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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