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Zapiski Nauchnykh Seminarov LOMI, 1989, Volume 178, Pages 5–22 (Mi znsl4674)  

Two classical theorems of function model theory via the coordinate-free approach

V. I. Vasyunin
Abstract: The aim of the paper is to present a new approach to the proof of two well-known theorems of Sz.-Hagy–Foiaş: the first one concerns the correspondence between invariant subspaces of a given contraction $T$ and regular factorizations of the characteristic function $\theta_T$ of $T$, the second one is the commutant lifting theorem. The proofs are based on the coordinate-free approach to the functional model. In other words, a concrete spectral representation of a minimal unitary dilation is not fixed. The essential point in the first theorem is an assertion in terms of functional mappings $\eta: L^2(F)\longmapsto\mathcal{H}$ ($\mathcal{H}$ is the space of a minimal unitary dilation $U$) equivalent to the existence of an invariant supspace of $T$. As to the lifting theorem, our approach provides us with a new parametrization of lifted operator that seems to be more natural than the known Sz.-Nagy–Foiaş parametrization.
English version:
Journal of Soviet Mathematics, 1992, Volume 61, Issue 2, Pages 1951–1962
DOI: https://doi.org/10.1007/BF01095661
Bibliographic databases:
Document Type: Article
UDC: 517.98
Language: Russian
Citation: V. I. Vasyunin, “Two classical theorems of function model theory via the coordinate-free approach”, Investigations on linear operators and function theory. Part 18, Zap. Nauchn. Sem. LOMI, 178, "Nauka", Leningrad. Otdel., Leningrad, 1989, 5–22; J. Soviet Math., 61:2 (1992), 1951–1962
Citation in format AMSBIB
\Bibitem{Vas89}
\by V.~I.~Vasyunin
\paper Two classical theorems of function model theory via the coordinate-free approach
\inbook Investigations on linear operators and function theory. Part~18
\serial Zap. Nauchn. Sem. LOMI
\yr 1989
\vol 178
\pages 5--22
\publ "Nauka", Leningrad. Otdel.
\publaddr Leningrad
\mathnet{http://mi.mathnet.ru/znsl4674}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=1037763}
\zmath{https://zbmath.org/?q=an:0784.47020|0734.47004}
\transl
\jour J. Soviet Math.
\yr 1992
\vol 61
\issue 2
\pages 1951--1962
\crossref{https://doi.org/10.1007/BF01095661}
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