aa r X i v : . [ m a t h . F A ] O c t A KATZNELSON-TZAFRIRI THEOREM FOR MEASURES
DAVID SEIFERT
Abstract.
This article generalises the well-known Katznelson-Tzafriritheorem for a C -semigroup T on a Banach space X , by removing theassumption that a certain measure in the original result be absolutelycontinuous. In an important special case the rate of decay is quantifiedin terms of the growth of the resolvent of the generator of T . Theseresults are closely related to ones obtained recently in the Hilbert spacesetting by Batty, Chill and Tomilov in [6]. The main new idea is to in-corporate an assumption on the non-analytic growth bound ζ ( T ) whichis equivalent to the assumption made in [6] if X is a Hilbert space. Introduction
The Katznelson-Tzafriri theorem is a cornerstone of the asymptotic theoryof operator semigroups. Given a function a ∈ L ( R ), define its Fouriertransform F a by(1.1) ( F a )( s ) = Z R e − i st a ( t ) d t for s ∈ R . Further, given a bounded C -semigroup T on a complex Banachspace X and a function a ∈ L ( R + ), define the operator b a ( T ) for x ∈ X by(1.2) b a ( T ) x = Z R + a ( t ) T ( t ) x d t. The version of the Katznelson-Tzafriri theorem that is of interest here cannow be stated as follows; see [17], [24] for proofs.
Theorem 1.1.
Let X be a complex Banach space and let T be a bounded C -semigroup on X with generator A . Suppose that i σ ( A ) ∩ R is of spectralsynthesis and let a ∈ L ( R + ) be such that ( F a )( s ) = 0 for all s ∈ i σ ( A ) ∩ R .Then k T ( t ) b a ( T ) k → as t → ∞ . Here a closed subset Λ of R is said to be of spectral synthesis if anyfunction a ∈ L ( R ) such that ( F a )( s ) = 0 for all s ∈ Λ can be approximatedin L -norm by elements of L ( R ) whose Fourier transforms vanish in an openneighbourhood of Λ. In the case where the underlying space is a Hilbert Mathematics Subject Classification.
Primary 47D06; Secondary 34G10, 34D05.
Key words and phrases. C -semigroup; asymptotics; Katznelson-Tzafriri theorem; non-analytic growth bound; rate of decay.This work was completed with financial support from the EPSRC. space, the following generalisation of Theorem 1.1 was recently obtained in[6, Theorem 6.14]. Theorem 1.2.
Let X be a complex Hilbert space and let T be a bounded C -semigroup on X with generator A . Suppose that i σ ( A ) ∩ R is of spectralsynthesis and that there exists R > such that { i s : | s | ≥ R } ⊂ ρ ( A ) and sup {k R (i s, A ) k : | s | ≥ R } < ∞ . Then, given any bounded Borel measure µ on R + such that ( F µ )( s ) = 0 for all s ∈ i σ ( A ) ∩ R , k T ( t ) b µ ( T ) k → as t → ∞ . Here the Fourier transform F µ and the operator b µ ( T ) are defined, for anybounded Borel measure on R and R + , respectively, and for any bounded C -semigroup T , by formulas analogous to (1.1) and (1.2), with the measure a ( t )d t replaced by d µ ( t ) in each case. Thus Theorem 1.2 generalises Theo-rem 1.1 in the Hilbert space setting by no longer requiring the measure µ tobe absolutely continuous with respect to Lebesgue measure. Remark 1.3.
It is shown in [22, Theorem 3.1] that the assumption ofspectral synthesis can be dropped in Theorem 1.1 when X is a Hilbert space.It remains open whether this is true also in the setting of Theorem 1.2.The purpose of this article is to extend Theorem 1.2 to Banach spaces,and this is achieved in Theorem 3.4 below. As it turns out, the condition inTheorem 1.2 on the resolvent operator needs to be replaced in the Banachspace case by a condition involving the so-called non-analytic growth bound ζ ( T ) of the C -semigroup T . This condition requires the semigroup to beclose in a certain asymptotic sense to an analytic operator-valued function,and consequently a C -semigroup T satisfying this condition will be saidto be asymptotically analytic . When X is a Hilbert space, this propertyis equivalent to the condition on the resolvent operator in Theorem 1.2.The main new idea in this paper is to use a characterisation of analyticanalyticity obtained in [5], which makes it possible to extend Theorem 1.2to the Banach space setting under this assumption.A particularly important instance of Theorem 1.2 arises when i σ ( A ) ∩ R = { } . In this case, the result shows that(1.3) lim t →∞ k T ( t ) AR (1 , A ) k = 0provided {k R (i , s, A ) k : | s | ≥ } is bounded, as can be seen by choosing µ = e − δ . Here δ denotes the Dirac mass at 0 and the function e is defined,for t ≥
0, by e ( t ) = e − t . This result, which provides the main motivationboth for the more general Theorem 1.2 and for the present paper, is ofespecial interest in connection with the abstract Cauchy problem associated KATZNELSON-TZAFRIRI THEOREM FOR MEASURES 3 with T , namely(1.4) ( ˙ u ( t ) = Au ( t ) , t ≥ u (0) = x, where x ∈ X . The solution of (1.4) is given by u ( t ) = T ( t ) x for all t ≥
