Model Theory and the mean ergodic theorem for abelian unitary actions

TL;DR

This paper investigates the relationship between model theory and the mean ergodic theorem for abelian groups of unitary operators, providing a new proof of Wiener's generalization.

Contribution

It introduces a novel connection between model theory and ergodic theory, extending von Neumann's mean ergodic theorem to general abelian unitary actions.

Findings

Established a link between model theory and ergodic theorems

Provided a new proof of Wiener's generalization

Extended the theorem to any abelian group of unitary transformations

Abstract

We explore connections between von Neumann's mean ergodic theorem and concepts of model theory. As an application we present a proof Wiener's generalization of von Neumann's result in which the group acting on the Hilbert space $\mathcal{H}$ is any abelian group of unitary transformations of $\mathcal{H}$.