Laplace Transforms, Non-Analytic Growth Bounds and C 0 -Semigroups

Laplace Transforms, Non-Analytic Growth Bounds and C -Semigroups Sachi Srivastava St. John s College University of Oxford A thesis submitted for the degree of Doctor of Philosophy Hilary 22

Laplace Transforms, Non-Analytic Growth Bounds and C -Semigroups Sachi Srivastava St. John s College University of Oxford A thesis submitted for the degree of Doctor of Philosophy Hilary 22 In this thesis, we study a non-analytic growth bound ζ(f) associated with an exponentially 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, ζ 1 (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 ω (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 regularisation 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, ζ 1 and κ extend to the operator-valued case also. For a C -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 solutions 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 ω α (T), α >, of a C -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 C -semigroups. In particular, we study the relations between the existence of certain approximations of the identity on the Banach space X and that of C -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............................. 1 2.2.1 Exponential growth bound....................... 1 2.2.2 The Laplace transform......................... 1 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 C -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........... 3 3.2.1 Reduction to the vector-valued case.................. 3 3.2.2 The C -semigroup case......................... 33 3.3 Essential holomorphy.............................. 39 3.4 A comparison of the critical growth bound and the non-analytic growth bound 4 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 C -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 16 6.1 Weak compactness and sun-reflexivity..................... 17 6.2 Approximations of the identity......................... 19 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 ), (ACP) u() = x, 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 u(s) ds D(A) and A u(s) ds = u(t) x (t ). 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 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 û(λ) D(A) and λû(λ) Aû(λ) = x, for Re λ sufficiently large. (ACP) is mildly well posed if and only if ρ(a) and (ACP) is classically well-posed, if and only if A generates a C -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 [2], 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 C -semigroup T is called uniformly exponentially stable if ω (T) <, where ω (T) is the exponential growth bound of T given by ω (T) = inf{ω R : sup e ωt T(t) < }. t T being uniformly exponentially stable is equivalent to lim T(t) =. t For a closed operator A, the spectral bound s(a) is given by s(a) = sup{re λ : 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 C n generated by A M n (C). Then ω (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(e ta ) = e tσ(a), t, for such semigroups. However, for general C -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 determined by the uniform growth bound ω (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)) \ {} = e tσ(a) holds [28], and therefore, so does Liapunov s stability theorem. The category of eventually norm-continuous semigroups includes all semigroups which are eventually compact, eventually differentiable or holomorphic. Building on preliminary work of Martinez and Mazon [37], Blake [11] introduced the concept of asymptotically norm-continuous semigroups 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 ω (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 further describe the asymptotic behaviour of strongly continuous semigroups. Among these are the growth bound ω 1 (T), which determines the exponential growth of classical solutions of (ACP), higher order analogues ω n (T), n N, which estimate the exponential growth of solutions of (ACP) with initial values in D(A n ), and the more general fractional growth bounds ω α (T), α. The pseudo-spectral bound s (A) is the abscissa of uniform boundedness of the resolvent while the n-th spectral bound s n (A) is the abscissa of polynomial boundedness of degree n of the resolvent. There is a large literature on the relations between these growth bounds associated with the semigroup T and the spectral bounds of the generator A. We refer to [4] and [2, Chapter 5] for surveys. Inequalities showing that the growth bounds are not less than spectral bounds are relatively easy to obtain compared with opposite inequalities. The first relation showing a spectral bound dominating a growth bound for arbitrary C -semigroups was ω 2 (T) s (A), obtained in [47]. Amongst the most striking results in this direction are the Gearhart-Prüss theorem establishing the equality ω (T) = s (A) for strongly continuous semigroups defined on Hilbert spaces [22], [44] and the theorem of Weis and Wrobel showing ω 1 (T) s (A) for semigroups on general Banach spaces [5]. The analogue of the Gearhart-Prüss Theorem for higher order bounds, involving the equality of ω n (T) and s n (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 ω crit (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 (A), the bounds of the spectrum and the pseudo-spectrum of A away from the real axis. The spectral bounds s (A) and s (A) may be considered as 3

