On the $(C,\alpha )$ Cesàro bounded operators

Elmouloudi Ed-dari¹

¹ Faculté des sciences Jean Perrin Université D'Artois SP 18 62307 Lens Cedex, France

Studia Mathematica, Tome 161 (2004) no. 2, pp. 163-175

Résumé

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.

Détail

DOI : 10.4064/sm161-2-4

Keywords: given linear operator complex banach space alpha mathbb alpha define nth ces mean order alpha powers alpha alpha sum n k alpha alpha sum usual ces mean necessary sufficient conditions alpha bounded operator alpha strongly weakly ergodic