Pairs of commuting isometries - I

TL;DR

This paper provides an explicit classification and construction method for pure pairs of commuting isometries, including invariants and comparisons with other representations, and studies their defect operators.

Contribution

It offers an explicit recipe for constructing pairs of commuting isometric multipliers and a complete set of unitary invariants, advancing the understanding of their structure.

Findings

Explicit construction method for pure pairs of commuting isometries

Complete set of joint unitary invariants identified

Comparison with other analytic representations conducted

Abstract

We present an explicit version of Berger, Coburn and Lebow's classification result for pure pairs of commuting isometries in the sense of an explicit recipe for constructing pairs of commuting isometric multipliers with precise coefficients. We describe a complete set of (joint) unitary invariants and compare the Berger, Coburn and Lebow's representations with other natural analytic representations of pure pairs of commuting isometries. Finally, we study the defect operators of pairs of commuting isometries.