0. Forgeneral x ∈ X , this function u : R + → X solves (1.4) only in the mild sense,however if x ∈ D ( A ) then u is a classical solution of (1.4). In particular, u is then differentiable with derivative ˙ u ( t ) = T ( t ) Ax for t ≥
0. Since theresolvent operator R (1 , A ) maps X onto D ( A ), (1.3) is simply another wayof saying that classical solutions of (1.4) have derivatives which decay to zerouniformly for all x ∈ D ( A ) with graph norm k x k + k Ax k ≤
1. Examplesof Cauchy problems in which i σ ( A ) ∩ R = { } arise naturally. Consider forinstance the problem ∂ u∂t − ∆ u = 0 , x ∈ Ω , t > ,∂u∂n + Z t a ( t − s ) ∂u∂t ( x, s ) d s = 0 , x ∈ ∂ Ω , t > , where Ω is a bounded open subset of R n for some n ≥ ∂u∂n denotes the outward normal derivative and a denotes asufficiently well-behaved function defined on R + . If u is interpreted as acous-tic pressure, the equation can be understood as modelling the evolution ofsound in a compressible medium with viscoelastic surface. It is shown in[13] that the associated semigroup generator A corresponding to this prob-lem has boundary spectrum σ ( A ) ∩ i R ⊂ { } and is in general non-empty;see also [1], [4] and [14]. As is shown in [6, Theorem 6.15], it is possible toobtain estimates on the rate of decay in (1.3) when X is a Hilbert space.Theorem 3.7 below extends this result to the Banach space setting for theclass of asymptotically analytic semigroups. For further background mate-rial on the problem at hand, including its connection with a closely relatedproblem having applications to damped wave equations, see [6, Section 6].The new results in this paper are contained in Section 3 below. First, Sec-tion 2 provides some context for these main results by providing an overviewof some of the results obtained in [6] concerning the rate of decay in the casewhere i σ ( A ) ∩ R = { } . The paper concludes with some remarks and openquestions, which are collected in Section 4. The notation used throughoutis standard except where introduced explicitly. In particular, given a closedlinear operator A on a complex Banach space X , the domain of A is de-noted by D ( A ), the spectrum and resolvent sets of A are denoted by σ ( A )and ρ ( A ), respectively, and for λ ∈ ρ ( A ) the resolvent operator is written as R ( λ, A ) = ( λ − A ) − . DAVID SEIFERT Background on the case of one-point boundary spectrum
Let T be a bounded C -semigroup with generator A on a complex Banachspace X and suppose that i σ ( A ) ∩ R = { } . This section collects togetherseveral results from [6] which relate the rate of decay of k T ( t ) AR (1 , A ) k as t → ∞ to the behaviour of k R (i s, A ) k as | s | →
0. In order to makethe relationship precise, it is convenient to have in place two pieces of non-standard notation. Thus a decreasing function m : (0 , → (0 , ∞ ) such that k R (i s, A ) k ≤ m ( | s | ) for 0 < | s | ≤ dominating function(for the resolvent of A ) . Likewise a decreasing function ω : R + → (0 , ∞ )such that k T ( t ) AR (1 , A ) k ≤ ω ( t ) for all t ≥ dominatingfunction (for T ) . The minimal dominating functions are given, for s ∈ (0 , t ≥
0, by(2.1) m ( s ) = sup (cid:8) k R (i r, A ) k : s ≤ | r | ≤ (cid:9) ,ω ( t ) = sup (cid:8) k T ( s ) AR (1 , A ) k : s ≥ t (cid:9) , respectively. The function m defined in (2.1) is continuous and in whatfollows the same will be assumed to be true of any dominating function m .In particular, any such dominating function m possesses a right-inverse m − defined on the range of m . On the other hand, given a dominating function ω for T such that ω ( t ) → t → ∞ , let the function ω ∗ : (0 , ∞ ) → R + begiven by(2.2) ω ∗ ( s ) = min (cid:8) t ≥ ω ( t ) ≤ s (cid:9) . Then ω ( ω ∗ ( s )) ≤ s for all s >
0, with equality for all s in the range of ω .For T and A as above, it follows from the elementary properties of re-solvent operators and the fact that 0 ∈ σ ( A ) that m ( s ) ≥ s − for anydominating function m for the resolvent of A . As is shown in [6, Exam-ple 6.7] by means of a simple direct sum construction, dominating functions ω for T need not satisfy any such lower bound and indeed can decay ar-bitrarily slowly, even for semigroups of normal operators on Hilbert space.On the other hand, the following result shows that cases in which a dom-inating function ω for T decays faster than t − must be precisely of thistype involving a direct sum. The proof, which relies on a simple spectralsplitting argument combined with a result from [18], can be found in [6,Theorem 6.9]. Theorem 2.1.
Let X be a complex Banach space and let T be a C -semigroup on X with generator A . Suppose that i σ ( A ) ∩ R = { } . Theneither lim sup t →∞ t k T ( t ) AR (1 , A ) k > or there exist closed T -invariant subspaces X and X of X such that X ⊂ Fix( T ) , the restriction A of A to X is invertible, and X = X ⊕ X . KATZNELSON-TZAFRIRI THEOREM FOR MEASURES 5
The following result provides an estimate on the size of k R (i s, A ) k forsmall and large values of | s | ; see [6, Theorem 6.10] for a proof. Theorem 2.2.
