On the asymptotic behaviour of a semigroup of linear operators

Transcription

1 On the asymptotic behaviour of a semigroup of linear operators J.M.A.M. van Neerven California Institute of Technology Alfred P. Sloan Laboratory of Mathematics Pasadena, CA 925, U.S.A. B. Straub 2 and L. Weis 2 Louisiana State University Department of Mathematics Baton Rouge, LA 783, U.S.A. Let T = {T (t} t be a C -semigroup on a Banach space X. In this paper, we study the relations between the abscissa ω L p(t of weak p-integrability of T ( p <, the abscissa ω p R (A of p-boundedness of the resolvent of the generator A of T ( p, and the growth bounds ω β (T, β, of T. Our main results are as follows. (i Let T be a C -semigroup on a B-convex Banach space such that the resolvent of its generator is uniformly bounded in the right half plane. Then ω ε (T < for some ε >. (ii Let T be a C -semigroup on L p whose resolvent is uniformly bounded in the right half plane. Then ω β (T < for all β > p p, p + p =. (iii Let p 2 and let T be a weakly L p -stable C -semigroup on a Banach space X. Then for all β > p we have ω β(t. Further, we give sufficient conditions in terms of ω q R (A for the existence of Lp -solutions and W,p -solutions ( p of the abstract Cauchy problem for a general class of operators A on X. 99 Mathematics Subject Classification: 47D3, 93D5 Keywords & phrases: C -semigroup, asymptotic stability, growth bound, fractional powers, Fourier type, B-convex, L p -solution, abstract Cauchy problem.. Introduction In this paper, we study the asymptotic behaviour of a C -semigroup T = {T (t} t on a complex Banach space X. For x X and n N, the growth bounds ω(x and ω n (T are defined by ω(x := inf{ω R : T (tx Me ωt for some M = M x and all t }; ω n (T := sup{ω(x : x D(A n }. Research supported by the Netherlands Organization for Scientific Research (NWO 2 Research supported by the Louisiana Education Quality Support Fund (LEQSF

2 2 Here, A is the generator of T and D(A is understood to be X. More generally, if the fractional powers of A are defined, for β we put ω β (T := sup{ω(x : x D(( A β }. Of particular interest are the uniform growth bound of T, that is ω (T, and the growth bound ω (T which characterizes the growth of the classical solutions of the corresponding abstract Cauchy problem. The growth bound ω (T equals the abscissa of improper convergence of the Laplace transform of T [Ne]: { ω (T = inf ω R : lim t t } e ωs T (sx ds exists for all x X. Similarly, ω (T is equal to the abscissa of absolute convergence of the Laplace transform of T; this is a special case of the Datko-Pazy theorem [Pa, Thm. 4.4.]. The spectral bound s(a is defined by s(a = sup{reλ : λ σ(a}. We always have s(a ω (T ω (T [Na, Cor. A.IV..5]; for positive C -semigroups on a Banach lattice there is the equality s(a = ω (T [Na, Thm. C.IV..3]. In general, however, both inequalities may be strict; see e.g. [Na, Ex. A.IV..6] or [Ne]. Therefore, in order to be able to say something about the asymptotic behaviour of T, it is not enough to know the location of the spectrum of A. For this reason, several other quantities have been studied. The abscissa ω L p(t ( p < of weak p-integrability of T is defined as { ω L p(t := inf ω R : e ωt x, T (tx p dt < for all x X and x X }. The abscissa ω p R (A ( p of p-boundedness of the resolvent R(λ, A = (λ A of A denotes the infimum of all ω R for which R(λ, Ax H p ({Reλ > ω}, X for all x X, that is, { ω p R {ω } } (A := inf R : sup R(t + is, Ax p ds : t > ω < for all x X ; for p = we take the sup-norm along the lines t+ir. We denote the resulting abscissa of uniform boundedness of R(λ, A by ω R (A rather than ωr (A. Similarly, the abscissa weak-ω p R (A ( p of weak p-boundedness of the resolvent denotes the infimum of all ω R for which x, R(λ, Ax H p ({Reλ > ω} for all x X and x X. It is well-known that for a C -semigroup on a Hilbert space we have ω R (A = ω (T = ω L p(t for all p <. The first equality is due to Gearhart [Ge]; an elegant short proof is given in [We]. The second equality is due to Huang and Kangsheng [HK] and, independently, Weiss [We]. In the Banach space case, several results are known about the relations between ω R (A and ω L (T on the one hand and the growth bounds ω n (T on the other hand: (i ω 2 (T ω R (A [Sl]; see also [Na, Thm. A.IV..9]; (ii ω (T ω L (T; this follows from [GVW, Prop..] in combination with [Na, Thm. A.IV..4]; (iii If X is B-convex, in particular if X is uniformly convex, then ω (T ω R (A [Wr]; (iv If X is a Banach lattice and T is positive, then s(a = ω (T = ω R (A [Na, p. 7]. It can happen that ω (T < ω R (A; an example can be found in [Wr]. Huang [Hu2] claimed that ω (T ω R (A = ω L (T holds for every C -semigroup on a Banach space X. His proofs,

