Linear Algebra and its Applications, ( ISI ), Volume (532), No (1), Year (2017-6) , Pages (127-139)

#### Title : ( More on operator monotone and operator convex functions of several variables )

Authors: Hamed Najafi ,

Citation: BibTeX | EndNote

#### Abstract

‎Let $C_1,C_2,\ldots,C_k$ be positive matrices in $M_n$ and $f$ be a continuous real-valued function on $[0,\infty)$‎. ‎In addition‎, ‎consider $\Phi$ as a positive linear functional on $M_n$ and define‎ ‎$$\phi(t_1,t_2,t_3,\ldots,t_k)=\Phi\left(f(t_1C_1+t_2C_2+t_3C_3+\ldots+t_kC_k)\right),$$‎ ‎as a $k$ variables continuous function on $[0,\infty) \times \ldots \times [0,\infty)$‎. ‎In this paper‎, ‎we show that if $f$ is an operator convex function of order $mn$‎, ‎then $\phi$ is a $k$ variables operator convex function of order $(n_1,\ldots,n_k)$ such that $m=n_1 n_2\ldots n_k$‎. ‎Also‎, ‎if $f$ is an operator monotone function of order $n^{k+1}$‎, ‎then $\phi$ is a $k$ variables operator monotone function of order $n$‎. ‎In particular‎, ‎if $f$ is a non-negative operator decreasing function on $[0,\infty)$‎, ‎then the function $t\rightarrow \Phi\left(f(A+tB)\right)$ is an operator decreasing and can be written as a Laplace transform of a positive measure‎.

#### Keywords

Operator monotone functions; Operator convex functions; BMV conjecture; Laplace transform
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@article{paperid:1063452,
author = {Najafi, Hamed},
title = {More on operator monotone and operator convex functions of several variables},
journal = {Linear Algebra and its Applications},
year = {2017},
volume = {532},
number = {1},
month = {June},
issn = {0024-3795},
pages = {127--139},
numpages = {12},
keywords = {Operator monotone functions; Operator convex functions; BMV conjecture; Laplace transform},
}