A Katznelson-Tzafriri type theorem for Ces\`aro bounded operators

TL;DR

This paper generalizes the Katznelson-Tzafriri theorem to Cesàro bounded operators of any order using a new functional calculus, and applies it to derive ergodic properties of Cesàro means.

Contribution

It introduces a novel functional calculus linking fractional Wiener algebras to bounded operators, extending the theorem to a broader class of operators.

Findings

Extended Katznelson-Tzafriri theorem to Cesàro bounded operators

Established a new functional calculus for fractional Wiener algebras

Derived ergodicity results for Cesàro means of bounded operators

Abstract

We extend the well-known Katznelson-Tzafriri theorem, originally posed for power-bounded operators, to the case of Ces\`aro 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 Ces\`aro boundedness. Finally, we apply the main theorem to get ergodicity results for the Ces\`aro means of bounded operators.