The Ritt property of subordinated operators in the group case

TL;DR

This paper investigates conditions under which subordinated operators in the group case are Ritt operators with bounded functional calculus, focusing on properties of the Banach space and the probability measure involved.

Contribution

It establishes new criteria for subordinated operators to be Ritt operators with bounded $H^$ calculus, based on the geometry of the Banach space and the measure's properties.

Findings

If $X$ is a UMD Banach lattice and $ u$ has bounded angular ratio, then $S(, u)$ is a Ritt operator with bounded $H^$ calculus.

If $ u$ is the square of a symmetric measure and $X$ is $K$-convex, then $S(, u)$ is a Ritt operator.

The assertion fails on non-$K$-convex spaces.

Abstract

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 $X$.