Abstract
In this paper, we present a functional model theorem for completely non-coisometric n-tuples of operators in the noncommutative variety mathcal{V}_{f,varphi,mathcal{I}}(mathcal{H}) in terms of constrained characteristic functions. As an application, we prove that the constrained characteristic function is a complete unitary invariant for this class of elements, which can be viewed as the noncommutative analogue of the classical Sz.-Nagy–Foiaş functional model for completely nonunitary contractions. On the other hand, we provide a Sarason-type commutant lifting theorem. Applying this result, we solve the Nevanlinna–Pick-type interpolation problem in our setting. Moreover, we also obtain a Beurling-type characterization of the joint invariant subspaces under the operators B_{1},ldots,B_{n}, where the n-tuple (B_{1},ldots,B_{n}) is the universal model associated with the abstract noncommutative variety mathcal{V}_{f,varphi,mathcal{I}}.
Highlights
In the last fifty years, the study of the closed operator unit ball B(H) := T ∈ B(H) : TT∗2 ≤1 has generated the celebrated Sz.-Nagy–Foiaş theory of contractions on Hilbert spaces
In 1963, Sz.-Nagy and Foiaş obtained an effective H∞functional calculus for completely nonunitary contractions on Hilbert spaces based on the existence of a unitary dilation of a contraction T
We present a functional model theorem for completely non-coisometric n-tuples of operators in the noncommutative variety Vf,φ,I(H) in terms of constrained characteristic functions
Summary
2 ≤1 has generated the celebrated Sz.-Nagy–Foiaş theory of contractions on Hilbert spaces. According to [22, 24] and [33], if f = Z and φ = Z, the corresponding domain Df ,φ(H) coincides with the closed operator unit ball [B(H)]–1 , the study of which has generated Sz.-Nagy–Foiaş theory of contractions. Zn), the corresponding domain Df ,φ(H) coincides with the closed operator unit n-ball [B(H)n]–1 , the study of which has generated a free analogue of Sz.-Nagy–Foiaş theory. We present a functional model theorem for completely non-coisometric n-tuples of operators in the noncommutative variety Vf ,φ,I(H) in terms of constrained characteristic functions. Applying this result, we prove that the constrained characteristic function is a complete unitary invariant for this class of elements.
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