J. Aust. Math. Soc.  74 (2003), 351-378
Two results about functional calculus on analytic UMD Banach spaces

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

Abstract
Let be a Banach space with the analytic UMD property, and let and be two commuting sectorial operators on which admit bounded functional calculi with respect to angles and satisfying . It was proved by Kalton and Weis that in this case, is closed. The first result of this paper is that under the same conditions, actually admits a bounded functional calculus. Our second result is that given a Banach space and a number , the derivation operator on the vector valued Hardy space admits a bounded functional calculus if and only if has the analytic UMD property. This is an `analytic' version of the well-known characterization of UMD by the boundedness of the functional calculus of the derivation operator on vector valued -spaces for (Dore-Venni, Hieber-Prüss, Prüss).
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