A condition equivalent to uniform ergodicity
Volume 167 / 2005
Studia Mathematica 167 (2005), 215-218
MSC: Primary 47A35.
DOI: 10.4064/sm167-3-2
Abstract
Let $T$ be a linear operator on a Banach space $X$ with $\mathop {\rm sup}_n \| T^n/n^w\| < \infty $ for some $0\le w < 1$. We show that the following conditions are equivalent: (i) $n^{-1}\sum _{k=0}^{n-1} T^k$ converges uniformly; (ii) ${\rm cl}\, (I -T)X = \{ z \in X : \mathop {\rm lim}_n\sum _{k=1}^n { T^kz/k}\hbox { exists} \} $.
