2 Rectangular Matrix Orthogonal Polynomials

Classical orthogonal polynomials and matrix polynomials being orthogonal with respect to some Hermitean positive deenite matrix of measures share several properties , e.g., three term recurrencies, Christooel{Darboux formulas; there are connections to the triangular decomposition of the (inverse) moment matrix and to eigenvalue{problems for the banded matrix of recurrence coeecients. Also, a connection between these orthogonal polynomials and the spectral analysis of selfadjoint operators is well{known. In this paper we study these aspects for rectangular matrix valued measures. In particular, we establish a caracterization in terms of matrix Pad e approximants of the spectrum and the resolvent set of a diierence operator, i.e., a not necessarily selfadjoint, but bounded operator which may be represented as a biinnnite banded matrix. As an application, we give convergence theorems of matrix Pad e approximants.