Let X be a complex Banach space and let T be a bounded C -semigroup on X with generator A . Suppose ω is a dominating functionfor T such that ω ( t ) → as t → ∞ , and let ω ∗ be defined as in (2.2) . Then i σ ( A ) ∩ R ⊂ { } , sup {k R (i s, A ) k : | s | ≥ } < ∞ and, for any c ∈ (0 , , k R (i s, A ) k = O (cid:18) | s | + ω ∗ ( cs ) (cid:19) as s → . This result in turn leads to the following bound on how fast the quantity k T ( t ) AR (1 , A ) k can decay as t → ∞ ; see [6, Corollary 6.11]. Corollary 2.3.
Let X be a complex Banach space and let T be a bounded C -semigroup on X with generator A . Suppose that i σ ( A ) ∩ R = { } andthat (2.3) lim s → max (cid:8) k sR (i s, A ) k , k sR ( − i s, A ) k (cid:9) = ∞ , and let m be the minimal dominating function for the resolvent of A definedin (2.1) . Then, given any right-inverse m − of m , there exist constants c, C > such that k T ( t ) AR (1 , A ) k ≥ cm − ( Ct ) for all sufficiently large t ≥ . Remark 2.4.
A similar argument to that used in [6, Theorem 6.10] showsthat, given any constant
K > M , where M is as above, there exists c ∈ (0 , k R (i s, A ) k ≤ K (cid:18) | s | + ω ∗ ( c | s | ) (cid:19) whenever | s | is sufficiently small. Thus the conclusion (2.3) in fact remainstrue if (2.3) is replaced by the weaker condition that L > M , where M isas above and L = lim inf s → max (cid:8) k sR (i s, A ) k , k sR ( − i s, A ) k (cid:9) . Taking A = 0 shows that the conclusion can become false when L = M ; seealso [6, Remark 6.12]The next result shows that it is not necessarily possible even for semi-groups of normal operators on a Hilbert space to obtain a correspondingupper bound in terms of m − for the quantity k T ( t ) AR (1 , A ) k as t → ∞ ;see [6, Theorem 6.13] for a proof. Theorem 2.5.
Let X be a complex Hilbert space and let T be a bounded C -semigroup of normal operators on X with generator A . Suppose that i σ ( A ) ∩ R = { } and that sup {k R (i s, A ) k : | s | ≥ } < ∞ . Furthermore, let DAVID SEIFERT m be the minimal dominating function for the resolvent of A and let m − be any right-inverse of m . Then, given any constant c > , k T ( t ) AR (1 , A ) k ≤ O ( m − ( ct )) as t → ∞ if and only if there exists a constant C > such that m ( s ) m ( t ) ≥ c log (cid:18) ts (cid:19) − C for < s, t ≤ . Remark 2.6.
For analogues of the above results in the discrete setting ofthe classical Katznelson-Tzafriri theorem see [21].3.
Main results
The objective now is to establish an upper bound on k T ( t ) AR (1 , A ) k as t → ∞ , where T is a bounded C -semigroup on a complex Banach space X whose generator A satisfies i σ ( A ) ∩ R = { } . A result of this type will beobtained in Theorem 3.7 below, providing a counterpart to Corollary 2.3.As in [6], Theorem 3.7 will follow from a more general result by refining itsproof in the special case at hand. This more general result is Theorem 3.4below, which is analogous to Theorem 1.2 but does not require X to be aHilbert space. On the other hand, the assumption on the resolvent operatorthat appears in the statement of Theorem 1.2 needs to be modified, and thisrequires some further notation.Given a complex Banach space X and a set Ω ⊂ C , a function S : Ω →B ( X ) will be said to be exponentially bounded if there exist constants C ≥ ω ∈ R such that k S ( λ ) k ≤ C e ω | λ | for all λ ∈ Ω. The space ofall exponentially bounded holomorphic B ( X )-valued functions on Ω will bedenoted by H (Ω; B ( X )). Further, given any exponentially bounded function S defined on (0 , ∞ ), let its growth bound ω ( S ) be given by ω ( S ) = inf (cid:8) ω ∈ R : k S ( t ) k ≤ M e ωt for some M ≥ t > (cid:9) . The non-analytic growth bound ζ ( T ) of a C -semigroup T is defined as(3.1) ζ ( T ) = inf (cid:8) ω ( T − S ) : S ∈ H (Σ θ ; B ( X )) for some θ > (cid:9) , where Σ θ = { λ ∈ C \{ } : | arg( λ ) | < θ } . Thus the non-analytic growthbound measures the degree to which T can, or rather cannot , be approxi-mated asymptotically by exponentially bounded analytic functions definedon certain sectors. It is clear that ζ ( T ) ≤ ω ( T ) for any C -semigroup T andthat ζ ( T ) = −∞ when T is analytic. It is shown in [5, Theorem 5.7] that ζ ( T ) = −∞ also if T is eventually differentiable or if T has an L p -resolventfor some p ∈ (1 , ∞ ), in the sense that there exist α ∈ R and β ≥ KATZNELSON-TZAFRIRI THEOREM FOR MEASURES 7 { λ ∈ C : Re λ ≥ α, | Im λ | ≥ β } ⊂ ρ ( A ) and Z | s |≥ β k R ( α + i s, A ) k p d s < ∞ . Furthermore, if ω ess ( T ) = inf (cid:8) ω ∈ R : k T ( t ) k ess ≤ M e ωt for some M ≥ t ≥ (cid:9) denotes the essential growth bound of T , where for Q ∈ B ( X ) k Q k ess = inf {k Q − K k : K ∈ B ( X ) is compact } , then ζ ( T ) ≤ ω ess ( T ); see [5, Proposition 5.3]. A C -semigroup T will be saidto be asymptotically analytic if ζ ( T ) < α ∈ R and β ≥
0, let Q α,β = { λ ∈ C : Re λ ≥ α, | Im λ | ≥ β } and,for any semigroup generator A , let s ∞ ( A ) = inf (cid:8) α ∈ R : Q α,β ⊂ ρ ( A ) and k R ( λ, A ) k is uniformlybounded on Q α,β for some β ≥ (cid:9) . It is shown in [5, Proposition 2.4] that s ∞ ( A ) ≤ ζ ( T ). When X is a Hilbertspace, the following non-analytic analogue of the Gearhart-Pr¨uss theoremholds; see [8, Example 3.12]. Theorem 3.1.
