Growth orders and ergodicity for absolutely Cesà ro bounded operators

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 α with 0<α≤1, then ‖Tn‖=o(nα), generalizing the result obtained for α=1. Moreover, if α>1, then ‖Tn‖=O(n). We apply such results to get stability properties for the Cesàro means of bounded operators.