Jensen and Minkowski inequalities for operator means and anti-norms

TL;DR

This paper extends Jensen and Minkowski inequalities to operator means and introduces anti-norms, providing new tools for matrix analysis and inequalities involving positive linear maps.

Contribution

It develops Jensen inequalities for operator means, extends Minkowski inequalities, and introduces anti-norms, a novel concept parallel to symmetric norms in matrix analysis.

Findings

Established Jensen inequalities for positive linear maps and operator means.

Extended Minkowski determinantal inequality.

Developed the theory of anti-norms and an interpolation theorem for Schur multiplication.

Abstract

Jensen inequalities for positive linear maps of Choi and Hansen-Pedersen type are established for a large class of operator/matrix means. These results are also extensions of the Minkowski determinantal inequality. To this end we develop the study of anti-norms, a notion parallel to the symmetric norms in matrix analysis, including functionals like Schatten q-norms for a parameter q<1 and the Minkowski functional. An interpolation theorem for the Schur multiplication is given in this setting. Two sections have been added to the previous version, devoted to means of severable variables and anti-norms.