Let X be a complex Hilbert space and let T be a C -semigroupon X with generator A . Then s ∞ ( A ) = ζ ( T ) . The proof of this result relies on Plancherel’s theorem and the equivalenceof conditions (i) and (iii) in the following result, which will be crucial in whatfollows; see [8, Theorem 3.6] for a proof. Here L ( R ; B ( X )) denotes the spaceof maps S : R → B ( X ) such that t S ( t ) x is Bochner measurable for all x ∈ X and such that there exists g ∈ L ( R ) with k S ( t ) k ≤ g ( t ) for almost all t ∈ R . Furthermore, given a semigroup generator A satisfying s ∞ ( A ) < φ ∈ C ∞ ( R ) will be said to be compatible with A if thereexists a bounded open interval I ⊂ R satisfying { s ∈ R : i s ∈ σ ( A ) } ⊂ I andif φ ( s ) = 0 for all s ∈ I and φ ( s ) = 1 when | s | is sufficiently large. Theorem 3.2.
Let X be a complex Banach space and let T be a C -semigroup on X with generator A . Suppose that ≤ p < ∞ . Then thefollowing are equivalent:(i) T is asymptotically analytic;(ii) s ∞ ( A ) < and, for any compatible function φ ∈ C ∞ ( R ) , thereexists a map S ∈ L ( R ; B ( X )) such that φ ( s ) R (i s, A ) = ( F S )( s ) forall s ∈ R ;(iii) s ∞ ( A ) < and, for any compatible function φ ∈ C ∞ ( R ) , the map s φ ( s ) R (i s, A ) is a Fourier multiplier on L p ( R ; X ) . Here φ ( s ) R (i s, A ) is taken to be zero whenever φ ( s ) = 0 and the Fouriertransform F is taken in the strong sense, so that, given S ∈ L ( R ; B ( X )) DAVID SEIFERT and s ∈ R , ( F S )( s ) x = Z R e − i st S ( t ) x d t for all x ∈ X . Note also that, even though the maps s φ ( s ) R (i s, A )depend on the choice of φ , the property of such a map being a Fouriermultiplier on L p ( R ; X ) is independent of this choice; see [8, Remark 2.2].It is now possible to extend Theorem 1.2 to the Banach space setting, asis done in Theorem 3.4 below. The proof of this result combines the methodused in [6] with Theorem 3.2 and the following simple lemma, which isprobably well known. Here S ( R ) denotes the space of Schwartz functionson R . Lemma 3.3.
Let X be a complex Banach space and let S ∈ L ( R ; X ) .(a) If µ is a bounded Borel measure on R , then µ ∗ S ∈ L ( R ; X ) .(b) If (3.2) Z R ρ ( t ) S ( t ) d t = 0 for all ρ ∈ S ( R ) , then S ( t ) = 0 for almost all t ∈ R . Proof . Part (a) is a special case of the vector-valued version of Fubini’stheorem; see for instance [3, Theorem 1.1.9] and [15, Chapter III, Section 11,Theorem 9]. For a direct proof, let the map F : R → X be given by F ( s, t ) = S ( t − s ) and note that, by Pettis’ theorem (see [3, Theorem 1.1.1]),there is no loss of generality in assuming that X is separable. Moreover, themap F is measurable with respect to the product measure of µ and theLebesgue measure on R , as can be seen for instance by another applicationof Pettis’ theorem and a simple approximation argument based on the factthat, given any Lebesgue measurable subset E of R , there exists a Borelmeasurable set E ′ ⊂ R such that the symmetric difference E △ E ′ is null.Since(3.3) Z R Z R k F ( s, t ) k d t d | µ | ( s ) < ∞ , it follows form the scalar-valued versions of Tonelli’s and Fubini’s theoremsthat ( s, t )
7→ k F ( s, t ) k is integrable over R with respect to the productmeasure and that R R k F ( s, t ) k d µ ( s ) exists for almost all t ∈ R . Hence sodoes R R F ( s, t ) d µ ( s ). Furthermore, for each φ ∈ X ∗ , the map t Z R φ ( F ( s, t )) d µ ( s ) = φ (cid:18)Z R F ( s, t ) d µ ( s ) (cid:19) is measurable on R . By Pettis’ theorem, the map µ ∗ S : t R R F ( s, t ) d µ ( s )is Bochner measurable on R , and the result now follows, since Z R k ( µ ∗ S )( t ) k d t ≤ Z R Z R k F ( s, t ) k d | µ | ( s ) d t < ∞ KATZNELSON-TZAFRIRI THEOREM FOR MEASURES 9 by the scalar-valued version of Fubini’s theorem and (3.3).The proof of (b) runs along similar lines. Once again, by Pettis’ theorem,it is possible to assume that X is separable. Let { x n : n ≥ } be a densesubset of X and, for each n ≥
