Operators and the norm continuity problem for semigroups

OPERATORS L1 (R+ ) → X AND THE NORM CONTINUITY PROBLEM FOR SEMIGROUPS RALPH CHILL AND YURI TOMILOV Abstract. We present a new method for constructing C0 -semigroups for which properties of the resolvent of the generator and continuity properties in operator topology are controlled simultaneously. It allows us to show that a) there exists a C0 -semigroup continuous in operator topology for no t ∈ [0, 1] such that the resolvent of its generator has a logarithmic decay at infinity along vertical lines; b) there exists a C0 -semigroup continuous in operator topology for no t ∈ R+ such that the resolvent of its generator has a decay along vertical lines arbitrarily close to a logarithmic one. These examples rule out any possibility of characterizing norm-continuity of semigroups on arbitrary Banach spaces in terms of resolvent decay. 1. Introduction The study of continuity properties of C0 -semigroups (T (t))t≥0 on a Banach space X in the uniform operator topology of L(X) (norm-continuity) has been initiated in [28] and attracted a considerable attention over the last decades; see in particular [2], [3], [4], [15], [16], [27], [30], [37], [43], [44]. The classes of immediately norm-continuous semigroups, of eventually normcontinuous semigroups, and of asymptotically norm-continuous semigroups (or, equivalently, semigroups norm continuous at infinity) emerged and were studied at depth during this period. The interest in these classes comes mainly from the fact that a condition of norm continuity of a semigroup implies a variant of the spectral mapping theorem, and thus asymptotic properties of a semigroup are essentially determined by the spectrum of the generator. One of the main issues in the study of norm-continuity is to characterize these classes in terms of the resolvent of the semigroup generator (or in other a priori terms). In particular, the so called norm-continuity problem for C0 -semigroups attributed to A. Pazy was a focus for a relevant research during the last two decades. Given a C0 -semigroup (T (t))t≥0 on a Banach space X, with generator A, the problem is to determine whether the resolvent decay condition (1.1) lim kR(ω + iβ, A)k = 0 for some ω ∈ R |β|→∞ Date: March 8, 2008. 2000 Mathematics Subject Classification. Primary 47D03; Secondary 44A10, 46G10, 46J25. Key words and phrases. C0 semigroup, norm continuity, resolvent, Banach algebra homomorphism, Laplace transform. Supported by the Marie Curie ”Transfer of Knowledge” programme, project ”TODEQ”. This research was started during a Research in Pair stay at the Mathematisches Forschungsinstitut Oberwolfach. Supports are gratefully acknowledged. 1 2 RALPH CHILL AND YURI TOMILOV implies that the semigroup is immediately norm-continuous, that is, norm-continuous for t > 0. The decay condition (1.1) is certainly necessary for immediate normcontinuity, by the fact that the resolvent of the generator is the Laplace transform of the semigroup, and by a simple application of the Lemma of RiemannLebesgue. Hence, the question is whether condition (1.1) characterizes immediate norm-continuity. The resolvent decay condition (1.1) does characterize immediate norm continuity if the underlying Banach space is a Hilbert space [44], [15], [43], [1, Theorem 3.13.2], or if it is an Lp space and the semigroup is positive, [27]. Only very recently, T. Matrai [37] constructed a counterexample showing that the answer to the norm continuity problem is negative in general. The generator in his example is an infinite direct sum of Jordan blocks on finite dimensional spaces. The infinite sum is equipped with an appropriate norm and the resulting Banach space is reflexive. This kind of counterexample going back to [45] has been used in the spectral theory of semigroups to show the failure of the spectral mapping theorems or certain relationships between semigroup growth bounds, see for example [1], [18]. We point out that the resolvent decay condition (1.1) implies that the resolvent exists and is uniformly bounded in a domain of the form Σϕ := {λ ∈ C : Re λ > −ϕ(|Im λ|)}, where ϕ ∈ C(R+ ) satisfies limβ→∞ ϕ(β) = ∞. It is known that the existence of the resolvent and its uniform boundedness in such a domain can imply regularity properties of the semigroup if the function ϕ is growing sufficiently fast: we recall corresponding results for analytic, immediately differentiable and eventually differentiable semigroups, [1, Theorem 3.7.11], [39, Theorems 4.7, 5.2]. It follows from the proofs of these results (which use the complex inversion formula for Laplace transforms) that there are similar results in the more general context of Laplace transforms of vector-valued functions; see, for example, [1, Theorem 2.6.1], [14, I.4.7, II.7.4], [42], [41]. One could therefore think of the following Laplace transform version of the norm-continuity problem: if the Laplace transform of a bounded scalar (or vector-valued) function extends analytically to a bounded function in some domain Σϕ , where ϕ ∈ C(R+ ) satisfies limβ→∞ ϕ(β) = ∞, is the function immediately or eventually continuous? It is relatively easy to give counterexamples to this Laplace transform version of the norm-continuity problem. It follows from the main result in this article (Theorem 4.2) that every counterexample to the norm-continuity problem for scalar functions yields a counterexample to the the norm-continuity problem for semigroups. As indicated in the title of this article, we approach the problem of normcontinuity via Banach algebra homomorphisms L1 (R+ ) → A. The connection between semigroups and such homomorphisms is well-known. We recall that to every bounded C0 -semigroup (T (t))t≥0 on a Banach space X one can associate an algebra homomorphism T : L1 (R+ ) → L(X) given by (1.2) Tg = Z 0 ∞ T (t)g(t) dt, g ∈ L1 (R+ ) (integral in the strong sense). OPERATORS L1 (R+ ) → X AND NORM CONTINUITY OF SEMIGROUPS 3 Conversely, every algebra homomorphism L1 (R+ ) → A is, after passing to an equivalent homomorphism, of this form; cf. Lemma 3.1 below. It is therefore natural to ask how regularity properties of the semigroup or the resolvent of its generator are encoded in the corresponding algebra homomorphism or its adjoint. We will discuss some of the connections in the first part of this article, partly in the context of general operators L1 (R+ ) → X. Then, given a function f ∈ L∞ (R+ ) such that its Laplace transform extends to a bounded analytic function on some domain Σϕ , we will show how to construct an algebra homomorphism T : L1 (R+ ) → L(X) which is represented (in the strong sense) by a C0 -semigroup such that f ∈ range T ∗ and such that the resolvent of the generator satisfies a precise decay estimate. In fact, the space X will be continuously embedded into L∞ (R+ ) and left-shift invariant, and the operator T will be represented by the leftshift semigroup on X. In this way, we will be able to show that the norm-continuity problem has a negative solution and at the same time we will be able to estimate the resolvent decay along vertical lines. It turns out that the decay kR(ω + iβ, A)k = O(1/ log |β|) implies eventual differentiability but not immediate norm-continuity, and that any slower decay does not imply eventual norm-continuity. The paper is organized as follows. In the second section, we remind some basic properties and definitions from theory of operators L1 → X needed in the sequel, and introduce the notion of a Riemann-Lebesgue operator. In the third section, we set up a framework of homomorphisms L1 → A and establish the relation to the norm continuity problem for semigroups. The main, fourth section, is devoted to the construction of Riemann-Lebesgue homomorphisms. Finally, in the fifth section, we apply the main result from the fourth section to give counterexamples to the norm-continuity problem. 