7 through 22.10, let B , a , m , V and O be as in Theorem 22.5. P r o p o s i t i o n 22.7. If G E B ( X ) commutes with all resolvents of B, then G I ( B ) C_ f ( B ) G , for all f E ~ L ( V ) . Theorem 22.8, Corollary 22.9 and Theorem 22.10 state in what sense the map f ~ f ( B ) is a functional calculus for B. In all these results, there are no growth restrictions on f. T h e o r e m 22.8. Suppose B is of a-type V, A ~_ V, f,g E 7-~L(V) and there exists 7 E R such that g, f g E 7~.y(V). Let n =_ max(O, [a + 7] + 2). Then I)(B n) C_ / ) ( g ( B ) ) N 7)(f(B)g(B)) and both g ( B ) ( B - A)-n and f ( B ) g ( B ) ( B - A) -• are in B ( X ) , with

f(B)g(B)(B-

1 fo o ( f g ) ( w ) ( w - B) -1 A) -k _ 27ri

dw

(w- a)k,

for Ml k > n, h E N. If f g E ~ L ( V ) , then I)(B n) C_ :D((fg)(B)), and

f ( B ) g ( B ) ( B - ~)-n

=

( f g ) ( B ) ( B - A)-'~.

By choosing 7 < - ( 1 + a) and using the fact that the generator of a C-regularized semigroup is automatically closed, we obtain large subsets of domains of our operators, a large class of bounded operators and a type of continuity of the map f ~ f ( B ) . C o r o l l a r y 22.9. Suppose f E ~~L(V), and there exists 7 < - ( 1 + a) such that g E 7G(V ). Then

(~) g B ) e B(X), with

1 /o o g(w)(w- B) -ldw; g(B) = 2~i 135

(b) there exists M.~ < oc such that, for all h E ~.~(V),

h(z)

[[h(B)l I _< M.Y~evSUp[~1~+ [z[)"y[; and (c) if fg E 7-L~(V), then Im(g(B)) C_D(f(B)), with

f ( B ) ( g ( B ) x ) = (fg)(B)x, Vx E X. Note that Corollary 22.9(a) implies that Definition 22.6 extends the Riesz-Dunford functional calculus for bounded operators; that is, if B and V are bounded, then Definition 22.6 is equivalent to (22.1). In the following theorem, there are no growth restrictions on f, fg or fig. 1 T h e o r e m 22.10. Let fk(z) = z ~, g~(z) - ~-z"

(a) fo(B) = I. (b) 9),(B) = ( A - B) -1 , for all A ~_ V. (c) If f, fg E ~L(V), and there exists "~ E R such that g E ~L,,(V), then f(B)g(B) C_(fg)(B). (d) If f19 E ~ L ( V ) , g E ?-{L,o(V) and g(B) E B(X), then

g(B)B C_Bg(B) C_(f~g)(B). (e) If p E 7-IL(V), where p is a polynomial, p(z) =_ E~=o ak zk, then p(B) = ~-~=o akBk, with D(p(B))= ])(Bk). (f) f ( B ) + g(B) C_ ( f + g)(B), for all f,g E 7-lL(Y), where D(I(B) + g(B)) = 7:)(f(B)) N 7)(g(B)). (g) I(B) + Ag(B) = ( f + Ag)(B), for all f E ~ L ( Y ) , g E 7-IL,o(V),A E

C, when g(B) E B(X). In (c), note that, if both g(B) and f ( B ) are in B(X), then (fg)(B) E B ( X ) and (fg)(B) = I(B)g(B), the usual algebra homomorphism property. Some of the previous results may be summarized by introducing a generalization of an .T-functional calculus, what we will call a C-regularized .T-functional calculus. 136

D e f i n i t i o n 22.11. Suppose A is an operator with nonempty C-resolvent (see Definition 3.6), ~" is a Banach algebra of complex-valued functions on 1 for a subset of the complex plane containing fo(z) = 1 and g~(z) - ~_~, some A E pc(A), and C is a bounded, injective operator that commutes with A. By a C-regularized ~-functional calculus for A we will mean a continuous linear map, A, from 3c into B ( X ) such that (1) h ( f o ) = C; (2) If g~ E 9c, then ~ E pc(A) and A(g~) = ()~ - A)-IC; and (3) A(fg)C = A ( / ) A ( g ) , for all f , g E 5v. Note that, when C = I, this is the definition of a ~'-functional calculus for A. Intuitively, A ( f ) = f(A)C. This m a y also be considered a generalization of a C-regularized semigroup, { W ( t ) ) , generated by A; the operator W(t) m a y be thought of as etAC. As with C-regularized semigroups, the idea is that one m a y have unbounded operators, but the "smoothing" operator, C, provides a uniform control over the unboundedness. C o r o l l a r y 22.12. Suppose B , a , m , V and 0 are as in Theorem 22.5 and f ( B ) is as in Definition 22.6. Then, for all A r V, there exists a (B - A)-m-regularized H~(V)-functionM calculus for B, defined by

1 A ( f ) - f ( B ) ( A - B) - m = 27ri

o f ( w ) ( w - B) -1 ( w - ) 0 m"

Iterating the proof of Theorem 22.8 gives us the following. C o r o l l a r y 22.13. Suppose {fi}i=lN Q 7-~L(V), 1-[i=lJfi E 7-~.y(V), for all j (1 + a), Theorem 22.15 implies that sp(B") C V", when B is of a-type V and (-oo,0] is disjoint from V. It is natural to ask if Theorem 22.8 could be used to define f(B), that is, define f(B) to be the closure of the operator, defined on :D(B'~), for n sufficiently large, by

1 fo o f ( w ) ( w - B ) - I ( B f(B)x - 27ri

A)nx ( w -dwA)n'

(22.17)

for ~ ~ V,x 6 7)(Bn). This is the same as asking w h e n / ) ( B n) is a core for f(B). The following theorem gives a sufficient condition for this to occur. This includes the hypotheses of Theorem 22.15 (Corollary 22.19). Clearly it would be impossible to use (22.17) when f 6 ~tL(V) is not polynomiMly bounded. Theorem 22.10(a) implies that, w h e n / ) ( B ) is not dense, we also would be unable to use (22.17) to define f(B) equivalently, even when f is polynomially bounded. T h e o r e m 22.18. Suppose 7)(B) is dense, B is of a-type V, there exists

7 6 tt such that f 6 ~L,,~(V) and p(f(B)) is nonempty. Then 7)(B n) is a core for f(B), for all n >_ [7 + a] + 2. C o r o l l a r y 22.19. Suppose 7 ) ( , ) i s dense, B is of a- type V and (~) 6 U.~(V), for some 7 < - ( 1 + a). Then I)(B k) is a core for f(B), for all k > [q + 1. R e m a r k 22.20. In general, the question of when 79(B n) is a core for f ( B ) , when f is polynomially bounded, is a special case of the following open question. If Ira(C) is dense and A generates a C-regularized semigroup, is C(7)(d)) a core for A (see [Dav-P])? For our examples, let LHP, RHP, So and H~ be as in Chapter XX; let V~ - {z 9 CllRe(z)] < e}. E x a m p l e 22.21: F R A C T I O N A L ERS AND LOGARITHMS

POWERS, IMAGINARY POW-

139

For arbitrary a _ - 1 , i f B is of a - t y p e S~, we m a y define, for Re(z) > 0, B z, and define log(B) as - i g ( B ) , where g(z) --- ilog(z), with Definition 22.6. In particular, for any real t, since z ~ z ~t is bounded on S~, we m a y also use Definition 22.6 to define the imaginary powers of B, B ~t. In fact, it is clear from the definition that ilog(B) is defined as the generator of the B - m - g r o u p { B i t B - ' ~ } t e R (m =_ [a] + 2). Traditionally, when B is of ( - 1 ) - t y p e S~, ~ < ~r, the imaginary powers of B are defined by taking the closure of

Bitx = fro

w i t ( w - B ) - l B 2 x 2 7 r idw w2,XeT)(B2),

(,)

S~,~

where Sr - Sr N {z E CIM > e},w < r < ~ and 9 is sufficiently small. By Proposition 3.10, if {Bit}teR is a strongly continuous group, then its generator is i log(B), as defined above. On a Hilbert space, when B is of ( - 1 ) - t y p e S~,w < ~, {B it }teR is a strongly continuous group if and only if there exists an H ~176 functional

calculus for B (see [M]). On a general Banach space, {Bit}~eR being a strongly continuous group has m a n y applications to partial differential equations and operator theory; see [Do-V1, 2], [Pr-Soh], [Y1, 2]. E x a m p l e 22.22: E X P O N E N T I A L S OF OPERATORS WITH S P E C T R U M IN A L E F T H A L F - P L A N E . Suppose B is of a - t y p e c + L H P , for some real c and m - [a] + 2. Then by Theorem 22.10(e) (see Definition 22.6), B generates an exponentially bounded (k - B) -m regularized semigroup, for all k :> c. Since the map z ~ e tz is bounded on c + L H P , for all t >_ 0, the unbounded operator e t s , yielding the solutions u(t,x) - etBx, of the abstract Cauchy problem (0.1), may be defined directly, by Definition 22.6. A consequence of Example 22.22 is that an operator with polynomially bounded resolvent outside a vertical strip will generate a C-regularized group. W h a t is more surprising is that the same is true when the spectrum is contained in a horizontal strip (see Example 22.24). E x a m p l e 22.23: E X P O N E N T I A L S OF OPERATORS WITH REAL SPECTRUM . Suppose the spectrum of B is contained in the real line and B is of a - t y p e H~, for all 9 > 0. For any s E R, we may define e ss, and hence the solutions of the abstract Cauchy problem (0.1), directly by Definition 22.6, since the map z ~ - e sz is in 7-LL(H~). 140

Note that here we are defining f ( B ) , where f is not polynomially bounded on the spectrum of B.

