Reducibility of Matrix Weights

JUAN A. TIRAO; IGNACIO N. ZURRIÁN

RAMANUJAN JOURNAL

SPRINGER

Lugar: Berlin; Año: 2018

1382-4090

In this paper, we discuss the notion of reducibility of matrix weights andintroduce a real vector space C R which encodes all information about the reducibilityof W . In particular, a weight W reduces if and only if there is a nonscalar matrix Tsuch that T W = W T ∗ . Also, we prove that reducibility can be studied by lookingat the commutant of the monic orthogonal polynomials or by looking at the coeffi-cients of the corresponding three-term recursion relation. A matrix weight may notbe expressible as direct sum of irreducible weights, but it is always equivalent to adirect sum of irreducible weights. We also establish that the decompositions of twoequivalent weights as sums of irreducible weights have the same number of terms andthat, 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 generalstructure. Finally, we make a change of emphasis by considering the reducibility ofpolynomials, instead of reducibility of matrix weights.