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

Part of: General theory of linear operators

Published online by Cambridge University Press:  09 April 2009

Christian Le Merdy
Christian Le Merdy
Affiliation:
Département de Mathématiques Université de Franche-Comté 25030 Besancon Cedex France e-mail: lemerdy@math.univ-fcomte.fr
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Abstract

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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 θ1 and θ2 satisfying θ1 + θ2 > 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 Hp (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 Lp-spaces Lp (R; X) for 1 < p < ∞ (Dore-Venni, Hieber-Prüss, Prüss).

MSC classification

Secondary: 47A60: Functional calculus
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 74 , Issue 3 , June 2003 , pp. 351 - 378
Copyright
Copyright © Australian Mathematical Society 2003

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