Functional calculus for C ₀ -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∈[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 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.