Functional calculus forC0-groups using type and cotype

Abstract

We study the functional calculus properties of generators of C 0groups under type and cotype assumptions on the underlying Banach space. In particular, we show the following. Let −iA generate a C 0 -group on a Banach space X with type p ∈ [1, 2] and cotype q ∈ [2, ∞). Then f (A) : (X, D(A)) 1 p − 1 q ,1 → X is bounded for each bounded holomorphic function f on a sufficiently large strip. As a corollary of this result, for sectorial operators we quantify the gap between bounded imaginary powers and a bounded H ∞ -calculus in terms of the type and cotype of the underlying Banach space. For cosine functions we obtain similar results as for C 0 -groups. We extend our theorems to R-bounded operator-valued calculi, and we give an application to the theory of rational approximation of C 0 -groups.2010 Mathematics Subject Classification. Primary 47A60; Secondary 47D03, 46B20, 42A45.