2. Operators L1 (R+ ) → X Operators L1 → X and their representations is a classical subject of both operator theory and geometric theory of Banach spaces. For a more or less complete account of basic properties of these operators one may consult [12], and a selection of more recent advances pertinent to our studies include [6], [7], [11], [23], [24], [26], [29], [31], [33]. The following representation of operators L1 (R+ ) → X by vector-valued Lipschitz continuous functions on R+ will be used in the sequel. We denote by Lip0 (R+ ; X) the Banach space of all Lipschitz continuous functions F : R+ → X satisfying F (0) = 0. Then, for every F ∈ Lip0 (R+ ; X) the operator TF : L1 (R+ ) → X given by the Stieltjes integral Z ∞ g(t) dF (t), g ∈ L1 (R+ ), (2.1) TF g := 0 is well defined and bounded, and it turns out that every bounded operator T : L1 (R+ ) → X is of this form. In fact, by the Riesz-Stieltjes representation theorem [1, Theorem 2.1.1], the operator F → TF is an (isometric) isomorphism from Lip0 (R+ ; X) onto L(L1 (R+ ), X). There are several analytic properties of operators L1 → X which have been defined and studied in the literature. Among them, we will recall Riesz representability and the (local) Dunford-Pettis property, and we introduce the RiemannLebesgue property. The first and the latter will be relevant for this article while 4 RALPH CHILL AND YURI TOMILOV the (local) Dunford-Pettis property is mentioned for reasons of comparison. Throughout the following, for every λ ∈ C and every t ∈ R+ , we define eλ (t) := e−λt . If λ belongs to the open right half-plane C+ , then eλ ∈ L1 (R+ ). Recall that an operator between two Banach spaces is called Dunford-Pettis or completely continuous if it maps weakly convergent sequences into norm convergent sequences. Definition 2.1. Let T : L1 (R+ ) → X be a bounded operator. (a) We call T Riesz representable, or simply representable, if there exists a function f ∈ L∞ (R+ ; X) such that Z (2.2) Tg = g(s)f (s) ds for every g ∈ L1 (R+ ). R+ (b) We call T locally Dunford-Pettis if for every measurable K ⊂ R+ of finite measure the restriction of T to L1 (K) is Dunford-Pettis. (c) We call T Riemann-Lebesgue if lim|β|→∞ kT (eiβ g)k = 0 for every g ∈ L1 (R+ ). The definitions of representable and local Dunford-Pettis operators clearly make sense on general L1 spaces. For some operator theoretical questions it may be more natural to consider operators on L1 (0, 1) or a similar L1 space. In the context of (bounded) C0 -semigroups, the space L1 (R+ ) is appropriate. We point out that if F ∈ Lip0 (R+ ; X) and if T = TF : L1 (R+ ) → X is repreRt sentable by some function f ∈ L∞ (R+ ; X), then necessarily F (t) = 0 f (s) ds. In fact, T = TF is representable if and only if the function F admits a Radon-Nikodym derivative in L∞ (R+ ; X). Proposition 2.2. Let T : L1 (R+ ) → X be a bounded operator. The following implications are true: T is weakly compact ⇓ T is representable ⇓ T is locally Dunford-Pettis ⇓ T is Riemann-Lebesgue. Proof. The first implication follows from [12, Theorem 12, p.75], while the second implication is a consequence of [12, Lemma 11, p.74, and Theorem 15, p.76]. Note that these results only deal with finite measure spaces whence the necessity to consider local Dunford-Pettis operators. In order to prove the last implication, one has to remark that for general T : L1 (R+ ) → X the space (2.3) E := {g ∈ L1 (R+ ) : lim kT (eiβ g)k = 0} is closed in L1 (R+ ). β→∞ Next, by the Lemma of Riemann-Lebesgue, for every g ∈ L1 (R+ ) one has w − limβ→∞ eiβ g = 0 in L1 (R+ ) and L1 (K), where K is any compact subset of R+ . OPERATORS L1 (R+ ) → X AND NORM CONTINUITY OF SEMIGROUPS 5 Hence, if T is locally Dunford-Pettis, then the space E contains all compactly supported functions in L1 (R+ ), and since this space is dense in L1 (R+ ), the operator T must be Riemann-Lebesgue.  The properties from Definition 2.1 have also been defined for Banach spaces instead of single operators. For example, a Banach space X has the Radon-Nikodym property if every operator T : L1 (R+ ) → X is representable, [12], or, equivalently, if every function in Lip0 (R+ ; X) admits a Radon-Nikodym derivative in L∞ (R+ ; X). Similarly, a Banach space X has the complete continuity property if every operator T : L1 (R+ ) → X is locally Dunford-Pettis. Note that the Dunford-Pettis property for Banach spaces has also been defined in the literature, but is different from the complete continuity property, [38, Definition 3.7.6]. Finally, a Banach space X has the Riemann-Lebesgue property if every operator T : L1 (R+ ) → X is Riemann-Lebesgue. The Riemann-Lebesgue property for Banach spaces has been defined only recently, [8], and Definition 2.1 is perhaps the first instance where the Riemann-Lebesgue property is defined for a single operator. It has been recently shown that the complete continuity property and the RiemannLebesgue property for Banach spaces are equivalent, [31]. It is therefore natural to ask whether a similar result holds for single operators. Problem 2.3. Is every Riemann-Lebesgue operator T : L1 (R+ ) → X already a local Dunford-Pettis operator? The following theorem gives a characterization of Riemann-Lebesgue operators using only exponential functions. Theorem 2.4. An operator T : L1 (R+ ) → X is a Riemann-Lebesgue operator if and only if lim|β|→∞ kT eω+iβ k = 0 for some/all ω > 0. Proof. Assume first that T : L1 (R+ ) → X is a bounded operator satisfying lim|β|→∞ kT eω+iβ k = 0 for some ω > 0. Let 0 < a < ω < b < ∞, and define the closed strip S := {λ ∈ C+ : a ≤ Re λ ≤ b}. The function f :S λ → X, 7 → T eλ , is bounded, continuous on S, and analytic in the interior of S. By a standard argument from complex function theory (involving Vitaly’s theorem) and the assumption we obtain lim kT (eα+iβ )k = 0, |β|→∞ for all α ∈ (a, b). Since a ∈ (0, ω) and b ∈ (ω, ∞) are arbitrary, the above equation is true for every α ∈ (0, ∞). Next, recall from (2.3) that the space of all g ∈ L1 (R+ ) such that lim|β|→∞ kT (eiβ g)k = 0 is closed in L1 (R+ ). By the preceding argument, this space contains the set {eα : α > 0}. Since this set is total in L1 (R+ ), by the Hahn-Banach theorem and by uniqueness of the Laplace transform, it therefore follows that T is a RiemannLebesgue operator. The other implication is trivial.  