Operator theory on noncommutative varieties, II

An $n$-tuple of operators $T:=[T_1,\ldots, T_n]$ on a Hilbert space $\mathcal{H}$ is called a $J$-constrained row contraction if $T_1T_1^*+\cdots + T_nT_n^*\leq I_\mathcal{H}$ and

$f(T_1,\ldots, T_n)=0,\quad f\in J,$

where $J$ is a WOT-closed two-sided ideal of the noncommutative analytic Toeplitz algebra $F_n^\infty$ and $f(T_1,\ldots, T_n)$ is defined using the $F_n^\infty$-functional calculus for row contractions.

We show that the constrained characteristic function $\Theta_{J,T}$ associated with $J$ and $T$ is a complete unitary invariant for $J$-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 $q$-commuting c.n.c. row contractions.