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Zapiski Nauchnykh Seminarov LOMI, 1989, Volume 178, Pages 5–22
(Mi znsl4674)
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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.
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
Linking options:
https://www.mathnet.ru/eng/znsl4674 https://www.mathnet.ru/eng/znsl/v178/p5
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Abstract page: | 97 | Full-text PDF : | 41 |
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