## Tohoku Mathematical Journal 2015 March SECOND SERIES VOL. 67, NO. 1

 Tohoku Math. J. 67 (2015), 39-50

Title ORDER OF OPERATORS DETERMINED BY OPERATOR MEAN

Author Masaru Nagisa and Mitsuru Uchiyama

(Received July 9, 2013, revised December 20, 2013)
Abstract. Let $\sigma$ be an operator mean and $f$ a non-constant operator monotone function on $(0,\infty)$ associated with $\sigma$. If operators $A, B$ satisfy $0\le A \le B$, then it holds that $Y \sigma (tA+X) \le Y \sigma (tB+X)$ for any non-negative real number $t$ and any positive, invertible operators $X,Y$. We show that the condition $Y \sigma (tA+X) \le Y \sigma (tB+X)$ for a sufficiently small $t>0$ implies $A \le B$ if and only if $X$ is a positive scalar multiple of $Y$ or the associated operator monotone function $f$ with $\sigma$ has the form $f(t) = (at+b)/(ct+d)$, where $a,b,c,d$ are real numbers satisfying $ad-bc>0$.

Mathematics Subject Classification. Primary 47A63; Secondary 15A39.

Key words and phrases. Matrix order, operator mean, operator monotone function, Schur product, Fréchet derivative.