E x a m p l e 22.24: E X P O N E N T I A L S A N D C - R E G U L A R I Z E D S E M I G R O U P S , S P E C T R U M IN A H O R I Z O N T A L S T R I P . More generally, if B is of a-type H~, for some e(0 < e < 1) (this is equivalent to iB generating an exponentially bounded ( i - B)-k-regularized group, for some k E N) we may write local solutions of the abstract Cauchy problem down directly, by defining e sB, for s < ~ , with Definition 22.6. We may also produce global solutions by using Definition 22.6 to define a C-regularized group generated by B. For t > 0, let ft(z) -- etZe -z2. Then by Corollary 22.9, ft(B) E B ( X ) and B f t ( B ) = f t ( B ) B = ~ f t ( B ) , for all t _> 0. This implies that {ft(B)}teIt is an f0(B)-regularized group generated by an extension of B (see Theorem 3.7). Since p(B) is nonempty, B itself is the generator (see Proposition 3.9). It may be shown that fo(B) = e -B2 Again using Corollary 22.9, it may be shown that Im(e -~B~) C Im(e -B2), for all c > 0, which implies that the set of initial data for which the abstract Cauchy problem has a solution is dense, whenever 79(B) is dense. It is also not hard to show that the C-regularized group generated by B extends to an entire group. E x a m p l e 22.25: E X P O N E N T I A L S OF OPERATORS WITH S P E C T R U M I N L E F T O R R I G H T S E C T O R S . It is well known that B is of a-type c - Se (a left sector), for some c E R, 8 < ~ , a _> - 1 , if and only if B generates an exponentially bounded holomorphic k-times integrated semigroup, for some k E N (see Chapter XXI). This means that B generates an exponentially bounded holomorphic ( A - B ) - k - r e g u l a r i z e d semigroup (see Chapter XVIII). When B is of a-type c + Se(a right sector), for some c E R, 0 < ~, we may, analogously to Example 22.24, define a C-regularized semigroup, {ft(B)}t>0, generated by B, with Definition 22.6, by letting ft(z) = ctZe -(z-c)r, for some r such that 1 < r < ~ , since ft will then decay exponentially on c + $6. This may be thought of as a result about reversibility of solutions of the abstract Cauchy problem, when sufficiently many of those solutions are analytic in a sector. E x a m p l e 22.26: C O S I N E S , S P E C T R U M I N V E R T I C A L If B is of a-type V~, for some ~ > 0, we may define cos(itB) and directly by Definition 22.6, since the maps z ~ cos(itz) and z ~ are bounded on V~, for all t E R. If m - [a] + 2, then Theorem 141

STRIP.

sin(itB) sin(itz) 22.8 im-

plies that (22.2), with A = B 2, has an exponentially bounded solution, for all initial d a t a in :D(Bm+2), given by u(t)= cos(itB)x + fo cos(isB)yds. In fact, {cos(itB)(A- B ) - m } t e R will be an exponentially bounded ( A B)-m-regularized cosine family generated by B 2, when IRe(A)I > e. E x a m p l e 22.27: C O S I N E S , S P E C T R U M IN HORIZONTAL S T R I P . For operators, B, with spectrum contained in a horizontal strip H~, as in Example 22.24, we m a y also define cos(itB) directly, for sufficiently smM1 Itl, and hence treat the second order abstract C a t c h y problem (22.2), with A = B 2. Or we may define a C-regularized cosine family, {gt(B)}teR, where gt(z) =- cos(itz)e-Z2; as in 22.26, this will produce solutions of (22.2), for all initial data in a dense set, when :D(B) is dense. As in Example 22.23, note that the map z ~ cos(itz) is not polynomially bounded on the spectrum of B. E x a m p l e 22.28: C O S I N E S , S P E C T R U M I N S E C T O R . When B is of a - t y p e c + So, for some real c,O < ~, as in Example 22.25, we m a y define a C-regularized cosine family generated by A = B 2, {ht(A)}teR, whereht(z) = cos(itz)e -(z-cY, where 1 < r < ~ , since ht will decay exponentially on c+ So, giving us solutions of (22.2), as in Example 22.26. It is clear that we may replace c + So by a rotation, A(c + So), for any complex A, by replacing e -(z-cY with e -[~(z-r Again, the map z ~ cos(itz) is not polynomially bounded on the spect r u m of B. E x a m p l e 22.29: C O S I N E S , S P E C T R U M IN SECTOR. When A is of a - t y p e ASo, for some complex A,0 < 7r, we may define a Cregularized cosine family generated by A, {ht(A)}teR, where ht(z) =--

cos(itv/-~)e - ( v / ~ Y , where ~

is chosen so as to map S~ into R H P , and 1 < r < 5 , since ht will decay exponentially on ASo, giving us solutions of (22.2), as in Example 22.26. E x a m p l e 22.30: T H E L A P L A C I A N . As a more specific example, we will apply our construction to B _ - / k , on LP(Rn), 1 _ p _ co (for 7?(B) see Example 22.2(2)). (a) It is known (see [Ril]) that, for 1 < p < c~, n = 1, there exists a continuous H~(So)-functional calculus for B, for all 0 > 0. Thus f ( B ) is defined and bounded, for all f C H~(So),O > 0, when 1 < p < c~,n = 1. For 1 _< p < oo and arbitrary n, our construction defines f ( B ) , for f E H~176 - 1), however, f ( B ) may not be bounded. Since B is of ( - 1 ) - t y p e ( S o - 1), f ( B ) ( 1 - B) -1 is bounded. Another way of saying this is that there exists a (1 - B ) - l - r e g u l a r i z e d H~176 - 1)-functional calculus for B (see Definition 22.11). 142

The function z ~ e -tz C H~176 for all 0 ~ ~ , t ~ 0, so this fits into the construction in [Rill, for 1 < p < o c , n = 1. In fact, it is well known that - B generates a bounded holomorphic strongly continuous semigroup, for 1 _< p < oc. For p = oc, - B does not generate a strongly continuous semigroup, but our construction guarantees that it generates a (1 + B ) - l - r e g u l a r i z e d semigroup. This enables us to treat the heat

equation u'(t) = (Au)(t), (t >_ O)u(O) : f, for f E LP(Rn), 1 _< p < o~. (b) The operator B is not of ( - 1 ) - t y p e He, for any e > 0. However, there exists n such that B is of n-type H~, for all e > 0. Thus, for f E H~176 we may construct f ( B ) such that f ( B ) ( 1 + B) -(n+2) is bounded. Another way of saying this is that there exists a (l+B)-(n+2)-regularized H~176 calculus for B, for all e > 0 (see Definition 22.11). This is also saying that 4-iB is as in Example 22.22 so that we may treat the SchrSdinger equation u'(t) = ( i A u ) ( t ) ( t C a ) , u(0) = f, for f C L v ( R ~ ) , I < p < oc. (c) The operator B also satisfies the hypotheses of Example 22.25. This allows us to treat the backwards heat equation,

u'(t) + (Au)(t) = 0, u(0) = f , oil LP(Rn), 1 _< p < oc; we may write down the solution as u(t) = ft(B)g, when f = e-(B-c)rg (see Example 22.25); this is a dense set, f , of initial data. (d) The fractional power B 89may be constructed so as to have the same spectral behaviour as B (this m a y be seen by using the functional calculus construction for commuting generators of bounded strongly continuous groups--see Chapter XII). Thus, as in Example 22.27 or 22.28, we m a y treat the Cauchy problem for the Laplace equation

u"(t) + (Au)(t) = O(t > 0), u(O) : f, u'(O) = g, on LP(Rn), 1 _~ p < oo. (e) As in (d), s i n c e / k = (iB 1):, Example 22.26 may be applied to the

wave equation u"(t) = ( A u ) ( t ) ( t > 0), u(0) : f, u'(0) = g, 143

on LP(Rn), 1 _< p _< co. E x a m p l e 22.31. In [Be1, 2], operators, B, of a-type Va = {x + iy I x < ]yl a} are considered, where 0 < a < 1. In effect, what is being constructed there is a C-regularized semigroup generated by B. For e > 0, let g~,t(z) -- etZe -~(-z)s, for a < b < 1. Then, as in the previous examples in this chapter, {ge,t(B)} is an e-~(-s)b-regularized semigroup generated by B. Example

22.32. A M O D I F I E D

-~u(t,x)O

HEAT

EQUATION.

Consider

= (~x)O 2u(t,x)(t > O,x e R), u(t,O)= f(t).

(22.33)

Instead of specifying an initial distribution, we are describing the change of t e m p e r a t u r e at x = 0, as time progresses. A natural choice for X is C0[0, co), the set of all continuous f : [0, co) C such t h a t l i m t ~ f(t) = O. Let A f =_ f', on X , with maximal domain. T h e n (22.33) becomes (22.2) (with x replacing t). Since A generates a b o u n d e d strongly continuous semigroup, it has a square root, B, such t h a t B is of ( - 1 ) - t y p e ( i - iS_~). Thus we may use Example 22.28 to deal with

(22.33). We m u s t begin our construction of f ( B ) with the constant function fo(w) = 1, verifying t h a t fo(B)(B - A)-m = (B - A)--m. This is the same as showing t h a t Wf(0), from T h e o r e m 22.5, equals (B - )0 -m L e m m a 22.34. Suppose B,)~, m, V and O are as in Theorem 22.5. Then,

for all k > m, dw (B -

_

o

- B)-I

(w-

P r o o f : Let D be a disc containing )% contained in the complement of 0 . Let F - O0 U OD. Our hypotheses on V imply t h a t there exist complex a,b such t h a t a + b[0, co) is contained in the complement of (V t_J D). Let g(z) - [b(z - a)]zz , where the fractional power is chosen so that I arg(g(w))l < 3, for all w C V. For t > 0, define

1 ~r e-tg(~)(w- B ) - I dw. T(t) - 2~ri By Cauchy's theorem, 2

B)-I dw.

iT(t) = f o o

144

Thus, for x in 7:)(B),

2riT(t)x = foo e-tg(to)(w- B + B = (~o e_ta(~ ) o

A)(w-

dw B)-'x -(~,- ~)

dw ) (.;- ~)

+ ( B - a) o e-'~(~)(w- B)-i~ (~;-- ~) = ( B - a)

o e-~'(to)(w- B)-~x ( ~ - ~)'

by a calculus of residues argument, since B is closed; continuing this argument k times gives

2riT(t)(B- )~)-k = ~o e-tg(~)(w- B)-I (w dw)~)k" 0

By dominated convergence, limt--.0 T(t)(B - A)-k exists and equals B)-I (to-~p" dw Hence, it is sufficient to show the following:

fO0( w _

] ~ T ( t ) x = x , W e V(Bm). 145

(,)

The calculation follows, for x 9 7)(Bin):

(w-)`)

B)-lz- x]

=

e -tg(~') [ ( w - )` + B - B ) ( w -

=

~-~.(~)(~,- B)-I(B- )`)x (~_)`------3

=

e - ~ g ( ~ ) ( w - )` + B - B ) ( w - B ) - I ( B - )`)x ( w - ) , ) 2

= [fre_tg(~ )

(w--)`)

(~-dw~)2 ] (B-)`)x

dw + fr e-tg(~')(w- B ) - I ( B - A)2x (w ----A): ~ 1 7 6

= 27ri ~

~

e -'"(z) Iz=:,(B- )`)ix

j----1

+

e-~"(~)(w- B)-I(B- )`)rex (w-)`)m"

Each term in the sum is divisible by t, hence goes to zero, as t does. The integral converges to zero, as t does, by dominated convergence and the residue theorem. This establishes (,) and the Lemma. | L e m m a 22.35. Suppose B,)`,m, V and 0 are as in Theorem 22.5, and h and g are bounded and holomorphic on V.

