Subadditivity of eigenvalue sums

Abstract

Abstract. Let f (t) be a nonnegative concave function on 0 ≤ t < ∞ with f (0) = 0, and let X, Y be n×n matrices. Then it is known that fwhere · 1 is the trace norm. We extend this result to all unitarily invariant norms and prove some inequalities of eigenvalue sums.

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

“…Therefore, we recapture a result from [8]: the map X −→ f (X) is subbaditive. For the trace-norm, this is Rotfel'd Theorem.…”

Operator inequalities of Jensen type

“…The authors of [9] used inequality (1.3) to obtain some operator inequalities. In particular, they gave a generalization of the Petrović operator inequality as follows: Some other operator extensions of (1.4) can be found in [1,2,11]. In this paper, as a continuation of [9], we extend inequality (1.3), refine (1.3) and improve some of our results in [9].…”

Matrix Analysis: Matrix Monotone Functions, Matrix Means, and Majorization

“…In this way, it turned out that the results in [20] (also [75]) are refined to those for each matrix order.…”

“…In the first half of this section, we prove the subadditivity (resp., superadditivity) inequality for f ðA þ BÞ and f ðAÞ þ f ðBÞ when f is a nonnegative concave (resp., convex) function on ½0; 1Þ and A; B We begin with the subadditivity inequality due to Ando and Zhan [9] in the case where f is an operator concave function. The proof was substantially simplified by Uchiyama [75] as presented below.…”