6 RALPH CHILL AND YURI TOMILOV Corollary 2.5. Let F ∈ Lip0 (R+ ; X), and let TF : L1 (R+ ) → X be the correspondc the Laplace-Stieltjes transform ing bounded operator given by (2.1). Denote by dF of F , that is, Z c (λ) := dF ∞ e−λt dF (t), λ ∈ C+ . 0 Then TF is a Riemann-Lebesgue operator if and only if c (ω + iβ) = 0 for some/all ω > 0. lim dF |β|→∞ Proof. This is a direct consequence from Theorem 2.4 and the definition of the representing function.  3. Algebra homomorphisms L1 (R+ ) → A and the norm-continuity problem In the following, we will equip L1 (R+ ) with the usual convolution product given by f ∗ g(t) = Z t f (t − s)g(s) ds, f, g ∈ L1 (R+ ). 0 Then L1 (R+ ) is a commutative Banach algebra with bounded approximate identity; for example, the net (λeλ )λր∞ is an approximate identity bounded by 1. Let A be a Banach algebra. If (a(t))t>0 ⊂ A is a uniformly bounded and continuous semigroup, then the operator T : L1 (R+ ) → A given by Z ∞ a(t)g(t) dt (3.1) Tg = 0 is an algebra homomorphism as one easily verifies. Conversely, if T : L1 (R+ ) → A is an algebra homomorphism, then T is represented as above, but (a(t))t>0 is a semigroup of multipliers on A and the integral is to be understood in the sense of the strong topology of the multiplier algebra M(A); see, for example, [10, Theorems 3.3 and 4.1], [40, Proposition 1.1]. We will state this result in a slightly different form, more convenient to us, using the notion of equivalent operators which we introduce here. We call two operators T : L1 (R+ ) → X and S : L1 (R+ ) → Y equivalent if there exist constants c1 , c2 ≥ 0 such that kT gkX ≤ c1 kSgkY ≤ c2 kT gkX for every g ∈ L1 (R+ ). It is easy to check that properties like weak compactness, representability, the local Dunford-Pettis property and the Riemann-Lebesgue property are invariant under equivalence, that is, for example, if T and S are equivalent, then T is representable if and only if S is representable; one may prove that if FT and FS are the representing Lipschitz functions, then FT is differentiable almost everywhere if and only if FS is differentiable almost everywhere (use that difference quotients are images of multiples of characteristic functions). We point out that two operators T and S are equivalent if and only if range T ∗ = range S ∗ ; compare with [17, Theorem 1]. The proof of the following lemma is similar to the proof of [9, Theorem 1] (see also [34, Corollary 4.3], [35, Theorem 10.1], [5, Theorem 1.1]). OPERATORS L1 (R+ ) → X AND NORM CONTINUITY OF SEMIGROUPS 7 Lemma 3.1. For every algebra homomorphism T : L1 (R+ ) → A there exists a Banach space X0 , an equivalent algebra homomorphism S : L1 (R+ ) → L(X0 ) and a uniformly bounded C0 -semigroup (S(t))t≥0 ⊂ L(X0 ) such that for every g ∈ L1 (R+ ) Z ∞ S(t)g(t) dt (integral in the strong sense). Sg = 0 If A ⊂ L(X) as a closed subspace, then X0 can be chosen to be a closed subspace of X. Proof. We first assume that A ⊂ L(X) as a closed subspace, and we put R(λ) := T eλ ∈ L(X) (λ ∈ C+ ). Since T is an algebra homomorphism, the function R is a pseudoresolvent, that is, R(λ) − R(µ) = (µ − λ)R(λ)R(µ) for every λ, µ ∈ C+ . This resolvent identity implies that range R(λ) is independent of λ ∈ C+ . We put k·kX X0 := range R(λ) ⊂ X. Clearly, X0 is invariant under R(λ), and since kλeλ kL1 = 1 for every λ > 0, we obtain the estimate (3.2) k(λR(λ))n kL(X0 ) ≤ k(λR(λ))n kL(X) ≤ kT k for every λ > 0, n ≥ 1. By using this estimate (with n = 1), for every x ∈ X and every λ ∈ C+ one obtains µR(λ)x − µR(µ)x = R(λ)x, µ→∞ µ−λ lim µR(µ)R(λ)x = lim µ→∞ which implies lim µR(µ)x = x for every x ∈ X0 . µ→∞ This relation and the resolvent identity imply that R(λ) is injective on X0 and the range of R(λ) is dense in X0 . As a consequence, there exists a densely defined, closed operator A on X0 such that (3.3) R(λ)x = R(λ, A)x for every λ ∈ C+ , x ∈ X0 . By (3.2) and the Hille-Yosida theorem, A is the generator of a uniformly bounded 1 C0 -semigroup R ∞ (S(t))t≥0 ⊂ L(X0 ). Let S : L (R+ ) → L(X0 ) be the operator defined by Sg = 0 S(t)g(t) dt, where the integral is understood in the strong sense. Let F ∈ Lip0 (R+ ; L(X)) be the function representing T (Riesz-Stieltjes representation). Then the equality (3.3) and the definition of R imply Z ∞ Z ∞ −λt e−λt S(t)x dt for every λ ∈ C+ , x ∈ X0 . e dF (t)x = 0 0 By the uniqueness of the Laplace-Stieltjes transform, we obtain Z t S(s)x ds = F (t)x for every t ≥ 0, x ∈ X0 , 0 and hence (Sg)(x) = (T g)(x) Clearly, this implies for every g ∈ L1 (R+ ), x ∈ X0 . kSgkL(X0 ) ≤ kT gkL(X) for every g ∈ L1 (R+ ). 8 RALPH CHILL AND YURI TOMILOV On the other hand, for every x ∈ X one has k(T g)(x)k = = ≤ lim kµR(µ)(T g)(x)k µ→∞ lim k(Sg)(µR(µ)x)k µ→∞ kSgkL(X0 ) sup kµR(µ)xk µ>0 ≤ kSgkL(X0 ) kT k kxk. The last two inequalities imply that T and S are equivalent. The general case can be reduced to the case A ⊂ L(X) in the following way. First of all, we may assume without loss of generality that range T is dense in A. Since L1 (R+ ) admits a bounded approximate identity, it is then easy to verify that also A admits a bounded approximate identity. From this one deduces that the natural embedding A → L(A), which to every element a ∈ A associates the multiplier Ma ∈ L(A) given by Ma b = ab, is an isomorphism onto its range.  Remark 3.2. It is in general an open problem to give conditions on an algebra homomorphism T : L1 (R+ ) → A which imply that there exists an equivalent algebra homomorphism S : L1 (R+ ) → L(X) on a Banach space X having additional properties, for example, being reflexive, being an Lp space etc. If T has dense range, this is essentially the problem of representing A as a closed subalgebra of L(X). The next lemma relates norm-continuity problem with the problem of representability of homomorphisms L1 (R+ ) → A. Lemma 3.3. An algebra homomorphism T : L1 (R+ ) → A is representable if and only if there exists a uniformly bounded and continuous semigroup (a(t))t>0 ⊂ A (no continuity condition in zero) such that T is represented by (3.1). Proof. The necessity part is trivial. So assume that T is representable by some a ∈ L∞ (R+ , A). Without loss of generality we may assume that T has dense range in A. By Lemma 3.1, there exists an equivalent algebra homomorphism S : L1 (R+ ) → L(A) which is (strongly) represented by a uniformly bounded C0 -semigroup (S(t))t≥0 ⊂ L(A). By the proof of Lemma 3.1, one even has Z ∞ Z ∞ a(t)b g(t) dt = S(t)b g(t) dt for every b ∈ A, g ∈ L1 (R+ ), 0 0 which in turn implies a(t)b = S(t)b for every b ∈ A and almost every t > 0. As a consequence, after changing a on a set of measure zero, (a(t))t>0 is a semigroup. Since S is also representable, the semigroup (S(t))t≥0 is measurable in L(A), and hence immediately norm continuous by [28, Theorem 9.3.1]. Since A admits a bounded approximate identity, one thus obtains that also (a(t))t>0 is norm continuous.  