(a) dz

(z:~)m] = (2~'i)

dz

o ( h g ) ( z ) ( z - B) -1 (z _ )`)2m"

(b) For all x 9 X, foo h(w)(w - B )-l x (~,-:9 "+dl~

9 /)(B), with

[ dw (B- )`) fJoo h(w)(w-B)-lx ( w - dw )`)m+l - Jooh(W)(w-B) -ix ( w - )`),n"

146

Proof: (a) Let F1 - 00. There exists another smooth curve, F2, contained in (V - O----). We calculate as follows, using the residue theorem,

dw dz [fooh(W)(w- B)-l (w: ~)m] [foog(Z)( z- B)-l (z : ~)m] = fir2 fir~ h(w)g(z)( wl-- z)

((z-- B) -1 - ( w - B) -1) ( z - dzA)TM (w--dwA)m

(w : -ff)m (w : z) ) g(z)(z -

B) -1 (z - A)TM

+ /r~ ( fr~ (z-----A) g(z) m (z-w))h(w)(w-B)-l dz (w-dwA)m ffr, (27ri)h(z) dz = - ~ - - ~ ) m g ( z ) ( z - B ) - I ( z - ~)m" w C OV, (B-A)(w-B)-lx = (B-w-kw-A)(w-B)-lx : (w-A)(w-B)-lx-x, (b) For

1 (B - A ) ( w - B)-lxll is 0 ((1 + Iwl)-~). Since B is so that I1~+~ closed, this implies that fooh(w)(w- B)-lx (~-),)"+~d~e :D(B), with ( B - A) if0 h(w)(w- B ) - l x dw O (W -- ~)m+l =

o h(~) [ ( ~ - ~ ) ( ~ -

=

o h(~)(~-

B)-I~

B ) - l x - x] (~ _ ~)m+l

(~_ a)m,

by the residue theorem. | P r o o f of L e m m a 22.4: There exists M < oc such that [[(w- A)-I[[ _< M(1 + [w[)a, Vw ~ V. 1 . Then ( z - A) -1 exiSts and Suppose w ~ V and [ z - w[ < 2M(l+[w[)a is given by ~k~__0(w- z)k(w -- A) -(k+l), so that k

]](z-A)-1][ ~ E

k=O

2M(l+[w[) a

(M(1 + ['w[)c~)(k§

-- 2M(1 + ]w[) ~.

(,) 147

Let K -- Uw~v{Z[[Z - w[ n. As in the proof of Theorem 22.5, it may be shown that the (B A)-(k+m)-regularized semigroup {Wg(t)(B-A)-k}t>o (generated by g(B)-see Proposition 3.10) is given by f

2riWg(t)(B - A) -k = /

JO0

e~g(W)(w - B) -1

dw

(w A)k+m"

Since IIg(w)(w - B) -111 is 0 ((1 + Iwl)~+'~), we may differentiate as follows, for x E X:

d 2ri-~Wg(t)(B

-

A)-kxlt=0 =

o ~d

(etg(w)]t=o)(w- B)-lx (w ,~)k+m

dw = foo [(W_ ;~)k] (w- B)-lX (w_ )O,~; g( w )

148

by Lemma 22.35(b), this equals

dw ( B - A) -m foog(W)(W- B)-ax (w - ~)k' which establishes (,), by the definition of g(B) as the generator of the (B - A)-m-regularized semigroup {Wg(t) }t>0. For t > 0, Lemma 22.35(a) and (,) now imply that

2riWi(t)g(B)(B- A)-k = fo o etl(w)g(w)(w- B ) - I (w _ dw A)k+m ' so that, as with the proof of (*), for x E X, we may differentiate: .d

2m-~Wl(t)g(B)(B- A)-kxlt=o :

0 f(w)g(w)(w-- B)-lx (w - A) k+m

= (B - ~)-~

fo o

f(~)9(~)(w

- B ) -1 ( ~ dw _ ~)~,

by Lemma 22.35(b). This implies that, for all x e X, g(B)(B - A)-kx e D(f(B)), so that ( B - A)-kx e l)(f(B)g(B)), with

dw

f(B)g(B)(B- A)-kx - 2~i 1 fo o (fg)(~)(~- B)-I~ ( w - ,X)k" By (.), this equals ( f g ) ( B ) ( B - A)-kx, when (fg) e ~lL(V). | Proof of T h e o r e m 22.10: (a) Wfo(t ) 1 foo e~( w B) -1 (~-~).~ d~ is clearly differentiable and equal to its own derivative, thus, by Lemma 22.34, D(fo(B)) = X, with fo(B)x = (B - A)mdWfo(t)xlt=o = x, for all -

xCX.

(b) W~ (t) = ~ Lo

e ~ ( ~ - = ) - l ( w - B ) -~

~= thus we may differen(~_~),~,

tiate as follows:

dwa~(t)_

z~)m+l = (A_B)_Iwg~ (t) ' 1 ~o et(X-~)-l(w-B)-a (w dw 2~ri 0

by Lemma 22.35(b). This implies that D(g~(B)) = X, with ga(B)x --

(B

- A)m~W~

(t)xl,_-0 = ( ~ - B ) - ' x .

149

(c) Let n _= [ a + 7 ] + 2

and fix A ~ V. By Theorem 22.8, g(B)(,~-B) -n E

B(X). By the definition of (fg)(B), to show t h a t f(B)g(B) C_ (fg)(B), it is sufficient to show t h a t

fot Wig(s ) (f(B)g(B)x) ds = Wfg(t)x - Wfa(O)x , for all x C 79 (f(B)g(B)), t >_O. Since B is closed, this is equivalent to

fotWjg(8) (f(B)g(B)(~ = wzg(t)(~-

B)-nx-

B)-nx) ds

(,)

Wzg(0)(~- B)-nx,

for all x E 79 ( f ( B ) g ( B ) ) , t >_ O. Fix x E 7 9 ( f ( B ) g ( B ) ) , t >_ O. By Proposition 22.7, Wfg(8)x is in the domain of ( f ( B ) g ( B ) ) , and f ( B ) g ( B ) W f g ( s ) x = W f a ( 8 ) f ( B ) g ( B ) x , for all s _> 0. Since g(B)()~ - B) -n is bounded and f ( B ) is closed, f(B)g(B)()~ - B) -n is closed. Thus,

y=-

Wfg(s)xdseD(f(B)g(B)(A-B)-n),

with

(w_~).~, Corollary 22.9 and

: S0o [So' L e m m a 22.35 imply that, for all r _> 0,

27riWl(r)g(B)()~ - B)-ny

dw = ~ooerf(w)g(w) [~oteS('fg)(W)ds] (w-B)-lx (w_ ~)2m+n 9 Since the r-derivative of the integrand is an L 1 function of w, uniformly in r, we m a y differentiate under the integral sign:

27rid wf(r)g(B )(,~ - B l- ny dw

-~-~ooerf(w' (et(fg)(w' -- 1) (w-U)-ix (w_)~)2m+n . 150

By Lemma 22.35(b), this implies that

I(B)g(B)()~-B)-ny 1 r dw = 2~ri(B - )om JO[O "(et(fa)(~')- 1~/(w- B ) - l x (w ),)~,~+~ =

B)-nx),

B)-nx-

proving (.), as desired. (d) It is clear from the definition of g(B) that it commutes with all resolvents of B; this implies that g(B)B C_Bg(B). As in (c), to show that Bg(B) C_(flg)(B), it is sufficient to show that

/o

tWhg(s)(Bg(B)x) ds = W h g ( t ) x - Wflg(O)x ,

for all x E 79 (Bg(B)),I >__O. Since B is closed, this is equivalent to showing that

fotWflg(s) (B()~ - B)-lg(B)x) ds = ()~- B) -1 (Whg(t)x - Whg(O)x ) , for A ~ V. This follows by a simple integration, after applying Lemma 22.35(b), Theorem 22.10(b) and Corollary 22.9(a) and writing B()~-B) -1 as ;~(;~ - B) -1 - I. (e) Let us write p(B)l for the operator defined by n

p(B)lx - E akBkx' ~D(P(B)I)

------

~)(Bk)"

k=O

We need to show that p(B)I = p(B). Note that, since p(V) is contained in a left half-plane, there exists complex A such that p(A) ~ p(V). This implies that there exist not necessarily distinct hi, outside V, such that n

p ( ~ ) - p(B)I = I I ( ~ i -

B).

i----1

Suppose x e / ) ( p ( B ) I ) . Then, by Lemma 22.35(b),

2~riWp(t)x = Joo / etp(w)(w-B)-I(P()~)-P(B)I)X 151

dw

[l-In=l('~i--W--)](~ -

-

W)m'

so that we may differentiate d

_ -

W,( t )xlt=o p(w) B)-I(P()') o I-ILI(A - w) ( w -

P(Bh)x2 ri(f-

= ( A - B)-mp(B)lx, by Lemmas 22.35(b) and 22.34. This implies that x E :D(p(B)), with p(B)x = p(B)lX, that is, p(B)I C_p(B). Since p(p(B)t) is nonempty, this implies that p(B)l = p(B) (see Proposition 3.9). (f) By Lemma 22.35(a), {Wi(t)Wg(t))t> o = {W/+g(t)(B- A)-m)t>0, the ( B - A)-~m-regularized semigroup generated by (f + g)(B). In general, if {Wi(t))t>0 is a Ci-regularized semigroup generated by Gi, for i = 1,2, and Wl(t)W~(s) = W2(s)Wl(t), for all s,t >_ O, then {Wl(t)W2(t))t>_o is a C1C2-regularized semigroup generated by an extension of (G1 + G2). (g) By (f), it is sufficient to show that Ag(B) = (Ag)(B). This is clear from Definition 22.6. | P r o o f of T h e o r e m 22.14: (a) ~ (b). By Theorem 22.10(a) and (c), G=_ f ( B ) (~)(B)C_ I. Since ( ~ ) ( B ) E B(X), G is closed. By Theorem 22.8, since ~ C 7-/~(V), G is densely defined; thus, taking closures on both sides of the inclusion implies that I ( B ) ( ~ ) (B) - G = I. Proposition 22.7 implies that ( 7 ) (B)f(B) C_I. (b) ~ (a). By Theorem 22.8, there exists n e N such that f(B) (~) (B)(A-

