Functions preserving operator means

Trung Hoa Dinh; Hiroyuki Osaka; Shuhei Wada

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arXiv · 1912.10378

Functions preserving operator means

Trung Hoa Dinh·Hiroyuki Osaka·Shuhei Wada

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Abstract

Let $σ$ 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 $σ$-subpreserving functions $f\in OM_+^1$ satisfying $$f(AσB) \le f(A)σf(B)$$ 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 $σ$-preserving functions $f$ satisfying $$f(AσB) = f(A)σf(B)$$ for all positive operators $A$ and $B$. In particular, when $\lim_{t\rightarrow 0} (1σt) =0$, the function $f$ preserves $σ$ if and only if $f$ and $1σt$ are representing functions for weighted harmonic means.

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Trung Hoa Dinh, Hiroyuki Osaka, Shuhei Wada. 2019-12-22. Functions preserving operator means. https://arxiv.org/abs/1912.10378

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