Laplace Transforms, Non-Analytic

Growth Bounds and C0-Semigroups

Sachi Srivastava

St. John’s College

University of Oxford

A thesis submitted for the degree of

Doctor of Philosophy

Hilary 2002

Laplace Transforms, Non-Analytic Growth Bounds and

C0-Semigroups

Sachi Srivastava

St. John’s College

University of Oxford

A thesis submitted for the degree of

Doctor of Philosophy

Hilary 2002

In this thesis, we study a non-analytic growth bound ζ(f) associated with an exponen-

tially bounded measurable function f : R+ → X, which measures the extent to which f can

be approximated by holomorphic functions. This growth bound is related to the location of

the domain of holomorphy of the Laplace transform of f far from the real axis. We study

the properties of ζ(f) as well as two associated abscissas, namely the non-analytic abscissa

of convergence, ζ1(f) and the non-analytic abscissa of absolute convergence κ(f). These

new bounds may be considered as non-analytic analogues of the exponential growth bound

ω0(f) and the abscissas of convergence and absolute convergence of the Laplace transform

of f, abs(f) and abs(‖f‖). Analogues of several well known relations involving the growth

bound and abscissas of convergence associated with f and abscissas of holomorphy of the

Laplace transform of f are established. We examine the behaviour of ζ under regularisa-

tion of f by convolution and obtain, in particular, estimates for the non-analytic growth

bound of the classical fractional integrals of f. The definitions of ζ, ζ1 and κ extend to the

operator-valued case also. For a C0-semigroup T of operators, ζ(T) is closely related to

the critical growth bound of T. We obtain a characterisation of the non-analytic growth

bound of T in terms of Fourier multiplier properties of the resolvent of the generator. Yet

another characterisation of ζ(T) is obtained in terms of the existence of unique mild solu-

tions of inhomogeneous Cauchy problems for which a non-resonance condition holds. We

apply our theory of non-analytic growth bounds to prove some results in which ζ(T) does

not appear explicitly; for example, we show that all the growth bounds ωα(T), α > 0, of a

C0-semigroup T coincide with the spectral bound s(A), provided the pseudo-spectrum is

of a particular shape. Lastly, we shift our focus from non-analytic bounds to sun-reflexivity

of a Banach space with respect to C0-semigroups. In particular, we study the relations

between the existence of certain approximations of the identity on the Banach space X and

that of C0-semigroups on X which make X sun-reflexive.

To my parents and my brother

Acknowledgements

I am indebted to Prof. C.J.K. Batty for his invaluable support and guidance

over the last few years. Without his help this work would not have been possible.

I would also like to thank Ralph Chill for some valuable discussions concerning

my work.

My study at Oxford was funded by the Commonwealth Scholarship Commission,

U.K. and I am grateful for their support. Also, I would like to thank the

Radhakrishnan Memorial Bequest for their financial support while I was writing

this thesis.

Contents

1 Introduction 1

1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 A new growth bound . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.3 Overview of thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2 Preliminaries 8

2.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.1.1 Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.1.2 Banach spaces and operators . . . . . . . . . . . . . . . . . . . . . . 8

2.1.3 Function spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2 Vector-valued functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2.1 Exponential growth bound . . . . . . . . . . . . . . . . . . . . . . . 10

2.2.2 The Laplace transform . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2.3 Convolutions and the Fourier transform . . . . . . . . . . . . . . . . 13

2.3 Operator-valued functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.3.1 Laplace and Fourier transforms for operator-valued functions . . . . 15

2.3.2 C0-semigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.3.3 Norm continuity and the critical growth bound . . . . . . . . . . . . 17

2.3.4 Adjoint semigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3 A non-analytic growth bound for Laplace transforms and semigroups of

operators 21

3.1 Introducing the non-analytic growth bound . . . . . . . . . . . . . . . . . . 22

3.2 The non-analytic bounds for operator-valued functions . . . . . . . . . . . 30

3.2.1 Reduction to the vector-valued case . . . . . . . . . . . . . . . . . . 30

3.2.2 The C0-semigroup case . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.3 Essential holomorphy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.4 A comparison of the critical growth bound and the non-analytic growth bound 40

iv

4 Fractional growth bounds 44

4.1 Convolutions and regularisations . . . . . . . . . . . . . . . . . . . . . . . . 44

4.2 Boundedness of convolutions and non-resonance conditions . . . . . . . . . 53

4.3 Fractional integrals and non-analytic growth bounds . . . . . . . . . . . . . 56

4.4 Fractional growth bounds for C0-semigroups . . . . . . . . . . . . . . . . . 66

4.5 Convexity and fractional bounds for vector-valued functions . . . . . . . . 69

5 Fourier multipliers and the non-analytic growth bound 73

5.1 A characterisation for ζ(T) . . . . . . . . . . . . . . . . . . . . . . . . . . 75

5.2 Perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

5.3 Inhomogeneous Cauchy problems . . . . . . . . . . . . . . . . . . . . . . . 88

6 Weak compactness, sun-reflexivity and approximations of the identity 106

6.1 Weak compactness and sun-reflexivity . . . . . . . . . . . . . . . . . . . . . 107

6.2 Approximations of the identity . . . . . . . . . . . . . . . . . . . . . . . . . 109

v

Chapter 1

Introduction

1.1 Background

Linear differential equations in Banach spaces are intimately connected with the theory

of one-parameter semigroups and vector-valued Laplace transforms. In fact, given a closed

linear operator A with dense domain D(A) ⊂ X, where X is a Banach space, the associated

abstract Cauchy problem u′(t) = Au(t) (t ≥ 0),

u(0) = x,(ACP)

is mildly well posed (that is, for each x ∈ X there exists a unique mild solution of (ACP))

if and only if the resolvent of A is a Laplace transform. This is equivalent to saying that A

generates a strongly continuous semigroup T on X, and then the mild solution of (ACP) is

given by u(t) = T(t)x.

Here, by a mild solution of (ACP) we mean a continuous function u defined on the

non-negative reals and taking values in X such that∫ t

0u(s) ds ∈ D(A) and A

∫ t

0u(s) ds = u(t)− x (t ≥ 0).

By a classical solution of (ACP) we mean a continuously differentiable, X-valued function

u defined on the non-negative reals such that u(t) ∈ D(A) for all t ≥ 0 and (ACP) holds.

The abstract Cauchy problem is classically well-posed if for each x ∈ D(A), there exists a

unique classical solution of (ACP). A mild solution u is a classical solution if and only if

u is continuously differentiable. If u is a continuous, Laplace transformable function, then

u is a mild solution of (ACP) if and only if u(λ) ∈ D(A) and λu(λ) − Au(λ) = x, for

Reλ sufficiently large. (ACP) is mildly well posed if and only if ρ(A) 6= ∅ and (ACP) is

classically well-posed, if and only if A generates a C0-semigroup.

These relations between solutions of differentiable equations and semigroups are the

primary reasons why semigroups of operators have been studied intensively. We refer the

1

reader to the books of Hille and Phillips [28], Engel and Nagel [20], Davies [17] and Pazy [42]

for the basic theory. The recent monograph by Arendt, Batty, Hieber and Neubrander [2]

is particularly useful for our purposes as it presents the theory of linear evolution equations

and semigroups via Laplace transforms methods.

For applications, it is useful to describe the properties of a semigroup in terms of its

generator, as this gives valuable information about the solutions of the well posed or mildly

well posed Cauchy problem even though the solutions may not be known explicitly, which

is usually the case. Of particular interest is the asymptotic behaviour of these solutions;

this has led to investigations into the behaviour of T(t) as t → ∞ and more generally to

the theory of asymptotics of strongly continuous semigroups. The starting point of this

theory is Liapunov’s stability theorem for matrices which characterises the ‘stability’ of the

semigroup generated by an n × n matrix A in terms of the location of its eigenvalues. A

C0-semigroup T is called uniformly exponentially stable if ω0(T) < 0, where ω0(T) is the

exponential growth bound of T given by

ω0(T) = infω ∈ R : supt≥0

e−ωt‖T(t)‖ <∞.

T being uniformly exponentially stable is equivalent to

limt→∞‖T(t)‖ = 0.

For a closed operator A, the spectral bound s(A) is given by

s(A) = supReλ : λ ∈ σ(A).

In terms of the exponential growth bound and the spectral bound Liapunov’s theorem may

be stated as follows.

Theorem 1.1.1. Let T be the semigroup on Cn generated by A ∈Mn(C). Then

ω0(T) = s(A).

The above theorem extends to semigroups generated by bounded operators A defined

on any Banach space. This is a direct consequence of the validity of the spectral mapping

Theorem σ(etA) = etσ(A), t ≥ 0, for such semigroups. However, for general C0-semigroups

the growth bound and the spectral bound do not necessarily coincide; in most cases this

failure of Liapunov’s stability theorem is due to the absence of any kind of spectral mapping

theorem.

The exponential growth of the mild solutions of a well posed Cauchy problem is de-

termined by the uniform growth bound ω0(T),T being the associated semigroup. Thus

Liapunov’s theorem implies that if A is bounded, then the exponential growth of the mild

2

solutions of (ACP) is determined by the location of the spectrum of A. In the case when A

is an unbounded operator, information about the location of the spectrum of A is no longer

enough, and additional assumptions are needed, either on the smoothness of T or on the

geometry of the underlying space X.

For eventually norm-continuous semigroups the spectral mapping theorem σ(T(t)) \0 = etσ(A) holds [28], and therefore, so does Liapunov’s stability theorem. The category

of eventually norm-continuous semigroups includes all semigroups which are eventually com-

pact, eventually differentiable or holomorphic. Building on preliminary work of Martinez

and Mazon [37], Blake [11] introduced the concept of asymptotically norm-continuous semi-

groups or semigroups which are norm-continuous at infinity. A spectral mapping theorem

for the peripheral spectrum holds for such semigroups and this is sufficient for deducing

that ω0(T) = s(A). All eventually norm-continuous semigroups with finite growth bounds

are asymptotically norm-continuous.

Several other growth bounds and spectral bounds have been introduced in order to fur-

ther describe the asymptotic behaviour of strongly continuous semigroups. Among these

are the growth bound ω1(T), which determines the exponential growth of classical solutions

of (ACP), higher order analogues ωn(T), n ∈ N, which estimate the exponential growth of

solutions of (ACP) with initial values in D(An), and the more general fractional growth

bounds ωα(T), α ≥ 0. The pseudo-spectral bound s0(A) is the abscissa of uniform bound-

edness of the resolvent while the n-th spectral bound sn(A) is the abscissa of polynomial

boundedness of degree n of the resolvent. There is a large literature on the relations be-

tween these growth bounds associated with the semigroup T and the spectral bounds of

the generator A. We refer to [40] and [2, Chapter 5] for surveys. Inequalities showing that

the growth bounds are not less than spectral bounds are relatively easy to obtain com-

pared with opposite inequalities. The first relation showing a spectral bound dominating a

growth bound for arbitrary C0-semigroups was ω2(T) ≤ s0(A), obtained in [47]. Amongst

the most striking results in this direction are the Gearhart-Pruss theorem establishing the

equality ω0(T) = s0(A) for strongly continuous semigroups defined on Hilbert spaces [22],

[44] and the theorem of Weis and Wrobel showing ω1(T) ≤ s0(A) for semigroups on general

Banach spaces [50]. The analogue of the Gearhart-Pruss Theorem for higher order bounds,

involving the equality of ωn(T) and sn(A), n ∈ N, for semigroups defined on Hilbert spaces

has been obtained in [51].

A new growth bound, the growth bound of local variation δ(T) or the critical growth

bound ωcrit(T) has recently been introduced in [11] and [38], building on ideas from [37]. It

measures the growth of the uniform local variation of mild solutions of the Cauchy problem,

and it is related to s∞(A) and s∞0 (A), the bounds of the spectrum and the pseudo-spectrum

of A away from the real axis. The spectral bounds s∞(A) and s∞0 (A) may be considered as

3

analogues of s(A) and s0(A) determining the existence and boundedness of the resolvents

in those parts of the right half-planes which are away from the real axis. There is an

analogue of the Gearhart-Pruss Theorem for these bounds ( δ(T) = s∞0 (A)) for semigroups

on Hilbert spaces [11]. Applications of the critical growth bound to perturbation theory

and to various evolution equations may be found in [9], [13], [14] and [15].

The standard growth and spectral bounds for semigroups are all special cases of bounds

and abscissas associated with vector or operator-valued functions on R+ and their Laplace

transforms. For example, for a strongly continuous semigroup T with generator A, the

spectral bounds s(A) and s0(A) are just the abscissa of holomorphy and boundedness ([2,

Section 1.4, Section 1.5] ) of the operator-valued function T : R+ → L(X) while ω1(T) is

the abscissa of convergence of the Laplace transform T of T. Most of the general results

also extend naturally to exponentially bounded functions, but some, like the Gearhart-Pruss

Theorem are confined to semigroups and/or depend on the geometry of the Banach space

in question. The Weis-Wrobel Theorem is an example of a semigroup result extending

to the case of exponentially bounded functions as shown by Blake [11, Theorem 6.5.9],[5].

However, none of the characterisations of the critical growth bound known so far extends

in a useful way to functions.

1.2 A new growth bound

In this thesis, we study a growth bound ζ(f) associated with an exponentially bounded

function defined on R+, which may be described in a sense, as the growth bound of f

modulo functions which are holomorphic and exponentially bounded in a sector about the

positive real axis. Therefore, we call this growth bound the non-analytic growth bound of f .

We work as far as possible in the general setting of vector-valued, exponentially bounded

functions defined on R+ and deduce results for semigroups as special cases.

ζ(f) may be thought of as the non-analytic analogue of the exponential growth bound

ω0(f) of f . In fact, it is related to the analytic behaviour of f away from the real axis

in much the same way as ω0(f) is related to f in the right half-planes of C. In particular,

hol∞0 (f) ≤ ζ(f), where hol∞0 (f) is the analogue of the spectral bound s∞0 (A) for functions.

We also introduce the non-analytic abscissas of convergence and absolute convergence, ζ1(f)

and κ(f) associated with f , which are again analogues of the abscissas of convergence and

absolute convergence, abs(f) and abs(‖f‖) of the Laplace transform of f . We obtain non-

analytic analogues of many of the relations between growth bounds and spectral bounds

for semigroups and their extensions to exponentially bounded, vector-valued functions. In

particular, such an analogue of Blake’s extension of the Weis-Wrobel Theorem to functions

is obtained.

4

The non-analytic growth bound coincides with the growth bound of non-integrability

of operator-valued functions on R+ defined in [10]. It has been established in [11], [10]

that the critical growth bound and the growth bound of non-integrability associated with a

strongly continuous semigroup T with generator A defined on a Hilbert space coincide with

the spectral bound s∞0 (A). Thus, we have an analogue of the Gearhart-Pruss theorem for

the non-analytic growth bound also. We derive higher order analogues of this result. For

many semigroups δ(T) = ζ(T) (we do not know of any semigroup for which they differ).

A comparison of the critical and non-analytic growth bounds shows that unlike the critical

growth bound, the concept of the non-analytic growth bound of exponentially bounded

vector-valued functions is a useful and interesting study in itself.

An examination of the behaviour of the non-analytic growth bound of an exponentially

bounded, measurable function f defined on R+, under regularisation by convolution yields

some interesting results. Besides establishing analogues of known results for growth bounds

of convolutions of f with φ, we obtain estimates for ζ(φ ∗ f) in terms of the growth bound

and certain abscissas of holomorphy of the Laplace transform of f, where φ is a locally

integrable complex-valued function defined on R satisfying certain additional conditions.

A particular case of these results is an estimate for the classical fractional integral of f.

This line of study builds up to the definition of the fractional non-analytic growth bounds

for a function f . In the case of a semigroup T, these bounds are the natural analogues of

the fractional growth bounds ωα(T). The relation between the fractional non-analytic and

uniform growth bounds of T yields a rather striking result: The uniform growth bounds

ωα(T) equal the spectral bound s(A) for all α > 0 provided the pseudo-spectrum of A is

of a particular shape.

A characterisation of the uniform growth bound in terms of the Fourier multiplier prop-

erties of the resolvent has been obtained in [27] and for higher order growth bounds in

[32]. In fact, Fourier multipliers have often been used to study stability and hyperbolicity

of strongly continuous semigroups (see [30], [34] and [49]). Using ideas from [32], [30] and

[34] we obtain a characterisation for ζ(T) for a strongly continuous semigroup T in terms

of the shape of the pseudo-spectrum of the generator A and Fourier multiplier properties

of functions of the form s 7→ φ(s)R(w + is,A) where w > s∞0 (A), R(λ,A) is the resolvent

of A and φ is a suitable smooth function. Thus, one is able to obtain information about

the non-analytic behaviour of solutions of (ACP) from the pseduo-spectrum and resolvents

of the operator A.

In [45] and [43] results have been obtained concerning existence of bounded solutions of

the inhomogeneous Cauchy problem

u′(t) = Au(t) + f(t) (t ∈ R), (1.1)

5

when f is a bounded function on R taking values in X, A is the generator of a C0-semigroup,

and a non-resonance condition between A and f is satisfied together with some assumptions

on the spectrum of A. We study (1.1) when f ∈ Lp(R,X), 1 ≤ p <∞ and A is any closed

operator. We obtain a necessary and sufficient condition for a function in Lp(R,X) to be

a mild solution of (1.1) for this case. If, in addition, A generates a C0-semigroup T and f

satisfies a non-resonance condition with respect to A then the existence of a unique mild

solution of (1.1) is closely related to ζ(T). In fact, we are able to obtain a necessary and

sufficient condition for ζ(T) < 0 in terms of the shape of the pseudo-spectrum of A and

the existence of such solutions. This characterisation is comparable to characterisations of

hyperbolicity of the C0-semigroup generated by A in terms of the existence of unique mild

solutions of (1.1), for every f ∈ Lp(R,X) ([33], [16, Section 4.3]) and for every bounded

and continuous f ([44, Theorem 4]). The result in [33] is proven in the context of non-

autonomous Cauchy problems. We refer the reader to [16] for the definitions and theory

concerning non-autonomous Cauchy problems.

Towards the end of the thesis, we digress from the subject of non-analytic growth bounds

and study the relations between sun-reflexivity of a Banach space with respect to a strongly

continuous semigroup and the existence of approximations of the identity on the space with

some special properties. This study is inspired by [46] where strong Feller semigroups and

approximations of the identity on C∗-algebras are studied.

1.3 Overview of thesis

In Chapter 2 we introduce some notations and collect well known results from the vast

literature on strongly continuous semigroups and vector-valued Laplace and Fourier trans-

forms. Definitions of some well known abscissas of holomorphy and boundedness of Laplace

transforms are recalled, new definitions added and properties of these abscissas deduced.

We begin the first section of Chapter 3 by introducing the non-analytic growth bounds

and abscissas of convergence of the Laplace transform of an exponentially bounded vector-

valued function defined on R+. Subsequently, equivalent descriptions of these bounds are

derived (Proposition 3.1.4) and their basic properties studied. Section 3.2 is devoted to the

particular case of operator-valued functions. In Subsection 3.2.1, we obtain descriptions of

the non-analytic bounds of operator valued functions T : R+ → L(X) in terms of similar

bounds for the orbit maps t 7→ T(t)x, x ∈ X. Subsection 3.2.2 deals specifically with

the non-analytic bounds for strongly continuous semigroups. A non-analytic version of

the Gearhart-Pruss theorem for higher orders on Hilbert spaces is obtained in Theorem

3.2.9. Analogous to the concepts of essential norm continuity and essential measurability

of C0-semigroups [48], we introduce essentially holomorphic C0-semigroups in Section 3.3.

6

In Section 3.4 we undertake a comparison of the critical and non-analytic growth bounds.

We look at several classes of semigroups for which a non-analytic analogue of the Gearhart-

Pruss theorem holds for arbitrary Banach spaces, so that the critical and non-analytic

growth bounds coincide.

In Chapter 4 we study the behaviour of ζ(f) under convolutions. Theorem 4.1.6 is

an analogue of Blake’s extension [10, Theorem 6.5.7] of the Weis-Wrobel result [50] to

functions. In Section 4.3, the fractional non-analytic growth bounds ζα(f), α > 0 of f are

introduced and estimates for these are obtained in terms of the uniform growth bound and

certain abscissas of holomorphy. In particular, we obtain estimates for the non-analytic

growth bounds of the Weyl and the Riemann-Liouville fractional integrals of exponentially

bounded measurable functions (Corollary 4.3.2 and Theorem 4.3.3). Section 4.4 is devoted

to the study of the fractional growth bounds of C0-semigroups. In Theorem 4.4.2 we show

that for a strongly continuous semigroup T all the growth bounds wα(T), α > 0 coincide

with the spectral bound s(A) of the generator A provided s∞0 (A) = −∞. Convexity of the

function α 7→ ζα(f) is studied in Section 4.5

We obtain characterisations of the non-analytic growth bound of a strongly continuous

semigroup in terms of some properties of the resolvent of the generator, in particular Fourier

multiplier properties, in Section 5.1. In Section 5.2 the effect on ζ(T) due to perturbations

of the generator of a semigroup T is studied. Section 5.3 brings out the connection between

ζ(T) and the existence of unique solutions of some inhomogeneous Cauchy problems.

In Chapter 6 we study relationships between sun-reflexivity and the existence of ap-

proximations of the identity on a Banach space. We prove, in particular, the existence of

Banach spaces admitting no strongly continuous semigroups with respect to which they are

sun-reflexive.

7

Chapter 2

Preliminaries

2.1 Notation

2.1.1 Sets

The symbols N,Z,R,C shall denote the natural numbers, integers, the real numbers and

the complex numbers respectively. The half-line [0,∞) will be denoted by R+ and the open

half-plane λ ∈ C : Reλ > 0 by C+. In general, for w ∈ R, we define the open half plane

Hw by

Hw = λ ∈ C : Reλ > w.

Further, for b ≥ 0, we define

Qw,b = λ ∈ C : Reλ ≥ w, | Imλ| ≥ b,Qow,b = λ ∈ C : Reλ > w, | Imλ| > b.

For b > 0, Qw,b is a pair of closed quadrants and Qow,b is its interior.

For θ > 0, we shall denote the sector of the complex plane of angle θ, containing the

positive reals by

Σθ = λ ∈ C : | arg λ| < θ.

2.1.2 Banach spaces and operators

Throughout this thesis, X shall be a complex Banach space and X∗ its dual. The Banach

algebra of bounded linear operators on X shall be denoted by L(X). For an unbounded

linear operator A on X, D(A),Ran(A), and Ker(A) shall denote the domain, range and

kernel of A respectively.

If B is a linear operator on X with domain D(B) and Y is a subspace of X containing

8

D(B) then the part of B in Y is the operator BY defined by

D(BY) := y ∈ D(B) : B(y) ∈ Y;BY(y) := By, y ∈ D(BY).

2.1.3 Function spaces

For 1 ≤ p <∞, let Lp(R,X) be the space of all Bochner measurable functions f : R −→ X

such that

‖f‖p :=

(∫

R‖f(t)‖p dt

) 1p

<∞.

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

‖f‖∞ := ess supt∈R ‖f(t)‖ <∞.

The conjugate index for p, 1 ≤ p <∞, shall be denoted by p′ so that 1p + 1

p′ = 1.

The space of locally integrable functions L1loc(R,X) is given by

L1loc(R,X) =

f : R −→ X such that f is Bochner measurable and

for every compact K ⊂ R,∫

K‖f(t)‖ dt <∞

.

We denote by C(R,X) the vector space of all continuous functions f : R −→ X. For

k ∈ N, Ck(R,X) will be the space of all k-times differentiable functions with continuous kth

derivative and C∞(R,X) :=⋂∞k=1C

k(R,X). BUC(R,X) will be the space of all bounded,

uniformly continuous functions defined on R and taking values in X.

Cc(R,X) and C∞c (R,X) shall denote the space of all functions with compact support in

C(R,X) and C∞(R,X), respectively. The space of functions in C(R,X) vanishing at infinity

will be denoted by C0(R,X). Further, S(R,X) shall denote the Schwartz space of functions

in C∞(R,X) which are rapidly decreasing. Then C∞c (R,X) ⊂ S(R,X) ⊂ Lp(R,X) and

C∞c (R,X) is dense in Lp(R,X) for 1 ≤ p <∞.If X = C, then we shall write S(R) in place of S(R,C), C∞(R) instead of C∞(R,X) and

so on.

A function f : R+ → X may be considered to be defined on the whole of R by setting

f = 0 on (−∞, 0). We shall denote by Lp(R+,X) the subspace of Lp(R,X) consisting

of functions which take the value 0 on (−∞, 0). C(R+,X) shall denote the space of all

continuous functions f : R+ → X.

9

2.2 Vector-valued functions

2.2.1 Exponential growth bound

The exponential growth bound of f : R+ −→ X is given by

ω0(f) = infw ∈ R : sup

t≥0‖e−wtf(t)‖ <∞

.

In this, and other similar definitions throughout the thesis, we allow the values ∞ and −∞according to the usual conventions. We say that f is exponentially bounded if ω0(f) <∞.

A function g : Σθ −→ X is said to be exponentially bounded if there exist constants M,w

such that ‖g(z)‖ ≤ Mew|z| (z ∈ Σθ). The restriction of g to (0,∞) may be exponentially

bounded even if g is not exponentially bounded on Σθ. However, by ω0(g) we shall always

mean the exponential growth bound of the restriction of g to R+ with g(0) = 0.

2.2.2 The Laplace transform

For f ∈ L1loc(R+,X) we define the abscissas of absolute convergence and convergence of the

Laplace transform of f [2, Section 1.4] by :

abs(‖f‖) = infω ∈ R :

∫ ∞

0e−ωt‖f(t)‖ dt <∞

;

abs(f) = inf

Reλ : limτ→∞

∫ τ

0e−λtf(t) dt exists

.

It is clear that

abs(f) ≤ abs(‖f‖) ≤ ω0(f).

We say that f is Laplace transformable if abs(f) <∞ and define the Laplace transform

of f by f where

f(λ) :=

∫ ∞

0e−λtf(t) dt := lim

τ→∞

∫ τ

0e−λtf(t) dt

whenever this limit exists. If

∫ ∞

0e−λtf(t) dt exists as a Bochner integral, then by the dom-

inated convergence theorem, it agrees with the definition above. We now record some well

known facts about the Laplace transform of a locally integrable function f with abs(f) <∞.For the proof of these we refer to [2, Sections 1.4 and 1.5]. :

1. The Laplace integral f(λ) converges if Reλ > abs(f) and diverges if Reλ < abs(f)

[2, Proposition 1.4.1].

2. abs(f) = ω0(F − F∞), where F (t) =∫ t

0 f(s) ds, F∞ = limt→∞ F (t) if the limit exists

and F∞ = 0 otherwise [2, Theorem 1.4.3].

10

3. If Reλ > max(abs(f), 0), then

F (λ) =f(λ)

λ.

[2, Corollary 1.6.5].

4. λ 7→ f(λ) defines a holomorphic function from Habs(f) into X and for n ∈ N ∪ 0,Reλ > abs(f),

f (n)(λ) =

∫ ∞

0e−λt(−t)nf(t) dt.

[2, Theorem 1.5.1].

5. If abs(‖f‖) <∞ then f is bounded on Hw whenever w > abs(‖f‖); indeed,

supλ∈Hw ‖f(λ)‖ ≤∫∞

0 e−wt‖f(t)‖ dt <∞.

In general, f may have a holomorphic extension to a bigger region than Habs(f). We shall

denote the extension of f by the same symbol. We now define the largest such region with

which we shall work :

D(f) := λ = α+ iη : f has a holomorphic extension to

Habs(f) ∪ β + is : α− ε < β, |s− η| < ε for some ε > 0

Then D(f) is a connected open set which is a union of horizontal line-segments extending

infinitely to the right, and f has a unique holomorphic extension (also denoted by f) to

D(f). Moreover, D(f) is the largest such set with these properties. Next we define some

abscissas of holomorphy and boundedness:

hol(f) = inf

ω ∈ R : Hω ⊂ D(f)

; (2.1)

hol0(f) = inf

ω ∈ R : Hω ⊂ D(f) and sup

Reλ>ω‖f(λ)‖ <∞

; (2.2)

hol∞(f) = inf

ω ∈ R : Qω,b ⊂ D(f) for some b ≥ 0

; (2.3)

hol∞0 (f) = inf

ω ∈ R : Qω,b ⊂ D(f) and

supλ∈Qω,b

‖f(λ)‖ <∞ for some b ≥ 0

; (2.4)

holn(f) = inf

ω ∈ R : Hω ⊂ D(f) and sup

Reλ>ω

‖f(λ)‖(1 + |λ|)n <∞

; (2.5)

hol∞n (f) = inf

ω ∈ R : Qω,b ⊂ D(f) and

supλ∈Qω,b

‖f(λ)‖(1 + |λ|)n <∞ for some b ≥ 0

. (2.6)

11

for n ∈ N.

Thus, hol(f) and hol0(f) are the abscissas of holomorphy and boundedness of f [2,

Section 1.5] and hol∞(f) and hol∞0 (f) are analogues which ignore horizontal strips in C.

For n ∈ N, holn(f) gives the minimal abscissa for the half-plane where f grows along vertical

lines not faster than the n-th power and hol∞n (f) is the corresponding analogue ignoring

horizontal strips.

It is clear from the definitions and the properties of the Laplace transform above, that

for a Laplace transformable function f : R+ → X,

hol∞(f) ≤ hol(f) ≤ abs(f); (2.7)

hol(f) ≤ hol0(f) ≤ abs(‖f‖); (2.8)

hol∞(f) ≤ hol∞0 (f) ≤ hol0(f); (2.9)

hol∞(f) ≤ hol∞n (f) ≤ hol∞0 (f). (2.10)

For an exponentially bounded function f we have

abs(f) ≤ hol0(f) (2.11)

[5]. However, (2.11) is false for some Laplace transformable functions [12]. If f is exponen-

tially bounded, then it is clear that for w > ω0(f), there is a constant M such that

‖f(λ)‖ ≤ M

(Reλ− w)(Reλ > w). (2.12)

For a Laplace transformable function the following holds:

Lemma 2.2.1. If w > abs(f), and 0 < θ < π2 , then f is bounded on Hw ∩ Σθ. Further,

hol0(f) = max(

hol∞0 (f),hol(f))

;

holn(f) = max(

hol∞n (f),hol(f)), n ∈ N.

Proof. Let 0 < θ < π2 and w > max(0, abs(f)). Since f(λ) = λF (λ) and abs(f) = ω0(F −

F∞), (2.12) applied to F − F∞ yields a constant M such that

‖f(λ)‖ ≤ M |λ|Reλ− w + ‖F∞‖.

Choosing ε > 0 such that 0 < ε < cos θ, we have

‖f(λ)‖ ≤ K, for all λ ∈ Hwε∩ Σθ,

where K = (cos θ − ε)−1 + ‖F∞‖ is a constant. Thus, the first statement follows on observ-

ing that f is bounded on compact subsets of C.

12

From the inequalities (2.8) and (2.9) it follows that

hol0(f) ≥ max(hol∞0 (f),hol(f)).

Suppose a ∈ R is such that max(hol∞0 (f), hol(f)) < a. Then f has a holomorphic extension

to Ha which is bounded on Qa,b for some b > 0. Let

S0 = λ ∈ C : a ≤ Reλ ≤ ω, | Imλ| ≤ b,

where ω > max(0, abs(f)). Then f is holomorphic on the compact set S0 and therefore

bounded. That f is bounded on Reλ > ω, | Imλ| ≤ b follows from the first part. There-

fore, supλ∈S0∪Qa,b ‖f(λ)‖ <∞. Thus, hol0(f) ≤ a. Hence, hol0(f) ≤ max(hol∞0 (f),hol(f)).

The corresponding result for holn(f) follows similarly.

We note here that for a Laplace transformable function f , supλ∈Qa,b ‖f(λ)‖ <∞ actually

implies, on using Lemma 2.2.1, that

supλ∈Hw∪Qa,b

‖f(λ)‖ <∞,

where w ∈ R is sufficiently large. We shall often use this fact without mention.

2.2.3 Convolutions and the Fourier transform

Given f : R→ X and g : R→ C, the convolution g ∗ f is defined by

(g ∗ f)(t) =

∫ ∞

−∞g(t− s)f(s) ds

whenever this integral exists as a Bochner integral. From Young’s inequality [2, Proposition

1.3.2] we have that if g ∈ Lp(R) and f ∈ Lq(R,X), then g ∗ f ∈ Lr(R,X) and

‖g ∗ f‖r ≤ ‖g‖p‖f‖q,

where 1 ≤ p, q, r ≤ ∞ and 1/p+ 1/q = 1 + 1/r.

By a mollifier we shall mean a sequence (gn)n∈N in L1(R) of the following form: g1 ∈L1(R) satisfies

∫

Rg1(t) dt = 1, and gn ∈ L1(R) is given by gn(t) = ng1(nt), for all t ∈ R and

n ∈ N. It is often convenient to choose such a sequence (gn) in C∞c (R) with gn ≥ 0 for all

n ∈ N. For such a mollifier (gn),

limn→∞

‖gn ∗ f − f‖p = 0,

for f ∈ Lp(R,X), 1 ≤ p <∞.

13

For f ∈ L1(R,X), the Fourier transform Ff , is defined by

(Ff)(s) :=

∫ ∞

−∞e−istf(t) dt,

and the conjugate Fourier transform, Ff is given by

(Ff)(s) :=

∫ ∞

−∞eistf(t) dt = (Ff)(−s).

We quote here some of the properties of the vector-valued Fourier transform that shall be

used frequently in the sequel. We refer to [2, Section 1.8] for the details. For f ∈ L1(R,X)

and g ∈ L1(R) we have

1. F(g ∗ f)(s) = (Fg)(s)(Ff)(s).

2.

∫ ∞

−∞g(t)(Ff)(t) dt =

∫ ∞

−∞(Fg)(t)f(t) dt.

3. (Riemann-Lebesgue Lemma) Ff ∈ C0(R,X).

4. (Inversion Theorem) If Ff ∈ L1(R,X), then f =1

2πF(Ff) a.e.

5. If X is a Hilbert space then we have Plancherel’s theorem that 1√2πF : L2(R,X) →

L2(R,X) is a unitary operator.

The next result makes use of the Riemann-Lebesgue Lemma to describe the behaviour

of the Laplace transform of an exponentially bounded function f along vertical lines to the

right of hol∞0 (f).

Lemma 2.2.2. Let f : R+ → X be measurable and exponentially bounded. Then

lims→±∞

f(α+ is) = 0,

for all α > hol∞0 (f).

Proof. Let α > β > hol∞0 (f). Then there exists b > 0 such that Qβ,b ⊂ D(f). Let (sn) be

any sequence such that sn ≥ b for all n and sn → ∞ as n → ∞. Let gn(z) = f(z + isn),

z ∈ Qβ,0. For Re z > ω0(f), let h(t) = e−ztf(t), t ∈ R+. Then h ∈ L1(R+,X) and it follows

from the Riemann-Lebesgue Lemma that

lims→∞

Fh(s) = lims→∞

f(z + is) = 0.

