Structure of operator algebras for matrix orthogonal polynomials

TL;DR

This paper investigates the algebraic structure of differential operators and eigenvalues associated with matrix-valued orthogonal polynomials, revealing how their centers behave under Darboux transformations and analyzing specific weight cases.

Contribution

It introduces a framework to analyze the algebraic structure of operator algebras for matrix orthogonal polynomials, including the behavior of centers under Darboux transformations.

Findings

The eigenvalue algebra Lambda(W) is isomorphic to D(W), simplifying analysis.

Explicit relationships between centers of Darboux-equivalent weights are established.

The structure is exemplified through reducible and irreducible matrix weights, including a Jacobi-type weight.

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.