Models for q-commutative tuples of isometries

Abstract

A pair of Hilbert space linear operators (V 1 , V 2 ) is said to be q-commutative, for a unimodular complex number qA 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 2 V * 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.