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 $S(\pi,\nu) = \int_{G} \pi(t)\,d\nu(t)\,\colon X\to X$. 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 $H^\infty$ 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 $X$.