2007/04/01 by Koenraad M. R. Audenaert, Audenaert, Koenraad M. R., Jaspal Singh Aujla +1
Mathematics · #15A60 #FOS: Mathematics #Functional Analysis (math.FA) #math.FA #msc:15A60
paper · pdf · doi:10.48550/arxiv.0704.0099
18 pages
arxiv created 2007/04/01 · arxiv updated 2009/12/01
For positive semidefinite matrices A and B, Ando and Zhan proved the inequalities ||| f(A)+f(B) ||| ≥ ||| f(A+B) ||| and ||| g(A)+g(B) ||| ≤ ||| g(A+B) |||, for any unitarily invariant norm, and for any non-negative operator monotone f on [0,∞) 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) ||| ≤ ||| f(|A-B|) |||, and ||| g(A)-g(B) ||| ≥ ||| 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≥ ||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.