Thus, if Re z > ω0(f), then gn(z) −→ 0 as n → ∞. Now (gn) is uniformly bounded on

Re z > β, Im z > 0. By Vitali’s Theorem [2, Theorem A.5], limn→∞ gn(z) = 0 for all

z ∈ Qoβ,0. In particular,

0 = limn→∞

gn(α) = limn→∞

f(α+ isn).

Since (sn) is arbitrary, we conclude that lims→∞ f(α+ is) = 0. Similarly, we can show that

as s→ −∞, f(α+ is)→ 0.

14

2.3 Operator-valued functions

Let T : R+−→L(X). We shall say that T is strongly continuous if it is continuous in the

strong operator topology, that is, the map t 7→ T(t)x from R+ to X is continuous for each

x ∈ X. If T is strongly continuous, then by the Uniform Boundedness Principle, it is also

locally bounded. However, it is not necessarily Bochner measurable; indeed, T is Bochner

measurable if and only if it is almost separably-valued in the norm topology, by Pettis’s

Theorem [2, Theorem 1.1.1 ]. Clearly, if T is uniformly continuous (that is, continuous

with respect to the norm topology) then it is measurable and also strongly continuous and

strong continuity of T implies continuity in the weak operator topology. On the other hand,

if Ω is an open set in C, then a function S : Ω−→L(X) is holomorphic if and only if it is

holomorphic in the weak operator topology (that is, 〈S(·)x, x∗〉 is holomorphic for all x ∈ X

and x∗ ∈ X∗ [2, Proposition A.3 ]).

In what follows, to say S : R+ → L(X) converges uniformly, strongly and weakly as

t → ∞ will refer to convergence in, respectively, the norm, strong operator and weak

operator topology.

2.3.1 Laplace and Fourier transforms for operator-valued functions

Next, we recall the formulation of the definitions and results of Section 2.2 when f is replaced

by a strongly continuous operator-valued function T : R+ → L(X). For such a function T,

let

∫ t

0e−λsT(s) ds denote the bounded operator

x 7→∫ t

0e−λsT(s)x ds.

Then

abs(T) := inf

Reλ :

∫ t

0e−λsT(s) ds converges strongly as t→∞

= supabs(T(·)x) : x ∈ X= infω0(S− S0) : S0 ∈ L(X)

where S(t)x =

∫ t

0T(s)x ds. We refer the reader to [2, Section 1.4] for the proof of the

above equalities as well as the other results that follow. Whenever Reλ > abs(T), the limit

limt→∞∫ t

0 e−λsT(s) ds exists in operator norm. If T : R+ → L(X) is strongly continuous

and abs(T) <∞, the Laplace integral of T is defined by

T(λ) :=

∫ ∞

0e−λsT(s) ds := lim

t→∞

∫ t

0e−λsT(s) ds (Reλ > abs(T)).

15

Then T : Habs(T) → L(X) is holomorphic and all the results mentioned in sub-section

2.2.2 for f hold for T also, with hol(T),hol0(T), hol∞(T),hol∞0 (T),holn(T) and hol∞n (T)

defined as in (2.1), (2.2), (2.3), (2.4), (2.5) and (2.6) respectively.

The Fourier transform of an operator valued function S : R → L(X), is defined in a

similar manner.

2.3.2 C0-semigroups

By a C0-semigroup defined on the Banach space X we shall mean a function T : R+ → L(X)

satisfying the following properties:

1. T is strongly continuous;

2. T(0) = I;

3. T(t+ s) = T(t)T(s) for t, s ∈ R+.

For background information on C0-semigroups we refer to the books [20], [42], [17] and [2].

The infinitesimal generator A of T is given by

D(A) =

x ∈ X : lim

t↓0T(t)x− x

texists

Ax = limt↓0

T(t)x− xt

(x ∈ D(A)).

A is a closed, densely defined operator and⋂∞n=1D(An) = X. The resolvent and spec-

trum of A shall be denoted by ρ(A) and σ(A). For λ ∈ ρ(A),R(λ,A) = (λ −A)−1 shall

denote the resolvent operator. From the definition of the resolvent, it is easy to deduce the

very useful resolvent identity :

R(λ,A)−R(µ,A) = (µ− λ)R(λ,A)R(µ,A) (λ, µ ∈ ρ(A)).

Any C0-semigroup is exponentially bounded [20, Proposition 5.5]. For Reλ > ω0(T),

R(λ,A)x = T(λ)x (x ∈ X).

In fact, C0-semigroups are exactly those strongly continuous operator-valued functions

whose Laplace transforms are resolvents [2, Theorem 3.1.7].

16

2.3.3 Norm continuity and the critical growth bound

Let T : R+ → L(X) be strongly continuous and exponentially bounded. A growth bound

δ(T) measuring the absence of norm continuity of T has been introduced in [10], [11]:

δ(T) := infω : there exists a norm continuous function T1 : R+ → L(X)

such that ω0(T−T1) < ω= infω : there exists an infinitely differentiable function T1 : R+ → L(X)

such that ω0(T−T1) < ω.

(2.13)

This is equal to the growth bound of local variation of T, which is defined to be ω0(fT)

where

fT(t) = lim suph↓0

‖T(t+ h)−T(t)‖.

[10, Theorem 2.3.7] (see also [11]). It is immediate from the definition that if T is norm

continuous on R+ or norm continuous on (α,∞) for some α > 0 then δ(T) = −∞.A C0-semigroup T : R+ → L(X) is said to be eventually norm continuous [20, Definition

II.4.17]) if there exists α ∈ R+ such that T : (α,∞)→ L(X) is norm continuous. If α may

be chosen to be 0 then T is called immediately norm continuous. If A is the generator of

an immediately norm continuous semigroup T then

lims→±∞

‖R(a+ is,A)‖ = 0 (2.14)

for all a > ω0(T) ([20, Theorem II.4.18 ]). This condition is sufficient for immediate norm

continuity if X is a Hilbert space [20, Theorem II.4.20],[52] or if T is a positive semigroup

defined on Lp(R), 1 < p <∞ [23]. It is not known whether (2.14) implies immediate norm

continuity for arbitrary C0-semigroups or not.

The semigroup T is said to be eventually compact (respectively, immediately compact)

if there is an α > 0 such that T(α) is compact (respectively, T(t) is compact for all t > 0).

T is eventually compact if and only T is eventually norm continuous and its generator has

compact resolvent [20, Theorem II.4.29 ].

T is called eventually differentiable [20, Definition II.4.13]), [42, Section 2.4] if there

exists an α ≥ 0 such that the map t 7→ T(t)x is differentiable on (α,∞) for every x ∈ X.

A C0-semigroup T is called analytic of angle θ ∈ (0, π2 ) ([20, Definition II.4.5]) if T has

a holomorphic extension to Σθ (also denoted by T) satisfying

• T(z1 + z2) = T(z1)T(z2) for all z1, z2 ∈ Σθ;

• limΣβ3z→0 T(z)x = x for all x ∈ X and 0 < β < θ.

17

All the classes of C0-semigroups mentioned above satisfy δ(T) = −∞. In [11, Definition

3.4] a class of C0-semigroups has been introduced which includes all C0-semigroups with

finite growth bound falling in any of the above classes: T is said to be asymptotically norm

continuous if δ(T) < ω0(T). Such a semigroup has been called norm continuous at infinity

in [37].

With Γt = λ ∈ C : eδ(T)t < |λ| the following version of the spectral mapping Theorem

holds for asymptotically norm continuous semigroups [10, Theorem 4.4.1], [11, Theorem

3.6].

σ (T(t)) ∩ Γt = etσ(A) ∩ Γt, (t > 0). (2.15)

In particular, ω0(T) = supReλ : λ ∈ σ(A). Since δ(T) = −∞ for eventually norm

continuous semigroups, (2.15) re-asserts the well known fact [20, Theorem IV.3.10 ] that

the spectral mapping theorem

σ(T(t)) \ 0 = etσ(A) (t > 0) (2.16)

holds for such semigroups.

For general C0-semigroups, (2.16) fails to hold. However, in [38], the critical spectrum

σcrit(T(t)) for a C0-semigroup T has been introduced, which yields a spectral mapping

theorem of the form

σ(T(t)) \ 0 = etσ(A) ∪ σcrit(T(t)) \ 0 (t > 0),

for all C0- semigroups. We recall the definition of the critical spectrum of a C0-semigroup

T [38, Definition 2.3]. Let `∞(X) be the Banach space of all bounded sequences in X,

endowed with the sup-norm ‖(xn)n∈N‖ := supn∈N ‖xn‖ and let T be the extension of T to

`∞(X), given by

T(t) ((xn)n∈N) := (T(t)xn)n∈N (t ≥ 0).

Then the space of strong continuity `∞T (X), of T given by

`∞T (X) :=

(xn)n∈N : limt→0

supn∈N‖T(t)xn − xn‖ = 0

is a closed and T-invariant subspace of `∞(X). On the quotient space X =`∞(X)

`∞T (X)define

the semigroup of bounded operators T by

T(t)x := (T(t)xn)n∈N + `∞T (X), for x := (xn)n∈N + `∞T (X).

Then the critical spectrum of the C0-semigroup T is defined by

σcrit(T(t)) := σ(T(t)) (t > 0),

18

and the critical growth bound ωcrit(T) of T is defined as

ωcrit(T) = ω0(T).

It has been shown in [38, Proposition 4.6] that the growth bound of local variation and the

critical growth bound coincide for a C0-semigroup, that is

δ(T) = ωcrit(T).

Due to this equality, henceforth we shall call the growth bound of local variation, δ(T), of

any exponentially bounded operator-valued function T the critical growth bound of T and

use the equivalent characterisations of this bound without mention.

2.3.4 Adjoint semigroups

We recall some standard definitions and facts concerning the adjoint of a C0-semigroup.

The details may be found in [39, Chapter 1, Chapter 2].

Let A be the generator of the C0-semigroup T. The adjoint semigroup T∗ on X∗,

given by T∗(t) = T(t)∗ is weak∗-continuous with weak∗-generator A∗, but is not necessarily

strongly continuous.

The semigroup dual of X with respect to T, denoted by X is defined as the linear

subspace of X∗ on which the adjoint semigroup T∗ acts in a strongly continuous way; i.e.

X = x∗ ∈ X∗ : limt ↓0‖ T∗(t)x∗ − x∗ ‖= 0.

Then X is a closed, weak∗-dense, T∗(t)-invariant linear subspace of X∗ and X = D(A∗).

We denote by T(t) the restriction of T∗(t) to X. Then (T(t))t≥0 defines a C0-semigroup

on X whose generator A is the part of A∗ in X.

Starting with the C0-semigroup T, the duality construction can be repeated. We

define T∗ to be the adjoint of T and write X for (X). T and A are defined

analogously.

The norm ‖ · ‖′ defined on X by

‖x‖′ := supx∈BX

∣∣〈x, x〉∣∣,

where BX is the closed unit ball of X, is an equivalent norm. In fact,

‖x‖′ ≤ ‖x‖ ≤M‖x‖′,

with M = lim supt↓0 ‖T(t)‖. Define the map j : X −→ X∗ by

〈jx, x〉 := 〈x, x〉.

19

Then jX ⊂ X; in fact j is an embedding, with M−1 ≤ ‖j‖ ≤ 1. We can, therefore,

identify X isomorphically with the closed subspace jX of X. Thus, T(t) is an extension

of T(t),A is an extension of A and D(A) = D(A) ∩X.

X is said to be -reflexive or sun-reflexive with respect to T if and only if j(X) = X.

20

Chapter 3

A non-analytic growth bound for

Laplace transforms and semigroups

of operators

Analytic C0-semigroups play an extremely important role in the theory of evolution equa-

tions. In this chapter, we study a growth bound which measures the non-analyticity of C0-

semigroups and more generally, of any vector-valued, exponentially bounded measurable

function. It turns out that the non-analytic behaviour of a vector-valued, exponentially

bounded measurable function is closely related to the integrability of some derivative of the

Laplace transform of the function along vertical lines.

In [10, Theorem 2.3.3] it is shown that integrability along a vertical line of some deriva-

tive of the Laplace transform of a strongly continuous and exponentially bounded function

T : R+ → L(X) is sufficient for norm continuity of T for t > 0. This idea has been used

to define a growth bound for T which measures the non-integrability of the Laplace trans-

form of T. We recall the relevant definitions and results from [10, Section 2.4.2], where

these are stated in the context of operator-valued functions T, but remain valid for general

vector-valued functions also. We state these for the general case.

A Bochner measurable function f : R+ → X is said to have an L1-Laplace transform,

(see [10]) if there exist r > 0, N ∈ N and C > 0 such that f has a holomorphic extension

to Q0,r and

supω≥0

∫

|s|≥r

∥∥∥f (N)(ω + is)∥∥∥ ds ≤ C.

For an exponentially bounded, Bochner measurable function f : R+ → X, the growth

21

bound for non-integrability of f is defined by:

ζ(f) = infw ∈ R : there exists f1 : R+ → X such that e−w·f1(·) has

L1-Laplace transform and ω0(f − f1) ≤ w.

(3.1)

The next theorem gives a very useful property of ζ(f).

Theorem 3.0.1. ([10, Proposition 2.4.6]) Let f : R+ → X be Bochner measurable. Suppose

that ζ(f) < 0, so that there exist f1, f2 : R+ → X and positive N, r and ε such that

f = f1 + f2, ω0(f2) < −ε and

supw≥−ε

∫

|s|>r

∥∥f1(N)

(w + is)∥∥ ds <∞.

Let w > ω0(f) and Γ be the path consisting of line segments joining −ε− ir, w− ir, w+ ir

and −ε+ ir in that order, and define

g1(t) :=1

2πi

∫

Γeλtf1(λ) dλ,

g2(t) := f(t)− g1(t).

Then ω0(g2) ≤ −ε.

3.1 Introducing the non-analytic growth bound

The critical growth bound of a strongly measurable, exponentially bounded function T :

R+ → L(X) measures the growth bound of T modulo L(X)-valued functions on R+ that

are norm continuous for t > 0 and equivalently, modulo L(X)-valued functions on R+ which

are infinitely differentiable. A natural question that arises in this context is whether or not

the growth bound of T modulo holomorphic, exponentially bounded L(X)-valued functions

is also equal to the critical growth bound of T. This motivates us to define a new growth

bound η(T) for T. As we shall see later, this definition extends in a useful way to the case

of arbitrary vector-valued functions, unlike the definition of the critical growth bound. We

make the definition in the most general setting useful for us.

Definition 3.1.1. Let f : R+ → X be Laplace transformable. We define

η(f) := infw ∈ R : there exist θ > 0 and an exponentially bounded, holomorphic

function g : Σθ → X such that ω0(f − g) < w.

It is clear from the definition that the growth bound η(f) measures how well the function

f can be approximated by exponentially bounded functions that are holomorphic on some

sector Σθ. However, there are other, relatively smaller classes of approximating functions

which can be considered, as will be shown in the next proposition. We shall need the

following definition:

22

Definition 3.1.2. Let f : R+ → X be a Laplace transformable function. If Qα,b ⊂ D(f)

we define for t ≥ 0,

fα,b(t) :=1

2πi

∫

Γα,b

etλf(λ)dλ,

where Γα,b is any path in D(f) from α− ib to α+ ib.

We immediately obtain, from the above definition, an estimate for the growth bound of

the functions fα,b.

Lemma 3.1.3. Let f : R+ → X be Laplace transformable and α ∈ R, b ≥ 0 be such that

Qα,b ⊂ D(f). Then,

ω0 (fα,b) ≤ max(α,hol(f)

), and

fα,b(µ) =1

2πi

∫

Γα,b

f(λ)

µ− λ dλ,

where Γα,b is any path in D(f) from α − ib to α + ib and Reµ lies to the right of Γα,b.

Further, if α1, b1 ∈ R, b1 ≥ 0 satisfy Qα1,b1 ⊂ D(f) then

abs(fα,b − fα1,b1

)≤ ω0

(fα,b − fα1,b1

)≤ max(α, α1).

Proof. Let Γ = [α− ib, γ− ib]∪ [γ− ib, γ+ ib]∪ [γ+ ib, α+ ib], where γ > max(α,hol(f)

).

Then,

fα,b(t) =1

2πi

∫

Γeλtf(λ) dλ,

and straightforward calculations show that

∥∥fα,b(t)∥∥ ≤ Cγeγt

for some constant Cγ . Consequently, ω0 (fα,b) ≤ γ, for all γ > max(α,hol((f)

). Therefore,

it follows that ω0(fα,b) ≤ max(α,hol((f)

). The claim concerning fα,b follows directly

from the definition of the Laplace transform of a function, on interchanging the order of

integration. We note that for α1, b1 as in the hypothesis,

fα,b(t)− fα1,b1(t) =1

2πi

∫

Γ2

eλtf(λ)dλ,

where Γ2 consists of two paths in λ ∈ D(f) : Reλ ≤ max(α, α1) joining α± ib to α1± ib1.

Therefore, ∥∥fα,b(t)− fα1,b1(t)∥∥ ≤ C exp(tmax(α, α1))

for some constant C, so that ω0(fα,b − fα1,b1) ≤ max(α, α1).

23

Observe that in the terminology introduced in Definition 3.1.2, Theorem 3.0.1 shows

that if f, f1 : R+ → X are exponentially bounded, Bochner measurable functions with

ω0(f − f1) < −ε for some ε > 0 and the function t 7→ e−εtf1(t) has L1-Laplace transform,

then ω0 (f − (f1)−ε,r) ≤ −ε, where r is as in the statement of the Theorem. The proof of

implication (2)⇒(4) in the following proposition depends mainly on this fact.

Proposition 3.1.4. Let f : R+ → X be Laplace transformable and let ω ∈ C. The following

are equivalent:

1. There exist θ > 0 and an exponentially bounded, holomorphic function g : Σθ → Csuch that ω0(f − g) < ω.

2. There is an exponentially bounded, measurable function g : R+ → X such that, for

each α < ω, there exists b ≥ 0 such that

(a) Qα,b ⊂ D(g);

(b) supλ∈Qα,b ‖g(λ)‖ <∞;

(c) supγ≥α∫|s|≥b

∥∥g(N)(γ + is)∥∥ ds <∞ (N = 1, 2...);

(d) ω0(f − g) < ω.

3. There exist an exponentially bounded, measurable function g : R+ → X and α < ω,

b ≥ 0, N ∈ N such that (2a), (2c) and (2d) hold.

4. There exist α < ω and b ≥ 0 such that

(a) Qα,b ⊂ D(f);

(b) supλ∈Qα,b ‖f(λ)‖ <∞;

(c) ω0(f − fα,b) < ω.

5. There exist α < ω and b ≥ 0 such that (4a) and (4c) hold.

6. There is an exponentially bounded, entire function g : C→ X such that ω0(f −g) < ω

and ω0(g) ≤ max(ω,hol(f)

).

Proof. (1) =⇒ (2): Suppose that g : Σθ → X is holomorphic and there exists w′ ∈ R such

that ‖g(z)‖ ≤ Mew′|z| (z ∈ Σθ). We may assume without loss of generality, that ω < w′.

By considering a smaller sector if required, we may also assume that 0 < θ ≤ π2 . Then

w′ + Σθ+(π/2) ⊂ D(g) and

‖g(λ)‖ ≤ C

|λ− w′|

24

for some constant C [2, Theorem 2.6.1]. In particular, (2a) and (2b) hold if b > (ω −α) cot(θ/2). Moreover, Cauchy’s integral formula for derivatives gives

∥∥∥g(N)(γ + is)∥∥∥ =

∥∥∥∥N !

2πi

∫

|λ−(γ+is)|=εs

g(λ)

(λ− (γ + is))N+1dλ

∥∥∥∥

≤ N !

εN |s|NC

|s|(1− ε)

whenever γ ≥ α, |s| ≥ b > (ω − α) cot(θ/2), N ∈ N, 0 < ε < 1 and ε is sufficiently small so

that the disc of radius εb and centre α+ ib is contained in ω + Σθ+π/2. This implies (2c).

(2) =⇒ (3),(4) =⇒ (5): This is trivial.

(2) =⇒ (4),(3) =⇒ (5): Suppose that g : R+ → C is exponentially bounded and

measurable, and α < ω, b ≥ 0 and N ≥ 0 are such that (2a), (2c) and (2d) hold. Take β

such that max(α, ω(f−g)) < β < ω. Then Qβ,b ⊂ D(f − g)∩D(g), and hence Qβ,b ⊂ D(f).

If (2b) holds then f is bounded on Qβ,b. By Lemma 3.1.3,

ω0

(fβ,b − gβ,b

)= ω0

((f − g)β,b

)≤ β.

The assumptions on g imply that t 7→ e−βtg(t) has L1-Laplace transform and ω0(f−g) < β.

Therefore, from Theorem 3.0.1, it follows that ω0(f − gβ,b) ≤ β. Since

ω0(f − fβ,b) ≤ max (ω0(f − gβ,b), ω0(gβ,b − fβ,b)) ,

it follows that ω0(f − fβ,b) ≤ β < ω.

(5) =⇒ (6): For z ∈ C, let

g(z) =1

2πi

∫

Γα,b

eλzf(λ)dλ.

Then g is entire and exponentially bounded, with g(t) = fα,b(t) (t ≥ 0). So (6) follows from

Lemma 3.1.3 and (4c).

(6) =⇒ (1) : This is trivial.

Remark 3.1.5. 1. If the conditions of Proposition 3.1.4 hold, then (4c) holds for every

α < ω and b ≥ 0 satisfying (4a). This follows from Lemma 3.1.3.

2. If (4a) holds then α ≥ hol∞(f); if (4b) also holds, then α ≥ hol∞0 (f). Conversely, if

α > hol∞(f) then there exists b ≥ 0 such that (4a) holds; if α > hol∞0 (f) and (4a)

holds, then (4b) holds.

3. Proposition 3.1.4 remains valid if ω0(f − g) is replaced by abs(‖f − g‖) in conditions

(1), (2), (3) and (6), and ω0(f − fα,b) is replaced by abs(‖f − fα,b‖) in conditions (4)

and (5).

25

4. Similarly, the equivalence of (1 ), (3 ), (5 ) and (6 ) remains valid if ω0(f−g) is replaced

by abs(f − g) throughout. In both these cases, the corresponding analogue of (4c)

holds for every α < ω and b ≥ 0.

From Proposition 3.1.4, we conclude

Corollary 3.1.6. The non-analytic growth bound and the growth bound for non-integrability

for any Laplace transformable function f : R+ → X coincide, that is, ζ(f) = η(f).

Given a Laplace transformable function f : R+ → X, we shall call ζ(f) the non-analytic

growth bound for f and use Definition 3.1.1 to describe this growth bound. It is clear from

the definition that ζ(f) <∞ if and only if f is exponentially bounded.

Analogous to the non-analytic growth bound of a function f , we may define the non-

analytic abscissa of convergence and the non-analytic abscissa of absolute convergence of f

in the following manner:

Definition 3.1.7. Let f : R+ → X be Laplace transformable. Let

ζ1(f) := infω ∈ R : there exists θ > 0 and an exponentially bounded,

holomorphic function g : Σθ → X, with abs(f − g) < ωκ(f) := infω ∈ R : there exists θ > 0 and an exponentially bounded,

holomorphic function g : Σθ → X, with abs(‖f − g‖) < ω

ζ1(f) is called the non-analytic abscissa of convergence of f while κ(f) is the non-analytic

abscissa of absolute convergence of f .

From Remark 3.1.5 we get equivalent characterisations for ζ1(f) and κ(f). It is imme-

diate from the definitions that

ζ1(f) ≤ κ(f) ≤ ζ(f) ≤ ω0(f). (3.2)

In the following Proposition, we note some basic properties of these bounds. (3.3) has

been proven, for exponentially bounded, operator-valued functions T : R+ → L(X), in [10,

Theorem 2.4.8], using the non-integrability growth bound description of ζ(T). The proof

we give here for a general vector-valued function f uses the characterisation of ζ(f) as the

non-analytic growth bound of f .

Proposition 3.1.8. Let f : R+ → C be Laplace transformable. Then

1. ζ(f) ≥ κ(f) ≥ hol∞0 (f) and κ(f) ≥ ζ1(f) ≥ hol∞(f);

26

2. Suppose ζ(f) <∞. Then

ω0(f) = max(ζ(f),hol(f)

); (3.3)

abs(‖f‖) = max(κ(f),hol(f)

); (3.4)

abs(f) = max(ζ1(f),hol(f)

). (3.5)

3. Let τ ≥ 0, α ∈ R and fτ (t) = f(t+ τ), g(t) = e−αtf(t) (t ≥ 0). Then

ζ(fτ ) = ζ(f), κ(fτ ) = κ(f), ζ1(fτ ) = ζ1(f),

ζ(g) = −α+ ζ(f), κ(g) = −α+ κ(f), ζ1(g) = −α+ ζ1(f).

Proof. (1 ): From Remark 3.1.5 (3 ), it follows that hol∞0 (f) ≤ κ(f). ζ1(f) ≥ hol∞(f) follows

from Remark 3.1.5 (4). The other two inequalities are clear.

(2 ): That max(ζ(f),hol(f)) ≤ ω0(f) is clear. Suppose max(ζ(f),hol(f)) < a. From

Proposition 3.1.4 (6), it follows that there is an exponentially bounded, entire function

g : C→ X such that ω0(f − g) < a and ω0(g) ≤ a. Since ω0(f) ≤ max(ω0(f − g), ω0(g)), it

follows that ω0(f) ≤ a. Thus, ω0(f) = max(ζ(f),hol(f)). The other equalities are similar.

(3 ): This follows from Proposition 3.1.4 (6).

The inequalities in Proposition 3.1.8, (1) can all be strict.

Corollary 3.1.9. 1. If hol0(f) < ω0(f), then hol∞0 (f) < ζ(f).

2. If hol0(f) < abs(‖f‖), then hol∞0 (f) < κ(f).

3. If hol0(f) < abs(f), then hol∞0 (f) < ζ1(f).

Proof. If hol0(f) < ω0(f), then from (2.8) and (3.3) it follows that ω0(f) = ζ(f). Therefore,

if hol∞0 (f) = ζ(f) this would imply hol0(f) < hol∞0 (f), contradicting Lemma 2.2.1. The

other implications follow similarly.

Examples 3.1.10. 1. Let f(t) = et sin et (t ≥ 0). Then ω0(f) = 1 = abs(|f |), abs(f) =

0 = hol0(f) and hol(f) = −∞, [2, Example 1.5.2]. Therefore, hol∞0 (f) = 0 and from

Proposition 3.1.8 it follows that ζ(f) = κ(f) = 1 while ζ1(f) = 0.

2. Bloch [12] has given an example of a Laplace transformable function f : R+ → Csuch that hol0(f) = −∞ and abs(f) = abs(|f |) = 0. Then hol∞0 (f) = hol(f) = −∞,and κ(f) = ζ1(f) = 0 by Proposition 3.1.8 (2). However, a function f satisfying

abs(f) > hol0(f) cannot be exponentially bounded [2, Theorem 4.4.13]. Therefore,

we have ζ(f) = ω0(f) =∞.

27

3. An example of an exponentially bounded function f : R+ → C with hol0(f) < ω0(f),

and therefore with hol∞0 (f) < ζ(f), can be obtained by taking f as in the proof of [4,

Proposition 2.1] with the choices km = e2m and qm = 2m. Explicitly,

f(t) =

e−mΦe2m+1(t− 2m) if 2m ≤ t < 2m+ 1,m = 1, 2, . . . ;

0 otherwise ,

where, for k > 2, 0 ≤ t < 1, Φk is given by

Φk(t) =

2

1− 2tif 0 ≤ t ≤ k − 2

2k − 2or

k

2k − 2≤ t < 1;

0 otherwise.

Then |Φk(t)| ≤ 2(k − 1) and

∣∣∣∣∫ 1

0Φk(t)e

−λt dt

∣∣∣∣ ≤ CeRe λ2k−2 (Reλ > 0)

[4, Proof of Proposition 2.1]. From these properties of Φk it is easy to deduce that

ω0(f) = 1/2 and hol0(f) ≤ 0, so that ζ(f) = 1/2 and hol∞0 (f) ≤ 0.

Also, the function g(t) = e2t + f(t) gives an example for which the strict inequality

hol∞0 (g) < ζ(g) < ω0(g) holds.

The bounds, ζ(f), κ(f) and ζ1(f) may be thought of as the non-analytic analogues of

the bounds ω0(f), abs(‖f‖) and abs(f), respectively, of a Laplace transformable function f .

(1) of the next theorem is an analogue of (2) on page 10, relating the non-analytic abscissa

of convergence of the function f to the non-analytic growth bound of its primitive F in the

same way as the abscissa abs(f) is related to the growth bound ω0(F ) of F .

Theorem 3.1.11. Let f : R+ → X be Laplace transformable and F (t) =

∫ t

0f(s) ds. Then

1. ζ(F ) = ζ1(f);

2. hol∞0 (F ) = hol∞1 (f).

Proof. (1): Suppose first that ζ(F ) < 0. Then there exist F1, F2 : R+ → X satisfying

ω0(F2) < 0, F1 entire, and exponentially bounded with ω0(F1) ≤ max(0,hol(F )) and

F = F1 + F2. (3.6)

Let fi = F′i , i = 1, 2. Then abs(fi) = ω0(Fi − Fi,∞) where

Fi,∞ =

limt→∞ Fi(t) if the limit exists,

0 otherwise,

28

for i = 1, 2. Since ω0(F2) < 0, F2,∞ = 0, so that abs(f2) = ω0(F2) < 0. From (3.6) it follows

on differentiating, that f = f1 + f2, with f1 holomorphic and exponentially bounded on C.

Therefore, ζ1(f) ≤ 0.

Now suppose ζ(F ) < α. Then, writing G(t) = e−αtF (t) and using the first part we have

that

G′(t) = g1(t) + g2(t), (t ≥ 0)

where g1 is exponentially bounded in a sector and abs(g2) < 0. But

G′(t) = −αe−αtF (t) + e−αtf(t). (3.7)

Also, ζ(F ) < α, so that there exist F1, F2 : R+ → X satisfying ω0(F2) < α, F1 exponentially

bounded and holomorphic in a sector and F = F1 + F2. This together with (3.7) implies

f(t) =(αF1(t) + eαtg1(t)

)+(αF2(t) + eαtg2(t)

).

Since abs(t 7→ αF2(t) + eαtg2(t)

)< α and t 7→

(αF1(t) + eαtg1(t)

)is exponentially bounded

in a sector, ζ1(f) ≤ α. Since α > ζ(F ) is arbitrary we conclude that ζ1(f) ≤ ζ(F ).

Conversely, suppose ζ1(f) < α. Then by definition, f = f1 +f2, where f1 is holomorphic

and exponentially bounded in a sector and abs(f2) < α. Let

Fi(t) =

∫ t

0fi(s) ds, (t ≥ 0)

for i = 1, 2. Then, F (t) = F1(t) + F2(t), t ≥ 0 and F1 is holomorphic and exponentially

bounded in a sector. Since we may write F (t) = F1(t) +F2,∞+F2(t)−F2,∞, it follows that

ζ(F ) ≤ α on noting that ω0(F2−F2,∞) = abs(f2) < α and t 7→ F1(t) +F2,∞ is holomorphic

and exponentially bounded in a sector. We conclude, therefore that ζ(F ) ≤ ζ1(f). Hence,

ζ1(f) = ζ(F ).

(2): Recall that if f(λ) and F (λ) both exist and Reλ > 0, then f(λ) = λF (λ). Suppose

hol∞0 (F ) < α. Then, there exists b > 0 such that Qα,b ⊂ D(F ) and supλ∈Qα,b ‖F (λ)‖ <∞.Therefore, the map λ 7→ λF (λ) is holomorphic in Qα,b. Therefore, f also has a holomorphic

extension to Qα,b which is equal to λF (λ). Thus Qα,b ⊂ D(f) and ‖f(λ)‖ = ‖λF (λ)‖ for

all λ ∈ Qα,b. Therefore,

supλ∈Qα,b

‖f(λ)‖(1 + |λ|) ≤ sup

λ∈Qα,b‖F (λ)‖ <∞.

Consequently, hol∞1 (f) ≤ α, so that hol∞0 (F ) ≥ hol∞1 (f). Similarly, since hol∞1 (f) < α im-

plies that Qα,b ⊂ D(f) for some b > 0 and supλ∈Qα,b

‖f(λ)‖(1 + |λ|) <∞, the map λ 7→ f(λ)

λ= F (λ)

has a holomorphic extension to Qα,b and supλ∈Qα,b ‖F (λ)‖ <∞. Thus, hol∞0 (F ) ≤ hol∞1 (f).

29

Corollary 3.1.12. For a Laplace transformable function f : R+ → X,

ζ1(f) ≥ hol∞1 (f).

Proof. Let F (t) =∫ t

0 f(s) ds (t ≥ 0). Then from (1 ), Proposition 3.1.8 and Theorem 3.1.11

it follows that ζ1(f) = ζ(F ) ≥ hol∞0 (F ) = hol∞1 (f).

3.2 The non-analytic bounds for operator-valued functions

We shall now study the growth bounds introduced in Section 3.1 in the context of operator-

valued, strongly continuous functions T : R+ :→ L(X). In this case, ζ(T) is defined as

ζ(T) = infω0(T− S) : S is holomorphic and exponentially bounded

from Σθ to L(X) for some θ > 0.

(3.8)

ζ1(T) and κ(T) have analogous definitions. Since Proposition 3.1.4 remains valid when

f is replaced by T, we may use the equivalent descriptions of ζ(T) and the other two

non-analytic bounds associated with T which it provides.

3.2.1 Reduction to the vector-valued case

Suppose that T : R+ → L(X) is strongly continuous and Laplace transformable. From

the uniform boundedness principle, and the definitions of the growth bound and abscissa

of convergence it follows that

ω0(T) = supω0(T(·)x) : x ∈ X= supω0(〈T(·)x, x∗〉) : x ∈ X, x∗ ∈ X∗, (3.9)

abs(T) = supabs(T(·)x) : x ∈ X= supabs(〈T(·)x, x∗〉) : x ∈ X, x∗ ∈ X∗. (3.10)

Further, using arguments with Taylor series and the uniform boundedness principle it has

been shown in [2, Proposition 1.5.5 ] that

hol(T) = sup

hol(T(·)x) : x ∈ X

= sup

hol(〈T(·)x, x∗〉) : x ∈ X, x∗ ∈ X∗.

and a straightforward application of the uniform boundedness principle gives the corre-

sponding result for hol0(T). It is not clear that hol∞(T) = sup

hol∞(T (·)x) : x ∈ X. We

now prove the corresponding fact for hol∞0 (T), ζ(T), ζ1(T) and hol∞n (T). In particular,

if T is a C0-semigroup, we express ζ1(T) in terms of the non-analytic growth bounds of

certain orbit maps.

