HOME > Table of Contents and Abstracts > Vol. 67, No. 1
Tohoku Mathematical Journal
SECOND SERIES VOL. 67, NO. 1
|Tohoku Math. J.|
67 (2015), 39-50
ORDER OF OPERATORS DETERMINED BY OPERATOR MEAN
Masaru Nagisa and Mitsuru Uchiyama
(Received July 9, 2013, revised December 20, 2013)
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.
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