Models for q-commutative tuples of isometries

TL;DR

This paper develops a functional model for $q$-commutative pairs of isometries, providing a complete unitary invariance and extending the concept to tuples, with special focus on shift operators.

Contribution

It introduces a new functional model for $q$-commutative isometries and characterizes doubly $q$-commutative pairs, extending the theory to operator tuples.

Findings

Functional model parametrized by Hilbert spaces and operators

Complete unitary invariance for $q$-commutative pairs

Characterization of doubly $q$-commutative pairs, especially shifts

Abstract

A pair of Hilbert space linear operators $(V_1,V_2)$ is said to be $q$-commutative, for a unimodular complex number $q$, if $V_1V_2=qV_2V_1$. A concrete functional model for $q$-commutative pairs of isometries is obtained. The functional model is parametrized by a collection of Hilbert spaces and operators acting on them. As a consequence, the collection serves as a complete unitary invariance for $q$-commutative pairs of isometries. A $q$-commutative operator pair $(V_1,V_2)$ is said to be doubly $q$-commutative, if in addition, it satisfies $V_2V_1^*=qV_1^*V_2$. Doubly $q$-commutative pairs of isometries are also characterized. Special attention is given to doubly $q$-commutative pairs of shift operators. The notion of $q$-commutativity is then naturally extended to the case of general tuples of operators to obtain a similar model for tuples of $q$-commutative isometries.