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Abstract

For a given linear operator $T$ in a complex Banach space $X$ and $\alpha \in {{\mathbb C}}$ with $\Re (\alpha )>0$, we define the $n$th Cesàro mean of order $\alpha $ of the powers of $T$ by $ M_{n}^{\alpha }=(A_{n}^{\alpha })^{-1} \sum _{k=0}^{n}A_{n-k}^{\alpha -1}T^{k}$. For $\alpha =1$, we find $M_{n}^{1}=(n+1)^{-1}\sum _{k=0}^{n}T^k$, the usual Cesàro mean. We give necessary and sufficient conditions for a $(C,\alpha )$ bounded operator to be $(C,\alpha )$ strongly (weakly) ergodic.