Remark 3.4. One can also prove that an algebra homomorphism T : L1 (R+ ) → A is weakly compact if and only if it is represented by a uniformly bounded and OPERATORS L1 (R+ ) → X AND NORM CONTINUITY OF SEMIGROUPS 9 continuous semigroup (a(t))t≥0 ⊂ A (continuity in 0 included!); compare with [20], [21]. Let T : L1 (R+ ) → L(X) be an algebra homomorphism which is represented in the strong sense by a uniformly bounded C0 -semigroup (T (t))t≥0 with generator A. By Lemma 3.3 above, (T (t))t≥0 is immediately norm continuous if and only if T is representable. By Theorem 2.4 and Corollary 2.5, the resolvent of A satisfies the resolvent decay condition (1.1) if and only if T is Riemann-Lebesgue. Hence, by Lemma 3.1, the norm-continuity problem can be reformulated in the following way. Problem 3.5 (Norm-continuity problem reformulated). If A is a Banach algebra and if T : L1 (R+ ) → A is a Riemann-Lebesgue algebra homomorphism, is T representable? We recall from the Introduction, that the norm-continuity problem has a negative answer in general, but that there are some positive answers in special cases. For example, by the representation theorem for C ∗ algebras as subalgebras of L(H) (H a Hilbert space), and by Lemma 3.1, the answer to Problem 3.5 is positive if A is a C ∗ algebra. This follows from the result in [44]. The fact that the answer to Problem 3.5 is in general negative follows from Matrai’s example [37]. The aim of the following section is to prove the existence of Riemann-Lebesgue homomorphisms and to deduce from this different counterexamples to Problem 3.5 for which it is possible to control the resolvent decay along vertical lines. At the same time, we are not able to answer the following variant of the normcontinuity problem. Observe that since Problem 3.5 has in general a negative answer, this variant and Problem 2.3 are not independent of each other. Problem 3.6 (Variant of the norm-continuity problem). If A is a Banach algebra and if T : L1 (R+ ) → A is a local Dunford-Pettis algebra homomorphism, is T representable? We finish this section by collecting some basic properties of algebra homomorphisms L1 (R+ ) → A and their adjoints which are needed in the sequel. For every g ∈ L1 (R+ ), h ∈ L∞ (R+ ) we define the adjoint convolution g ⊛ h ∈ L (R+ ) by Z ∞ ∞ g(s)h(t + s) ds. g ⊛ h(t) = 0 With this definition, for every f , g ∈ L1 (R+ ) and every h ∈ L∞ (R+ ) we have the identities f ⊛ (g ⊛ h) = (f ∗ g) ⊛ h and hf ∗ g, hiL1 ,L∞ = hf, g ⊛ hiL1 ,L∞ , which will be frequently used in the following. The second identity explains the name of the product ⊛. From this identity one can also deduce that ⊛ is separately continuous on (L1 , weak) × (L∞ , weak∗ ) with values in (L∞ , weak∗ ). Whenever X is some Banach space, we denote by BX the closed unit ball in X. 10 RALPH CHILL AND YURI TOMILOV Lemma 3.7. Let T : L1 (R+ ) → A be an algebra homomorphism. Then the following are true: (a) The set T ∗ BA∗ ⊂ L∞ (R+ ) is non-empty, convex, weak∗ compact and T ∗ BA∗ = −T ∗ BA∗ . (b) If kT k = 1, then for every g ∈ BL1 and every h ∈ T ∗ BA∗ ⊂ L∞ (R+ ) one has g ⊛ h ∈ T ∗ BA∗ . In particular, range T ∗ is invariant under adjoint convolution. (c) If T is representable, then range T ∗ ⊂ C(0, ∞). (d) If T is represented (in the strong sense) by a bounded C0 -semigroup which is norm-continuous for t > t0 , then every function in range T ∗ is continuous on (t0 , ∞). Proof. The properties in (a) are actually true for general bounded linear operators T and do not depend on the spaces L1 (R+ ) and A. The weak∗ compactness follows from Banach-Alaoglu and the other properties are true for any unit ball in a Banach space. In order to prove (b), let g ∈ BL1 and h = T ∗ a∗ ∈ T ∗ BA∗ for some a∗ ∈ BA∗ . Since T is an algebra homomorphism, for every f ∈ L1 (R+ ), hf, g ⊛ T ∗ a∗ iL1 ,L∞ = hf ∗ g, T ∗ a∗ iL1 ,L∞ = hT f T g, a∗ iA,A∗ = hT f, T g a∗ iA,A∗ , so that g ⊛ T ∗ a∗ = T ∗ (T g a∗ ). However, kT g a∗ kA∗ ≤ kT gkA ka∗ kA∗ ≤ 1, so that (b) is proved. If T is representable, then, by Lemma 3.3, there exists a bounded norm-continuous semigroup (a(t))t>0 ⊂ A such that Z ∞ a(t)g(t) dt, g ∈ L1 (R+ ). Tg = 0 ∗ ∗ Hence, for every a ∈ A and every g ∈ L1 (R+ ), hg, T ∗ a∗ iL1 ,L∞ = hT g, a∗ iA,A∗ Z ∞ g(t)ha(t), a∗ iA,A∗ dt. = 0 ∗ ∗ ∗ ∈ C(0, ∞) so that (c) is proved. This implies T a = ha(·), a i The last assertion is very similar to (c), if we use in addition that L1 (R+ ) is the direct sum of L1 (0, t0 ) and L1 (t0 , ∞) (and L∞ (R+ ) is the direct sum of the corresponding duals).  A,A∗ 4. Construction of Riemann-Lebesgue homomorphisms This section is devoted to the main result of this article: we will describe a procedure how to construct Riemann-Lebesgue algebra homomorphisms T : L1 (R+ ) → A for which one can estimate the norm decay of the pseudoresolvent (T eλ )λ∈C+ along vertical lines. In the following, for every function ϕ ∈ C(R+ ) we define the domain Σϕ := {λ ∈ C : Re λ > −ϕ(|Im λ|)}. OPERATORS L1 (R+ ) → X AND NORM CONTINUITY OF SEMIGROUPS 11 The domains Σϕ are symmetric with respect to the real axis. Domains of the form Σϕ with limβ→∞ ϕ(β) = ∞ play an important role in connection with RiemannLebesgue algebra homomorphisms. The following lemma contains a necessary condition for algebra homomorphisms to be Riemann-Lebesgue. Lemma 4.1. If A is a Banach algebra and if T : L1 (R+ ) → A is a RiemannLebesgue algebra homomorphism, then there exists a function ϕ ∈ C(R+ ) satisfying limβ→∞ ϕ(β) = ∞ such that for every f ∈ range T ∗ the Laplace transform fˆ extends to a bounded analytic function on Σϕ . Proof. Assume that T : L1 (R+ ) → A is a Riemann-Lebesgue homomorphism. Then lim|β|→∞ kT e2+iβ k = 0. Developping the pseudoresolvent λ 7→ T eλ in a power series at the points 2 + iβ with β ∈ R, one easily verifies that this pseudoresolvent extends to a bounded analytic function in some domain Σϕ , where ϕ is as in the statement. If f = T ∗ a∗ ∈ range T ∗ , then for every λ ∈ C+ one has fˆ(λ) = heλ , f iL1 ,L∞ = hT eλ , a∗ iA,A∗ , and therefore the Laplace transform fˆ extends to a bounded analytic function on Σϕ .  The main result in this section asserts that the converse statement to Lemma 4.1 is also true. Throughout the following, we put for every k ∈ Z λk := 2 + ik, and if ϕ ∈ C(R+ ) is a given nonnegative function, then we also put dk := dist (λk , ∂Σϕ ). It will not be necessary to make the dependence of dk on ϕ explicit in the notation since the function ϕ will always be clear from the context. Theorem 4.2. Let f ∈ L∞ (R+ ) be a function such that its Laplace transform fˆ extends to a bounded analytic function in some domain Σϕ , where ϕ ∈ C(R+ ) satisfies inf ϕ > 0. Then there exists a Banach space X which embeds continuously into L∞ (R+ ) and which is left-shift invariant such that (i) the left-shift semigroup (T (t))t≥0 on X is bounded and strongly continuous, (ii) the resolvent of the generator A satisfies the decay estimate (4.1) kR(λk , A)kL(X) ≤ C log dk dk for every k ∈ Z, (iii) if T : L1 (R+ ) → L(X) is the algebra homomorphism which is represented (in a strong sense) by (T (t))t≥0 , then f ∈ range T ∗ , and (iv) the following inclusion holds: X ⊂ L1 (R+ ) ⊛ f (L∞ ,weak∗ ) If, in addition, the function C+ λ → L∞ (R+ ), 7 → eλ ⊛ f, , 12 RALPH CHILL AND YURI TOMILOV extends analytically to Σϕ and if there exists some r ∈ (0, 1) such that (4.2) sup keλ ⊛ f k∞ ≤ C λ∈B(λk ,r dk ) 1 dk for every k ∈ Z. then the space