The Ritt property of subordinated operators in the group case

Let G be a locally compact abelian group, let ν be a regular probability measure on G, let X be a Banach space, let π:G→B(X) be a bounded strongly continuous representation. Consider the average (or subordinated) operator S(π,ν)=∫Gπ(t)dν(t):X→X. We show that if X is a UMD Banach lattice and ν has bounded angular ratio, then S(π,ν) is a Ritt operator with a bounded H∞ functional calculus. Next we show that if ν is the square of a symmetric probability measure and X is K-convex, then S(π,ν) is a Ritt operator. We further show that this assertion is false on any non K-convex space X.