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A pair of Hilbert space linear operators $(V_1,V_2)$ is said to be$q$-commutative, for a unimodular complex number $q$, if $V_1V_2=qV_2V_1$. Aconcrete functional model for $q$-commutative pairs of isometries is obtained.The functional model is parametrized by a collection of Hilbert spaces andoperators acting on them. As a consequence, the collection serves as a completeunitary invariance for $q$-commutative pairs of isometries. A $q$-commutativeoperator 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 isometriesare also characterized. Special attention is given to doubly $q$-commutativepairs of shift operators. The notion of $q$-commutativity is then naturallyextended to the case of general tuples of operators to obtain a similar modelfor tuples of $q$-commutative isometries.

Models for q-commutative tuples of isometries