analogues of s(a) and s (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-Prüss Theorem for these bounds ( δ(t) = s (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 s (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 ω 1 (T) is the abscissa of convergence of the Laplace transform ˆT of T. Most of the general results also extend naturally to exponentially bounded functions, but some, like the Gearhart-Prüss 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 ω (f) of f. In fact, it is related to the analytic behaviour of ˆf away from the real axis in much the same way as ω (f) is related to ˆf in the right half-planes of C. In particular, hol ( ˆf) ζ(f), where hol ( ˆf) is the analogue of the spectral bound s (A) for functions. We also introduce the non-analytic abscissas of convergence and absolute convergence, ζ 1 (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 nonanalytic 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 [1]. It has been established in [11], [1] 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 (A). Thus, we have an analogue of the Gearhart-Prüss 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 ω α (T). The relation between the fractional non-analytic and uniform growth bounds of T yields a rather striking result: The uniform growth bounds ω α (T) equal the spectral bound s(a) for all α > provided the pseudo-spectrum of A is of a particular shape. A characterisation of the uniform growth bound in terms of the Fourier multiplier properties 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 [3], [34] and [49]). Using ideas from [32], [3] 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 φ(s)r(w + is, A) where w > s (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 C -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 L p (R, X), 1 p < and A is any closed operator. We obtain a necessary and sufficient condition for a function in L p (R, X) to be a mild solution of (1.1) for this case. If, in addition, A generates a C -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) < 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 C -semigroup generated by A in terms of the existence of unique mild solutions of (1.1), for every f L p (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 nonautonomous 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 transforms. 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 vectorvalued 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 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-Prüss 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 C -semigroups [48], we introduce essentially holomorphic C -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- Prüss 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 [1, Theorem 6.5.7] of the Weis-Wrobel result [5] to functions. In Section 4.3, the fractional non-analytic growth bounds ζ α (f), α > 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 C -semigroups. In Theorem 4.4.2 we show that for a strongly continuous semigroup T all the growth bounds w α (T), α > coincide with the spectral bound s(a) of the generator A provided s (A) =. Convexity of the function α ζ α (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 approximations 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 [, ) will be denoted by R + and the open half-plane {λ C : Re λ > } by C +. In general, for w R, we define the open half plane H w by H w = {λ C : Re λ > w}. Further, for b, we define Q w,b = {λ C : Re λ w, Im λ b}, Q o w,b = {λ C : Re λ > w, Im λ > b}. For b >, Q w,b is a pair of closed quadrants and Q o w,b is its interior. For θ >, we shall denote the sector of the complex plane of angle θ, containing the positive reals by Σ θ = {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 B Y defined by 2.1.3 Function spaces D(B Y ) := {y D(B) : B(y) Y}; B Y (y) := By, y D(B Y ). For 1 p <, let L p (R, X) be the space of all Bochner measurable functions f : R X such that ( f p := f(t) p dt R ) 1 p <. Let L (R, X) be the space of all Bochner measurable functions f : R X such that f := ess sup t R f(t) <. The conjugate index for p, 1 p <, shall be denoted by p so that 1 p + 1 p = 1. The space of locally integrable functions L 1 loc (R, X) is given by L 1 loc {f (R, X) = : R X such that f is Bochner measurable and } for every compact K R, f(t) dt <. K We denote by C(R, X) the vector space of all continuous functions f : R X. For k N, C k (R, X) will be the space of all k-times differentiable functions with continuous kth derivative and C (R, X) := k=1 Ck (R, X). BUC(R, X) will be the space of all bounded, uniformly continuous functions defined on R and taking values in X. C c (R, X) and Cc (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 C (R, X). Further, S(R, X) shall denote the Schwartz space of functions in C (R, X) which are rapidly decreasing. Then C c C c (R, X) is dense in L p (R, X) for 1 p <. so on. (R, X) S(R, X) L p (R, X) and If X = C, then we shall write S(R) in place of S(R, C), C (R) instead of C (R, X) and A function f : R + X may be considered to be defined on the whole of R by setting f = on (, ). We shall denote by L p (R +, X) the subspace of L p (R, X) consisting of functions which take the value on (, ). 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 ω (f) = inf { w R : sup e wt f(t) < }. 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 ω (f) <. A function g : Σ θ X is said to be exponentially bounded if there exist constants M, w such that g(z) Me w z (z Σ θ ). The restriction of g to (, ) may be exponentially bounded even if g is not exponentially bounded on Σ θ. However, by ω (g) we shall always mean the exponential growth bound of the restriction of g to R + with g() =. 2.2.2 The Laplace transform For f L 1 loc (R +, X) we define the abscissas of absolute convergence and convergence of the Laplace transform of f [2, Section 1.4] by : It is clear that abs( f ) = inf { ω R : abs(f) = inf { Re λ : lim τ e ωt f(t) dt < } ; τ abs(f) abs( f ) ω (f). e λt f(t) dt exists }. We say that f is Laplace transformable if abs(f) < and define the Laplace transform of f by ˆf where ˆf(λ) := whenever this limit exists. If τ e λt f(t) dt := lim e λt f(t) dt τ e λt f(t) dt exists as a Bochner integral, then by the dominated 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 ˆf(λ) converges if Re λ > abs(f) and diverges if Re λ < abs(f) [2, Proposition 1.4.1]. 2. abs(f) = ω (F F ), where F (t) = f(s) ds, F = lim t F (t) if the limit exists and F = otherwise [2, Theorem 1.4.3]. 1

