A matrix subadditivity inequality for f(A+B) and f(A)+f(B)

Abstract

In 1999 Ando and Zhan proved a subadditivity inequality for operator concave functions. We extend it to all concave functions: Given positive semidefinite matrices A, B and a non-negative concave function f on [0, ∞),for all symmetric norms (in particular for all Schatten p-norms). The case f (t) = √ t is connected to some block-matrix inequalities, for instance the operator norm inequalityfor any partitioned Hermitian matrix.

Unitary orbits of Hermitian operators with convex or concave functions

“…for all symmetric norms. The case of operator concave functions is given in [2] and the general case is established in [14]; see also [12] for further results. Concerning differences, the following inequality holds…”

Non-commutative Clarkson Inequalities for n-Tuples of Operators

“…In place of a detailed proof of the lemma, it suffices to note that part (a) of this lemma is an extension of Theorem 2.3 in [2] for n-tuples of operators (see, also [12, Theorem 1]), while a stronger version of part (b) of the lemma can be obtained by invoking an argument similar to that used in the proof of Proposition 4.1 in [17] (see, also [2, Theorem 2.9]). Parts (c) and (d) of the lemma have been recently given in [15] and [8], respectively. Henceforth, we assume that every function is continuous.…”

Submajorisation inequalities for convex and concave functions of sums of measurable operators

“…We remark that the preceding proof is adapted from [4] Proof. We may assume that 0 ≤ x, y ∈ M. For each n = 1, 2, .…”

“…In the special case that M = M n (C) equipped with standard trace, the preceding theorem is due to Bourin and Uchiyama [4].…”

“…One step in Kosem's argument is to notice that this submajorisation is true for functions which are operator convex and that this assertion is a direct consequence of the Theorem of Ando-Zhan. Finally, the Ando-Zhan theorem itself has been extended to arbitrary non-negative increasing concave functions by Bourin and Uchiyama [4]. Interestingly enough, their proof depends, in turn, on the convexity inequality of Kosem. The aim of the present paper is to show that each of the submajorisation inequalities cited above continue to hold in the more general setting of measurable operators affiliated with a semifinite von Neumann algebra.…”