3 however, depend on a lemma on the extension of H 2 -functions, to which we present a counterexample below. Therefore it remains an open question whether the relations ω (T ω R (A and/or ω R (A = ω L (T hold in general. In Section, we show that the inequalities ω (T ω L p(t and ω R (A ω L p(t hold for each p <. Section 2 gives some basic definitions concerning fractional powers of unbounded operators. Section 3 is devoted to some Paley-Wiener type lemmas concerning ω p R (A and weak-ωp R (A. The main result in Section 4 is as follows. If the resolvent of a C -semigroup T on X is uniformly bounded in the right half plane, then ω β (T < for all β > p p ( p + p = ; here p is the so-called Fourier type of the Banach space X. In particular, we somewhat improve Slemrod s theorem (statement (i above by showing that ω +ε (T < if the resolvent is uniformly bounded in the right half plane. Also, if T is a C -semigroup on L p, p <, with uniformly bounded resolvent in the right half plane, we obtain that ω β (T < for all β > p p, ( p + p =. It T is a positive C -semigroup on L p and the spectral bound s(a of its generator is negative, then the resolvent is automatically bounded in the right half plane. For this case, the third named author has recently proved [W] that s(a = ω (T. In Section 5 we study the abscissa weak-ω p R (A. It is shown that ω (A weak-ω p R (A for all p <. Also, if the resolvent is uniformly bounded in the right half plane, then x, T ( x L p (R + for all x D(A and x X and all 2 p <. Note that the cases p = in these two results would imply one of the claims of Huang. In the final Section 6, we extend some of the techniques of Section 3 in order to study the abstract Cauchy problem u (t = Au(t (t, u( = x, for a general class of operators A. Extending definitions of types of solutions given by Beals [Be] and Pazy [Pa], we discuss the existence of unique L p -solutions and W,p -solutions in relation to the abscissae ω q R (A. We conclude this introduction with a counterexample to Huang s lemma. We need the following terminology. For w R and p, let H p (w denote the Banach space of analytic functions in the open halfplane {Reλ > w} such that ( sup f(t + is p ds <. t>w We will write H p for H p (. Huang s lemma can be formulated as follows. Suppose δ > and f is an analytic function in the halfplane {Reλ > δ}. If moreover f H 2 (w for some w > and ( lim sup f(re iθ =, (. r θ π 2 then f H 2. The counterexample below was constructed by A. Poltoratski and I. Binder. 3 Example.. For t, let { e f (t = n, if t [n, n + e 2n ]; n =, 2,...;, otherwise. Then e ( εt f (t L (R + and e εt f (t L 2 (R + for each ε >, but f L 2 (R +. Let f be the Laplace transform of f, f(s = e st f (t dt.

4 4 Then f exists and is analytic in {Res > }, and, by the Paley-Wiener theorem, f H 2 (ε for all ε > but f H 2. We claim that (. holds. To this end, let η >. Then we use that ( { sup f(re iθ : θ π } 2, Re(reiθ > η = lim r by the general theory of H 2 functions (e.g., [Ho, Ch. 8]; also, ( { sup f(re iθ : θ π } 2, Re(reiθ η = lim r by the Riemann-Lebesgue lemma.. The abscissae ω L p(t Let T be a C -semigroup on a Banach space X and let p <. We say that T is weakly L p - stable if for each x X and x X we have x, T ( x L p (R +. We start with the observation that for a weakly L p -stable semigroup on a Banach space X, there exists a constant M such that ( x, T (tx p p dt M x x, for all x X, x X, (. cf. [We]. In fact, we have the following more general situation. Let X, Y, and Z be Banach spaces and let S : X Y Z be a separately continuous bilinear map. For x X define S x : Y Z, S x y := S(x, y. Then each S x is bounded by the continuity in the Y -variable. Using the continuity in the X-variable, it is easy to see that the map x S x is closed, and hence bounded by the closed graph theorem. It follows that S(x, y S x y K x y. Now let µ be a positive σ-finite Borel measure on a locally compact Hausdorff space Ω, let E be a Banach function space over (Ω, µ, and let X be a Banach space. Assume that S : Ω L(X is a mapping such that for each x X and x X, the function S x,x (ω := x, S(ωx, ω Ω, belongs to E. Consider the map S : X X E, defined by S(x, x := S x,x. We claim that this bilinear map is separately continuous. Indeed, fix x X. We will show that S x : X E, defined by S x x := S(x, x, is closed. To this end, let x n x in X and S x x n f in E. Since Cauchy sequences in Banach function spaces have pointwise a.e. convergent subsequences [Za], and since (S x x n (ω (S x x(ω for all ω, it follows that S x x = f, proving closedness. Therefore, each operator S x is bounded. Similarly, each operator S x : X E, S x x := S(x, x, is bounded. Therefore, S is separately continuous as claimed. From the above discussion we see that there is a constant K such that for all x X and x X, x, S( x E K x x. (.2 Applied to E = L p (R +, we obtain (.. The following lemma is an easy generalization of [GVW, Prop..] and [We, Prop. 2.2].

5 Lemma.. Let p < and suppose T is a weakly L p -stable C -semigroup on X. (i Let < p <. If R + = n E n, where (E n R + is a sequence of disjoint sets each of which is bounded, and if f L p (R + ( p + =, then for all x X the improper integral p f(tt (tx dt := lim f(tt (tx dt n n k= E k exists and is independent of the choice of (E n. Moreover, for all x X we have x, f(tt (tx dt = f(t x, T (tx dt, the second integral being in the Lebesgue sense. (ii Let p =. If f L (R + and lim t f(t =, then for all x X the improper integral τ f(tt (tx dt := lim τ f(tt (tx dt exists. (iii Let p <. For all x X and Reλ > we have λ ϱ(a and R(λ, Ax = e λt T (tx dt. 5 Proof: (i For bounded sets E R + define µ(e := f(tt (tx dt, the integral being Bochner. E For m n define E nm := m k=n E k and let α nm := E nm f(t p dt. Then we have lim n,m α nm =. Therefore, for x X be arbitrary, we have Hence, x, m µ(e k = x, µ(e nm = f(t x, T (tx dt E nm ( ( x, T (tx p p dt f(t p p dt M x x α E nm Enm k=n m µ(e k M x α k=n This proves that the series n µ(e n is convergent. Denoting the limit by µ(r +, by the dominated convergence theorem we have p nm. p n,m. x, µ(r + = n x, µ(e n = f(t x, T (tx dt. This proves the second formula and shows that µ(r + is independent of the particular choice of the (E n. (ii In this case we have the estimate τ τ f(tt (tx dt sup x τ τ ( f(t x, T (tx dt M x sup f(t. τ t τ (iii This can be proved either directly or by an analytic continuation argument: for each x, the map λ e λt T (tx dt is analytic for Reλ > and coincides with R(λ, Ax for Reλ > ω (T. ////