3. If Re λ > max(abs(f), ), then [2, Corollary 1.6.5]. ˆF (λ) = ˆf(λ) λ. 4. λ ˆf(λ) defines a holomorphic function from H abs(f) into X and for n N {}, Re λ > abs(f), [2, Theorem 1.5.1]. ˆf (n) (λ) = e λt ( t) n f(t) dt. 5. If abs( f ) < then ˆf is bounded on H w whenever w > abs( f ); indeed, sup λ Hw ˆf(λ) e wt f(t) dt <. In general, ˆf may have a holomorphic extension to a bigger region than Habs(f). We shall denote the extension of ˆf by the same symbol. We now define the largest such region with which we shall work : D( ˆf) := {λ = α + iη : ˆf has a holomorphic extension to H abs(f) {β + is : α ɛ < β, s η < ɛ} for some ɛ > } Then D( ˆf) is a connected open set which is a union of horizontal line-segments extending infinitely to the right, and ˆf has a unique holomorphic extension (also denoted by ˆf) to D( ˆf). Moreover, D( ˆf) is the largest such set with these properties. Next we define some abscissas of holomorphy and boundedness: { } hol( ˆf) = inf ω R : H ω D( ˆf) ; (2.1) { } hol ( ˆf) = inf ω R : H ω D( ˆf) and sup ˆf(λ) < ; (2.2) { Re λ>ω } hol ( ˆf) = inf ω R : Q ω,b D( ˆf) for some b ; (2.3) hol ( ˆf) = inf { ω R : Q ω,b D( ˆf) and } sup ˆf(λ) < for some b λ Q ω,b ; (2.4) { hol n ( ˆf) = inf ω R : H ω D( ˆf) and { hol n ( ˆf) = inf ω R : Q ω,b D( ˆf) and sup λ Q ω,b sup Re λ>ω ˆf(λ) } (1 + λ ) n < ; (2.5) ˆf(λ) } < for some b. (2.6) (1 + λ ) n 11

for n N. Thus, hol( ˆf) and hol ( ˆf) are the abscissas of holomorphy and boundedness of ˆf [2, Section 1.5] and hol ( ˆf) and hol ( ˆf) are analogues which ignore horizontal strips in C. For n N, hol n ( ˆf) gives the minimal abscissa for the half-plane where ˆf grows along vertical lines not faster than the n-th power and hol n ( ˆf) 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 ( ˆf) hol( ˆf) abs(f); (2.7) hol( ˆf) hol ( ˆf) abs( f ); (2.8) hol ( ˆf) hol ( ˆf) hol ( ˆf); (2.9) hol ( ˆf) hol n ( ˆf) hol ( ˆf). (2.1) For an exponentially bounded function f we have abs(f) hol ( ˆf) (2.11) [5]. However, (2.11) is false for some Laplace transformable functions [12]. If f is exponentially bounded, then it is clear that for w > ω (f), there is a constant M such that ˆf(λ) M (Re λ w) (Re λ > w). (2.12) For a Laplace transformable function the following holds: Lemma 2.2.1. If w > abs(f), and < θ < π 2, then ˆf is bounded on H w Σ θ. Further, hol ( ˆf) ( = max hol ( ˆf), hol( ˆf) ) ; hol n ( ˆf) ( = max hol n ( ˆf), hol( ˆf) ), n N. Proof. Let < θ < π 2 and w > max(, abs(f)). Since ˆf(λ) = λ ˆF (λ) and abs(f) = ω (F F ), (2.12) applied to F F yields a constant M such that ˆf(λ) Choosing ɛ > such that < ɛ < cos θ, we have M λ Re λ w + F. ˆf(λ) K, for all λ H w ɛ Σ θ, where K = (cos θ ɛ) 1 + F is a constant. Thus, the first statement follows on observing that ˆf is bounded on compact subsets of C. 12

