Subadditivity of eigenvalue sums

Abstract

Let f ( t ) f(t) be a nonnegative concave function on 0 ≤ t > ∞ 0 \leq t >\infty with f ( 0 ) = 0 f(0)=0 , and let X , Y X, Y be n × n n\times n matrices. Then it is known that ‖ f ( | X + Y | ) ‖ 1 ≤ ‖ f ( | X | ) ‖ 1 + ‖ f ( | Y | ) ‖ 1 \Vert f(|X+Y|)\Vert _1\leq \Vert f(|X|)\Vert _1 +\Vert f(|Y|)\Vert _1 , where ‖ ⋅ ‖ 1 \Vert \cdot \Vert _1 is the trace norm. We extend this result to all unitarily invariant norms and prove some inequalities of eigenvalue sums.