FUNCTIONAL CALCULUS FOR C0-GROUPS USING TYPE AND COTYPE

We study the functional calculus properties of generators of C-0-groups 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 is an element of [1, 2] and cotype q is an element of [2, infinity). Then f (A): (X, D (A))(1/p -1/q,1) -> X is bounded for each bounded holomorphic function f on a p sufficiently large strip. As a corollary of this result, for sectorial operators, we quantify the gap between bounded imaginary powers and a bounded H-infinity-calculus in terms of the type and the 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.