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1 STRUCTURE OF THE SPACES OF MATRIX MONOTONE FUNCTIONS AND OF MATRIX CONVEX FUNCTIONS AND JENSEN’S TYPE INEQUALITY FOR OPERATORS

By Hiroyuki Osaka and Jun Tomiyama

Abstract

Abstract. Let n ∈ N and Mn be the algebra of n × n matrices. We call a function f matrix monotone of order n or n-monotone in short whenever the inequality f(a) ≤ f(b) holds for every pair of selfadjoint matrices a, b ∈ Mn such that a ≤ b and all eigenvalues of a and b are contained in I. Matrix convex (concave) functions on I are similarily defined. The spaces for n-monotone functions and n-conve

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