From the inequalities (2.8) and (2.9) it follows that hol ( ˆf) max(hol ( ˆf), hol( ˆf)). Suppose a R is such that max(hol ( ˆf), hol( ˆf)) < a. Then ˆf has a holomorphic extension to H a which is bounded on Q a,b for some b >. Let S = {λ C : a Re λ ω, Im λ b}, where ω > max(, abs(f)). Then ˆf is holomorphic on the compact set S and therefore bounded. That ˆf is bounded on {Re λ > ω, Im λ b} follows from the first part. Therefore, sup λ S Q a,b ˆf(λ) <. Thus, hol ( ˆf) a. Hence, hol ( ˆf) max ( hol ( ˆf), hol( ˆf)). The corresponding result for hol n ( ˆf) follows similarly. We note here that for a Laplace transformable function f, sup λ Qa,b ˆf(λ) < actually implies, on using Lemma 2.2.1, that sup λ H w Q a,b ˆf(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 L p (R) and f L q (R, X), then g f L r (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 (g n ) n N in L 1 (R) of the following form: g 1 L 1 (R) satisfies g 1 (t) dt = 1, and g n L 1 (R) is given by g n (t) = ng 1 (nt), for all t R and R n N. It is often convenient to choose such a sequence (g n ) in C c n N. For such a mollifier (g n ), for f L p (R, X), 1 p <. lim g n f f p =, n (R) with g n for all 13

For f L 1 (R, X), the Fourier transform Ff, is defined by (Ff)(s) := and the conjugate Fourier transform, Ff is given by ( Ff)(s) := e ist f(t) dt, e ist f(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 L 1 (R, X) and g L 1 (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 C (R, X). 4. (Inversion Theorem) If Ff L 1 (R, X), then f = 1 2π F(Ff) a.e. 5. If X is a Hilbert space then we have Plancherel s theorem that 1 2π F : L 2 (R, X) L 2 (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 ( ˆf). Lemma 2.2.2. Let f : R + X be measurable and exponentially bounded. Then for all α > hol ( ˆf). lim ˆf(α + is) =, s ± Proof. Let α > β > hol ( ˆf). Then there exists b > such that Q β,b D( ˆf). Let (s n ) be any sequence such that s n b for all n and s n as n. Let g n (z) = ˆf(z + is n ), z Q β,. For Re z > ω (f), let h(t) = e zt f(t), t R +. Then h L 1 (R +, X) and it follows from the Riemann-Lebesgue Lemma that lim Fh(s) = lim ˆf(z + is) =. s s Thus, if Re z > ω (f), then g n (z) as n. Now (g n ) is uniformly bounded on {Re z > β, Im z > }. By Vitali s Theorem [2, Theorem A.5], lim n g n (z) = for all z Q o β,. In particular, = lim g n(α) = lim ˆf(α + is n ). n n Since (s n ) is arbitrary, we conclude that lim s ˆf(α + is) =. Similarly, we can show that as s, ˆf(α + is). 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 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 e λs T(s) ds denote the bounded operator Then abs(t) := inf { Re λ : x = sup{abs(t( )x) : x X} e λs T(s)x ds. = inf{ω (S S ) : S L(X)} } e λs T(s) ds converges strongly as t where S(t)x = T(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 Re λ > abs(t), the limit lim t e λs T(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 ˆT(λ) := e λs T(s) ds := lim t e λs T(s) ds (Re λ > abs(t)). 15