30

Theorem 3.2.1. Let Ω be a non-empty, connected, open subset of C and S : Ω → L(X)

be a bounded holomorphic function. Let (Ωn)n≥1 be a sequence of connected open subsets

of C such that Ωn ∩ Ω is non-empty for each n. Suppose that, for each x ∈ X there exists

nx ≥ 1 and a bounded holomorphic function Hx : Ωnx → X such that Hx(λ) = S(λ)x for all

λ ∈ Ωn ∩Ω. Then there exists N ≥ 1 and a bounded holomorphic function U : ΩN → L(X)

such that U(λ) = S(λ) for all λ ∈ ΩN ∩ Ω.

Proof. For n ≥ 1 and k ≥ 1, let

Xn,k =

x ∈ X : there is a holomorphic function H : Ωn → X such that

H(λ) = S(λ)x, λ ∈ Ωn ∩ Ω and supλ∈Ωn

‖H(λ)‖ ≤ k.

By assumption, X =⋃∞n,k=1 Xn,k. We show that each Xn,k is closed in X. Let (xr)r≥1 be

a sequence in Xn,k converging to some x ∈ X. For each r ≥ 1 there exists a holomorphic

function Hr : Ωn → X such that

‖Hr(λ)‖ ≤ k (λ ∈ Ωn), Hr(λ) = S(λ)xr (λ ∈ Ωn ∩ Ω).

Therefore, limr→∞Hr(λ) = S(λ)x (λ ∈ Ωn ∩ Ω). Since Hr : r ≥ 1 is uniformly bounded,

Vitali’s Theorem implies that H(λ) := limr→∞Hr(λ) exists for all λ ∈ Ωn and defines

a holomorphic function H : Ωn → X. Then ‖H(λ)‖ ≤ k for all λ ∈ Ωn and H(λ) =

S(λ)x (λ ∈ Ωn ∩Ω). Thus, x ∈ Xn,k. Using Baire’s Category Theorem, we find an N ≥ 1,

k ≥ 1, x0 ∈ X and an ε > 0 such that ‖x− x0‖ < ε implies x ∈ Xn,k . So for each such x,

there is a unique holomorphic function Hx : ΩN → X such that

‖Hx(λ)‖ ≤ k (λ ∈ ΩN ) and Hx(λ) = S(λ)x (λ ∈ ΩN ∩ Ω).

Thus, if ‖x− x0‖ < ε, there is a holomorphic function Hx−x0 := Hx −Hx0 : ΩN → X such

that

‖Hx−x0(λ)‖ ≤ 2k (λ ∈ ΩN ) and Hx−x0(λ) = S(λ)(x− x0) (λ ∈ Ω ∩ ΩN ).

Therefore, for any y ∈ X, we have that there is a unique holomorphic function, Hy : ΩN →X such that ‖Hy(λ)‖ ≤ 4k‖y‖/ε (λ ∈ ΩN ) and Hy(λ) = S(λ)y (λ ∈ ΩN ∩ Ω). So we may

define

U(λ)x = Hx(λ) (λ ∈ ΩN , x ∈ X).

U(λ) is linear, by uniqueness of holomorphic extensions. This is the required bounded

holomorphic function.

Corollary 3.2.2. Let T : R+ → L(X) be strongly continuous and Laplace transformable.

Then

31

1. hol∞0 (T) = sup

hol∞0 (T(·)x) : x ∈ X

;

2. ζ(T) = supζ(T(·)x) : x ∈ X

.

Proof. Clearly, hol∞0 (T) ≥ hol∞0 (T(·)x) and ζ(T) ≥ ζ(T(·)x) for all x ∈ X. Suppose that

w > suphol∞0 (T(·)x) : x ∈ X. Let Ω = Habs(T) and Ωn = Qw−n−1,n. By assumption,

for each x ∈ X, there exists nx ≥ 1 and a bounded holomorphic function on Ωnx which

agrees with T(·)x on Ωnx ∩ Ω. By Theorem 3.2.1, there exists an N ≥ 1 and a bounded

holomorphic function U : ΩN → L(X) agreeing with T on Ω ∩ ΩN . Hence, hol∞0 (T) ≤ w.

Now, let w > supζ(T(·)x) : x ∈ X. Since hol∞0 (T(·)x) ≤ ζ(T(·)x) for all x ∈ X it

follows from the first part that w > hol∞0 (T). Take α such that hol∞0 (T) < α < w. Choose

b such that Qα,b ⊂ D(T), so that Tα,b : R+ → L(X) is defined. Since ζ(T(·)x) < w, we

have from Proposition 3.1.4,

supt≥0

∥∥e−wt(T(t)−Tα,b(t))x∥∥ <∞

for each x ∈ X. By the Uniform Boundedness Principle,

supt≥0

∥∥e−wt(T(t)−Tα,b(t))∥∥ <∞.

Thus, ζ(T) ≤ w. This proves the second part.

Corollary 3.2.3. Let f : R+ → X be Laplace transformable. Then

1. hol∞0 (f) = suphol∞0 (x∗ f) : x∗ ∈ X∗;

2. ζ(f) = supζ(x∗ f) : x∗ ∈ X∗.

Proof. These results follow by the same proofs as Theorem 3.2.1 and Corollary 3.2.2.

Remark 3.2.4. Theorem 3.2.1 remains true if ‘bounded holomorphic function’ is replaced

by ‘linearly bounded holomorphic function’ or more generally by ‘polynomially bounded

function’.

Therefore, the following generalisation of the above result also holds:

Corollary 3.2.5. Let T : R+ → L(X) be strongly continuous and Laplace transformable.

Then

1. hol∞n (T) = suphol∞n (T(·)x) : x ∈ X;

2. ζ1(T) = supζ1(T(·)x) : x ∈ X;

Proof. The first part follows exactly along the lines of Corollary 3.2.2, on noting Remark

3.2.4. (2 ) follows from Theorem 3.1.11 (1 ) and Corollary 3.2.2 (2 ).

32

3.2.2 The C0-semigroup case

Let T be a C0-semigroup on X with generator A. We recall the definitions of the various

spectral bounds of A:

s(A) = sup Reλ : λ ∈ σ(A) ;

s0(A) = inf

ω > s(A) : there exists Cω such that ‖R(λ,A)‖ ≤ Cω

whenever Reλ > ω

;

s∞(A) = inf

ω ∈ R : Qω,b ⊂ ρ(A) for some b ≥ 0

;

s∞0 (A) = inf

ω ∈ R : Qω,b ⊂ ρ(A) and sup

λ∈Qω,b‖R(λ,A)‖ <∞ for some b ≥ 0

;

sn(A) = inf

ω ∈ R : Hω ⊂ ρ(A) and sup

Reλ>ω

‖R(λ,A)‖(1 + |λ|)n <∞

;

s∞n (A) = inf

ω ∈ R : Qω,b ⊂ ρ(A) and sup

λ∈Qω,b

‖R(λ,A)‖(1 + |λ|)n <∞ for some b ≥ 0

,

for n ∈ N.

The precise relation between these spectral bounds and the abscissas associated with

the Laplace transform of T is given by the following equations:

s(A) = hol(T), s0(A) = hol0(T), sn(A) = holn(T),

s∞(A) = hol∞(T), s∞0 (A) = hol∞0 (T), s∞n (A) = hol∞n (T).

The first two of these equations follow from [2, Theorem 5.1.4] and the proof of the others

works along similar lines.

From [2, Proposition 5.1.6] we also have

abs(T) = ω1(T) = supω(T(·)x) : x ∈ D(A)

= ω (T(·)R(λ,A)) ,(3.11)

where λ ∈ ρ(A). Therefore, for the semigroup T : R+ → L(X) the results in (3.3) of

Proposition 3.1.8 read as

ω0(T) = max(ζ(T), s(A)),

ω1(T) = max(ζ1(T), s(A)). (3.12)

Moreover, we can write ζ1(T) in terms of the non-analytic growth bounds of the orbit

functions ux(t) = T(t)(x), thus obtaining an analogue of (3.11).

33

Theorem 3.2.6. Let T : R→ L(X) be a C0-semigroup with generator A. Then

ζ1(T) = ζ(T(·)R(λ,A)

)

= supζ(T(·)x) : x ∈ D(A)

,

where λ ∈ ρ(A) satisfies Reλ > ω0(T).

Proof. Since D(A) = Ran(R(λ,A)), for all λ ∈ ρ(A), we have using (2 ) of Corollary 3.2.2

supζ(T(·)x) : x ∈ D(A)

= sup

ζ(T(·)R(λ,A) ) : y ∈ X

= ζ(T(·)R(λ,A)).

For Reλ > ω0, and x ∈ X,

T(t)R(λ,A) x =

∫ ∞

0e−λsT(t+ s)x ds

= eλt(

R(λ,A) x−∫ t

0e−λsT(s)x ds

).

Observing that ζ(t 7→ eλtR(λ,A)x

)= −∞, we have

ζ(T(·)R(λ,A)x

)= Reλ+ ζ

(t 7→

∫ t

0U(s)x ds

)

= Reλ+ ζ1

(U(·)x

)

= ζ1

(T(·)x

),

where U(s) = e−λsT(s) is the rescaled semigroup. So, it follows from Corollary 3.2.5 that

ζ(T(·)R(λ,A) ) = supζ(T(·)R(λ,A) y) : y ∈ X= supζ1(T(·)x) : x ∈ X= ζ1(T).

In [11, Corollary 3.3] it has been shown that s∞0 (A) ≤ δ(T) and strict inequality may

hold ([11, Proposition 3.7]). Thus for a C0-semigroup T on X,

s∞0 (A) ≤ δ(T) ≤ ζ(T). (3.13)

If X is a Hilbert space then actually equality holds in (3.13). This follows from [11, Lemma

4.3] where it is shown that if s∞0 (A) < α then there exists b > 0 such that ω0(T − T1) ≤α, where

T1(t)x =1

2πi

∫

Γα,b

eµtR(µ,A) x dµ,

34

Γα,b consists of line segments joining α − ib, ω − ib, ω + ib and α + ib successively, and

ω > max(α, ω0(T)). Since T1 has a holomorphic, exponentially bounded extension to a

sector Σθ, we have ζ(T) ≤ α. So, for a C0-semigroup on a Hilbert space we have

s∞0 (A) = δ(T) = ζ(T). (3.14)

This result may be considered as the non-analytic analogue of the famous Gearhart-

Pruss theorem [2, Theorem 5.2.1]:

Theorem 3.2.7. (Gearhart-Pruss) For a C0-semigroup T on a Hilbert space X with gen-

erator A the uniform exponential growth bound and the pseudo-spectral bound coincide, that

is

ω0(T) = s0(A).

It is well known that such a theorem does not hold for individual semigroup orbits [2,

Example 5.2.3 ] and in fact, the same is true of its non-analytic analogue. More precisely,

an analogue of (3.14) does not hold for individual orbits of semigroups defined on Hilbert

spaces. Indeed, if T is a C0-semigroup on a Hilbert space with hol∞0 (ux) = ζ(ux), where

ux(t) = T(t)x, then we must have

hol0(ux) = ω0(ux).

This follows from the inequality hol∞0 (ux) ≤ hol0(ux) ≤ ω0(ux) and Proposition 3.1.8 (2).

This would imply that an analogue of Gearhart-Pruss Theorem [2, Theorem 5.2.1] holds for

individual semigroup orbits, thus yielding a contradiction.

In [51] higher order analogues of the Gearhart-Pruss Theorem have been established,

specifically, the equalities

ωn(T) = sn(A) (n ∈ N),

where T is a C0-semigroup with generator A, defined on a Hilbert space X. We shall obtain

the corresponding result for the non-analytic abscissa of convergence ζ1(T). For this, we

need a lemma, with a proof very similar to [11, Lemma 4.2]. We state the lemma without

proof.

Lemma 3.2.8. Let T be a C0-semigroup defined on the Hilbert space X. Let α,w, b,K ∈ R,α < w, b > 0 be such that Q := Qα,b ∩ λ : Reλ ≤ w ⊂ ρ(A) and

supλ∈Q

∥∥∥∥R(λ,A)

(1 + |λ|)

∥∥∥∥ ≤ K.

Then for x ∈ X we have

lim|s|→∞

∥∥∥∥R(µ+ is,A)x

µ+ is

∥∥∥∥ = 0, (3.15)

35

the limit being uniform with respect to µ ∈ [α,w] and

lim|s|→∞

∫ ω+is

α+is

eλtR(λ,A)x

λdλ = 0. (3.16)

Theorem 3.2.9. For a C0-semigroup T defined on a Hilbert space X, with generator A,

ζ1(T) = s∞1 (A).

Proof. We first consider the case when ω0(T) < 0. Let s∞1 (A) < α. We may assume here,

without loss of generality, that α < 0. By the definition of s∞1 (A), there exist b > 0 and a

constant K such that Qα,b ⊂ ρ(A) and

‖R(a+ is,A)‖ ≤ K(1 + |s|), a ≥ α, |s| ≥ b.

Let ω > max(α, ω0(T)) and ω 6= 0. From the Inversion theorem [20, Theorem III.5.14], we

have ∫ t

0T(s)x ds = lim

M→∞1

2πi

∫ ω+iM

ω−iM

eλt

λR(λ,A)x dλ, (3.17)

for t ≥ 0, and all x ∈ X. Define Sα,b,ω(t) by

Sα,b,ω(t)x =1

2πi

∫

Γ

eλtR(λ,A)x

λdλ,

where Γ is the path consisting of line segments [α−ib, ω−ib], [ω−ib, ω+ib] and [ω+ib, α+ib].

Let φ ∈ X∗. Then

φ

(∫ t

0T(s)(x)ds

)= lim

M→∞1

2πi

∫ ω+iM

ω−iM

eλtφ(R(λ,A)x)

λdλ

=1

2πi

∫ ω+ib

ω−ib

eλtφ(R(λ,A)x)

λdλ

+ limM→∞

1

2πi

(∫ ω−ib

ω−iM+

∫ ω+iM

ω+ib

)eλtφ(R(λ,A)x)

λdλ. (3.18)

For M > b, using Cauchy’s Theorem, we have

∫ ω+iM

ω+ib

eλtφ(R(λ,A)x)

λdλ =

(∫ α+ib

ω+ib+

∫ α+iM

α+ib

)eλtφ(R(λ,A)x)

λdλ

+

∫ ω+iM

α+iM

eλtφ(R(λ,A)x)

λdλ.

Using Lemma 3.2.8, we have that

limM→∞

∫ ω+iM

α+iM

eλtφ(R(λ,A)x)

λdλ = 0.

36

Consequently, (3.18) yields

φ

(∫ t

0T(s)x ds− Sα,b,ω(t)x

)

= limM→∞

1

2πi

(∫ α+iM

α+ib+

∫ α−ib

α−iM

)eλtφ(R(λ,A)x)

λdλ. (3.19)

Applying the resolvent identity, we have for M > b,

∣∣∣∣∫ α+iM

α+ib

eλt

λφ(R(λ,A)x) dλ

∣∣∣∣

=

∣∣∣∣∣

∫ M

b

e(α+is)tφ(R(ω + is,A)x+ (ω − α)R(ω + is,A)R(α+ is,A)x)

α+ isds

∣∣∣∣∣

≤ eαt∫ ∞

−∞

∣∣∣∣φ(R(ω + is,A)x)

α+ is

∣∣∣∣ ds

+(ω − α)eα t∫

|s|>b

‖R(ω + is,A)∗φ‖‖R(α+ is,A)x‖|α+ is| ds

≤ eαt(∫ ∞

−∞|φ(R(ω + is,A)x)|2 ds

) 12(∫ ∞

−∞

1

|α+ is|2 ds) 1

2

+(ω − α)eαt(∫ ∞

−∞‖R(ω + is,A)∗φ‖2 ds

) 12

(∫

|s|>b

‖R(α+ is,A)x‖2|α+ is|2 ds

) 12

≤ Ceαt(1 + ω − α)‖φ‖ ‖x‖, (3.20)

where C is a constant depending on α and b. The final step depends on the following

observations. By Plancherel’s Theorem, for ω > ω0(T),

∫ ∞

−∞‖R(ω + is,A)x‖2 ds = 2π

∫ ∞

0‖e−ωtT(t)x‖2 dt

≤ C ′‖x‖2,

where C ′ is a constant and a similar estimate holds for

∫ ∞

−∞‖R(ω + is,A)∗φ‖2 ds.Moreover,

∫

|s|>b

‖R(ω + is,A)x‖2|α+ is|2 ds ≤ C ′ ‖x‖

2

b2

while∫

|s|>b

‖R(ω + is,A)R(α+ is,A)x‖2|α+ is|2 ds

≤∫

|s|>b

K2(1 + |s|)2

α2 + s2‖R(ω + is,A)x‖2 ds

≤ C ′′‖x‖2,

where C ′′ is a constant depending on α and b.

37

It follows then from the resolvent identity R(α+is,A)x = R(ω+is,A)x+(ω−α)R(ω+

is,A)R(α+ is,A)x, that

∫

|s|>b

‖R(α+ is,A)x‖2|α+ is|2 ds ≤ C ′′‖x‖2.

We obtain estimates similar to (3.20) for the other integral on the right hand side of

(3.19), to arrive at

∣∣∣∣φ(∫ t

0T(s)(x) ds− Sα,b,ω(t)x

)∣∣∣∣ ≤ Keαt‖φ‖‖x‖, (3.21)

for x ∈ X. Since (3.21) holds for all φ ∈ X∗ and x ∈ X, we have

∥∥∥∥∫ t

0T(s) ds− Sα,b,ω(t)

∥∥∥∥ ≤ Keαt. (3.22)

For x ∈ X let

S2(t)x =

∫ t

0T(s)x ds− Sα,b,ω(t)x. (3.23)

Then (3.22) implies that ω0(S2) ≤ α. Further, differentiating (3.23) with respect to t

yields

T(t)x =1

2πi

∫

ΓeλtR(λ,A)x dλ+ T2(t)x,

where T2(t)x is the derivative of S2(t)x, so that

S2(t)x =

∫ t

0T2(s)x ds+ S2(0)x.

Therefore,

abs(T2) ≤ ω0(S2) ≤ α.

Setting T1(t) =1

2πi

∫

ΓeλtR(λ,A) dλ we have that T1 is holomorphic and exponentially

bounded in a sector and abs(T − T1) ≤ α. Hence, ζ1(T) ≤ α. Thus s∞1 (A) ≥ ζ1(T) and

from Corollary 3.1.12 it follows that ζ1(T) = s∞1 (A) if ω0(T) < 0. For the general case, the

result follows by rescaling.

Example 3.2.10. We consider Example 5.1.10 in [2]. Here X is the Hilbert space given by

X :=

x = (xn)n≥1 : xn ∈ Cn,

∞∑

n=1

‖xn‖2 <∞,

‖x‖ :=

( ∞∑

n=1

‖xn‖2)1/2

, (3.24)

38

where the norm on Cn is the Euclidean norm. Let Bn = (β(n)i,j )1≤i,j≤n be the n× n matrix

with β(n)i,i+1 = 1 for 1 ≤ i < n, β

(n)i,j = 0 otherwise, and let An = i2nIn + Bn. Let A be the

operator on X defined by

D(A) =

x ∈ X :

∞∑

n=1

22n‖xn‖2 <∞,

Ax = (Anxn)n≥1.

Let T be the C0-semigroup generated by A. It is shown in [2], that

ω0(T) = 1, ω1(T) =1

2, σ(A) = i2n : n ≥ 1

and s0(A) = 1. Also, s∞(A) = s(A) = 0. Since X is a Hilbert space, s1(A) = ω1(T) [51,

Theorem 1.4]. From (3.12) it follows that ζ1(T) = ω1(T). Hence, s(A) = s∞(A) = 0 <

s∞1 (A) = s1(A) = ζ1(T) = ω1(T) = 12 < s0(A) = ζ(T) = ω(T) = 1.

3.3 Essential holomorphy

Thieme [48, Definition 2.6] has introduced essentially norm-continuous and essentially

norm-measurable C0-semigroups, generalising both eventually norm-continuous semigroups

and essentially compact semigroups. In [11, Definition 2.1] these definitions have been

modified to make them applicable to the more general situation of exponentially bounded

families T : R+ → L(X). Continuing in the same spirit, we introduce the concept of essen-

tial holomorphy. In the context of C0-semigroups, this may be thought of as a generalisation

of analytic C0-semigroups.

Definition 3.3.1. Let T : R+ → L(X) be exponentially bounded and strongly continuous

and let β > 0. T is said to be essentially holomorphic (of type β) if for each α such that

0 < α < β there exist an exponentially bounded, holomorphic function T1 : Σθ → L(X) for

some θ > 0 with ω0(T−T1) ≤ ω0(T)− α.

As is evident from the definition, there is a very close relation between essential holo-

morphy of the function T and its non-analytic growth bound ζ(T). Precisely, we have

Theorem 3.3.2. Let T : R+ → L(X) be exponentially bounded and strongly continuous.

Then the following are equivalent:

1. T is essentially holomorphic of type β;

2. ζ(T) ≤ ω0(T)− β.

39

By analogy with the definition of an asymptotically norm-continuous C0-semigroup we

can define a C0-semigroup T to be asymptotically holomorphic if ζ(T) < ω0(T). It is

immediate that T is essentially holomorphic if and only if it is asymptotically holomorphic.

We recall here that a C0-semigroup T is called essentially norm-measurable (respectively,

essentially norm-continuous) if there exists a decomposition of T as in Definition 3.3.1

with T1 being norm-measurable (respectively, norm-continuous) instead of holomorphic

and exponentially bounded in a sector and the corresponding condition on ω0(T − T1) is

satisfied. It is obvious that

essential holomorphy ⇒ essential norm-continuity ⇒ essential norm-measurability.

(3.25)

These concepts are equivalent if the underlying space is a Hilbert space. That essential

norm-measurability ⇒ essential norm-continuity for Hilbert spaces has been shown in [10,

Theorem 3.4.3] and essential norm-continuity⇒ essential holomorphy follows from the fact

that δ(T) = ζ(T) for C0-semigroups on Hilbert spaces. We do not know whether the

converse implications in (3.25) hold in general.

Example 3.3.3. Let T1 be a norm-continuous semigroup on X with generator A1 and

T2 be a C0 -semigroup on a Banach space Y with generator A2 satisfying ω0(T1) >

ω0(T2) > −∞. Then the C0-semigroup U = T1⊕T2 defined on X⊕

Y has growth bound

ω0(U) = ω0(T1). Further, since ζ(T1) = −∞, ζ(U) = ζ(T2) ≤ ω0(T2). Therefore, we have

−∞ < ζ(U) < ω0(U). Thus U is essentially holomorphic.

3.4 A comparison of the critical growth bound and the non-

analytic growth bound

For an exponentially bounded, strongly continuous function T : R+ → L(X) the non-

analytic growth bound ζ(T) may be thought of as the growth bound of T modulo operator-

valued functions which are exponentially bounded and holomorphic on some sector, whereas

the critical growth bound δ(T) is the growth bound of T modulo exponentially bounded and

norm continuous operator-valued functions defined on R+. It is clear from the definitions

in (2.13) and (3.8) that

δ(T) ≤ ζ(T) ≤ ω0(T).

None of the known characterisations of the critical growth bound extend in a meaningful

way to general vector-valued functions f : R+ → X, unlike in the case of the non-analytic

growth bound. In particular, any continuous function f : R+ → X would have its critical

growth bound as −∞ while the various abscissas of holomorphy of f may or may not be

40

finite. As an example, consider the function f : R+ → C, given by

f(t) =∞∑

n=1

e(α+in)t

n2, α < 0.

Then f is continuous so that its critical growth bound is −∞, but hol∞0 (f) = hol(f) = α =

ω0(f). Thus, the critical growth bound does not give any information about the abscissas

of holomorphy. Such a situation cannot arise in the case of the non-analytic growth bound.

In fact, Proposition 3.1.8, establishes definite relations between hol∞0 (f),hol(f) and ζ(f).

For the example above, it follows that ζ(f) = α.

Even for general operator-valued functions T : R+ → L(X) the critical growth bound

δ(T) may not determine the value of hol∞0 (T). In fact, the following corollary of Theorem

3.1.11 and Theorem 3.2.9 shows that for certain once integrated semigroups S defined on a

Hilbert space, there may be no relation between the critical growth bound and hol∞0 (S).

Corollary 3.4.1. Let T be a C0-semigroup defined on a Hilbert space X with generator A

satisfying s∞(A) > −∞. Let S denote the once integrated semigroup obtained from T, that

is S(t)x =

∫ t

0T(s)x ds (x ∈ X). Then δ(S) = −∞ while −∞ < hol∞0 (S) = ζ(S).

Proof. We first note that such a semigroup exists. Suppose if possible, δ(S) = ζ(S). Now

δ(S) is equal to −∞ since S is norm continuous. Further, by Theorem 3.1.11 and Theorem

3.2.9,

ζ(S) = ζ1(T) = s∞1 (A) = hol∞1(T) = hol∞0 (S).

Moreover, s∞1 (A) ≥ s∞(A) > −∞. Thus ζ(S) > −∞.

As mentioned before, the equality s∞0 (A) = δ(T) = ζ(T) holds for C0-semigroups

on Hilbert spaces. We do not know whether the equality δ(T) = ζ(T) holds for all C0-

semigroups defined on general Banach spaces. However, it is known that the two growth

bounds and the spectral bound coincide for C0-semigroups that fall under the categories

mentioned below. We remark that in all these cases, actually the equality

s∞0 (A) = ζ(T) = −∞

holds. These classes are:

• ([10, Theorem 6.6.4]) Eventually differentiable semigroups.

• ([10, Theorem 6.6.3]) C0-semigroups with an Lp-resolvent for some p ∈ (1,∞). A C0-

semigroup T with generator A is said to have an Lp-resolvent (see [7]) if there exists

w ∈ R and b ≥ 0 such that Qw,b ⊂ ρ(A) and∫

|s|≥b‖R(w + is,A)‖p ds <∞.

41

• ([10, Proposition 4.5.7]) Eventually compact semigroups.

A description of the critical growth bound δ(T) of a C0-semigroup T in terms of the

spectral radius of certain operators has been obtained in [6] using Banach algebra techniques.

It is similar to the description of ζ(T) given by (4) of Proposition 3.1.4, stating that given

a C0-semigroup T, and ω ∈ R,

δ(T) < ω ⇐⇒ for each α < ω, b ≥ 0 satisfying Qα,b ⊂ ρ(A),

ω0 (r (T(·)−Tα,b(·))) < ω.

Here r(B) denotes the spectral radius of a bounded operator B on X. From the above

description of the critical growth bound and Proposition 3.1.4 it follows that δ(T) = ζ(T)

for any C0-semigroup T for which the spectral radius and the norm of the operators t 7→T(t)−Tα,b coincide whenever α ∈ R, b ≥ 0 satisfy Qα,b ⊂ ρ(A).

In particular, multiplier semigroups satisfy this condition. Therefore, we have

Theorem 3.4.2. Let T(t) be a C0-semigroup on X = C0(Ω), where Ω is a locally compact

space, given by

T(t)(f)(s) = eq(s)f(s) (f ∈ X, s ∈ Ω),

where q : Ω −→ C is a continuous function satisfying

sups∈Ω

Re q(s) <∞.

Then ζ(T) = δ(T) = s∞0 (A) = s∞(A).

Using the next result, we can add another category of C0-semigroups to the class for

which the critical and non-analytic growth bounds coincide:

Theorem 3.4.3. ([10, Theorem 6.6.1]) Let T be a C0-semigroup with generator A. If

δ(T) < a < ω0(T) and a+ iR ⊂ ρ(A), then ζ(T) < a.

Theorem 3.4.4. If T is a C0-semigroup such that its generator A has compact resolvents

then δ(T) = ζ(T). Also, if B is a bounded operator and S is the C0-semigroup generated

by A + B then δ(S) = ζ(S).

Proof. If δ(T) = ω0(T), then the result is true anyway. So we suppose that δ(T) < ω0(T).

Let A denote the generator of the semigroup. Then, if R(λ,A) is compact for λ ∈ ρ(A)

we have that the spectrum of R(λ,A) consists of a countable number of eigenvalues with

zero as the only possible limit point. Therefore, σ(A) consists of only countably many

eigenvalues, with infinity as the only possible limit point. So

σ(A)⋂λ ∈ C : δ(T) < Reλ < ω0(T)

42

is countable. From Theorem 3.4.3 it follows that δ(T) ≥ ζ(T). The resolvent of A + B is

compact if that of A is. Therefore, the second claim follows from the first part.

The equality s∞0 (A) = ζ(T) for a C0-semigroup on a Hilbert space depends on the

L2-integrability of the maps s 7→ R(ω+ is,A)x, ω > ω0(T), x ∈ X. Therefore, the following

generalisation holds:

Theorem 3.4.5. Let A be the generator of a C0-semigroup T on X and suppose there

exists ω > ω0(T) such that for each x ∈ X and each x∗ ∈ X∗, the following conditions are

satisfied:

∫ ∞

−∞‖R(ω + is,A)x‖2 ds <∞;

∫ ∞

−∞‖R(ω + is,A)∗x∗‖2 ds <∞.

Then s∞0 (A) = δ(T) = ζ(T).

Proof. The proof works on exactly the same lines as [11, Lemma 4.3].

It is clear that for any Laplace transformable function f , the non-analytic abscissa of

convergence ζ1(f) relates to the non-analytic growth bound ζ(f) in the same way as the

abscissa of convergence, abs(f) relates to the exponential growth bound ω0(f). This is true,

in particular for a C0-semigroup T. One would like to be able to define an abscissa δ1(T)

corresponding to the bound δ(T) in a manner similar to ζ1(T) and ζ(T). However, such

an attempt is unsuccessful. In fact, if we make the definition:

δ1(T) = infω ∈ R : T = T1 + T2, where T1 is norm

continuous and abs(T2) < ω,

then we have that δ1(T) = −∞ for every C0-semigroup T on X. To see this, let T be a C0-

semigroup and S its once integrated semigroup. Since S is norm continuous, δ(S) = −∞.

So for n ∈ N, there exists S1,S2 : R+ −→ X such that S1 is norm differentiable and

ω0(S2) < −n. Consequently, T = T1 + T2 where ddtSi(t)(x) = Ti(t)(x), i = 1, 2. This

means that δ1(T) < −n and since n ∈ N was arbitrary, it follows that δ1(T) = −∞.

43

Chapter 4

Fractional growth bounds

4.1 Convolutions and regularisations

In this section, we give some estimates for the non-analytic growth bound when a function

f is regularised by convolution. If f : R+ → X is exponentially bounded and measurable

and φ : R+ → C is in L1loc(R+) , then the convolution φ ∗ f is Laplace transformable with

abs(φ ∗ f) ≤ max(abs(φ), abs(‖f‖));φ ∗ f(λ) = φ(λ)f(λ),

whenever Reλ > max(abs(φ), abs(‖f‖)) [2, Proposition 1.6.4]. Therefore,

hol∞0 (φ ∗ f) ≤ max(

hol∞0 (φ),hol∞0 (f )).

Also, trivial estimates show that

ω0(φ ∗ f) ≤ max(ω0(φ), ω0(f)). (4.1)

We shall obtain the corresponding estimate for ζ(φ ∗ f) and give some sharper estimates

when φ is more regular. For this we require a basic estimate, derived in the next theorem

by closely following the strategy of [8, Corollary 2.2], [2, Proposition 4.4.11].

Theorem 4.1.1. Let f : R+ → X be an exponentially bounded and measurable function such

that f has a bounded, holomorphic extension to Q0,b for some b ≥ 0. Suppose φ : R −→ Cis a measurable function and the following conditions hold:

1.

∫ ∞

0eωt|φ(t)| dt <∞, for some ω > max(0, ω0(f));

2.

∫ 0

−∞|φ(t)| dt <∞;

3. Fφ ∈ L1(R).

44

Let Γ0,b be a path joining −ib to ib in λ ∈ D(f ) : 0 ≤ Reλ ≤ ω, and set

φ(−λ) =

∫ ∞

−∞eλtφ(t) dt (0 ≤ Reλ ≤ ω).

Then ∥∥∥∥∥

∫ ∞

0φ(t)f(t) dt− 1

2πi

∫

Γ0,b

f(λ)φ(−λ) dλ

∥∥∥∥∥ ≤ C‖Fφ‖1, (4.2)

where C = 12π sup

‖f(λ)‖ : λ ∈ Q0,b

.

Proof. We first assume that φ has compact support and Fφ ∈ L1(R). Suppose that ω >

max(0, ω0(f)) satisfies condition (1) of the hypothesis and 0 < α < ω is fixed. Now

t 7→ e−ωtf(t) ∈ L1(R+,X) and its Fourier transform is s 7→ f (ω + is). Let ψ ∈ C∞c (R)

with

ψ(t) = e(ω−α)t (t ∈ suppφ).

Then (Fφ) ∗ (Fψ) ∈ L1(R) and

F(Fψ ∗ Fφ)(t) = 4π2ψ(t)φ(t)

= 4π2e(ω−α)tφ(t)

for all t ∈ R. Therefore, F(Fψ ∗ Fφ) ∈ L1(R) and

(Fψ ∗ Fφ)(s) = F−1(4π2e(ω−α)·φ)(s)

= 2π

∫ ∞

−∞e(is+ω−α)tφ(t) dt

= 2πφ(−ω + α− is)

for all s ∈ R. Using [2, Theorem 1.8.1] we therefore obtain

∫ ∞

0e−αtf(t)φ(t) dt =

∫ ∞

0e−ωtf(t)e(ω−α)tφ(t) dt

=1

4π2

∫ ∞

0e−ωtf(t)F(Fψ ∗ Fφ)(t) dt

=1

2π

∫ ∞

−∞f (ω + is)φ(α− (ω + is)) ds.

Now we consider the integral ∫

Γ1

f (λ)φ(α− λ) dλ,

where Γ1 is the rectangle with vertices α− ir, ω − ir, ω − ib, α− ib and r > b. We have

∫ ω−ib

ω−irf(λ)φ(α− λ) dλ =

(∫ α−ir

ω−ir+

∫ α−ib

α−ir+

∫ ω−ib

α−ib

)(f(λ)φ(α− λ)) dλ.

45

Consider the first integral on the right above. It is given by∫ α

ωf(η − ir)φ(α− η + ir) dη.

For α < η < ω,

φ(α− η + ir) =

∫ ∞

−∞e−(α−η)tφ(t)e−irt dt

→ 0,

as r →∞ by the Riemann-Lebesgue lemma, since s 7→ e−(α−η)sφ(s) ∈ L1(R). Moreover,

|φ(α− η + ir)| ≤ K,

whenever r > b, α < η < ω, where

K =

(∫ 0

−∞|φ(t)| dt+

∫ ∞

0eωt|φ(t)| dt

)

is a constant independent of η and r. Therefore, by the dominated convergence theorem it

follows that

limr→∞

∫ ω

αf(η − ir)φ(α− η + ir) dη = 0.

Thus, ∫ ω−ib

ω−i∞f (λ)φ(α− λ)dλ =

(∫ α−ib

α−i∞+

∫ ω−ib

α−ib

)f (λ)φ(α− λ) dλ.

