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

publication date

  • November 2015

start page

  • 305

end page

  • 344

volume

  • 485

international standard serial number (ISSN)

  • 0024-3795

electronic international standard serial number (EISSN)

  • 1873-1856

abstract

  • 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.

keywords

  • orthogonal polynomials on the unit circle; ggt matrix; cmv matrix; fundamental matrix; canonical linear spectral transformations