Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Contractions with rank one defect operators and truncated CMV matrices   The main issue we address in the present paper are the new models for
completely non-unitary contractions with rank one defect operators acting on
some Hilbert space of dimension $N\leq\infty$. This model complements nicely
the well-known models of Liv$\rm{\check{s}}$ic and Sz.-Nagy--Foias. We show
that each such an operator is unitarily equivalent to some truncated CMV matrix
obtained from the ``full'' CMV matrix by deleting the first row and the first
column, and acting in $\ell^2$ ($\dC^N$). This result can be viewed as a
nonunitary version of the famous characterization of unitary operators with a
simple spectrum due to Cantero, Moral and Vel\'azquez. It is shown that another
functional model for contractions with rank one defect operators takes the form
of the compression $f(\zeta)\to P_\cK (\zeta f(\zeta))$ on the Hilbert space
$L^2(\dT,d\mu)$ with a probability measure $\mu$ onto the subspace
$\cK=L^2(\dT,d\mu)\ominus \dC$. We develop direct and inverse spectral analysis
for finite and semi-infinite truncated CMV matrices. In particular, we study
the problem of reconstruction of such matrices from their spectrum or the mixed
spectral data involving Schur parameters. The uniqueness theorem for recovered
truncated CMV matrix from the given mixed spectral data is established. In this
part the paper is closely related to the results of Hochstadt and
Gesztesy--Simon obtained for finite self-adjoint Jacobi matrices.

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The categories of this paper are: astro-ph
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The categories of this paper are: cmp-lg
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The categories of this paper are: hep-ex
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The categories of this paper are: eess
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The categories of this paper are: cond-mat
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The categories of this paper are: adap-org
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The categories of this paper are: supr-con
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The categories of this paper are: comp-gas
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The categories of this paper are: alg-geom
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The categories of this paper are: quant-ph
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The categories of this paper are: stat
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The categories of this paper are: plasm-ph
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The categories of this paper are: econ
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The categories of this paper are: funct-an
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The categories of this paper are: atom-ph
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The categories of this paper are: chem-ph