Unitary orbits of Hermitian operators with convex or concave functions

Abstract

This short but self-contained survey presents a number of elegant matrix/operator inequalities for general convex or concave functions, obtained with a unitary orbit technique. Jensen-, sub- or superadditivity-type inequalities are considered. Some of them are substitutes to classical inequalities (Choi, Davis, Hansen–Pedersen) for operator convex or concave functions. Various trace, norm and determinantal inequalities are derived. Combined with an interesting decomposition for positive semi-definite matrices, several results for partitioned matrices are also obtained.