Note on the Structure of the Spaces of Matrix Monotone Functions

Abstract

Let n ∈ N and M _n 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 ∈ M _n such that a ≤ b and all eigenvalues of a and b are contained in I. The spaces for n-monotone functions is written as P _n (I). For each n ∈ N and a finite interval I we define the class C _n (I) by the set of all positive real-valued continuous functions f over I such that f(I ^∘) ⊂ (0, ∞) and for any subset S ⊂ I ^∘ there exists a positive Pick function h on (0, ∞) interpolating f on S. Then we characterize C _n ([0, 1)) by an operator inequality. Moreover we show that for each nC _2n ([0, ∞)) ⊊ P _n ⁺([0, ∞)).