The Ritt property of subordinated operators in the group case

Let $G$ be a locally compact abelian group, let $\nu$ be a regular probability measure on $G$, let $X$ be a Banach space, let $\pi\colon G\to B(X)$ be a bounded strongly continuous representation. Consider the average (or subordinated) operator $S(\pi ,\nu )={\int }_G\pi (t)d\nu (t)\text{\hspace{0.17em}⁣}:X\to X$. We show that if $X$ is a UMD Banach lattice and $\nu$ has bounded angular ratio, then $S(\pi ,\nu )$ is a Ritt operator with a bounded $H^{\mathrm{\infty }}$ functional calculus. Next we show that if $\nu$ is the square of a symmetric probability measure and $X$ is $K$-convex, then $S(\pi ,\nu )$ is a Ritt operator. We further show that this assertion is false on any non $K$-convex space $X$.