1, let φ n ∈ X ∗ be such that k φ n k = 1 and φ n ( x n ) = k x n k . The existence of such functionals is a consequence of theHahn-Banach theorem. Applying φ n to both sides of (3.2) shows that Z R ρ ( t ) φ n ( S ( t )) d t = 0for all ρ ∈ S ( R ), and hence, for each n ≥
1, there exists a null subset E n of R such that φ n ( S ( t )) = 0 for all t ∈ R \ E n . Let E = S n ≥ E n , noting that E itself is null, and suppose that t ∈ R \ E . Given ε >
0, there exists n ≥ k x n − S ( t ) k < ε/
2. Thus k S ( t ) k ≤ k x n k + k x n − S ( t ) k < φ n ( x n − S ( t )) + ε < ε. Since ε > S ( t ) = 0 for all t ∈ R \ E . (cid:3) The next result, then, is an unquantified version of the Katznelson-Tzafriritheorem for measures which generalises Theorem 1.2 to the Banach spacesetting.
Theorem 3.4.
Let X be a complex Banach space and let T be a boundedasymptotically analytic C -semigroup X with generator A . Suppose that i σ ( A ) ∩ R is of spectral synthesis and let µ be any bounded Borel measure on R + such that ( F µ )( s ) = 0 for all s ∈ i σ ( A ) ∩ R . Then k T ( t ) b µ ( T ) k → as t → ∞ .Proof. Since s ∞ ( A ) ≤ ζ ( T ), the assumption of asymptotic analyticity im-plies that { i s : | s | ≥ R } ⊂ ρ ( A ) and that sup {k R (i s, A ) k : | s | ≥ R } < ∞ forsome sufficiently large R >
0. Fix a function ϕ ∈ C ∞ c ( R ) such that ϕ ( s ) = 1for | s | ≤ R and let ψ = F − ϕ . Then ψ ∈ S ( R ), since C ∞ c ( R ) ⊂ S ( R ) and F maps S ( R ) bijectively onto itself. Furthermore, let the bounded Borelmeasures ν and ξ on R be defined by ν = µ ∗ ψ and ξ = µ ∗ ( δ − ψ ), so that µ = ν + ξ , and let the functions F, G : R → B ( X ) be given by(3.4) F ( t ) = Z R T ( s + t ) d ν ( s ) and G ( t ) = Z R T ( s + t ) d ξ ( s ) . Here the semigroup has been extended by zero on ( −∞ ,
0) and the integralsare to be understood in the strong sense. It is clear that T ( t ) b µ ( T ) = F ( t ) + G ( t ) for all t ≥
0, and hence the result will follow once it has been establishedthat both k F ( t ) k → k G ( t ) k → t → ∞ .Consider first the function F . Since the measure ν is absolutely continu-ous with respect to Lebesgue measure on R , there exists a function a ∈ L ( R )such that d ν ( t ) = a ( t )d t , and hence ( F a )( s ) = ( F µ )( s ) ϕ ( s ) for all s ∈ R . Inparticular, F a vanishes on i σ ( A ) ∩ R . By assumption this set is of spectral synthesis, and hence there exist functions a n ∈ L ( R ), n ≥
1, such that F a n vanishes in a neighbourhood of i σ ( A ) ∩ R for each n ≥ k a n − a k → n → ∞ . Since functions with compactly supported Fourier transform aredense in L ( R ), there is no loss of generality in assuming that each F a n hascompact support. By the dominated convergence theorem and Parseval’sidentity, Z R a n ( s ) T ( t + s ) d s = lim α → Z R a n ( s − t )e − αs T ( s ) d s = lim α → π Z R e i st ( F a n )( − s ) R ( α + i s, A ) d s = 12 π Z R e i st ( F a n )( − s ) R (i s, A ) d s for all t ∈ R and n ≥ . Since the last integral exists as a Bochner integralin B ( X ), (cid:13)(cid:13)(cid:13)(cid:13)Z R a n ( s ) T ( t + s ) d s (cid:13)(cid:13)(cid:13)(cid:13) → t → ∞ by the Riemann-Lebesgue Lemma. Now (cid:13)(cid:13)(cid:13)(cid:13) F ( t ) − Z R a n ( s ) T ( t + s ) d s (cid:13)(cid:13)(cid:13)(cid:13) ≤ M k a − a n k for all t ∈ R , and hence (cid:13)(cid:13)(cid:13)(cid:13) F ( t ) − Z R a n ( s ) T ( t + s ) d s (cid:13)(cid:13)(cid:13)(cid:13) → n → ∞ , uniformly for t ∈ R . It follows that k F ( t ) k → t → ∞ .Now consider the function G . Since the map φ ∈ C ∞ ( R ) given by φ ( s ) =1 − ϕ ( − s ) is compatible with A , it follows from Theorem 3.2 that thereexists a map S ∈ L ( R ; B ( X )) such that φ ( s ) R (i s, A ) = ( F S )( s ) for all s ∈ R . Writing µ ′ for the bounded Borel measure on ( −∞ ,
