Growth orders and ergodicity for absolutely Cesàro bounded operators

Abstract

In this paper, we extend the concept of absolutely Cesàro boundedness to the fractional case. We construct a weighted shift operator belonging to this class of operators, and we prove that if $T$ is an absolutely Cesàro bounded operator of order $\alpha$ with $0<\alpha \le 1,$ then $\Vert T^n\Vert =o(n^{\alpha })$, generalizing the result obtained for $\alpha =1$. Moreover, if $\alpha >1$, then $\Vert T^n\Vert =O(n)$. We apply such results to get stability properties for the Cesàro means of bounded operators.