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 $| T^n|=o(n^{\alpha})$,generalizingtheresultobtainedfor$\alpha =1$. Moreover, if $\alpha >1$, then $\mathrm{\parallel }{T}^n\mathrm{\parallel }=O(n)$. We apply such results to get stability properties for the Ces`aro means of bounded operators.