Algebraic and analytic properties associated with spectral transformations of matrix orthogonal polynomials (2019)
The main purpose of this work is the study of algebraic and analytic properties associated with spectral transformations of sequences of matrix orthogonal polynomials with respect to a matrix measure supported either on the unit circle or on the real line. In both frameworks, we study some properties related with a perturbation of a sequence of matrix moments. We extend to the matrix case some algebraic and analytic properties of matrix orthogonal polynomials on the unit circle, that are known in the scalar case, associated with the Uvarov and Christofeeel matrix transformations of a Hermitian matrix measure supported on the unit circle. We also study properties of block Hessenberg and block CMV matrices associated with sequences of matrix orthonormal polynomials, under certain transformations of the corresponding matrix measure. In addition, we extend to the matrix case some properties of the Szego transformation and then use these properties to analyze its effect on some spectral transformations, focusing on the relationship between the matrix-valued Stieltjes and Carathéodory functions.