Wold Decomposition on Odometer Semigroups

TL;DR

This paper extends the Wold decomposition to isometric and Nica-covariant representations of the odometer semigroup, generalizing previous results for commuting isometries and pairs of doubly commuting isometries.

Contribution

It introduces a new Wold-type decomposition for a broader class of semigroup representations, expanding the theoretical framework in operator theory.

Findings

Established a Wold-type decomposition for odometer semigroup representations

Generalized previous decompositions for commuting isometries

Enhanced understanding of Nica-covariant representations

Abstract

We establish a Wold-type decomposition for isometric and isometric Nica-covariant representations of the odometer semigroup. These generalize the Wold-type decomposition for commuting pairs of isometries due to Popovici and for pairs of doubly commuting isometries due to S{\l}oci\'nski.