B) -n = (A - B)-n(A ~ V), so that (~) (B)(A - B) -n = (f(B)) -1 (A B) -'~. Since ( ~ ) (B)is closed and :D(B n) is dense, we may take closures of both sides to conclude that ( } ) ( B ) = (f(B)) -1 E B(X). II P r o o f of T h e o r e m 22.15: This follows from Theorem 22.14 and Corol( f - ~ ) (~) and the first term is lary 22.9(a), since, for A ~ V, I-~1 bounded on O, for O as in Lemma 22.4. | For Theorem 22.18, we will need the following ([Ka, Problem IH-5.19]). =

L e m m a 22.36. Suppose l) C_ l)(A),r e p(A), and ( r - A)(I)) is dense. Then I) is a core for A. P r o o f of T h e o r e m 22.18: It is sufficient to assume 0 E p(f(B)). To see that (f(B)) (/)(Bk)) is dense, note that, by Theorem 22.8, there exists 152

m E N such that

v). This implies that (f(B))(~D(Bk)) contains ~)(Bm+k), which is dense, since :D(B) is dense. By Lemma 22.36, D(B k) is a core for f(B). ]

153

XXIII. SPECTRAL CONDITIONS GUARANTEEING SOLUTIONS OF THE ABSTRACT CAUCHY PROBLEM It is natural and very common to ask what conditions on the resolvent of A, (w - A) -1, will guarantee that the abstract Canchy problems (0.1) and (22.2) have solutions, for all initial data in dense sets. In this chapter, we characterize subsets, V, of the complex plane, with the property that, whenever A is of a-type V (Definition 20.1) and densely defined, then the abstract Cauchy problem has a solution, for all initial data, x, in a dense set (Theorem 23.6). We give a similar result for (22.2) (Theorem 23.11). These theorems reduce the operator theoretic question, of when solutions of (0.1) and (22.2) exist, to the complex analytic question of when there exist holomorphic g, on V, such that z ~-* etZg(z) is bounded, for all t > 0 (for (0.1)), or when there exist holomorphic g such that z ~-~ cos(itz)g(z) is bounded, for all real t (for (22.2)). D e f i n i t i o n 23.1. If V is as in Chapter XX, we will say that V is an a-spectral dense solution set if, whenever A is of a-type V and densely defined, then the abstract Cauchy problem (0.1) has a solution, for all initial data in a dense set. The simplest sufficient condition for being an a-spectral solution set is independent of a. We will see that, in general, this property is not independent of a (see Theorem 23.6, Example 23.7). D e f i n i t i o n 23.2. We will denote by E v the set of holomorphic functions g on V that decay more rapidly than any exponential, {gl z ~-* etZg(z) E H~176 for all t > 0}. T h e o r e m 23.3. Suppose the constant function fo(z) - 1 is the pointwise limit o f a bounded sequence from E v . Then V is an a-spectral dense solution set. The sufficient condition of Theorem 23.3 is not necessary, because when an operator is of a-type V, the V is not minimal, that is, we can always find a smaller set, O, such that A is of a-type 0 (Lemma 23.12). It is necessary and sufficient that there exist sufficiently many g E E o , whenever O is a subset of V that is asymptotically close enough (Theorem 23.6). The closeness depends on a (see Examples 23.5 and 23.7). The following definition measures how asymptotically close to V this subset, O, must be. D e f i n i t i o n 23.4. The set of complex numbers 0 is an a-interior subset of V if O is as in Chapter XX, O C_ V and d ( O , { z ~ VIIzl < R}) is

o((1 + 154

E x a m p l e 23.5. Let V - R H P - {z E C IRe(z) > 0}. Then, for all O < ~, So =- {reir > 0,1r I < 0} is a (-1)-interior subset of V, but not a 0-interior subset (or an a-interior subset, for any a > - 1 ) . For any e > 0, e + R H P is a 0-interior subset of V, but not an a-interior subset, for any a > 0. An example of a 1-interior subset of V would be {x + iylz > (1 + lu[)-'}. More generally, for any a >__ - 1 , Oa,k, from Example 20.2(3), is an a-interior subset of V, for all k > 0, and is not a E-interior subset of V, for any/~ > a. As suggested by this example, the following relationship between Definition 23.4 and 20.1 is not hard to see. 0 is an a-interior subset of V if and only if the operator n f ( z ) =_ z f ( z ) , on LP(O), 1 _< p ~ oo, is an operator of a-type V. T h e o r e m 23.6. The following are equivalent. (a) The set V is an a-spectral dense solution set. (b) For all a-interior subsets, O, of V, the constant function fo(z) - 1 is the pointwise limit of a bounded sequence from Eo. (c) For all a-interior subsets, O, of V, Eo is uniformly dense in H~176 E x a m p l e 23.7. Let V =_ R H P , as in Example 23.5. Then V is a (-1)spectral dense solution set, but is not a 0-spectral dense solution set. To see this, note first that any (-1)-interior subset, O, of V is conrained in S0, for some 9 < ~, thus g~(z) =_ e -~zr , where r is chosen so that 1 < r < ~0, and e > 0 is arbitrary, satisfies Theorem 23.6(b). Thus V is a (-1)-spectral dense solution set, by Theorem 23.6. To see that V is not a 0-spectral dense solution set, consider A m 1 "k ~-~, on X - {f e C[0,1]lf(0 ) = 0}, 7)(A) =_ { f e X l f ' e X } . Then it is not hard to show that A is of 0-type V, since 1 - A generates a bounded strongly continuous semigroup, but the abstract Cauchy problem (0.1) has no nontrivial solutions. Essentially the same proofs give the same results for second order problems (22.2), with e ~" replaced by cos(itz). D e f i n i t i o n 23.8. If V is as in Chapter XXII, we will say that V is a second order a-spectral dense solution set if, whenever B is of a-type V and densely defined, then (22.2), with A = B 2, has a solution, for all initial data in a dense set. D e f i n i t i o n 23.9. We will denote by Cv the set of all holomorphic functions g on V such that the map z ~ cos(itz)g(z) 6 H~176 for all t >_ 0. 155

T h e o r e m 23.10. Suppose the constant function fo(z) = 1 is the pointwise limit of a bounded sequence from Cv. Then V is a second order a-spectral dense solution set. T h e o r e m 23.11. The following are equivalent. (a) The set V is a second order a-spectral dense solution set. (b) For all a-interior subsets, O, of V, the constant function fo(z) - 1 is the pointwise limit of a bounded sequence from Co. (c) For all a-interior subsets, O, of V, Co is uniformly dense in H~176 The proof of Lemma 22.4 proves the following, which is well known for a = - 1 , V = So (see [Baiak]). L e m m a 23.12. Suppose A is of a-type V. Then there exists an a-interior subset, O, of V, such that A is of a-type O. P r o o f o f T h e o r e m 23.3: Suppose A is of a-type V. Let m - [a] + 2. Let {gk}k~176be a bounded sequence from E v that converges to f0 pointwise. Fix A r V. For t >_ O,x e X , k e N, let ht,k(Z) =-- (A-z)-(m+DetZgk(z) and let Uk,~(t) =- ht,k(A)z. By Coronary 22.9 and Theorem 22.10(d), uk,x(t) is a continuously differentiable function of t and duk,~(t) = A(u},~(t)), for all t _> 0. Since :D(A"n+l) is dense, all that remains is to show that (A - A)-('n+l)x = limk--.oo uk,~(0). This follows by dominated convergence, Corollary 22.9(a) and Theorem 22.10(b). 1 P r o o f o f T h e o r e m 23.6: (b) --* (a). Suppose A is of a-type V and densely defined. By Lemma 23.12, there exists an a-interior subset, O, of V, such that A is of a-type O. Assertion (a) now follows from Theorem 23.3. (a) --* (c). Let O be an a-interior subset of V. Let ( A f ) ( z ) - z f ( z ) , on H~176 with maximal domain. Then A is of a-type V, thus (0.1) has a solution, t ~-. u(t,g), for all g in a dense set, D. It is dear that u has the form (u(t,g))(z) = etZg(z), for t _> 0,z E O. Thus g G Eo, for all g e :/), as desired. (c) --* (b) is obvious. 1

Example 23.13. For any real c,a >_ -1, V =- ( c + L t l P ) is an a-spectral dense solution set, since f0(z) - 1 E Ev. In fact, it is known that an operator of a-type V generates an [a] + 2-times integrated semigroup and a (A - A)-([~l+2)-regularized semigroup (see Chapter XXI). E x a m p l e 23.14. For any a :> - 1 , e > 0, the vertical strip V~ -- {z E CIIRe(z)l < e} is a second order a-spectral dense solution set, since f0 e Cv, (see [A-Ke]). 156

E x a m p l e 23.15. For any real c, a > - 1 , 0 < 0 < ~, ~ V = c + So, where So - {reir162] < 0}, is both an a-spectral dense solution set and a second order a-spectral dense solution set. To see this, choose r such that 1 < r < ~ . For any e > 0, let gE(z) =-. e -~zr. Then it may be shown that, for all e > 0, g~ is in both E v and Cv. Since lim~..,0 g~(z) = 1, for all z E V, we may apply Theorems 23.6 and 23.11. As mentioned in Example 22.25, this may be considered a result about reversibility of solutions. We discussed S~ = R H P in Example 23.7. _

E x a m p l e 23.16. For any c > 0, a >__- 1 , the horizontal strip He = ~z E C I IIm(z)l 0. 7f

E x a m p l e 23.17. For 8 < u a > - 1 , c E R, the same choice of ge as in the previous example shows that the double sector (c + S0) t.I (c - S0) is both an a-spectral dense solution set and a second order a-spectral dense solution set. E x a m p l e 23.18. In [Bel, 2], operators of a-type Va -- ( z + i y { z < ly[ a} are considered. In the language of this chapter, it is shown there that V~ is an a spectral dense solution set, whenever 0 < a < 1. This follows from Theorem 23.6, by considering gE(z) = e -e(-z)*, for a < b < 1, and showing that this is in Ev,,, for all e > 0; in fact, this is the key to the construction in [Bel]. In [Bell it is also shown that Y - {x + i y l x < lyl[log(1 + lyl)] -1 } is not a (-1)-spectral dense solution set. This also follows from Theorem 23.6, after showing that there exists no g in Eo.

