A Katznelson-Tzafriri type theorem for Cesàro bounded operators

Abstract

We extend the well-known Katznelson-Tzafriri theorem, originally posed for power-bounded operators, to the case of Cesàro bounded operators of any order $\alpha >0.$ For this purpose, we use a functional calculus between a new class of fractional Wiener algebras and the algebra of bounded linear operators, whose existence is characterized by the Cesàro boundedness. Finally, we apply the main theorem to get ergodicity results for the Cesàro means of bounded operators.