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Minorant properties Weisenhofer, Stephen

Abstract

We study the minorant property and the positive minorant property for norms on spaces of matrices and norms on spaces of functions. A matrix is said to be a majorant of another if all the entries in the first matrix are greater than or equal to the absolute values of the corresponding entries in the second matrix. The Cp norm of a matrix is the tP norm of its singular values. The space of n x n matrices, with this norm, is said to have the minorant property provided that the norm of each nonnegative matrix is greater than or equal to the norm of every matrix that it majorizes. Similarly, if the norm of each nonnegative matrix is greater than or equal to the norm of every nonnegative matrix that it majorizes, then the space of matrices is said to have the positive minorant property. It is easy to verify that these properties hold if p is even. We show that the positive minorant property fails on n x n matrices with the Cp norm when 0 <p < 2(n — 1) and p is not even. We also consider versions of the minorant properties for LP-spaces of function on some commutative topological groups. We show that the positive minorant property fails on the cyclic group of order n when 0 <p < 2[(n — 1)/2] and p is not even. This relation between p and n is different from the one for matrices, and yet both seem to be optimal. We also show that the minorant property fails on cyclic groups when p and n are suitably related. We completely determine the combinations of p and n for which the minorant properties hold on some low-order groups. For compact abelian groups, we show that the positive minorant property is equivalent to a certain function operating on the class of positive definite functions. We deduce that if LP(G) has the positive minorant property, then so does LP+2 (G) .

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