Operator theory on noncommutative varieties, II

Abstract

An n n -tuple of operators T := [ T 1 , … , T n ] T:=[T_1,\ldots , T_n] on a Hilbert space H \mathcal {H} is called a J J -constrained row contraction if T 1 T 1 ∗ + ⋯ + T n T n ∗ ≤ I H T_1T_1^*+\cdots + T_nT_n^*\leq I_\mathcal {H} and \[ f ( T 1 , … , T n ) = 0 , f ∈ J , f(T_1,\ldots , T_n)=0,\quad f\in J, \] where J J is a WOT-closed two-sided ideal of the noncommutative analytic Toeplitz algebra F n ∞ F_n^\infty and f ( T 1 , … , T n ) f(T_1,\ldots , T_n) is defined using the F n ∞ F_n^\infty –functional calculus for row contractions. We show that the constrained characteristic function Θ J , T \Theta _{J,T} associated with J J and T T is a complete unitary invariant for J J -constrained completely non-coisometric (c.n.c.) row contractions. We also provide a model for this class of row contractions in terms of the constrained characteristic functions. In particular, we obtain a model theory for q q -commuting c.n.c. row contractions.