Mean ergodicity vs weak almost periodicity
Volume 248 / 2019
Studia Mathematica 248 (2019), 45-56
MSC: Primary 47B65; Secondary 47A35, 46B42, 46A45.
DOI: 10.4064/sm170918-20-3
Published online: 22 February 2019
Abstract
We provide explicit examples of positive and power-bounded operators on $c_0$ and $\ell ^\infty $ which are mean ergodic but not weakly almost periodic. As a consequence we prove that a countably order complete Banach lattice on which every positive and power-bounded mean ergodic operator is weakly almost periodic is necessarily a KB-space. This answers several open questions from the literature. Finally, we prove that if $T$ is a positive mean ergodic operator with zero fixed space on an arbitrary Banach lattice, then so is every power of $T$.
