The algebra $\mathcal{D}(W)$ via strong Darboux transformations

The Matrix Bochner Problem aims to classify weight matrices $W$ such that the algebra $\mathcal D(W)$, of all differential operators that have a sequence of matrix-valued orthogonal polynomials for $W$ as eigenfunctions, contains a second-order differential operator. In \cite{CY18} it is proven that, under certain assumptions, the solutions to the Matrix Bochner Problem can be obtained through a noncommutative bispectral Darboux transformation of some classical scalar weights. The main aim of this paper is to introduce the concept of strong Darboux transformation among weight matrices and explore the relationship between the algebras $\mathcal{D}(W)$ and $\mathcal{D}(\widetilde{W})$ when $\widetilde{W}$ is a strong Darboux transformation of $W$. Starting from a direct sum of classical scalar weights $\widetilde W$, and leveraging our complete knowledge of the algebra of $\mathcal D(\widetilde W)$, we can easily determine the algebra $\mathcal D(W)$ of a weight $W$ that is a strong Darboux transformation of $\widetilde W$.