Functional calculus for $C_{0}$-groups using (co)type

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 $p\in[1,2]$ and cotype $q\in[2,\infty)$. Then $A$ has a bounded $\mathcal{H}^{\infty}$-calculus from $\mathrm{D}_{A}(\tfrac{1}{p}-\tfrac{1}{q},1)$ to $X$, i.e. $f(A):\mathrm{D}_{A}(\tfrac{1}{p}-\tfrac{1}{q},1)\to X$ is bounded for each bounded holomorphic function $f$ on a sufficiently large strip. As a corollary of our main theorem, for sectorial operators we quantify the gap between bounded imaginary powers and a bounded $\mathcal{H}^{\infty}$-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 results to $R$-bounded operator-valued calculi, and we give an application to the theory of rational approximation of $C_{0}$-groups.