ASYMPTOTIC REGULARITY OF LINEAR POWER BOUNDED OPERATORS

Abstract

Let $T$ be a linear power bounded operator on a Banach space $X$ and let $S_{\lambda} = (1−\lambda)I + \lambda T$ be the averaged map of $T$, where $\lambda \in (0,1)$. It is shown that $S_{\lambda}$ is asymptotically regular on $X$; that is, $\lim_{n \to \infty} \| S_{\lambda}^{n}x - S_{\lambda}^{n+1}x \| = 0$ for every $x \in X$. Hence the sequence $\{S_{\lambda}^{n}x\}$ converges strongly provided it has a weak cluster point.