Structure of operator algebras for matrix orthogonal polynomials

Abstract: In this paper, we study the structure of the differential operator algebra ( \mathcal{D}(W) ) and its associated eigenvalue algebra ( \Lambda(W) ) for matrix-valued orthogonal polynomials. While ( \Lambda(W) ) is isomorphic to ( \mathcal{D}(W) ), its simpler framework allows us to efficiently derive strong results about ( \mathcal{D}(W) ) and its center ( \mathcal{Z}(W) ). We analyze the behavior of the center under Darboux transformations, establishing explicit relationships between the centers of Darboux-equivalent weights. These results are illustrated through the study of both reducible and irreducible matrix weights, including a detailed analysis of an irreducible Jacobi-type weight.