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: G -> B(X) be a bounded strongly continuous representation. Consider the average (or subordinated) operator S(pi, nu) = integral(G), pi(t)d nu(t) : X -> X. We show that if Xis a UMD Banach lattice and nu has bounded angular ratio, then S(pi, nu) is a Ritt operator with a bounded H-infinity 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. (C) 2018 Elsevier Inc. All rights reserved.