Mean ergodic weighted shifts on K\"othe echelon spaces

TL;DR

This paper establishes criteria for mean ergodicity, power boundedness, and topologizability of weighted shift operators on K"othe echelon spaces, with applications to classical operators on function spaces.

Contribution

It provides necessary and sufficient conditions based on weights and K"othe matrices, extending understanding of operator behavior on these spaces.

Findings

Criteria for mean ergodicity, power boundedness, topologizability derived

Conditions applied to power series spaces for detailed characterization

Analysis of classical operators demonstrates practical relevance

Abstract

Necessary and sufficient conditions are given for mean ergodicity, power boundedness, and topologizability for weighted backward shift and weighted forward shift operators, respectively, on K\"othe echelon spaces in terms of the weight sequence and the K\"othe matrix. These conditions are evaluated for the special case of power series spaces which allow for a characterization of said properties in many cases. In order to demonstrate the applicability of our conditions, we study the above properties for several classical operators on certain function spaces.