Mean ergodicity vs weak almost periodicity

TL;DR

The paper constructs explicit examples of positive, power-bounded operators that are mean ergodic but not weakly almost periodic, and explores their implications in Banach lattice theory.

Contribution

It provides explicit examples distinguishing mean ergodicity from weak almost periodicity and characterizes KB-spaces via operator properties.

Findings

Examples of operators on $c_0$ and $\\ell^\\infty$ that are mean ergodic but not weakly almost periodic.

Characterization of KB-spaces through properties of positive, power-bounded mean ergodic operators.

Proof that powers of zero-fixed-point mean ergodic operators remain zero-fixed-point on Banach lattices.

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 $T$.