S{\l}oci\'nski-Wold decompositions for row-isometries

TL;DR

This paper extends Wold decomposition results to row-isometries with specific commutation relations, generalizing known results for doubly-commuting cases and using Lebesgue decomposition techniques.

Contribution

It provides new sufficient conditions for Wold decompositions of row-isometries beyond doubly-commuting cases, based on Lebesgue decomposition.

Findings

Established analogous Wold decomposition results for certain row-isometries.

Provided sufficient conditions for Wold decomposition using Lebesgue decomposition.

Extended known results to broader classes of row-isometries with specific commutation relations.

Abstract

S{\l}oci\'nski gave sufficient conditions for commuting isometries to have a nice Wold-like decomposition. In this note we provide analogous results for row-isometries satisfying certain commutation relations. Other than known results for doubly-commuting row-isometries, we provide sufficient condtions for a Wold decomposition based on the Lebesgue decomposition of the row-isometries.