Almost weak polynomial stability of operators

TL;DR

This paper explores the relationship between almost weak stability and almost weak polynomial stability of operators on Banach spaces, establishing conditions under which one implies the other and providing counterexamples.

Contribution

It demonstrates that on Hilbert spaces, almost weak stability implies almost weak polynomial stability for contractions, and provides explicit counterexamples in other spaces.

Findings

On Hilbert spaces, the implication holds for contractions.

Counterexamples show the implication fails in some $C_0$ spaces.

Application to convergence of polynomial multiple ergodic averages.

Abstract

We investigate whether almost weak stability of an operator $T$ on a Banach space $X$ implies its almost weak polynomial stability. We show, using a modified version of the van der Corput Lemma that if $X$ is a Hilbert space and $T$ a contraction, then the implication holds. On the other hand, based on a TDS arising from a two dimensional ODE, we give an explicit example of a contraction on a $C_0$ space that is almost weakly stable, but its appropriate polynomial powers fail to converge weakly to zero along a subsequence of density 1. Finally we provide an application to convergence of polynomial multiple ergodic averages.