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.
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Literature cited
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Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 178, pp. 5–22, 1989.
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Vasyunin, V.I. Two classical theorems of function model theory in coordinate-free presentation. J Math Sci 61, 1951–1962 (1992). https://doi.org/10.1007/BF01095661
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DOI: https://doi.org/10.1007/BF01095661