Functional calculus for C0-groups using type and cotype

Abstract

We study the functional calculus properties of generators of C0-groups under type and cotype assumptions on the underlying Banach space. In particular, we show the following. Let −iA generate a C0-group on a Banach space X with type p∈[1, 2] and cotype q∈[2,∞) ⁠. Then f(A):(X,D(A))1p−1q,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 the cotype of the underlying Banach space. For cosine functions, we obtain similar results as for C0-groups. We extend our theorems to R-bounded operator-valued calculi, and we give an application to the theory of rational approximation of C0-groups Show more