Functions preserving operator means

Abstract

Let $\sigma$ be a non-trivial operator mean in the sense of Kubo and Ando, and let $O{M}_{+}^1$ the set of normalized positive operator monotone functions on $(0,\infty )$. In this paper, we study class of $\sigma$-subpreserving functions $f\in O{M}_{+}^1$ satisfying $$f(A\sigma B)\le f(A)\sigma f(B)$$ for all 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 $f$ satisfying $$f(A\sigma B)=f(A)\sigma f(B)$$ for all positive operators $A$ and $B$. In particular, when ${\lim }_{t\to 0}(1\sigma t)=0$, the function $f$ preserves $\sigma$ if and only if $f$ and $1\sigma t$ are representing functions for weighted harmonic means.