Isometric dilations and $H^\infty$ calculus for bounded analytic semigroups and Ritt operators

We show that any bounded analytic semigroup on $L^p$ (with $1<p<\infty$) whose negative generator admits a bounded $H^{\infty }(\Sigma _\theta )$ functional calculus for some $\theta \in (0,\frac {\pi }{2})$ can be dilated into a bounded analytic semigroup $(R_t)_{t\geq 0}$ on a bigger $L^p$-space in such a way that $R_t$ is a positive contraction for any $t\geq 0$. We also establish a discrete analogue for Ritt operators and consider the case when $L^p$-spaces are replaced by more general Banach spaces. In connection with these functional calculus issues, we study isometric dilations of bounded continuous representations of amenable groups on Banach spaces and establish various generalizations of Dixmier's unitarization theorem.