Doubly twisted near-isometries: Classification and a Wold-type decomposition

TL;DR

This paper introduces doubly twisted near-isometries, establishing conditions for Wold-type decompositions, their uniqueness, and providing an analytic model, thus generalizing and extending prior results in operator theory.

Contribution

It generalizes the concept of doubly commuting near-isometries by defining doubly twisted near-isometries and characterizes their Wold-type decompositions and unitary equivalence.

Findings

Every doubly twisted near-isometry admits a Wold-type decomposition.

Necessary and sufficient conditions for the existence of such decompositions are established.

An explicit analytic model for doubly twisted near-isometries is constructed.

Abstract

We introduce and study doubly twisted near-isometries. A doubly twisted near-isometry is a tuple of near-isometries satisfying certain relations determined by a prescribed family of unitaries, thereby generalizing the notion of doubly commuting near-isometries. We establish necessary and sufficient conditions for a tuple of near-isometries to admit a Wold-type decomposition and prove that the existence of such a decomposition automatically ensures its uniqueness by providing an explicit description of the summands. Furthermore, we show that every doubly twisted near-isometry admits a Wold-type decomposition. We also characterize unitary equivalence within the class of doubly twisted near-isometries and construct an analytic model for them. Several examples are included to highlight the distinctions between our results and the corresponding results in the setting of doubly twisted isometries.