On Ando's inequalities for convex and concave functions

Abstract

For positive semidefinite matrices $A$ and $B$, Ando and Zhan proved the inequalities $|||f(A)+f(B)|||\ge |||f(A+B)|||$ and $|||g(A)+g(B)|||\le |||g(A+B)|||$, for any unitarily invariant norm, and for any non-negative operator monotone $f$ on $[0,\infty )$ with inverse function $g$. These inequalities have very recently been generalised to non-negative concave functions $f$ and non-negative convex functions $g$, by Bourin and Uchiyama, and Kosem, respectively. In this paper we consider the related question whether the inequalities $|||f(A)-f(B)|||\le |||f(|A-B|)|||$, and $|||g(A)-g(B)|||\ge |||g(|A-B|)|||$, obtained by Ando, for operator monotone $f$ with inverse $g$, also have a similar generalisation to non-negative concave $f$ and convex $g$. We answer exactly this question, in the negative for general matrices, and affirmatively in the special case when $A\ge ||B||$. In the course of this work, we introduce the novel notion of $Y$-dominated majorisation between the spectra of two Hermitian matrices, where $Y$ is itself a Hermitian matrix, and prove a certain property of this relation that allows to strengthen the results of Bourin-Uchiyama and Kosem, mentioned above.

18 pages