Some basic facts on orthogonal rational matrix functions on the unit circle

We consider spaces of (square) matrix functions each entry of which is a rational (complex-valued) function with prescribed poles. In particular, the poles of these rational functions are not located on the unit circle of the complex plane. This fact will be necessary in a way, since based on a given nonnegative Hermitian matrix Borel measure on the unit circle, the spaces will be equipped simultaneously with left and right matrix inner products via integration. Essentially, we study some special systems of orthogonal rational matrix functions in that context and some basics on the systems in question will be presented. The bottom line is, we will see that larger parts of the classical theory of orthogonal polynomials on the unit circle can be extended to this more general situation. Among other things, we point out a construction of orthogonal rational matrix functions via reproducing kernels, the important role of Christoffel-Darboux formulae, and a characterization of the orthogonal systems via specific recurrence relations. In doing so, an essential feature is marked by an inherent (but far from self-evident) interplay between objects associated with the left and right versions of matrix inner products. The talk is based on a joint work with Bernd Fritzsche and Bernd Kirstein.Talk in the "Coloquio del Posgrado en Ciencias Matematicas" at the "Universidad Michoacana de San Nicolas de Hidalgo" in Morelia (Mexico)status: publishe