Two Results About H ∞ Functional Calculus on Analytic umd Banach Spaces

Source abstract

Abstract Let X be a Banach space with the analytic UMD property, and let A and B be two commuting sectorial operators on X which admit bounded H ∞ functional calculi with respect to angles θ 1 and θ 2 satisfying θ 1 + θ 2 > π. It was proved by Kalton and Weis that in this case, A + B is closed. The first result of this paper is that under the same conditions, A + B actually admits a bounded H ∞ functional calculus. Our second result is that given a Banach space X and a number 1 ≦ p < ∞, the derivation operator on the vector valued Hardy space H p ( R; X ) admits a bounded H ∞ functional calculus if and only if X has the analytic UMD property. This is an ‘analytic’ version of the well-known characterization of UMD by the boundedness of the H ∞ functional calculus of the derivation operator on vector valued L p -spaces L p (R; X) for 1 < p < ∞ (Dore-Venni, Hieber-Prüss, Prüss).