Mean ergodicity vs weak almost periodicity

Abstract

We provide explicit examples of positive and power-bounded operators on c 0 and ℓ ∞ 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 .

Long-term analysis of positive operator semigroups via asymptotic domination

“…Recall that it might happen that a positive operator T on a Banach lattice is mean ergodic, while not all the powers of T are mean ergodic. Indeed, Sine [37] constructed an example of a Koopman operator T on the space of continuous functions on a certain compact Hausdorff space such that T is mean ergodic, while T 2 is not; further counterexamples can be found in [10] and in [21].…”