X can be chosen in such a way that the resolvent satisfies the stronger estimate 1 kR(λk , A)kL(X) ≤ C for every k ∈ Z, dk Remark 4.3. The condition inf ϕ > 0 in the above theorem simplifies the proof in some places but is not essential. Moreover, the it can always be achieved by rescaling the function f or the semigroup (T (t))t≥0 . The important points in the above theorem are the statements that the resolvent decay condition (1.1) is satisfied as soon as limβ→∞ ϕ(β) = ∞, and that at the same time f ∈ range T ∗ . Thus, if we are able to find a function f ∈ L∞ (R+ ) such that its Laplace transform fˆ extends to a bounded analytic function on Σϕ , where ϕ ∈ C(R+ ) satisfies limβ→∞ ϕ(β) = ∞, and such that f is not continuous on (0, ∞), then the RiemannLebesgue operator from Theorem 4.2 is not representable by Lemma 3.7 (c), that is, the semigroup (T (t))t≥0 is not immediately norm continuous (Lemma 3.3). In other words, the existence of such a function f solves the norm-continuity problem. It is straightforward to check that the characteristic function f = 1[0,1] provides such an example. This and an other example will be discussed in Section 5. For these examples it will be of substantial interest that Theorem 4.2 also gives an estimate of the resolvent R(·, A) along vertical lines, in terms of the Laplace transform fˆ, the decay of the function λ 7→ eλ ⊛ f and the growth of the function ϕ. We point out that a decay condition weaker than (4.2) is always true, as we will prove in Lemma 4.14 below. We do not know whether the decay condition (4.2) is always satisfied. The rest of this section will be devoted to the proof of Theorem 4.2, that is, to the construction of the Banach space X and the algebra homomorphism T : L1 (R+ ) → L(X). The space X will be the closed subspace of an appropriate subspace M of L∞ (R+ ); we will first construct M be constructing its unit ball. Lemma 4.4. Let (fn ) ⊂ L∞ (R+ ) be a bounded sequence and define the set (L∞ ,weak∗ ) X gn ⊛ fn : (gn ) ∈ Bl1 (L1 (R+ )) ⊂ L∞ (R+ ). (4.3) BM := n Then: (a) The set BM is non-empty, convex, weak∗ compact and BM = −BM . (b) For every g ∈ BL1 and every h ∈ BM one has g ⊛ h ∈ BM . (c) For every n one has fn ∈ BM . Proof. The properties in (a) are either trivial or easy to check. P Next, let g ∈ BL1 and h ∈ BM . Assume first that h = n gn ⊛ fn for some sequence (gn ) ∈ Bl1 (L1 (R+ )) . Then X X g⊛h= g ⊛ (gn ⊛ fn ) = (g ∗ gn ) ⊛ fn . n n OPERATORS L1 (R+ ) → X AND NORM CONTINUITY OF SEMIGROUPS 13 Since (g ∗ gn ) ∈ Bl1 (L1 (R+ )) , this implies g ⊛ h ∈ BM . P For general h ∈ BM , by the definition of BM , there exists a net (hα ) ⊂ BM , hα = n gnα ⊛ fn for some (gnα ) ∈ Bl1 (L1 (R+ )) , such that w∗ − limα hα = h. However, then g ⊛ h = w∗ − limα g ⊛ hα as one easily verifies. Since BM is weak∗ closed, we have proved (b). By definition of BM , one has g ⊛ fn ∈ BM for every g ∈ BL1 (R+ ) . Taking an approximate unit (gj ) in BL1 (R+ ) , one easily shows w∗ − limj gj ⊛ fn = fn . Since BM is weak∗ closed, this proves (c).  For a bounded sequence (fn ) ⊂ L∞ (R+ ) we define the set BM ⊂ L∞ (R+ ) as in (4.3), and then we put (4.4) M := R+ BM . Then M is a (in general nonclosed) subspace of L∞ (R+ ) and becomes a normed space when it is equipped with the Minkowski norm khkM := inf{λ > 0 : h ∈ λ BM }. When M is equipped with this Minkowski norm, then BM is the unit ball of M , and there is no ambiguity with our previously introduced notation. Moreover, M embeds continuously into L∞ (R+ ). By a result by Dixmier, M is dual space, and in particular M is a Banach space, [13]. To be more precise, consider the natural embedding S : L1 (R+ ) → M ∗ given by hSg, miM ∗ ,M := hg, miL1 ,L∞ , (4.5) and let M∗ := range S (4.6) M∗ . Then we have the following result; the short proof follows [32, Proof of Theorem 1]. Lemma 4.5. The space M is isometrically isomorphic to M∗ ∗ , that is, to the dual of M∗ . Proof. The key point is the fact that, by construction, BM is weak∗ compact in L∞ (R+ ). By the definition of the operator S and by the definition of the space M∗ , this implies that the unit ball BM is compact with respect to the σ(M, M∗ ) topology. Consider the contraction J : M → M∗ ∗ which maps every m ∈ M to the functional Jm ∈ M∗ ∗ given by hJm, m∗ iM∗ ∗ ,M∗ := hm, m∗ iM,M ∗ . The space M∗ separates the points in M because the space L1 (R+ ) separates the points in M ⊂ L∞ (R+ ). Therefore, the operator J is injective. Next, let m∗∗ ∈ M∗ ∗ and assume for simplicity that km∗∗ kM∗ ∗ = 1. By HahnBanach, we may consider m∗∗ also as an element in BM ∗∗ . By Goldstine’s theorem, there exists a net (mα ) ⊂ BM which converges to m∗∗ in σ(M ∗∗ , M ∗ ). Since BM is compact with respect to the σ(M, M∗ ) topology, there exists m ∈ BM such that hm, m∗ iM,M ∗ = lim hmα , m∗ iM,M ∗ = hm∗∗ , m∗ iM∗ ∗ ,M∗ for every m∗ ∈ M∗ . α Hence, Jm = m∗∗ and kmkM ≤ 1 = kJmkM∗ ∗ . We have thus proved that J is also surjective and isometric.  14 RALPH CHILL AND YURI TOMILOV Remark 4.6. Using only the definitions of the operators J and S, it is straightforward to verify that S ∗ J is the natural embedding of M into L∞ (R+ ) and that (4.7) range S ∗ = M. Lemma 4.7. The space M is an L1 (R+ ) module in a natural way: for every g ∈ L1 (R+ ) and every m ∈ M (⊂ L∞ (R+ )) the adjoint convolution g ⊛ m belongs to M and kg ⊛ mkM ≤ kgkL1 kmkM . Proof. By Lemma 4.4 (b), for every nonzero g ∈ L1 (R+ ) and m ∈ M one has g m  kgk 1 ⊛ kmkM ∈ BM . The claim follows immediately. L In the following, we will always consider M as an L1 (R+ ) module via the adjoint convolution. Note that together with M also the dual space M ∗ is an L1 (R+ ) module if for every g ∈ L1 (R+ ) and every m∗ ∈ M ∗ we define the product g ∗ m∗ ∈ M ∗ by hg ∗ m∗ , miM ∗ ,M := hm∗ , g ⊛ miM ∗ ,M , m ∈ M. We use again the notation ∗ for the adjoint of the adjoint convolution. If M = L∞ (R+ ), then the product ∗ coincides with the usual convolution in L1 (R+ ) ⊂ L∞ (R+ )∗ and there is no ambiguity in the notation. Lemma 4.8. The natural embedding S : L1 (R+ ) → M ∗ defined in (4.5) is an L1 (R+ ) module homomorphism. The space M∗ is an L1 (R+ ) submodule of M ∗ . Proof. For every g, h ∈ L1 (R+ ) and every m ∈ M one has hS(g ∗ h), miM ∗ ,M = = = = hg ∗ h, miL1 ,L∞ hh, g ⊛ miL1 ,L∞ hSh, g ⊛ miM ∗ ,M hg ∗ Sh, miM ∗ ,M . Since this equality holds for every m ∈ M , this proves S(g ∗ h) = g ∗ Sh for every g, h ∈ L1 (R+ ), and therefore S is an L1 (R+ ) module homomorphism. At the same time, this equality proves that the closure of the range is a submodule of M ∗ .  