Mean ergodic composition operators in spaces of homogeneous polynomials

TL;DR

This paper investigates the ergodic properties of composition operators on spaces of homogeneous polynomials over Banach spaces, revealing different behaviors depending on the topology used, with some operators being uniformly mean ergodic under one topology but not the other.

Contribution

It provides a detailed analysis of mean ergodic composition operators in polynomial spaces, highlighting the impact of topology on their dynamical properties and offering new examples.

Findings

Uniformly mean ergodic for compact convergence topology

No direct relation between mean ergodicity and power boundedness in norm topology

Several illustrative examples provided

Abstract

We study some dynamical properties of composition operators defined on the space $\mathcal{P}(^m X)$ of $m$-homogeneous polynomials on a Banach space $X$ when $\mathcal{P}(^m X)$ is endowed with two different topologies: the one of uniform convergence on compact sets and the one defined by the usual norm. The situation is quite different for both topologies: while in the case of uniform convergence on compact sets every power bounded composition operator is uniformly mean ergodic, for the topology of the norm there is no relation between the latter properties. Several examples are given.