This is a preliminary version; the final version appeared inConferenze del Seminario di Matematica dell’Universita di Bari 259 (1995), 27-60.

LAPLACE TRANSFORM METHODS FOREVOLUTION EQUATIONS

BORIS BAUMER and FRANK NEUBRANDER *Louisiana State University, Baton Rouge, USA

The Laplace transform theory of Banach space valued functions and the field of evolutionequations presently offer many research problems which are of general interest in mathe-matical analysis. Although there is a longstanding relationship between the two fields, itwas not until recently that the basic transform principles for Banach space valued functionscould be formulated in a satisfactory way. In Section 1 we will give an introduction tosome aspects of the mathematical theory of the Banach space valued Laplace transform.

In contrast to semigroup methods, where the spectral and regularity assumptions onthe characteristic operator are rather restrictive, the Laplace transform method applies towellposed and illposed linear evolution problems alike. No a priori spectral assumptionshave to be made, the characteristic operators might not be closed or closable, might notbe densely defined and could be multivalued. The only assumptions needed are linearityand relative closedness. This class of operators, which includes any sum, product or limitof closed operators, will be discussed in Section 2.

Besides new theoretical concepts and a unified approach to many aspects of semigrouptheory, Laplace transform theory provides analytic tools for many of the basic problemsin the theory of evolution equations. We will demonstrate this in Section 3. There we willinvestigate existence, uniqueness and regularity of solutions of linear evolution equations.

The Laplace transform part of this paper builds on results of W. Arendt (1987, 1991),W. Arendt, M. Hieber and F. Neubrander (1994), B. Hennig and F. Neubrander (1993), F.Neubrander (1994, 1994b) and M. Sova (1979, 1980). In applying the Laplace transformmethods to evolution equations we follow ideas developed in papers by Yu. I. Lyubich(1966), M. Hieber, A. Holderrieth, and F. Neubrander (1991), B. Baumer and F. Neubran-der (1994), and F. Neubrander (1994).

* supported by Louisiana Education Quality Support Fund

1

1. THE LAPLACE TRANSFORM

The Laplace transform has a long history, dating back to L. Euler’s paper ‘De Con-

structione Aequationum’ from 1737. Since then it has been widely used in mathematics,in particular in ordinary differential, difference and functional equations. An informativedescription of the contributions of mathematicians like Euler, Lagrange, Laplace, Fourier,Poisson, Cauchy, Abel, Liouville, Boole, Riemann, Pincherle, Amaldi, Tricomi, Picard,Mellin, Borel, Heaviside, Bateman, Titchmarsh, Bernstein, Doetsch, Widder and manyothers can be found in two historical surveys by M. Deakin (1981, 1982).

The article ‘Studio della trasformazione di Laplace e della sua inversa dal punto

di vista dei funzionali analitici’ by the Italian mathematician Sylvia Martis in Biddau(1933) provides an interesting overall account of the theory prior to D.V. Widder’s and G.Doetsch’s contributions. D.V. Widder’s books ‘The Laplace Transform’ (1941) and ‘An

Introduction to Transform Theory’ (1971) as well as G. Doetsch’s ‘Theorie und Anwen-

dung der Laplace-Transformation’ (1937) and his monumental, three volumed ‘Handbuch

der Laplace-Transformation’ (1950, 1955, 1956) remain useful and modern works. Theyare still among the best introductions to the subject.

A first look at Laplace transform theory for functions with values in a Banach spaceX is contained in E. Hille’s monograph ‘Functional Analysis and Semi-Groups’ from 1948.In trying to extend the results of the classical numerical theory of the Laplace transform

(L) r(λ) =∫ ∞

0

e−λtf(t) dt (λ > 0)

to Banach space valued functions, E. Hille remarks on several occasions that this can bedone if X is reflexive, but not in general. In fact, it was shown by S. Zaidman (1960)(see also W. Arendt (1987) or Theorem 1.6 below) that some important results of classicalLaplace transform theory extend to a Banach space X if and only if X has the Radon-Nikodym property (such as reflexive spaces). This “negative” result led to the developmentof special Laplace transform theories on arbitrary Banach spaces for functions with ad-ditional algebraic properties; the most prominent one being the theory of C0-semigroups.There, the link between the generator A and the semigroup T is given via the Laplacetransform

(λI −A)−1x =∫ ∞

0

e−λtT (t)x dt (x ∈ X).

The crucial algebraic property which made it possible to extend classical transform resultsto this abstract setting is the semigroup law T (t+ s) = T (t)T (s), (t, s ≥ 0). In the revisedversion of E. Hille’s monograph from 1957, E. Hille and R.S. Phillips comment as followson the algebraic methods used to solve evolution equations:

2

“Thus in keeping in spirit of the times the algebraic tools now play a major role andare introduced early in the book; they lead to a more satisfactory operational calculus andspectral theory... On the other hand, the Laplace-Stieltjes transform methods, used by Hillefor such purposes, have not been replaced but rather supplemented by the new tools.”

In more recent monographs on semigroup theory the role of the Laplace transform isreduced to a footnote at best and almost every proof is based on the semigroup propertyT (t+ s) = T (t)T (s). The major disadvantage of the “algebraic approach” to linear evolu-tion equations becomes obvious if one compares the mathematical theories associated withthem (for example, semigroup theories, cosine families, the theory of integro-differentialequations, etc.). It is striking how similar the results and techniques are. Still, without aLaplace transform theory for arbitrary Banach space valued functions at hand, every typeof linear evolution equation required its own theory because the algebraic properties of theoperator families involved change. In recent years, in search of a general analytic principlebehind all these theories, the Laplace transform has been reconsidered for that reason.

The first major result in this direction is due to M. Sova (1979). His analytic Laplacerepresentation theorem (see Theorem 1.16 below) is behind every generation result foranalytic solution families of linear evolution equations. The real breakthrough came in1987 with W. Arendt’s paper on ‘Vector valued Laplace transforms and Cauchy problems’.He showed that Widder’s Theorem, one of the main theoretical results of the classicaltheory, extends to arbitrary Banach spaces if the Laplace transform is replaced by theLaplace-Stieltjes transform

(LS) r(λ) =∫ ∞

0

e−λt dF (t) = limτ→∞

∫ τ

0

e−λt dF (t) (λ > 0)

where F ∈ Lip0([0,∞);X); i.e., F (0) = 0 and ‖F‖Lip := supt,s≥0‖F (t)−F (s)‖

|t−s| < ∞.

In order to see how the Laplace-Stieltjes transform relates to the Laplace trans-form, let f ∈ L∞([0,∞);X). Then F (t) :=

∫ t

0f(s) ds is in Lip0([0,∞);X). In this case

the Laplace-Stieltjes transform reduces to the Laplace transform; i.e.,∫ ∞0

e−λt dF (t) =∫ ∞0

e−λtf(t) dt. We recall that a space X is said to have the Radon-Nikodym property ifthe fundamental theorem of calculus holds; i.e., if any absolutely continuous function isthe antiderivative of a Bochner integrable function. Hence, if a space X has the Radon-Nikodym property, then the Laplace-Stieltjes transform reduces to the Laplace transform.Otherwise it is a proper generalization. As it turned out, it is the generalization needed indealing with Banach space valued functions.

3

Clearly, the Laplace transform can also be considered for f ∈ Lp((0,∞);X) and theLaplace-Stieltjes transform for functions F of bounded p-variation or semivariation (see,for example, W. Arendt (1993), J. Pruss (1993), P. Vieten and L. Weis (1993), and L.Weis (1993)). However, we will restrict the discussion of L and LS to the case wheref ∈ L∞([0,∞);X) and F ∈ Lip0([0,∞);X). These special cases are relatively easy todeal with and have immediate and important applications to evolution equations.

The key to Laplace transform theory is the Riesz-Stieltjes representation of boundedlinear operators from L1(0,∞) into X . The proof of this well known observation is takenfrom F. Neubrander (1994). We will denote by χ[0,t] the characteristic function of theinterval [0, t].

Theorem 1.1 (Riesz-Stieltjes Representation). There exists an isometric iso-

morphism RS : Lip0([0,∞);X) → L(L1(0,∞), X) given by RS(F ) = T where Tg :=∫ ∞0

g(t) dF (t) for all continuous functions g ∈ L1(0,∞) and Tχ[0,t] = F (t), for all t ≥ 0.

PROOF. Let F ∈ Lip0([0,∞);X) and let D be the linear span of the functions withcompact support which are either continuous or of bounded variation. For such functionswe can define an operator TF on the subspace D by TF g :=

∫ ∞0

g(s) dF (s) and it is easyto see that for such g we have that ‖TF g‖ ≤ ‖F‖Lip‖g‖1. It follows from F (0) = 0 thatTFχ[0,t] =

∫ t

0dF (s) = F (t) for all t ≥ 0. Since D is dense in L1(0,∞), there exists a unique

extension of TF in L(L1(0,∞), X), denoted by the same symbol. Clearly, ‖TF ‖ ≤ ‖F‖Lipand, if TF = 0, then TFχ[0,t] = F (t) = 0 for all t ≥ 0. This shows that the linear mappingRS : F → TF is one-to-one. Moreover, if g is a continuous function in L1(0,∞), theng = limt→∞ gχ[0,t] in L1(0,∞). Thus,

TF g = limt→∞ TF (gχ[0,t]) = lim

t→∞

∫ t

0

g(s) dF (s) =∫ ∞

0

g(s) dF (s).

