Power bounded weighted composition operators on function spaces defined by local properties

TL;DR

This paper investigates the power boundedness and ergodic properties of (weighted) composition operators on various function spaces defined by local properties, providing characterizations for their behavior on kernels of differential operators.

Contribution

It offers new characterizations of when (weighted) composition operators are power bounded, mean ergodic, and generators of semigroups on specific function spaces, extending understanding in operator theory.

Findings

Characterization of power bounded composition operators on differential operator kernels

Conditions for mean ergodicity and topologizability of these operators

Identification of generators of strongly continuous semigroups on function spaces

Abstract

We study power boundedness and related properties such as mean ergodicity for (weighted) composition operators on function spaces defined by local properties. As a main application of our general approach we characterize when (weighted) composition operators are power bounded, topologizable, and (uniformly) mean ergodic on kernels of certain linear partial differential operators including elliptic operators as well as non-degenrate parabolic operators. Moreover, under mild assumptions on the weight and the symbol we give a characterisation of those weighted composition operators on the Fr\'echet space of continuous functions on a locally compact, $\sigma$-compact, non-compact Hausdorff space which are generators of strongly continuous semigroups on these spaces.