Mean ergodic weighted shifts on K\"othe echelon spaces

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.