6 6 By arguing exactly as in [We], it follows from Lemma. (iii that ω R (A <. Also, we noted in the introduction that the abscissa of improper convergence of the Laplace transform of T coincides with ω (T. We thus recover the following result due to Weiss [We]: Theorem.2. Let T be a C -semigroup on a Banach space X and let p <. (i ω R (A ω L p(t; (ii ω (T ω L p(t. For certain values of p, part (ii will be improved later on (Theorem 5.5. Recall that a map f : (Ω, µ X is Pettis integrable if x, f( L (µ for all x X and for each measurable H Ω there is an element x H X such that x, x H = H x, f(ω dµ(ω, for all x X. If in Lemma. (i and (ii we replace R + by an arbitrary measurable subset of R +, the argument shows that in fact the function f(tt (tx is Pettis integrable. We have ω (T < if and only if the maps t f(tt (tx are Bochner integrable: Proposition.3. Let T be weakly L p -stable for some p <. Then ω (T < if and only if f(tt (tx is Bochner integrable for all x X and f L p (R + ( p + =. p Proof: We only have to prove the if part. Let x X be arbitrary and fixed and assume that f(tt (tx is Bochner integrable for all f L p (R +. We claim that t T (tx is in L p (R + ; X. If not, then T (tx p dt =. Define g n (t := T (tx χ [,n] (t. Then (g n is an unbounded sequence in L p (R +. By the uniform boundedness theorem, there is a φ L p (R + such that the scalar sequence ( n ( φ, g n n = φ(t T (tx dt n is unbounded. This contradicts the Bochner integrability of t φ(t T (tx. The proposition now follows from the Datko-Pazy theorem. //// We already observed that the growth bound ω (T coincides with the abscissa of improper convergence of the Laplace transform of T, and that ω (T coincides with the abscissa of absolute convergence of the Laplace transform of T. Therefore it is of some interest to consider the abscissa of Pettis integrability. For an arbitrary C -semigroup T on a Banach space X and p < we always have ω (T ω P ettis (T ω L p(t ω (T; the second inequality follows by applying Lemma. to the function f(t = e λt, λ >, and the first follows from the countable additivity of the indefinite Pettis integral, cf. [DU, Ch. 2]. Theorem.4. Let T be a positive C -semigroup on a Banach lattice E. Then the following assertions are equivalent: (i s(a < ; (ii ω (T < ; (iii T is weakly L -stable; (iv T is Pettis integrable; (v ω R (A <.

7 Proof: The equivalence of (i, (ii and (v was already mentioned in the introduction. Assume (i. By [Na, Thm. A.IV..4 and Thm. C.IV..3], for each x X there exists a R(x X such that τ lim T (tx dt = R(x. τ In particular, the integrals on the left hand are uniformly bounded with respect to τ, say by a constant M(x. Then for any τ >, x X and x X we have τ τ τ x, T (tx dt = x, T (tx dt x T (tx dt x M(x. Since τ is arbitrary, it follows that T is weakly L -stable. The implication (iii (v follows from Theorem.2. If T is Pettis integrable, by definition T is weakly L -stable. This gives (iv (iii. Finally, suppose T is weakly L -stable. Because of the equivalence (i (iii, also the semigroup defined by U(t := e εt T (t is weakly L -stable for some small ε >. Therefore, t T (tx = e εt U(tx is Pettis integrable by the observation preceding this theorem. //// If T is bounded and one of the equivalent hypotheses in the theorem is fulfilled, then obviously T is weakly L p -stable for all p <. The following example shows that a weakly L -stable semigroup can have strictly positive growth bound ω (T. In particular, T need not be bounded. This partially answers a question raised in [We]; note the contrast with the Hilbert space case. Also, the example shows that a bounded weakly L p -stable semigroup can have growth bound zero. Example.5. Let < p < and let X = L p (R + L (R +, e t dt. With the norm f := f Lp (R + + f L (R +,e t dt, X is a Banach lattice. Define T (tf(s := f(t + s. Then T is a positive C -semigroup with T (t = for all t and s(a = ; this is proved in [GVW]. Put S(t := e 2 t T (t. Then ω(t =, ω(s = 2, and by Theorem.4 and the remark after it, T is weakly L p -stable for all p < and S is weakly L -stable. As in [GVW], this example can be modified to give a counterexample on a reflexive Banach lattice Fractional powers of closed operators In this section we state some basic facts and definitions which we will use in the sequel. The two lemmas below are probably not new, but in view of the many different definitions of fractional powers in the literature and the difficulty to find references for the lemmas, we include their proofs. A closed, densely defined linear operator A on a Banach space X is called sectorial if (, ϱ(a and there is a constant K such that R(λ, A K (λ + for all λ >. (2. For a sectorial operator A, fractional powers of A are defined. If β = we define ( A β as usual and for < β < we use the (real representation of the bounded operator ( A β given by ( A β x := sin βπ π r β R(r, Ax dr (x X.

8 8 For any n Z, the fractional power ( A n β is then defined by ( A n β x = ( A n ( A β x for x D(( A n β = {x X : ( A β x D(A n }. The operators ( A β (β R are closed, injective, and satisfy the semigroup property ( A α ( A β = ( A α+β if α β or α, β or α, β. In particular, we have that ( A β ( A β x = x for every β R and x D(( A β ; in other words, ( A β is the inverse of the operator ( A β. We have inclusions D(( A β D(( A α if β α. For details, we refer to [Ko]. If the resolvent satisfies (2., then there exist constants d, C > and < ϕ < π such that R(λ, A C( + λ for all λ in the sector {λ C : arg λ ϕ} {λ C : λ d}; this follows from an easy resolvent expansion argument. Hence, for every β > and x X, the integral ( λ β R(λ, Ax dλ, (2.2 Γ where µ β is defined in terms of the principal branch of the logarithm, exists as a Bochner integral. Here, Γ = Γ(ϕ, d = Γ ( (ϕ, d Γ (2 (ϕ, d Γ (3 (ϕ, d denotes the upwards oriented curve defined by Γ ( (ϕ, d = {λ C : λ d, arg λ = ϕ}; Γ (2 (ϕ, d = {λ C : λ = d, arg λ > ϕ}; Γ (3 (ϕ, d = {λ C : λ d, arg λ = ϕ}. Note that we use the argument function with values in ( π, π]. By Cauchy s theorem, the integral (2.2 is equal to the curve integral over Γ( ϕ, d for any < ϕ ϕ and < d d. Letting the curve Γ collaps into the real line, we obtain the following. Lemma 2.. For every β > and x X, the integral (2.2 equals ( A β x. Proof: If β N, the statement follows by Cauchy s integral formula applied to the right half plane. In the case that < β <, consider for < θ ϕ the curve Γ(θ = {re iθ : r > }. Using that ( λ β R(λ, Ax dλ =, lim d Γ (2 (θ,d it follows from Cauchy s theorem that ( λ β R(λ, Ax dλ = Γ = Γ(θ sin βπ = π ( λ β R(λ, Ax dλ ( λ β R(λ, Ax dλ Γ( θ (re i(θ π β R(re iθ, Ax e iθ dr (re i( θ+π β R(re iθ, Ax e iθ dr r β (e iπβ e iπβ R(r, Ax dr as θ r β R(r, Ax dr = ( A β x. (2.3

