Functions preserving operator means

TL;DR

This paper investigates functions that preserve or subpreserve operator means, providing criteria for triviality and characterizations of such functions, especially in relation to weighted harmonic means.

Contribution

It offers new criteria and characterizations for functions that preserve or subpreserve operator means, extending understanding of their structure and properties.

Findings

Criteria for trivial functions $f(t)=1$ or $f(t)=t$

Characterizations of $\sigma$-preserving functions

Connection to weighted harmonic means

Abstract

Let $\sigma$ be a non-trivial operator mean in the sense of Kubo and Ando, and let $OM_+^1$ the set of normalized positive operator monotone functions on $(0, \infty)$. In this paper, we study class of $\sigma$-subpreserving functions $f\in OM_+^1$ 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 $\lim_{t\rightarrow 0} (1\sigma t) =0$, the function $f$ preserves $\sigma$ if and only if $f$ and $1\sigma t$ are representing functions for weighted harmonic means.