Reflexive Banach spaces with all power-bounded operators almost periodic

TL;DR

This paper studies power-bounded operators on certain reflexive Banach spaces, showing they are almost periodic and analyzing their ergodic properties, with implications for operators on specific constructed spaces.

Contribution

It demonstrates that power-bounded operators on a class of reflexive Banach spaces are almost periodic and extends this to operators on subspaces of a special constructed space with invariant subspaces.

Findings

Power-bounded operators are almost periodic on these spaces.

Weakly mixing operators are stable with convergent powers.

All operators on subspaces of X-ISP are almost periodic.

Abstract

We analyze the ergodic properties of power-bounded operators on a reflexive Banach space of the form "scalar plus compact-power", and show that they are almost periodic (all the orbits are conditionally compact). If such an operator is weakly mixing, then it is stable (its powers converge in the strong operator topology).Let X-ISP be the separable reflexive indecomposable Banach space constructed by Argyros and Motakis, in which every operator has an invariant subspace. We conclude that every power-bounded operator on a closed subspace of X-ISP is almost periodic.