Growth orders and ergodicity for absolutely Ces\`aro bounded operators

In this paper, we extend the concept of absolutely Ces\`aro 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 Ces\`{a}ro bounded operator of order $\alpha$ with $0<\alpha\le 1,$ then $\| T^n\|=o(n^{\alpha})$, generalizing the result obtained for $\alpha =1$. Moreover, if $\alpha > 1$, then $\|T^{n}\|= O(n)$. We apply such results to get stability properties for the Ces\`aro means of bounded operators.