Some Mean and Uniform Ergodic Type Theorems

Abstract

Let $X$ be a Banach space and $T\in B(X)$. Cohen determined a class of regular infinite matrices $A=(a_{nk})$ for which $ L_n:=\sum\limits_{k=1}^\infty a_{nk}T^k$ converges strongly to an element invariant under $T$. In the present paper we study $A$-mean and $A$-uniform ergodic type results when $A=(a_{nk})$ is a regular infinite matrix satisfying Cohen's uniformity condition $\lim\limits_j\sum\limits_{k=j}^\infty|a_{n,k+1}-a_{nk}|=0$, uniformly in $n$.