9 The convergence is a consequence of Lebesgue s convergence theorem, the integrands in the integrals being dominated by the function r Cr β ( + r. For β >, β N, we use that 9 R(λ, Ax ( λ [β] = R(λ, A( A [β] x + [β] k= ( A k x ( λ [β] k, where [β] denotes the unique integer such that [β] β < [β] +. The integrals in ( λ β R(λ, Ax dλ Γ = Γ ( λ β+[β] R(λ, A( A [β] x dλ + [β] k= ( λ β+k dλ ( A k x Γ exist as Bochner integrals. Cauchy s theorem applied to the left half plane yields that ( λ β+k dλ = for every k [β]. Γ Since < β [β] <, it follows by (2.3 that ( λ β R(λ, Ax dλ = ( A [β] ( A β+[β] x = ( A β x. //// Γ Under certain conditions, the complex representation (2.2 of ( A β x can be changed to a curve integral over a line. Lemma 2.2. Let a < and let A be a sectorial operator. Assume that the resolvent of A is uniformly bounded in {λ C : Reλ a}. Then for all β > and x D(A we have ( A β x = ( λ β R(λ, Ax dλ. Proof: For all λ {Reλ a}, λ, we have R(λ, Ax = Reλ=a λ [R(λ, A(λ Ax + R(λ, AAx] ( x + M Ax, (2.4 λ where M = sup{ R(λ, A : Reλ a}. Hence the lemma follows from Lemma 2. and Cauchy s theorem. //// 3. The abscissae ω p R (A and Paley-Wiener type lemmas for semigroups Let A be a closed linear operator on a Banach space X, let p and let w R. We say that the resolvent of A belongs to H p (w if {Reλ > w} ϱ(a and for each x X we have { } sup R(t + is, Ax p ds : t > w < ;

10 in case p =, as usual, we take the sup-norm along the lines t + ir. It is not hard to see, cf. the argument at the beginning of Section, that for each t > w there exists a constant M t such that ( R(t + is, Ax p p ds M t x, for all x X. (3. Moreover, for all t > w, B t := sup{ R(λ, A : Reλ t} <. (3.2 In fact, for functions f H p (w, X and t > w we have the estimate sup f(t + is s R f (π(t w Hp (w,x, w < w < t (3.3 p cf. [HP, Thm ]. Applying this to f(λ = R(λ, Ax, by the uniform boundedness theorem we obtain (3.2. We define ω p R (A to be the abscissa ω p R (A = inf{w R : A has H p (w-resolvent}. If the resolvent fails to be in H p (w for all w R, we put ω p R (A =. As before, we write ω R (A instead of ωr (A for the abscissa of uniform boundedness of the resolvent of A. By (3.2, ω R (A ω p R (A for all p. We say that the resolvent of A belongs to H p (w weakly if for all x X and x X the function λ x, R(λ, Ax belongs to H p (w. We denote by weak-ω p R (A the corresponding abscissa, i.e. the infimum of all w such that R(λ, A belongs to H p (w weakly. If R(λ, A belongs to H p (w weakly and t > w, there is a constant M t such that ( x, R(t + is, Ax p p ds M t x x, for all x X, x X. (3.4 This follows from the observations at the beginning of Section. Moreover, for all t > w we have B t := sup{ R(λ, A : Reλ t} <. In particular, ω R (A = weak-ω R (A weak-ω p R (A for all p. Our next aim is to prove that the functions ω p R (A and weak-ωp R (A are non-increasing in p. Let L (R, X denote the closure in L (R, X of the set of all X-valued step functions of the form N n= x nχ Fn, where x n X and F n R is measurable and has finite Lebesgue measure. Clearly, C (R, X L (R, X. Also, it is well-known [HP] that the restriction of an f H p (w, X to each line t + ir belongs to L (R, X ( p <, t > w. The basic fact we will use is that for all p < and < θ <, the complex interpolation method gives ( L p (R, X, L (R, X = [θ] Lq (R, X, (3.5 where /q = ( θ/p [BL, Thm. 5..2]. Proposition 3.. The functions ω p R (A and weak-ωp R (A are non-increasing in p. Moreover, for all p we have either ω p R (A = or ωp R (A = ω R(A.