157

XXIV. POLYNOMIALS OF GENERATORS

In this chapter, we prove a type mapping theorem (see Chapter XX) for polynomials; that is, we show that, if A is of n-type ~, then p(A) is of n-type p(~2), for any polynomial p. When p maps [~ into an appropriate sector, this allows us to use the results of Chapter XXI to conclude that p(A) generates an exponentially bounded holomorphic k-times integrated semigroup. Our definition of type ~2, in Chapter XX, involves only growth conditions on the resolvent at infinity: To obtain a bounded strongly continuous holomorphic semigroup, we also need growth conditions at zero (see Lemma 24.12). If p is a polynomial, p(z) = ~-~'~N=0ckzki then we define N

p(A) - ~ ckA k, D(p(A)) =- V(AN). k=O

T h e o r e m 24.1. Suppose A is of n-type ['l and p is a polynomial. Then p(A) is of n-type p(f~). T h e o r e m 24.2. Suppose A is of (k - 1)-type [~, ~ > 0 > O, and q is a polynomial such that q(~) C_ Se. Then (a) the operator - q ( A ) generates an exponentially bounded holomorphic ( k + 1)-times integrated semigroup of angle (~ - 0); (b) if D( q( A ) ) is dense, then -q( A ) generates an exponentially bounded holomorphic k-times integrated semigroup of angle ( ~ - 0); and (c) if k = O, D(q(A)) is dense, and q(O) = O, then -q(A) generates a bounded holomorphic strongly continuous semigroup of angle

-o) C o r o l l a r y 24.3. Suppose - A generates a bounded holomorphic strongly

continuous semigroup of angle 8, and n( ~ - 8 ) < "i" ~ Then - A n generates a bounded holomorphic strongly continuous semigroup ofang/e ~ - n ( ~ - 0 ) .

Remark 24.4. Note that this is saying that, if A is of type 0 (see 20.2(7)), then A ~ is of type nO. 158

T h e o r e m 24.5. Suppose - A generates an exponentially bounded holo-morphic k-times integrated semigroup of angle 0, p(t) = t" + q(t), where q is a polynomial of degree less than n, and n(~ - 8) < ~. Then (a) the operator -p( A ) generates an exponentially bounded holomorphic (k + 1)-times integrated semigroup of angle ~ - n( ~ - 0); (b) if:D(p(A)) is dense, then -p( A ) generates an exponentially bounded holomorphic k-times integrated semigroup of angle ~ - n(~ - 0); and (c) i l k = O, then -p(A) generates a strongly continuous holomorphic s e m i g r o u p o f angle

- n(

- o).

C o r o l l a r y 24.6. Suppose p is a polynomial with positive leading coefficient. (a) I f - A generates a strongly continuous holomorphic semigroup of angle ~, then -p( A ) generates a strongly continuous holomorphic semigroup of angle r (b) I f - A generates an exponentially bounded holomorphic k-times integrated semigroup of angle ~, then -p(A) generates an exponentially bounded bolomorphic ( k + 1)-times integrated semigroup of angle ~ The following theorem would follow from Theorem 24.5 and the fact that the square of the generator of a strongly continuous group generates a strongly continuous holomorphic semigroup, if q(t) contained only even powers of t. Theorem 24.7 is more general, in that q may be any polynomial of degree less than 2n. T h e o r e m 24.7. Suppose iA generates a strongly continuous group, and p(t) = t 2n + q(t), where q is a polynomial of degree less than 2n. Then -p( A ) generates a strongly continuous holomorphic semigroup of angle E 2"

The same argument, using Theorem 24.2(a) and (b), instead of (c), gives the following. T h e o r e m 24.8. Suppose both iA and - i A generate exponentially bounded k-times integrated semigroups and p is as in Theorem 24.7. Then (a) -p(A) generates an exponentially bounded holomorphic (k + 1)times integrated semigroup of angle ~. (b) Ifl)(p( A ) ) is dense, then -p( A ) generates an exponentially bounded holomorphic k-times integrated semigroup of angle ~.

159

C o r o l l a r y 24.9. Suppose p is an arbitrary polynomial. (a) If both iA and - i A generate exponentially bounded k-times integrated semigroups, then _[p[2(A) generates an exponentially bounded holomorphic (k + 1)-times integrated semigroup of angle K 2"

(b) I~, in addition to (a), ~(Ipl2(A)) is dense, then -IPl2(A) generates an exponentially bounded holomorphic k-times integrated semigroup of angle -i" ~ (c) If iA generates a strongly continuous group, then -[p[2(A) generates a strongly continuous holomorphic semigroup of angle ~. E x a m p l e 24.10. Let A = iddz, on LP(R) (1 _< p _< c~), with maximal domain, and B = ( - - 1 ) n ( d ) 2~ + q(i d ) = A 2" + q(A), where q is a polynomial of degree less than 2n. (a) If 1 < p < ce, then B generates a strongly continuous holomorphic semigroup of angle ~. (b) If p = ee, then B generates an exponentially bounded holomorphic once-integrated semigroup of angle ~. E x a m p l e 24.11. Let A = A, the Laplacian, on LP(Rn)(1 < p < oo), with domain as in Example 20.2(2), or the Laplacian on LP(f~), where is a bounded open set in R '~ with smooth boundary, with :D(A) -= W2'p(f~)NW~'P(f~). Let p be a polynomial with positive leading coefficient.

(a) If 1 _< p < ~ , then -p(A) generates a strongly continuous holomorphic semigroup of angle ~. (b) If p = c~, then -p(A) generates an exponentially bounded holomorphic once-integrated semigroup of angle ~. L e m m a 24.12. Suppose sp(A) C_ K and there exists finite M such that

II(w- A)-lll _< Mlwl " ,

Vw~K,

and p is an N 'h degree polynomial such that p(O) = O. Then sp(p(A)) C_ p( g ) , with II(z- P(A))-llJ < MNI~I ",

for ~J z r V(K). Proof: Let V =- p( K). 160

Suppose z is not in V. Let {wj)~= 1 be the (not necessarily distinct) zeroes of z - p(w), that is, N z -

=

1-I(w

-

V complex w.

w),

j=l

We have

N

z - p(A) = H ( w j - A). j=l

For any j, since p(wj) = z is not in V, wj is not in K. Thus, for 1 < j g N, (wj - A) is invertible, and II(wj - A)-I[[ _< U[wjl k-1. Thus, z - p(A) is invertible, and we obtain the following upper bound for ( z - p ( A ) ) -1. N

I1(~-P(A))-all-< IX II(w~- A)-~II j=l N

< MN

H

IwJl k-x

j=l -

-

MNIzlk-a.

| P r o o f of T h e o r e m 24.2: Parts (a) and (b) follow from Lemma 24.12 and Theorem 21.13. Part (c) follows from Lemma 24.12 and the fact that, when an operator B is densely defined, then - B generates a bounded strongly continuous holomorphic semigroup if and only if there exists 8 < such that sp(B) C_ S--o and {[[z(z - B)-IH [z • S-o0}is bounded. | P r o o f o f C o r o l l a r y 24.3: For this proof, let us write BHS for bounded strongly continuous holomorphic semigroup. Suppose { > r > n(~ - 0). Then, since - A generates a BHS of angle 0, the spectrum of A is contained in S~, and { l l z ( z - A)-lll I z ~ S ~ ) is bounded. Let q(t) = t n. Since A generates a BHS, :P(q(A))is dense. Also q ( S ~ ) is contained in Sr so by Theorem 24.2(c), - A n = - q ( A ) generates a BHS of angle ~ - r whenever ~ > r > n(~ - 0). This implies that - A ~ generates a BHS of angle ~ - n(~ - 0). | tt

tt

In order to apply Theorem 24.2 to more general polynomials of other generators, we need some elementary lemmas. 161

L e m m a 24.13. Suppose E is a subset of the complex plane, and 0 > O. Then lim sup{]arg(z)l l z E E, lzl = R } < 8 R----~oo

if and only i f for all r > 0, there exists real cr such that E is contained in cv; + S~.

Proof: Suppose the lira inequality holds, and r > 0. There exists finite M such that larg(z)l < r when z is in E and Izl > M. Thus, E is contained in Sr U {z E C I Izl < M}, which may be shown to be contained in - M ( 1 + cot r + Sr Conversely, suppose that, for all r > 8, there exists real c~ such that E is contained in cr + S~. For any r _< 7r, it is not hard to see that lira sup(larg ( z ) l l z e co + S~0, Izl- R}

R---* oo

equals r Thus limn__.~ sup{larg(z)[[ z ~ E, which concludes the proof. |

Izl

= R} _< r for all r > 0,

L e m m a 24.14. I f p(t) = t n + q(t), where q is a polynomial o f degree less than n, 8 >_ 0 and c is real, then, for all r > nO, there exists rea/c~ such that p(c + Se) is contained in c O + S~. Proof:

Clearly limlzl_+oo v(c+z)zn= 1. Thus,

lim sup{larg(z)l I z e p(c + So),

R---*oo

Izl-- R}

= lira sup{larg(p(c + z))l I z e So, Izl = R) R---* oo

=

lim sup{larg(zn)liz e So, Izl = R}

R--~co

-= nO.

Applying Lemma 24.13 now gives the result. | L e m m a 24.15. Suppose K equMs - c "t- So U c - Se, where c and 0 are nonnegative, and p(t) = t 2'~ + q(t), where q is a polynomial o f degree less than 2n. Then, for at1 r > 2n8, there exists real cr such that p ( K ) is contained in c,) + S O. P r o o f : Let K + = - c + S0. Since K = K + O - K +, it is sufficient, by Lemma 24.13, to show that l i m ~ o o sup{larg(p(z))ll z e + K +} = 2n; this follows exactly as in the proof of Lemma 24.14. It P r o o f o f T h e o r e m 24.5: Suppose ~ > r > n(~ - 0). Choose r such that ~ > r > ~ - 0. Since - A generates an exponentially bounded 162

holomorphic b-times integrated semigroup of angle 8, there exists real c such that the spectrum of A is contained in c + S~, and {llwl-k(w A)-'II I w c + S,) is bounded. By Lemma 24.14, there exists real cr such that p(c + Sr is contained in cr + Sr By Theorem 24.2(a), c,~I - p(A) generates an exponentially bounded holomorphic (k + 1)-times integrated semigroup of angle ~ - r Thus, whenever ~ > r > n(~ - O ) , - p ( A ) generates an exponentially bounded holomorphic (k + 1)-times integrated semigroup of angle { - r This implies (a). The same argument, using Theorem 24.2(b), implies (b). For (c), note that, since - A generates a strongly continuous semigroup, D(p(A)) is dense. Thus the argument above, with Theorem 24.2(c), implies (c). | P r o o f of T h e o r e m 24.7: Suppose ~ > r > 0. Since iA generates a strongly continuous group, there exists positive r such that the spectrum of A is contained in the horizontal strip {z E C I IIm(z)l < r), with {lllm(z)(z- A)-lll I IIm(z)l > T} bounded. Let c = r c o t ( ~ ) , If = ( - c + S ~ ) U (c- S ~ ) . Since {z E C I IIm(z)l < r} is contained in If, and {lim-~l I z ~ K} is bounded, it follows that {llz(z- A)-~ll l z r K} is bounded. By Lemma 24.15, there exists real cr such that p(K) is contained in cr + Sr Since iA generates a strongly continuous group, D(p(A)) is dense. By Theorem 24.2(c), c r p(A) generates a bounded strongly continuous holomorphic semigroup of angle ~ - r Thus, for any positive r -p(A) generates a strongly continuous holomorphic semigroup of angle ~ - r so that -p(A) generates a strongly continuous holomorphic semigroup of augle ~. |

163

XXV. ITER.ATED ABSTtZACT CAUCHY PROBLEMS

Pelhaps the most unified treatment of higher order abstract Cauchy problems is via the iterated abstract Cauchy problem

(a)

-~ - nk

u(t) = O (t >_ O)

k=l

(b) ~"-~)(01 = ~i

(25.1)

(1 < i < n).

