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Conservative discrete time-invariant systems and block operator CMV matrices


It is well known that an operator-valued function $\Theta$ from the Schur class ${\bf S}(\mathfrak M,\mathfrak N)$, where $\mathfrak M$ and $\mathfrak N$ are separable Hilbert spaces, can be realized as a transfer function of a simple conservative discrete time-invariant linear system. The known realizations involve the function $\Theta$ itself, the Hardy spaces or the reproducing kernel Hilbert spaces. On the other hand, as in the classical scalar case, the Schur class operator-valued function is uniquely determined by its so-called "Schur para-me ers". In this paper we construct simple conservative realizations using the Schur parameters only. It turns out that the unitary operators corresponding to the systems take the form of five diagonal block operator matrices, which are analogs of Cantero-Moral-Vel\'azquez (CMV) matrices appeared recently in the theory of scalar orthogonal polynomials on the unit circle. We obtain new models given by truncated block operator CMV matrices for an arbitrary completely non-unitary contraction. It is shown that the minimal unitary dilations of a contraction in a Hilbert space and the minimal Naimark dilations of a semi-spectral operator measure on the unit circle can also be expressed by means of block operator CMV matrices.

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TitleConservative discrete time-invariant systems and block operator CMV matrices
SourceMethods Funct. Anal. Topology, Vol. 15 (2009), no. 3, 201-236
MathSciNet   MR2567307
CopyrightThe Author(s) 2009 (CC BY-SA)

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Yury Arlinskii
Department of Mathematical Analysis, East Ukrainian National University, 20-A Kvartal Molodizhny, Lugansk, 91034, Ukraine

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Yury Arlinskiĭ, Conservative discrete time-invariant systems and block operator CMV matrices, Methods Funct. Anal. Topology 15 (2009), no. 3, 201-236.


@article {MFAT509,
    AUTHOR = {Arlinskiĭ, Yury},
     TITLE = {Conservative discrete time-invariant systems and block operator CMV matrices},
   JOURNAL = {Methods Funct. Anal. Topology},
  FJOURNAL = {Methods of Functional Analysis and Topology},
    VOLUME = {15},
      YEAR = {2009},
    NUMBER = {3},
     PAGES = {201-236},
      ISSN = {1029-3531},
  MRNUMBER = {MR2567307},
       URL = {},

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