Reducibility of matrix weights

Abstract

In this paper, we discuss the notion of reducibility of matrix weights and introduce a real vector space CR which encodes all information about the reducibility of W. In particular, a weight W reduces if and only if there is a nonscalar matrix T such that TW= WT∗. Also, we prove that reducibility can be studied by looking at the commutant of the monic orthogonal polynomials or by looking at the coefficients of the corresponding three-term recursion relation. A matrix weight may not be expressible as direct sum of irreducible weights, but it is always equivalent to a direct sum of irreducible weights. We also establish that the decompositions of two equivalent weights as sums of irreducible weights have the same number of terms and that, up to a permutation, they are equivalent. We consider the algebra of right-hand-side matrix differential operators D(W) of a reducible weight W, giving its general structure. Finally, we make a change of emphasis by considering the reducibility of polynomials, instead of reducibility of matrix weights.