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.