Operator theory on noncommutative varieties. II. (English) Zbl 1119.47012
Let \(J\) be a WOT-closed two-sided ideal of the noncommutative analytic Toeplitz algebra \(F_n^\infty\) (introduced by the author in [G. Popescu, Math. Scand. 68, No. 2, 292–304 (1991; Zbl 0774.46033)] in connection with a multivariable noncommutative von Neumann inequality). A completely noncoisometric (c.n.c.) row contraction \(T=[T_1,\dots,T_n]\) on a Hilbert space is called \(J\)-constrained if \(f(T_1,\dots,T_n)=0,\;f\in J,\) where \(f(T_1,\dots,T_n)\) is defined using the \(F_n^\infty\)-functional calculus for c.n.c. row contractions (see G. Popescu [Mich. Math. J. 42, No. 2, 345–356 (1995; Zbl 0876.47016)]).
The paper under review is a sequel to Part I which appeared in [Indiana Univ. Math. J. 55, No. 2, 389–442 (2006; Zbl 1104.47013)] and where the author introduced and investigated a constrained characteristic function \(\Theta_{J,T}\) associated with \(J\) and \(T\). Certain results concerning the constrained Poison kernels \(K_{J,T}\) corresponding to \(J\) and \(T\) are presented. The constrained characteristic function is proved to be a complete unitary invariant and provides a model for the class of constrained c.n.c.row contractions. The results apply, in particular, to c.n.c.row contractions which are \(q\)-commuting , i.e., to c.n.c.row contractions \(T=[T_1,\dots,T_n]\) which are subject to the constraints \(T_iT_j=q_{ij}T_jT_i,\;1\leq i<j\leq n,\) where \(q_{ij}\in{\mathbb C}\).
The paper under review is a sequel to Part I which appeared in [Indiana Univ. Math. J. 55, No. 2, 389–442 (2006; Zbl 1104.47013)] and where the author introduced and investigated a constrained characteristic function \(\Theta_{J,T}\) associated with \(J\) and \(T\). Certain results concerning the constrained Poison kernels \(K_{J,T}\) corresponding to \(J\) and \(T\) are presented. The constrained characteristic function is proved to be a complete unitary invariant and provides a model for the class of constrained c.n.c.row contractions. The results apply, in particular, to c.n.c.row contractions which are \(q\)-commuting , i.e., to c.n.c.row contractions \(T=[T_1,\dots,T_n]\) which are subject to the constraints \(T_iT_j=q_{ij}T_jT_i,\;1\leq i<j\leq n,\) where \(q_{ij}\in{\mathbb C}\).
Reviewer: Dan E. Popovici (Timişoara)
MSC:
| 47A20 | Dilations, extensions, compressions of linear operators |
| 47A13 | Several-variable operator theory (spectral, Fredholm, etc.) |
| 47A56 | Functions whose values are linear operators (operator- and matrix-valued functions, etc., including analytic and meromorphic ones) |
| 47A63 | Linear operator inequalities |
Keywords:
multivariable operator theory; noncommutative variety; characteristic function; model theory; row contraction; constrained shift; Poisson kernel; Fock space; unitary invariant; von Neumann inequalityReferences:
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