Relative asymptotics for orthogonal matrix polynomials with respect to a perturbed matrix measure on the unit circle

Abstract

Given a positive definite matrix measure Ω supported on the unit circle T, then main purpose of this paper is to study the asymptotic behavior of $L_n(\tilde{\Omega}) L_n(\Omega) -1}$ and $\Phi_n(z, \tilde{\Omega}) \Phi_n(z, \tilde{\Omega}) -1}$ where $\tilde{\Omega}(z) = \Omega(z) + M \delta ( z - w)$, $ 1$, M is a positive definite matrix and δ is the Dirac matrix measure. Here, Ln(·) means the leading coefficient of the orthonormal matrix polynomials Φn(z; •).

Finally, we deduce the asymptotic behavior of $\Phi_n(omega, \tilde{\Omega}) \Phi_n(omega, \Omega)$ in the case when M=I.