11 Proof: Fix p q. We claim that H p (w, X H q (w + ε, X for all ε >. By (3.2, H p (w, X H (w + ε, X for all ε >, so we may assume that p < q <. Let δ ε > be arbitrary and let < θ < be such that /q = ( θ/p. Define the linear operators T (p δ Then T (p δ : H p (w, X L p (R, X and T ( δ : H p (w, X L (R, X by f f w+δ+ir. and T ( δ (πδ p by (3.3. By complex interpolation, it follows that f w+δ+ir Lq (R,X (πδ θ p f Hp (R,X (πε q p pq f Hp (R,X. This proves that H p (w, X H q (w + ε; X. Applying this to the functions λ R(λ, Ax shows that ω q R (A ωp R (A. So far, we have proved that p ω p R (A is non-increasing. Now suppose that for some fixed p we have ω p R (A <. By (3.2, we see that ω R(A ω p R (A; we will prove that the converse inequality also holds. Let b > ω p R (A, let ω R(A t b and let t t b be arbitrary and fixed. Let B t be defined by (3.2. By the resolvent equation, for all s R we have R(t + is, A = [I + (b tr(t + is, A]R(b + is, A ( + B t (b t R(b + is, A. Consequently, ( p ( ( + B t (b t R(b + is, Ax p p ds. R(t + is, Ax p ds Since t t b is arbitrary and the resolvent is in H p (b, this shows that the resolvent is in H p (t. Finally, we have to prove the inequality weak-ω q R (A weak-ωp R (A. This is done similarly as in the strong case, using the spaces H p (w and the functions λ x, R(λ, Ax. //// Recall that a Banach space X has Fourier type p for some p 2, if the Fourier transform F extends to a bounded operator p + p F : L p (R, X L p (R, X, =, between the Lebesgue-Bochner spaces L p (R, X and L p (R, X. In other words, it is assumed that the vector-valued Hausdorff-Young theorem holds for the exponent p. Every Banach space has Fourier type but only a Hilbert space has Fourier type 2 [Kw]. The Banach space L r (Ω, µ has Fourier type min{r, r } [Pe]. Every B-convex (in particular every uniformly convex space has a Fourier type for some p with < p 2 [Bo]. Recall that a complex Banach space is B-convex if it does not contain the spaces ln uniformly, or equivalently, if it is of type p for some < p 2. For more information we refer to [Pi]. From (3.5 and the Riemann-Lebesgue Lemma one sees that if a Banach space has Fourier type p for some p 2, then it has Fourier type q for all q p. Now let A be a sectorial operator on X with ω R (A <, and fix ω R (A < a < and x D(A. By (2.4, for all β > the integral (Sx(t := e λt ( λ β R(λ, Ax dλ, (3.6 Reλ=a where µ β is defined in terms of the principle branch of the logarithm, exists as a Bochner integral. By Cauchy s theorem, the integral does not depend on the value of a. Moreover, (2.4 shows that there is a constant C a, which only depends on A and a, such that We also define (Sx(t C a e at x D(A. (3.7 (S a x(t := e at (Sx(t. (3.8 Lemma 3.2. Assume X has Fourier type p for some p 2 and assume A is a sectorial operator with ω q R (A < for some p q. Then ω R(A <, and for all β > p q and a > ω q R (A, formula (3.8 extends to a bounded linear operator S a : X L p (R +, X.

12 2 Proof: Fix x D(A. Since ω R (A ω q R (A <, the definition of (Sx(t makes sense. Without loss of generality, we may assume ω q R (A < a <. Then (Sx(t = eat e ist g(s ds = eat 2π 2π F(g(t where g(s = ( a + is β R(a is, Ax. Since r := p q < β, we have by (3., ( g(s p ds p ( a is βr r ds ( R(a is, Ax q ds q C x. Since X has Fourier type p, for all x D(A we get e a( (Sx( L p (R +,X C x. Therefore, S a extends to a bounded linear operator S a : X L p (R +, X. //// It is easy to see that a generator whose resolvent is uniformly bounded in the open right half plane (or equivalently, ω R (A < ; cf. Lemma 4. is sectorial. In that case, the integral (3.6 can be expressed in terms of T: Lemma 3.3. Let A be the generator of a C -semigroup T with ω R (A <. Then for all β >, ω R (A < a < and x D(A we have (Sx(t = T (t( A β x for all t. Proof: First let x D(A 2. By Cauchy s theorem, e λt ( λ β dλ =. (3.9 Reλ=a Using this and the equation λ AR(λ, Ax = R(λ, Ax x λ, we see that (Sx(t = e λt ( λ β R(λ, A( Ax dλ. Reλ=a Since Ax D(A, we may differentiate under the integral sign and obtain d dt (Sx(t = λe λt ( λ β R(λ, A( Ax dλ Reλ=a ( = A e λt ( λ β R(λ, A( Ax dλ = A((Sx(t. Reλ=a To obtain the second identity, we used λ = (λ A + A, (3.9, and the closedness of A. Also, (Sx( = ( A β x by Lemma 2.2. We have shown that (Sx( is the solution to the abstract Cauchy problem d u(t = Au(t dt u( = ( A β x. By the uniqueness of classical solutions, we must have (Sx(t = T (t( A β x. Next, let x D(A and choose a sequence (x n D(A 2 such that x n x in the graph norm of D(A. Let t be fixed. By (3.7, (Sx n (t (Sx(t in X. The boundedness of ( A β implies that also (Sx n (t = T (t( A β x n T (t( A β x, and the lemma follows. ////

13 The above two lemmas will now be used to derive two Paley-Wiener type results, giving L p - stability of certain orbits of a semigroup in terms of H r -abscissae of its Laplace transform, i.e. of the resolvent. Lemma 3.4. Suppose X has Fourier type p for some p 2 and let p q. Furthermore, assume A is the generator of a C -semigroup T satisfying ω q R (A <. Then for all β > p q, a > ω q R (A, and x D(( Aβ, the map t e at T (tx belongs to L p (R +, X. Proof: Without loss of generality, assume ω q R (A < a <. Let y D(( A β be arbitrary and put x := ( A β y. Choose a sequence x n x with x n D(A for each n. By Lemma 3.2, S a x n S a x in L p (R +, X. By taking a subsequence if necessary, we may assume that (S a x n (t (S a x(t for almost all t. For any such t, by Lemma 3.3 we have e at T (t( A β x = lim n e at T (t( A β x n = lim n (S ax n (t = (S a x(t. Therefore, e at T ( y is equal a.e. to the function (S a x( L p (R +, X. //// We will now give the weak analogue of Lemma 3.4. Lemma 3.5. Let p 2 and p q. Furthermore, let A be the generator of a C - semigroup T satisfying weak-ω q R (A <. Then for all β > p q, a > weak-ωq R (A, x D(( Aβ and x X, the map t e at x, T (tx belongs to L p (R +. Proof: Without loss of generality, assume weak-ω q R (A < a <. Using (3.4 and the Hausdorff-Young theorem, the argument of Lemma 3.2 can be modified to show that e a( x, (Sx( L p (R + C x x, x D(A, x X. (3. Now fix x X. Define S a,x : D(A L p (R + by (S a,x x(t := x, (S a x(t. It follows from (3. that S a,x extends to a bounded operator S a,x : X L p (R +. Let y D(( A β and x X be arbitrary and put x := ( A β y. Choose a sequence x n x with x n D(A for each n. Then S a,x x n S a,x x in L p (R +. By taking a subsequence if necessary, we may assume that (S a,x x n (t (S a,x x(t for almost all t. For any such t, we have e at x, T (t( A β x = lim n e at x, T (t( A β x n = lim n x, (S a x n (t = lim n (S a,x x n(t = (S a,x x(t. Therefore, the function e a( x, T ( y is equal a.e. to the function S a,x x L p (R +. //// 3 4. The abscissae ω p R (A and stability In this section, we will apply Lemma 3.4 to obtain stability results for C -semigroups. The first lemma, along with its proof, is taken from [Hu]. Lemma 4.. Let A be a closed operator on a Banach space X. If the resolvent of A is uniformly bounded in the open right half plane, then ω R (A <.

