laplace transforms, non-analytic growth bounds and c0
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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.
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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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