On the Szegő transformation for some spectral perturbations of matrix measures

Given a normalized measure μ supported on [−1,1], a measure σ supported on the unit circle can be defined by using the so-called Szegő transformation. This also establishes a well known relation between the corresponding Stieltjes and Carathéodory functions. In this contribution, we obtain a generalization to the matrix case of this relation for matrix measures μ and σ, and such generalization is used to obtain necessary conditions such that some spectral perturbations applied to μ are preserved under the matrix Szegő transformation.