0] satisfying µ ′ ( E ) = µ ( − E ) for any Borel subset E of ( −∞ , µ ′ ∗ S ∈ L ( R ; B ( X )) . For ρ ∈ S ( R ) and x ∈ X , it followsby Fubini’s theorem, the dominated convergence theorem and Parseval’s KATZNELSON-TZAFRIRI THEOREM FOR MEASURES 11 identity that Z R ρ ( t ) G ( t ) x d t = Z R ρ ( t ) Z R T ( s + t ) x d ξ ( s ) d t = lim α → Z R Z R ρ ( t − s )e − αt T ( t ) x d t d ξ ( s )= lim α → Z R Z R e i st ( F − ρ )( t ) R ( α + i t, A ) x d t d ξ ( s )= lim α → Z R ( F − ρ )( t )( F µ )( − t )(1 − ϕ ( − t )) R ( α + i t, A ) x d t = Z R ( F − ρ )( t )( F µ )( − t )( F S )( t ) x d t = Z R ρ ( t ) S µ ( t ) x d t, where S µ = µ ′ ∗ S . Fix x ∈ X and, for t ∈ R , let S x ∈ L ( R ; X ) be given by S x ( t ) = G ( t ) x − S µ ( t ) x . Then S x ( t ) = 0 for almost all t ∈ R by part (b) ofLemma 3.3, and hence G ( · ) x ∈ L ( R ; X ). Since the map Φ : X → L ( R ; X )given by Φ( x ) = G ( · ) x has closed graph, it follows from the closed graphtheorem that, for some constant C > Z R k G ( t ) x k d t ≤ C k x k for all x ∈ X . Now, given any x ∈ X and t ≥ Z t T ( t − s ) G ( s ) x d s = Z t Z R − Z [ − t, − s ) ! T ( t + r ) x d ξ ( r )d s = tG ( t ) x + Z [ − t, rT ( t + r ) x d ξ ( r )and hence, for t > G ( t ) x = 1 t Z t T ( t − s ) G ( s ) x d s − Z [ − t, rT ( t + r ) x d ξ ( r ) ! . By (3.5),(3.7) (cid:13)(cid:13)(cid:13)(cid:13)Z t T ( t − s ) G ( s ) d s (cid:13)(cid:13)(cid:13)(cid:13) ≤ M C, where M = sup {k T ( t ) k : t ≥ } , and the dominated convergence theoremgives (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)Z [ − t, rt T ( t + r ) d ξ ( r ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) → t → ∞ . Thus k G ( t ) k → t → ∞ and the result follows. (cid:3) Remark 3.5.
As observed in [8, Remark 3.8], the map S ∈ L ( R ; B ( X )) inthe above proof can be written down explicitly. Indeed, given x ∈ X , S ( t ) x = T ( t ) x − T R ( t ) x − π i Z | s |≥ R e i st ϕ ( − s ) R (i s, A ) x d s for almost all t ∈ R . Here ϕ and R are as above, the semigroup T has againbeen extended by zero on ( −∞ ,
0) and T R ( t ) = 12 π i Z Γ R e λt R ( λ, A ) d λ for any path Γ R that connects the points ± i R and otherwise lies in { λ ∈ C :Re λ > } . By Cauchy’s theorem, the definition of T R is independent of thechoice of the contour Γ R .As stated near the beginning of this section, there are several importantcases in which a semigroup is known to be asymptotically analytic. The firstpart of the following corollary corresponds to [6, Theorem 6.14]. Note thata C -semigroup T is said to be uniformly exponentially balancing if thereexist ω ∈ R and a non-zero P ∈ B ( X ) such that k e − ωt T ( t ) − P k → t → ∞ , and recall that T is said to be quasi-compact if ω ess ( T ) < ω ( T ). Corollary 3.6.
Let X be a complex Banach space and let T be a bounded C -semigroup X with generator A . Suppose that i σ ( A ) ∩ R is of spectral syn-thesis, and let µ be any bounded Borel measure on R + such that ( F µ )( s ) = 0 for all s ∈ i σ ( A ) ∩ R . Then lim t →∞ k T ( t ) b µ ( T ) k = 0 provided one of the following conditions is satisfied:(i) X is a Hilbert space and there exists R > such that { i s : | s | ≥ R } ⊂ ρ ( A ) and sup {k R (i s, A ) k : | s | ≥ R } < ∞ ;(ii) T is uniformly exponentially balancing;(iii) T is quasi-compact;(iv) T has L p -resolvent for some p ∈ (1 , ∞ ) ;(v) T is eventually differentiable.Proof. It suffices to show that in each of the cases T is asymptotically an-alytic, since the result will then follow from Theorem 3.4. In the first casethis is a consequence of Theorem 3.1, since the assumption on the resolventimplies, by a simple Neumann series argument, that s ∞ ( A ) <
0. In thesecond case the claim follows from the fact that for uniformly exponentiallybalancing semigroups ζ ( T ) < ω ( T ) (see [9, Proposition 4.5.9] and also [23]),while in case (iii) it is a consequence of the estimate ζ ( T ) ≤ ω ess ( T ). If either(iv) or (v) holds, then ζ ( T ) = −∞ , as mentioned above. (cid:3) The final result of this section is a quantified version of Theorem 3.4 in thespecial case where, in the notation of Section 1, µ = e − δ so that b µ ( T ) = KATZNELSON-TZAFRIRI THEOREM FOR MEASURES 13 AR (1 , A ). This result is a generalisation to the Banach space setting of [6,Theorem 6.15] and provides a counterpart to Corollary 2.3. It is possibleto formulate versions of the result in which the assumption of asymptoticanalyticity is replaced by one of the conditions (i)–(v) of Corollary 3.6. Theorem 3.7.