We omit the proof of the following lemma which is straightforward. Lemma 4.9. Let M∗ be defined as in (4.6). The natural embedding T∗ : L1 (R+ ) → L(M∗ ) given by T∗ g(m∗ ) := g ∗ m∗ , m∗ ∈ M∗ , is an algebra homomorphism. The following lemma allows us to calculate kT∗ gkL(M∗ ) in terms of the sequence (fn ). Lemma 4.10. Let (fn ) ⊂ L∞ (R+ ) be a bounded sequence, let M∗ be defined as in (4.6), and let T∗ : L1 (R+ ) → L(M∗ ) be the induced algebra homomorphism from Lemma 4.9. Then, for every g ∈ L1 (R+ ), one has (4.8) kT∗ gkL(M∗ ) = sup kg ⊛ fn kM . n OPERATORS L1 (R+ ) → X AND NORM CONTINUITY OF SEMIGROUPS 15 Proof. Let g ∈ L1 (R+ ). Then, by the definition of T∗ , S, and by the definition of M∗ , kT∗ gkL(M∗ ) = sup m∗ ∈BM∗ = kg ∗ m∗ kM ∗ sup |hg ∗ m∗ , miM ∗ ,M | sup m∗ ∈BM∗ m∈BM = sup sup |hg ∗ Sh, miM ∗ ,M |. h∈L1 (R+ ) m∈BM kShkM ∗ ≤1 Since S is an L1 (R+ ) module homomorphism, and by the definition of S, kT∗ gkL(M∗ ) = sup sup |hS(g ∗ h), miM ∗ ,M | h∈L1 (R+ ) m∈BM kShkM ∗ ≤1 = sup |hg ∗ h, miL1 ,L∞ |. sup h∈L1 (R+ ) m∈BM kShkM ∗ ≤1 P Since, by definition, { n gn ⊛ fn : (gn ) ∈ Bl1 (L1 (R+ )) } is weak∗ dense in BM (with respect to the weak∗ topology in L∞ (R+ )), and since M∗ is norming for M by Lemma 4.5, we can continue to compute X kT∗ gkL(M∗ ) = sup sup |hg ∗ h, gn ⊛ fn iL1 ,L∞ | h∈L1 (R+ ) (gn )∈Bl1 (L1 (R+ )) kShkM ∗ ≤1 = sup sup |hh, h∈L1 (R+ ) (gn )∈Bl1 (L1 (R+ )) kShkM ∗ ≤1 = sup sup sup (gn )∈Bl1 (L1 (R = k + )) sup kg ⊛ fn kM X X gn ⊛ (g ⊛ fn )iL1 ,L∞ | n |hSh, h∈L1 (R+ ) (gn )∈Bl1 (L1 (R+ )) kShkM ∗ ≤1 = n X gn ⊛ (g ⊛ fn )iM ∗ ,M | n gn ⊛ (g ⊛ fn )kM n n The claim is proved.  The operator T∗ from Lemma 4.9 will be equivalent to the operator we are looking for in Theorem 4.2. However, so far we have not said anything about the sequence (fn ) ⊂ L∞ (R+ ) which served for the construction of M , and which allows us by Lemma 4.10 to obtain the desired resolvent estimate in Theorem 4.2. It remains to explain how the sequence (fn ) is constructed in order to prove Theorem 4.2. For the moment being, let f ∈ L∞ (R+ ) be a fixed function, and suppose that the Laplace transform fˆ extends analytically to a bounded function on Σϕ , where ϕ ∈ C(R+ ) satisfies inf ϕ > 0. In order to simplify the notation, we define for every k ∈ Z λk := 2 + ik, dk := dist (λk , ∂Σϕ ), and ẽk = eλk , 16 RALPH CHILL AND YURI TOMILOV and we will choose numbers ck > 0 depending on the functions λ 7→ eλ ⊛f and ϕ; see Lemma 4.11 below for the precise definition of ck . We define inductively for n ≥ 1 and k ∈ Zn+1 ẽk := ẽk̄ ∗ ẽkn+1 = ẽk1 ∗ . . . ∗ ẽkn+1 and ck := ck̄ · ckn+1 = ck1 · . . . · ckn+1 , n where k̄ ∈ Z is such that k = (k̄, kn+1 ). Then, for every n ≥ 1 and every k ∈ Zn we put (4.9) fk := ẽk ⊛ f ẽk ∗ . . . ∗ ẽkn = 1 ⊛ f. ck ck1 · . . . · ckn Finally, we set f∞ := f, and I := {∞} ∪ [ Zn , n≥1 and we will define the unit ball BM , the space M and the space X starting from the family (fk )k∈I . Lemma 4.11. Let f ∈ L∞ (R+ ) be such that the Laplace transform fˆ extends to a bounded analytic function in Σϕ , where ϕ ∈ C(R+ ) satisfies inf ϕ > 0. Let r ∈ (0, 41 ) be arbitrary. For every k ∈ Z we put (4.10) ck = 4 log dk . r dk Then the family (fk )k∈I given by (4.9) is bounded in L∞ (R+ ). The same is true if the condition (4.2) is satisfied and if we then put, for every k ∈ Z, 4 1 . ck = r dk The proof of Lemma 4.11 is based on the following series of four lemmas. The statement and the proof of the following lemma should be compared to [1, Lemmas 4.6.6, 4.7.9]. Lemma 4.12. Let f ∈ L∞ (R+ ) be such that the Laplace transform fˆ extends to a bounded analytic function in Σϕ , where ϕ ∈ C(R+ ) satisfies inf ϕ > 0. Then also the function λ 7→ eλ ⊛ f , C+ 7→ BU C(R+ ) extends to a bounded analytic function in Σϕ . Proof. For every t ∈ R+ and every λ ∈ C+ one has Z ∞ e−λs f (t + s) ds eλ ⊛ f (t) = 0 Z t λt ˆ eλ(t−s) f (s) ds. = e f (λ) − 0 OPERATORS L1 (R+ ) → X AND NORM CONTINUITY OF SEMIGROUPS From this identity we obtain first that for every eλ ⊛ f (t) extends to an analytic function on Σϕ , t ∈ R+ and every λ ∈ Σϕ the estimate   kf k∞ |Re λ| (4.11) |eλ ⊛ f (t)| ≤  kf k∞ + kfˆk∞ |Re λ| 17 fixed t ∈ R+ the function λ 7→ and we obtain second for every if Re λ > 0, if Re λ < 0. By assumption, there exists 0 < α ≤ 1 such that inf ϕ > α. The above estimate immediately yields 2 kf k∞ sup sup |eλ ⊛ f (t)| ≤ + kfˆk∞ . α λ∈Σϕ t∈R+ |Re λ|≥ α 2 In order to show that the function eλ ⊛ f (t) is bounded in the strip {λ ∈ C : |Re λ| ≤ α2 } (with a bound independent of t ∈ R+ ) we can argue as follows. For every β ∈ R, by the maximum principle and by the estimate (4.11), (λ − iβ)2
sup eλ ⊛ f (t) 1 + α2 |λ−iβ|≤α = eλ ⊛ f (t) 1 + sup |λ−iβ|=α 5 kf k∞ + 5 kfˆk∞ . α Hence, for every t ∈ R+ and every β ∈ R (λ − iβ)2
α2 ≤ sup |λ−iβ|≤ α 2 |eλ ⊛ f (t)| ≤ 7 kf k∞ + 7 kfˆk∞ , α which yields the desired estimate in the strip {λ ∈ C : |Re λ| < obtain sup sup |eλ ⊛ f (t)| < ∞, α 2 }. So we finally λ∈Σϕ t∈R+ and in particular the function λ 7→ eλ ⊛f is bounded on Σϕ with values in BC(R+ ). Pointwise analyticity and uniform boundedness imply, by [1, Proposition A.3], that the function λ 7→ eλ ⊛ f is bounded and analytic on Σϕ with values in BC(R+ ). Since eλ ⊛ f ∈ BU C(R+ ) for every λ ∈ C+ , by the identity theorem for analytic functions (see, for example, the version in [1, Proposition A.2]), we finally obtain the claim.  The main argument in the proof of the following lemma (the two constants theorem) is also used in [30, Proof of Theorem 5.3], but the following lemma gives a better estimate. Lemma 4.13. Let X be some Banach space and let ϕ ∈ C(R+ ) be a nonnegative function. Let h : Σϕ → X be a bounded analytic function satisfying the estimate C for every λ ∈ C+ and some C ≥ 0. kh(λ)k ≤ Re λ Then for every r ∈ (0, 41 ) there exists Cr ≥ 0 such that for every k ∈ Z kh(λ)k ≤ Cr sup d λ∈B(λk ,r logkd ) k log dk . dk 18 RALPH CHILL AND YURI TOMILOV Proof. We may assume that the constant C from the hypothesis satisfies C ≥ khk∞ . Fix r ∈ (0, 41 ). We may in the following consider only those k ∈ Z for which 4 < log dk . For the other k, the estimate in the claim becomes trivial if the constant Cr is chosen sufficiently large. Let Ω := {λ ∈ C : |Re λ|, |Im λ| < 1}, and let Γ0 := {λ ∈ ∂Ω : Re λ = 1} and Γ1 := ∂Ω \ Γ0 . By the two constants theorem, for every analytic function g : Ω → X having a continuous extension to Ω̄ and satisfying the boundary estimate kg(λ)k ≤ Ci if λ ∈ Γi (i = 0, 1), one has the estimate w(λ) kg(λ)k ≤ C0 1−w(λ) C1 for every λ ∈ Ω, where w = wΩ (·, Γ0 ) is the harmonic measure of Γ0 with respect to Ω, that is, w : Ω → [0, 1] is the harmonic function satisfying w = 1 on Γ0 and w = 0 on Γ1 . For every k ∈ Z with log dk > 4 we apply this two constants theorem to the function given by g(λ) = h(λk + 1 dk )), (λ − 1 + 4 log dk λ ∈ Ω̄, which satisfies by assumption the estimates ( C logdkdk if λ ∈ Γ0 , and kg(λ)k ≤ C if λ ∈ Γ1 . We then obtain log dk
w(λ) for every λ ∈ Ω. dk By the Schwarz reflection principle, the function 1 − w extends to a harmonic function in the rectangle {λ ∈ C : −1 < Re λ < 3, |Im λ| < 1}, and in particular the function w is continuously differentiable there. We can therefore find a constant c > 0 such that cr inf . w(λ) ≥ 1 − log dk λ∈B(1− log1d ,4r log1d ) k k kg(λ)k ≤ C Combining the preceding two estimates, we obtain sup λ∈B(λk ,r kh(λ)k = dk log dk ) kg(λ)k sup λ∈B(1− log1d ,4r k ≤ C 1 log dk ) sup λ∈B(1− log1d ,4r k 1 log dk ) log dk
1− logcrdk dk log log d cr(1− log d k ) log dk k = Ce dk log dk . ≤ Cr dk log dk
