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

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 Theta 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.The authors would like to thank Professor Bernhard Beckermann and Professor RogerA. Horn for valuable and insightful discussions about congruence relations. We also thank the suggestions by the referees which have contributed to improve substantially the presentation of the manuscript. The work of the first author (FM) was partially sup-ported by Dirección General de Política Científica y Tecnológica, Ministerio de Economía y Competitividad (MINECO) of Spain, under grant MTM2012-36732-C03-01. The sec-ond author (NS) thanks Alexander von Humboldt Foundation for the support and the Department of Mathematics, Universidad Carlos III de Madrid, for its constant support and friendly atmosphere during the period January–July 2014 when the manuscript was finished