ORDER OF OPERATORS DETERMINED BY OPERATOR MEAN

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 $0 \leq A \leq B$, then it holds that $Y \sigma (tA+X) \leq Y \sigma (tB+X)$ for any non-negative real number $t$ and any positive, invertible operators $X,Y$. We show that the condition $ Y \sigma (tA+X) \leq Y \sigma (tB+X)$ for a sufficiently small $t > 0$ implies $A \leq B$ 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 $ad-bc > 0$.