2008/05/14 by Hiroyuki Osaka, Jun Tomiyama, Osaka, Hiroyuki +1
Mathematics · #26A48 #26A51 #Advanced Banach Space Theory #FOS: Mathematics #Holomorphic and Operator Theory #Mathematical Inequalities and Applications #Operator Algebras (math.OA) #math.OA #msc:26A48 #msc:26A51
paper · pdf · doi:10.48550/arxiv.0805.1996
arxiv created 2008/05/14 · openalex publication_date 2008/05/14 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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-convex functions are written as Pn(I) and Kn(I). In this note we discuss several assertions at each leven n for which we regard themas the problems of double piling structure of those sequences \Pn(I)\n∈\N and \Kn(I)\n∈\N. In order to see clear insight of the aspect of the problems, however, we choose the following three main assertions among them and discuss their mutual dependence: \beginenumerate \item[(i)] f(0)≤ 0 and f is n-convex in [0,α), \item[(ii)] For each matrix a with its spectrum in [0,α) and a contraction c in the matrix algebra Mn, f(c⋆a c)≤ c⋆f(a)c, \item[(iii)] The functon g(t)/t is n-monotone in (0,α). \endenumerate In particular, we show that for any n ∈ \N two conditions (ii) and (iii) are equivalent.