OPUC, CMV matrices and perturbations of measures supported on the unit circle

Abstract

a b s t r a c t Keywords:Orthogonal polynomials on the unit circle GGT matrix CMV matrix Fundamental matrix Canonical linear spectral transformations Let us consider a Hermitian linear functional defined on the linear space of Laurent polynomials with complex coefficients. In the literature, canonical spectral transformations of this functional are studied. The aim of this research is focused on perturbations of Hermitian linear functionals associated with a positive Borel measure supported on the unit circle. Some algebraic properties of the perturbed measure are pointed out in a constructive way. We discuss the corresponding sequences of orthogonal polynomials as well as the connection between the associated Verblunsky coefficients. Then, the structure of the Θ matrices of the perturbed linear functionals, which is the main tool for the comparison of their corresponding CMV matrices, is deeply analyzed. From the comparison between different CMV matrices, other families of perturbed Verblunsky coefficients will be considered. We introduce a new matrix, named Fundamental matrix, that is a tridiagonal symmetric unitary matrix, containing basic information about the family of orthogonal polynomials. However, we show that it is connected to another family of orthogonal polynomials through the Takagi decomposition.

A dichotomy result about Hessenberg matrices associated with measures in the unit circle

“…The Hessenberg matrix is the representation of the multiplication operator by z in terms of the orthonormal polynomials sequence that is known in the literature as Geronimus, Gragg and Teplyaev (GGT) matrix and CMV matrix from Laurent orthogonal polynomials. The properties of these matrices can be used to obtain properties of the associate measure; for example, matrix methods have been used to obtain quadrature formulas on the unit circle in Bultheel et al and in Marcellán and Shayanfarb, properties of the perturbed measure were obtained.…”

On Szegő's theorem for a nonclassical case

“…The condition a n = 0 is equivalent to the invertibility of all N × N leading principal submatrices of the moment matrix of the corresponding measure. In the setting of OPUC, it is the condition |α n | = 1 that is similarly necessary and sufficient for invertibility of leading principal submatrices of the corresponding moment matrix (see [22,Theorem 1.5.11] or [17] 1 ). Therefore, this is the analogous condition we impose in (1.14).…”