Operator theory on noncommutative varieties, II

Popescu, Gelu

doi:10.1090/S0002-9939-07-08719-9

Operator theory on noncommutative varieties, II
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Proc. Amer. Math. Soc. 135 (2007), 2151-2164 Request permission

Abstract:

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.

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