Functions preserving operator means

Abstract

Let $\sigma$ be a non-trivial operator mean in the sense of Kubo and Ando, and let $OM_+^1$ be the set of normalized positive operator monotone functions on $(0, \infty )$. In this paper, we study the class of $\sigma$-subpreserving functions $f\in OM_+^1$ satisfying

$$\begin{aligned} f(A\sigma B) \le f(A)\sigma f(B) \end{aligned}$$

for all invertible positive operators A and B. We provide some criteria for f to be trivial, i.e., $f(t)=1$ or $f(t)=t$. We also establish characterizations of $\sigma$-preserving functions $f\in OM_+^1$ satisfying

$$\begin{aligned} f(A\sigma B) = f(A)\sigma f(B) \end{aligned}$$

for all invertible positive operators A and B. In particular, when $\lim _{t\rightarrow 0} (1\sigma t) =0$, the function $f\in OM_+^1\backslash \{1,t\}$ preserves $\sigma$ if and only if f and $1\sigma t$ are representing functions for a weighted harmonic mean.