In this chapter, we show that (25.1) has a unique solution, for all zl in a large set, whenever Ak generates a regularized semigroup, for 1 < k < n. 0 The empty product 1-Ik=l ( d _ Ak) denotes the identity operator. Corollary 25.7 guarantees solutions for a dense set of initial data. By a solution of (25.1) we mean (as in [S]) u such that Ak)u(t) E C1[0,oo) n :D(Aj+I), for t > 0, 0 _< j < n, satisfying (25.1).

II~=l(d/dt-

D e f i n i t i o n 25.2. We will say that (25.1) is nicely solvable with respect to C if there exists a unique solution, for all xi E Cn(:P), where

:P-- ~{:P(A~lAk~...Ak,~) l ki # kj when i # j, 1 _o; (2) for 1 o is a C-regularized semigroup generated by A, f : [0,co) --. 1re(C) has the property that t ~-~ C-~ f(t) is continuously differentiable, and x E C (/)(a)). Then

u'(t) = A(u(t)) + f(t)(t > 0), u(O) = x

(*)

has the unique solution u(t) = W ( t ) C - l x +

W ( t - 8)C-1/(s)ds.

(**)

Proof: Suppose u is given by (**). By Theorem 3.4, for all t > 0, l o W ( t - s)C-l f(s)ds e D(A), with

A (fotW(t-

s)C-l f ( s ) d s ) = W(t)C -l f(O) - f(t)

+ so that

I'

w ( t - s)C-lI'(~)ds,

a (~(t))= w ' ( t ) c - l ~ + w ( t ) c -if(o) - I(t) +

I'

w ( t - . ) c -II'(s)d~,

which a calculation shows to be equal to u'(t) - f(t). For uniqueness, suppose u satisfies (*). Then, for 0 < s < t, ~ W ( t s)u(s) = W ( t - s ) f ( s ) , s o that W(O)u(t)-W(t)u(O)= f 2 W ( t - s ) f ( s ) d s , or

Cu(t) = C W ( t ) c - l x +

W ( t - , ) c - l f(8)d,

so that, since C is injective, u is as in (**). | 172

,

Suppose, for i = 1, 2, that Ai generates a C-regularized semigroup {Wi(t)}t>o and Wl(s)W2(t) = W2(t)Wx(s), for all s,t > O.

L e m m a 25.27.

Then, for a~l t > 0,~ e V(A~) n V(A2),

f o t W l ( t - s)W2(s)(A2 Proof."

Ax)z

= C (W2(t)- Wl(t))z.

We calculate:

fot W l ( t - s)W2(s)(A2 - A1)xds = Jot -~s d (Wl(t- s)W2(s)x) ds = w , ( o ) w 2 ( t ) ~ - w,(t)w2(o)~ = c ( w ~ ( t ) - w,(t))~. I P r o o f o f T h e o r e m 25.4: For 0 _< j < n, let

vj(t)=--'-- I'I ( d - - A k ) u ( t ) . k=l

We will show, by induction on k, that, for 0 < k < n, there exists

I]

( A , - A , ) m'''

such that

(()) v

I n - k < rai,l 0, since (r - Ak)-IC is injective. Since R is injective, v(t) = 0, for all t >__0, as desired. | P r o o f o f T h e o r e m 25.11: Since hA has dense range and generates a bounded strongly continuous semigroup, it is injective. Let (-)~A)~ be the unique n th root of ( - h A ) such that - ( - A A ) ~ generates a bounded strongly continuous holomorphic semigroup of angle ~ ( 1 Let

9

-

1

s

~

.

/-

B y Th

rem S.2,

generates an entire e - ( - ) ~ A ) ~ -regularized group {W(z)}z~c; thus for each k, Ak generates an e -(-xA) aLn -regularized semigroup, { W ( ( - $1 ) -.1. e ~ k-. t t)}t>0. The factorization (25.10) and Theorem 25.3 now imply the nice solvability of (25.10). To show the existence of the dense set, ~, we may apply Corollary 25.7, once we verify that Im(e-(-XA) ~ ) is dense. If a < f~ < ~ where ~ - # is the angle of the holomorphic semigroup generated by - ( - ~ A ) - 1, then it may be shown, with the usual Cauchy integral formula construction of fractional powers, that for all e > 0, Im(e -'(-xa)~ ) C_ Im(e-(-xA) ~/').

Since {e-e(-~A)~ }e_>0 is

a strongly continuous semigroup, this implies that Im(e -(-)'A)"].) is dense. The d'Alembert form of the solution follows from Corollary 25.7. The estimates of I]w(l) (t) II follow from Theorem 25.6 and the fact that each entry of (.A,~)-1 has the form c~,j(A -1. )k,.j, where ei,j is a complex number, and kij is an integer between 0 and n - 1. II 174

XXVI. EQUIPARTITION

OF ENERGY

In this chapter, we consider

w"(t) = B2(w(t)) w(0) = xl,

(t > 0),

(26.1)

w'(0) = =2.

D e f i n i t i o n 26.2. As in [19], we will say t h a t a solution, u, of (26.1), admits sharp equipartition of energy if lim K(t) = 1, t--,oo P(t) where K(t), the kinetic energy, is defined to be IIr potential energy, is defined to be IIBu(t)ll 2.

2 and P(t), the

R e m a r k s 26.3. Let u = u(t,x) (respectively, v = v(t,x)) be the position (respectively, velocity) vector describing a wave in R 3. Under some natural assumptions, u and v both satisfy the wave equation W t t -~ A W

for t e R,= e R 3. If I1" II denotes the n o r m in L2(R3), then

K(t)

=

IIw,(t,.)ll 2, P(t)= IIV=~(t,.)ll =

are the kinetic and potential energies (respectively the potential and kinetic energies) at time t for w = u (respectively w = v). In Maxwell's equations in a vacuum, the kinetic energy of the electric vector is the potential energy of the magnetic vector and vice versa, cf. [Go-S1, p. 404]. T h u s "kinetic" and "potential" energies are nice suggestive terms, but even in concrete physical contexts one should be careful in using them. T h u s our use of these terms in the nonphysical context of this section should not be taken to have physical implications. T h e o r e m 26.4. Suppose I m ( C ) is dense, B is injective, has dense range and generates a C-regularized group {W(t)}te• such that limt-.oo W ( t ) z = O, for all x E X . Then there exists a dense subspace, 5 , of X 2, such that for all ~ E 5 , there exists a unique solution of (26.1) that admits sharp equipartition of energy. 175

C o r o l l a r y 26.5. Suppose B has dense range and generates a bounded strongly continuous holomorphic semigroup. Then there exists a dense subspace, ~, of X 2, such that for a/l ~ 9 there exists a unique solution of (26.1) that admits sharp equipartition of energy. E x a m p l e 26.6. The Canchy problem for the Laplace equation admits sharp equipartition of energy. More precisely, for 1 _~ p < oo, D as in (25.13), there exists a dense subspace, ~, of (LP(D)) ~, such that for all (fl, f2) E/~, there exists a unique solution of (25.13) that admits sharp equipartition of energy. This is a consequence of Corollary 26.5, by choosing B to be a square root of - A , on LP(D), that generates a bounded strongly continuous holomorphic semigroup. Since/k has dense range, B does also. More generally, since the holomorphic semigroup generated by B is of angle ~, we could replace/% by A/k, in (25.13), for any complex A whose argument is less than ~r. All these assertions are also true for (25.14). R e m a r k 26.7. It is interesting that in 26.4, 26.5 and 26.6, both K(t) and P(t) may go to infinity, as t goes to infinity, although their ratio converges to one. Consider, for example, B f ( s ) - 8f(3), on X - C0[0, oo), with maximal domain. Then w(t)(s) = etSyl(s ) + e-tay2(s), for some Yl,Y2 E X , so that g ( t ) and P(t) are greater than or equal to sup,_>0 ]set~yl(s)lProof of Theorem 26.4: First, note that, for any x 6 X, {HW(-t)~ll},>0 is bounded below. This may be seen by supposing, for the sake of contradiction, that there exists a sequence of real numbers {tk},x 9 X, such that tk --~ 00 and IIW(-t ,)xll--, o. Then, since C2x = W(tk)W(--tk)x and C is injective, this would imply that I[W(tk)ll--* oo, which contradicts the stability of {W(t)}t>o. As in the proof of Theorem 25.11, we factor (26.1), w"(t)-

=

d

- B)( d + B)w(t),

and apply Corollary 25.7, to obtain a dense subspace ~ such that, for all 9 ~, there exists a solution of (26.1) given by = w(t)y

+ w(-t)y2,

Thus u'(t) = W(t)Byl - W ( - t ) B y 2 , and Bu(t) = W(t)By, + W ( - t ) B y 2 , so that, since ]]W(t)Byl]] --~ 0 and ]]W(-t)By2]]

for some yi 9 X.

is bounded below, as t ~ c~, u admits sharp eqnipartition of energy. | 176

P r o o f of C o r o l l a r y 26.5: By Theorem 8.2, there exists a > 1 such that B generates an e-(-B)~-regularized group. By Lemma 8.8, the image of e - ( - B ) " is dense. Since B generates a bounded strongly continuous semigroup, and has dense range, B is injective. Thus we may apply Theorem 26.4. |