14 4 Proof: Let R(λ, A M for all Reλ >. Put σ := (2M. Then for all ν R and µ < λ such that λ µ < σ, the series R µ+iν := (λ µ n R(λ + iν, A n+ (4. n= converges absolutely. Therefore the function z R z is a analytic operator valued function for Rez > σ, and it coincides with R(z, A whenever z ϱ(a. We claim that actually R z = R(z, A for all Rez > σ. Suppose the contrary. Then there is a z σ(a {Rez > σ}. But if we let z n z inside ϱ(a {Rez > σ}, we see that R(z n, A, hence also R zn, tends to, a contradicition to the analyticity of z R z. Finally, the uniform boundedness of the resolvent on the half plane Reλ > σ follows immediately from (4.. //// The next lemma gives a sufficient condition in order that ω q R (A < for certain values of q. Lemma 4.2. Suppose X has Fourier type p for some p 2, and suppose A is the generator of a C -semigroup T on X. If the resolvent of A is uniformly bounded in the open right half plane, then ω p R (A = ω R(A < ( p + p =. Proof: First we note that by Lemma 4., ω R (A <. Choose b so that T (t Ce b 2 t for some C < and all t >. Fix t b. Since X has Fourier type p and R(t + is, Ax = e isξ ( e tξ T (ξx dξ, we get ( R(t + is, Ax p ds p F Lp (R,X L p (R,X ( F Lp (R,X L p (R,X ( e tξ T (ξx p p dξ (4.2 e 2 bpξ dξ p x. This shows that ω p R (A b, and therefore ωp R (A = ω R(A by Proposition 3.. //// Before proceeding with the main result of this section, let us observe the following simple consequence of this lemma and Proposition 3.. Corollary 4.3. If A generates a C -semigroup on a Banach space with Fourier type p for some p 2, then for all p q one has ω p R (A = weak ωp R (A = ωq R (A = weak ωq R (A = ω R (A. If T is a C -semigroup with generator A on a Banach space X whose resolvent R(λ, A is uniformly bounded in the open right half plane, then by the Hille-Yosida theorem A is a densely defined sectorial operator. Consequently, the fractional powers of A are defined. Recall from the introduction the notation ω β (T = sup{ω(x : x D(( A β }. Theorem 4.4. Assume that X has Fourier type p for some p 2, and that A generates a C - semigroup T whose resolvent is uniformly bounded in the right half plane. Then for all β > p p one has ω β (T <.

15 Proof: By Lemma 4., ω R (A <. Lemma 4.2 yields that ω p R (A = ω R(A <. Now fix an arbitrary x D(( A β. Applying Lemma 3.4 with q = p, for β > p p and a > ω R (A we get that e at T ( x is equal a.e. to (S a (( A β x( L p (R +, X. By a standard argument (cf. [Pa, Thm. 4.4.], this implies that e a( T ( x is bounded. Hence, ω(x a. //// Corollary 4.5. Let A be the generator of a C -semigroup on a Banach space X. If the resolvent of A is uniformly bounded in the right half plane, then ω +ε (T < for every ε >. Indeed, this follows from Lemma 4., Theorem 4.4 and the fact that every Banach space has Fourier type. In fact, the corollary is an immediate consequence of (3.7 and Lemmas 3.3 and 4.. This result can be generalized considerably, cf. [NS]. As we observed after Lemma., a weakly L p -stable semigroup has uniformly bounded resolvent in a right half plane {Re λ > ε}. Combining this with Corollary 4.5 it follows that ω +ε (T < for all ε > if T is a weakly L p -stable C -semigroup for some p <. Theorem 4.4 implies a refinement of a stability result of Wrobel [Wr]. He proved that ω (T < for C -semigroups on B-convex Banach spaces whose generators A have uniformly bounded resolvents in the right half plane. Theorem 4.6. Let A be the generator of a C -semigroups on a B-convex Banach space X. If the resolvent of A is uniformly bounded in the right half plane, then there exists an ε > such that ω ε (T <. This follows from the fact that X has Fourier type p for some < p 2 and Theorem 4.4. Theorem 4.7. Let A be the generator of a positive C -semigroup on a space L p, p <. Then s(a = ω q R (A for all max{p, p } q. If s(a <, then for all β > p p we have ω β (T <. This follows from the facts that L p is of Fourier type min{p, p } and that for positive semigroups one knows that s(a = ω R (A. Of course, Theorem 4.7 is true for non-positive semigroups as well, provided we replace s(a by ω R (A Stability of weakly L p -stable semigroups In this section we will apply Lemma 3.5 to obtain stability results for C -semigroups. Theorem 5.. Let T be a C -semigroup on a Banach space X. (i If weak-ω q R (A < for some q, then for all β > q we have ω β(t weakω q R (A <. In particular, ω (T weak-ω q R (A for all q <. (ii If the resolvent is uniformly bounded in the open right half plane and if 2 q <, then for all β > q, x D(( Aβ and x X, the map x, T ( x belongs to L q (R +. In particular, if x D(A, then for all x X we have x, T ( x L q (R + for all 2 q <. Proof: (i For all weak-ω q R (A < a <, Lemma 3.5 (with p = shows that the function x, e a( T ( x is bounded for all x D(( A β and x X. Therefore, ω β (A a by the uniform boundedness theorem. (ii Apply Lemma 3.5 (with q =. //// We will now prepare some lemmas for a somewhat less trivial application of Lemma 3.5.

