Operator Theory on Noncommutative Varieties
نویسنده
چکیده
We develop a dilation theory for row contractions T := [T1, . . . , Tn] subject to constraints such as p(T1, . . . , Tn) = 0, p ∈ P , where P is a set of noncommutative polynomials. The model n-tuple is the universal row contraction [B1, . . . , Bn] satisfying the same constraints as T , which turns out to be, in a certain sense, the maximal constrained piece of the n-tuple [S1, . . . , Sn] of left creation operators on the full Fock space on n generators. The theory is based on a class of constrained Poisson kernels associated with T and representations of the C-algebra generated by B1, . . . , Bn and the identity. Under natural conditions on the constraints we have uniqueness for the minimal dilation. A characteristic function ΘT is associated with any (constrained) row contraction T and it is proved that I −ΘTΘ ∗ T = KT K ∗ T , where KT is the (constrained) Poisson kernel of T . Consequently, for pure constrained row contractions, we show that the characteristic function is a complete unitary invariant and provide a model. We show that the curvature invariant and Euler characteristic asssociated with a Hilbert module generated by an arbitrary (resp. commuting) row contraction T can be expressed only in terms of the (resp. constrained) characteristic function of T . We provide a commutant lifting theorem for pure constrained row contractions and obtain a Nevanlinna-Pick interpolation result in our setting.
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تاریخ انتشار 2007