We can similarly deal with the integral over [ω + ib, ω + i∞) to obtain,

∫ ∞

0e−αtf(t)φ(t) dt =

1

2π

∫

Reλ=ωf (λ)φ(α− λ) dλ

=

(∫ α−ib

α−i∞+

∫

Γα,b

+

∫ α+i∞

α+ib

)(f (λ)φ(α− λ)) dλ. (4.3)

Further,

∥∥∥∥∫ α−ib

α−i∞f (λ)φ(α− λ) dλ

∥∥∥∥+

∥∥∥∥∫ α+i∞

α+ibf (λ)φ(α− λ) dλ

∥∥∥∥

≤ C∫ α−ib

α−i∞|φ(α− λ)| dλ+ C

∫ α+i∞

α+ib|φ(α− λ)| dλ

= C

∫ −b

−∞|φ(−is)| ds+ C

∫ ∞

b|φ(−is)| ds

≤ C‖Fφ‖1. (4.4)

Therefore, combining (4.3) and (4.4), we obtain,∥∥∥∥∥

∫ ∞

0e−αtf(t)φ(t) dt− 1

2πi

∫

Γα,b

f (λ)φ(α− λ) dλ

∥∥∥∥∥ ≤C

2π‖Fφ‖1. (4.5)

46

Then using the Dominated Convergence Theorem we have that

supλ∈Γ0,b

|φ(α− λ)− φ(−λ)| −→ 0

as α ↓ 0. Taking the limit as α ↓ 0 in (4.5) we obtain

∥∥∥∥∥

∫ ∞

0f(t)φ(t) dt−

∫

Γ0,b

f(λ)φ(−λ)dλ

∥∥∥∥∥ ≤C

2π‖Fφ‖1. (4.6)

Now consider the case when φ is any function in L1(R) with Fφ ∈ L1(R) and there

exists an ω > max(0, ω0(f)) such that t 7→ eωtφ(t) ∈ L1(R). Let ψ ∈ C∞c (R) be any

function satisfying 0 ≤ ψ ≤ 1, ψ(0) = 1, and∫∞−∞ ψ(s) ds = 1. Let ψn(t) = ψ( tn) (t ∈ R)

and φn(t) = φ(t)ψn(t). Then (2π)−1(Fψn)(s) = (2π)−1n(Fψ)(ns), which forms a mollifier

(see Subsection 2.2.3). Therefore, Fφn = (2π)−1Fφ ∗ Fψn −→ Fφ in L1(R) as n → ∞.Applying (4.6) to the functions φn and noting that

‖∫

Γ0,b

f(λ)φn(−λ) dλ−∫

Γ0,b

f(λ)φ(−λ) dλ‖ −→ 0

we get (4.6) for a general φ.

Remark 4.1.2. We can make the following observations concerning the proof of the above

theorem.

1. The conditions (1 ) and (2 ) together, in the hypotheses of Theorem 4.1.1 may be

reformulated as φ ∈ L1(R) and eω·φ ∈ L1(R) for some ω > max(0, ω0(f)).

2. The above theorem also remains true if we replace ‖Fφ‖1 by∫|s|>b |Fφ(s)| ds in (4.2).

This is due to the fact that (4.4 ) remains valid if ‖Fφ‖1 is replaced by∫|s|>b |Fφ(s)| ds.

We now state our basic result in a form which allows regularisations by functions defined

on R.

Theorem 4.1.3. Let f : R −→ X be an exponentially bounded measurable function. Let

φ : R −→ X be locally integrable, and suppose that there exist C > 0, ω > hol∞0 (f ),

γ > max(ω, ω0(f)) and α ∈ (0, 1] such that

1.

∫ ∞

0|φ(s)|e−ωs ds <∞;

2.

∫ 0

−∞|φ(s)|e−γs ds <∞;

3.

∣∣∣∣∫ ∞

−∞φ(s)e−(ω+iη)s ds

∣∣∣∣ ≤C

|η|α (η ∈ R);

47

4.

∫ ∞

−∞|φ(s)− φ(s− h)|e−γs ds ≤ Chα (0 < h < 1).

Then φ ∗ f is defined on R+ and

ζ(φ ∗ f) ≤ (1− α)γ + αω.

Proof. The fact that (φ ∗ f)(t) exists follows immediately from the assumption (2). First

we assume that ω = 0 and hol∞0 (f ) < 0 so that f has a bounded holomorphic extension

to Q0,b for some b ≥ 1. Let t ≥ 0 and δ ∈ (0, 1]. Consider the function φδ : R −→ C defined

by

φδ(s) =1

δ(φ ∗ χ(0,δ))(t− s) =

1

δ

∫ t−s

t−s−δφ(r) dr.

Then ∫ ∞

0φδ(s)f(s) ds =

1

δ(φ ∗ χ(0,δ) ∗ f)(t)

and by an application of Fubini’s theorem,

φδ(−λ) =1

δ

∫ ∞

−∞eλs∫ t−s

t−s−δφ(r) dr ds

=1

δλ

∫ ∞

−∞(1− e−λδ)eλ(t−r)φ(r) dr

= eλt(

1− e−λδλδ

)φ(λ).

In particular,

(Fφδ)(s) = φδ(is)

= e−isteiδs2

(sin(sδ/2)

δs/2

)(Fφ)(−s).

If α ∈ (0, 1), we therefore have from assumption (3) that

‖Fφδ‖1 ≤∫ ∞

−∞

∣∣∣∣sin(δs/2)

δs/2

∣∣∣∣C

sαds

= 2C

(2

δ

)1−α ∫ ∞

0

| sin s|s1+α

ds

=C1

δ1−α ,

where C1 is a constant depending on C and α only. If α = 1, we have

∫

|s|≥b|(Fφδ)(s)| ds ≤ 2

(∫ 1/δ

1

C

sds+

∫ ∞

1/δ

2C

δs2ds

)

≤ C1(1 + | log δ|).

48

Consequently, for any α ∈ (0, 1], Theorem 4.1.1 and Remark 4.1.2 give

∥∥∥∥1

δ(φ ∗ χ(0,δ) ∗ f)(t) − 1

2πi

∫

Γ0,b

eλt(

1− e−λδλδ

)φ(λ)f(λ) dλ

∥∥∥∥ ≤ C21 + | log δ|δ1−α , (4.7)

for some constant C2. Since γ > ω0(f), there exists M such that ‖f(t)‖ ≤ Meγt (t ≥ 0).

Therefore,∥∥∥∥

1

δ(φ ∗ χ(0,δ) ∗ f)(t)− (φ ∗ f)(t)

∥∥∥∥

=

∥∥∥∥∫ t

−∞f(t− s)

(1

δ

∫ s

s−δφ(r) dr − φ(s)

)ds

∥∥∥∥

≤∫ t

−∞Meγ(t−s) 1

δ

∫ δ

0|φ(s− r)− φ(s)| dr ds

≤ M

δ

∫ δ

0eγt(∫ ∞

−∞e−γs|φ(s− r)− φ(s)| ds

)dr

≤ Meγt1

δ

∫ δ

0Crα dr

= C3eγtδα, (4.8)

where C3 is some constant and we have used assumption (4) to get the penultimate inequal-

ity. Further,∥∥∥∥∥

∫

Γ0,b

eλtφ(λ)f(λ) dλ−∫

Γ0,b

eλt(

1− e−λδλδ

)φ(λ)f(λ) dλ

∥∥∥∥∥

=

∥∥∥∥∥

∫

Γ0,b

eλt(e−λδ − (1− λδ)

λδ

)φ(λ)f(λ) dλ

∥∥∥∥∥≤ C4e

γtδ, (4.9)

where C4 is some constant, since we may choose Γ0,b and c so that Reλ ≤ γ and

|e−λδ − (1− λδ)| ≤ c|λ|δ2

for all λ ∈ Γ0,b. Combining the estimates (4.7), (4.8) and (4.9), we obtain∥∥∥∥∥(φ ∗ f)(t)− 1

2πi

∫

Γ0,b

eλtφ(λ)f(λ) dλ

∥∥∥∥∥

≤∥∥∥∥

1

δ(φ ∗ χ(0,δ) ∗ f)(t)− (φ ∗ f)(t)

∥∥∥∥

+

∥∥∥∥∥1

δ(φ ∗ χ(0,δ) ∗ f)(t)− 1

2πi

∫

Γ0,b

eλt(

1− e−λδλδ

)φ(λ)f(λ) dλ

∥∥∥∥∥

+1

2π

∥∥∥∥∥

∫

Γ0,b

eλtφ(λ)f(λ) dλ−∫

Γ0,b

eλt(

1− e−λδλδ

)φ(λ)f(λ) dλ

∥∥∥∥∥

≤ C5

(δαeγt +

1 + | log δ|δ1−α + δeγt

)

49

for each δ ∈ (0, 1] and t ≥ 0. Choosing δ = e−γt in the above inequality gives∥∥∥∥∥(φ ∗ f)(t)− 1

2πi

∫

Γ0,b

eλtφ(λ)f(λ) dλ

∥∥∥∥∥ ≤ C6(1 + γt)e(1−α)γt.

Since the function z 7→∫

Γ0,b

eλzφ(λ)f(λ) dλ is entire and exponentially bounded on C we

have

ζ(φ ∗ f) ≤ (1− α)γ.

For the general case, that is, when hol∞0 (f ) > ω consider the functions

fω(t) = e−ωtf(t), φω(t) = e−ωtφ(t) (t ≥ 0).

Then fω and φω satisfy the assumptions of the special case above, with γ replaced by γ−ω.Hence,

ζ(φω ∗ fω) ≤ (1− α)(γ − ω).

Since (φω ∗ fω)(t) = e−ωt(φ ∗ f)(t), we have using (3) of Proposition 3.1.8,

ζ(φ ∗ f) ≤ (1− α)γ + αω.

If f : R+ → X is regularised by a complex-valued function which is supported on the

half-line and satisfies certain conditions, then we have

Corollary 4.1.4. Suppose that f : R+ → X is exponentially bounded and measurable. Let

φ : R+ → C be locally integrable, and suppose that there exists α ∈ (0, 1] such that

1.

∫ ∞

0|φ(s)|e−ωs ds <∞ for all ω > hol∞0 (f );

2. For each ω > hol∞0 (f ), there exists Cω such that

∫ ∞

0|φ(s)− φ(s− h)|e−ωs ds ≤ Cωhα (0 < h < 1).

Then φ ∗ f is defined on R+ and

ζ(φ ∗ f) ≤ (1− α)ω0(f) + α hol∞0 (f ).

Proof. Let ω > hol∞0 (f ) and γ > max(ω, ω0(f)). Given µ with |µ| > π, let h = π/|µ|.Then, if µ > 0, we may write

∫ ∞

0φ(s)e−(ω+iµ)s ds = −

∫ ∞

0φ(s)e−ωs−iµ(s+π/µ) ds

= −∫ ∞

0φ(s− h)e−ω(s−h)−iµs ds.

50

If µ < 0, we get a similar expression. Therefore,∣∣∣∣∫ ∞

0φ(s)e−(ω+iµ)s ds

∣∣∣∣

=1

2

∣∣∣∣∫ ∞

0

((φ(s)− φ(s− h))e−ωs + (e−ωh − 1)φ(s− h)e−ω(s−h)

)e−iµs ds

∣∣∣∣

≤ 1

2

(Cωh

α +∣∣∣e−ωh − 1

∣∣∣∫ ∞

0|φ(s)|e−ωs ds

)

≤ 1

2(Cωh

α + C ′ωh)

≤ C

|µ|α ,

where C is a constant depending on ω. Thus assumption (3 ) of Theorem 4.1.3 is satisfied.

That the other three assumptions of Theorem 4.1.3 are satisfied is immediate from our

hypotheses here. Therefore,

ζ(φ ∗ f) ≤ (1− α)γ + αω.

Letting ω ↓ hol∞0 (f ) and γ ↓ ω0(f) we obtain the required result.

Corollary 4.1.5. Suppose that f : R+ → X is exponentially bounded and measurable.

Let φ : R+ → C be locally integrable, and suppose that there exist C > 0, β > ω0(f) and

α ∈ (0, 1] such that

1.

∫ ∞

0|φ(s)|eβs ds <∞;

2.

∫ ∞

−h|φ(s)− φ(s+ h)|eβs ds ≤ Chα (0 < h < 1).

Let Fφ(t) =

∫ ∞

0φ(s)f(s+ t) ds (t ≥ 0). Then

ζ(Fφ) ≤ (1− α)ω0(f) + α hol∞0 (f ).

Proof. Let φ(t) = φ(−t) (t ≤ 0) and φ(t) = 0 (t > 0). Choose ω and γ such that hol∞0 (f ) <

ω < β and max(ω, ω0(f)) < γ < β. Then

∫ 0

−∞|φ(s)|e−γs ds =

∫ ∞

0|φ(s)|eγs ds <∞

so that φ satisfies assumptions (1 ) and (2 ) of Theorem 4.1.3. Since∣∣∣∣∫ ∞

−∞e−(ω+iη)sφ(s) ds

∣∣∣∣ =

∣∣∣∣∫ ∞

0e(ω+iη)sφ(s) ds

∣∣∣∣ ,

arguments similar to those in the proof of Corollary 4.1.4 show that φ satisfies assumption

(3 ) of Theorem 4.1.3. The last assumption of this theorem is also valid for φ as∫ ∞

−∞|φ(s)− φ(s− h)|e−γs ds =

∫ ∞

−h|φ(s)− φ(s+ h)|eγs ds.

51

Consequently, we have from Theorem 4.1.3,

ζ(φ ∗ f) ≤ (1− α)γ + αω.

Since Fφ = φ ∗ f the required result is obtained by letting γ ↓ ω0(f) and ω ↓ hol∞0 (f ).

The following is an analogue of (2.11). Example 3.1.10 (2 ), shows that this result fails

for some Laplace transformable functions. Thus the assumption that f be exponentially

bounded cannot be relaxed.

Theorem 4.1.6. If f : R+ → X is an exponentially bounded, measurable function, then

ζ1(f) = ζ(F ) ≤ hol∞0 (f),

where F (t) =

∫ t

0f(s) ds.

Proof. From Theorem 3.1.11, ζ1(f) = ζ(F ). Let ω > max(

0,hol∞0 (f ))

and φ = χR+ .

Then φ satisfies all the assumptions of Theorem 4.1.3, with α = 1. Since F (t) = (φ ∗f)(t) (t ≥ 0), an application of Theorem 4.1.3 shows that ζ1(f) = ζ(F ) ≤ ω. Hence, ζ1(f) ≤max

(0,hol∞0 (f )

). Now take ω > hol∞0 (f ) and let fω(t) = e−ωtf(t). Since hol∞0 (fω) =

hol∞0 (f )− ω < 0, the above discussion shows that ζ1(fω) < 0. Thus,

ζ1(f) = ω + ζ1(fω) ≤ ω.

Letting ω ↓ hol∞0 (f ) we get the required result.

Since the non-analytic growth bound is in a sense a measure of non-analyticity, regular-

ising a function f : R+ → X by convolving with an analytic function should reduce ζ(f).

Indeed, we have,

Theorem 4.1.7. Let ψ : Σβ → C be exponentially bounded and holomorphic, where 0 <

β ≤ π2 , and suppose that

∫ 1

0|ψ′(s)| ds <∞. If f : R+ → X is measurable and exponentially

bounded, then

ζ(ψ ∗ f) ≤ hol∞0 (f) ≤ ζ(f).

Proof. The assumptions imply that ψ ∗ f is continuously differentiable. Moreover,

((ψ ∗ f)′)(λ) = λψ(λ)f(λ) (4.10)

for Reλ sufficiently large. Suppose that |ψ(z)| ≤ Meω|z| for all z ∈ Σβ . From [2, Theorem

2.6.1] , we have that ψ(λ) has a holomorphic extension to ω + Σ β+π2

and

supλ∈ω+Σβ+π

2

‖(λ− ω)ψ(λ)‖ <∞.

52

Therefore, λ 7→ λψ(λ) has a bounded holomorphic extension to ω + Σ β+π2. Thus, (4.10)

implies that hol∞0 ( (ψ ∗ f)′) ≤ hol∞0 (f ). Then an application of Theorem 4.1.6 yields

ζ(ψ ∗ f) = ζ1((ψ ∗ f)′) ≤ hol∞0 ( (ψ ∗ f)′) ≤ hol∞0 (f ).

Remark 4.1.8. The following variation of Theorem 4.1.7 is valid with an identical proof:

If f : Σβ → X is an exponentially bounded, holomorphic function with

∫ 1

0‖f ′(t)‖ dt <∞

and ψ : R+ → C is exponentially bounded and measurable, then ζ(ψ∗f) ≤ hol∞0 (ψ) ≤ ζ(ψ).

As an analogue of (4.1) we have,

Theorem 4.1.9. Let f : R+ → X and ψ : R+ → C be exponentially bounded and measur-

able. Then

ζ(ψ ∗ f) ≤ max(ζ(ψ), ζ(f)).

Proof. Let g : C→ X and φ : C→ C be exponentially bounded, entire functions. We write

ψ ∗ f = φ ∗ f + (ψ − φ) ∗ g + (ψ − φ) ∗ (f − g).

By Theorem 4.1.7, ζ(φ ∗ f) ≤ ζ(f) ≤ ω0(f − g) and by Remark 4.1.8,

ζ((ψ − φ) ∗ g) ≤ ζ(ψ − φ) ≤ ω0(ψ − φ).

Further,

ζ((ψ − φ) ∗ (f − g)) ≤ ω0((ψ − φ) ∗ (f − g)) ≤ max(ω0(ψ − φ), ω0(f − g)).

Therefore,

ζ(ψ ∗ f) ≤ max(ω0(ψ − φ), ω0(f − g)).

Taking the infimum over all possible choices of φ and g yields the result.

4.2 Boundedness of convolutions and non-resonance condi-

tions

In [7], several results have been given showing that the convolution ψ ∗ f of two bounded

measurable functions ψ and f is bounded if a certain non-resonance condition is satisfied

and any of various supplementary assumptions hold. The half-line spectrum sp(f), ([2,

Sections 4.4, 4.7], [7]), of a bounded measurable function f : R+ → X is defined to be:

sp(f) =µ ∈ R : iµ /∈ D(f )

.

53

The non-resonance condition in this context is that the intersection of the half-line spectra of

f and ψ is empty. One of the supplementary conditions used in [7, Theorem 4.1] corresponds

to f having an L1 Laplace transform with N ≥ 2 in Theorem 3.0.1. Explicitly, [7, Theorem

4.1] states that if for the function f ∈ L∞(R+,X) there exist b, k ≥ 2 such that sp(f) ⊂(−b, b) and ∫

|η|≥b

∥∥∥f (k)(iη)∥∥∥ dη <∞, (4.11)

and ψ ∈ L∞(R+) is such that sp(ψ)∩ sp(f) is empty, then ψ ∗ f is bounded; and condition

(4.11) corresponds to f having an L1 Laplace transform with N ≥ 2. Using this, in [10,

Theorem 2.4.12], boundedness of the convolution ψ ∗ f when f : R+ → X and ψ : R+ → Csatisfy the non-resonance condition is deduced under a simple assumption on the growth

bound of non-integrability of f . In fact, it is enough to impose the assumption on the

non-analytic abscissa κ(f):

Theorem 4.2.1. [10, Theorem 2.4.12] Let f : R+ → X and ψ : R+ → C be bounded mea-

surable functions. Suppose that sp(f)∩ sp(ψ) is empty and κ(f) < 0. Then the convolution

ψ ∗ f is bounded.

The same idea as in the proof of [10, Theorem 2.4.12] yields, on using Fubini’s theorem,

Corollary 4.2.2. Let f ∈ L∞(R+,X) and ψ : R+ → C be bounded and Lipschitz continu-

ous. Suppose that sp(f) ∩ sp(ψ) is empty. If ζ1(f) < 0, then ψ ∗ f ∈ L∞(R+,X).

Proof. Let f ∈ L∞(R+,X) with ζ1(f) < 0. Then, there exist α < 0 and b ≥ 0 such that

Qα,b ⊂ D(f) and f = fα,b + f2 with abs(f2) < 0. Suppose that ψ is bounded and Lipschitz

and sp(f) ∩ sp(ψ) is empty. Recall from Proposition 3.1.4 that fα,b is given by

fα,b(t) =1

2πi

∫

Γα,b

eλtf(λ) dλ,

where Γα,b is any path in D(f) from α− ib to α+ ib. From Lemma 3.1.2, we have

fα,b(µ) =1

2πi

∫

Γα,b

f(λ)

µ− λ dλ (4.12)

if Reµ is sufficiently large. The right-hand side of (4.12) defines a holomorphic function of

µ on C \ Γα,b. Given iη ∈ D(f), we may assume that Γα,b does not pass to the right of iη.

This shows that iη ∈ D(fα,b). Hence, sp(fα,b) ⊂ sp(f), and in particular, sp(fα,b)∩ sp(ψ) is

empty. Further,∣∣∣f (2)α,b(iη)

∣∣∣ =

∣∣∣∣∣1

π

∫

Γα,b

f(λ)

(iη − λ)3dλ

∣∣∣∣∣ = O(|η|−3)

54

as |η| → ∞. Thus, by [7, Theorem 4.1], ψ ∗ fα,b is bounded. Let F2(t) =

∫ t

0f2(s) ds and

F∞ =

limt→∞ F (t) if the limit exists,

0 otherwise.

Then ω0(F2 − F∞) = abs(f2) < 0, so ψ′ ∗ (F2 − F∞) is bounded. Since

(ψ′ ∗ F2)(t) = (ψ ∗ f2)(t)− ψ(0)F2(t) (t ∈ R+),

by Fubini’s Theorem, and

(ψ′ ∗ 1)(t) = ψ(t)− ψ(0) (t ∈ R+),

where 1 is the function taking the constant value 1, we conclude that

(ψ′ ∗ (F2 − F∞))(t) = F∞(ψ(0)− ψ(t)) + (ψ ∗ f2)(t)− ψ(0)F2(t),

for all t ∈ R+. Thus, (ψ ∗ f2) is in L∞(R+,X). Therefore, we conclude that ψ ∗ f =

ψ ∗ fα,b + ψ ∗ f2 is in L∞(R+,X).

In [7, Theorem 5.1] it has been shown that if f ∈ L∞(R+,X), φ ∈ L∞(R+) satisfy the

non-resonance condition, and ψ ∈ L1(R) with Fψ ∈ C2c (R), then φ ∗ ψ ∗ f is bounded. We

present a comparable result in the following corollary.

Corollary 4.2.3. Let f ∈ L∞(R+,X), φ ∈ L∞(R+) and suppose that hol∞0 (f ) < 0 and

sp(f)∩ sp(φ) is empty. Let ψ : R+ → C be locally integrable, and suppose there exist ω < 0,

C > 0 and α ∈ (0, 1] such that

1.

∫ ∞

0|ψ(s)|e−ωs ds <∞;

2.

∫ ∞

0|ψ(s)− ψ(s− h)|e−ωs ds ≤ Chα (0 < h < 1).

Then φ ∗ ψ ∗ f is bounded.

Proof. It follows from Theorem 4.1.3 and the proof of Corollary 4.1.4 that ζ(ψ ∗ f) ≤(1−α)γ+αω for every γ > 0. Choose ε > 0 such that ε < −αω. For γ =

−αω − ε1− α this gives

ζ(ψ ∗ f) ≤ −ε < 0. From assumption (1), abs(|ψ|) ≤ ω < 0. Therefore, sp(ψ ∗ f) ⊂ sp(f)

and the result follows on applying Theorem 4.2.1 to ψ ∗ f and φ.

55

4.3 Fractional integrals and non-analytic growth bounds

We apply the results estimating the non-analytic growth bound of functions of the type ψ∗fobtained in Section 4.1, to some special cases. In particular we are able to obtain estimates

for the non-analytic growth bound of the classical Riemann-Liouville fractional integral of

an exponentially bounded measurable function. Recall from [21, Chapter XIII ] that the

Riemann-Liouville fractional integral of order α of an exponentially bounded measurable

function f : R+ → X is given by

(Rαf)(t) :=

∫ t

0Γ(α)−1(t− s)α−1f(s) ds (t ≥ 0),

for α > 0, where Γ(α) is the usual Γ function given by

Γ(z) =

∫ ∞

0tz−1e−t dt (Re z > 0).

The Weyl fractional integral of f of order α is given by

(Wαf)(t) := Γ(α)−1

∫ ∞

t(s− t)α−1f(s) ds

= Γ(α)−1

∫ ∞

0sα−1f(s+ t) ds (α > 0)

whenever the integral exists.

For µ ∈ R and α > 0, define

ψα,µ(s) =

Γ(α)−1sα−1e−µs (s > 0),

0 (s ≤ 0).(4.13)

It is easy to see that for Reλ > −µ,

ψα,µ(λ) =1

(λ+ µ)α.

Thus, for α > 0, β > 0 and Reλ > −µ,

(ψα,µ ∗ ψβ,µ)ˆ(λ) = ψα,µ(λ)ψβ,µ(λ)

=1

(λ+ µ)α+β

= ψα+β,µ(λ).

Therefore, for α, β > 0, µ ∈ R,

ψα,µ ∗ ψβ,µ = ψα+β,µ. (4.14)

For an exponentially bounded and measurable function f : R+ → X the convolution

ψα,µ ∗ f, where ψα,µ is as in (4.13), exists for all α > 0 and µ ∈ R and is given by

(ψα,µ ∗ f)(t) = Γ(α)−1

∫ t

0sα−1e−µsf(t− s) ds (t ≥ 0).

56

If µ = 0, we shall write ψα := ψα,0. Writing fµ(t) = eµtf(t), it is easy to see that

(ψα,µ ∗ f)(t) = e−µt (Rαfµ) (t).

If µ > ω0(f) then the convolution ψα,µ ∗f, where ψα,µ(s) = ψα,µ(−s) is well defined and

we have

(ψα,µ ∗ f)(t) = Γ(α)−1

∫ ∞

0ψα,µ(s)f(t+ s) ds

= Γ(α)−1

∫ ∞

0sα−1e−µsf(t+ s) ds

= eµt (Wαf−µ) (t).

Thus, ψα ∗ f and ψα ∗ f (when the latter exists) are respectively the Riemann-Liouville

and Weyl fractional integrals of the exponentially bounded, measurable function f .

Definition 4.3.1. Let f : R+ → X be an exponentially bounded measurable function,

µ > ω0(f) and α > 0. We define the growth bounds ωα,µ(f) and ζα,µ(f) by

ωα,µ(f) := ω0(ψα,µ ∗ f) = ω0

(t 7→

∫ ∞

0ψα,µ(s)f(t+ s) ds

)

ζα,µ(f) := ζ(ψα,µ ∗ f) = ζ

(t 7→

∫ ∞

0ψα,µ(s)f(t+ s) ds

).

It follows immediately from the definitions that

ζα,µ(f) ≤ ωα,µ(f) ≤ ω0(f). (4.15)

An application of Corollary 4.1.5 gives the following estimate of ζα,µ(f) in terms of ω0(f)

and hol∞0 (f ). An improvement in this estimate is obtained later, in Proposition 4.3.8 (see

also Remark 4.3.4 and Remark 4.5.2).

Corollary 4.3.2. If f : R+ → X is exponentially bounded and measurable then

ζα,µ(f) ≤ (1− α)ω0(f) + α hol∞0 (f ),

for µ > ω0(f) and α ∈ (0, 1].

Proof. We check that ψα,µ satisfies the assumptions of Corollary 4.1.5 for ω0(f) < β < µ.

For such a choice of β,∫∞

0 sα−1e−(µ−β)s ds < ∞, so assumption (1 ) of Corollary 4.1.5 is

valid. Clearly, if α = 1, assumption (2 ) holds as well. Now suppose α ∈ (0, 1). Then,

putting η = µ− β we can write

Γ(α)

∫ ∞

−h|ψα,µ(s)− ψα,µ(s+ h)|eβs ds ≤ I1 + I2 + I3 + I4, (4.16)

57

where 0 < h < 1 and

I1 =

∫ ∞

he−ηs(s+ h)α−1|1− e−µh| ds

≤ |1− e−µh|∫ ∞

he−ηssα−1 ds

≤ hα

η(e|µ| − 1)e−ηh

≤ Cµhα;

I2 =

∫ ∞

he−ηs|sα−1 − (s+ h)α−1| ds

≤ h

∫ ∞

h(1− α)e−ηssα−2 ds

≤ C ′hα;

I3 =

∫ h

0eβs|sα−1e−µs − (s+ h)α−1e−µ(s+h)| ds

≤ C

∫ h

0sα−1e−ηs ds

≤ Chα

α

I4 =

∫ 0

−h(s+ h)α−1e−µ(s+h)+βs ds

≤ e−ηh∫ h

0sα−1e−ηs ds

≤ C ′′hα

α,

Cµ, C,C′, C ′′ being constants. Therefore, assumption (2 ) of Corollary 4.1.5 remains valid

for ψα,µ. This implies that

ζ

(t 7→

∫ ∞

0ψα,µ(s)f(t+ s) ds

)≤ (1− α)ω0(f) + α hol∞0 (f ).

Thus, for an exponentially bounded, measurable function f with ω0(f) < 0, we can

obtain from the above corollary, an estimate for the non-analytic growth bound of ψα ∗ f,the Weyl fractional integral of f . We now obtain an estimate for the non-analytic growth

bound of the Riemann-Liouville fractional integral, ψα ∗ f of an exponentially bounded,

measurable function f , similar to the estimate obtained in Theorem 4.3.2, in terms of the

growth bound ω0(f) and hol∞0 (f ).

Theorem 4.3.3. Let f be an exponentially bounded, measurable function and α ∈ (0, 1].

Then

ζ(ψα ∗ f) ≤ (1− α)ω0(f) + α hol∞0 (f ). (4.17)

58

Proof. If hol∞0 (f ) > 0, then the assumptions of Corollary 4.1.4 are satisfied and there-

fore (4.17) holds in this case. For α = 1, (4.17) follows from Theorem 4.1.6. Let α ∈(0, 1) and µ ∈ R. Then ψα − ψα,µ satisfies the conditions of Theorem 4.1.7. Therefore,

ζ ((ψα − ψα,µ) ∗ f) ≤ hol∞0 (f ). Take ω > max(−µ,hol∞0 (f )

)and γ > max(ω, ω0(f)).

Then calculations similar to those in the proof of Theorem 4.3.2 show that all the assump-

tions of Theorem 4.1.3 are satisfied by ψα,µ. Thus,

ζ(ψα,µ ∗ f) ≤ (1− α)γ + αω0(f).

Therefore,

ζ(ψα ∗ f) ≤ max(ζ(ψα,µ ∗ f),hol∞0 (f )

)

≤ (1− α)γ + αω. (4.18)

Taking the infimum over all possible choices of µ, ω and γ yields (4.17).

Remark 4.3.4. It is possible to further improve the estimates for ζα,µ(f) (µ sufficiently

large) and ζ(ψα ∗f) obtained in Theorem 4.3.2 and Theorem 4.3.3. First consider ζ(ψα ∗f).

Note that if g is holomorphic and exponentially bounded in a sector, then since

(ψα ∗ g)(λ) =g(λ)

λα,

for Reλ sufficiently large, it follows from [2, Theorem 2.6.1] that ψα ∗g is holomorphic and

exponentially bounded in a sector. Therefore, ζ(ψα ∗ g) = −∞. Hence, if f : R+ → X is

an exponentially bounded and measurable function and g is holomorphic and exponentially

bounded in a sector, we have on applying Theorem 4.3.3 to f − g,

ζ(ψα ∗ f) ≤ max (ζ(ψα ∗ (f − g)), ζ(ψα ∗ g))

= ζ(ψα ∗ (f − g))

≤ (1− α)ω0(f − g) + α hol∞0(f − g

)

≤ (1− α)ω0(f − g) + α hol∞0 (f). (4.19)

Taking the infimum over all possible choices of g in (4.19) we obtain, for 0 < α ≤ 1, and

f : R+ → X exponentially bounded and measurable,

ζ(ψα ∗ f) ≤ (1− α)ζ(f) + α hol∞0 (f). (4.20)

Next we consider ζα,µ(f) = ζ(ψα,µ ∗ f), µ > ω0(f), α > 0. Let g be entire and exponentially

bounded in a sector, with ω0(g) < µ. Then ψα,µ ∗ g is also exponentially bounded and

entire. Thus, using Theorem 4.3.2, Proposition 3.1.4 (6) and arguments similar to those in

the above paragraph,

ζα,µ(f) ≤ (1− α)ζ(f) + α hol∞0 (f). (4.21)

59

Observe that

ζα,µ(f) = ζ(ψα,µ ∗ f) for µ > ω0(f), and α = 1. (4.22)

Indeed, for an exponentially bounded, measurable function f : R+ → X

ζ1,µ(f) = ζ

(t 7→

∫ ∞

0e−µsf(s+ t) ds

)

= ζ

(t 7→ eµt

∫ ∞

te−µsf(s) ds

)

= µ+ ζ

(t 7→

(∫ ∞

0e−µsf(s) ds−

∫ t

0e−µsf(s) ds

))

= µ+ ζ

(t 7→

∫ t

0e−µsf(s) ds

)

= µ+ ζ1(t 7→ e−µtf(t))

= µ− µ+ ζ1(f)

= ζ(F ),

where F (t) =

∫ t

0f(s) ds and we have used Theorem 4.1.6 to obtain the last equality. More-

over,

ζ(ψ1,µ ∗ f) = ζ

(t 7→

∫ t

0e−µ(t−s)f(s) ds

)

= ζ

(t 7→ e−µt

∫ t

0eµrf(r) dr

)

= −µ+ ζ1(g)

= ζ1(f)

where g(t) = eµtf(t). Thus, for all µ > ω0(f),

ζ1,µ(f) = ζ1(f) = ζ(ψ1,µ ∗ f). (4.23)

We do not know whether (4.22) is true for α ∈ (0, 1) or not. Next, we prove that ζ(ψα,µ ∗f)

is independent of the choice of µ ∈ R for all α > 0. For α = 1, this has already been shown

in (4.23).

Theorem 4.3.5. Let f : R+ → X be an exponentially bounded, measurable function, α > 0

and µ1, µ2 ∈ R. Then

ζ(ψα,µ1 ∗ f) = ζ(ψα,µ2 ∗ f).

Proof. Let gi = ψα,µi ∗ f, i = 1, 2 and ei = ψα,µi , i = 1, 2. We will show that there exists

a function h : R+ → C, which extends to an entire, exponentially bounded function on C,and satisfies the equation

e1(t)− e2(t) = (e2 ∗ h)(t) (t ≥ 0). (4.24)

60

Then g1 − g2 = h ∗ g2, so that ζ(g1) ≤ max(ζ(g2), ζ(h ∗ g2)). Since g2 is an exponentially

bounded, measurable function, it will follow from Theorem 4.1.7 that ζ(h ∗ g2) ≤ ζ(g2).

Therefore, ζ(g1) ≤ ζ(g2). Interchanging the roles of e1 and e2 then gives the required

result. To establish the existence of such an h we observe that the Laplace transform of

such a function must satisfy, for Reλ > max(−µ1,−µ2, µ2 − 2µ1),

h(λ) =e1(λ)− e2(λ)

e2(λ)

=

(µ2 + λ

µ1 + λ

)α− 1

=

(1 +

C

µ1 + λ

)α− 1, (C = µ2 − µ1)

=∞∑

n=1

α(α− 1) . . . (α− n+ 1)Cn

n!(µ1 + λ)n, (4.25)

the series on the right-hand side above being absolutely convergent. Let

hn(t) = e−µ1tα(α− 1) . . . (α− n+ 1)Cntn−1

n!(n− 1)!(n = 1, 2...; t ≥ 0).