To prove that RS is onto, let T ∈ L(L1(0,∞), X). Define F (t) := Tχ[0,t]. Then ‖F (t) −F (s)‖ = ‖Tχ[s,t]‖ ≤ ‖T‖‖χ[s,t]‖1 = ‖T‖|t−s|. So F ∈ Lip0([0,∞);X) , ‖F‖Lip ≤ ‖T‖, andTχ[0,t] = F (t) =

∫ t0dF (s) = TFχ[0,t]. Since the characteristic functions χ[0,t], t ≥ 0 are

total in L1(0,∞), we conclude that T = TF and ‖TF ‖ = ‖F‖Lip for all F ∈ Lip0([0,∞);X).♦

4

The Riesz-Stieltjes Representation Theorem is crucial for the following reason. Con-sidering the Laplace-Stieltjes transform LS we want to know how properties of F and itstransform r affect each other. Observe that

r(λ) = TF e−λ and F (t) = TFχ[0,t]

for all λ, t > 0, where e−λ denotes the exponential function t → e−λt. Thus, if one knowsthe function F , then the operator TF is determined on the set of characteristic functions,which is total in L1(0,∞). Hence TF , and in particular TF e−λ = r(λ), is completelydetermined. Conversely, any information on r(λ) for λ > 0 translates into information onTF on the set of exponential functions, which is also total in L1(0,∞) (see below). Thus,r determines the properties of TF and, in particular, of TFχ[0,t] = F (t), (t ≥ 0).

Lemma 1.2. Let λn (n ∈ IN) be a sequence of distinct complex numbers with Reλn ≥γ > 0 for some γ > 0. If

(1.1)N∑n=1

1 − |λn − 1||λn + 1| → ∞

as N → ∞, then the exponential functions e−λnare total in L1(0,∞).

PROOF. a) Let γ < n0 ∈ IN. Clearly, the monomials t → 1n0tn/n0 (n ∈ IN0) are

total in C[0, 1] and thus in L1(0, 1). Since Φ : L1(0, 1) → L1(0,∞) defined by Φ(g)(t) :=n0e

−n0tg(e−n0t) is an isometric isomorphism which maps the monomials onto the set E ofexponential functions e−n (n ≥ n0), it follows that E is total in L1(0,∞).

b) Let H be the closure of the linear span of the functions e−λn(n ∈ IN) in L1(0, 1).

If H contains the exponential functions e−n (n ≥ n0), the claim follows from a). Assumethat there exists m ≥ n0 such that e−m ∈ H. By the Hahn-Banach theorem there existsφ ∈ L∞(0,∞) such that φ(H) ≡ 0 and φ(e−m) = 0. Let 0 < β < γ. The functionλ → Ψ(λ) := φ(e−λ) =

∫ ∞0

e−λtφ(t) dt is analytic and bounded for Reλ ≥ β and thefunction ρ : z → 1+z

1−z + β is a conformal map between the unit disk and the halfplaneReλ > β. Define h(z) := Ψ(ρ(z)) and µn := λn−β−1

λn−β+1. Then h is analytic and bounded

on the unit disk and h(µn) = Ψ(λn) = φ(e−λn) = 0 for all n ∈ IN. Let an := 1 − |µn| and

bn := 1− |λn−1||λn+1| . Then

anbn

=|λn + 1|

|λn − β + 1| ·|λn − β + 1|2 − |λn − β − 1|2|λn − β + 1| + |λn − β − 1| · |λn + 1|+ |λn − 1|

|λn + 1|2 − |λn − 1|2

=|λn + 1|

|λn − β + 1| ·Reλn − β

Reλn· |λn + 1|+ |λn − 1||λn − β + 1|+ |λn − β − 1| .

5

Since |λn + 1| ≥ |λn − β ± 1| and |λn − 1| ≥ |λn + 1| − 2 it follows that

anbn

≥ Reλn − β

Reλn· |λn + 1|+ |λn − 1||λn − β + 1|+ |λn − β − 1|

≥ Reλn − β

Reλn· 2|λn + 1| − 2

2|λn + 1|≥ (1 − β

γ)(1− 1

1 + γ) > 0.

Since∑N

n=1 bn → ∞ as N → ∞, we obtain that∑N

n=1 an =∑N

n=1 1 − |µn| → ∞ asN → ∞. Thus h(µ) = 0 for |µ| < 1 (see, for example, W. Rudin (1987), Theorem 15.23).But this implies that Ψ(λ) = 0 for Reλ > β contradicting Ψ(m) = 0. Thus H = L1(0,∞).

♦

Typical examples of sequences satisfying (1.1) are equidistant sequences λn = a +nb (a, b > 0) or convergent sequences λn → λ, (Reλ > 0). Examples of sequences not

satisfying (1.1) are given by λn = nα, (α > 1) and λn = 1 + in.

Combining Lemma 1.2 with the Riesz-Stieltjes Representation 1.1 we obtain immedi-ately the following uniqueness theorem for the Laplace-Stieltjes transform (see Yu-ChengShen (1947) or G. Doetsch (1950), Satz 2.9.6).

Corollary 1.3 (Uniqueness). Let F ∈ Lip0([0,∞);X). If∫ ∞0

e−λnt dF (t) = 0 for a

sequence of complex numbers λn satisfying (1.1), then F ≡ 0.

Next we determine the range of the Laplace-Stieltjes transform. Let F ∈ Lip0([0,∞);X).Then r(λ) =

∫ ∞0

e−λt dF (t) is analytic and

(1.2) r(n)(λ) =∫ ∞

0

e−λt(−t)ndF (t)

for n ∈ IN0 and Reλ > 0. For a proof, see for example, G. Doetsch (1950), Satz 3.2.1 orW. Arendt, M. Hieber and F. Neubrander (1995). To obtain an estimate for the Taylorcoefficients of r let x∗ ∈ X∗ and define fx∗(t) := 〈F (t), x∗〉. Then fx∗ ∈ Lip0([0,∞)) and‖fx∗‖Lip ≤ ‖F‖Lip‖x∗‖. It follows that fx∗ is differentiable a.e. and fx∗(t) =

∫ t0f ′x∗(s) ds

for all t ≥ 0. Clearly, ‖f ′x∗(t)‖ ≤ ‖F‖Lip‖x∗‖ a.e. and, by (1.2),

〈λk+1 1k!r(k)(λ), x∗〉 = λk+1 1

k!

∫ ∞

0

e−λt(−t)kf ′x∗(t) dt.

6

Since λn+1∫ ∞0

e−λt tn

n! dt = 1 (λ > 0, n ∈ IN0) we obtain |〈λk+1 1k!r

(k)(λ), x∗〉| ≤ ‖F‖Lip‖x∗‖for all x∗ ∈ X∗ and λ > 0. Thus

(1.3) ‖r‖W := supk∈IN0

supλ>0

‖λk+1 1k!r(k)(λ)‖ ≤ ‖F‖Lip.

Let C∞W ((0,∞);X) := r ∈ C∞((0,∞);X) : ‖r‖W < ∞. Then the Widder space

C∞W ((0,∞);X) is a Banach space and we obtain the following crucial result due to D.V.

Widder (1936) for the numerical case and due to W. Arendt (1987) for the vector-valuedcase. The following proof is a modernized version of Widder’s classical proof. It is takenfrom F. Neubrander (1994).

Widder’s Theorem 1.4. The Laplace-Stieltjes transform LS : Lip0([0,∞);X) →C∞W ((0,∞);X) is an isometric isomorphism.

PROOF. It follows from (1.3) that LS maps Lip0([0,∞);X) into C∞W ((0,∞);X) and

that ‖LS(F )‖W = ‖r‖W ≤ ‖F‖Lip. It follows from Corollary 1.3 that LS is one-to-one. Itremains to be shown that LS is onto. Let r ∈ C∞

W ((0,∞);X). Define Tk ∈ L(L1(0,∞), X)by

Tkf :=∫ ∞

0

f(t)(−1)k1k!

(k

t

)k+1

r(k)

(k

t

)dt.

Then ‖Tk‖ ≤ ‖r‖W for all k ∈ IN0 and

Tke−λ = (−1)k1k!

∫ ∞

0

e−λt(k

t)k+1 r(k)(

k

t) dt.

Using change of variables and integration by parts one obtains (with some work; for details,see D.V. Widder (1941) or B. Hennig and F. Neubrander (1993)),

Tke−λ = u

∫ ∞

0

1k!

(k

u

)k+1

e−kt

u tkh(t) dt

where h(t) := 1t r(

1t ) and 1/u := λ. The functions

ρk(t) :=1k!

(k

u

)k+1

e−kt

u tk (t ≥ 0)

7

are “approximate identities”; i.e., they are positive, their L1-norm is one and for all ε > 0and all open intervals I ⊂ [0,∞) containing u we have

∫t/∈I ρk(t) dt < ε for all sufficiently

large k. Since h is continuous in u > 0 and ‖h‖∞ ≤ ‖r‖W , it follows from

(1.4)‖∫ ∞

0

ρk(t)h(t) dt− h(u)‖ = ‖∫ ∞

0

ρk(t) (h(t) − h(u)) dt‖

≤ 2‖h‖∞∫t/∈I

ρk(t) dt + supt∈I

‖h(t) − h(u)‖

that∫ ∞0

ρk(t)h(t) dt → h(u) as k → ∞. Hence,

Tke−λ −→ uh(u) = r(1u) = r(λ)

as k → ∞. By Lemma 1.2, the set of exponential functions e−λ : λ > 0 is total inL1(0,∞). Thus, by the Theorem of Banach-Steinhaus, there exists a bounded operatorT ∈ L(L1(0,∞), X) with ‖T‖ ≤ ‖r‖W such that Tkf → Tf for all f ∈ L1(0,∞). Inparticular,

r(λ) = limk→∞

Tke−λ = Te−λ.

