Mean ergodic type theorems

Abstract

Let $T$ be a bounded linear operator on a Banach space $X$. Replacing the Ces\`{a}ro matrix by a regular matrix $A=(a_{nj})$ Cohen studied a mean ergodic theorem. In the present paper we extend his result by taking a sequence of infinite matrices $\mathcal{A}=(A^{(i)})$ that contains both convergence and almost convergence. This result also yields an $\mathcal{A}$-ergodic decomposition. When $T$ is power bounded we give a characterization for $T$ to be $\mathcal{A}$-ergodic.