w(λ) dk ≤ C The claim is proved.  OPERATORS L1 (R+ ) → X AND NORM CONTINUITY OF SEMIGROUPS 19 Lemma 4.14. Let f ∈ L∞ (R+ ) be such that the Laplace transform fˆ extends to a bounded analytic function in Σϕ , where ϕ ∈ C(R)+ satisfies inf ϕ > 0. Then for every r ∈ (0, 41 ) there exists Cr ≥ 0 such that keλ ⊛ f k∞ ≤ Cr sup d λ∈B(λk ,r logkd ) log dk dk for every k ∈ Z. k Proof. Since keλ ⊛f k∞ ≤ ReC λ for every λ ∈ C+ , this lemma is a direct consequence of Lemma 4.12 and Lemma 4.13.  The following is a consequence of the resolvent identity and should probably be known. We will give the easy proof here. Lemma 4.15. For every n ≥ 1, every λ1 , . . . , λn ∈ C+ , and every closed path Γ ⊂ C+ such that λ1 , . . . , λn are in the interior of Γ one has Z (−1)n+1 eλ (4.12) eλ1 ∗ . . . ∗ eλn = dλ. 2πi (λ − λ ) · . . . · (λ − λn ) 1 Γ Proof. The proof goes by induction on n. If n = 1, then the formula (4.12) is just Cauchy’s integral formula. So assume that the formula (4.12) is true for some n ≥ 1. Let λ1 , . . . , λn , λn+1 ∈ C+ , and let Γ ⊂ C+ be a closed path such that λ1 , . . . , λn , λn+1 are in the interior of Γ. Then, by the resolvent identity and the induction hypothesis, Z eλ ∗ eλn+1 (−1)n+1 dλ eλ1 ∗ . . . ∗ eλn ∗ eλn+1 = 2πi Γ (λ − λ1 ) · . . . · (λ − λn ) Z (−1)n+2 eλ = dλ + 2πi Γ (λ − λ1 ) · . . . · (λ − λn ) (λ − λn+1 ) Z (−1)n+1 1 +eλn+1 dλ. 2πi Γ (λ − λ1 ) · . . . · (λ − λn ) (λ − λn+1 ) For the induction step it suffices to show that the second integral on the right-hand side of this equality vanishes. In order to see that this integral vanishes, we replace the path Γ by a circle centered in 0 and having radius R > 0 large enough, without changing the value of the integral. A simple estimate then shows that Z 1 dλ = O(R−n ) as R → ∞. (λ − λ ) · . . . · (λ − λ ) (λ − λ ) 1 n n+1 |λ|=R Since the left-hand side is independent of R and since n ≥ 1, by letting R → ∞, we obtain that the integral above is zero.  Proof of Lemma 4.11. It will be convenient in this proof to define the function h(s) := logs s , s ≥ 2. Then ck = 4 1 r h(dk ) for every k ∈ Z, where r ∈ (0, 41 ) is fixed as in the assumption. Let Cr ≥ 0 be as in Lemma 4.14. We will show that sup kfk k∞ ≤ Cr . k∈I,k6=∞ The proof goes by induction on n. 20 RALPH CHILL AND YURI TOMILOV By Lemma 4.14, for every k ∈ Z, kẽk ⊛ f k∞ ≤ Cr ≤ Cr ck h(dk ) by the definition of ck and since r ≤ 1, and therefore kfk k∞ ≤ Cr for every k ∈ Z. Next, we assume that there exists n ≥ 1 such that kfk k∞ ≤ Cr for every k ∈ Zn . Let k = (kν )1≤ν≤n+1 ∈ Zn+1 . Assume first that there exist 1 ≤ ν, µ ≤ n + 1 such that (4.13) r (h(dkν ) + h(dkµ )). 4 |kν − kµ | > There exists k̃ ∈ I such that fk = ẽkν ∗ ẽkµ ⊛ fk̃ , ckν · ckµ and therefore, by the resolvent identity, by the induction hypothesis, by the definition of ck , and by (4.13), kfk k∞ ẽkν ⊛ fk̃ − ẽkµ ⊛ fk̃ (kµ − kν )ckν · ckµ ∞ 1 1
1 ≤ Cr + |kν − kµ | ckν ckµ
1 r = Cr h(dkν ) + h(dkµ ) |kν − kµ | 4 ≤ Cr . = Hence, we may suppose that (4.14) |kν − kµ | ≤ r (h(dkν ) + h(dkµ )) 4 for every 1 ≤ ν, µ ≤ n + 1. After a permutation of the indices, we may assume in addition that h(dk1 ) = max 1≤ν≤n+1 h(dkν ). From the estimate |λkν − λk1 | = |kν − k1 | ≤ 2r h(dk1 ) we obtain λkν ∈ B(λ1 , 3r 4 h(dk1 )) for every 1 ≤ ν ≤ n + 1. As a consequence, by Lemma 4.15, and since λ 7→ eλ ⊛ f extends to a bounded analytic function on B(λk1 , h(dk1 )), (4.15) Z eλ ⊛ f 1 dλ. ẽk ⊛f = (eλk1 ∗ . . . ∗eλkn+1 )⊛f = 2πi ∂B(λk1 , 3r4 h(dk1 )) (λ − λk1 ) · · · · · (λ − λkn+1 ) OPERATORS L1 (R+ ) → X AND NORM CONTINUITY OF SEMIGROUPS 21 Note that for every λ ∈ ∂B(λ1 , 3r 4 h(dk1 )) and every 1 ≤ ν ≤ n + 1 one has |λ − λkν | ≥ |λ − λk1 | − |λk1 − λkν | 3r r ≥ h(dk1 ) − (h(dk1 ) + h(dkν )) 4 4 r ≥ h(dk1 ) 4 r 1 ≥ h(dkν ) = 4 ckν or 1 ≤ ckν . |λ − λkν | This inequality, the equality (4.15), and the decay condition from Lemma 4.14 yield kẽk ⊛ f k∞ ≤ 3r h(dk1 ) 4 sup λ∈∂B(λk1 , 3r 4 h(dk1 )) keλ ⊛ f k∞ ck1 · . . . · ckn+1 ≤ Cr ck1 · . . . · ckn+1 = Cr ck . This implies kfk k∞ ≤ Cr for every k ∈ Zn+1 , and by induction, the first claim is proved. If the estimate (4.2) holds and if ck = 4r d1k , then one may repeat the above proof replacing the function h by the function h(s) := s.  We are ready to prove Theorem 4.2. Proof of Theorem 4.2. Let f ∈ L∞ (R+ ) and ϕ ∈ C(R+ ) be as in the hypothesis. Define the numbers ck > 0 as in Lemma 4.11 (depending on whether the condition (4.2) holds or not), and let the family (fk )k∈I be defined as in (4.9). By Lemma 4.11, the family (fk )k∈I is uniformly bounded in L∞ (R+ ). Define the unit ball BM , and the spaces M and M∗ as above. We recall that the space M embeds continuously into L∞ (R+ ), and that by construction M ⊂ L1 (R+ ) ⊛ f (L∞ ,weak∗ ) . Let T∗ : L1 (R+ ) → L(M∗ ) be the algebra homomorphism defined in Lemma 4.9, and let T : L1 (R+ ) → L(M ) be the algebra homomorphism given by T g(m) := g ⊛ m, m ∈ M . kT gkL(M ) = kT∗ gkL(M∗ ) for every g ∈ L1 (R+ ). Clearly, 22 RALPH CHILL AND YURI TOMILOV By Lemma 4.10, and since supk∈I kfk kM ≤ 1 by Lemma 4.4 (c), for every k ∈ Z, kT eλk kL(M ) = kT∗ eλk kL(M∗ ) = sup keλk ⊛ fk̄ kM k̄∈I = sup kẽk ⊛ k̄∈I ẽk̄ ⊛ f kM ck̄ ẽ(k̄,k) ⊛ f ck kM c(k̄,k) = sup k = sup kf(k̄,k) ck kM k̄∈I k̄∈I ≤ ck . By the definition of ck (see Lemma 4.11), this leads to the estimate ( C logdkdk or (4.16) kT eλk kL(M ) ≤ for every k ∈ Z, C d1k depending on whether the condition (4.2) holds or not. By Lemma 3.1, after replacing the space M by a closed subspace X, if necessary, we can assume that the homomorphism T is represented (in the strong sense) by a C0 -semigroup (T (t))t≥0 ∈ L(X). Since T was defined by adjoint convolution, it follows that the semigroup (T (t))t≥0 is the left-shift semigroup on X. If A is the generator of this semigroup, then the estimate (4.16) implies ( C logdkdk or kR(λk , A)kL(X) ≤ for every k ∈ Z, C d1k depending on whether the condition (4.2) holds or not. It remains to show that f ∈ range T ∗ . For every g ∈ L1 (R+ ) we can estimate kSgkM ∗ sup kS(g ∗ h)kM ∗ = h∈BL1 = sup kg ∗ ShkM ∗ h∈BL1 = kSk sup kg ∗ h∈BL1 ≤ kSk Sh kM ∗ kSk kg ∗ ShkM ∗ sup h∈L1 kShkM ∗ ≤1 = kSk sup m∗ ∈BM∗ kg ∗ m∗ kM ∗ = kSk kT gkL(M∗ ) . In other words, there is a bounded operator R : range T k·kL(M∗ ) → M∗ ⊂ M ∗ such that S = RT . Hence, T ∗ R∗ = S ∗ : M ∗∗ → L∞ (R+ ), which implies (4.17) range S ∗ ⊂ range T ∗ . On the other hand, we recall from (4.7) that range S ∗ = M . Since f = f∞ ∈ BM by Lemma 4.4 (c), we thus obtain f ∈ range T ∗ . Theorem 4.2 is completely proved.  