Let X be a complex Banach space and let T be a boundedasymptotically analytic C -semigroup X with generator A . Suppose that i σ ( A ) ∩ R = { } . Then k T ( t ) AR (1 , A ) k → as t → ∞ . In fact, given anydominating function m for the resolvent of A , any right-inverse m − of m and any ε ∈ (0 , , (3.8) k T ( t ) AR (1 , A ) k = O (cid:0) m − ( t − ε ) (cid:1) as t → ∞ .Proof. Since the set { } is of spectral synthesis, the first statement followsimmediately from Theorem 3.4 applied to the bounded Borel measure on R + given by µ = e − δ , whose Fourier transform is given by( F µ )( s ) = − i s s . The quantified statement follows from a modification of the argumentgiven in the proof of Theorem 3.4. Fix a function ϕ ∈ C ∞ c ( R ) such that ϕ ( s ) = 1 for | s | ≤ ψ ∈ S ( R ) be given by ψ = F − ϕ . As inthe proof of Theorem 3.4, let the bounded Borel measures ν and ξ on R be defined by ν = µ ∗ ψ and ξ = µ ∗ ( δ − ψ ). Then µ = ν + ξ and T ( t ) b µ ( T ) = F ( t ) + G ( t ) for all t ≥
0, where
F, G : R → B ( X ) are as definedin (3.4). As before, the measure ν is absolutely continuous with respect toLebesgue measure on R and hence there exists a function a ∈ L ( R ) suchthat d ν ( t ) = a ( t )d t . In fact,(3.9) a ( t ) = Z ∞ ψ ( t − s )e − s d s − ψ ( t )for almost all t ∈ R . As in the proof of [6, Theorem 6.15], a refinement ofthe argument used in the proof of Theorem 3.4 to show that k F ( t ) k → t → ∞ in fact gives that, for any integer k ≥ k F ( t ) k = O (cid:18) t ε ( k +1) − + m − ( t − ε ) (cid:19) as t → ∞ .In order to obtain an estimate on k G ( t ) k as t → ∞ , note first that, since µ is supported in R + , ξ coincides with − ν on ( −∞ , M = sup {k T ( t ) k : t ≥ } , (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)Z [ − t, rT ( t + r ) d ξ ( r ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ M Z ( −∞ , | r | d | ξ | ( r ) for all t ≥ t ≥ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)Z [ − t, rT ( t + r ) d ξ ( r ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ M Z −∞ | r | (cid:12)(cid:12)(cid:12)(cid:12) ψ ( r ) − Z ∞ ψ ( r − s )e − s d s (cid:12)(cid:12)(cid:12)(cid:12) d r < ∞ . It follows from (3.6) and (3.7) that(3.11) k G ( t ) k = O ( t − )as t → ∞ . To conclude the proof, choose k ≥ ε ( k +1) ≥ m − ( t − ε ) ≥ t ε − ≥ t − for t ≥
1, it follows from (3.10) and (3.11)that (3.8) holds. (cid:3)
Remark 3.8.
From the point of view of decay rates the case i σ ( A ) ∩ R = ∅ ,which could be dealt with in a similar way, is of limited interest in Theo-rem 3.7. Indeed, [5, Proposition 2.4] shows that ω ( T ) = max { ζ ( T ) , s ( A ) } ,and hence the rate of decay in such cases is necessarily at least exponentialif T is asymptotically analytic.4. Concluding remarks
This final section contains some remarks and open questions. The firstconcerns the rate of decay obtained in Theorem 3.7, the rest relate to theproperty of asymptotic analyticity of a C -semigroup T . Throughout, X will be a complex Banach space and T a bounded C -semigroup on X withgenerator A .4.1. Non-optimality of the rate of decay.
Theorem 2.5 shows that onecannot in general hope for an upper bound in terms of m − ( Ct ) for anyconstant C > m ( s ) grows very rapidly as s →
0+ but that itis not optimal in general. In the special case where m ( s ) = Cs − α for some C > α ≥
1, it is shown by a completely different method in [6,Theorem 7.6] that the right-hand side of (3.8) can be replaced by the optimal O ( t − /α ) when X is a Hilbert space. Results such as [7, Theorem 1.5], [12,Theorem 4.1] and [19, Proposition 3.1] (see also [21] for analogous resultsin the discrete setting) suggest that for general Banach spaces it might bepossible to obtain an estimate differing only by a logarithmic factor fromthis polynomial rate of decay rather than having the wrong power as in(3.8). It remains an open question whether this is indeed the case.4.2. Necessity of asymptotic analyticity.
