Structure of operator algebras for matrix orthogonal polynomials

2025/02/22 by Parisi, Ignacio Bono, Pacharoni, Inés · 2 citations
#33C45 #34L05 #34L10 #42C05 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Operator Algebras (math.OA)

paper · doi:10.48550/arxiv.2502.16070

Abstract

In this paper, we study the structure of the differential operator algebra \( D(W) \) and its associated eigenvalue algebra \( Λ(W) \) for matrix-valued orthogonal polynomials. While \( Λ(W) \) is isomorphic to \( D(W) \), its simpler framework allows us to efficiently derive strong results about \( D(W) \) and its center \( Z(W) \). 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.

Cited by

• Using quasi-Darboux transformations to construct exceptional matrix polynomials
• Matrix-Valued Hermite and Laguerre polynomials via Quadratic Transformation

Related

• Explicit construction of matrix-valued orthogonal polynomials of arbitrary size
• Algebraic interpretation of discrete families of matrix valued orthogonal polynomials
• Lie algebras of differential operators for Matrix valued Laguerre type polynomials
• Symmetric Jacobi Polynomials on a Triangle and Their Spectral Algebra
• Darboux equivalence for matrix-valued orthogonal polynomials