It follows from the Riesz-Stieltjes Representation Theorem 1.1 that there exists F ∈Lip0([0,∞);X) with ‖F‖Lip = ‖T‖ ≤ ‖r‖W such that Tg =

∫ ∞0g(t) dF (t) for all con-

tinuous g ∈ L1(0,∞). Hence, for all λ > 0,

r(λ) = Te−λ =∫ ∞

0

e−λt dF (t). ♦

Closely related to Widder’s Theorem is the following inversion of the Laplace transformwhich has been widely used in applications to differential equations.

Theorem 1.5 (Post-Widder Inversion). Let f ∈ L1loc([0,∞);X). Assume that there

exists ω ∈ IR such that r(λ) =∫ ∞0

e−λtf(t) dt exists for all λ > ω. Then for all continuity

points t > 0 of f ,

f(t) = limk→∞

(−1)k1k!

(k

t

)k+1

r(k)(k

t).

8

PROOF. Since we will make no use of this result in the applications below, we will verify itonly for bounded continuous functions f . For a proof of the general case see, for example,G. Doetsch (1950), Satz 8.2.1 or W. Arendt, M. Hieber and F. Neubrander (1995). If f isbounded and continuous, then it follows from (1.4) that

f(t) = limk→∞

1k!

(k

t

)k+1 ∫ ∞

0

e−ks

t skf(s)ds.

Now the statement follows from (1.2). ♦

If F : [a, b] → X is Lipschitz continuous and g : [a, b] → C continuous, then g and F

are Riemann-Stieltjes integrable with respect to each other and∫ b

ag(t) dF (t) = F (b)g(b)−

F (a)g(a) − ∫ b

aF (t) dg(t). Thus, for F ∈ Lip0([0,∞);X) and λ > 0 the Laplace-Stieltjes

transform equals the Laplace-Carson transform, or λ-multiplied Laplace transform; i.e.,

r(λ) =∫ ∞

0

e−λt dF (t) = λ

∫ ∞

0

e−λtF (t) dt.

Combining this with the Post-Widder inversion formula we obtain that the inverse Laplace-Stieltjes transform L−1

S : C∞W ((0,∞);X) → Lip0([0,∞);X) is given by

F (t) = limk→∞

(−1)k1k!

(k

t

)k+1 (r(λ)λ

)(k)

λ=k/t

= limk→∞

k∑j=0

(−1)j1j!

(k

t

)j

r(j)

(k

t

).

Studying whether or not Widder’s original proof of the surjectivity of LS extends to Banachspace valued functions, E. Hille (1948) remarks that Widder’s

“ ... argument seems to hold for any abstract space in which bounded sets are weaklycompact, but it is not clear in the present writing whether or not his conditions are alwayssufficient for the existence of the representation.”

This statement was made precise by S. Zaidman (1960) and W. Arendt (1987). Theresult is the following characterization of Banach spaces with the Radon-Nikodym property.

9

Theorem 1.6 (Riesz Representation). Let X be a Banach space. The following are

equivalent.

(i) The antiderivative operator I : f → F , F (t) :=∫ t0f(s) ds is an isometric isomorphism

between L∞([0,∞);X) and Lip0([0,∞);X).

(ii) The Riesz operator R : f → Tf , Tfg :=∫ ∞0g(t)f(t) dt is an isometric isomorphism

between L∞([0,∞);X) and L(L1(0,∞), X).

(iii) The Laplace transform L : f → r, r(λ) :=∫ ∞0

e−λtf(t) dt is an isometric isomorphism

between L∞([0,∞);X) and C∞W ((0,∞);X).

PROOF. As shown above, the Riesz-Stieltjes operator RS : F → TF , TF g =∫ ∞0

g(t)dF (t)is an isometric isomorphism between Lip0([0,∞);X) and L(L1(0,∞), X) and the Laplace-Stieltjes transform LS : F → r, r(λ) =

∫ ∞0

e−λt dF (t) is an isometric isomorphism betweenLip0([0,∞);X) and C∞

W ((0,∞);X). Now the statement follows from the fact that L =LS · I and R = RS · I on L∞([0,∞);X). ♦

Let X = L∞[0,∞) and f(t) := χ[0,t] (t ≥ 0). Since ‖f(t)−f(s)‖ = 1 for all t = s, therange of f does not contain a countable dense set. Thus, by Pettis’ Theorem (see E. Hilleand R.S. Phillips (1957), Theorem 3.5.3), f is not measurable. However, f is Riemannintegrable. In fact,

F (t) := (R)∫ t

0

f(s) ds = lim|π|→0

n∑i=1

χ[0,ξi](si − si−1) = (t− ·)χ[0,t]

for all t ≥ 0. It is easy to see that F is Lipschitz continuous. However, although F

is Lipschitz continuous and an indefinite Riemann integral, F is nowhere differentiable.Again, it is easy to check that 1

h(F (t + h) − F (t)) does not converge in X as h → 0 for

any t ≥ 0. Since F ∈ C0[0,∞) ⊂ L∞[0,∞), it follows that C0 and L∞-spaces do not havethe Radon-Nikodym property. Since L∞[0,∞) ⊂ L(Lp[0,∞)) for 1 ≤ p ≤ ∞ it followsthat L(Lp[0,∞)) does not have the Radon-Nikodym property. In fact, there seems to beno known infinite dimensional Banach space X for which L(X) has the Radon-Nikodymproperty.

10

The function F : [0,∞) → c0, F (t) := ( 1n sin(nt))n∈IN is another example of a Lips-

chitz continuous function which is nowhere differentiable. Thus c0 and any Banach spacecontaining it does not have the Radon-Nikodym property.

Examples of spaces with the Radon-Nikodym property are reflexive Banach spaces andspaces with a boundedly complete Schauder basis (like .1). For a proof of this statementand a thorough discussion of the Radon-Nikodym property, see J. Diestel and J.J. Uhl(1977).

As we will see in Section 3, the inversion formulas for the Laplace and Laplace-Stieltjes transforms play a fundamental role in applications. Compared to the Post-Widderinversion, it is remarkable that in the following inversion formula (due to G. Doetsch (1950),Satz 8.1.1) only the values of r(n) for large n ∈ IN are needed and that the convergence isuniform for all t ≥ 0.

Theorem 1.7 (Phragmen-Doetsch Inversion). Let F ∈ Lip0([0,∞);X) and r(λ) =∫ ∞0

e−λt dF (t). Then

‖F (t)−∞∑j=1

(−1)j+1

j!etnjr(nj)‖ ≤ c

n‖r‖W

for all t ≥ 0 and n ∈ IN, where c ≈ 1.0159....

PROOF. By the Riesz-Stieltjes Representation Theorem 1.1 and Widder’s Theorem1.3, there exists T ∈ L(L1(0,∞), X) such that r(λ) =

∫ ∞0

e−λt dF (t) = Te−λ (λ > 0),Tχ[0,t] = F (t) (t ≥ 0) and ‖T‖ = ‖r‖W = ‖F‖Lip. Thus

‖F (t)−∞∑j=1

(−1)j+1

j!etnjr(nj)‖ ≤ ‖T‖ ‖χ[0,t] −

∞∑j=1

(−1)j+1

j!etnje−nj‖1.

Define pn,t := 1 − e−ente−n(·)

=∑∞

j=1(−1)j+1

j! etnje−nj . Then

‖χ(0,t]−pn,t‖1 =∫ t

0

|pn,t(s)−1| ds+∫ ∞

t

|pn,t(s)| ds =∫ t

0

e−en(t−s)

ds+∫ ∞

t

1−e−en(t−s)

ds

=1n

∫ ent

1

1ue−u du+

1n

∫ 1

0

1 − e−u

udu ≤ 1

n

[∫ ∞

1

1ue−u du+

∫ 1

0

1 − e−u

udu

]

for all t ≥ 0 and n ∈ IN. Now the claim follows from the fact that∫ ∞1

1ue

−u du +∫ 1

01−e−u

u du = −2Ei(−1) + γ ≈ 1.0159... where Ei(z) is the exponential integral andγ is Euler’s constant; see N.N. Lebedev (1972), 3.1. ♦

11

The Phragmen-Doetsch inversion of the Laplace-Stieltjes transform shows that any r ∈C∞W ((0,∞);X) (and thus also the unique F ∈ Lip0([0,∞);X) with r(λ) =

∫ ∞0

e−λt dF (t)(λ > 0)) is uniquely determined by the sequence xn := r(n) (n ∈ IN). We ask now theconverse question: i.e. to characterize those sequences (xn)n∈IN in a Banach space X forwhich there exists r ∈ C∞

W ((0,∞);X) (or equivalently F ∈ Lip0([0,∞);X)) such that

xn = r(n) =∫ ∞

0

e−λt dF (t) =∫ 1

0

tn dH(t)

(where H(t) := −F (− ln t)). This special “Hausdorff moment problem” can be solved,among others, by adapting H. Hahn’s (1927) method to solve the moment problem (see,for example, IV.5 in K. Yosida (1971) or Theorem 2.7.6 in E. Hille and R.S. Phillips(1957)).

Theorem 1.8 (Moment Theorem). Let M > 0 and xn ∈ X (n ∈ IN). The following

are equivalent.

(i) There exists T ∈ L(L1(0,∞), X) with ‖T‖ ≤ M and xn = T (e−n) for all n ∈ IN.

(ii) There exists F ∈ Lip0([0,∞);X) with ‖F‖Lip ≤ M and xn =∫ ∞0

e−nt dF (t) for all

n ∈ IN.

(iii) There exists r ∈ C∞W ((0,∞);X) with ‖r‖W ≤ M and xn = r(n) for all n ∈ IN.

(iv) ‖∑ni=1 λixi‖ ≤ M‖∑n

i=1 λie−i‖L1 for all n ∈ IN and λi ∈ C.

