Constrained characteristic functions, multivariable interpolation, and invariant subspaces

Abstract

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 $\mathcal{V}_{f,\varphi,\mathcal{I}}$ .

Keywords

• Noncommutative Poisson transform
• Characteristic function
• Universal model
• Interpolation