Darboux transformations and the algebra $\mathcal{D}(W)$

The problem of finding weight matrices $W(x)$ of size $N \times N$ such that the associated sequence of matrix-valued orthogonal polynomials are eigenfunctions of a second-order matrix differential operator is known as the Matrix Bochner Problem, and it is closely related to Darboux transformations of some differential operators. This paper aims to study Darboux transformations between weight matrices and to establish a direct connection with the structure of the algebra $\mathcal D(W)$ of all differential operators that have a sequence of matrix-valued orthogonal polynomials with respect to $W$ as eigenfunctions.