PROOF. The equivalence of (i),(ii) and (iii) follows from the Riesz-Stieltjes represen-tation and Widder’s Theorem. The implication (i)⇒(iv) follows from ‖∑n

i=1 λixi‖ ≤‖T‖ ‖∑n

i=1 λie−i‖L1 .To prove (iv)⇒(i), let z =

∑ni=1 λie−i ∈ spane−i, i ∈ IN =: E . If z =

∑ni=1 λie−i =∑m

j=1 µje−j , then ‖∑ni=1 λixi −

∑mj=1 µjxj‖ = ‖∑l

k=1 γkxk‖ ≤ M‖∑lk=1 γke−k‖L1 =

M‖∑ni=1 λie−i −

∑mj=1 µje−j‖L1 = 0. Thus we can define on E an operator T by T (z) :=∑n

i=1 λixi, where z =∑n

i=1 λie−i. Now (i) follows from ‖T (z)‖ ≤ M‖z‖L1 and the densityof E . ♦

We remark that any r ∈ C∞W ((0,∞);X) has a Laplace-Stieltjes representation and thus

an analytic extension for Reλ > 0 which we denote by the same symbol. Widder’s growthconditions “supλ>0 ‖λk+1 1

k!r(k)(λ)‖ ≤ M ” translate via Cauchy’s Integral Theorem to the

following complex conditions (see also M. Sova (1980)).

12

Theorem 1.9 (Complex Widder Conditions). Let M > 0 and let r : Reλ > 0 → X

be an analytic function with limλ→∞ ‖r(λ)‖ = 0 and supReλ>γ ‖r(λ)‖ < ∞ for all γ > 0.The following are equivalent.

(i) r ∈ C∞W ((0,∞);X), with ‖r‖W ≤ M.

(ii) For some, or equivalently, for all k0 ∈ IN0,

supλ>0

∥∥∥∥λk+1 1k!r(k)(λ)

∥∥∥∥ ≤ M for all k ≥ k0.

(iii) For some, or equivalently, for all k0 ∈ IN0,

supγ>0

sups>0

∥∥∥∥ 12π

∫ ∞

−∞

r(γ + it)(1− ist)k+2

dt

∥∥∥∥ ≤ M for all k ≥ k0.

PROOF. Clearly, (i) implies (ii). First we show that (ii) implies (i). This follows fromthe observation that

supλ>0

‖λk0+1 1k0!

r(k0)(λ)‖ ≤ M

for some k0 ∈ IN implies that

supλ>0

‖λk+1 1k!r(k)(λ)‖ ≤ M

for all 0 ≤ k ≤ k0 − 1. In order to prove this statement define

rk(λ) :=(−1)k+1

k!

∫ ∞

λ

(λ− µ)kr(k0)(µ) dµ (λ > 0).

These integrals exist absolutely and

‖rk(λ)‖ ≤ M k0!k!

∫ ∞

λ

∥∥∥∥(λ− µ)k

µk0+1

∥∥∥∥ dµ =M(k0 − k − 1)!

λk0−k .

Since r(k0)k0−1 = r(k0) it follows that (rk0−1 − r)(k0) = 0. Thus rk0−1 − r is a polynomial with

limλ→∞ rk0−1(λ) − r(λ) = 0. Hence r = rk0−1 and

‖r(k)(λ)‖ = ‖rk0−k−1(λ)‖ ≤ M k!λk+1

for all k < k0.Next we show the equivalence of (ii) and (iii). Let λ > γ > 0, n ∈ IN and let Γn be the

counterclockwise oriented path consisting of the line segments Γn,1 := γ+ i[−n, n], Γn,2 :=

13

γ + n + i[−n, n], Γn,3 := [γ, γ + n] + in, Γn,4 := [γ, γ + n] − in. By Cauchy’s IntegralTheorem we obtain

r(k+1)(λ) =(k + 1)!

2πi

∫Γn

r(z)(z − λ)k+2

dz (k ∈ IN0).

For n → ∞ the boundedness of r for Reλ ≥ γ implies that

r(k+1)(λ) = −(k + 1)!2πi

∫γ+iIR

r(z)(z − λ)k+2

dz.

Hence

(λ− γ)k+2 1(k + 1)!

r(k+1)(λ) = (−1)k+1 (γ − λ)k+2

2πi

∫γ+iIR

r(z)(z − λ)k+2

dz

= (−1)k+1 (γ − λ)k+2

2π

∫ ∞

−∞

r(γ + it)(γ + it− λ)k+2

dt.

=(−1)k+1

2π

∫ ∞

−∞

r(γ + it)(1 − ist)k+2

dt,

where s := 1λ−γ . Thus (ii) implies (iii) and (iii) implies that ‖(λ−γ)k+2 1

(k+1)!r(k+1)(λ)‖ ≤

M for all γ > 0 and all k ≥ k0. Taking γ → 0 we obtain that (iii) implies (ii). ♦

The following inversion theorem uses the fact that any r ∈ C∞W ((w,∞);X) is a

Laplace-Stieltjes transform and therefore has an analytic extension to Reλ > 0 whichwill be denoted by the same symbol.

Theorem 1.10 (Complex Inversion). The inverse Laplace-Stieltjes transform L−1S :

C∞W ((0,∞);X) → Lip0([0,∞);X) is given by

L−1S (r)(t) = F (t) = lim

n→∞

∫ 1+in

1−ineλt

r(λ)λ

dλ,

where the limit is uniform on compact intervals.

14

PROOF. By the Riesz-Stieltjes Representation Theorem 1.1 and Widder’s Theorem 1.3,there exist F ∈ Lip0([0,∞);X) and T ∈ L(L1(0,∞), X) such that r(λ) =

∫ ∞0

e−λt dF (t) =Te−λ (λ > 0) and Tχ[0,t] = F (t) (t ≥ 0). Then

‖F (t)−∫ 1+in

1−ineλt

r(λ)λ

dλ‖ ≤ ‖T‖‖χ[0,t] −∫ 1+in

1−ineλt

e−λλ

dλ‖1.

Now it is an exercise to show that the functions∫ 1+in

1−in eλt e−λ

λdλ converge towards the

characteristic function χ[0,t] in L1(0,∞) as n → ∞ uniformly (in t) on compact intervals(see F. Neubrander (1994b)). ♦

Other consequences of the Riesz-Stieltjes Representation Theorem are Laplace trans-form versions of Trotter-type approximation theorems of semigroup theory (for further the-orems of this type, see B. Hennig and F. Neubrander (1993) and F. Neubrander (1994a)).

Theorem 1.11 (Approximation Theorem). Let Fn ∈ Lip0([0,∞);X) with ‖Fn‖Lip ≤M for all n ∈ IN and rn(λ) =

∫ ∞0

e−λt dFn(t), (λ > 0). The following are equivalent.

(i) There exist a, b > 0 such that limn→∞ rn(a+ kb) exists for all k ∈ IN0.

(ii) There exists F ∈ Lip0([0,∞);X) with ‖F‖Lip ≤ M such that limn→∞ rn(λ) =∫ ∞0

e−λtdF (t) uniformly (in λ > 0) on compact sets.

(iii) limn→∞ Fn(t) exists for all t ≥ 0.(iv) There exists F ∈ Lip0([0,∞);X) with ‖F‖Lip ≤ M such that Fn(t) → F (t) uniformly

(in t ≥ 0) on compact sets.

PROOF. By the Riesz-Stieltjes Representation Theorem there exist Tn ∈ L(L1(0,∞), X)with ‖Tn‖ ≤ M such that Tne−λ = rn(λ) and Tnχ[0,t] = Fn(t) for all n ∈ IN, all t ≥ 0and all λ > 0. Each of the statements imply that the operators Tn converge on a totalsubset of L1(0,∞). By the theorem of Banach-Steinhaus there exists T ∈ L(L1(0,∞), X)such that Tn → T as n → ∞. Let b > 0. Then the set of characteristic functionsχb := χ[0,t] : 0 ≤ t ≤ b and the set of exponential functions Eb := e−λ : 1

b≤ λ ≤ b

are compact in L1(0,∞). Hence, the uniformly bounded sequence Tn converges uniformlyon χb and Eb (H.H. Schaefer (1980), Theorem III.4.5). Now the statement follows fromTheorem 1.1. ♦

15

Applying vector-valued Laplace-Stieltjes transform theory to evolution equations, itis desirable to allow for Lipschitz functions with arbitrary exponential growth. This canbe done by using the following shifting procedure.

Remark 1.12 (The Laplace Transform of Functions with Exponential Growth).

For w ∈ IR let Lipw([0,∞);X) be the space of all functions G : [0,∞) → X withG(0) = 0 and ‖G(t) −G(s)‖ ≤ M

∫ tsewr dr for all 0 ≤ s ≤ t and some constant M . Then

Lipw([0,∞), X) is a Banach space with norm

‖G‖Lip(w) := infM : ‖G(t) −G(s)‖ ≤ M

∫ t

s

ewr dr for all 0 ≤ s ≤ t,

and Iw defined by IwG(t) :=∫ t0e−ws dG(s) is an isometric isomorphism between the spaces

Lipw([0,∞);X) and Lip0([0,∞);X).

For w ∈ IR let C∞W ((w,∞);X) be the Banach space of all functions r ∈ C∞((w,∞);X)

with norm

‖r‖W,w := supk∈IN0

supλ>w

‖ 1k!

(λ− w)k+1r(k)(λ)‖ < ∞.

The shift Swr(λ) := r(λ − w) is an isometric isomorphism between C∞W ((0,∞);X) and

C∞W ((w,∞);X). Hence, the Laplace-Stieltjes transform Sw LS Iw is an isometric isomor-

phism between Lipw([0,∞);X) and C∞W ((w,∞);X) and all theorems mentioned so far in

this section can be rephrased for these spaces. ♦

In general, it is often impossible to verify the Widder growth conditions

supλ>ω

‖ 1k!r(k)(λ)‖ ≤ M

(λ− ω)k+1(k ∈ IN0)

or the equivalent complex conditions (Theorem 1.10), whereas the growth of r in the wholecomplex halfplane can be estimated. In these cases one can use the following representationtheorem.

