Structure of the spaces of matrix monotone functions and of matrix convex functions and Jensen's type inequality for operators

TL;DR

This paper explores the structure of matrix monotone and convex functions, examining their properties and relationships through inequalities and operator functions, with a focus on the mutual dependence of key assertions at each matrix size n.

Contribution

It establishes the equivalence between certain matrix inequalities and monotonicity conditions for functions, deepening understanding of the structure of matrix monotone and convex functions.

Findings

Conditions (ii) and (iii) are equivalent for all n

Provides insight into the double piling structure of matrix function spaces

Clarifies the relationship between matrix convexity and monotonicity

Abstract

Let $n \in \N$ and $M_n$ be the algebra of $n \times n$ matrices. We call a function $f$ matrix monotone of order $n$ or $n$-monotone in short whenever the inequality $f(a) \leq f(b)$ holds for every pair of selfadjoint matrices $a, b \in M_n$ such that $a \leq b$ 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 $P_n(I)$ and $K_n(I)$. 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 $\{P_n(I)\}_{n\in\N}$ and $\{K_n(I)\}_{n\in\N}$. 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: \begin{enumerate} \item[(i)] $f(0)\leq 0$ and $f$ is $n$-convex in $[0,\alpha)$, \item[(ii)] For each matrix $a$ with its spectrum in $[0,\alpha)$ and a contraction $c$ in the matrix algebra $M_n$, \[ f(c^{\star}a c)\leq c^{\star}f(a)c, \] \item[(iii)] The functon $g(t)/t$ is $n$-monotone in $(0,\alpha)$. \end{enumerate} In particular, we show that for any $n \in \N$ two conditions $(ii)$ and $(iii)$ are equivalent.