The Algebra of Differential Operators Associated to a Weight Matrix

Abstract

Given a weight matrix W(x) of size N on the real line one constructs a sequence of matrix valued orthogonal polynomials, {P n } n≥0. We study the algebra $${\mathcal{D}}(W)$$ of differential operators D with matrix coefficients such that P n D = Λ n P n , with Λ n in the algebra A of N × N complex matrices. We study certain representations of this algebra, prove that it is a *-algebra and give a precise description of its isomorphic image inside the algebra $${A^{{N}_0}}$$ .