177

XXVII. SIMULTANEOUS

SOLUTION

SPACE

Suppose {A~}~el is a family of closed operators on a Banach space. As we did in Chapters IV and V with a single operator, we wish to construct maximal continuously embedded subspaces on which, for all a, Am generates a strongly continuous semigroup. This will be the simultaneous solution space for {A~}~eI. In this chapter, we will consider the case where the simultaneous solution space is a locally convex space; as in Chapter IV, this space has both an operatortheoretic maximality (see Theorem 27.5) and a "pointwise" maximality (see Remark 27.4). In the next chapter, we will, as in Chapter V, consider the case where the simultaneous solution space is a Banach space. T e r m i n o l o g y 27.1. Throughout this chapter and the next two chapters, A will be a collection of closed operators {A~}~eI. Since our operators will not commute, in general, we must specify the order of a product of operators: n

1-I Gk -= G 1 G 2 " " G ~ . k---1 0

The e m p t y product of operators ~ k = l Gk will mean the identity operator. For a single closed operator, we introduced, in Definition 4.6, its solution space, and gave it a Frechet space topology, with respect to which the operator generated a strongly continuous locally equicontinuous semigroup (Theorem 4.8). Let us write Z(A) for the solution space of the closed operator A. W h e n the solutions of the abstract Cauchy problem (0.1) are unique, that is, there are no nontrivial solutions when x = 0, we may then define, for t > 0, the operator e tA by

etAx = u(t,x), 1)(e tA) = Z(A). There is no reason to believe that general.

will be closed, or even closable, in

e tA

D e f i n i t i o n 27.2. We will write Z(A) for the simultaneous solution space for .A, which we define to be the set of all x in n

k--1

178

where the intersection is taken over all finite sequences ~ - < - < ak > nk = l ' satisfying the following:

tk >~=1,

(1) n

IIxll.a,8,~,8 =-- sup{ll( H e tkA~k )xll ltk e [ak,bk]} k=l

is finite, for all finite sequences of nonnegative real numbers if, and all finite sequences 8, from I; and (2) the m a p t ~ etA~x, from [0, o c ) i n t o (:D(A), II 11~4,~,g,a)is continuous, for all a e I, if, b, (~. We give Z(.A) the locally convex topology generated by the seminorms R e m a r k s 27.3. For a single operator, t h a t is, A = {A}, for some closed operator A, both (1) and (2) of Definition 27.2 are automatically satisfied. Another natural choice of seminorms would be n

llxll , = ll(I-[ e"A~ )xll, k=l

for tk >_ O, ak E I. Completeness of the space and local eqtficontinuity of the semigroups seem to fail here; also, there is little hope of obtaining a Frechet space, even when I is countable (see Theorem 27.5). R e m a r k 27.4. Informally, :D(.A) m a y be described in terms of the abstract Cauchy problem, as follows. In order t h a t x0 be in :D(A), one wants to be able to first find a mild solution, ul, of (0.1), for A = A~I, x = x0; then, for x = xl - ul(tl,xO), one wants a mild solution u2, of (0.1), for A = Aa2 , etc. This means we are following one solution curve, then starting another solution curve at an arbitrary point on the original solution curve, and continuing this process indefinitely. T h e o r e m 27.5. Suppose that, for all a E I, the abstract Cauchy problem (0.1), with A = A ~ , x = O, has no nontrivial solutions. Then (1) Z ( A ) is a sequentially complete locally convex space;

(2) z(•)

x;

(3) for all a E I, A~lz(ct ) generates a strongly continuous locally equicontinuous semigroup; and (4) Z ( A ) is maximal-unique, that is, if Y is a locally convex space satisfying (2) and (3), then Y ~-~ Z ( A ) . 179

If I is countable, then Z(,4) is a Frechet space. Proof: Properties (2) and (3) are clear from the definition of Z ( A ) (Definition 27.2); for equicontinuity, note that, for a0 G to _< b0,

Ile~~176176

= sup{ll(II ~*~A~

Itk

e [ak,bk](1 is a Cauchy sequence in Z ( A ) . T h e n there exists x E X such t h a t x~ --. x in X. We will show, by induction on m, the number of coordinates in i ' a n d ~, that for all finite sequences ffand c~,

and

m

lim II(~I e t ' A ~ )(x~ -- x)ll : 0.

j-~oo

(27.6)

k=l

For m = 0 (27.6) clearly holds. Suppose (27.6) is valid for all sequences of length m. Given t l , . . . , tm+l ~ 0, O~1,..., O~m-I-1 E I, j E N, let m

B -_ A ~ + I ,

m

y~ - ( I I e~Ao~ )~j, y _ ( i i k=l

~Ao~)~.

k=l

Since < xn > is Cauchy in Z(.A), the functions t ~-* etByj, from [ 0 , ~ ) into X , are uniformly Cauchy on compact subsets of [0, oo), thus converge uniformly on compact subsets to a continuous r : [0, 0o) ~ X. Note t h a t r For a n y j E N , t _ > 0 , B

(/0

e sB yj

ds

)

= e tB yj -

y j;

thus, since B is closed and the convergence of e ~Byj is uniform on [0,t],

B

(/o

r

d~

)

= r

- y.

Because of the uniqueness of the mild solutions of (0.1), this is saying t h a t y E D(etS), and etBy = r so that etB(yj -- y) ~ O, for all t > 0. This concludes the induction, proving (27.6). 180

To see that Ilzll~,~,~,~ is finite, for any d,b',c~, first note that, since < x,~ > is Cauchy, there exists a constant Kg,~,a such that Ilx,~llA,~,~,a < K~,~.,u for all n E N. Thus, by (27.6), for any tk E [ak,bk], m

m

II(H etkA~k )xll = .lim II(H etkA~k )xjll _ K~,g, s, k=l

3-.-~ o o

k=l

so that Ilxll~,~,~,~ _< K~,~,~ < oo. Next, we will show that II(x. - ~)11~,~,~,~ converges to 0, as n ~ oo. Fix e > 0 and choose g so that II(xn - xj)H.~,~,~, ~ < e, for all n , j > N. Forn> N, m

k=l

m

9)11 = 3.lim II(H d~A~ )(x~ - ~DII, --..~OO k=l

for any tk E [ak,bk], by (27.6), thus is less than or equal to e. Taking suprema, we conclude that ]](xn - x)llA,g,~,a < e, for all n > N. To prove (1), all that remains is to show that x satisfies (2) of Definition 27.2. Fix ff, b,g. From the definitions of the seminorms, it is not hard to see that, for a E I, the maps t ~ etA~xn, from [0,oo) into (7)(,4),11 I[.~,g,g,a), converge uniformly to etA~x, as n ~ oo, on any bounded subset of [0, oo). Since xn E Z(A), for all n, this implies that the map t ~ e tA~ x, from [0, oo) into (7)(A), II I[~t,~,;,~), is continuous, as desired. For (4), suppose Y is a locally convex space, topologized by the seminorms {l[ II~}~eJ, satisfying (2) and (3). Since Y ~ X , there exist/~1, ...,fin E J, Cl, ...cn > 0 such that n

I1~11_< ~ e~llxllz,, v~ e Y.

(27.7)

{---1

By the local equicontinuity, for any g, b, g, for 1 < i < n, there exists

{(4..~N(i) C 1 f d . . l N(i) t"',JJj=l -- ~, t~tOJj=l C [0, O0) such that

N(O

sup{ll(1-I

e'~A~

Irk e

[ak,bk]} < E

k=l

di,Jllxll~,,~, W: e Y. (27.8)

j=l

Since Y ~-. X , it follows that Y C 79(A), with X~

181

(27.9)

for all x E Y. Assertions (27.7), (27.8) and (27.9) now imply t h a t

0 is a strongly continuous semigroup, for all a E I, it follows that, for any ~ E ] , the m a p

t~ ~-*sup{ll (~-I et~A~) xiipitk E [ak'bk](l < k < is continuous. Since Y ~ X , the m a p t ~-+ etA~ox, from [ 0 , ~ ) into (:D(A), II IIJt,~,g,a) is continuous, as desired. Finally, if I is countable, then we may, in the definitions of the seminorms in Definition 27.2, restrict ourselves to g, b"equal to finite sequences of rational numbers, so that Z(A) is a sequentially complete locally convex space topologized by a countable family of seminorms, making it a Frechet space. I

182

XXVIII. EXPONENTIALLY BOUNDED SOLUTION SPACE

SIMULTANEOUS

Let .A - {A~}~ei be, as in the previous chapter, an arbitrary family of closed operators. In this chapter, we wish to construct a maximal continuously embedded Banach space, Z, such that A~lz generates a strongly continuous semigroup, for all a. We need, as in Chapter V, to restrict ourselves to exponentially bounded solutions. For a closed operator A, we will denote by Z~p(A) the exponentially bounded solution space for A, the set of all x in Z(A) (see Terminology 27.1) for which the mild solution of the abstract Cauchy problem (0.1) t ~-* u(t,x), from [0, oc) into X, is exponentially bounded. D e f i n i t i o n 28.1. Suppose A is a closed operator such that (r - A) is injective, for r large; that is, there exists r0 E R such that (r - A) is injective, for all r > r0. By Proposition 2.9, for any x E Ze::p(A), the exponentially bounded mild solution of (0.1) is unique. We then define, for t >_ 0, the operator (etA)exp by

(etA)~,pX = u(t, X), I)((etA)~xp) -- Zexp(A). D e f i n i t i o n 28.2.

We will write Ze~p(A) for the exponentially bounded

simultaneous solution space for A n

k=l

over all finite sequences D e f i n i t i o n 28.3.


~=1,

< a k >~=1"

We will write It] =__~-~=1 tk.