16

Theorem 1.13 (Complex Representation Theorem). Let q : Reλ > ω ≥ 0 → X

be analytic and supReλ>ω ‖λq(λ)‖ < ∞. Then, for all b > 0 there exists f ∈ C([0,∞);X)with supt>0 ‖e−ωt f(t)

tb‖ < ∞ such that q(λ) = λb

∫ ∞0

e−λtf(t) dt for Reλ > ω.

PROOF. For ε > ω we define f(t) := 12πi

limn→∞∫ ε+inε−in eλt q(λ)

λb dλ. Observe that theintegral is absolutely convergent and that it converges uniformly for t ∈ [0, m] (m ∈ IN).Hence f ∈ C([0,∞);X). Let Γε,R be the path ε+ i(−∞,−R]∪ ε+Rei[−

π2 ,

π2 ] ∪ ε+ i[R,∞).

It follows from Cauchy’s integral theorem that the definition of f is independent of ε andthat

f(t) =1

2πi

∫Γε,R

eλtq(λ)λb

dλ =12π

∫ −R

−∞e(ε+ir)t

q(ε+ ir)(ε+ ir)b

dr

+12π

∫ π2

− π2

e(ε+Reiα)t q(ε+Reiα)

(ε+Reiα)bReiα dα+

12π

∫ ∞

R

e(ε+ir)tq(ε+ ir)(ε+ ir)b

dr

for all R > 0. Hence, for all t ≥ 0,

‖f(t)‖ ≤ M

πeεt

∫ ∞

R

1rb+1

dr +M

2π

∫ π2

−π2

1Rb

e(ε+R cosα)t dα

≤ M

πb

1Rb

eεt +M

π

1Rb

eεt∫ π

2

0

eRt cosα dα.

Choosing R = 1t, we obtain for all ε > ω that ‖f(t)‖ ≤ Ctbeεt for some C > 0. Hence,

‖f(t)‖ ≤ Ctbeωt. For Reλ > ω we choose ω < ε < Reλ and let ΓR be the path consisting ofthe halfcircle ε+Re−i[−

π2 ,

π2 ] and the line segment ε+ i[−R,R]. Then the residue theorem

and Cauchy’s theorem imply that

∫ ∞

0

e−λtf(t) dt =∫ ∞

0

e−λt1

2πi

∫ ε+i∞

ε−i∞eµt

q(µ)µb

dµ dt

=1

2πi

∫ ε+i∞

ε−i∞

1λ− µ

1µb

q(µ) dµ

= limR→∞

12πi

∫ΓR

1λ− µ

1µb

q(µ) dµ

=1λb

q(λ).

♦

17

The following is an immediate consequence of the Complex Representation Theorem1.13. We say that a function q is polynomially bounded for λ ∈ Ω if there exists apolynomial p such that ‖q(λ)‖ ≤ p(|λ|) for λ ∈ Ω. Similarly, q is said to be exponentiallybounded if there exists ω ∈ IR and M > 0 such that ‖q(λ)‖ ≤ Meω|λ| for all λ ∈ Ω.

Corollary 1.14. Let q be a function with values in X . The following are equivalent.

(i) q is analytic and polynomially bounded for some half-plane Reλ > ω.(ii) There exists b ≥ 0 and an exponentially bounded f ∈ C([0,∞);X) such that q(λ) =

λb∫ ∞0

e−λtf(t) dt for some half-line λ > ω′.

As another consequence of Theorem 1.13 we obtain the following representation resultdue to J. Pruss (1993).

Corollary 1.15. Let q : Reλ > 0 → X be analytic. If there exists M > 0 such

that ‖λq(λ)‖ ≤ M and ‖λ2q′(λ)‖ ≤ M for Reλ > 0, then there exists a bounded function

f ∈ C((0,∞);X) such that q(λ) =∫ ∞0

e−λtf(t) dt for Reλ > 0.

PROOF. It follows from the Complex Representation Theorem 1.13 that there are func-tions fi ∈ C([0,∞);X), (i ∈ 0, 1) and C > 0 such that ‖fi(t)‖ ≤ Ct for t > 0,

q(λ) = λ

∫ ∞

0

e−λtf0(t) dt, and λq′(λ) = λ

∫ ∞

0

e−λtf1(t) dt

for Reλ > 0. Hence

q′(λ) =∫ ∞

0

e−λtf0(t) dt− λ

∫ ∞

0

e−λttf0(t) dt =∫ ∞

0

e−λtf1(t) dt.

Integrating by parts yields

λ

∫ ∞

0

e−λt[∫ t

0

f0(s) ds− tf0(t)]dt = λ

∫ ∞

0

e−λt∫ t

0

f1(s) ds dt.

Since the Laplace transform is one-to-one, it follows that tf0(t) =∫ t

0f0(s) ds−

∫ t0f1(s) ds.

Thus f0 ∈ C1((0,∞);X) and tf ′0(t) = −f1(t). Therefore ‖f ′

0(t)‖ ≤ C for all t > 0 and

q(λ) = λ

∫ ∞

0

e−λtf0(t) dt =∫ ∞

0

e−λtf ′0(t) dt

if Reλ > 0. ♦

18

We conclude this section with M. Sova’s characterization of those functions q : (ω,∞) → X

which are Laplace transforms of analytic, exponentially bounded functions defined on asector. For a proof see M. Sova (1979), F. Neubrander (1989, 1994b), W. Arendt (1991) orW. Arendt, M. Hieber, F. Neubrander (1995). For 0 < α ≤ π we denote by Σα the opensector Σα := reiγ : r > 0,−α < γ < α and by Σα its closure.

Theorem 1.16 (Analytic Representation Theorem). Let 0 < α ≤ π2, ω ∈ IR and

q : (ω,∞) → X . The following are equivalent.

(i) There exists an analytic function f : Σα → X such that supz∈Σβ‖e−ωzf(z)‖ < ∞ for

all 0 < β < α and q(λ) =∫ ∞0

e−λtf(t) dt for all λ > ω.

(ii) The function q is analytic in the sector ω+Σα+ π2and supλ∈ω+Σγ+ π

2‖(λ−ω)q(λ)‖ < ∞

for all 0 < γ < α.

Moreover, for all 0 < β < α there exists a constant Cβ > 0 such that

(1.5) ‖zkf (k)(z)‖ ≤ CβeωRez(|ω||z|+ 1)k

for all z ∈ Σβ . ♦

Let f and q be as in the previous theorem and assume that that ‖(λ− ω)q(λ)‖ ≤ M

for all λ ∈ ω + Σα+ π2. Let ω′ < ω. It is easy to see that if q is analytic for all Reλ > ω′,

then there exists 0 < γ < α such that q is analytic on ω′ +Σγ+ π2

and ‖(λ−ω′)q(λ)‖ ≤ M .Thus, by (1.5), for all k ∈ IN0, there exists a constant C > 0 such that

‖f (k)(t)‖ ≤ Ceω′t(|ω′| + 1

t)k for all t > 0.

This shows that the exponential growth (at infinity) of the function f and its derivativesis determined by the abscissa of analyticity of its Laplace transform q, i.e., infω ∈ IR : qextends analytically to Reλ > ω.

19

2. EVOLUTION EQUATIONS, CAUCHY PROBLEMS AND CLOSED OPERATORS

To simplify the demonstration of the Laplace transform method, we think of a linearevolution equation as a system of equations which can be written as an abstract Cauchyproblem

(ACP ) u′(t) ∈ Au(t) ; u(0) = x ; 0 ≤ t < T (T ≤ ∞)

where A is a linear, but in general unbounded, not closed, not densely defined and evenmultivalued operator on some Banach space X . It is clear from applications to partialdifferential equations that A is in general unbounded with possibly nondense domain. Asa typical example of a problem where A might be not closed, consider an inhomogeneous

initial boundary value problem for an integro-differential equation; i.e.,

w′(t) = A0w(t) +∫ t

0

b(t− s)Bw(s) ds + f(t), w(0) = w0, Cw(t) = g(t).

Here A0, B are operators on a Banach space X0; C has its domain in X0 and maps in aBanach space Y ; w0 ∈ X0, f ∈ F ⊂ L1

loc([0,∞);X0), and g ∈ G ⊂ L1loc([0,∞);Y ) (where

F and G are Banach function spaces). This equation can be written as an (ACP ) onX = X0 ×F × G with initial value x = (w0, f, g), and

A =

A0 δ0 0

b(·)B DF 00 0 DG

,

where DF , DG denote the first derivative operators on F and G, and

D(A) = (x0, f, g) : x0 ∈ D(A0) ∩D(B) ∩D(C) , f ∈ D(DF ) , g ∈ D(DG) , Cx0 = g(0).

As for many other abstract systems of equations (i.e., if A is an operator matrix onproducts of Banach spaces), the operator A might not be closed. And even if the operatorA is closable, we might not want to switch to its closure Ac, because much of the basicinformation on the underlying evolution problem is contained in the domain of A and wouldbe lost by considering the larger and often difficult to describe domain of the closure Ac.

