A model theory for q-commuting contractive tuples

Abstract

A contractive tuple is a tuple (T₁, . . . , T_d) of operators on a common Hilbert space such that (0.1) T₁T^∗₁ + + T_dT^∗_d ≤ 1. It is said to be q-commuting if T_jT_i = q_ijT_iT_j for all 1 ≤ i < j ≤ d, where qij , 1 ≤ i < j ≤ d are complex numbers. These are higher-dimensional and non-commutative generalizations of a contraction. A particular example of this is the q-commuting shift. In this note, we investigate model theory for q-commuting contractive tuples using representations of the q-commuting shift.