For x E Ze~p(A),w E It, define n

Ilxll ,

sup(lle- l (1-I e

)xllltk

O, k 9 I , ( n - 1) e

N).

k=l

The space Z~(.A) is defined to be the set of all x E Zexp(M) such that (1) Hx[[.a,~ < oc; and (2) The map s ~ esA~x, from [0, o c ) i n t o (Zexv(A), 11 II.a,~)is continuous, for all a C I. 183

We will also write Z~(A) for the normed vector space (Z~(A), II I1~,~) R e m a r k 28.4. When A = {A}, a single operator, then Zo(A) equals the set of all x for which (0.1) has a bounded, uniformly continuous mild solution (see Chapter V). Theorem 28.5. Suppose that, for all a 9 I, there exists r~ 9 I~ such

that (r - A~) is injective, for all r > r~. Then, for any real w, (1) Z~(.4) is a Banach space; (2) Z~(A) ~ X; (3) for all a 9 I, A~[z,,(.a) generates a strongly continuous semigroup, with IleCAo'z < )ll < e ~', for all t > O; and (4) Z is maximal-unique, that is, i f Y is a normed vector space satisfying (2) and (3), then r ~-+ Z. E x a m p l e 28.6. It is clear, by maximality, that Z0({A~, A2 }) is contained in Z0(A1) N Z0(A2), for any pair of closed operators A1, A2. We give here an example of commuting closed operators A1 and A2 such that Zo({A1,A2}) ~ Zo(A1)AZo(A2). This will be a two dimensional version of the "bumpy translation" space of Example 4.17. For fl a region in t t 2 , write B U C ( ~ ) for the set of all bounded uniformly continuous complex-valued functions on fl, B U C I ( ~ ) for the set of all f 9 B U C ( a ) such that ~ and ~ exist and are in BUC(fl). Let A1 - o , the generator of left-translation,

(e~Al f ) ( x , y ) = f ( x + t,y), A2 = 0-~, the generator of downward translation,

(etA~f)(x,y) ==f ( x , y + t), both with maximal domains. We let both these operators act on a space where translation is not strongly continuous, X =_BUC(R 2) N BUCI({(x,Y) e R21x, y _< 0}). Then it is not hard to see that Zo({A1,A2}) = BUCI(R2), while Zo(A1)NZo(A2) = B U C ( R 2 ) N B U C I ( { ( x , y ) 9 l:t 2 Ix ~_ 0 or y < 0}). Suppose that (w, oc) C_ p(A,), for all a 9 I. We may then use the resolvents to construct Z~(A). Without loss of generality, suppose w = 0. Then we may mimic the construction and proof in [K2] as follows. 184

Let Y(`4) be the set of all x E X such t h a t n

Ilzlly(~) - sup{ll(II ),k()~k- A.~)-X)xlll)~k > 0,~k 9 s

1) 9 N}

k= l

is finite. Then, using the maximality of our construction in this chapter, we have the following. Theorem

28.7. Suppose (0, oo) C_ p(A~), for all a 9 I.

(1) Zo(A) equals the closure, in Y ( A ) , of n T ) ( A a , [Y(A)), where the intersection is taken over all finite sequences < ak >C_ I.

(2) Ilzllgo(~) = lizllY(~), ~or all x 9 g0(A). We should mention that, for m a n y interesting examples, the resolvent condition of this section will not be satisfied; for example, the matrices of operators in C h a p t e r XXX. P r o o f o f T h e o r e m 28.5: W i t h o u t loss of generality, we m a y assume w = 0 (otherwise work with ( A , - ~o)). Properties (2) and (3) are clear from the definition of II I1~,0 and (2) of Definition 28.3. To prove (1), suppose < x,~ > is a Cauchy sequence in Zo(A). T h e same a r g u m e n t as in the proof of T h e o r e m 27.5, with uniform convergence on compact subsets of [0,oo) replaced by uniform convergence on [0, oc), shows t h a t there exists X E N~ )

e t~A~k \k--1

, /

with the intersection taken over all finite sequences ~, (7, such t h a t II(xn z)ll~,0 converges to 0, as n ~ oc. All t h a t remains is to show t h a t x satisfies (2) of Definition 28.3. It is clear from the definition of the n o r m t h a t Ile'A~(z~ - x)ll~,0 < II(z~ x)]I.4,0, for all s > 0 , n 9 N , a 9 I. T h u s the functions s ~ esA'~x,~, from [0,~) ~ (zexp(`4),ll I1~,0), converge uniformly to esA'~x. Since x,~ 9 Zo(A), for all n, this implies t h a t s ~ esA"x, from [0, cr into (zexp(,4), II I1~,0), is continuous, as desired. For (4), suppose Y is a normed vector space satisfying (2) and (3). There exists a constant M such t h a t Ilxll ___ MIIzllr, for all x 9 Y. For any ~, (7, x 9 Y, m

m

I1(1-I ~Ao~)zll < MII(1-I et'A"~lr)xlly < MIIxllg" k----1

k----1

185

Thus, IIx[[.a,o w , M > 0, there exists a simultaneous (1 + [B[2) -~existence family for .AM that is O(eUlt3). T h e o r e m 30.3. Let

.4 =_ {(Pi,j(B)) I (pi,j ) 9 5r(w,N)}.

Suppose, > } ( g

1) +

Then there exists a Banach space W such that ([/)(IBF)]) m ~

W ~

X",

and for all A 9 A , AIw generates a strongly continuous semigroup. T h e o r e m 30.4. For any M > 0, let AM be the set of all (pi,j(B)) such that (Pi,j) 9 Jr(w) and

ID~p~,j(~)I

< MeI~I,vx e R '~, 1 _< i , j w , M > 0, there exists a simultaneous e-IBI-existence family for .AM that is O(e~'l~). T h e o r e m 30.5. Let .A = {(P,,j(B))I(P~,i) 9 f ( w ) } . Then there exists a Banach space W such that ([Xm(e-IBI)]) m ~ W ~ X m,

and for M1 A 9 .A, A I w generates a strongly continuous semigroup. Theorem

30.6. Let

.A = {(Pi,j(B))i'~= 1 I (pi,j(x))i,'~= 1 is symmetric for all x 9 R n } . Then there exists a Banach space W such that ([Im(e-IBI]) m ~ W ~ X ~ , 192

and for all A E A, s E N, i(A[w) ~ generates a strongly continuous group. E x a m p l e 30.7. Note that .4, from Theorem 30.6, includes all symmetric hyperbolic systems, that is, operators A of the form n

A = EMjDj, j=l

for some constant m • m complex-valued symmetric matrices .M j, on (LP(l:tn))m(1 _< p < oc),C0(Rn) m, etc. More generally, it includes operators of the form

A = E .A4jDj + fl/[0, j=l

(30.8)

for some constant m • m complex-valued symmetric matrices A/Ij(0 _( j < n), by choosing n

(p,,~(x)),,5=l ----~ M T j + M0, j=l

for x E It' ~. Thus, on the dense Banach subspace W, of (LP(Rn))m(1 _< p < oc), or (C0(Rn)) m, of Theorem 30.6, all the abstract Cauchy problems (1.1) cotresponding to operators as in (30.8), are simultaneously well-posed. This includes the wave equation, Maxwell's equations, the abstract Cauchy problem corresponding to the Dirac operator, and many other famous partial differential equations; see [Go2, Section II.9] and [Gi-S] for more examples. If we remove the desire for exponential bounded~less (hence replace a Banach subspace by a Frechet subspace), we obtain a much more sweeping statement. T h e o r e m 30.9. The collection of all matrices of polynomials o r b has a simultaneous e-IBI-existence family. In particular, all higher order problems of the form n--1

u~)(t) =

~p~(B)u(~)(t)

(t >__ 0), ul~)(0) = x~ (0 < ~ < n-

1);

l---0

this includes constant-coefficient partial differential equations of arbitrarily high order; are well-posed in the sense of Frechet spaces. We will need the following generalization of Lemma 13.13. 193

Lemma30.10. There exists finite K such that

"((~e(P"J"(x))'~"=l)g(x))"BCk(R~,B(Cm)) _< K sup [

sup

sup

xER ~' Llo is a family of closed operators and 9 denotes convolution on [0,0o). Note that, by choosing B(t) - 1, we obtain a general second order abstract Cauchy problem; more generally, by choosing B to be an operator-valued polynomial, n

B(t) _=Z tkBk' k=O

(**) becomes an arbitrary higher order abstract Cauchy problem. See [dL4] for an outline of existence families for (**), including HilleYosida type theorems. See also [A-Ke], [dL6], and [dL-K], for results related to (**).

III.

Regularized semigroups were introduced by Da Prato ([D1]). They were independently introduced by Davies and Pang ([Dav-P]), where they were called C-semigroups. There are many problems with the latter terminology; for example, C sounds like Co, which is a shorthand for strongly 202

continuous; also, "C-semigroup" obscures the fact that a specific C is being presented, that is, it implies "C-regularized semigroup, for some C," instead of "C-regularized semigroup, for this specific C," which is more accurate and much more informative. Thus we have returned to the terminology of Da Prato. The recent interest in the subject was started by the introduction of integrated semigroups (see Chapter XVIII) in [A1]. Thus the definition of generator given in [Dav-P] involves a Laplace transform (see (*) in the notes for Chapter II). Independently of [D1], we introduced the more direct definition of the generator in 3.1, involving differentiating at zero (this is the definition in [D1]), as is done to define the generator of a strongly continuous semigroup, in [dL13]. (We should mention here that [dL13] was submitted in 1987.) In [D1] and [Dav-P], it is assumed that the image of C is dense; in [Dav-P], it is assumed that the regularized semigroup is exponentially bounded. The generality of Chapter III first appeared in [dL13]. Some other contributers to regularized semigroups are Pang ([P]), Neubrander, nieber and Holderrieth ([Hi-Ho-Neu]), Lumer [Lu], Miyadera and Tanaka ([T1]), Keyantuo ([Key]), Zheng and Lei ([Z-Leil, 2, 3]), Li and Shaw ([Li], [Li-Shl, 2]) and Ha, Kim and Kim ([Ha-Kim-Kim]). Most of the work of Miyadera and Tanaka may be found in IT1], thus we are, for convenience, referring the reader to [T1] for these results. Their work is not in the same spirit as this book, and there is very little overlap between the material of this book and the work in IT1]. Much of their focus is on what they call the complete infinitesimal generator, G, which is defined as follows. If {W(t)}~__0 is a C-regularized semigroup, then denote by Dw the set of all x E Ira(C) such that the map t ~ W ( t ) C - l x is differentiable at t = 0. It is clear that :Dw is contained in the domain of the generator of {W(t)}t>0. Then G is defined to be the closure of AIvw, where A is the generator of {W(t)}t>o. In many ways, this appears to be a difficult object to work with. It seems more like the generator of the semigroup of unbounded operators {C -1W(t)}t>_o; as we have commented before, the real power of the operator-theoretic approach to the abstract Cauchy problem relies on obtaining bounded operators to work with. Theorems 3.4, 3.5 and 3.7 are from [dL13]. Analogues of Theorems 3.4, 3.5 and 3.7 may be found in [D1], [Dav-P] and IT1]. Proposition 3.9 appears in [dLT]. Proposition 3.10 is from [dL15]. Special cases of Proposition 3.11 and Corollary 3.12 first appear in IT1]. Theorems 3.13 and 3.14 are new. In [Dav-P], the following appears. O p e n Q u e s t i o n . If Ira(C) is dense and A generates an exponentially 203

bounded C-regularized semigroup, is C (:D(A)) a core for A? It is shown in [Dav-P] that the answer to this open question is "yes" when C is accretive, or when (a, oo) C_p(A). It is clear how to define a local regularized semigroup {W(t)}o