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Tohoku Mathematical Journal
2015
March
SECOND SERIES VOL. 67, NO. 1
Tohoku Math. J.
67 (2015), 3950

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 nonconstant 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 nonnegative 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 $adbc>0$.
Mathematics Subject Classification.
Primary 47A63; Secondary 15A39.
Key words and phrases.
Matrix order, operator mean, operator monotone function, Schur product, Fréchet derivative.


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