Orthogonal decompositions and twisted isometries II

TL;DR

This paper classifies tuples of isometries that admit von Neumann-Wold decomposition, introduces twisted isometries, and generalizes orthogonal decompositions to broader classes of isometries, unifying existing results.

Contribution

It introduces the concept of twisted isometries and proves orthogonal decompositions for these, extending classical results to non-commuting and twisted cases.

Findings

Classification of isometries with von Neumann-Wold decomposition

Introduction of twisted isometries and their orthogonal decompositions

Unification of known orthogonal decomposition results

Abstract

We classify tuples of (not necessarily commuting) isometries that admit von Neumann-Wold decomposition. We introduce the notion of twisted isometries for tuples of isometries and prove the existence of orthogonal decomposition for such tuples. The former classification is partially inspired by a result that was observed more than three decades ago by Gaspar and Suciu. And the latter result generalizes Popovici's orthogonal decompositions for pairs of commuting isometries to general tuples of twisted isometries which also includes the case of tuples of commuting isometries. Our results unify all the known orthogonal decomposition related results in the literature.