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# Conservative discrete time-invariant systems and block operator CMV matrices

### Abstract

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.

### Article Information

 Title Conservative discrete time-invariant systems and block operator CMV matrices Source Methods Funct. Anal. Topology, Vol. 15 (2009), no. 3, 201-236 MathSciNet MR2567307 Copyright The Author(s) 2009 (CC BY-SA)

### Authors Information

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

### Citation Example

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

### BibTex

@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},
URL = {http://mfat.imath.kiev.ua/article/?id=509},
}