Then the series∑∞

n=1 hn(t) is absolutely convergent for each t ≥ 0. Set

h(t) =∞∑

n=1

hn(t), t ≥ 0.

Then ω0(h) ≤ max(−µ1,−µ2, µ2 − 2µ1) = −µ1 + |µ2 − µ1|. Moreover, for Reλ > −µ1 +

|µ2 − µ1|,

h(λ) =∞∑

n=1

α(α− 1) . . . (α− n+ 1)Cn

n!(µ1 + λ)n,

and h has an entire, exponentially bounded extension. Thus the proof is complete.

We note here that even though the corresponding non-analytic growth bound does not

depend on µ, the exponential growth bound ω0 (ψα,µ ∗ f) is not necessarily independent of

µ ∈ R. For example, if f ≡ 1, and α = 1, then (ψα,µ ∗ f)(t) = 1µ(1− e−µt), for µ ∈ R.

In the proof of the next Theorem, given a function g : R+ → C, the function g will be

given by

g(s) =

g(−s) (s ≤ 0)

0 (s > 0).

Theorem 4.3.6. Let f : R+ → X be an exponentially bounded, measurable function and

γ > 0. Then ζ(t →

∫∞0 ψγ,µ(s)f(s + t) ds

)= ζγ,µ(f) is independent of the choice of

µ > ω0(f).

61

Proof. First suppose that ω0(f) = 0. Let µ1 > µ2/2 > 0. Following the same notation as in

Theorem 4.3.5, we have ω0(h) ≤ −µ1 + |µ2 − µ1| < 0 and h satisfies all the conditions for

the function φ in Corollary 4.1.5, with α = 1, β ∈ (0,−ω0(h)) with respect to the function

G2, where

Gi(t) =

∫ ∞

0ψγ,µi(s)f(s+ t) ds = (ei ∗ f)(t),

and ei = ψγ,µi , i = 1, 2. Moreover, (4.24) implies that G1(t) − G2(t) = (e2 ∗ h) ∗ f. Since

(e2 ∗ h) = e2 ∗ hG1(t)−G2(t) =

∫ ∞

0h(s)G2(s+ t) ds. (4.26)

Therefore, applying Corollary 4.1.5 to G2 and h, we obtain

ζ

(t 7→

∫ ∞

0h(s)G2(s+ t) ds

)≤ hol∞0 (G2) ≤ ζ(G2).

Since

ζ(G1) ≤ max

(ζ(G2), ζ

(t 7→

∫ ∞

0h(s)G2(s+ t) ds

)),

we may conclude that

ζγ,µ1(f) ≤ ζγ,µ2(f), if µ1 > µ2/2. (4.27)

Thus, if 0 < µ1/2 < µ2 < 2µ1,

ζγ,µ1(f) = ζγ,µ2(f). (4.28)

Repeated application of this equality yields

ζγ,µ1(f) = ζγ,µ2(f), for all µ1, µ2 > 0.

Now consider the case when ω0(f) is arbitrary, but ω0(f) > −∞. Setting a = ω0(f)

and applying the result obtained in the above paragraph to the function e−a·f we have for

µ1, µ2 > 0,

ζγ,µ1(e−a·f) = ζγ,µ2(e−a·f),

hence

ζγ,µ1+a(f) = ζγ,µ2+a(f), for all µ1, µ2 > 0.

Therefore,

ζγ,µ1(f) = ζγ,µ2(f) for all µ1, µ2 > a.

If ω0(f) = −∞, then ζγ,µ2(f) = −∞ = ζγ,µ2(f).

Unlike ω0(ψγ,µ ∗ f), ωγ,µ(f) is independent of the choice of µ > ω0(f) for any expo-

nentially bounded, measurable function f. We prove this in the following, using the same

strategy as employed in Theorem 4.3.6.

62

Theorem 4.3.7. Let f : R+ → X be exponentially bounded and measurable and α > 0.

Then ωα,µ(f) is independent of the choice of µ > ω0(f).

Proof. We shall use the same notation as in Theorem 4.3.6. First suppose ω0(f) = 0. Then,

for µ1 > µ2/2 > 0, we have from (4.26),

ω0(G1) ≤ max

(ω0(G2), ω0

(t 7→

∫ ∞

0h(s)G2(t+ s) ds

)). (4.29)

Since ω0(h) ≤ −µ1 + |µ2 − µ1|, and ω0(G2) ≤ 0, it is easy to see that

ω0

(t 7→

∫ ∞

0h(s)G2(t+ s) ds

)≤ ω0(G2).

It follows then from (4.29) that

ωα,µ1(f) = ω0(G1) = ω0(G2) = ωα,µ2(f) (µ1/2 < µ2 < 2µ1). (4.30)

Using (4.30) repeatedly we therefore obtain ωα,µ1(f) = ωα,µ2(f), µ1, µ2 > 0, if ω0(f) = 0.

The other cases may be treated as in Theorem 4.3.6.

We collect some properties of the the bounds ζα,µ(f) and ζ(ψα,µ ∗ f) in the following

proposition. Note that (1) of Proposition 4.3.8 improves the estimates obtained in Corollary

4.3.2 and Corollary 4.3.3.

Proposition 4.3.8. Suppose f : R+ → X is an exponentially bounded, measurable function.

Let µ > ω0(f) and ν ∈ R. Then the following hold:

1. For α ∈ (0, 1],

ζα,µ(f) ≤ (1− α)ζ(f) + α hol∞0 (f);

ζ(ψα,ν ∗ f) ≤ (1− α)ζ(f) + α hol∞0 (f).

2. ζα,µ(f) ≤ ζ(f), and ζ(ψα,ν ∗ f) ≤ ζ(f), for all α > 0.

3. The maps α 7→ ζ(ψα,ν ∗ f), α 7→ ζα,µ(f) and α 7→ ωα,µ(f) are decreasing on (0,∞).

Proof. (1): Theorem 4.3.5 and Theorem 4.3.6 show that ζ(ψα,ν ∗ f) and ζα,µ(f) do not

depend on the choice of ν ∈ R and µ > ω0(f), respectively. Therefore, (1) follows from

(4.20) and (4.21).

(2): Since hol∞0 (f) ≤ ζ(f), it follows immediately from (1) that ζ(ψα,ν ∗ f) ≤ ζ(f) for

α ∈ (0, 1]. Let β ∈ (1, 2]. Then we can choose α1 ∈ (0, 1] such that β − α1 ∈ (0, 1]. Using

(4.14) we have,

ζ(ψβ,ν ∗ f) = ζ(ψβ−α1,ν ∗ (ψα1,ν ∗ f))

≤ ζ(ψα1,ν ∗ f)

≤ ζ(f).

63

Iterating this process, we obtain

ζ(ψα,ν ∗ f) ≤ ζ(f) for all α > 0. (4.31)

Similarly, we have from (1) that ζα,µ(f) ≤ ζ(f) for α ∈ (0, 1]. Moreover, straightforward

calculations show that for α, γ > 0,

(ψγ,µ ∗ (ψα,µ ∗ f))(t) =

∫ ∞

t(ψγ,µ ∗ ψα,µ)(s− t)f(s) ds

=

∫ ∞

t(ψγ+α,µ)(s− t)f(s) ds

= (ψγ+α,µ ∗ f)(t). (4.32)

Therefore, arguments similar to those in the previous paragraph yield

ζα,µ(f) ≤ ζ(f), for all α > 0. (4.33)

(3): Let β > α > 0. Using (4.14) and (4.31) we have

ζ(ψβ,ν ∗ f) = ζ(ψβ−α,ν ∗ (ψα,ν ∗ f))

≤ ζ(ψα,ν ∗ f).

Using (4.32) and (4.33), we have for β > α > 0,

ζβ,µ(f) = ζ(ψβ−α+α,µ ∗ f)

= ζ(ψβ−α,µ ∗ (ψα,µ ∗ f))

= ζβ−α,µ(ψα,µ ∗ f)

≤ ζ(ψα,µ ∗ f)

= ζα,µ(f).

Similarly, since ωα,µ(f) ≤ ω0(f) for all α > 0, µ > ω0(f) it follows that ωβ,µ(f) ≤ ωα,µ(f).

Both ζα,µ(f) and α → ζ(ψα,µ ∗ f) behave well under translations and rescaling. More

specifically, we have,

Corollary 4.3.9. Let f : R → X be an exponentially bounded, measurable function and

α > 0. Then the following hold:

1. For ω ∈ R and µ > max(ω0(f),−ω + ω0(f)),

ζα,µ(e−ω·f

)= −ω + ζα,µ(f);

64

2. For ω, µ ∈ R,ζ(ψα,µ ∗ e−ω·f

)= −ω + ζ(ψα,µ ∗ f);

3. For τ ≥ 0, ζα,µ(fτ ) = ζα,µ(f) (µ > ω0(f)) and ζ(ψα,µ ∗ fτ ) = ζ(ψα,µ ∗ f) (µ ∈ R),

where fτ (t) = f(t+ τ) (t ≥ 0.

Proof. (1) : Let ω ∈ R and µ > max(ω0(f),−ω, ω0(f)). Then from the definition of ζα,µ,

ζα,µ(e−ω·f

)= ζ

(t 7→

∫ ∞

0ψα,µ(s)e−ω(s+t)f(s+ t) ds

)

= ζ

(t 7→ e−ωt

∫ ∞

0ψα,µ+ω(s)f(s+ t) ds

)

= −ω + ζα,µ+ω(f)

= −ω + ζα,µ(f),

where we have used Theorem 4.3.6 to obtain the last equality.

(2): This follows similarly from the definition, on making use of Theorem 4.3.5.

(3) : Writing hτ (t) = h(t+ τ) (t ≥ 0), for any function h : R+ → X, we see that

ζα,µ(fτ ) = ζ

(t 7→

∫ ∞

0ψα,µ(s)f(t+ τ + s) ds

)

= ζ((ψα,µ ∗ f

)τ)

= ζ(ψα,µ ∗ f),

where the last equality follows from Proposition 3.1.8 (3).

For t ≥ 0, we may write

(ψα,µ ∗ fτ )(t) =

∫ t

0ψα,µ(s)f(t+ τ − s) ds

=

∫ t+τ

0ψα,µ(s)f(t+ τ − s) ds−

∫ t+τ

tψα,µ(s)f(t+ τ − s) ds

= (ψα,µ ∗ f)(t+ τ)−∫ τ

0ψα,µ(t+ τ − s)f(s) ds

= (ψα,µ ∗ f)τ (t)−∫ τ

0ψα,µ(t+ τ − s)f(s) ds.

Since the function t 7→∫ τ

0 ψα,µ(t+ τ − s)f(s) ds extends to an exponentially bounded,

holomorphic function on a sector, we conclude that

ζ(ψα,µ ∗ fτ ) = ζ((ψα,µ ∗ f)τ ) = ζ(ψα,µ ∗ f).

For an exponentially bounded, measurable function f , since both ωα,µ(f) and ζα,µ(f)

are independent of the choice of µ > ω0(f), we can make the following definitions:

65

Definition 4.3.10. Let f : R+ → X be a measurable and exponentially bounded function

and α > 0. Fix µ > ω0(f). We define ωα(f) := ωα,µ(f) to be the fractional growth bound

of f of order α and ζα(f) := ζα,µ(f) to be the non-analytic fractional growth bound of f of

order α.

4.4 Fractional growth bounds for C0-semigroups

In this section, we take a closer look at the quantity ζα,µ(T) for α > 0 and µ ∈ R in the

particular case when T is a C0-semigroup. Throughout this section T shall denote a C0-

semigroup defined on X with generator A unless otherwise stated. It turns out that these

bounds are related to the non-analytic growth bound of T in much the same way as the

fractional growth bounds ωα(T) of T are related to the growth bound of T. We first recall

some definitions:

Let µ > ω0(T). For α > 0 the fractional powers R(µ,A)α are given by

R(µ,A)αx :=sinπα

π

∫ ∞

0t−αR(t+ µ,A)x dt

=1

Γ(α)

∫ ∞

0tα−1e−µtT(t)x dt, (4.34)

for all x ∈ X. The integrals on the right-hand side are absolutely convergent for all x ∈ X

and define injective bounded linear operators on X. Then the operators (µ −A)α, α > 0

are defined as the inverse of R(µ,A)α with domain D((µ − A)α) = Ran(R(µ,A)α). For

details of fractional powers see [41, Section 2], [2, Section 3.8] and [31]. Observe here that

R(µ,A)αx = (Wα(e−µ·T)) (0)x.

The fractional growth bounds ωα(T) associated with the semigroup T are defined by

ωα(T) = ω0 (T(·)R(µ,A)α) (4.35)

= sup ω0 (T(·)R(µ,A)αx) : x ∈ X= sup ω0(T(·)x) : x ∈ D((µ−A)α) .

It has been shown in [31] that for µ1, µ2 > ω0(T),

D((µ1 −A)α) = D((µ2 −A)α), (4.36)

for each α > 0. It follows therefore, that the growth bounds ωα(T) are independent of the

choice of µ > ω0(T).

From Theorem 4.3.7 we can deduce a proof for this fact without resorting to (4.36).

Indeed, using (4.34) we obtain

ω0(T(·)R(µ,A)α) = ωα,µ(T),

66

and Theorem 4.3.7 shows that ωα,µ(T) is independent of µ > ω0(f).

Analogous to the description of ωα(T) in (4.35) we may consider the non-analytic growth

bounds of the function t 7→ T(t)R(µ,A)α for α > 0 where µ > ω0(T). (2 ) of Corollary

3.2.2 yields

ζ (T(·)R(µ,A)α) = supζ (T(·)R(µ,A)αx) : x ∈ X

. (4.37)

For x ∈ X, µ > ω0(f) and α > 0 it follows from (4.34), for all t ≥ 0,

T(t)R(µ,A)αx =1

Γ(α)

∫ ∞

0sα−1e−µsT(t+ s)x ds

=

∫ ∞

0ψα,µ(s)T(t+ s)x ds

= (ψα,µ ∗T(·)x)(t).

Therefore, from Theorem 4.3.6 it follows that for each x ∈ X, ζ (T(·)R(µ,A)αx) and thus

ζ (T(·)R(µ,A)α) is independent of the choice of µ > ω0(T).

Another way of determining that ζ (T(·)R(µ,A)αx) is independent of the choice of

µ > ω0(T), is as follows: Let µ1, µ2 > ω0(T) and α > 0. Then Ran(R(µ1,A)α) =

Ran(R(µ2,A)α), in view of (4.36). Thus C := (µ2 − A)αR(µ1,A)α is a closed opera-

tor, and by the Closed Graph Theorem, C ∈ L(X). Further, R(µ1,A)α = CR(µ2,A)α.

Therefore, for x ∈ X,

ζ(T(·)R(µ1,A)αx) = ζ(CT(·)R(µ2,A)αx)

≤ ζ(T(·)R(µ2,A)αx).

Interchanging the roles of µ1 and µ2 shows that for each x ∈ X, ζ (T(·)R(µ,A)αx) is

independent of µ.

We define the fractional non-analytic growth bound of order α, ζα(T), α > 0 associated

with a C0-semigroup T by

ζα(T) := ζ (T(·)R(µ,A)α)

= supζα,µ(T(·)x) : x ∈ X

= ζα,µ(T),

where µ > ω0(T). It is clear from the above definition and (4.37) that for µ > ω0(T),

ζα(T) = supζ (T(·)x) : x ∈ D((µ−A)α)

. (4.38)

Since D((µ−A)α) does not depend on the choice of µ this gives another proof for the fact

that ζα(T) is independent of µ.

In the discussion so far, we have established that ζα(T) and ωα(T) are respectively the

non-analytic and exponential growth bounds of the operator-valued function obtained on

67

convolving the semigroup T with functions of the type ψα,µ. Therefore, Corollary 4.3.2 and

(3.3) yield corresponding relations for ζα(T) and ωα(T). These are recorded in the following

proposition along with other basic properties of ζα(T).

Proposition 4.4.1. Let T be a C0-semigroup with generator A and α ≥ 0. Then

1. ζα(T) ≤ ωα(T) ≤ ω0(T);

2. α 7→ ζα(T) is decreasing on [0,∞);

3. s∞(A) ≤ ζα(T) ≤ ζ(T);

4. ζα(T) ≤ (1− α)ζ(T) + αs∞0 (A) (0 ≤ α ≤ 1);

5. ωα(T) = max(ζα(T), s(A)),

where ζ0(T) = ζ(T).

Proof. (1 ),(2 ) and (4 ): These are special cases of the corresponding properties already

discussed for exponentially bounded, measurable functions in Proposition 4.3.8.

(3 ): From (1 ) of Proposition 3.1.8 we have, for µ > ω0(T),

ζα(T) = ζ (T(·)R(µ,A)α) ≥ hol∞(T(·)R(µ,A)α

)= s∞(A).

(5 ): This follows from (3.3) of Proposition 3.1.8 on noting that hol(T(·)R(µ,A)α

)=

s(A) for all α > 0.

Weis and Wrobel [50] showed that the map α 7→ ωα(T) is convex on [0,∞) and contin-

uous on (0,∞) and that ω1(T) ≤ s0(A). It follows then that

ωα(T) ≤ (1− α)ω0(T) + αs0(A) (0 < α ≤ 1).

The following result gives a sharper inequality :

Theorem 4.4.2. Let T be a C0-semigroup on X with generator A. Then,

ωα(T) ≤ max ((1− α)ω0(T) + αs∞0 (A), s(A)) (0 < α ≤ 1).

In particular, if s∞0 (A) = −∞ then ωα(T) = s(A), for all α ∈ (0,∞).

Proof. This follows immediately on combining (4) and (5) of Proposition 4.4.1.

Our results in Proposition 4.4.1 (5) and Theorem 4.4.2 improve those obtained in [10,

Theorem 6.4.6 and Corollary 6.4.5] respectively where the techniques of [8] have been used

to show that if s∞0 (A) = −∞, then ω 12(T) = s(A) and that for ω > ω0(T) and α ∈ (1

2 , 1],

ζ(R(ω,A)αT(·)) ≤ ω0(T) + (1− 2α)(ω0(T)− s∞0 (A)).

68

We remark that Theorem 4.4.2 cannot be extended to the interval [0,∞). In fact, [24,

Example 4.2] gives a semigroup of positive operators satisfying ω0(T) = 0 while s∞0 (A) =

s(A) = −∞ where the semigroup T is given by T(t)(f)(s) = f(t + s), f ∈ X1 ∩ X2,

X1 = Lq(R+), and X2 = Lp(R+, ept2dt) with 1 < p < q < ∞. The generator of this

semigroup has empty spectrum.

Analogous to the convexity of ωα(T) in α we obtain a result concerning the convexity

of the maps α 7→ ζα(T). The strategy of the proof is similar to that of [49, Lemma 3.5].

Theorem 4.4.3. Let T be a C0-semigroup on X with generator A. Then the map α 7→ζα(T) from [0,∞)→ R is convex.

Proof. Let 0 ≤ α < β and 0 < θ < 1. Put γ = (1−θ)α+θβ. Take µ > ω0(T). Let a > ζα(T),

b > ζβ(T). From (4) of Proposition 4.4.1 it follows that s∞(A) < min(a, b). Choose l,m ≥ 0

such that and Ql,m ⊂ ρ(A). Then there is a constant M such that

‖T(t)R(µ,A)α −Tl,m(t)R(µ,A)α‖ ≤ Meat∥∥∥T(t)R(µ,A)β −Tl,m(t)R(µ,A)β∥∥∥ ≤ Mebt

for all t ≥ 0.

For x ∈ X, the convexity estimate [31, Theorem 8.1] gives

‖T(t)R(µ,A)γx−Tl,m(t)R(µ,A)γx‖

≤ K ‖R(µ,A)α (T(t)−Tl,m(t))x‖1−θ∥∥∥R(µ,A)β (T(t)−Tl,m(t))x

∥∥∥θ

(4.39)

≤ KMe((1−θ)a+θb)t‖x‖,

where K is a constant. Thus, ζγ(T) ≤ (1 − θ)a + θb whenever a > ζα(T) and b > ζβ(T)

and the result follows.

Remark 4.4.4. From the proof of Theorem 4.4.3, in particular (4.39), it follows that the

map α 7→ ζα (T(·)x) is convex on (0,∞) for each x ∈ X.

4.5 Convexity and fractional bounds for vector-valued func-

tions

From Theorem 4.4.3 we can deduce the corresponding result about convexity of the map

α 7→ ζα(f).

Theorem 4.5.1. Let f : R+ → X be exponentially bounded and measurable. Then α 7→ζα(f) is convex on (0,∞).

69

Proof. Let f : R+ → X be measurable, with ω0(f) < ∞ and choose w > ω0(f). Suppose

first that f is continuous. Define

Cw :=g ∈ C(R+,X) : lim

t→∞e−wtg(t) = 0

.

Then Cw is a Banach space with norm given by ‖g‖ = supt≥0 ‖e−wtg(t)‖ and f ∈ Cw. Let

S : R+ → L(Cw) denote the shift C0-semigroup given by

(S(t)g)(s) = g(s+ t).

Then, ζ(g) = ζ(S(·)g) for all g ∈ Cw. Indeed, if ζ(S(·)g) < a, then there exists b > 0 such

that Qa,b ⊂ D((S(·)g), supλ∈Qa,b ‖(S(·)g)(λ)‖ <∞ and

S(t)g =1

2π

∫

Γa,b

eλt(S(·)g)(λ) dλ+ h2(t) (t ≥ 0), (4.40)

where h2 : R+ → Cw, with ω0(h2) < a. If λ ∈ D(S(·)g), then λ ∈ D(g) and (S(·)g)(λ)(0) =

g(λ). Therefore, from (4.40) we have

g(t) = (S(t)g)(0)

=1

2πi

∫

Γa,b

eλtg(λ) dλ+ h2(t)(0) (t ≥ 0).

The map z 7→ 12πi

∫Γa,b

eλz g(λ) dλ is analytic and exponentially bounded in a sector Σθ, θ >

0, and t 7→ h2(t)(0) is a map from R+ → X, satisfying

‖h2(t)(0)‖ ≤ ‖h2(t)‖ ≤Meat,

for some constant M. Thus, ζ(g) ≤ a.Conversely, ζ(S(·)g) ≤ ω0(S) = w. Suppose that ζ(S(·)g) < a < w. Then there exists

b > 0 such that Qa,b ⊂ D(g), supλ∈Qa,b ‖g(λ)‖ <∞, and

g(t) =1

2πi

∫

Γa,b

eλtg(λ) dλ+ g2(t) (t ≥ 0),

with ω0(g2) < a. The path Γa,b may be chosen to be contained in the half-plane λ : Reλ <

w. Thus, for every s ≥ 0, we may write, for t ≥ 0,

(S(t)g)(s) =1

2πi

∫

Γa,b

eλ(t+s)g(λ) dλ+ (S(t)g2)(s). (4.41)

Further, ∫

Γa,b

eλ(t+s)g(λ) dλ =

∫

Γa,b

eλteλ(s)g(λ) dλ,

where, eλ(s) := eλs (s ∈ R+). Thus, from (4.41),

S(t)g =1

2πi

∫

Γa,b

eλteλg(λ) dλ+ S(t)g2.

70

For z ∈ C, the integral 12πi

∫Γa,b

eλzeλg(λ) dλ exists as a Bochner integral in Cw, and this

defines an exponentially bounded entire function with values in Cw. Moreover,

‖S(t)g2‖ = sups≥0

e−ws‖g2(s+ t)‖

= sups≥0

e(a−w)seate−a(s+t)‖g2(s+ t)‖

≤ eat sups≥0

e−as‖g2(s)‖.

Hence, ω0(S(t)g2) ≤ a, so ζ(S(·)g) ≤ a. Thus ζ(g) = ζ(S(·)g), for all g ∈ Cw.Let α > 0 and µ > ω0(f). Straightforward calculations show that

(ψα,µ ∗ S(·)f)(t) = S(t)(ψα,µ ∗ f).

Noting that ψα,µ ∗ f ∈ Cw, it follows that

ζα(f) = ζ(ψα,µ ∗ f) = ζ(S(·)(ψα,µ ∗ f)) = ζα(S(·)f).

Then the result is a consequence of Remark 4.4.4 for such an f. Now consider the general

case. For any ε > 0, and µ > ω0(f), ψε,µ ∗ f is continuous (by the Dominated Convergence

Theorem) and ω0(ψε,µ ∗ f) < w. Thus, ψε,µ ∗ f ∈ Cw. Moreover,

ψα,µ ∗ (ψε,µ ∗ f) = ψα+ε,µ ∗ f, α > 0.

It follows from the first case that α 7→ ψα+ε,µ ∗ f is convex on (0,∞) and the result follows.

Remark 4.5.2. We can improve the estimate for ζα,µ(f) obtained in Corollary 4.3.2 using

the result obtained above. We claim that for f : R+ → X exponentially bounded and

measurable, µ > ω0(f) and α ∈ (0, 1],

ζα,µ(f) ≤ (1− α)ζ(f) + αζ1(f).

Indeed, due to convexity of α 7→ ζα,µ(f), we have for α ∈ (0, 1],

ζα,µ(f) ≤ (1− α)ζ0,µ(f) + αζ1,µ(f)

= (1− α)ζ(f) + αζ1(f)

≤ (1− α)ζ(f) + α hol∞0 (f).

Here the last inequality follows from Theorem 4.1.6. Thus, we have an alternative proof of

Corollary 4.3.2.

71

Remark 4.5.3. The technique used in Theorem 4.5.1 to deduce convexity of α 7→ζα(f), α > 0 when f is an exponentially bounded, measurable function taking values in

X, from the corresponding result for orbits of C0-semigroups, may also be employed to

give another proof of the fact that ζα,µ(f) does not depend on µ > ω0(f) (see Theo-

rem 4.3.6). Recall that on page 67, it has been shown that if T is a C0-semigroup, then

ζ(T(·)R(µ,A)αx) is independent of the choice of µ > ω0(T) using two different methods.

One of these methods does not make use of the properties of ζα,µ(f).

72

Chapter 5

Fourier multipliers and the

non-analytic growth bound

As we have mentioned before, if T is a C0-semigroup defined on a Hilbert space X then

ω0(T) < 0 if and only if s0(A) < 0 ([2, Theorem 5.2.1]). This result does not extend

to arbitrary Banach spaces. In [32] it has been shown that the above statement is true

on arbitrary Banach spaces if the condition s0(A) < 0 is supplemented by the condition

that the resolvent restricted to the imaginary axis is a Fourier multiplier for Lp(R,X) for

some p ∈ [1,∞). Analogously, as we have discussed in subsection 3.2.2, the assertion that

ζ(T) < 0 if and only if s∞0 (A) < 0 for a semigroup T defined on a Hilbert space may

not hold for arbitrary Banach spaces. We look for supplementary Fourier multiplier-type

conditions, similar to the ones in the case of the exponential growth bound, which together

with s∞0 (A) < 0 would ensure that ζ(T) < 0 for a C0-semigroup defined on any Banach

space.

Fourier multipliers have been used extensively for characterising stability and hyper-

bolicity of strongly continuous semigroups. M. Hieber [27] has given a characterisation of

uniform stability in terms of these concepts while in [32], exponential growth bounds of

higher orders for a C0-semigroup have been described in terms of Fourier multiplier prop-

erties of the resolvent of its generator. Amongst the first results connecting the theory of

Fourier multipliers to stability and hyperbolicity of C0-semigroups were those by Kaashoek

and S. Verduyn Lunel [30]. L. Weis in [49] used Fourier multiplier properties of the resolvent

on Besov spaces to give alternative proofs of some stability results. An extensive study of

the relation between Fourier multipliers and hyperbolicity has been done by Y. Latushkin

and R. Shvydkoy [34]. Here we study the relation between the non-analytic growth bound

of a semigroup and Fourier multipliers, drawing on ideas from [34] and [30]. The results

obtained in the first section lead on to a characterisation of this growth bound in terms

73

of mild solutions of certain inhomogeneous Cauchy problems. This characterisation of the

non-analytic growth bound is comparable to the characterisation of hyperbolicity obtained

in [33], [16, Section 4.3], using the theory of evolution families.

We first recall some definitions and results. For ψi ∈ S(R) and xi ∈ X, i = 1, 2, ..., n,

the element Ψ =∑n

i=1 ψi ⊗ xi in S(R,X) will be given by

Ψ(t) =

n∑

i=1

(ψi ⊗ xi)(t) =

n∑

i=1

ψi(t)xi (t ∈ R).

We define the space L1s(R,L(X)) by

L1s(R,L(X)) = T : R −→ L(X) such that T(·)x is Bochner measurable for all x ∈ X

and there exists g ∈ L1(R) with ‖T(t)‖ ≤ g(t),

and the space FL1s(R,L(X)) by setting

FL1s(R,L(X)) =

FT : T ∈ L1

s(R,L(X)).

Here, the Fourier transform is taken in the strong operator topology. Thus, for T ∈L1s(R,L(X)), FT is given by

(FT)(s)x =

∫ ∞

−∞e−istT(t)x dt.

A function m ∈ L∞(R,L(X)) is called a Fourier multiplier on Lp(R,X), where 1 ≤ p <∞, if for all f ∈ S(R,X),

M(f) = F−1(m(·)Ff) ∈ Lp(R,X)

and there exists a constant C such that

‖M(f)‖p ≤ C‖f‖p.

Here, F−1f is to be interpreted as 12π Ff. Then M extends to a bounded linear operator on

Lp(R,X). If m is a Fourier multiplier on L1(R,X), it follows from the density of S(R,X)

in L1(R,X), that

F(M(f)) = m(·)Ff (f ∈ L1(R,X)). (5.1)

It is immediate then that

suppF(M(f)) ⊂ suppFf.

For 1 ≤ p < ∞, we shall denote the space of all Fourier multipliers on Lp(R,X) by

Mp(X) with the usual identification of functions which coincide a.e. We put

‖m‖Mp(X) := ‖M‖, (5.2)

74

where the norm of M refers to its norm as a bounded linear operator on Lp(R,X). The

space Mp(X) equipped with the norm defined in (5.2) is a Banach algebra.

It is easy to see that if m ∈ FL1s(R,L(X)) then m is a Fourier multiplier on Lp(R,X)

for all 1 ≤ p <∞ . Indeed, if m = FS, for some S ∈ L1s(R,L(X)) then, for any f ∈ S(R,X)

we find, on using the operator-valued version of Young’s inequality ( see [2, Remark 1.3.8]),

that

M(f) = F−1(FSFf)

= S ∗ f∈ Lp(R,X) (5.3)

and

‖S ∗ f‖p ≤ ‖f‖p∫ ∞

−∞‖S(t)‖ dt, (5.4)

for 1 ≤ p <∞.

For the scalar-valued case, the well known Mikhlin’s Theorem, [29, Theorem 7.9.5]

states that if a function m ∈ C1(R \ 0) is bounded and the function s 7→ sm′(s) is also

bounded, then m is Fourier multiplier on Lp(R), 1 < p < ∞. For p = 1, a similar result

for m ∈ C1(R), involving the boundedness of s 7→ sj+εm(j)(s), j = 0, 1 for some ε > 0 was

given by Hieber[2, Prop. 8.2.3]. We shall need a related theorem for operators, due to H.

Amann [1, Corollary 4.4] (see also [26]).

Theorem 5.0.4. Let m ∈ C2(R,L(X)) satisfy the following conditions for some ε > 0:

supt∈R‖tε+jm(j)(t)‖ <∞, j = 0, 1, 2.

Then m ∈ FL1(R,L(X)) and m is a Fourier multiplier on Lp(R,X), 1 ≤ p <∞.

If the underlying space X is a Hilbert space, then every m ∈ L∞(R,L(X)) is a Fourier

multiplier on L2(R,X). This is a direct consequence of Plancherel’s Theorem.

5.1 A characterisation for ζ(T)

In this section we use ideas from [30] and [34] to study the non-analytic growth bound

of a semigroup in terms of Fourier multiplier properties of the resolvent of its generator.

In Lemmas 5.1.3, 5.1.4 and 5.1.6 we record some properties of the Fourier multiplier m,

on Lp(R,X) and the associated bounded operator M, when m is of a particular form,

namely the resolvent multiplied by an appropriate smooth function. Lemma 5.1.4 and 5.1.6

are based on ideas used in [34, Theorem 2.7]. We first introduce some notation. Let T

be a C0-semigroup with generator A. Given α > s∞0 (A), we shall say that φ ∈ C∞(R)

75

satisfies condition (Pα) if there exists a bounded open subset Uα of R containing the set

s ∈ R : α+ is ∈ σ(A) and such that

φ(s) =

0 (s ∈ Uα),

1 (|s| sufficiently large ).(5.5)

Further, given α > s∞0 (A) and b ≥ 0, such that Qα,b ⊂ ρ(A), a function φ ∈ C∞(R) shall

satisfy property (Pα,b) if there exists b1 > b with

φ(s) =

0 (|s| ≤ b)1 (|s| > b1).

(5.6)

Clearly, if φ satisfies (Pα,b) then it satisfies (Pα). Next, we describe the functions mα ∈L1(R,L(X)). For α > s∞0 (A), and φ satisfying property (Pα) we set

mα(s) = φ(s)R(α+ is,A), (5.7)

with the understanding that mα(s) = 0 whenever φ(s) = 0. Where the meaning is clear, we

shall often write just m(s) instead of m0(s).

We shall be investigating the links between the non-analytic growth bound ζ(T) and

the Fourier multiplier properties of the functions mα. It turns out that the behaviour of

mα as a Fourier multiplier is independent of the choice of φ satisfying property (Pα). This

fact will play an important role in the subsequent theory and we record it as

Remark 5.1.1. Let α > s∞0 (A) and φ1, φ2 satisfy (Pα). If mα(s) = φ1(s)R(α+ is,A) is a

Fourier multiplier on Lp(R,X), 1 ≤ p <∞, then mα, given by mα(s) = φ2(s)R(α+is,A) is

also a Fourier multiplier on Lp(R,X). Indeed, mα−mα ∈ C∞c (R,L(X)) ⊂ FS(R,L(X)) ⊂FL1(R,L(X)), and therefore, is a Fourier multiplier. In fact, if mα is in FL1

s(R,L(X)),

then so is mα.

In [34, Theorem 2.7], the case when φ ≡ 1 has been dealt with. It is shown there that

T is hyperbolic if and only if iR ⊂ ρ(A) and m(s) = R(is,A) is a Fourier multiplier on

Lp(R,X).

The following lemma is technical in nature and is needed to prove the main result of

this section. It reduces to Lemma 2.4 of [34], when φ(s) = 1 for all s ∈ R.

Lemma 5.1.2. Suppose that there exists b > 0 such that Q0,b ⊂ ρ(A) and

supλ∈Q0,b

‖R(λ,A)‖ <∞.