Then ˆT : H abs(t) L(X) is holomorphic and all the results mentioned in sub-section 2.2.2 for ˆf hold for ˆT also, with hol( ˆT), hol ( ˆT), hol ( ˆT), hol ( ˆT), hol n ( ˆT) and hol n ( ˆT) 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 C -semigroups By a C -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() = I; 3. T(t + s) = T(t)T(s) for t, s R +. For background information on C -semigroups we refer to the books [2], [42], [17] and [2]. The infinitesimal generator A of T is given by { } T(t)x x D(A) = x X : lim exists t t T(t)x x Ax = lim (x D(A)). t t A is a closed, densely defined operator and n=1 D(An ) = X. The resolvent and spectrum 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 C -semigroup is exponentially bounded [2, Proposition 5.5]. For Re λ > ω (T), R(λ, A)x = ˆT(λ)x (x X). In fact, C -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 [1], [11]: δ(t) := inf{ω : there exists a norm continuous function T 1 : R + L(X) such that ω (T T 1 ) < ω} = inf{ω : there exists an infinitely differentiable function T 1 : R + L(X) such that ω (T T 1 ) < ω}. (2.13) This is equal to the growth bound of local variation of T, which is defined to be ω (f T ) where f T (t) = lim sup T(t + h) T(t). h [1, 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 > then δ(t) =. A C -semigroup T : R + L(X) is said to be eventually norm continuous [2, 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 then T is called immediately norm continuous. If A is the generator of an immediately norm continuous semigroup T then lim R(a + is, A) = (2.14) s ± for all a > ω (T) ([2, Theorem II.4.18 ]). This condition is sufficient for immediate norm continuity if X is a Hilbert space [2, Theorem II.4.2],[52] or if T is a positive semigroup defined on L p (R), 1 < p < [23]. It is not known whether (2.14) implies immediate norm continuity for arbitrary C -semigroups or not. The semigroup T is said to be eventually compact (respectively, immediately compact) if there is an α > such that T(α) is compact (respectively, T(t) is compact for all t > ). T is eventually compact if and only T is eventually norm continuous and its generator has compact resolvent [2, Theorem II.4.29 ]. T is called eventually differentiable [2, Definition II.4.13]), [42, Section 2.4] if there exists an α such that the map t T(t)x is differentiable on (α, ) for every x X. A C -semigroup T is called analytic of angle θ (, π 2 ) ([2, Definition II.4.5]) if T has a holomorphic extension to Σ θ (also denoted by T) satisfying T(z 1 + z 2 ) = T(z 1 )T(z 2 ) for all z 1, z 2 Σ θ ; lim Σβ z T(z)x = x for all x X and < β < θ. 17

All the classes of C -semigroups mentioned above satisfy δ(t) =. In [11, Definition 3.4] a class of C -semigroups has been introduced which includes all C -semigroups with finite growth bound falling in any of the above classes: T is said to be asymptotically norm continuous if δ(t) < ω (T). Such a semigroup has been called norm continuous at infinity in [37]. With Γ t = {λ C : e δ(t)t < λ } the following version of the spectral mapping Theorem holds for asymptotically norm continuous semigroups [1, Theorem 4.4.1], [11, Theorem 3.6]. σ (T(t)) Γ t = e tσ(a) Γ t, (t > ). (2.15) In particular, ω (T) = sup{re λ : λ σ(a)}. Since δ(t) = for eventually norm continuous semigroups, (2.15) re-asserts the well known fact [2, Theorem IV.3.1 ] that the spectral mapping theorem holds for such semigroups. σ(t(t)) \ {} = e tσ(a) (t > ) (2.16) For general C -semigroups, (2.16) fails to hold. However, in [38], the critical spectrum σ crit (T(t)) for a C -semigroup T has been introduced, which yields a spectral mapping theorem of the form σ(t(t)) \ {} = e tσ(a) σ crit (T(t)) \ {} (t > ), for all C - semigroups. We recall the definition of the critical spectrum of a C -semigroup T [38, Definition 2.3]. Let l (X) be the Banach space of all bounded sequences in X, endowed with the sup-norm (x n ) n N := sup n N x n and let T be the extension of T to l (X), given by T(t) ((x n ) n N ) := (T(t)x n ) n N (t ). Then the space of strong continuity l T (X), of T given by l T (X) := { (x n ) n N : lim sup t n N T(t)x n x n = } is a closed and T-invariant subspace of l (X). On the quotient space ˇX = l (X) l define T (X) the semigroup of bounded operators Ť by Ť(t)ˇx := (T(t)x n ) n N + l T (X), for ˇx := (x n ) n N + l T (X). Then the critical spectrum of the C -semigroup T is defined by σ crit (T(t)) := σ(ť(t)) (t > ), 18

