Contractions with rank one defect operators and truncated CMV matrices

Abstract

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.