76

Let φ ∈ C∞(R) satisfy (P0,b). Let x ∈ X, Φ ∈ S(R), t ∈ R and τ > 0. Then,

∫

Reistφ(s)T(τ)R(is,A)xFΦ(s) ds =

∫

Reis(t+τ)φ(s)R(is,A)xFΦ(s) ds

− 2π

∫ τ

0T(u)xΦ(t+ τ − u) du

+ 2π

∫ τ

0T(u)x(ψ ∗ Φ)(t+ τ − u) du,

where ψ = F−1(1− φ).

Proof. Sinced

dt(e−istT(t)R(is,A)x) = −e−istT(t)x, integrating with respect to t gives

e−isτT(τ)R(is,A)x = R(is,A)x−∫ τ

0e−istT(t)x dt,

for |s| > b. Making use of the above we have,

∫

Reistφ(s)T(τ)R(is,A)xFΦ(s) ds =

∫

Rei(t+τ)sφ(s)R(is,A)xFΦ(s) ds

−∫

Rei(t+τ)sφ(s)

∫ τ

0e−isuT(u)xFΦ(s) du ds

=

∫

Rei(t+τ)sφ(s)R(is,A)xFΦ(s) ds

−∫

R

∫ τ

0ei(t+τ−u)sT(u)xFΦ(s) du ds

+

∫

R

∫ τ

0ei(t+τ−u)sT(u)x(1− φ(s))FΦ(s) du ds

=

∫

Rei(t+τ)sφ(s)R(is,A)xFΦ(s) ds

− 2π

∫ τ

0T(u)xΦ(t+ τ − u) du

+ 2π

∫ τ

0T(u)x(ψ ∗ Φ)(t+ τ − u) du,

where ψ = F−1(1− φ).

Lemma 5.1.3. Let 1 ≤ p <∞. Suppose there exists b > 0 such that

1. Q0,b ⊂ ρ(A), supλ∈Q0,b‖R(λ,A)‖ <∞ and

2. m is a Fourier multiplier on Lp(R,X), where

m(s) = φ(s)R(is,A) (s ∈ R)

and φ ∈ C∞(R) satisfies (P0,b).

77

Then, there exists an ε > 0 such that Q−ε,b ⊂ ρ(A), supλ∈Q−ε,b ‖R(λ,A)‖ < ∞, and

whenever |α| < ε, mα is a Fourier multiplier on Lp(R,X), where

mα(s) = φ(s)R(α+ is,A) (s ∈ R).

Proof. Let C1 = supλ∈Q0,b‖R(λ,A)‖. The condition on the resolvent of A ensures that C1

is finite. Set

ε1 = min

(1

2C1 sup|φ(s)| : s ∈ R ,1

2C1

).

Then, Q−ε1,b ⊂ ρ(A) and supλ∈Q−ε1,b ‖R(λ,A)‖ < ∞. Further, for |α| < ε1, the Neumann

series ∞∑

n=0

(−1)nαnφ(s)n+1R(is,A)n+1 = φ(s)R(αφ(s) + is,A)

is uniformly convergent for all s ∈ R, where both the sides of the above equality are taken to

be zero when φ(s) = 0. Since s 7→m(s) = φ(s)R(is,A) is a Fourier multiplier on Lp(R,X),

so is s 7→ φ(s)n+1R(is,A)n+1 and therefore so is

s 7→∞∑

n=0

(−1)nαnφ(s)n+1R(is,A)n+1, (5.8)

provided |α| < ε where

ε = min

(1

‖m‖ , ε1).

Here, ‖m‖ = ‖m‖Mp(X). Therefore we may conclude that s 7→ φ(s)R(αφ(s) + is,A) is a

Fourier multiplier on Lp(R,X) for |α| < ε.

Define F : R −→ L(X) by

F(s) = φ(s)R(αφ(s) + is,A)− φ(s)R(α+ is,A)

=

0 (|s| < b)

αφ(s)(1− φ(s))R(αφ(s) + is,A)R(α+ is,A) (b ≤ |s| ≤ b1)

0 (|s| > b1).

So F ∈ C∞c (R,L(X)) and is therefore a Fourier multiplier on Lp(R,X). Thus,

s 7→ φ(s)R(α+ is,A) = φ(s)R(αφ(s) + is,A)− F(s)

is also a Fourier multiplier on Lp(R,X) for all α ∈ R satisfying |α| < ε.

Lemma 5.1.4. Let 1 ≤ p < ∞. Suppose there exists b > 0 such that conditions (1) and

(2) of Lemma 5.1.3 hold. Then the bounded operator M on Lp(R,X) associated with the

Fourier multiplier m is a bounded map from Lp(R,X) to L∞(R,X).

78

Proof. Let 1 ≤ p <∞. We have to show that the bounded map M : Lp(R,X) −→ Lp(R,X),

M(f) = F−1(mFf), f ∈ S(R,X) maps Lp(R,X) to L∞(R,X). Let

Φ =

n∑

i=1

Φi ⊗ xi (xi ∈ X, Φi ∈ S(R)).

Then Φ is in Lp(R,X). Since m is a Fourier multiplier on Lp(R,X), M(Φ) ∈ Lp(R,X) and

there is a constant K such that

‖M(Φ)‖p ≤ K‖Φ‖p.

Now M(Φ) ∈ Lp(R,X) implies that for each n ∈ N, there exists s ∈ [n, n+ 1] such that

‖M(Φ)(s)‖X ≤ ‖M(Φ)‖p ≤ K‖Φ‖p

i.e. for each n ∈ N, there exists s ∈ [n, n+ 1] such that

‖F−1(φR(i·,A)FΦ)(s)‖X ≤ K‖Φ‖p.

Let t ∈ [0, 2]. Then,

‖T(t)(F−1(φR(i·,A)FΦ)(s))‖ ≤ supu∈[0,2]

‖T(u)‖‖F−1(φR(i·,A)FΦ)(s)‖

≤ C‖Φ‖p, (5.9)

where C = supu∈[0,2] ‖T(u)‖K is a constant. Using Lemma 5.1.2, we have

2πT(t)(F−1(φR(i·,A)FΦ)(s)) =

∫

ReisτT(t)φ(τ)R(iτ,A)FΦ(τ) dτ

= M(Φ)(s+ t)− 2π

∫ t

0T(u)Φ(s+ t− u) du

+ 2π

∫ t

0T(u)(ψ ∗ Φ)(s+ t− u) du (5.10)

where ψ = F−1(1− φ). From (5.10) and (5.9) it follows that

‖M(Φ)(s+ t)‖ ≤ 2π‖∫ t

0T(u)Φ(s+ t− u)du‖

+ 2π‖∫ t

0T(u)(ψ ∗ Φ)(s+ t− u)du‖+ 2πC‖Φ‖p

≤ 2π

(∫ 2

0‖T(u)‖p′du

) 1p′(‖Φ‖p + ‖ψ ∗ Φ‖p

)+ 2πC‖Φ‖p

≤ C‖Φ‖p,

where C is a constant. We have used Holder’s inequality to obtain the second last estimate

for 1 < p <∞ with p′ such that 1/p+ 1/p′ = 1, and for p = 1,(∫ 2

0 ‖T(u)‖p′du) 1p′

is to be

interpreted as supt∈[0,2] ‖T(u)‖. By varying the choice of t, n we obtain :

‖M(Φ)(u)‖ ≤ C‖Φ‖p, for all u ∈ R.

79

Therefore, M(Φ) ∈ L∞(R,X). This holds for all such Φ and by a density argument it follows

that M maps Lp(R,X) into L∞(R,X) and is bounded.

Remark 5.1.5. We observe here that if the operator M is bounded from Lr(R,X) to

Lq(R,X) for some r, q satisfying 1 ≤ r, q < ∞ then arguments almost identical to those

above yield that M maps Lr(R,X) to L∞(R,X).

Lemma 5.1.6. Let A be the generator of a C0-semigroup T and 1 ≤ p <∞. Suppose there

exists b > 0 satisfying conditions (1) and (2) of Lemma 5.1.3. Then the associated operator

M on Lp(R,X) is bounded from L1(R,X) to L∞(R,X).

Proof. For p = 1, this is immediate from Lemma 5.1.4. Assume that 1 < p < ∞ and let q

be such that 1p + 1

q = 1. Since M : Lp(R,X) −→ Lp(R,X) is bounded, so is the adjoint map

M∗ : Lp(R,X)∗ −→ Lp(R,X)∗. Recall from Subsection 2.3.4 that X is the subspace of X∗

on which the restriction T of the adjoint semigroup T∗ is strongly continuous. There is a

natural isometric embedding of Lq(R,X∗) and hence of Lq(R,X) in Lp(R,X)∗, [19, Page

98, Chapter 4]. Similarly, Lp(R,X∗) is isometrically embedded in Lq(R,X)∗. Since X is

isomorphically embedded in X∗ (see Subsection 2.3.4) it follows that ‖ · ‖′p, given by

‖f‖′p = sup|〈f, g〉 : g ∈ Lq(R,X), ‖g‖q = 1

, (5.11)

defines an equivalent norm on Lp(R,X). Further, M, the restriction of M∗ to the space

Lq(R,X), is given by

M(g) = F(φ(·)R(i·,A)F−1g),

for g ∈ S(R,X). Therefore, M is a bounded operator from Lq(R,X) to itself. Then

Lemma 5.1.4 with M replaced by M,T by the semigroup T,X by X and p by q implies

that M : Lq(R,X) −→ L∞(R,X) is bounded. Therefore, for any f ∈ S(R,X), it follows

on using (5.11) that

‖M(f)‖′p = sup|〈M(f), g〉| : g ∈ Lq(R,X), ‖g‖q = 1

= sup|〈f,Mg〉| : g ∈ Lq(R,X), ‖g‖q = 1

≤ ‖f‖1 sup‖Mg‖∞ : g ∈ Lq(R,X), ‖g‖q = 1

≤ C‖f‖1,

where C is a constant. Consequently, M is a bounded operator from L1(R,X) to Lp(R,X).

Therefore, it follows from Remark 5.1.5 that M : L1(R,X) −→ L∞(R,X) is bounded.

We now state the main result of this section. It is comparable to [34, Theorem 2.7],

where it has been shown that a strongly continuous semigroup is hyperbolic if and only if

the map s 7→ R(is,A) is a Fourier multiplier on Lp(R,X) for some/all p, 1 ≤ p < ∞. We

prove the corresponding result for the non-analytic growth bound.

80

Theorem 5.1.7. Let 1 ≤ p < ∞. For a C0-semigroup T with generator A the following

are equivalent:

1. ζ(T) < 0;

2. κ(T) < 0;

3. There exists b ≥ 0 such that Q0,b ⊂ ρ(A), supλ∈Q0,b‖R(λ,A)‖ < ∞ and m ∈

FL1s(R,L(X)), where

m(s) = φ(s)R(is,A) (s ∈ R),

for some φ ∈ C∞(R) satisfying (P0);

4. There exist b ≥ 0 such that Q0,b ⊂ ρ(A), supλ∈Q0,b‖R(λ,A)‖ <∞ and m is a Fourier

multiplier on Lp(R,X), where

m(s) = φ(s)R(is,A) (s ∈ R),

for some φ ∈ C∞(R) satisfying (P0);

5. There exist b > 0 and ε′ > 0 such that Q−ε′,b ⊂ ρ(A), supλ∈Q−ε′,b ‖R(λ,A)‖ <∞ and

for each x ∈ X, x∗ ∈ X∗ and Φ ∈ S(R)

|〈rα,Φ〉| ≤ Kα‖x‖‖x∗‖‖FΦ‖1

for each α, |α| < ε′ and some constant Kα where

rα(s) =

〈x∗, φ(s)R(α+ is,A)x〉 (|s| > b)

0 (|s| ≤ b)

for some φ ∈ C∞(R) satisfying (Pα,b). Here,

〈rα,Φ〉 =

∫

Rrα(s)Φ(s) ds.

Remark 5.1.8. We note here that if the equivalent conditions of Theorem 5.1.7 hold then

conditions (3 ) and (4 ) in Theorem 5.1.7 hold for all φ in view of Remark 5.1.1.

Proof of Theorem 5.1.7 . (1) =⇒ (2): This follows immediately from the definitions of ζ(T)

and κ(T).

(2) =⇒ (3): Since κ(T) < 0 there exist ε > 0 and b > 0 such that Q−ε,b ⊂ ρ(A),

supλ∈Q−ε,b ‖R(λ,A)‖ <∞ and

T(t) = T−ε,b(t) + T2(t) (t ≥ 0),

81

with abs(‖T2‖) < 0. We may choose b in such a way that Q0,b ⊂ ρ(A) and

supλ∈Q0,b

‖R(λ,A)‖ <∞.

In view of Remark 5.1.1 it is enough to prove the assertion for φ ∈ C∞(R) satisfying (P0,b).

Suppose then that φ ∈ C∞(R) and there exists b1 > b such that (5.6) hold. For x ∈ X we

can write,

φ(s)R(is,A)x = φ(s)T−ε,b(is)x+ φ(s)T2(is)x

where φ(s)T−ε,b(is) is taken to be zero whenever φ(s) is zero. Putting

m1(s)x =

φ(s)T−ε,b(is)x (|s| > b)

0 (|s| ≤ b),m2(s)x = (1− φ(s))T2(is)x,

m3(s)x = T2(is)x,

we have

m(s) = m1(s)−m2(s) + m3(s).

Since the function t 7→ T−ε,b(t) has an exponentially bounded, holomorphic extension to

some sector Σβ, 0 < β ≤ π2 , there exists w ∈ R such that

supz∈Σβ

‖e−wzT−ε,b(z)‖ <∞.

It follows then from [2, Theorem 2.6.1 ] that

supλ∈w+Σγ+π

2

‖(λ− w)j+1T(j)−ε,b(λ)‖ <∞, (j = 0, 1, 2)

for 0 < γ < β. Since b may be chosen such that is ∈ w + Σγ+π2

for |s| > b, there is a

constant C such that

‖(is− w)j+1T(j)−ε,b(is)‖ ≤ C (j = 0, 1, 2), (5.12)

for |s| > b. Since φ ∈ C∞(R), m1 : R −→ L(X) is a smooth, operator-valued function.

Further, for any µ, 0 < µ < 1 we have, using (5.12) for j = 0 and 1,

sups∈R‖s1+µm

(1)1 (s)‖ ≤ sup

b<|s|<b1|sµφ(1)(s)| sup

|s|>b‖sT−ε,b(is)‖

+ sup|s|>b|φ(s)| sup

|s|>b‖sµ+1T

(1)−ε,b(is)‖

≤ C1 sup|s|>b

|s||is− w| + C2 sup

|s|>b

|s|1+µ

|is− w|2

<∞,

82

where C1, C2 are constants. In fact, using similar arguments we find

sups∈R‖sj+µm(j)

1 (s)‖ <∞ (j = 0, 1, 2).

Therefore, Theorem 5.0.4 applied to m1 allows us to conclude that

m1 ∈ FL1(R,L(X)).

Due to the particular choice of φ, m2 has compact support and is obviously smooth.

Therefore, m2 ∈ C∞c (R,L(X)) ⊂ S(R,L(X)). Since FS(R,L(X)) = S(R,L(X)), we con-

clude that m2 ∈ FL1(R,L(X)).

By the hypothesis, abs(‖T2‖) < 0, so that T2 ∈ L1s(R,L(X)) and m3(s) = (FT2)(s),

where T2(t) = 0 for t < 0. Therefore, m3 ∈ FL1s(R,L(X)). Thus the linear combination

m = m1 −m2 + m3 ∈ FL1s(R,L(X)).

(3) =⇒ (4) : (3) implies that there exists S ∈ L1s(R,L(X)) satisfying

m(s) = φ(s)R(is,A) = FS(s).

Therefore, it follows from (5.3) that m is a Fourier multiplier on Lp(R,X) .

(4) =⇒ (5): It follows from (4) that there is a bounded linear map M : Lp(R,X) −→Lp(R,X) with

M(f) = F−1(mFf),

for f ∈ S(R,X), where m(s) = φ(s)R(is,A), (s ∈ R) and φ ∈ C∞(R) satisfies (P0,b). From

Lemma 5.1.6, it follows that M is a bounded map from L1(R,X) to L∞(R,X), that is, there

exists K > 0 such that

‖M(f)‖∞ ≤ K‖f‖1. (5.13)

Fix x ∈ X. Let Φ ∈ S(R). Then F−1Φ⊗ x ∈ S(R,X) and

‖M(F−1Φ⊗ x)‖∞ ≤ K‖F−1Φ‖1‖x‖.

Note that

M(F−1Φ⊗ x) = F−1(φR(i·,A)Φ⊗ x), (5.14)

which is continuous as φR(i·,A)Φ⊗ x ∈ L1(R,X). Therefore,

‖M(F−1Φ⊗ x)(0)‖ ≤ K‖F−1Φ‖1‖x‖. (5.15)

It follows from (5.14) and (5.15) that∥∥∥∥∫

Rφ(s)R(is,A)xΦ(s) ds

∥∥∥∥ = 2π‖M(F−1Φ⊗ x)(0)‖

≤ 2πK‖F−1Φ‖1‖x‖= K‖FΦ‖1‖x‖,

83

for any x ∈ X. For x∗ ∈ X∗, and

r(s) = 〈x∗, φ(s)R(is,A)x〉,

we therefore have, ∣∣∣∣∫

Rr(s)Φ(s) ds

∣∣∣∣ ≤ C1‖x‖‖x∗‖‖FΦ‖1. (5.16)

Since m is a Fourier multiplier on Lp(R,X), using Lemma 5.1.3 we can find an ε > 0 such

that Q−ε,b ⊂ ρ(A) and supλ∈Q−ε,b ‖R(λ,A)‖ < ∞ and mα, given by mα(s) = φ(s)R(α +

is,A) is a Fourier multiplier on Lp(R,X), whenever |α| < ε. Let α be such that |α| < ε and

consider the rescaled semigroup e−α·T(·). Since mα is a Fourier multiplier on Lp(R,X) and

mα(s) = φ(s)R(is,−α+ A), where −α+ A is the generator of the rescaled semigroup, the

previous argument applied to Mα and e−α·T(·) yields the required inequality for rα.

(5) =⇒ (1): From (5 ) we have that s∞0 (A) < 0. So, for x ∈ X,

lims→∞

‖R(w + is,A)x‖ = 0

uniformly for w ∈ [0, a] for each a > 0 [40, Lemma 1.3.2]. Hence the complex inversion

theorem for Laplace transforms and Cauchy’s Theorem yield the following formula for T(t)x

when t ≥ 0:

T(t)x = T0,b(t)x+1

2π(C, 1)

∫

|s|>beistR(is,A)x ds,

where the Cesaro-convergence of the integral is uniform for t ∈ [0, a] for each a > 0 (see [2,

Theorem 2.3.4] and [40, Theorem 1.3.3]). Let

S(t)x = T(t)x−T0,b(t)x−1

2π

∫

b≤|s|≤b1eist(1− φ(s))R(is,A)x ds

=1

2π(C, 1)

∫

|s|>beistφ(s)R(is,A)x ds.

For Φ ∈ C∞c (R) with supp Φ ⊂ R+, the uniformity of the Cesaro-convergence gives

∫ ∞

0S(t)xΦ(t) dt = lim

N−→∞

∫ ∞

0

1

2π

∫ N

−N

(1− |s|

N

)eistφ(s)R(is,A)x dsΦ(t) dt

= limN−→∞

∫ N

−N

(1− |s|

N

)φ(s)R(is,A)xF−1Φ(s) ds

=

∫ ∞

−∞φ(s)R(is,A)xF−1Φ(s) ds. (5.17)

The assumption (5) implies that for x∗ ∈ X∗,∣∣∣∣∫ ∞

0〈x∗,S(t)x〉Φ(t) dt

∣∣∣∣ = |〈r0,F−1Φ〉|

≤ K0‖x‖‖x∗‖‖Φ‖1

84

for all x ∈ X. Since t 7→ 〈x∗,S(t)x〉 is continuous, this implies that

∣∣〈x∗,S(t)x〉∣∣ ≤ K0‖x‖‖x∗‖,

for all t ≥ 0. Hence

‖S(t)x‖ ≤ K0‖x‖.

Further, since ∥∥∥∥1

2π

∫

b≤|s|≤b1eist(1− φ(s))R(is,A)x ds

∥∥∥∥ ≤ C‖x‖

for some constant C, we conclude that

‖T(t)x−T0,b(t)x‖ ≤ K0‖x‖

for some constant K0 and all x ∈ X. It follows that

‖T(t)−T0,b(t)‖ ≤ K0

so that ζ(T) ≤ 0.

Now, take α ∈ R : −ε < α < 0. Applying the above procedure to the rescaled semigroup

e−α·T(·) and making use of the hypothesis on rα we obtain ζ(T) ≤ α < 0.

Remark 5.1.9. If the equivalent conditions of Theorem 5.1.7 hold then m = FS where

S ∈ L1s(R,L(X)) is given by

S(t)x =

T(t)x−T0,b(t)x−1

2πi

∫

b≤|s|≤b1eist(1− φ(s))R(is,A)x ds (t ≥ 0)

−T0,b(t)x−1

2πi

∫

b≤|s|≤b1eist(1− φ(s))R(is,A)x ds (t < 0).

For t ≥ 0 this follows from (5.17). For t < 0 one has

0 = T0,b(t)x+1

2π(C, 1)

∫

|s|≥beistR(is,A)x dt

where the Cesaro-convergence is uniform on compact subsets of (−∞, 0), and a similar

argument to the proof of (5) =⇒ (1) above leads to the conclusion.

For a C0-semigroup T, it is trivial that ω0(T) = abs(‖T‖) (see e.g. [2, Prop. 5.1.1]).

Theorem 5.1.7 gives the following analogue of this result for ζ(T). We do not know of any

direct proof of this fact.

Corollary 5.1.10. Let T be a C0-semigroup on X. Then

ζ(T) = κ(T).

85

As an immediate consequence of Theorem 5.1.7, we have

Corollary 5.1.11. For a C0-semigroup T with generator A defined on X,

s∞0 (A) = δ(T) = ζ(T)⇐⇒ for each α > s∞0 (A), s 7→ φ(s)R(α+ is) is a Fourier

multiplier on Lp(R,X) for some p ∈ [1,∞), for some

φ ∈ C∞(R) satisfying (Pα).

Remark 5.1.12. As already noted, if the equivalent conditions of Theorem 5.1.7 hold, then

condition (2 ) holds whenever φ ∈ C∞(R), satisfies (P0). Let 1 ≤ p <∞. Assume now that

iR ⊂ ρ(A). Then Theorem 5.1.7 gives

ζ(T) < 0⇐⇒ R(i·,A) is a Fourier multiplier on Lp(R,X) and s∞0 (A) < 0. (5.18)

Latushkin and Shvydkoy [34, Theorem 2.7] have shown that

T is hyperbolic ⇐⇒ R(i·,A) is a Fourier multiplier on Lp(R,X).

Therefore, (5.18) reduces to

ζ(T) < 0⇐⇒ s∞0 (A) < 0 and T is hyperbolic.

However, using completely different methods it has been shown in [10, Theorem 3.5.2] that

if T is hyperbolic, then ζ(T) < 0 ⇐⇒ s∞0 (A) < 0. Further, [10, Proposition 4.5.5] states

that if T is a C0-semigroup and δ(T) < 0 then T is hyperbolic if and only if iR ⊂ ρ(A).

Thus (5.18) is just a reformulation of known results in the case when iR ⊂ ρ(A).

Remark 5.1.13. From Theorem 5.1.7 we are able to obtain a different proof for the fact

that s∞0 (A) = δ(T) = ζ(T) for a strongly continuous semigroup on a Hilbert space. This

analogue of the Gearhart-Pruss Theorem [see page 35] was first shown by M. Blake [10], [11,

Lemma 4.3]. The alternative proof works in the following way: Suppose that s∞0 (A) < 0.

Then the first part of condition (4 ) of Theorem 5.1.7 is satisfied. Since X is a Hilbert

space, the function m defined in hypothesis (4) of Theorem 5.1.7, being bounded, is a

Fourier multiplier on L2(R,X). So it follows from Theorem 5.1.7 that ζ(T) < 0. A rescal-

ing argument then implies that s∞0 (A) ≥ ζ(T). For any strongly continuous semigroup,

s∞0 (A) ≤ δ(T) ≤ ζ(T). Therefore, the required equality holds.

5.2 Perturbations

The technique used in Lemma 5.1.3 enables us to deduce how the non-analytic growth

bound of a C0-semigroup behaves under small bounded perturbations. We have,

86

Theorem 5.2.1. ζ(·) is upper semi-continuous with respect to bounded perturbations, that

is, if A generates a C0-semigroup T and ε > 0 then there exists δ > 0 such that ζ(S) ≤ζ(T) + ε whenever S is a C0-semigroup generated by A + B where B is a bounded operator

such that ‖B‖ < δ.

Proof. Let A be the generator of a C0-semigroup T. We note that it is enough to show that

if ζ(T) < 0, then there exists a δ > 0 such that for every bounded operator B with ‖B‖ < δ,

ζ(S) < 0, S being the semigroup obtained on perturbing A by B. So, suppose ζ(T) < 0.

Then using Theorem 5.1.7 we can find a b ≥ 0 and an ε0 > 0 such that Q−ε0,b ⊂ ρ(A),

supλ∈Q−ε0,b ‖R(λ,A)‖ <∞ and m is a Fourier multiplier on L1(R,X) where

m(s) = φ(s)R(is,A) (s ∈ R)

for some φ ∈ C∞(R) satisfying (P0,b). We may also assume that 0 ≤ φ ≤ 1. Let w > ω0(T),

K = sup‖R(λ,A)‖ : λ ∈ Q−ε0,b ∪ µ : Reµ > w

and B be a bounded operator on X with ‖B‖ ≤ 12K . Let S be the C0-semigroup generated

by A + B. Then Q−ε0,b ∪ µ : Reµ > w ⊂ ρ(A + B) and

sup‖R(λ,A + B)‖ : λ ∈ Q−ε0,b ∪ µ : Reµ > w ≤ 2K <∞.

This is due to the fact that for any λ ∈ Q−ε0,b ∪ µ : Reµ > w

‖R(λ,A)B‖ ≤ 1

2,

so that (I−R(λ,A)B) is invertible (see [10, Lemma 5.2.1]). Further, for all such λ

R(λ,A + B) = (I−R(λ,A)B)−1R(λ,A)

=∞∑

n=0

(−1)n(R(λ,A)B)nR(λ,A). (5.19)

Also, for any s ∈ R we have

φ(s) (I− φ(s)R(is,A)B)−1 R(is,A) =∞∑

n=0

(−1)nφ(s)n+1(R(is,A)B)nR(is,A), (5.20)

where, both sides of the equality are zero for |s| < b. Since s 7→ φ(s)R(is,A) is a Fourier

multiplier on L1(R,X), so is the map s 7→ φ(s)R(is,A)B. Therefore, for each n, the map

s 7→ φ(s)n(R(is,A)B)n is also a Fourier multiplier on L1(R,X). Thus,

s 7→ (I− φ(s)R(is,A)B)−1 φ(s)R(is,A)

87

is a Fourier multiplier on L1(R,X) provided ‖φ(·)R(i·,A)B‖M1(X) < 1. Recall from Theo-

rem 5.1.7 that under the present hypothesis, there is an operator U ∈ FL1s(R,L(X)) such

that φ(s)R(is,A) = (FU)(s). Then it is easy to see that

‖φ(·)R(i·,A)B‖M1(X) ≤ ‖U‖1‖B‖,

where ‖U‖1 =

∫ ∞

−∞‖U(t)‖ dt. Let

δ = min

(1

2K,

1

‖U‖1

).

Then, for ‖B‖ < δ, the map s 7→ (I− φ(s)R(is,A)B)−1 φ(s)R(is,A) is a Fourier multiplier

on L1(R,X). Define F : R −→ L(X) by

F(s) = (I− φ(s)R(is,A)B)−1 φ(s)R(is,A)− φ(s)R(is,A + B)

=

0 (|s| < b)

(I− φ(s)R(is,A)B)−1φ(s)R(is,A)− φ(s)R(is,A + B) (b ≤ |s| ≤ b1)

0 (|s| > b1).

So F ∈ C∞c (R,L(X)) and is therefore a Fourier multiplier on L1(R,X). Thus, s 7→φ(s)R(is,A + B) is a Fourier multiplier on L1(R,X) provided ‖B‖ < δ. From Theorem

5.1.7 it then follows that ζ(S) < 0 as required.

Neither ω0(·) nor δ(·) is lower semi-continuous under bounded perturbations of C0-

semigroups. Examples exhibiting this have been given by M. Blake [10, Example 5.3.4].

It turns out that ζ(·) is not lower semi-continuous under bounded perturbations of C0-

semigroups either. In fact, [10, Example 5.3.4] works for this case also.

5.3 Inhomogeneous Cauchy problems

A characterisation of the non-analytic growth bound in terms of the existence of unique

mild solutions of certain inhomogeneous Cauchy problems on R, under some conditions, is

presented in this section. By an inhomogeneous Cauchy problem on R we mean an equation

u′(t) = Au(t) + f(t) (t ∈ R). (ACPf )

Here A is a closed operator on X, and f ∈ L1loc(R,X).

For f ∈ L1loc(R,X), a function u ∈ L1

loc(R,X) is called a mild solution of (ACPf ) if

∫ t

su(τ) dτ ∈ D(A), (5.21)

u(t) = u(s) + A

∫ t

su(τ) dτ +

∫ t

sf(τ) dτ, (5.22)

88

for almost all t, s (t ≥ s). We observe here that then (5.22) is true for almost all (s, t) ∈ R2

(with the usual convention about∫ ba = −

∫ ab ). We define a classical solution as a C1-

function u such that u(t) ∈ D(A), for all t and such that (ACPf ) is valid. For f continuous,

a continuous mild solution u is a classical solution if and only if u ∈ C1(R,X). In the special

case when A is the generator of a C0- semigroup, we have

Lemma 5.3.1. Let u be a mild solution of (ACPf ) in L1loc(R,X) and f ∈ L1

loc(R,X). If A

is the generator of a C0-semigroup then there exists a unique continuous function u, such

that u = u a.e. and

u(t) = T(t− s)u(s) +

∫ t

sT(t− r)f(r) dr (5.23)

whenever t ≥ s.

Proof. Let u be a mild solution of (ACPf ), where f ∈ L1loc(R,X). First assume that both

f, u are Laplace transformable. Then (5.21) and (5.22) hold for almost all (t, s) ∈ R2.

Suppose first that s = 0 so that

∫ t

0u(τ) dτ ∈ D(A) and

u(t) = u(0) + A

∫ t

0u(τ) dτ +

∫ t

0f(τ) dτ,

for almost all t ≥ 0. Taking Laplace transforms, we have for Reλ sufficiently large,

u(λ) =u(0)

λ+

Au(λ)

λ+f(λ)

λ

u(λ) = R(λ,A)(u(0) + f(λ)

).

Let v0(t) = T(t)u(0) +

∫ t

0T(t− r)f(r) dr (t ≥ 0). Then for Reλ sufficiently large,

v0(λ) = u(λ).

Therefore, by the Uniqueness Theorem, u = v0 a.e., that is,

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

∫ t

0T(t− τ)f(τ) dτ

for almost all t ≥ 0. For s not necessarily 0, we obtain, on applying the previous case to

appropriate translates of u and f,

u(t) = T(t− s)u(s) +

∫ t

sT(t− τ)f(τ) dτ

for almost all s ∈ R and for almost all t ≥ s. Define, for s ∈ R,

us(t) = T(t− s)u(s) +

∫ t

sT(t− τ)f(τ) dτ (t ≥ s).

89

Then us is continuous on [s,∞). Further, for almost all (s1, s2), s1 < s2,

us1(t) = us2(t) = u(t), for almost all t ∈ [s2,∞).

By continuity, for almost all s1, s2, us1(t) = us2(t) for all t ∈ [s2,∞). Define

u(t) = us(t), for such s.

This is independent of the choice of s (excluding a null set). Therefore, u is continuous and

u(t) = u(t) a.e. Further,

u(t) = T(t− s)u(s) +

∫ t

sT(t− τ)f(τ) dτ

for almost all t ≥ s, and therefore, by continuity for all t ≥ s.For the general case, fix τ > 0. Replacing f, u by the functions g, v respectively, where

g(t) =

f(t) (t < τ)

0 (t ≥ τ);

v(t) =

u(t) (t < τ)

T(t− τ)u(τ) (t ≥ τ).

Then the previous case applies to g and v and we obtain the required result.

In what follows, we shall be considering (ACPf ) with the function f ∈ Lp(R,X) and A

satisfying a condition of non-resonance. To make this precise, we recall the definition of the

Carleman transform and Carleman spectrum of a function f ∈ Lp(R,X) (Sections 4.6 and

4.8 of [2]).

Let f ∈ Lp(R,X), 1 ≤ p ≤ ∞. The Carleman transform f of f is defined by

f(λ) =

∫ ∞

0e−λtf(t) dt (Reλ > 0)

−∫ ∞

0eλtf(−t) dt (Reλ < 0).

Then f is a holomorphic function defined on C \ iR. We use the same symbol for the

Carleman transform and the Laplace transform. This will not lead to any confusion. A

point iη ∈ iR is called regular for f if there exists an open neighbourhood V of iη in Cand a holomorphic function h : V −→ X such that h(λ) = f(λ) for all λ ∈ V \ iR. The

Carleman spectrum spc(f) of f , is defined by

spc(f) = η ∈ R : iη is not regular for f.

A closed operator A defined on X is said to have no resonance with f ∈ Lp(R,X) (or,

A, f satisfy a condition of non-resonance) if i spc(f) ∩ σ(A) is empty.

90

For f ∈ Lp(R,X) we define Ff as a linear mapping from S(R) into X by

〈Φ,Ff〉 =

∫

Rf(t)(FΦ)(t) dt (Φ ∈ S(R)).

The support of Ff is defined by

suppFf =η ∈ R : for all ε > 0, there exists Φ ∈ S(R) such that

supp Φ ⊂ (η − ε, η + ε) and 〈Φ,Ff〉 6= 0. (5.24)

This is consistent with the definition of Fourier transforms of distributions. From [2, The-

orem 4.8.1], we have for all f ∈ Lp(R,X)

spc(f) = suppFf. (5.25)

If f ∈ L1(R,X) then

spc(f) = suppFf = s ∈ R : (Ff)(s) 6= 0−. (5.26)

Here Ff is the usual function. For f ∈ Lp(R,X) and Φ ∈ S(R) we have from [2, Remark

4.8.6],

spc(f ∗ Φ) ⊂ spc(f) ∩ suppFΦ. (5.27)

Further, using the idea in the first part of the proof of [2, Theorem 4.8.1], it is easy to

deduce the following.

Lemma 5.3.2. If f ∈ Lp(R,X) and Φ ∈ S(R) are such that spc(f)∩ supp Φ is empty, then

〈Φ,Ff〉 = 0.

