J. Aust. Math. Soc.
74 (2003), 351378

Two results about functional calculus on analytic UMD Banach spaces


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 wellknown
characterization of UMD by the boundedness of
the functional calculus of the derivation operator
on vector valued spaces for (DoreVenni, HieberPrüss, Prüss).

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