On Ascertaining Inductively the Dimension of the Joint Kernel of Certain Commuting Linear Operators

ISSN： 0196-8858

Source： Advances in Applied Mathematics, Vol.17, Iss.3, 1996-09, pp. : 209-250

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Abstract

Given an index set X , a collection B of subsets of X (all of the same cardinality), and a collection { l x } x∈ X of commuting linear maps on some linear space, the family of linear operators whose joint kernel K = K ( B ) is sought consists of all l A≔∏ a∈ A l a with A any subset of X which intersects every B∈ B . The goal is to establish conditions, on B and l , which ensure that dim K ( B )=[summation operator] B∈ B dim K ({ B }), or, at least, one or the other of the two inequalities contained in this equality. Concrete instances of this problem arise in box spline theory, and specific conditions on l were given by Dahmen and Micchelli for the case that B consists of the bases of a matroid. We give a new approach to this problem and establish the inequalities and the equality under various rather weak conditions on B and l . These conditions involve the solvability of certain linear systems of the form l b ?=phi b , b∈ B , with B∈ B , and the existence of "placeable" elements of X , i.e., of x∈ X for which every B∈ B not containing x has all but one element in common with some B '∈ B containing x .