It follows from Lemma 5.3.2, (5.24) and (5.25) that if E is a closed subset of R then

spc(f) ⊂ E ⇐⇒ Φ ∈ S(R), supp Φ ∩ E = ∅implies that 〈Φ,Ff〉 = 0. (5.28)

Throughout this section, given two functions g, h defined on R, we shall use 〈g, h〉 to

denote ∫

Rg(s)h(s) ds

whenever this integral makes sense. In the expression 〈·, ·〉 the first coordinate will usually

be reserved for either a scalar or an operator-valued function.

Remark 5.3.3. Given Φ ∈ S(R,L(X)), and f ∈ Lp(R,X), the integral

∫

R(FΦ)(t)f(t) dt

is well defined. Thus, given f ∈ Lp(R,X), we may define the linear mapping Ff from

S(R,L(X)) into X by

〈Φ,Ff〉 =

∫

R(FΦ)(t)f(t) dt.

91

It is easy to see that Ff extends the definition of Ff given above. In fact, we may

define supp Ff analogously to the definition in (5.24). Further, a proof identical to that of

[2, Theorem 4.8.1] shows that

suppFf = spc(f) = supp Ff,

for all f ∈ Lp(R,X). The corresponding versions of Lemma 5.3.2 and (5.28) also hold.

From now on we shall denote Ff also by Ff. This will not lead to any confusion since the

choice of Φ in 〈Φ,Ff〉 clearly indicates whether we are referring to the linear map from

S(R,L(X)) or from S(R).

If m is a Fourier multiplier on Lp(R,X), and M is the associated bounded operator on

Lp(R,X), then the following analogue of (5.1) holds for all Φ ∈ S(R)

〈Φ,F(Mf)〉 = 〈F(Φm), f〉 (f ∈ S(R,X)). (5.29)

Due to the density of S(R,X) in Lp(R,X), (5.29) holds for all f ∈ Lp(R,X), if m is smooth

and has bounded derivatives. Then, from the definition of suppFf(= supp Ff), it follows

that

suppF(Mf) ⊂ suppFf. (5.30)

With this framework in the background, we proceed with our study of the relation

between the existence of unique mild solutions of (ACPf ), f ∈ Lp(R,X), 1 ≤ p < ∞, and

the non-analytic growth bound. We shall deal with the case p = 1 first because the proofs

are a little simpler. In order to obtain our characterisation of ζ(T) in terms of mild solutions

of (ACPf ) in this case, we need a result which may be considered as the analogue on R of

the well known result characterising continuous mild solutions of (ACPf ) on R+ in terms

of Laplace transforms [2, Theorem 3.1.3]. For the proof, we shall make use of the following

analogue of [2, Proposition 1.7.6]. This is probably well known and we omit the proof which

is very similar to [2, Proposition 1.7.6].

Lemma 5.3.4. Let A be a closed linear operator on X. For f, g ∈ L1(R,X), the following

are equivalent:

1. Ff(s) ∈ D(A) and AFf(s) = Fg(s), for all s ∈ R.

2. f(t) ∈ D(A) and Af(t) = g(t) a.e. on R.

Lemma 5.3.5. Let f ∈ L1(R,X) and consider the inhomogeneous Cauchy problem

u′(t) = Au(t) + f(t) (ACPf )

where A is a closed linear operator on X. Suppose u ∈ L1(R,X). Then u is a mild solution

of (ACPf ) if and only if (Fu)(s) ∈ D(A) and for all s ∈ R,

(is−A)Fu(s) = Ff(s). (5.31)

92

Proof. Suppose first that u, f ∈ L1(R,X) and (5.31) holds. We show that u satisfies (5.21)

and (5.22). Let (ρn) be a sequence of functions in C∞c (R) which forms a mollifier. Then

limn→∞

‖ρn ∗ u− u‖1 = 0,

limn→∞

‖ρn ∗ f − f‖1 = 0.

From (5.31) it follows that

(is−A)F(ρn ∗ u)(s) = (is−A)Fρn(s)Fu(s)

= Fρn(s)Ff(s)

= F(ρn ∗ f)(s), for all n ∈ N. (5.32)

Since ρn ∗ u ∈ C∞(R,X) ∩ L1(R,X), and (ρn ∗ u)′ ∈ L1(R,X), (5.32) may be rewritten as

F(ρn ∗ u)′(s)−AF(ρn ∗ u)(s) = F(ρn ∗ f)(s) for all s ∈ R.

Then Lemma 5.3.4 implies that for each n ∈ N, (ρn ∗ u)(t) ∈ D(A) and

(ρn ∗ u)′(t)−A(ρn ∗ u)(t) = (ρn ∗ f)(t), (5.33)

a.e. on R. Since ρn ∗ u, (ρn ∗ u)′ and ρn ∗ f are continuous and A is closed, (5.33) is true

for all t ∈ R. So for each n ∈ N, ρn ∗ u is a classical solution of (ACPf ). It is therefore also

a mild solution of (ACPf ) so that

∫ t

s(ρn ∗ u)(τ) dτ ∈ D(A),

and whenever t ≥ s,

(ρn ∗ u)(t) = (ρn ∗ u)(s) + A

∫ t

s(ρn ∗ u)(τ) dτ +

∫ t

s(ρn ∗ f)(τ) dτ. (5.34)

Since A is closed, letting n→∞ through a subsequence in (5.34) yields

∫ t

su(τ) dτ ∈ D(A)

and

u(t) = u(s) + A

∫ t

su(τ) dτ +

∫ t

sf(τ) dτ,

for almost all t, s (t ≥ s) as required.

Conversely, suppose that u is a mild solution. Then, for almost all s, (5.21) and (5.22)

hold for almost all t. Assuming first that this is true for s = 0, we have, for almost all t in

R,

u(t) = u(0) + A

∫ t

0u(τ) dτ +

∫ t

0f(τ) dτ. (5.35)

93

Let u+ = u R+, f+ = f R+

and v(t) = u(−t), g(t) = f(−t), for all t ≥ 0. Then, using

(5.35) and [2, Theorem 3.1.3] we have that for Reλ > 0, u+(λ) ∈ D(A) and

u+(λ) =u(0)

λ+

Au+(λ)

λ+f+(λ)

λ.

Letting Reλ→ 0, we obtain that∫ ∞

0e−iτtu(t) dt ∈ D(A)

and

(iτ −A)

∫ ∞

0e−iτtu(t) dt = u(0) +

∫ ∞

0e−iτtf(t) dt, (5.36)

for all τ ∈ R. Also, (5.35) gives

v(t) = u(0)−A

∫ t

0v(τ) dτ −

∫ t

0g(τ) dτ,

for almost all t ≥ 0. Taking Laplace transforms, we have, as before, that∫ ∞

0eiτtv(t) dt ∈ D(A)

and

(iτ −A)

∫ ∞

0eiτtv(t) dt = −u(0) +

∫ ∞

0eiτtg(t) dt, (5.37)

for all τ ∈ R. Adding (5.36) and (5.37) yields Fu(τ) ∈ D(A) and

(iτ −A)Fu(τ) = Ff(τ),

for all τ ∈ R.If s 6= 0, then the previous case may be applied to u(t) = u(t+ s) and f(t) = f(t+ s).

Then, the required result follows immediately .

Theorem 5.3.6. The following are equivalent for a C0-semigroup T:

1. ζ(T) < 0;

2. s∞0 (A) < 0 and for all f ∈ L1(R,X) with i spc(f)∩σ(A) empty, there exists a unique

mild solution u ∈ L1(R,X) of (ACPf ) such that i spc(u) ∩ σ(A) is empty.

Proof. (1) =⇒ (2) : From Theorem 5.1.7, (1 ) implies that s∞0 (A) < 0. Let f ∈ L1(R,X),

with i spc(f) ∩ σ(A) empty. Choose φ ∈ C∞(R) satisfying condition (P0) and such that

φ = 1 near spc(f).

Then, from Theorem 5.1.7, and Remark 5.1.8 it follows that

M : L1(R,X) −→ L1(R,X)

M(g) = F−1(φ(·)R(i·,A)Fg) (g ∈ S(R,X))

94

is a bounded operator. Let u = M(f). Using (5.1), we have that

Fu(s) = φ(s)R(is,A)Ff(s),

for all s ∈ R, and suppFu ⊂ suppFf . Therefore,

i spc(u) ∩ σ(A) ⊂ i spc(f) ∩ σ(A) = ∅.

Further, for s ∈ suppFf,Fu(s) = R(is,A)Ff(s). Therefore, Fu(s) ∈ D(A) and

(is−A)Fu(s) = Ff(s). (5.38)

On the other hand, if s 6∈ suppFf, then Fu(s) = 0 = Ff(s), so that (5.38) remains true.

Therefore, for all s ∈ R(is−A)Fu(s) = Ff(s)

and Lemma 5.3.5 allows us to conclude that u is a mild solution of (ACPf ). If u1 is another

mild solution of (ACPf ) in L1(R,X) with i spc(u1) ∩ σ(A) empty and (is − A)Fu1(s) =

Ff(s) for all s ∈ R, then Fu = Fu1 on R. It therefore follows that u(t) = u1(t) almost

everywhere, i.e. u = u1 in L1(R,X).

(2) =⇒ (1) : Since s∞0 (A) < 0 we can find b > 0 such that

Q0,b ⊂ ρ(A) and supλ∈Q0,b

‖R(λ,A)‖ <∞.

Choose φ ∈ C∞(R) satisfying condition (P0,b). Note that iR ∩ σ(A) ⊂ (−ib, ib). Let

Y =f ∈ L1(R,X) : spc(f) ⊂ R \ (−b, b)

=f ∈ L1(R,X) : Ff = 0 on (−b, b)

.

Then Y is closed in L1(R,X). For f ∈ Y, i spc(f) ∩ σ(A) is empty. Therefore, hypothesis

(2 ) implies the existence of a unique uf ∈ L1(R,X) with i spc(uf ) ∩ σ(A) empty, which is

a mild solution of (ACPf ). Further, from Lemma 5.3.5

(is−A)Fuf (s) = Ff(s) (s ∈ R). (5.39)

We show next that the linear map B : Y −→ L1(R,X), given by

B(f) = uf

is closed. Suppose (fn) ⊂ Y converges to f in L1(R,X) and B(fn) converges to some

u ∈ L1(R,X). Since for each n, B(fn) = ufn is the unique mild solution of (ACPfn),

∫ t

sufn(τ) dτ ∈ D(A)

95

and

ufn(t) = ufn(s) + A

∫ t

sufn(τ) dτ +

∫ t

sfn(τ) dτ (5.40)

for almost all s, t ∈ R and all n ∈ N. Since A is closed, letting n→∞ in (5.40) through a

subsequence, we have that

∫ t

su(τ) dτ ∈ D(A) and

u(t) = u(s) + A

∫ t

su(τ) dτ +

∫ t

sf(τ) dτ

for almost all s, t ∈ R. This implies that u is a mild solution of (ACPf ). If η ∈ spc(ufn) =

suppFufn , then iη ∈ ρ(A). Therefore, it follows from (5.39) that i spc(ufn) ⊂ i spc(fn) ⊂ R\(−b, b). Since ufn converges to u in L1(R,X), it follows that spc(u) ⊂ R\ (−b, b). Therefore,

i spc(u) ∩ σ(A) is empty. If v is any other mild solution of (ACPf ) with i spc(v) ∩ σ(A)

empty, then it follows from Lemma 5.3.5 that Fu = Fv. Therefore, u = v in L1(R,X).

Thus, u ∈ L1(R,X) is the unique mild solution corresponding to f given by (2) so that

B(f) = u. Since B is a closed linear map between two Banach spaces, it follows from the

Closed Graph Theorem that B is bounded on Y, i.e. there is a constant K such that

‖B(f)‖1 ≤ K‖f‖1 (f ∈ Y). (5.41)

Now consider any g ∈ L1(R,X). Set fg = g−F−1(1− φ) ∗ g. Then Ffg = φFg, so that

fg ∈ Y. Thus, from the above discussion, it follows that there is a unique ufg ∈ L1(R,X)

with i spc(ufg) ∩ σ(A) empty, which is a mild solution of (ACPfg), and satisfies

(is−A)Fufg(s) = Ffg(s) (s ∈ R),

‖ufg‖1 ≤ K‖fg‖1.

If is ∈ ρ(A), then Fufg(s) = φ(s)R(is,A)Fg(s). On the other hand if is ∈ σ(A), then

s 6∈ spc(ufg) so that Fufg(s) = 0. Also, φ(s)R(is,A)Fg(s) = 0 since s ∈ (−b, b). Therefore,

Fufg(s) = φ(s)R(is,A)Fg(s), for all s ∈ R.

If g ∈ S(R,X), then the function s 7→ φ(s)R(is,A)Fg(s) is in S(R,X) so that

ufg = F−1(φR(i·,A)Fg).

Further,

‖ufg‖1 ≤ K(‖g‖1 + ‖F−1(1− φ) ∗ g‖1)

≤ K1‖g‖1.

Setting M(g) = ufg one sees that M : L1(R,X) −→ L1(R,X) is a bounded linear operator.

Therefore, φR(i·,A) is a Fourier multiplier on L1(R,X). From Theorem 5.1.7 we have that

ζ(T) < 0.

96

Now we turn towards the analogue of Theorem 5.3.6 for Lp(R,X), (1 < p < ∞). The

following Lemma corresponds to Lemma 5.3.5 with (5.31) replaced by a distributional equa-

tion.

Lemma 5.3.7. Let A be a closed operator on X and u, f ∈ Lp(R,X), (1 ≤ p <∞). Then

1. The following are equivalent:

(a) u is a mild solution of (ACPf );

(b) For all Φ ∈ S(R), and λ ∈ ρ(A),

〈F(Φ(i · −A)R(λ,A)), u〉 = 〈FΦ,R(λ,A)f〉; (5.42)

(c) For all Φ ∈ S(R,L(X)), and λ ∈ ρ(A), (5.42) holds.

2. If in addition, both spc(f) ∩ σ(A) and spc(u) ∩ σ(A) are empty, and σ(A) ∩ iR is

compact then the conditions in (1) are also equivalent to :

〈FΦ, u〉 = 〈F(ΦR(i·,A)), f〉 (5.43)

for Φ ∈ S(R) (or in S(R,L(X))) with i supp Φ ⊂ ρ(A).

Proof. (1). (1a) =⇒ (1b) : Let u be a mild solution of (ACPf ). As in the proof of Lemma

5.3.5, we may assume without loss of generality, that

u(t) = u(0) + A

∫ t

0u(τ) dτ +

∫ t

0f(τ) dτ,

for almost all t ∈ R. Let Φ ∈ S(R) and u+, f+ denote the restrictions of u, f to the

non-negative reals, respectively. Further, set

u(t) =

∫ t

0u+(r) dr and f(t) =

∫ t

0f+(r) dr (t ≥ 0).

Then, for a fixed λ ∈ ρ(A) and almost all t ≥ 0 we have,

R(λ,A)u+(t) = R(λ,A)u(0) + AR(λ,A)u(t) + R(λ,A)f(t).

Therefore,

〈FΦ,R(λ,A)u+〉 =

∫ ∞

0FΦ(s)R(λ,A)u(0) ds+ 〈FΦ,AR(λ,A)u〉

+ 〈FΦ,R(λ,A)f〉.(5.44)

97

Using Fubini’s theorem and the fact that Φ ∈ S(R) we have

〈F(i · Φ),R(λ,A)u〉 =

∫ ∞

0F(i · Φ)(s)R(λ,A)u(s) ds

=

∫ ∞

0

∫ s

0F(i · Φ)(s)R(λ,A)u+(t) dt ds

=

∫ ∞

0

∫ ∞

tF(i · Φ)(s)R(λ,A)u+(t) ds dt

= −∫ ∞

0

∫ ∞

t(F(Φ))′ (s)R(λ,A)u+(t) ds dt

= 〈FΦ,R(λ,A)u+〉.

Similarly,

〈F(i · Φ),R(λ,A)f〉 = 〈FΦ,R(λ,A)f+〉.

The above remains true if R(λ,A) is replaced by AR(λ,A). On substituting i ·Φ in place

of Φ in (5.44), one obtains

〈F(i · Φ),R(λ,A)u+〉 =

∫ ∞

0F(i · Φ)(s)R(λ,A)u(0) ds+ 〈FΦ,AR(λ,A)u+〉

+ 〈FΦ,R(λ,A)f+〉.(5.45)

By considering similarly the restrictions u− and f− of u, f respectively, to the negative real

axis we obtain

〈F(i · Φ),R(λ,A)u−〉 =

∫ 0

−∞F(i · Φ)(s)R(λ,A)u(0) ds+ 〈FΦ,AR(λ,A)u−〉

+ 〈FΦ,R(λ,A)f−〉.(5.46)

Adding (5.45) and (5.46) and noting that

∫

RF(i · Φ)(s) ds = 0, yields

〈F(i · Φ),R(λ,A)u〉 = 〈FΦ,AR(λ,A)u〉+ 〈FΦ,R(λ,A)f〉. (5.47)

Since AR(λ,A) and R(λ,A) are bounded operators,

〈FΦ,AR(λ,A)u〉 = F(ΦAR(λ,A)), u〉

and

〈F(i · Φ),R(λ,A)u〉 = 〈F(i ·ΦR(λ,A)), u〉.

Therefore, (5.47) may be rewritten as

〈F(Φ(i · −A)R(λ,A)), u〉 = 〈FΦ,R(λ,A)f〉

as required.

98

(1b) =⇒(1a): Let (gn) ⊂ C∞c (R) be a mollifier. Then gn ∗ u ∈ Lp(R,X) ∩ C∞(R,X),

for all n. For any Φ ∈ S(R) and h ∈ S(R,X), by Fubini’s Theorem,

〈FΦ, gn ∗ h〉 = 〈F(FgnΦ), h〉. (5.48)

Using the density of S(R,X) in Lp(R,X) and the Dominated Convergence Theorem we see

that (5.48) in fact holds for all h ∈ Lp(R,X). Let λ ∈ ρ(A). Applying (5.42), with Φ

replaced by (Fgn)Φ, and using (5.48) we arrive at

〈F(i · (Fgn)Φ),R(λ,A)u〉 = 〈F(FgnΦ),AR(λ,A)u〉+ 〈F(FgnΦ),R(λ,A)f〉

= 〈FΦ,AR(λ,A)(gn ∗ u)〉+ 〈FΦ,R(λ,A)(gn ∗ f)〉.

(5.49)

Also, making use of (5.48), we have

〈FΦ, (gn ∗ u)′〉 = −〈(FΦ)′, gn ∗ u〉= −〈F(−i · Φ), gn ∗ u〉= 〈F(i · (Fgn)Φ), u〉. (5.50)

Therefore, from (5.50) it follows that

〈FΦ,R(λ,A)(gn ∗ u)′〉 = 〈F(i · (Fgn)Φ),R(λ,A)u〉.

Combining this with (5.49), we have

〈FΦ,R(λ,A)(gn ∗ u)′〉 = 〈FΦ,AR(λ,A)(gn ∗ u)〉+ 〈FΦ,R(λ,A)(gn ∗ f)〉. (5.51)

Since F [S(R)] = S(R), and all the terms are continuous functions, we have

R(λ,A)(gn ∗ u)′(s) = R(λ,A)(gn ∗ f)(s) + AR(λ,A)(gn ∗ u)(s)

for all s ∈ R. As the first two terms of the above equation lie in D(A), AR(λ,A)(gn ∗u)(s)

is in D(A) so that (gn ∗ u)(s) ∈ D(A) and AR(λ,A)(gn ∗ u)(s) = R(λ,A)A(gn ∗ u)(s).

Using the injectivity of R(λ,A), we conclude that

(gn ∗ u)′(s) = A(gn ∗ u)(s) + (gn ∗ f)(s) for all s ∈ R.

So, gn ∗u is a classical solution of (ACPf ). Therefore, it is also a mild solution and satisfies

∫ t

s(gn ∗ u)(τ) dτ ∈ D(A)

99

and

(gn ∗ u)(t) = (gn ∗ u)(s) + A

∫ t

s(gn ∗ u)(τ) dτ +

∫ t

s(gn ∗ f)(τ) dτ (5.52)

for all t, s ∈ R. Since gn ∗u −→ u in Lp(R,X) and gn ∗f −→ f in Lp(R,X), letting n −→∞in (5.52) through a subsequence, we have

∫ t

su(τ) dτ ∈ D(A) and

u(t) = u(s) + A

∫ t

su(τ) dτ +

∫ t

sf(τ) dτ for almost all t, s ∈ R.

Therefore, u is a mild solution of (ACPf ).

(1b) =⇒ (1c): First suppose that Φ = Φ0⊗U, where Φ0 ∈ S(R), U ∈ L(X). Since (5.42)

holds for Φ0, it also holds for this Φ. Therefore, by linearity, (5.42) holds for Φ = Σni=1Φi⊗Ui,

Φi ∈ S(R), Ui ∈ L(X), i = 1, 2, ..., n. Given Φ ∈ S(R,L(X)), we can thus find a sequence

Φn of functions of this form such that

‖F(Φn − Φ)‖Lp′ (R,L(X)) −→ 0 as n −→∞;

‖F(i · Φn − i · Φ)‖Lp′ (R,L(X)) −→ 0 as n −→∞.

Note that (5.42) may be rewritten as

〈F(i · Φ),R(λ,A)u〉 − 〈FΦ,AR(λ,A)u〉 = 〈FΦ,R(λ,A)u〉.

Since both sides of the above equation are continuous in the Lp′(R,L(X)) norm, (5.42)

holds for all Φ ∈ S(R,L(X)).

(1c) =⇒ (1b): This is immediate.

(2): Let i spc(f) ∩ σ(A) and i spc(u) ∩ σ(A) be empty. First suppose that (5.43) holds

for all Φ ∈ S(R) with i supp Φ ⊂ ρ(A). Let Ψ ∈ S(R) with i supp Ψ ⊂ ρ(A). Applying

(5.43) with Φ = i ·Ψ, we have for λ ∈ ρ(A),

〈F(i ·Ψ),R(λ,A)u〉 = 〈F(i ·ΨR(i·,A)),R(λ,A)f〉= 〈F(Ψ(Id + AR(i·,A))),R(λ,A)f〉= 〈FΨ,R(λ,A)f〉+ F(ΨAR(i·,A)),R(λ,A)f〉= 〈FΨ,R(λ,A)f〉+ 〈F(ΨR(i·,A)),AR(λ,A)f〉= 〈FΨ,R(λ,A)f〉+ 〈FΨ,AR(λ,A)u〉, (5.53)

where the last equality is obtained by applying the bounded operator AR(λ,A) on both

sides of equation (5.43). For Ψ ∈ S(R) with supp Ψ ∩ (spc(f) ∪ spc(u)) empty, we have on

using Lemma 5.3.2,

〈FΨ,R(λ,A)f〉 = 0;

〈F(i ·Ψ),R(λ,A)u〉 = 0;

〈FΨ,AR(λ,A)u〉 = 0.

100

Since i(spc(f)∪ spc(u))∩σ(A) is empty, there exist disjoint open sets U1, U2 in R such that

spc(f) ∪ spc(u) ⊂ U1 and σ(A) ∩ iR ⊂ iU2. Let g ∈ C∞(R), 0 ≤ g ≤ 1 satisfy

g(s) =

0 (s near spc(f) ∪ spc(u) or near ±∞),

1 (s ∈ R \ U1).

For Ψ ∈ S(R), writing Ψ = gΨ+(1−g)Ψ, we see that supp(gΨ) ⊂ R\ (spc(f)∪ spc(u)) and

i supp((1−g)Ψ) ⊂ iU1 ⊂ ρ(A)∩iR. Thus every Ψ ∈ S(R) may be written as Ψ1,Ψ2 ∈ S(R)

with supp Ψ1 ∩ (spc(f) ∪ spc(u)) empty and i supp Ψ2 ⊂ ρ(A). Therefore, (5.53) holds for

all Ψ ∈ S(R), so that (1b) is true.

Next suppose that the equivalent conditions of (1) hold. Let Ψ ∈ S(R), with i supp Ψ ⊂ρ(A). Applying ( 1c) with Φ = ΨR(i·,A) and using

isΨ(s)R(is,A) = Ψ(s) + Ψ(s)AR(is,A)

we have,

〈FΨ,R(λ,A)u〉+ 〈F(ΨAR(i·,A)),R(λ,A)u〉 = 〈F(ΨR(i·,A)),AR(λ,A)u〉+ 〈F(ΨR(i·,A)),R(λ,A)f〉.

Since 〈F(ΨR(i·,A)),AR(λ,A)u〉 = 〈F(ΨAR(i·,A)),R(λ,A)u〉,

〈FΨ,R(λ,A)u〉 = 〈F(ΨR(i·,A)),R(λ,A)f〉.

The injectivity of R(λ,A) implies

〈FΨ, u〉 = 〈F(ΨR(i·,A)), f〉.

The claim for Ψ ∈ S(R,L(X)) follows along the same lines as (1b) =⇒ (1c).

We are now in a position to prove the Lp-version of Theorem 5.3.6.

Theorem 5.3.8. Let 1 ≤ p <∞. The following are equivalent for a C0-semigroup T :

1. ζ(T) < 0;

2. s∞0 (A) < 0 and for all f ∈ Lp(R,X) with i spc(f)∩σ(A) empty, there exists a unique

mild solution u ∈ Lp(R,X) of (ACPf ) such that i spc(u) ∩ σ(A) is empty.

Proof. (1) =⇒ (2) : Let f ∈ Lp(R,X) with i spc(f)∩σ(A) empty. Since ζ(T) < 0, s∞0 (A) <

0. Choose φ ∈ C∞(R) satisfying (P0) such that φ = 1 near spc(f). Then from Remark 5.1.8

it follows that M : S(R,X) −→ S(R,X) given by

M(g) = F−1(φ(·)R(i·,A)Fg) (g ∈ S(R,X)),

101

extends to a bounded linear operator on Lp(R,X). Let M(f) = uf , f ∈ Lp(R,X). Then,

using (5.29) we have, for all Φ ∈ S(R),

〈FΦ, uf 〉 = 〈F(φR(i·,A)Φ), f〉 (5.54)

and suppFuf ⊂ suppFf . Thus spc(uf ) ⊂ spc(f) so that i spc(uf )∩σ(A) is empty. Further,

for Ψ ∈ S(R) with i supp Ψ ⊂ ρ(A),

〈FΨ, uf 〉 = 〈ΨφR(i·,A),Ff〉= 〈ΨR(i·,A),Ff〉 − 〈ΨR(i·,A),F(F−1(1− φ) ∗ f)〉= 〈ΨR(i·,A),Ff〉 (5.55)

= 〈F(Ψ(·)R(i·,A)), f〉. (5.56)

Here we have used the following to obtain the second last inequality: From (5.27), it follows

that spc(F−1(1− φ) ∗ f

)⊂ spc(f) ∩ supp(1 − φ) which is empty due to the particular

choice of φ. Thus, from [2, Theorem 4.8.2] we have F−1(1− φ) ∗ f = 0. It follows then that

〈ΨR(i·,A),F(F−1(1− φ) ∗ f)〉 = 0.

An application of Lemma 5.3.7 then shows that uf is a mild solution of (ACPf ). If vf

is another mild solution of (ACPf ) with i spc(vf )∩σ(A) empty then from Lemma 5.3.7 we

have that

〈FΨ, uf 〉 = 〈F(Ψ(·)R(i·,A)), f〉= 〈FΨ, vf 〉

(5.57)

for all Ψ ∈ S(R) with i supp Ψ ⊂ ρ(A). On the other hand, if Ψ ∈ S(R) is such that

supp Ψ∩ (spc(uf ) ∪ spc(vf )) is empty, then the sets spc(uf )∩ supp Ψ, and spc(vf )∩ supp Ψ

are both empty, so that

〈FΨ, uf 〉 = 0 = 〈FΨ, vf 〉,

by Lemma 5.3.2. Writing Ψ ∈ S(R) as Ψ1 + Ψ2, where Ψ1 ∈ S(R), supp Ψ1 ⊂ ρ(A) and

Ψ2 ∈ S(R) is such that supp Ψ2∩(spc(uf ) ∪ spc(vf )) is empty, we have that 〈Ψ, uf 〉 = 〈Ψ, vf 〉for all Ψ ∈ S(R). Therefore, we conclude that uf = vf .

(2) =⇒ (1): From (2) it follows that there is a b > 0 such that

Q0,b ⊂ ρ(A) and supλ∈Q0,b

‖R(λ,A)‖ <∞.

Then σ(A) ∩ iR ⊂ (−ib, ib). Let φ ∈ C∞(R) satisfy (P0,b). Let

Y =f ∈ Lp(R,X) : spc(f) ⊂ R \ (−b, b)

.

Then Y is closed in Lp(R,X). As in the proof of Theorem 5.3.6, we have a linear map

B : Y −→ Lp(R,X) given by

B(f) = uf ,

102

where uf is the unique mild solution of (ACPf ) such that i spc(uf ) ∩ σ(A) is empty. From

Lemma 5.3.7 we have, for all Ψ ∈ S(R) with i supp Ψ ⊂ ρ(A)

〈FΨ, uf 〉 = 〈F(ΨR(i·,A)), f〉. (5.58)

Note that (5.58) implies that spc(uf ) ⊂ spc(f), f ∈ Y. Indeed, if i supp Ψ∩(σ(A)∪ i spc(f))

is empty, then 〈F(ΨR(i·,A)), f〉 = 0, by the operator-valued version of Lemma 5.3.2 (see

Remark 5.3.3). Therefore (5.58) implies that 〈FΨ, uf 〉 = 0. This implies that i spc(uf ) ⊂σ(A) ∪ i spc(f) (see (5.28)), and hence spc(uf ) ⊂ spc(f).

We shall now show that B is closed. Then it will follow from the Closed Graph Theorem

that B is bounded. Suppose (fn) ⊂ Y converges to f in Lp(R,X) and B(fn) = ufn converges

to some u in Lp(R,X). Using the same arguments as in the proof of (2) =⇒ (1) of Theorem

5.3.6 we conclude that u is a mild solution of (ACPf ).

Now spc(fn) ⊂ R \ (−b, b), so that from Lemma 5.3.2 it follows that 〈FΨ, fn〉 = 0 for all

Ψ ∈ S(R) with supp Ψ ⊂ (−b, b). Since 〈FΨ, fn〉 converges to 〈FΨ, f〉 as n→∞, Ψ ∈ S(R),

it follows that 〈FΨ, f〉 = 0 for all Ψ ∈ S(R) with supp Ψ ⊂ (−b, b). It follows then from

(5.28) that spc(f) ⊂ R \ (−b, b). Therefore, f ∈ Y.

Similarly, since spc(ufn) ⊂ spc(fn) ⊂ R \ (−b, b) and ufn converges to u in Lp(R,X) it

follows that spc(u) ⊂ R \ (−b, b). Therefore, i spc(u) ∩ σ(A) is empty. The uniqueness of

u follows from arguments similar to those used in the proof of (1) =⇒ (2) above. Thus

B(f) = u.

Let g ∈ Lp(R,X) be arbitrary. Then fg = g−F−1(1−φ)∗g ∈ Y. Indeed, for Ψ ∈ S(R),

with supp Ψ ⊂ (−b, b) we have on applying (5.48) (with gn = F−1(1− φ) and h = g),

〈FΨ,F−1(1− φ) ∗ g〉 = 〈F((1− φ)Ψ), g〉= 〈FΨ, g〉.

Thus, 〈FΨ, fg〉 = 0 for all Ψ ∈ S(R), with supp Ψ ⊂ (−b, b). From (5.28) it follows that

spc(fg) ⊂ R \ (−b, b) so that fg ⊂ Y. Therefore, there is a unique mild solution ufg = B(fg)

of (ACPf ) with i spc(ufg) ∩ σ(A) empty, which satisfies (5.58). Set M(g) = ufg . Then M

is a well defined linear map on Lp(R,X). Since B is bounded on Y, there exist constants

K and K1 such that

‖M(g)‖p = ‖ufg‖p= ‖B(fg)‖≤ K‖f‖p= K‖g −F−1(1− φ) ∗ g‖p≤ K1‖g‖p.

103

Suppose that g ∈ S(R,X). Then, for Ψ ∈ S(R) with i supp Ψ ⊂ ρ(A)

〈FΨ,M(g)〉 = 〈F(ΨR(i·,A)), fg〉= 〈ΨR(i·,A),Ffg〉= 〈ΨR(i·,A), φFg〉= 〈Ψ, φR(i·,A)Fg〉= 〈FΨ,F−1(φR(i·,A)Fg)〉.

Further, if Ψ ∈ S(R) is such that supp Ψ ⊂ [−b, b] then

〈FΨ,M(g)〉 = 0 = 〈FΨ,F−1(φR(i·,A)Fg)〉.

The first equality in the above follows from Lemma 5.3.2 on noting that spc(M(g)) ⊂spc(fg) ⊂ R\(−b, b), so that supp Ψ∩spc(M(g)) = ∅. Since any Ψ ∈ S(R) may be expressed

as a sum of Ψ1,Ψ2 ∈ S(R), with i supp Ψ1 ⊂ ρ(A) and supp Ψ2 ⊂ [−b, b], it follows that

〈FΨ,M(g)〉 = 〈FΨ,F−1(φR(i·,A)Fg)〉

for all Ψ ∈ S(R). We conclude therefore that

M(g) = F−1(φR(i·,A)Fg)

for all g ∈ S(R,X). Thus s 7→ φ(s)R(is,A) is a Fourier multiplier on Lp(R,X). An

application of Theorem 5.1.7 yields ζ(T) < 0.

We have used (2) of Lemma 5.3.7 to prove the above result. However, Theorem 5.3.8

can be proved, alternatively, by just making use of (1) of Lemma 5.3.7. We do not include

the details here. The basic strategy of both the proofs, of course, is the same.

Remark 5.3.9. Let 1 ≤ p < ∞ and T be a C0-semigroup with generator A such that

σ(A) ∩ iR is empty. Then Theorem 5.3.8 gives

ζ(T) < 0⇐⇒ s∞0 (A) < 0 and for all f ∈ Lp(R,X) there exists

a unique mild solution of (ACPf ) in Lp(R,X).

Latushkin and Shvydkoy’s result [34, Theorem 2.7 ] relating Fourier multiplier properties

of the resolvent and hyperbolicity of the semigroup, (see Remark 5.1.12) and the proof of

Theorem 5.3.8 also shows that

T is hyperbolic ⇐⇒ For all f ∈ Lp(R,X) there exists a unique

mild solution of (ACPf ) in Lp(R,X).(5.59)

104

(5.59) has been proven by Latushkin, Randolph and Schnaubelt ([33, Theorem 2.1], [16,

Theorem 4.33] ) in the more general setting of evolution families. A result similar to (5.59)

was first established by Pruss [44]. The precise result there was proven under the assumption

that the function f and the corresponding mild solution of (ACPf ) are both continuous.

105

Chapter 6

Weak compactness, sun-reflexivity

and approximations of the identity

We examine in this chapter the relation between the existence of a certain type of approx-

imation of the identity on a Banach space X and the sun-reflexivity of X with respect to

some C0-semigroup. Sauvageot [46] has shown that if X is a C∗-algebra, a weakly compact

approximation of the identity exists if and only if X admits a C0-semigroup T, such that

the canonical extension of T(t) maps X∗∗ into the multiplier algebra of X. We study these

concepts for general Banach spaces and relate them to sun-reflexivity of the space with

respect to some semigroup.

