Unitary orbits of Hermitian operators with convex or concave functions

TL;DR

This survey explores matrix inequalities involving convex and concave functions using unitary orbit techniques, providing new inequalities and generalizations of classical results for operators and matrices.

Contribution

It introduces a unified approach to derive matrix inequalities for convex and concave functions via unitary orbits, extending classical inequalities and applying to partitioned matrices.

Findings

Derived new Jensen-type inequalities for operators.

Extended classical inequalities like Choi, Davis, Hansen-Pedersen.

Established trace, norm, and determinantal inequalities for matrices.

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 super-additivity 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.