and the critical growth bound ω crit (T) of T is defined as ω crit (T) = ω (Ť). It has been shown in [38, Proposition 4.6] that the growth bound of local variation and the critical growth bound coincide for a C -semigroup, that is δ(t) = ω crit (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 C -semigroup. The details may be found in [39, Chapter 1, Chapter 2]. Let A be the generator of the C -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 X = {x X : lim t T (t)x x = }. acts in a strongly continuous way; i.e. 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 defines a C -semigroup on X whose generator A is the part of A in X. Starting with the C -semigroup T, the duality construction can be repeated. 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 := sup x, x, x B X We where B X is the closed unit ball of X, is an equivalent norm. In fact, x x M x, with M = lim sup t 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. 2

Chapter 3 A non-analytic growth bound for Laplace transforms and semigroups of operators Analytic C -semigroups play an extremely important role in the theory of evolution equations. In this chapter, we study a growth bound which measures the non-analyticity of C - 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 [1, Theorem 2.3.3] it is shown that integrability along a vertical line of some derivative 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 >. This idea has been used to define a growth bound for T which measures the non-integrability of the Laplace transform of T. We recall the relevant definitions and results from [1, 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 L 1 -Laplace transform, (see [1]) if there exist r >, N N and C > such that ˆf has a holomorphic extension to Q,r and sup ˆf (N) (ω + is) ds C. ω s r For an exponentially bounded, Bochner measurable function f : R + X, the growth 21

bound for non-integrability of f is defined by: ζ(f) = inf { w R : there exists f 1 : R + X such that e w f 1 ( ) has L 1 -Laplace transform and ω (f f 1 ) w }. (3.1) The next theorem gives a very useful property of ζ(f). Theorem 3..1. ([1, Proposition 2.4.6]) Let f : R + X be Bochner measurable. Suppose that ζ(f) <, so that there exist f 1, f 2 : R + X and positive N, r and ɛ such that f = f 1 + f 2, ω (f 2 ) < ɛ and sup w ɛ s >r ˆf1 (N) (w + is) ds <. Let w > ω (f) and Γ be the path consisting of line segments joining ɛ ir, w ir, w + ir and ɛ + ir in that order, and define Then ω (g 2 ) ɛ. g 1 (t) := 1 2πi Γ g 2 (t) := f(t) g 1 (t). e λt ˆf1 (λ) dλ, 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 > 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) := inf { w R : there exist θ > and an exponentially bounded, holomorphic function g : Σ θ X such that ω (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 Σ θ. 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 Q α,b D( ˆf) we define for t, f α,b (t) := 1 e tλ ˆf(λ)dλ, 2πi Γ α,b where Γ α,b is any path in D( ˆf) from α ib to α + ib. We immediately obtain, from the above definition, an estimate for the growth bound of the functions f α,b. Lemma 3.1.3. Let f : R + X be Laplace transformable and α R, b be such that Q α,b D( ˆf). Then, ω (f α,b ) max ( α, hol( ˆf) ˆf α,b (µ) = 1 2πi Γ α,b ), and ˆf(λ) µ λ dλ, where Γ α,b is any path in D( ˆf) from α ib to α + ib and Re µ lies to the right of Γ α,b. Further, if α 1, b 1 R, b 1 satisfy Q α1,b 1 D( ˆf) then abs ( f α,b f α1,b 1 ) ω ( fα,b f α1,b 1 ) max(α, α1 ). ( Proof. Let Γ = [α ib, γ ib] [γ ib, γ + ib] [γ + ib, α + ib], where γ > max α, hol( ˆf) ). Then, f α,b (t) = 1 e λt ˆf(λ) dλ, 2πi Γ and straightforward calculations show that f α,b (t) C γ e γt ( for some constant C γ. Consequently, ω (f α,b ) γ, for all γ > max α, hol(( ˆf) ). Therefore, ( it follows that ω (f α,b ) max α, hol(( ˆf) ). The claim concerning ˆf α,b follows directly from the definition of the Laplace transform of a function, on interchanging the order of integration. We note that for α 1, b 1 as in the hypothesis, ˆf α,b (t) ˆf α1,b 1 (t) = 1 e λt ˆf(λ)dλ, 2πi Γ 2 where Γ 2 consists of two paths in {λ D( ˆf) : Re λ max(α, α 1 )} joining α ± ib to α 1 ± ib 1. Therefore, f α,b (t) f α1,b 1 (t) C exp(t max(α, α 1 )) for some constant C, so that ω (f α,b f α1,b 1 ) max(α, α 1 ). 23

