Two classical theorems of function model theory in coordinate-free presentation

Abstract

This paper is devoted to the presentation, within the framework of a coordinate-free model, of two known Sz. Nagy—Foiaş theorems: The first one deals with the correspondence between the invariant subspaces of a contraction T and the regular factorizations of its characteristic function θT, while the second one is the commutant lifting theorem. The proofs are based on a coordinate-free approach to the model. In the first theorem an essential point is the singling out of the role of functional imbeddings and the formulation of a criterion for the existence of an invariant subspace in terms of a functional imbedding of a special form. As far as the commutant lifting theorem is concerned, our approach enables us to give a parametrization of the lifted operators with the aid of one free parameter instead of two dependent ones, as done by Sz.-Nagy and Foiaş.