Before starting on the main theme, we record a small observation concerning the non-

analytic growth bound and the adjoint of a C0-semigroup:

Remark 6.0.10. For a C0-semigroup T, with generator A, we have

1. ζ(T) = ζ(T);

2. δ(T) = δ(T);

3. s∞0 (A) = s∞0 (A).

For the adjoint C0-semigroup T with generator A the following hold [40, Theorem

1.4.2]: ρ(A) = ρ(A) and R(λ,A) = R(λ,A) for all λ ∈ ρ(A). Also, ‖R(λ,A)‖ ≤‖R(λ,A)‖ ≤ ‖R(λ,A)‖. Thus, if ζ(T) < β, then there exist α < β and b > 0 such that

for all t ≥ 0, T(t) = Tα,b(t) + T2(t), with ω0(T2) < β. It follows on taking adjoints, that

T(t) = Tα,b(t) + T2(t) (t ≥ 0).

It is easy to see that

Tα,b(t) =1

2πi

∫

Γα,b

eλtR(λ,A) dλ.

106

Since ω0(T2 ) ≤ ω0(T2), it follows that ζ(T) < β. On the other hand, if ζ(T) < β

then applying the above argument to the C0-semigroup T instead of T, we have that

ζ(T) < β so that there exist α < β and b > 0 such that

T(t) =

∫

Γα,b

eλtR(λ,A) dλ + S(t),

with ω0(S) < β. Since T,A may be considered as extensions of T and A respectively,

taking restrictions of the operators in the above equation to X, we have that ζ(T) < β.

The other two equalities also follow easily from the definitions.

6.1 Weak compactness and sun-reflexivity

We recall some standard definitions and facts concerning the adjoint of a C0-semigroup and

weakly compact operators. The details may be found in [39, Chapter 1, Chapter 2].

A strongly continuous semigroup T on X is called weakly compact if T(t) is weakly

compact for all t > 0, that is, T(t) maps bounded subsets of X to relatively weakly compact

subsets of X. Gantmacher’s theorem asserts that a bounded operator S is weakly compact

on X if and only if S∗∗X∗∗ ⊂ X.

Given a C0-semigroup T on X, the locally convex topology on X generated by the

semi-normspx : x ∈ X

,where px(x) = |〈x, x〉| (x ∈ X),

is denoted by σ(X,X).

A useful way of identifying σ(X,X)-continuous operators is given by [39, Proposition

2.4.3]:

Proposition 6.1.1. If S commutes with T(t) for each t > 0 then S is σ(X,X)-continuous.

Analogous to Gantmacher’s Theorem for weak compactness of a bounded operator on

X, we have the following characterisation of σ(X,X)-compactness of a bounded operator

on X [39, Theorem 2.4.2] :

Theorem 6.1.2. A σ(X,X)-continuous operator S on X is σ(X,X)-compact if and

only if S∗X∗ ⊂ jX.

We quote now some results from [39] which bring out the relation between -reflexivity

of the space with respect to a given C0-semigroup and the compactness of the semigroup and

resolvents of the generator with respect to the different topologies. The following theorem

gives a very useful characterisation of -reflexivity [39, Theorem 2.5.2].

Theorem 6.1.3. Let (T(t))t ≥ 0 be a C0-semigroup of operators on X with generator A.

Then the following are equivalent:

107

1. X is -reflexive with respect to T;

2. R(λ,A) is σ(X,X)-compact for some λ ∈ ρ(A);

3. R(λ,A) is weakly compact for some λ ∈ ρ(A).

From the resolvent identity it follows that if R(λ,A) is weakly compact for one λ ∈ ρ(A),

then R(λ,A) is weakly compact for all λ ∈ ρ(A) and the same is true for σ(X,X)-

compactness of R(λ,A).

A Banach space X is said to have the Dunford-Pettis Property (DPP) if every weakly

compact operator B : X −→ Y, Y being any Banach space, maps relatively weakly compact

subsets to relatively compact subsets. It is easy to see that if B is a weakly compact

operator on a space with (DPP) then B2 is compact. A use of the resolvent identity yields

the following corollary [39, Corollary 2.5.4]:

Corollary 6.1.4. If X has the Dunford-Pettis property, then X is - reflexive if and only

if R(λ,A) is compact for some/ all λ ∈ ρ(A).

A semigroup T is said to be strongly continuous for t > 0 or C>0 if

limt↓0‖T(s+ t)x−T(s)x‖ = 0

holds for all s > 0 and x ∈ X.

If T is weakly compact, then R(λ,A) is weakly compact for each λ ∈ ρ(A) and it follows

from Theorem 6.1.3 that X is -reflexive. The converse is not always true, but we have

from [39, Corollary 5.2.9]

Theorem 6.1.5. Let (T(t))t ≥ 0 be a C0-semigroup of bounded linear operators with gen-

erator A. Then, the following are equivalent:

1. T is weakly compact;

2. X is -reflexive and T∗∗ is C>0.

To the above set of results we can add an analogue of Theorem 6.1.5 for semigroups

that are σ(X,X)-compact:

Theorem 6.1.6. Let (T(t))t ≥ 0 be a C0-semigroup of bounded linear operators on X. The

following are equivalent:

1. T(t) is σ(X,X)- compact for all t > 0;

2. X is -reflexive and T∗ is C>0.

108

Proof. Let (T(t))t ≥ 0 be a C0-semigroup on X which is σ(X,X)-compact. We show that

then R(λ,A) is σ(X,X)-compact. In view of Theorem 6.1.2 and Proposition 6.1.1 it is

enough to show that R(λ,A)∗X∗ ⊂ jX. We note here that the map s 7→ T∗(s)x∗

is continuous on (0,∞) for each x∗ ∈ X∗. This is because σ(X,X)-compactness of T

implies, by Theorem 6.1.2, that T∗(s)X∗ ⊂ jX ⊂ X for all s > 0. So, for s > 0,

limt↓0‖ T∗(t)(T∗(s)x∗) − T∗(s)x∗ ‖= 0.

This means that for all s > 0,

limt↓0‖T∗(t+ s)x∗ −T∗(s)x∗‖ = 0. (6.1)

From (6.1), it also follows that the map s 7→ T∗(s)x∗ is measurable on (0,∞). Thus, for

Reλ > ω0(T) and y ∈ X, we have

〈R(λ,A)∗x∗, y〉 = 〈x∗,R(λ,A)y〉= 〈x∗,R(λ,A)y〉

= 〈x∗,∫ ∞

0e−λsT(s)y ds〉

= 〈∫ ∞

0e−λsT∗(s)x∗ ds, y〉.

Since T∗(s)x∗ ∈ jX for all s > 0 we have that R(λ,A)∗X∗ ⊂ jX. Thus R(λ,A) is

σ(X,X)-compact and it follows from Theorem 6.1.3 that X is -reflexive with respect to

the semigroup T. Further, (6.1) says precisely that T∗ is C>0.

Next assume that X is -reflexive, i.e. jX = X, and T∗ is C>0. To show T is

σ(X,X)-compact, it is enough to show that T∗(s)X∗ ⊂ jX = X for all s > 0. Let

T∗(s)x∗ ∈ T∗(s)X∗. Since T∗ is C>0 we have that

‖ T∗(t+ s)x∗ −T∗(s)x∗ ‖−→ 0 as t→ 0,

for all s > 0 and x∗ ∈ X∗. In other words, we have, for all s > 0,

‖ T∗(t)(T∗(s)x∗)−T∗(s)x∗ ‖−→ 0 as t→ 0.

Therefore, T∗(s)x∗ ∈ X, for all s > 0.

6.2 Approximations of the identity

We bring out in this section some connections between the existence of a certain type of

approximations of the identity on a Banach space and the existence of a semigroup with

respect to which the space is -reflexive. The results obtained in this process show the

109

existence of separable Banach spaces which cannot be sun-reflexive with respect to any

strongly continuous semigroup.

The ideas for some of the proofs are inspired by those in [46], where the existence of

approximations of the identity are studied in the context of C∗-algebras admitting ‘Strong

Feller semigroups’. A bounded operator B defined on a C∗-algebra X is said to have the

strong Feller property [46, Definition 3.2] if

B∗∗(X∗∗) ⊂M(X) := x ∈ X∗∗ : xy ∈ X, yx ∈ X, for all y ∈ X

and a C0-semigroup T on a C∗-algebra X is called a strong Feller semigroup if for each

t > 0, T(t) has the strong Feller property. Since X ⊂ M(X) for any C∗-algebra X, from

Gantmacher’s Theorem it follows that every weakly compact operator on a C∗-algebra has

the strong Feller property.

Recall (see e.g. [36, Section 1.g]) that a Banach space X is said to have the compact

approximation property or CAP if for every compact set K ⊂ X and ε > 0, there exists a

compact operator B ∈ L(X) satisfying

‖Bx− x‖ ≤ ε, for all x ∈ K. (6.2)

If the compact operators above may be chosen to have norm less than or equal to 1, then X

is said to have the metric compact approximation property or the metric CAP. Analogously,

we may define the weakly compact approximation property or WCAP for a Banach space.

We shall say that a Banach space X has the WCAP if for every ε > 0 and every compact

subset K of X there exists a weakly compact operator B ∈ L(X) satisfying (6.2). If the

weakly compact operators may be chosen to have norm less than or equal to one, then X

is said to have the metric WCAP. The weakly compact approximation property has been

considered by Grønbæk and Willis [25] and Lima and Nygaard [35] among others.

We shall call a net (Bα) of bounded linear operators on X an approximation of the

identity if

limα‖Bαx− x‖ = 0, for all x ∈ X.

It follows immediately from the definitions that if X admits a uniformly bounded com-

pact (respectively, weakly compact) approximation of the identity, then X has the CAP

(respectively, WCAP).

The next Lemma, taken from [46], is technical in nature. It is integral to the proof of

the main result of this section and therefore, for the sake of completeness, we include the

proof of the Lemma here.

Lemma 6.2.1. There exists a universal constant K, such that if X is a Banach space and

B is a bounded linear operator on X with ‖ B ‖≤ 1 then we have

‖ (B− I)et(B−I) ‖ ≤ Kt−1/2 (t ∈ R+).

110

Proof. Let t > 0 be fixed and let n = nt denote the integer part of t. Then we have

(I−B)etB =

n∑

k=0

(1− k

t

) tkBk

k!−

∞∑

k=n+1

(kt− 1) tkBk

k!.

With B = I in the above formula, we obtain

n∑

k=0

(1− k

t

) tkk!

=∞∑

k=n+1

(kt− 1) tkk!.

So, for any B with ‖B‖ ≤ 1,

‖(I−B)etB‖ ≤n∑

k=0

(1− k

t

) tkk!

+∞∑

k=n+1

(kt− 1) tkk!

= 2n∑

k=0

(1− k

t

) tkk!

= 2tn

n!.

Therefore,

‖ (B− I)et(B−I) ‖ ≤ 2e−ttn

n!

≤ Kt−1/2,

where the estimate in the last line of the above, has been obtained as follows: By Stirling’s

formula, there exists t0 > 1 such that, for t ≥ t0

n! ≥√

2πne−n−1nn

where n = nt. Setting t = n+ θ, 0 < θ < 1, we therefore have for t ≥ t0, n = nt

2e−ttn

n!≤ 2

e−(n+θ)(n+ θ)n√2πne−n−1nn

= 2e−θ+1

√2πn

(1 +

θ

n

)n

≤ 2e√2πn

(1 +

1

n

)n

≤ 2e2

√2πn

≤ 2e2

√2π(t− 1)

.

Thus we can choose K such that 2e2√2π(t−1)

≤ Kt−1/2 for t ≥ t0 and 2e2t ≤ Kt−1/2 for

t < t0.

111

Let(etBk

)t≥0

(k ∈ N), be a family of commuting contraction semigroups. In [3], the

conditions under which the infinite product∏∞k=1 e

tBk converges and defines a C0-semigroup

have been investigated. We quote [3, Proposition 2.7]:

Theorem 6.2.2. Let Tk : k ∈ N be a commuting family of contraction semigroups (or

groups) on X with generator Bk, (k ∈ N). Assume that the space

D1 :=

x ∈

⋂

k∈ND(Bk) :

∞∑

k=1

‖Bkx‖ <∞

is dense in X. Then the semigroup ( respectively, group) product∏∞k=1 Tk exists, in the

sense that for each x ∈ X,( ∞∏

k=1

Tk(t)

)x := lim

n→∞

n∏

k=1

Tk(t)x

converges uniformly on compact subsets of [0,∞). Define B on D1 by Bx =∑∞

k=1 Bkx.

Then B is closable and B is the generator of the product semigroup.

Taking Bk = Ak − I, k ∈ N where Ak are as below, yields the first part of the next

result. However, we present a proof for it which is essentially similar to that of [46, Lemma

2.3 ], where the result has been proven for C∗-algebras.

Lemma 6.2.3. Let Ann∈N be a sequence of contractions on X and suppose that

1. An is a commuting family and

2. the subspace D1 = x ∈ X :∑∞

n=1 ‖ x−An(x) ‖ < ∞ is dense in X.

Then there exists a contractive semigroup (T(t))t ≥0, on X such that, for any t ∈ R+, the

sequence Tn(t) given by

Tn(t) = et(A1 +A2 +...An−nI) (n ∈ N),

converges to T(t) in the strong operator topology . Moreover, for any t > 0 we have,

limn→∞

∥∥∥∥(

I− (A1 + ...+ An)

n

)T(t)

∥∥∥∥ = 0.

Proof. For n,m ∈ N, m ≥ n, set

∆nm = (m− n)I− (An+1 + ...+ Am).

Then we may write

Tm(t) = et(A1+A2+...An−nI)+t(An+1...+Am)−t(m−n)I

= Tn(t) exp(−t∆nm).

112

Note that both Tn(t) and exp(−t∆nm) are contractive on X. For x ∈ D1, we have

‖Tm(t)x−Tn(t)x‖ = ‖Tn(t)(exp(−t∆nm)x− x) ‖≤ ‖e−t∆nmx− x‖

=

∥∥∥∥∫ t

0

d

dse−s∆nmx ds

∥∥∥∥

≤∫ t

0

∥∥∥∥d

dse−s∆nmx

∥∥∥∥ ds

≤∫ t

0‖∆nmx‖ ds

= t‖∆nmx‖. (6.3)

By the definition of D1, ‖∆nmx‖ tends to 0 as n → ∞, uniformly in m. Therefore, the

sequence Tn(t)xn∈N is Cauchy for any x ∈ D1. By the norm density of D1, this property

extends to all x ∈ X. Therefore, we can define the pointwise norm limit semigroup (T(t))t≥0

by

T(t)x = limn→∞

Tn(t)x (x ∈ X, t ≥ 0).

Further, applying the inequality obtained in (6.3) above, with n = 0 we have for x ∈ D1,

‖ T(t)x− x‖ = limm‖ Tm(t)x− x‖

= limm‖et(A1 +A2 +...Am−mI)x− x‖

≤ t limm‖ (A1 + ...+ Am −mI)x‖

→ 0 as t→ 0.

Therefore, the semigroup T is strongly continuous.

Now, T(t) can be written as

T(t)x = limm

Tn(t) (e−t∆nmx

).

Therefore, from Lemma 6.2.1 it follows that

∥∥∥∥(

I− 1

n(A1 + ...+ An)

)T(t)

∥∥∥∥ ≤∥∥∥∥(

I− 1

n(A1 + ...+ An)

)Tn(t)

∥∥∥∥

=

∥∥∥∥(

I− 1

n(A1 + ...+ An)

) ent( 1

n(A1+...+An))−I)

∥∥∥∥≤ K(nt)−1/2

→ 0 as n → ∞.

113

Corollary 6.2.4. Using the same notation as in Lemma 6.2.3 we have, for x ∈ X, and

Reλ > 0,

R(λ,A)x =

∫ ∞

0e−λsT(s)x ds

= limn→∞

[λI− (A1 + A2 + ...An − nI)]−1(x),

= limn−→∞

R(λ,Bn)(x)

where A is the generator of the C0-semigroup T and Bn = A1 + A2 + ...+ An− nI, n ∈ N.

Proof. This follows immediately from the first Trotter-Kato Approximation Theorem [20,

Theorem III.4.8], on noting, from the proof of Lemma 6.2.3, that the convergence

Tn(t)x −→ T(t)x, x ∈ X,

is uniform on compact subsets of [0,∞).

The following lemma is an alteration of [46, Lemma 2.4]. The proof of [46, Lemma 2.4]

does not seem to work with the particular choice of the finite dimensional subspaces cited

in the statement. We present the result here with the necessary modifications and in the

context of Banach spaces.

Lemma 6.2.5. Let An be a sequence of contractions on the separable Banach space X and

xnn∈N be a dense set in X. Suppose that for each n ∈ N,

‖(I−An)(x)‖ ≤ 2−n‖x‖,

for any x ∈ Dn, where Dn is the finite dimensional subspace of X generated by the set(

k∏

i=1

Ari

)(xl) : 0 ≤ ri ≤ n(1 ≤ i ≤ k), 0 ≤ l ≤ n, 0 ≤ k ≤ n2

and A0 = I. For n ∈ N, let Bn = A1+A2+...+An−nI. Then the sequence(etBn

)t≥0

n∈N

of C0-semigroups on X converges in the resolvent sense towards a C0-semigroup (T(t))t≥0,

that is,

limn→∞

R (µ,Bn) (x) = R (µ,A) (x), (6.4)

for any x ∈ X and µ ∈ (0,∞), where A is the generator of T.

Proof. Let Θn = 1n (A1 + A2 + ...+ An) . Then, n (I−Θn) = −Bn. For n, p > 0, 0 < l < n,

and α > 0 we have

(I−An+p) (I− αBn)−1 (xl) = (I−An+p)1

1 + nα

∞∑

k=0

(nα

1 + nα

)kΘkn(xl)

= E1 + E2,

114

where,

E1 = (I−An+p)1

1 + nα

(n+p)2∑

k=0

(nα

1 + nα

)kΘkn(xl)

=1

1 + nα

(n+p)2∑

k=0

(nα

1 + nα

)k(I−An+p) Θk

n(xl),

E2 = (I−An+p)1

1 + nα

∞∑

k=(n+p)2+1

(nα

1 + nα

)kΘkn(xl).

For 0 ≤ k ≤ (n+ p)2, each of the terms Θkn(xl) ∈ Dn+p. Therefore, the assumptions imply

that

‖E1‖ ≤1

1 + nα

(n+p)2∑

k=0

(nα

1 + nα

)k2−n−p‖xl‖

≤ 2−n−p+1‖xl‖.

Further, since ‖Θn‖ ≤ 1 for each n ∈ N,

‖E2‖ ≤2

1 + nα‖xl‖

∞∑

k=(n+p)2+1

(nα

1 + nα

)k

≤ 2

(nα

1 + nα

)(n+p)2+1

‖xl‖.

Fix α0 > 0 and α ∈ (0, α0]. Since limn→∞(

nα1+nα

)n= e−1/α, there exists n0 ∈ N such that

for all n ≥ n0 , one has (nα

1 + nα

)n≤ e−1/2α0

and (nα

1 + nα

)(n+p)2+1

≤(

nα

1 + nα

)n(n+p)

≤ e−(n+p)/2α0.

Thus, ∥∥∥(I−An+p) (I− αBn)−1 (xl)∥∥∥ ≤ 2‖xl‖

(2−n−p + e−(n+p)/2α0

)

For fixed l and m > n > max(n0, l), we therefore have

∥∥∥(I− αBm) (I− αBn)−1 (xl)− xl∥∥∥ = α

∥∥∥∥∥∥

m−n∑

p=1

(I−An+p) (I− αBn)−1 (xl)

∥∥∥∥∥∥

≤ C(

2−n + e−n/2α0

),

where C is a constant depending only on α and ‖xl‖. Since Bm generates a contractive

C0-semigroup, (I− αBm)−1 is a contraction. Therefore,

∥∥(I− αBm)−1(xl)− (I− αBn)−1(xl)∥∥ ≤ C

(2−n + e−n/2α0

), (6.5)

115

so that n→ (I− αBn)−1(xl) is a Cauchy sequence for any l ∈ N. It follows by density that

n→ (I− αBn)−1(x) is Cauchy for each x ∈ X. Thus,

Rα(x) = limn→∞

(I− αBn)−1(x) (6.6)

exists for each x ∈ X. Further, for α > 0 and fixed n ∈ N, the equality

(I− αBn)−1(y) = α(I− αBn)−1Bny + y (y ∈ D(Bn)),

together with density of D(Bn) in X yields, for fixed n ∈ N

limα↓0

(I− αBn)−1(x) = x (x ∈ X). (6.7)

From (6.5), it is clear that the convergence in (6.6) is uniform for α ∈ (0, α0) for fixed

x = xl, l ∈ N. Hence,

limα→0

Rα(xl) = xl. (6.8)

By density, (6.8) is true for all x ∈ X. Further, the family Rαα>0 satisfies

αRα − βRβ = (α− β)RαRβ (α, β > 0).

It follows that

Ker(Rα) = Ker(Rβ)

Ran(Rα) = Ran(Rβ)

for all α, β > 0. From (6.8) it then follows that Rα is one to one andD = Ran(Rα) is dense in

X for all α > 0. Then A =1

α

(I−R−1

α

)is a closed, densely defined operator, independent of

α. Further, for µ > 0, µR (µ,A) = R 1µ, which is a contraction. Therefore, the Hille-Yosida

Theorem [20, II.3.5] implies that A generates a C0-semigroup T of contractions. Then (6.6)

yields (6.4).

We combine the -reflexivity results quoted previously with some of the ideas from [46,

Lemma 4.2] to obtain :

Theorem 6.2.6. The following are equivalent for a separable Banach space X:

1. There exists a contractive C0-semigroup on X, with respect to which X is -reflexive;

2. There exists a contractive C0-semigroup on X whose generator has weakly compact

resolvents;

3. There exists a weakly compact, commuting, contractive approximation of the identity

on X;

116

4. There exists a weakly compact, contractive approximation of the identity on X;

5. There exists a weakly compact, contractive C0-semigroup on X.

Proof. (1)⇐⇒ (2): This follows immediately from Theorem 6.1.3.

(2) =⇒ (3) : Let A be the generator of the weakly compact contractive C0-semigroup

T. Then, by the Hille-Yosida Generation Theorem [20, Theorem II.3.5 ] (0,∞) ⊂ ρ(A)

and ‖λR(λ,A)‖ ≤ 1 for all λ ∈ (0,∞). Using the same arguments as in (6.7), we have for

y ∈ D(A),

‖nR(n,A)y − y‖ −→ 0 as n −→∞.

Since ‖nR(n,A)‖ is uniformly bounded, it follows that ‖nR(n,A)y−y‖ −→ 0 for all y ∈ X.

Thus, An := nR(n,A) is a commuting, contractive, weakly compact approximation of the

identity.

(3) =⇒ (5) : Let An be a weakly compact, commuting, approximation of the identity,

and suppose that ‖An‖ ≤ 1, for all n ∈ N. By replacing, if required, the given sequence by

a subsequence, one may assume that An is such that

D1 = x ∈ X :

∞∑

n=1

‖x−An(x)‖ <∞ (6.9)

is dense in X. To arrive at such a subsequence, we reason as follows: Let a1, a2, a3... be a

countable dense set in X. Since An form an approximation of the identity on X, we can

find n1 ∈ N such that

‖a1 −An1(a1)‖ < 1

2

We continue inductively, choosing at the kth step, nk > nk−1 such that

‖aj −Ank(aj)‖ ≤1

2k(j = 1, 2, ..., k).

Then Ank forms a subsequence satisfying (6.9).

By Lemma 6.2.3 there exists a semigroup (T(t))t≥0 which is strongly continuous, and

limn→∞

∥∥∥∥(

I− (A1 + ...+ An)

n

)T(t)

∥∥∥∥ = 0,

for each t > 0. So, for each positive real number t,T(t) is the norm limit of a sequence of

weakly compact operators. Hence, it is itself weakly compact. Also, one notes from Lemma

6.2.3 that ‖T(t)‖ ≤ 1 for all t ≥ 0. So (T(t))t≥0 is a contractive, weakly compact semigroup

defined on X.

(5) =⇒ (1): From Theorem 6.1.5, it follows that if T is a weakly compact semigroup on

X then X is -reflexive with respect to it.

(3) =⇒ (4) : This is trivial.

117

(4) =⇒ (2): Let An be a sequence of weakly compact contractions which form an

approximation of the identity of X. Suppose that xll∈N is a countable dense subset of X

and let Dn, n ∈ N, be the finite dimensional subspaces of X defined in the statement of

Lemma 6.2.5. Then for each k ∈ N, the convergence ‖An(x)−x‖ → 0 as n→∞ is uniform

for all unit vectors x ∈ Dk. Thus, we may assume, by replacing An by an appropriate

subsequence if required, that the sequence An satisfies the assumptions of Lemma 6.2.5. It

follows from that Lemma that there exists a C0-semigroup T of contractions, with generator

A such that for each µ > 0,

limn→∞

(µI− (A1 + A2 + ...+ An − nI))−1(x) = R(µ,A)x

for any x ∈ X. Now let Bn and Θn be as in the proof of Lemma 6.2.5 and ∆nm = Bm−Bn,

m > n. Fix β > 0. For x ∈ X, define

xnm =

(I− 1

βBn

)−1(I− 1

n+ β∆nm

)−1

(x)

= β(n+ β)R(β,Bn)R(n+ β,∆nm)x.

Then, straightforward computations show that

x =

(I− 1

n+ β∆nm

)(I− 1

βBn

)xnm

=

(I− Bm

β+

n

nβ + β2∆nmΘn

)xnm,

so that

R(β,Bm)x =1

β

(xnm + R(β,Bm)

n

n+ β∆nmΘnxnm

)

= (n+ β)R(β,Bn)R(n+ β,∆nm)x

+ nR(β,Bm)∆nmΘnR(β,Bn)R(n+ β,∆nm)x

and

R(β,Bm)x− (n+ β)R(β,Bn)R(n+ β,∆nm)x (6.10)

= nR(β,Bm) (Bm −Bn) R(β,Bn)ΘnR(n+ β,∆nm)x

= n (R(β,Bm)−R(β,Bn)) ΘnR(n+ β,∆nm)x, (6.11)

for all x ∈ X. For each n ∈ N, applying Lemma 6.2.5 to the sequence An+1,An+2, ..., we

obtain a contractive C0-semigroup Tn with generator Gn such that

limm→∞

R(α,∆nm)y = R(α,Gn)y,

118

for each y ∈ X and α > 0. In particular, this is true for α = n+β, β > 0. It follows therefore,

on letting m→∞ in (6.10) that

R(β,A)− (n+ β)R(β,Bn)R(n+ β,Gn) (6.12)

= n (R(β,A)−R(β,Bn)) ΘnR(n+ β,Gn). (6.13)

Since ‖βR(β,Bn)‖ ≤ 1 for β > 0, we have

‖(I−Θn)R(β,Bn)‖ =1

n‖BnR(β,Bn)‖ ≤ 2

n.

Using (6.12) we therefore obtain

∥∥R(β,A)− n (R(β,A)−R(β,Bn)) ΘnR(n+ β,Gn)

− (n+ β)ΘnR(β,Bn)R(n+ β,Gn)∥∥

= ‖(n+ β)R(β,Bn)R(n+ β,Gn)− (n+ β)ΘnR(β,Bn)R(n+ β,Gn)‖≤ ‖(I−Θn)R(β,Bn)‖‖(n+ β)R(n+ β,Gn)‖≤ 2

n.

It follows that R(β,A) is the limit in norm of the sequence of bounded operators Φnn∈Nwhere, for n ∈ N,

Φn = n (R(β,A)−R(β,Bn)) ΘnR(n+ β,Gn)− (n+ β)ΘnR(β,Bn)R(n+ β,Gn).

Since Θn is weakly compact for each n ∈ N, so is Φn. This implies that R(β,A) is weakly

compact, being the limit in norm of weakly compact operators. Thus, we have obtained a

C0-semigroup T whose generator A has weakly compact resolvents, and (2) is true.

It is possible to obtain a version of Theorem 6.2.6 for C0-semigroups which are not

necessarily contractive. The price to pay for this is a more complicated condition concerning

the existence of an approximation of the identity. We have,

Corollary 6.2.7. Let X be a separable Banach space. Then the following are equivalent:

1. There exists a C0-semigroup on X, with respect to which X is -reflexive;

2. There exists a C0-semigroup on X whose generator has weakly compact resolvents;

3. There exists a weakly compact, commuting, approximation of the identity, An on

X, such that

An1An2 ...Ank : k ∈ N, ni ∈ N (6.14)

is bounded;

119

4. There exists a weakly compact, approximation of the identity An on X such that

An1An2 ...Ank : k ∈ N, ni ∈ N

is bounded;

5. There exists a weakly compact C0-semigroup on X.

Proof. If any of (1), (2) or (5) is true for a C0-semigroup T, then by considering the rescaled

C0-semigroup e−ω·T(·), ω > ω0(T) on the Banach space X equipped with the equivalent

norm ||| · |||, given by

|||x||| = supt≥0‖e−ωtT(t)x‖, (6.15)

we obtain the corresponding condition for a contractive C0-semigroup. Then it follows from

Theorem 6.2.6 and rescaling and renorming in the reverse direction, that (1), (2) and (5)

are equivalent.

(2) =⇒ (3) : Let T be a C0-semigroup whose generator A has weakly compact resolvents.

Let S(t) = e−ωtT(t), ω > ω0(T). Then S is a contractive C0-semigroup on (X, ||| · |||) , where

||| · ||| is as in (6.15), with generator B = A − ω, and B has weakly compact resolvents.

From the proof of (2) =⇒ (3) in Theorem 6.2.6 there is a weakly compact approximation

of the identity An such that |||An||| ≤ 1, and then An1An2 ...Ank : k ∈ N, n1 ∈ N is

bounded with respect to ‖ · ‖. Thus (3) holds.

(3) =⇒ (4) : This is obvious.

(4) =⇒ (5) : Let An be a weakly compact approximation of the identity on X satis-

fying the assumption in (4). We define a new norm ‖ · ‖b on X by setting,

‖x‖b := sup ‖An1An2 ...Ank(x)‖ : k ∈ N, ni ∈ N, i = 1, 2, ..., k .

It is easy to see that ‖ · ‖b defines an equivalent norm on X and then An forms a weakly

compact contractive approximation of the identity of X. It follows from Theorem 6.2.6 that

there exists a C0-semigroup on X which is weakly compact.

We note here that in Theorem 6.2.6, (5) =⇒ (2) may be proven directly in the following

manner: For Reλ > ω0(T),

R(λ,A)2 =

∫ ∞

0e−λtT(t)R(λ,A) dt,

the integral existing as a L(X)-valued Bochner integral. Since weakly compact operators

form a norm closed ideal of L(X), weak compactness of T implies that of R(λ,A)2. For

any µ ∈ ρ(A), we have, on making use of the resolvent identity (see proof of [39, Corollary

2.5.4], that

limλ→∞

‖(λR(λ,A))2R(µ,A)−R(µ,A)‖ = 0.

120

Therefore, R(µ,A) is also weakly compact, being the limit in norm of weakly compact

operators. Thus, in proving the implication (5) =⇒ (2) we have just used the fact that

weakly compact operators form a norm closed two sided ideal of L(X). In fact, in the proof

of Theorem 6.2.6, this is the only property of weakly compact operators that has been

used, except for the implications (5) =⇒ (1). Therefore, we may deduce the following from

Corollary 6.2.7:

Corollary 6.2.8. Let X be a separable Banach space and B ⊂ L(X) be a norm closed, two

sided ideal of L(X). Then the following are equivalent:

1. There exists a C0-semigroup on X whose generator has resolvents in B;

2. There exists a commuting approximation of the identity, An on X, such that An ∈ Bfor each n ∈ N and

An1An2 ...Ank : k ∈ N, ni ∈ N (6.16)

is bounded.

3. There exists an approximation of the identity An on X such that An ∈ B for each

n ∈ N and

An1An2 ...Ank : k ∈ N, ni ∈ N

is bounded.

4. There exists a C0-semigroup T on X such that T(t) ∈ B for all t > 0.

Remark 6.2.9. We collect here some observations concerning the results obtained so far

in this section.

1. In [46], Sauvageot has proven an analogue of Corollary 6.2.8 for strong Feller, com-

pletely positive contractions on a C∗-algebra. More precisely, it has been shown [46,

Proposition 4.1] that on a separable C∗-algebra the following properties are equivalent:

• There exists a strong Feller, completely positive approximation of the identity of

X;

• There exists a strong Feller C0-semigroup on X of completely positive contrac-

tions.

Note that strong Feller, completely positive contractions defined on a C∗- algebra are

not, in general, associated with a closed two sided ideal of L(X). Therefore, in [46],

additional C∗-algebraic methods had to employed to prove the required results. The

basic strategy, however, was the same.

121

2. As a particular case of Corollary 6.2.8, we may take the ideal of compact operators

on X. Then the equivalent conditions of Corollary 6.2.8 imply that X is -reflexive

with respect to a C0-semigroup. If additionally, the separable Banach space X has

(DPP), then the reverse implication also holds.

3. In general it is not true that X is -reflexive if and only if T is weakly compact. As

an example, one may consider the rotation group T defined on C(Γ), Γ the unit circle,

defined by

T(t)f(eiθ) = f(ei(θ+t)).

Then C(Γ) = L1(Γ) and C(Γ) = C(Γ), so that C(Γ) is -reflexive with respect to

T [39, Example 1.3.9], but T is not weakly compact. However, in view of Theorem

6.2.6, we may conclude that if X is -reflexive with respect to a C0-semigroup then

it at least admits a weakly compact C0-semigroup.

We recall here that a Banach space X with (DPP) is said to have the hereditary Dunford

Pettis property (DPh) if every closed subspace of X also has (DPP).

Theorem 6.2.10. There exist separable Banach spaces which do not admit bounded, weakly

compact, approximations of the identity and therefore, there exist separable Banach spaces

which are not sun-reflexive with respect to any C0-semigroup.

Proof. It is known that `1 has (DPh), [18, Page 26] and that there exists a closed subspace

X of `1 which does not have the compact approximation property [36, Theorem 1.g.4 ].

Suppose Bn is a bounded, weakly compact, approximation of the identity on X. Then

B2n is also an approximation of the identity on X. Further, since X has (DPP), the

operators B2n are compact for each n ∈ N. Thus, X has CAP, which is a contradiction.

Now suppose that there exists a C0-semigroup on X with respect to which X is sun-

reflexive. Since X is separable, by Corollary 6.2.7 there exists a weakly compact approxima-

tion of the identity on X satisfying (6.14); in particular, this weakly compact approximation

of the identity is bounded. The discussion in the previous paragraph shows that this is not

possible. Thus, we may conclude that X cannot be sun-reflexive with respect to any C0-

semigroup.

122

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