Observe that in the terminology introduced in Definition 3.1.2, Theorem 3..1 shows that if f, f 1 : R + X are exponentially bounded, Bochner measurable functions with ω (f f 1 ) < ɛ for some ɛ > and the function t e ɛt f 1 (t) has L 1 -Laplace transform, then ω (f (f 1 ) ɛ,r ) ɛ, 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 > and an exponentially bounded, holomorphic function g : Σ θ C such that ω (f g) < ω. 2. There is an exponentially bounded, measurable function g : R + X such that, for each α < ω, there exists b such that (a) Q α,b D(ĝ); (b) sup λ Qα,b ĝ(λ) < ; (c) sup γ α s b ĝ(n) (γ + is) ds < (N = 1, 2...); (d) ω (f g) < ω. 3. There exist an exponentially bounded, measurable function g : R + X and α < ω, b, N N such that (2a), (2c) and (2d) hold. 4. There exist α < ω and b such that (a) Q α,b D( ˆf); (b) sup λ Qα,b ˆf(λ) < ; (c) ω (f f α,b ) < ω. 5. There exist α < ω and b such that (4a) and (4c) hold. 6. There is an exponentially bounded, entire function g : C X such that ω (f g) < ω ( and ω (g) max ω, hol( ˆf) ). Proof. (1) = (2): Suppose that g : Σ θ X is holomorphic and there exists w R such that g(z) Me w z (z Σ θ ). We may assume without loss of generality, that ω < w. By considering a smaller sector if required, we may also assume that < θ π 2. Then w + Σ θ+(π/2) D(ĝ) and ĝ(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 ĝ (N) (γ + is) = N! ĝ(λ) 2πi (λ (γ + is)) λ (γ+is) =ɛs N! C ɛ N s N s (1 ɛ) N+1 dλ whenever γ α, s b > (ω α) cot(θ/2), N N, < ɛ < 1 and ɛ is sufficiently small so that the disc of radius ɛb and centre α + ib is contained in ω + Σ θ+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 and N are such that (2a), (2c) and (2d) hold. Take β such that max(α, ω(f g)) < β < ω. Then Q β,b D( f g) D(ĝ), and hence Q β,b D( ˆf). If (2b) holds then ˆf is bounded on Q β,b. By Lemma 3.1.3, ω ( fβ,b g β,b ) = ω ( (f g)β,b ) β. The assumptions on g imply that t e βt g(t) has L 1 -Laplace transform and ω (f g) < β. Therefore, from Theorem 3..1, it follows that ω (f g β,b ) β. Since it follows that ω (f f β,b ) β < ω. (5) = (6): For z C, let ω (f f β,b ) max (ω (f g β,b ), ω (g β,b f β,b )), g(z) = 1 e λz ˆf(λ)dλ. 2πi Γ α,b Then g is entire and exponentially bounded, with g(t) = f α,b (t) (t ). 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 satisfying (4a). This follows from Lemma 3.1.3. 2. If (4a) holds then α hol ( ˆf); if (4b) also holds, then α hol ( ˆf). Conversely, if α > hol ( ˆf) then there exists b such that (4a) holds; if α > hol ( ˆf) and (4a) holds, then (4b) holds. 3. Proposition 3.1.4 remains valid if ω (f g) is replaced by abs( f g ) in conditions (1), (2), (3) and (6), and ω (f f α,b ) is replaced by abs( f f α,b ) in conditions (4) and (5). 25