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On Ando's inequalities for convex and concave functions

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Jaspal Singh Aujla
Department of Applied Mathematics, National Institute of Technology, Jalandhar 144011, Punjab, India
Abstract 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 nonnegative 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. Key words: Matrix norm inequality, Convex functions, Majorization. 1991 MSC: 15A60
On Ando’s inequalities for convex and concave functions 2007
Koenraad M.R. Audenaert
Institute for Mathematical Sciences, Imperial College London, 53 Prince’s Gate, London SW7 2PG, United Kingdom
be reported in Section 6. On the more constructive side, the Proposition also allows to strengthen the results of Bourin-Uchiyama and Kosem mentioned above. This is the topic of the final Section, along with a few other applications.
The conditions on f are satisfied by every operator concave function f with f (0) = 0. Inequality (4) was generalised by Kosem [7] to non-negative convex functions g on [0, ∞), with g (0) = 0. Inequality (3) was generalised very recently to any non-negative concave function on [0, ∞) by Bourin and Uchiyama [5], who also gave a simpler proof of Kosem’s result. In the same spirit, we consider the question whether inequalities (1) and (2) can also be generalised to non-negative concave f and convex g , respectively. After introducing the necessary prerequisites in Section 2, we present our main results concerning this question in Section 3. Regrettably, most of our results are negative answers, and we give counterexamples to this generalisation. The answer is even negative for the special case A ≥ B , although the apparent hardness of finding counterexamples had led us temporarily into believing that in that case the generalisation might actually hold. All is not bad news, however. In Section 4 we answer the question affirmatively in the special case when A ≥ ||B ||. In Section 5, we introduce the novel notion of Y -dominated majorisation between the spectra of two Hermitian matrices, where Y is itself a Hermitian matrix. We prove a certain property of this relation, namely Proposition 3, which we subsequently use, first in a rather destructive fashion. To wit, the Proposition has been instrumental in finally discovering a counterexample to the generalisation of (1) for A ≥ B ; this will 2
In a later paper [2], Ando and Zhan proved the related inequalities (with the same conditions on f and g ) |||f (A) + f (B )||| ≥ |||f (A + B )|||, |||g (A) + g (B )||| ≤ |||g (A + B )|||. (3) (4)
Email addresses: k.audenaert@ (Koenraad M.R. Audenaert), aujlajs@nitj.ac.in (Jaspal Singh Aujla).
Preprint submitted to Elsevier
5 February 2008, 2:24
2
Preliminaries
In this Section, we introduce the notations and necessary prerequisites; a more detailed exposition can be found, e.g. in [4]. We will use the abbreviations LHS and RHS for left-hand side and right-hand side, respectively. We denote the vector of diagonal entries of a matrix A by Diag(A). We denote the absolute value by | · |, both for scalars and for matrices. For matrices this is defined as |A| := (A∗ A)1/2 . Similarly, we denote the positive part of a real scalar or Hermitian matrix by (·)+ , and define it by A+ := (A + |A|)/2. In this paper, we are mainly concerned with monotonously increasing convex and concave functions from R to R. Kosem noted in [7] that any such function can be approximated by a sum of angle functions x → ax + b(x − x0 )+ , where a ≥ 0, and b > 0 for a convex angle function (b < 0 for a concave one). We are also concerned with the unitarily invariant (UI) matrix norms, which we denote by ||| · |||, and which are defined in terms of the singular values σj (·) of the matrix only. We adopt the customary convention that the singular values are sorted in non-increasing order: σ1 ≥ σ2 ≥ . . . ≥ σd . Special cases of these norms are the operator norm || · ||, which is just equal to the largest singular value σ1 (·), and the Ky Fan norms || · ||(k) , which are defined as the sum of the k largest singular values:

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1
Introduction
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xT x = 1 (or
1):
(1)
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convex and quasi convex

convex and quasi convex


separable
function.
• Why does Theorem 5 require that each φi be strictly concave to ensure that f is strictly concave, while Theorem 4 requires only one fi be strictly concave to ensure that f is is strictly concave?
i=1
i=1
This establishes that f is concave. If some fi is strictly concave and αi > 0, then the inequality is strict.
Since a constant function is concave, Theorem 4 implies:
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