16 6 Lemma 5.2. Let p 2 and let T be a weakly L p -stable C -semigroup on X. Then the resolvent belongs to H p weakly. Proof: Let t > and s R be arbitrary. Let x X and x X be arbitrary. By Lemma. (iii we have x, R(t + is, Ax = e (e isξ tξ x, T (ξx dξ = (Fg(s, (5. where F is the Fourier transform and g(ξ = χ R + (ξe tξ x, T (ξx. Note that a priori we obtain the integral in (5. as an improper integral, but the conditions on T and t ensure that the integral is actually a Lebesgue integral. Let K = K x,x := x, T ( x Lp (R +. By the Hausdorff-Young theorem, there is a constant C p such that x, R(t + i(, Ax L p (R C p g( L p (R C pk. Since t > is arbitrary, the lemma follows. //// Lemma 5.3. [Ko, Thm. 6.4] Let A be a sectorial operator. For all ω and β R we have D(( A β = D(( (A ω β. For our purposes, it is in fact sufficient to know that for all ε >, D(( A β+ε D(( (A ω β. Lemma 5.4. Let p 2, and let A be the generator of a weakly L q -stable C -semigroup T for some q 2. Then for all β > p q, ( q + q =, x D(( A β, x X, and ε >, the map t e εt x, T (tx belongs to L p (R +. Proof: Let x D(( A β. By Lemma 5.2, the resolvent R(λ, A belongs to H q weakly, so R(λ, A 2ε belongs to H q ( 2ε weakly. By Lemma 5.3, x D(( (A 2ε β. Therefore we can apply Lemma 3.5 to the semigroup T 2ε (t := e 2εt T (t and a = ε and find that t e εt x, T (tx belongs to L p (R +. //// Now we are in a position to prove the main result of this section. Theorem 5.5. Let T be a C -semigroup which is weakly L q -stable for some q 2. Then for all β > q we have ω β(t. Proof: An application of Lemma 5.4, with p =, shows that t e εt x, T (tx is bounded for all x D(( A β, x X, and ε >. The theorem follows from the uniform boundedness theorem. //// 6. Existence of L p -solutions In this section we apply the ideas of Section 3 to the abstract Cauchy problem. Let A be a sectorial operator on a Banach space X of Fourier type p for some p 2. Assume that there is a q with p q such that ω q R (A <. Let β > p q and ωq R (A < a <, and consider the map S a of Lemma 3.2.

17 Lemma 6.. and For all x X t e as (S a x(s ds D(A for almost all t >, (6. ( t e at (S a x(t = A e as (S a x(s ds Proof: Fix x D(A. We have y t := t e as (S a x(s ds = t 7 + ( A β x for almost all t >. (6.2 (Sx(s ds = Reλ=a λ (eλt ( λ β R(λ, Ax dλ. Since λ AR(λ, Ax = R(λ, Ax x λ, the integrals in z t := Reλ=a λ (eλt ( λ β AR(λ, Ax dλ = e λt ( λ β R(λ, Ax dλ ( λ β R(λ, Ax dλ Reλ=a Reλ=a + (e λt ( λ β xdλ Reλ=a exist as Bochner integrals. The closedness of A implies that y t D(A and Ay t = z t. The last integral in (6.3 is zero by Cauchy s theorem. Then (6.3 and Lemma 2.2 yield that ( t (Sx(t = A (Sx(s ds + ( λ β R(λ, Ax dλ Reλ=a ( t = A e as (S a x(s ds + ( A β x. This proves the lemma in case x D(A. For an arbitrary x X, choose x n D(A with x n x in X. Using Lemma 3.2, we see that that (S a x n (S a x in the norm of L p (R +, X and therefore, for all t >, t e as (S a x n (s ds t e as (S a x(s ds. By taking a subsequence if necessary, we may assume that (S a x n (t (S a x(t almost everywhere. Hence, for almost all t, e at (S a x n (t ( A β x n e at (S a x(t ( A β x. Since (6. and (6.2 hold for all x n and A is closed, we see that (6. and (6.2 hold for x. //// Consider the abstract Cauchy problem (ACP u (t = Au(t (t, u( = x. (a A function u : R + X is called a L p w -solution of (ACP if e w( u( L p (R +, X and ( t u(t = A u(s ds + x for almost all t > and u( = x. (6.3

