On ergodic operator means in Banach spaces

TL;DR

This paper develops a general framework for ergodic theorems involving operator means in Banach spaces, extending known results and introducing new methods that apply broadly, including to classical Cesaro means.

Contribution

It introduces a unified approach to ergodic theorems for operator means in Banach spaces, generalizing and strengthening existing results.

Findings

Ergodic theorems hold under mild regularity conditions for a broad class of operator means.

New growth estimates are established for these operator means.

The methods provide a novel perspective, applicable even to classical Cesaro means.

Abstract

We consider a large class of operator means and prove that a number of ergodic theorems, as well as growth estimates known for particular cases, continue to hold in the general context under fairly mild regularity conditions. The methods developed in the paper not only yield a new approach based on a general point of view, but also lead to results that are new, even in the context of the classical Cesaro means.