Let T : H 1 (R) → H 1 (R) be a bounded Fourier multiplier on the analytic Hardy space H 1 (R) ⊂ L 1 (R) and let m ∈ L ∞ (R + ) be its symbol, that is, T (h) = m h for all h ∈ H 1 (R). Let S 1 be the Banach space of all trace class operators on ℓ 2 . We show that T admits a bounded tensor extension T ⊗I S1 : H 1 (R; S 1 ) → H 1 (R; S 1 ) if and only if there exist a Hilbert space H and two functions α, β ∈ L ∞ (R + ; H) such that m(s + t) = α(t), β(s) H for almost every (s, t) ∈ R 2 + . Such Fourier multipliers are called S 1 -bounded and we let M S 1 (H 1 (R)) denote the Banach space of all S 1 -bounded Fourier multipliers. Next we apply this result to functional calculus estimates, in two steps. First we introduce a new Banach algebra A 0,S 1 (C + ) of bounded analytic functions on C + = z ∈ C : Re(z) > 0 and show that its dual space coincides with M S 1 (H 1 (R)). Second, given any bounded C 0 -semigroup (T t ) t≥0 on Hilbert space, and any b ∈ L 1 (R + ), we establish an estimate ∞
In this article we discuss the notion of γ-H ∞ -bounded calculus, strong γ-m-H ∞bounded calculus on half-plane and weak-γ-Gomilko-Shi-Feng condition and give a connection between them. Then we state a characterization of generation of γ-bounded C 0 -semigroup in K-convex space, which leads to a version of Gearhart-Prüss on K-convex space.2010 Mathematics Subject Classification. 47A60, 47D06. Key words and phrases. γ-boundedness, K-convex space, H ∞ -calculus, Half-plane type operators. This work is supported by the French "Investissements d'Avenir" program, project ISITE-BFC (contract ANR-15-IDEX-03).Definition 2.1. Let A be a closed and densely defined operator on X. We will say thatWe will say that a C 0 -semigroup (T t ) t≥0 is of type ω ∈ R if there exists a constant C > 0 such that T t ≤ Ce −ωt for all t ≥ 0. Note that such ω always exists [17, Theorem 2.2 section 1.2]. It follows from the Laplace formula (see e.g. [17, Formula (7.1) section 1.7]) that if −A generates a C 0 -semigroup (T t ) t≥0 of type ω, then A is an operator of half-plane type ω (in fact A is even strong half-plane in the sense of [4, Definition 2.1]).Throughout the rest of this subsection, we let A be an operator of half-plane type ω. For any α ≤ ω, let H ∞ (R α ) be the space of all bounded analytic functions f : R α → C, equipped with the norm f H ∞ (Rα) := sup z∈Rα |f (z)|. Then H ∞ (R α ) is a Banach algebra. Next we consider the auxiliary space E(R α ) := {f ∈ H ∞ (R α ) : ∃s > 0, f (z) = O(|z| −(1+s) ) as |z| → ∞}.
In this article we study bounded operators T on Banach space X which satisfy the discrete Gomilko Shi-Feng conditionWe show that it is equivalent to a certain derivative bounded functional calculus and also to a bounded functional calculus relative to Besov space. Also on Hilbert space discrete Gomilko Shi-Feng condition is equivalent to power-boundedness. Finally we discuss the last equivalence on general Banach space involving the concept of γ-boundedness.
Let D be a Schauder decomposition on some Banach space X. We prove that if D is not R-Schauder, then there exists a Ritt operator T ∈ B(X) which is a multiplier with respect to D, such that the set {T n : n ≥ 0} is not R-bounded. Likewise we prove that there exists a bounded sectorial operator A of type 0 on X which is a multiplier with respect to D, such that the set {e −tA : t ≥ 0} is not R-bounded.2000 Mathematics Subject Classification: 47A99, 46B15.R-boundedness plays a prominent role in the study of sectorial operators and Ritt operators. Namely the notions of R-sectorial operators and R-Ritt operators have been instrumental in the development of H ∞ -functional calculus, square function estimates and applications to maximal regularity and to many other aspects of the harmonic analysis of semigroups (in either the continuous or the discrete case).The existence of sectorial operators which are not R-sectorial was discovered by Kalton and Lancien in their paper solving the L p -maximal regularity problem [6]. The existence of Ritt operators which are not R-Ritt was established a bit later by Portal [14]. More recently, Fackler [4] extended the work of Kalton-Lancien in various directions. In contrast with [6], which focused on existence results, [4] supplied explicit constructions of sectorial operators which are not R-sectorial. Further it is easy to derive from the latter paper explicit constructions of Ritt operators which are not R-Ritt. In [4,6,14], sectorial operators which are not R-sectorial (resp. Ritt operators which are not R-Ritt) are defined as multipliers with respect to Schauder decompositions having various "bad" properties. In particular, these Schauder decompositions cannot be R-Schauder (see Lemma 0.2).The aim of this note is two-fold. First we show that given any Schauder decomposition D which is not R-Schauder, one can define a sectorial operator A which is a multiplier with respect to D and which is not R-sectorial (resp. a Ritt operator T which is a multiplier with respect to D and which is not R-Ritt). Second we strengthen these negative results in both cases by showing that A can be chosen bounded and such that {e −tA : t ≥ 0} is not R-bounded, whereas T is taken such that {T n : n ≥ 0} is not R-bounded.
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