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,∞), ‖ f ( A + B ) ‖ ⩽ ‖ f ( A ) + f ( B ) ‖ 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 inequality A X ∗ X B ∞ ⩽ max X | ‖ ∞ ; ‖ | B | + | X ∗ | ‖ ∞ } for any partitioned Hermitian matrix.