Structure of operator algebras for matrix orthogonal polynomials

In this paper, we study the structure of the differential operator algebra $\mathcal{D}(W)$ and its associated eigenvalue algebra $\mathrm{\Lambda }(W)$ for matrix-valued orthogonal polynomials. While $\mathrm{\Lambda }(W)$ 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.