Structure of the spaces of matrix monotone functions and of matrix convex functions and Jensen's type inequality for operators
| dc.creator | Osaka, Hiroyuki | |
| dc.creator | Tomiyama, Jun | |
| dc.date | 2008-05-14 | |
| dc.date.accessioned | 2026-07-07T09:38:47Z | |
| dc.date.available | 2026-07-07T09:38:47Z | |
| dc.description | 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 $[0,α)$, \item[(ii)] For each matrix $a$ with its spectrum in $[0,α)$ 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 $(0,α)$. \end{enumerate} In particular, we show that for any $n \in \N$ two conditions $(ii)$ and $(iii)$ are equivalent. | |
| dc.identifier | https://arxiv.org/abs/0805.1996 | |
| dc.identifier | http://arxiv.org/abs/0805.1996 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/160938 | |
| dc.subject | Operator Algebras | |
| dc.subject | 26A48, 26A51 | |
| dc.title | Structure of the spaces of matrix monotone functions and of matrix convex functions and Jensen's type inequality for operators | |
| dc.type | text |