20

Illposed problems connected with evolution equations (backwards equations, observa-tion and inverse problems) lead to abstract Cauchy problems where the operator A has“bad” spectral properties (like the pointspectrum pσ(A) covering a right or upper half-plane). For example, consider the heat equation wt(r, t) = wrr(r, t) in an infinitely longbar. Suppose we observe the temperature and flux of heat at the point r = 0 over one timeunit and that the initial temperature distribution is unknown. We would like to estimatefrom our observations the initial temperature distribution w(r, 0) for 0 ≤ r < ε. This leadsto

wt(r, t) = wrr(r, t) , w(0, t) = f(t) , wr(0, t) = g(t) , t ∈ [0, 1] , 0 < r < ε,

where f, g ∈ C[0, 1]. Interchanging t and r we obtain the initial value problem

wr(t, r) = wtt(t, r) , w(0, r) = f(r) , wt(0, r) = g(r) , r ∈ [0, 1] , 0 ≤ t < ε,

where f, g ∈ C[0, 1], and where we ask for the value of w(t, 0) for 0 ≤ t < ε. Clearly,this initial value problem can be written as an (ACP ) if we choose X := C[0, 1]× C[0, 1],

x := (f, g), and A :=(

0 ID 0

), where D denotes the first derivative on C[0, 1]. Since the

point spectrum of D covers the whole complex plane, the same holds for A. For furtherexamples of illposed Cauchy problems see, for example, S. Agmon and L. Nirenberg (1963).

As an example of a linear evolution equation which leads to a multivalued linearoperator A, consider the degenerate Cauchy problem

(DCP ) Bu′(t) = A0u(t) , u(0) = x,

where A0, B are linear operators with domains in a Banach space X and ranges in aBanach space Y . Clearly, (DCP) can be written as an abstract Cauchy problem u′(t) ∈Au(t) , u(0) = x on the space X , where A := B−1A0 ⊂ X × X is given by (x, y) :x ∈ D(A0), y ∈ D(B), A0x = By. Then A is a possibly multivalued linear operator (i.e.,A(µx+ y) = µA(x) +A(y) for x, y ∈ D(A), µ ∈ C), whose graph is, in general, not closedin X ×X .

It is one of the major advantages of the Laplace transform method that it applies toall of the problems above. With it we can study abstract Cauchy problems (ACP ) for awide variety of operators A, including sums, products and limits of closed operators whichare not necessarily closed or closable, operators which are not densely defined, and thosewhere the point spectrum pσ(A) covers C.

21

For simplicity, we will include in the following survey only the case of single valuedoperators and refer to C. Knuckles and F. Neubrander (1994) for the multivalued case.For other related approaches to illposed Cauchy problems see I. Cioranescu and G. Lumer(1994), R. deLaubenfels (1994), and G. Lumer (1994).

In order to demonstrate the Laplace transform method for evolution equations, it isconvenient to consider relatively closed operators (see S. Agmon and L. Nirenberg (1963),S.R. Caradus (1973), and B. Baumer and F. Neubrander (1994)). This class of linearoperators is quite large. It contains basically all linear operators appearing in applicationsand imposes just enough continuity on the operator A such that A commutes with theBochner and Stieltjes integral for sufficiently regular functions.

If a Banach space Z is continuously embedded in a Banach space X , then we will usethe notation Z → X .

Definition 2.1. A linear operator A is called relatively closed in a Banach space X if

there exists an auxiliary Banach space XA such that D(A) ⊂ XA → X and A is closed in

XA ×X . A relatively closed operator will also be called (XA → X)-closed.

Examples 2.2 (Sums and Products). An important class of examples of relativelyclosed operators are sums and products of closed operators A,B. In general, the sumS := A + B with domain D(S) = D(A) ∩ D(B) and the composition C := BA withdomain D(C) = x ∈ D(A) : Ax ∈ D(B) will not be closed or closable in X . However,both S and C are relatively closed if we choose as XS and XC the Banach space [D(A)]with the graph norm ‖x‖A := ‖x‖ + ‖Ax‖.

We remark that S and C can be relatively closed even if the operators A,B are notclosed themselves. As example take a pair of jointly closed operators A,B on a Banachspace X ; i.e., D(A) ∩ D(B) xn → x, and Axn → y1, and Bxn → y2 implies that x ∈D(A)∩D(B), Ax = y1, and Bx = y2 (see also N. Sauer (1982)). Then XS := [D(A)∩D(B)]with norm ‖x‖XS

:= ‖x‖ + ‖Ax‖ + ‖Bx‖ is a Banach space and D(S) ⊂ XS → X . It iseasy to see that the sum S is (XS → X)-closed. ♦

22

As an example of an operator which is not closable in X, but relatively closed, weconsider the following composition of closed operators. Let A be the first derivative onX = C[0, 1] with maximal domain and let B be the bounded operator Bf := f(0)g,where 0 = g ∈ X . As seen above, the composition Cf := BAf = f ′(0)g with domainD(C) = D(A) = C1[0, 1] is relatively closed. However, since there is a sequence fn ∈ D(C)with fn → 0 and f ′

n(0) = 1, it follows from Cfn = f ′n(0)g = g = 0 that C is not closable;

i.e., the closure of the graph of C in X × X which is given by X × Cg is not the graphof a single-valued operator. Because the multivalued closure of C does not contain anyinformation about the original operator, and because closedness is absolutely necessary formost operations, it is necessary to consider the graph as a subset of XA ×X .

In this example one might consider the domain D(Cmax) := f ∈ X : f ′(0) existsinstead of C1[0, 1]. Define bounded operators on X by Atf := f(t)−f(0)

tg. Then, for each

f ∈ D(Cmax) one has that Atf → Cmaxf as t → 0. Because Cmax is the pointwise limitof closed operators, it follows from the next theorem that it is relatively closed (for proofsof the following theorems see B. Baumer and F. Neubrander (1994)).

Theorem 2.3. For all t ∈ I := (0, 1] let At be an (XAt→ X)-closed operator. Suppose

there exists a Banach space Y such that Y → XAtfor all t ∈ I. Then Ax := limt→0 Atx

with D(A) = x ∈ ⋂t∈I D(At) ∩ Y : limt→0 Atx exists is (XA → X)-closed, where

XA = x ∈ ⋂t∈I D(At) ∩ Y : ‖x‖XA

:= ‖x‖Y + supt∈I ‖Atx‖ < ∞.

Theorem 2.4. For all n ∈ IN let An be a (Xn → X)-closed operator. Suppose there

exists a Banach space Y such that Y → Xn for all n ∈ IN. Then Ax :=∑∞

n=0 Anx

with D(A) = x ∈ ⋂n∈IN D(An) ∩ Y :

∑∞n=0 Anx exists is (XA → X)-closed, where

XA = x ∈ ⋂n∈IN D(An) ∩ Y : ‖x‖XA

= ‖x‖Y + supn∈IN ‖∑ni=1 Aix‖ < ∞.

Theorem 2.5. For all n ∈ IN let An be a (Xn → X)-closed operator. Then for all n ∈ IN,

Cnx := An · · ·A0x with D(Cn) = x ∈ D(A0) : Ak · · ·A0x ∈ D(Ak+1) for all 0 ≤ k ≤n − 1 is (XCN

→ X)-closed, where XCn:= x ∈ D(A0) : Ak · · ·A0x ∈ D(Ak+1) for all

0 ≤ k ≤ n− 2, An−1 · · ·A0x ∈ Xn and ‖x‖Cn:= ‖x‖X0 + sup0≤k≤n−1 ‖Ak · · ·A0x‖Xk+1 <

∞. If there exists a Banach space Y such that Y → XCnfor all n ∈ IN, then Cx :=

limn→∞Cnx with D(C) := x ∈ ⋂n∈IN D(Cn) ∩ Y : limn→∞ Cnx exists is (XC → X)-

closed, where XC := x ∈ ⋂n∈IN D(Cn) ∩ Y : ‖x‖XC

:= ‖x‖Y + supn∈IN ‖Cnx‖ < ∞.

23

Clearly, most linear operators that appear in applications can be decomposed intosums, products and/or limits of relatively closed operators. In order to find linear operatorswhich are not relatively closed one may consider “small” domains. For example, considerthe identity operator I on X = C[0, 1] with the polynomials P as its domain. Clearly, I isclosable. Assume it were relatively closed. Then there exists a Banach space XI such thatthe graph G = (p, p) : p ∈ P of I is closed in XI ×X . Thus G is a complete metric spaceof first category, which is a contradiction to Baire’s theorem. It follows that the operatorI with domain P is not relatively closed.

The following is a classical result of functional analysis due to E. Hille. For a proof,see E. Hille and R.S. Phillips (1957), Theorem 3.7.12.

Proposition 2.6. Let A be a (XA → X)-closed operator. Assume that u(·) maps an

interval I into D(A). If u(·) : I → XA and Au(·) : I → X are (improperly) Bochner

integrable, then∫Iu(t) dt ∈ D(A) and

∫IAu(t) dt = A

∫Iu(t)dt .

3. THE LAPLACE TRANSFORM METHOD

Let A be a relatively closed operator on a Banach space X , x ∈ X and 0 < T ≤ ∞.In this section we will study the abstract Cauchy problem

(ACP ) u′(t) = Au(t); u(0) = x; 0 ≤ t < T

in terms of the asymptotic characteristic equation

(CE) (kI − A)yk = x− ak (k0 ≤ k ∈ IN),

where ak is a sequence with exponential decay −T in X ; i.e., lim supk→∞1k ln ‖ak‖ ≤ −T .

If T = ∞, we set ak = 0 for all k ∈ IN. To obtain necessary and sufficient conditions forthe existence of solutions, we include the integral equations

(ACPn) v(t) = A

∫ t

0

v(s) ds+tn

n!x ; 0 ≤ t < T

in our considerations (n ∈ IN0). Formally, the (n + 1)-times integrated Cauchy problem

(ACPn) can be obtained by integrating (ACP ) (n+1)-times from 0 to t. Then a solutionv(·) of (ACPn) can be thought of as the n-th normalized antiderivative

u[n](t) :=∫ t

0

(t− s)n−1

(n− 1)!u(s) ds

of a solution u(·) of (ACP ).