OPERATORS L1 (R+ ) → X AND NORM CONTINUITY OF SEMIGROUPS 23 Remark 4.16. It would be interesting to understand the geometric structure of the spaces M and X, for example, whether they might be UMD spaces or spaces having nontrivial Fourier type. 5. The norm continuity problem In this section we present two examples which imply that the norm continuity problem has a negative answer. In these two examples emphasis will be put on precise decay estimates for the resolvent along vertical lines. Before stating the two examples, we recall the following known result; see [39, Theorem 4.9], [19, Theorem 4.1.3]. Lemma 5.1. Let A be the generator of a bounded C0 -semigroup (T (t))t≥0 . Then the following are true: (i) If 1 kR(2 + iβ, A)k = o( ) as |β| → ∞, log |β| then the semigroup (T (t))t≥0 is immediately differentiable. (ii) If 1 kR(2 + iβ, A)k = O( ) as |β| → ∞, log |β| then the semigroup (T (t))t≥0 is eventually differentiable. The following is our first counterexample to the norm continuity problem. Theorem 5.2. There exists a Banach space X and a uniformly bounded C0 semigroup (T (t))t≥0 ⊂ L(X) with generator A such that: (i) the resolvent satisfies the estimate 1 ) as |β| → ∞, kR(2 + iβ, A)k = O( log |β| and in particular the resolvent satisfies the resolvent decay condition (1.1), (ii) T (1) = 0, that is, the semigroup (T (t))t≥0 is nilpotent, and (iii) whenever t0 ∈ [0, 1), then the semigroup (T (t))t≥0 is not norm-continuous for t > t0 . Proof. Let f = 1[0,1] be the characteristic function of the interval [0, 1]. Since f has compact support, the Laplace transform fˆ and also the function λ 7→ eλ ⊛ f extend to entire functions, and for every λ ∈ C \ {0} and every t ∈ R+ , ( 1 −λ(1−t) ) if 0 ≤ t ≤ 1, λ (1 − e eλ ⊛ f (t) = 0 if t > 1. Hence, for every λ ∈ C \ {0}, (5.1) keλ ⊛ f k∞ ≤   2 e−Re λ |λ|  2 1 |λ| if Re λ < 0, if Re λ ≥ 0, λ 6= 0. Let ϕ ∈ C(R+ ) be the function given by ϕ(β) = log+ (β), where log + β ≥ 0, is the positive part of the logarithm. Clearly limβ→∞ ϕ(β) = +∞. 24 RALPH CHILL AND YURI TOMILOV It follows from (5.1) that sup |fˆ(λ)| < ∞, λ∈Σϕ so that f satisfies the hypothesis of Theorem 4.2. Moreover, it is easy to see that for the function ϕ one has 2+ 1 1 log+ |k| ≤ dk ≤ 2 + log+ |k| for every k ∈ Z, 8 4 where, as before, we put dk := dist (λk , ∂Σϕ ) and λk = 2 + ik. From (5.1) one therefore also obtains for every r ∈ (0, 1) the estimate sup keλ ⊛ f k∞ ≤ 2 max{e−Re λ , 1} |λ| λ∈B(λk ,r dk ) ≤ 2 er log |k| |k| − 41 log+ |k| λ∈B(λk ,r dk ) sup ≤ C |k|−(1−r) , and in particular sup keλ ⊛ f k∞ ≤ Cr λ∈B(λk ,r dk ) 1 dk for every k ∈ Z. This means that the function f satisfies the decay condition (4.2) from Theorem 4.2. By Theorem 4.2, there exists a left-shift invariant Banach space X ֒→ L∞ (R+ ) such that the resolvent of the generator A of the left-shift semigroup (which is strongly continuous on X) satisfies the decay estimate kR(2 + ik, A)kL(X) ≤ C 1 1 + log+ |k| for every k ∈ Z, 1 1 + log+ |β| for every β ∈ R, By the resolvent identity, this implies kR(2 + iβ, A)kL(X) ≤ C ′ so that the resolvent satisfies the resolvent decay condition (1.1). Moreover, if T : L1 (R+ ) → L(X) is the algebra homomorphism which is represented (in the strong sense) by the left-shift semigroup, then f = 1[0,1] ∈ range T ∗ . In particular, by Lemma 3.7 (d), the semigroup can not be continuous for t > t0 whenever t0 ∈ [0, 1). On the other hand, it follows from Theorem 4.2 (iv) that every function in X is supported in the interval [0, 1] so that the left-shift semigroup on X vanishes for t ≥ 0.  In the second example we show that there are also C0 -semigroups which are never norm-continuous, whose generator satisfies the resolvent decay condition (1.1), and the decay of the resolvent along vertical lines is even arbitrarily close to a logarithmic decay. Note that, by Lemma 5.1, a logarithmic decay as in Theorem 5.2 (i) is not possible for semigroups which are not eventually norm-continuous, that is, norm-continuous for t > t0 . OPERATORS L1 (R+ ) → X AND NORM CONTINUITY OF SEMIGROUPS 25 Theorem 5.3. Let h ∈ C(R+ ) be a positive, increasing and unbounded function such that also the function log+ /h is increasing and unbounded. Then there exists a Banach space X and a uniformly bounded C0 -semigroup (T (t))t≥0 ⊂ L(X) with generator A such that: (i) the resolvent satisfies the estimate kR(2 + iβ, A)k = O( h(|β|) ) log |β| as |β| → ∞, and in particular the resolvent satisfies the resolvent decay condition (1.1), and (ii) the semigroup (T (t))t≥0 is not eventually norm-continuous. Proof. Since the function log+ /h is increasing and unbounded, then also the function s → s1/h(s) = elog s/h(s) is increasing and unbounded for s ≥ 1. We may assume that this function is strictly increasing for s ≥ 1. In particular, the function ψ given by 1 (5.2) ψ(s1/h(s) ) := h(s), s ≥ 1, 4 is well-defined, increasing and unbounded. Choose coefficients an > 0 such that 1 ≥ an ≥ an+1 and such that ∞ X (5.3) an rn+1 ≤ rψ(r) for every r ≥ 1; n=0 it is an exercise to show that such coefficients exist (see also [36, Problem 2, p.1]). We put ∞ X an 1[n,n+1] . f= n=0 Then clearly f ∈ L∞ (R+ ) and it follows from (5.3) that the function λ 7→ eλ ⊛ f extends to an entire function. It is straightforward to show that for every λ ∈ C\{0} and every t ∈ R+ one has eλ ⊛ f (t) = ∞ X eλt e−λ([t]−t) − 1 (1 − e−λ ) an e−λn − a[t] λ λ n=[t] ∞ X e 1 − e−λ([t]+1−t) (1 − e−λ ) . an e−λn + a[t] λ λ λt = n=[t]+1 If Re λ < 0, this yields the estimate |eλ ⊛ f (t)| ≤ ≤ ≤ ∞ 2 2 X an+[t] e−Re λ(n+1−t) + |λ| n=0 |λ| ∞
2 X an e−Re λ(n+1) + 1 |λ| n=0
2 −Re λψ(e−Re λ ) e +2 , |λ| where in the second line we have used the fact that the sequence (an ) is decreasing, and in the third line we have used the estimate (5.3). If Re λ ≥ 0, then we obtain 26 RALPH CHILL AND YURI TOMILOV the estimate |eλ ⊛ f (t)| ≤ C |λ| for some C ≥ 1, so that keλ ⊛ f k∞ ≤
C −Re λψ(e−Re λ ) e + 2 for every λ ∈ C \ {0}. |λ| Let ϕ(β) := log+ β/h(β), β ≥ 0. By assumption, limβ→∞ ϕ(β) = ∞. Moreover, for every β ∈ R, sup keλ ⊛ f k∞ ≤ λ∈B(2+iβ,2+ϕ(β))
C −Re λψ(e−Re λ ) e +2 λ∈B(2+iβ,2+ϕ(β)) |λ| sup ϕ(|β|) ) eϕ(|β|)ψ(e +2 ≤ C . |β| − ϕ(|β|) For all β large enough we have ϕ(|β|) ≤ |β|. Moreover, if |β| ≥ 1, then eϕ(|β|)ψ(e ϕ(|β|) ) = |β|ψ(|β| 1/h(|β|) )/h(|β|) 1 = |β| 4 , by definition of the function ψ. Hence, if β is large enough, then sup 3 keλ ⊛ f k∞ ≤ 2C |β|− 4 . λ∈B(2+iβ,2+ϕ(β)) In particular, sup keλ ⊛ f k∞ < ∞. λ∈Σϕ Moreover, if we let, as before, λk = 2 + ik and dk = dist (λk , ∂Σϕ ), then 1 (ϕ(|k|) + 2) ≤ dk ≤ ϕ(|k|) + 2 for every k ∈ Z large enough 2 and sup keλ ⊛ f k∞ ≤ C λ∈B(λk ,dk ) 1 dk for every k ∈ Z. By Theorem 4.2, there exists a left-shift invariant Banach space X ֒→ L∞ (R+ ) such that the left-shift semigroup (T (t))t≥0 ⊂ L(X) is strongly continuous and such that the resolvent of the generator A satisfies the estimate kR(2 + ik, A)k ≤ C h(|k|) log |k| for every k ∈ Z. h(|β|) log |β| for every β ∈ R, By the resolvent identity, this implies kR(2 + iβ, A)k ≤ C ′ so that the resolvent decay condition (1.1) is satisfied. Moreover, still by Theorem 4.2, if T : L1 (R+ ) → L(X) is the algebra homomorphism which is represented (in the strong sense) by the semigroup (T (t))t≥0 , then f ∈ range T ∗ . Since the function is not continuous on any interval of the form (t0 , ∞), by Lemma 3.7 (d), the semigroup (T (t))t≥0 can not be eventually norm-continuous.  OPERATORS L1 (R+ ) → X AND NORM CONTINUITY OF SEMIGROUPS 27 References 1. W. Arendt, C. J. K. Batty, M. Hieber, and F. 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Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 23 (1975), no. 8, 895–898. Université Paul Verlaine - Metz, Laboratoire de Mathématiques et Applications de Metz - CNRS, UMR 7122, Bât. A, Ile du Saulcy, 57045 Metz Cedex 1, France E-mail address: [email protected] Faculty of Mathematics and Computer Science, Nicolas Copernicus University, ul. Chopina 12/18, 87-100 Torun, Poland E-mail address: [email protected]