Theorem 2.2 and a simpleNeumann series argument show that (1.3) implies s ∞ ( A ) <
0. If X is aHilbert space, it follows from Theorem 3.1 that T is asymptotically analyticwhenever (1.3) holds. The same conclusion holds if T is hyperbolic (see forinstance [16, Section V.1.c] for background), since in this case T is asymp-totically analytic if and only if s ∞ ( A ) <
0, by [9, Proposition 3.5.2]. It
KATZNELSON-TZAFRIRI THEOREM FOR MEASURES 15 remains unknown whether (1.3) implies asymptotic analyticity for general C -semigroups acting on arbitrary Banach spaces.This question is related to a further open question which concerns thenon-analytic growth bound ζ ( T ) and another growth bound associated with T . Indeed, allowing the function S in (3.1) to vary over all functions S :(0 , ∞ ) → B ( X ) which are continuous with respect to the norm topologyon B ( X ) gives the definition of δ ( T ), which is referred to sometimes as the critical growth bound . It is clear from the definitions that δ ( T ) ≤ ζ ( T ).Furthermore, s ∞ ( A ) ≤ δ ( T ) so that, if X is a Hilbert space, δ ( T ) = ζ ( T ) = s ∞ ( A ) by Theorem 3.1. These results along with other cases in which δ ( T )and ζ ( T ) coincide are contained in [5, Section 5], but it remains open whetherthis is always the case. The following result shows that if it were known that ζ ( T ) = δ ( T ) at least when i σ ( A ) ∩ R = { } , then asymptotic analyticitywould be not only sufficient but also necessary for (1.3) to hold in this case.Since s ∞ ( A ) ≤ δ ( T ), the result strengthens the above conclusion derivedfrom Theorem 2.2 that s ∞ ( A ) < Proposition 4.1.
Let X be a complex Banach space and let T be a bounded C -semigroup on X with generator A . Suppose that (1.3) holds. Then δ ( T ) < .Proof. Note that, for t ≥ T ( t ) = T ( t ) R (1 , A ) − T ( t ) AR (1 , A ) . The first term on the right-hand side is continuous in t with respect to thenorm topology on B ( X ). Letting D ( t ) = lim sup h → k T ( t + h ) − T ( t ) k for t ≥
0, it follows that D ( t ) ≤ lim sup h → k ( T ( t + h ) − T ( t )) AR (1 , A ) k ≤ D (0) k T ( t ) AR (1 , A ) k , and hence D ( t ) → t → ∞ by the assumption that (1.3) holds. Since D is submultiplicative by [10, Proposition 3.5], it follows that ω ( D ) <
0. Butby [5, Proposition 5.1] ω ( D ) = δ ( T ), which gives the result. (cid:3) Combining this observation with [9, Corollary 4.5.6], it follows that anybounded C -semigroup T which satisfies (1.3) and whose generator A sat-isfies i σ ( A ) ∩ ( R + i ω ) = ∅ for some ω ∈ ( δ ( T ) ,
0) must be asymptoticallyanalytic. It follows from Theorem 2.2 that this is true in particular if { } isan isolated point of σ ( A ).4.3. Towards an alternative characterisation of asymptotic analyt-icity.
The following result is proved in [10, Theorem 3.6]; see also [20,Proposition 4.3] and [2]. Here, given ω ∈ R , Ω ω denotes the set { λ ∈ C : | λ | > e ω } . Theorem 4.2.
Let X be a complex Banach space and let T be a C -semigroup on X with generator A . Then σ ( T ( t )) ∩ Ω tδ ( T ) = exp( tσ ( A )) ∩ Ω tδ ( T ) for all t > . Now, letting s ∞ ( A ) = inf (cid:8) α ∈ R : Q α,β ⊂ ρ ( A ) for some β ≥ (cid:9) , it is clear that s ∞ ( A ) ≤ s ∞ ( A ) and hence s ∞ ( A ) ≤ ζ ( T ). It follows fromTheorem 4.2 that − ∈ ρ ( T ( t )) for all sufficiently small t > T is asymp-totically analytic, since in this case δ ( T ) <
0. The following partial converseis proved in [9, Sections 4.3 and 4.5]; see also [11].
Theorem 4.3.
Let X be a complex Banach space and let T be a bounded C -semigroup on X with generator A . Suppose there exists a non-null set I ⊂ R + such that − ∈ ρ ( T ( t )) for all t ∈ I . Then s ∞ ( A ) < . In particular, when combined with Theorem 3.1 this shows that if T is abounded C -semigroup and X is a Hilbert space, then T is asymptoticallyanalytic if and only if − ∈ ρ ( T ( t )) for all sufficiently small t >
0. Ona general Banach space, no such characterisation is known. Nevertheless,Theorems 4.2 and 4.3 suggest at least loosely that the property of asymptoticanalyticity of a C -semigroup T is connected with some notion of regularityof the operators T ( t ) for small t > Acknowledgements
The author would like to express his thanks to Professor C.J.K. Battyfor his guidance and an observation that led to Proposition 4.1, and to theanonymous referee for his or her helpful remarks.
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