18 8 We say that u is a L p -solution if it is a L p w-solution for some w R. (b The function u : R + X is called a Ww,p-solution of (ACP if e w( u( W,p (R +, X and u (t = Au(t for almost all t > and u( = x. By a W,p -solution, we mean a Ww,p -solution for some w R. Definition (a says essentially that a L p -solution is a mild solution, but a L p -solution is not necessarily continuous. By part (b, it follows in particular that a W,p -solution is a strong solution in the sense of Pazy, see [Pa, p. 9]. The name L p -solution is motivated by the following notion of Beals [Be]. (c u : R + X is a L 2 -solution in the sense of Beals if u( is locally L 2, and for each y D(A the function y, u( is equal a.e. to a locally absolutely continuous function u y (t with u y ( = y d, x, dt u y (t = A y, u y (t. Proposition 6.2. For each q 2, a L q -solution u of (ACP is a L 2 -solution in the sense of Beals. Proof: Let u be a L q -solution of (ACP. Clearly, u( is locally L 2. Let y D(A and define u y (t := A y, t u(s ds + y, x. Then u y ( = y, x, u y ( is locally absolutely continuous with d dt u y (t = A y, u(t, and u y (t = y, u(t a.e. //// Theorem 6.3. Let A be a sectorial operator on a Banach space X of Fourier type p for some p 2. Let p q and assume that ω q R (A <. Then for all w > ωq R (A and β > p q the following assertions hold. (a (ACP has a L p w -solution for every x D(( A β. (b (ACP has a Ww,p -solution for every x D(( A β+. Proof: (a Fix x D(( A β. By scaling A, we may assume that ω q R (A < w < ; after doing so, Lemma 5.3 guarantees that still x D(( A β. Put y = ( A β x and u(t = e wt (S w y(t where S w is the map of Lemma 3.2. Then e w( u( L p (R +, X and ( t ( t u(t = e wt (S w y(t = A e ws (S w y(s ds + ( A β y = A u(s ds + x a.e. (b Fix x D(( A β+. After rescaling A as in (i we have x D(( A β+. Put y = ( A β+ x and u(t = t ews (S w y(s ds + x. Then e wt d dt u(t = (S wy(t belongs to L p (R +, X. It follows easily from (3.6 that A S w = S w A. Also, since x = A ( A β y, it follows from ((6.2 that e wt u(t = A (S w y(t = (S w (A y(t belongs to L p (R +, X. It follows that also d dt e wt u(t belongs to L p (R +, X. Furthermore, u( = x and, since ( A β y = Ax, //// ( d t dt u(t = e wt (S w y(t = A e ws (S w y(s ds + ( A β y ( t = A e ws (S w y(s ds + x = Au(t a.e.

19 If X has Fourier type p, then ω p R (A ω (T, where p + p =. Therefore, the condition ω q R (A < is automatically satisfied for all p q. Beals [Be] proves the following. If A is a densely defined closed operator on a Hilbert space H, such that the resolvent is uniformly bounded in some right half plane, then for each x D(A, the problem (ACP has a unique L 2 -solution in the sense of Beals. If in addition we assume that A is sectorial we can improve this as follows: Corollary 6.4. Let A be a sectorial operator on a B-convex Banach space X. Then there exists an ε > such that for each x D(( A ε, the problem (ACP has a unique L 2 -solution in the sense of Beals. Proof: As we observed at the beginning of this section, X is of Fourier type p for some < p 2. Therefore the existence assertions follow by combining the previous two results. Uniqueness of L 2 -solutions is proved in [Be]. //// Corollary 6.5. Let A be a sectorial operator on a Hilbert space H. Then for each β > 2 and x D(( A β, the problem (ACP has a unique L 2 -solution in the sense of Beals. Moreover, if ω q R < for some q 2, then this solution is globally L2. Proof: We only need to prove the globality assertion, which follows from Theorem 6.3. //// Acknowledgement - We would like to thank George Weiss for some valuable comments and Aleksei Poltoratski and Ilia Binder for pointing out to us Example References [Be] R. Beals, On the abstract Cauchy problem, J. Funct. Anal. (972, [BL] J. Bergh and J. Löfström, Interpolation Spaces, An Introduction, Grundlehren 223, Springer- Verlag, Berlin-Heidelberg-New York (976. [Bo] J. Bourgain, Vector-valued Hausdorff-inequalities and applications, Springer Lect. Notes in Math. 37 (986/87, [DU] J. Diestel and J.J. Uhl, Vector Measures, Math. Surveys nr. 5, Amer. Math. Soc., Providence, R.I. (977. [Ge] L. Gearhart, Spectral theory for contraction semigroups on Hilbert spaces, Trans. Am. Math. Soc. 236 (978, [GVW] G. Greiner, J. Voigt, and M. Wolff, On the spectral bound of the generator of semigroups of positive operators, J. Operator Th. 5 (98, [HP] E. Hille and R.S. Phillips, Functional Analysis and Semi-groups, Amer. Math. Soc. Colloq. Publ., vol. 3, Amer. Math. Soc., rev. ed., Providence, R.I. (957. [Ho] K. Hoffman, Banach Spaces of Analytic Functions, Englewood Cliffs, New York (962. [Hu] F. Huang, Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces, Ann. Diff. Eq. (985, [Hu2] F. Huang, Exponential stability of linear systems in Banach spaces, Chin. Ann. of Math. B (989,

20 2 [HK] F. Huang and L. Kangsheng, A problem of exponential stability for linear dynamical systems in Hilbert spaces, Kexue Tongbao 33 (988, [Ko] H. Komatsu, Fractional powers of operators, Pacific J. Math. 9 (966, [Kw] S. Kwapien, Isomorphic characterizations of inner product spaces by orthogonal series with vector valued coefficients, Studia Math. 44 (972, [Na] R. Nagel (ed., One-parameter Semigroups of Positive Operators, Springer Lect. Math. 84 (986. Notes in [NS] J.M.A.M. van Neerven and B. Straub, On the existence and growth of classical solutions of the abstract Cauchy problem for a generator of an α-times integrated semigroup, submitted. [Ne] F. Neubrander, Laplace transform and asymptotic behavior of strongly continuous semigroups, Houston Math. J. 2 (986, [Pa] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, Berlin-Heidelberg-New York-Tokyo (983. [Pe] J. Peetre, Sur la transformation de Fourier des fonctions à valeurs vectorielles, Rend. Sem. Mat. Univ. Padova 42 (969, [Pi] G. Pisier, Factorisation of Linear Operators and Geometry of Banach Spaces, CBMS, Regional Conference Series, No. 6, AMS, Providence (986. [Sl] M. Slemrod, Asymptotic behavior of C -semigroups as determined by the spectrum of the generator, Indiana Univ. Math. J. 25 (976, [W] L. Weis, Stability of positive semigroups on L p (µ, to appear in: Proc. Am. Math. Soc. [We] G. Weiss, Weak L p -stability of a linear semigroup on a Hilbert space implies exponential stability, J. Diff. Eq. 76 (988, [Wr] V. Wrobel, Asymptotic behavior of C -semigroups in B-convex spaces, Indiana Univ. Math. J. 38 (989, -4. [Za] A.C. Zaanen, Integration, 2nd ed., North Holland (967.