On Ando's inequalities for convex and concave functions

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dc.creatorAudenaert, Koenraad M. R.
dc.creatorAujla, Jaspal Singh
dc.date2007-04-01
dc.date.accessioned2026-07-07T07:54:20Z
dc.date.available2026-07-07T07:54:20Z
dc.descriptionFor 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.
dc.description18 pages
dc.identifierhttps://arxiv.org/abs/0704.0099
dc.identifierhttp://arxiv.org/abs/0704.0099
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/126568
dc.subjectFunctional Analysis
dc.subject15A60
dc.titleOn Ando's inequalities for convex and concave functions
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