24

For v ∈ C([0, T ]);X) let vT be the 0-continuation of v to IR+; i.e., vT := v on [0, T ]and vT := 0 on (T,∞). For v ∈ L1

loc([0,∞);X) we denote by absX(v) the infimum of alla ∈ IR for which the Laplace transform

v(λ) :=∫ ∞

0

e−λtv(t) dt = limr→∞

∫ r

0

e−λtv(t) dt

of v exists for Reλ > a. If ω ≥ 0, then absX(v) ≤ ω if and only if ωX(v[1]) ≤ ω, where theexponential growth bound ωX(v[1]) is defined to be the infimum of the numbers ω′ ∈ IR forwhich there exists a constant M such that ‖v[1](t)‖X ≤ Meω

′t for all t ≥ 0 (see G. Doetsch(1950), Satz 7 [2.2]). Notice that absX(vT ) = ωX(vT ) = −∞ if T < ∞. We denote by INω

the set of all integers larger than ω.

The following lemma is the foundation of the Laplace transform method. It showsthat the Laplace transform allows us to switch freely between the Cauchy problem andthe asymptotic characteristic equation for any relatively closed operator A (see B. Baumerand F. Neubrander (1994)). Notice that the following lemma contains also a uniquenessresult.

Lemma 3.1 (Fundamental Lemma). Let A be (XA → X)-closed, x ∈ X , n ∈ IN0,

0 < T ≤ ∞ and ω ≥ 0. Let v ∈ C([0, T ];X) with v[1] ∈ C([0, T ];XA), absX(vT ) ≤ ω and

absXA(v[1]T ) ≤ ω. The following are equivalent.

(i)∫ t

0v(s) ds ∈ D(A) and v(t) = A

∫ t0v(s) ds+ tn

n!x for all t ∈ [0, T ].

(ii) vT (λ) ∈ D(A) and (λI−A)λnvT (λ) = x−e−λT(λnv(T ) +

∑n−1i=0

(λT )i

i!x)if Reλ > ω.

(iii) vT (k) ∈ D(A) for all k ∈ INω and there exists a sequence ak of exponential decay −T

such that (kI − A)knvT (k) = x− ak for all k ∈ INω.

PROOF. We outline the proof given in B. Baumer and F. Neubrander (1994). Assume(i) holds. If Reλ > ω, then (ii) follows from

vT (λ) =∫ T

0

e−λtv(t) dt =∫ T

0

e−λtAv[1] (t) dt+∫ T

0

e−λttn

n!x dt

= − 1λe−λT v(T ) +

1λAvT (λ) +

1λn+1

x− e−λT

λn+1

n−1∑i=0

(λT )i

i!x.

Clearly, (ii) implies (iii).

25

Assume (iii) holds. Let ω′ > ω. Then v[2]T ∈ Lipω′([0,∞);X), v[3]

T ∈ Lipω′([0,∞);XA)and

y(λ) := λnvT (λ) = λn+1

∫ ∞

0

e−λtdv[2]T (t) = λn+2

∫ ∞

0

e−λtdv[3]T (t).

Since limk→∞∑∞

j=1(−1)j+1

j! etkj 1(kj)n+2 akj = 0 and Ay(k) = ky(k)−x+ak for all sufficiently

large k ∈ IN it follows from the (XA → X)-closedness of A that v[3](t) ∈ D(A) and

Av[3](t) = limk→∞

∞∑j=1

(−1)j+1

j!etkj

Ay(kj)(kj)n+2

= limk→∞

∞∑j=1

(−1)j+1

j!etkj

(y(kj)

(kj)n+1− x

(kj)n+2+

akj(kj)n+2

)

= v[2]T (t) − tn+2

(n+ 2)!x

Now (i) follows from the (XA → X)-closedness of A. ♦

As the Fundamental Lemma 3.1 shows, a solution of (ACP ) or (ACPn) can only existon [0, T ] if the initial value x is in the T -approximate range of the operators λI − A ;i.e., there exists a sequence ak ∈ X of exponential decay −T and a sequence yk such thatyk ∈ D(A) and

(3.1) (kI −A)yk = x− ak

for all sufficiently large k ∈ IN. If there exists a function y which is analytic and polyno-mially bounded in a right half-plane such that y(k) = yk for all sufficiently large k ∈ IN,then (3.1) is also sufficient for the existence of a solution of (ACPn) for some n ∈ IN0 (seeB. Baumer and F. Neubrander (1994)).

Theorem 3.2 (Existence and Uniqueness). Let A be a (XA → X)-closed operator,

x ∈ X , and 0 < T ≤ ∞. The following are equivalent.

(i) There exists n ∈ IN0 and v ∈ C([0, T ];X) with v[1] ∈ C([0, T ];XA), absX(vT ) < ∞,

absXA(v[1]T ) < ∞,

∫ t

0v(s) ds ∈ D(A) and

v(t) = A

∫ t

0

v(s) ds+tn

n!x for all t ∈ [0, T ].

(ii) There exist a sequence ak in X of exponential decay −T and, for some ω ∈ IR, an

analytic, polynomially bounded y : Reλ > ω → XA such that y(k) ∈ D(A) and

(kI −A)y(k) = x− ak for all sufficiently large k ∈ IN.

26

Now let T = ∞. It follows from (3.1) that

(3.2) x ∈⋂

Reλ>ω

Im (λI − A)

is a necessary range condition for the existence of a global, Laplace transformable solutionof (ACPn) for some n ∈ IN0.

Assume now that (3.2) holds; i.e., there exists y(λ) ∈ D(A) such that (λI−A)y(λ) = x

for Reλ > ω. Then, by the Fundamental Lemma 3.1, (ACPn) has a solution for somen ∈ IN0 if and only if 1

λn y(λ) = v(λ) for some v ∈ C([0,∞);X) with v[1] ∈ C([0,∞);XA),absX(v) < ∞ and absXA

(v[1]) < ∞. Given a local resolvent λ → y(λ) solving the char-acteristic equation (λI − A)y(λ) = x for some x ∈ X , any of the Laplace RepresentationTheorems 1.4., 1.8 and 1.9 will lead to a Hille-Yosida type existence result. For example,Widder’s Theorem 1.4 leads to the following local version of the Hille Yosida theorem.

Corollary 3.3 (Local Hille-Yosida Theorem). Let A be (XA → X)-closed, x ∈ X ,

n ∈ IN and ω ≥ 0. The following are equivalent.

(i) There exists a solution v ∈ Lipω([0,∞);X) of (ACPn) with v[1] ∈ Lipω([0,∞);XA).(ii) There exists y ∈ C∞((ω,∞);XA) and M > 0 such that

(a) (λI − A)y(λ) = x for all λ ∈ IN , λ > ω,

(b) supλ>ω ‖ 1k!

(λ− ω)(k+1)(

1λn−1 y(λ)

)(k) ‖X ≤ M for all k ∈ IN0 ,

(c) supλ>ω ‖ 1k!

(λ− ω)(k+1)(

1λn y(λ)

)(k) ‖XA≤ M for all k ∈ IN0.

Moreover, if XA = X , then condition (c) can be dropped.

PROOF. Assume (i) holds. For Reλ > ω define y(λ) := λnv(λ). Then 1λn−1 y(λ) = λv(λ)

and 1λn y(λ) = λv[1](λ) for Reλ > ω. The statement (a) in (ii) follows from the Fundamental

Lemma 3.1 and the statements (b) and (c) in (ii) follow from Widder’s Theorem 1.4.Assume (ii) holds. It follows from Widder’s Theorem that there exist v ∈ Lipω([0,∞);X)and w ∈ Lipω([0,∞);XA) such that 1

λn−1 y(λ) = λv(λ) and 1λn y(λ) = λw(λ) for all λ > ω.

Because 1λn y(λ) = v(λ) = λv[1](λ) for all λ > ω, it follows from the Uniqueness Theorem

of Laplace transform theory that v[1] = w. Since λn(λI − A)v(λ) = (λI − A)y(λ) = x forall λ ∈ IN, λ > ω, the statement (i) follows from the Fundamental Lemma 3.1. ♦

27

If a closed operator A is ω−dissipative, i.e.,

‖(λI − A)x‖ ≥ (λ− ω)‖x‖

for all λ > ω and for all x ∈ D(A), the growth conditions on y in the corollary aboveare automatically satisfied. In fact, more can be said. It follows from the closedness andω−dissipativity of A that for λ > ω and xn ∈ D(A)

(3.3) (λI −A)xn → z implies xn → x ∈ D(A) and (λI −A)x = z.

As mentioned above, it is necessary for the existence of global, Laplace transformablesolutions for the abstract Cauchy problem that the initial data x is an element of

X :=⋂λ>ω

Im(λI −A).

Let X xn → x and λ > ω. Then there exists yn ∈ D(A) such that (λI−A)yn = xn → x.By (3.3), there exists y ∈ D(A) with (λI−A)y = x. Thus X is closed in X whenever (3.3)holds.

Let A be closed and ω−dissipative and let x ∈ X. Then there exists y : (ω,∞) → D(A)such that

(λI − A)y(λ) = x and ‖y(λ)‖ ≤ 1λ− ω

‖x‖for all λ > ω. It follows from

y(λ0) = (λI − A)(y(λ)− y(λ0)

λ0 − λ

)(λ > ω)

that y(λ) ∈ D(A) ∩ X for all λ > ω. Let A be the part of A in X; i.e., D(A) =x ∈ D(A) ∩ X : Ax ∈ X and Ax = Ax. Since (λI − A)(λy(λ) − x) = Ax for allx ∈ D(A)∩X, we obtain that D(A) = D(A)∩X . Furthermore, it follows that y(λ) ∈ D(A)and (λI − A)y(λ) = x for all x ∈ X . Thus λI − A is one-to-one (by ω−dissipativity), ontoX, and ‖R(λ, A)x‖ = ‖y(λ‖ ≤ 1

λ−ω ‖x‖ for all x ∈ X . This, the Fundamental Lemma 3.1and Widder’s Theorem 1.4 imply the following.

