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

TL;DR

This paper demonstrates how bounded analytic semigroups with certain functional calculus properties can be dilated into positive contractions on larger spaces, extending to Ritt operators and general Banach spaces.

Contribution

It introduces a dilation technique for bounded analytic semigroups with $H^$ calculus and extends the theory to Ritt operators and Banach spaces.

Findings

Dilations of analytic semigroups into positive contractions on larger $L^p$-spaces.

Discrete analogue established for Ritt operators.

Generalizations of Dixmier's unitarization theorem for Banach space representations.

Abstract

We show that any bounded analytic semigroup on $L^p$ (with $1<p<\infty$) whose negative generator admits a bounded $H^{\infty}$ functional calculus with respect to some angle $< \pi/2$ 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 $t$. 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.