Constrained characteristic functions, multivariable interpolation, and invariant subspaces

Abstract In this paper, we present a functional model theorem for completely non-coisometric n-tuples of operators in the noncommutative variety V f, φ, I ( H ) $\mathcal{V}_{f,\varphi,\mathcal{I}}(\mathcal{H})$ 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, …, B n $B_{1},\ldots,B_{n}$, where the n-tuple ( B 1, …, B n ) $(B_{1},\ldots,B_{n})$ is the universal model associated with the abstract noncommutative variety V f, φ, I $\mathcal{V}_{f,\varphi,\mathcal{I}}$.

Abstract licence: CC BY-NC-ND

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