Theorem 3.4 (Dissipative Operators). Let A be closed and ω−dissipative. Then

(a) X :=⋂λ>ω Im(λI − A) is a closed subspace of X .

(b) A leaves X invariant; i.e., if x ∈ D(A)∩ X then Ax ∈ X. Let A denote the part of A

in X . Then D(A) = D(A) ∩ X , (ω,∞) ⊂ ρ(A) and ‖R(λ, A)‖ ≤ 1λ−ω (λ > ω).

In particular, if A is densely defined, then A generates a semigroup on X.

28

A similar result can be proved for closed, inverse positive operators on Banach latticeswith order continuous norm (see B. Baumer and F. Neubrander (1994)).

We conclude this section with the following local analytic generation theorem.

Corollary 3.5 (Analytic Solutions). Let A be (XA → X)-closed, 0 < α ≤ π2and ω ∈

IR. Assume that y(λ) ∈ D(A) and (λI −A)y(λ) = x for all λ > ω. If y : ω + Σα+π2→ XA

is analytic and supλ∈ω+Σγ+ π2‖(λ − ω)y(λ)‖XA

< ∞ for all 0 < γ < α, then there exists

an analytic function u : Σα → XA such that supz∈Σβ‖e−ωzu(z)‖ < ∞ for all 0 < β < α,∫ t

0u(s) ds ∈ D(A) and

u(t) = A

∫ t

0

u(s) ds+ x for all t > 0.

PROOF. The statement follows from the Fundamental Lemma 3.1 and the AnalyticRepresentation Theorem 1.16. ♦

In this section we have indicated how the Laplace transform method can be used tofind mild or integrated ‘solutions’ of u′(t) = Au(t), u(0) = x for a particular x ∈ X . Theresults mentioned are only a small sample of those possible. The Laplace transform methodcan also be used to study approximation problems (with approximation theorems similarto the one of Section 1) as well as the asymptotics of the solutions (with the Abelian andTauberian theorems of Laplace transform theory, see for example W. Arendt (1991) orW. Arendt and C.J.K. Batty (1988)). We have demonstrated how the method works forthe abstract Cauchy problem. However, Laplace transform techniques can also be applieddirectly to evolution equations without reducing them to abstract Cauchy problems. Al-though the direct approach to linear evolution equations has some advantages, the methodsone uses and the problems one encounters remain, in principle, unchanged.

Clearly, the Laplace transform method also allows us to characterize those operatorsA for which (ACP ) has mild or integrated solutions for all x ∈ X (i.e., generators of C0-semigroups or integrated semigroups). A more detailed introduction to Laplace transformtheory, a Laplace transform approach to strongly continuous and integrated semigroups,and applications of these classes of semigroups to partial differential equations can befound in a forthcoming monograph on Laplace Transforms and Evolution Equations byW. Arendt, M. Hieber and F. Neubrander (1995).

29

REFERENCES

1. S. Agmon and L. Nirenberg (1963). Properties of solutions of ordinary differentialequations in Banach space. Comm. Pure Appl. Math. 16, 121-239.

2. W. Arendt (1987). Vector valued Laplace transforms and Cauchy problems. Israel J.Math. 59, 327-352.

3. W. Arendt (1991). Linear Evolution Equations. Lecture Notes, Universitat Zurich.

4. W. Arendt (1993). Vector valued versions of some representation theorems in realanalysis. Preprint.

5. W. Arendt, C.J.K. Batty (1988). Tauberian theorems and stability of one-parametersemigroups. Trans. Amer. Math. Soc. 306, 837-852.

6. W. Arendt, M. Hieber and F. Neubrander (1995). The Laplace Transform in BanachSpaces and Evolution Equations. Monograph in preparation.

7. B. Baumer and F. Neubrander (1994). Relatively closed operators and abstract dif-ferential equations. Preprint. A preliminary version can be found in the LSU SeminarNotes in Functional Analysis and PDEs. Baton Rouge, 1993.

8. S.R. Caradus (1973). Semiclosed operators. Pacific J. Math. 44, 75-79.

9. I. Cioranescu and G. Lumer (1994). Regularization of evolution equations via ker-nels K(t), K-evolution operators and convoluted semigroups, generation theorems,Preprint.

10. M. Deakin (1981). The development of the Laplace transform 1737-1937, I. Euler toSpitzer, 1737-1880. Arch. Hist. Math., 346-390.

11. M. Deakin (1982). The development of the Laplace transform 1737-1937, II. Poincareto Doetsch, 1780-1937. Arch. Hist. Math., 351-381.

12. R. deLaubenfels (1994). Existence Families, Functional Calculus and Evolution Equa-tions. Lect. Notes Math. 1570, Springer Verlag.

13. J. Diestel, J.J. Uhl (1977). Vector Measures. Amer. Math Soc., Providence, RhodeIsland.

14. G. Doetsch (1937). Theorie und Anwendung der Laplace-Transformation. VerlagJulius Springer, Berlin.

15. G. Doetsch (1950). Handbuch der Laplace-Transformation I. Verlag Birkhauser, Basel.

16. G. Doetsch (1955). Handbuch der Laplace-Transformation II. Verlag Birkhauser,Basel.

17. G. Doetsch (1956). Handbuch der Laplace-Transformation III. Verlag Birkhauser,Basel.

30

18. B. Hennig and F. Neubrander (1993). On representations, inversions and approxima-tions of Laplace transforms in Banach spaces. Applicable Analysis 49, 151-170.

19. M. Hieber, A. Holderrieth and F. Neubrander (1992)). Regularized semigroups andsystems of linear partial differential equations. Ann. Scuola Norm. Sup. Pisa 19,363-379.

20. E. Hille (1948). Functional Analysis and Semi-Groups. Amer. Math Soc., Providence,Rhode Island.

21. E. Hille and R.S. Phillips (1957). Functional Analysis and Semi-Groups. Amer. Math.Soc. Providence, Rhode Island.

22. C. Knuckles and F. Neubrander (1994). Remarks on the Cauchy problem for multi-valued linear operators. Partial Differential Equations, Models in Physics and Biology.G. Lumer, S. Nicaise, B.-W. Schulze (eds.). Mathematical Research 82, AkademieVerlag.

23. N.N. Lebedev (1972). Special Functions and their Applications. Dover Publications,New York.

24. G. Lumer (1994). Evolution equations. Solutions for irregular evolution problems viageneralized solutions and generalized initial values. Applications to periodic shockmodels. Annales Saraviensis 5, 1-102

25. Yu. I. Lyubich (1966). The classical and local Laplace transformation in an abstractCauchy problem. Uspehi Mat. Nauk 21, 3-51.

26. Sylvia Martis in Biddau (1933). Studio della transformazione di Laplace e della suainversa dal punto di vista dei funzionali analitici. Rendiconti del circolo matematicodi Palermo LVII, 1-70.

27. F. Neubrander (1989). Abstract elliptic operators, analytic interpolation semigroups,and Laplace transforms of analytic functions. Semesterbericht Funktionalanalysis,Universitat Tubingen.

28. F. Neubrander (1994). The Laplace-Stieltjes transform in Banach spaces and abstractCauchy problems. In: Proceedings of the ‘3rd International Workshop Conferenceon Evolution Equations, Control Theory, and Biomathematics’, Han-sur-Lesse, Ph.Clement and G. Lumer (eds.), Lecture Notes in Pure and Applied Math. 155 (417-431), Marcel Dekker.

29. F. Neubrander (1994b), Laplace transforms and evolution equations. Lecture Notes,Louisiana State University.

30. J. Pruss (1993). Evolutionary Integral Equations and Applications. Verlag Birk-hauser, Basel - Boston - Berlin.

31

31. W. Rudin (1987). Real and Complex Analysis (3rd printing). McGraw-Hill Book Co.,New York.

32. H.H. Schaefer (1980). Topological Vector Spaces (4th printing). Springer, New York- Heidelberg - Berlin.

33. N. Sauer (1982). Linear evolution equations in two Banach spaces. Proc. Royal Soc.Edinburgh 91A, 287-303.

34. M. Sova (1979). The Laplace transform of analytic vector-valued functions. Casopispro pestovani matematiky 104, 267-280.

35. M. Sova (1980). Relations between real and complex properties of the Laplace trans-form. Casopis pro pestovani matematiky 105, 111-119.

36. P. Vieten and L. Weis (1993). Functions of bounded semivariation and classical inte-gral transforms. Preprint.

37. Yu-Cheng Shen (1947). The identical vanishing of the Laplace integral. Duke Math.J. 14, 967-973.

38. L. Weis (1993). The inversion of the Laplace transform in Lp(X)-spaces. In: Pro-ceedings of the Conference on Differential Equations in Banach Spaces, Bologna 1991;North Holland.

39. D.V. Widder (1936). A classification of generating functions. Trans. Amer. Math.Soc. 39, 244-298.

40. D.V. Widder (1941). The Laplace Transform. Princeton University Press, Princeton,New Jersey.

41. D.V. Widder (1971). An Introduction to Transform Theory. Academic Press, NewYork.

42. S. Zaidman (1960). Sur un theoreme de I. Miyadera concernant la representation desfunctions vectorielles par des integrales de Laplace. Tohoku Math. J. 12, 47-51.

Boris Baumer, Frank Neubrander, Dept. of Mathematics, Louisiana State University,Baton Rouge, LA, 70803, USA ;E-mail: baeumer@marais.math.lsu.edu and neubrand@marais.math.lsu.edu ;Phone : 504-388-1612 (office) ; 504-767-3415 (home); Fax: 504-388-4276.

32