Generalized Log-Majorization and Multivariate Trace Inequalities

TL;DR

This paper extends multivariate trace inequalities to all unitarily invariant norms using a generalized concept of log-majorization, broadening their applicability in matrix analysis.

Contribution

It introduces a generalized log-majorization framework that enables the extension of existing matrix inequalities to a wider class of norms.

Findings

Multivariate inequalities now hold for all unitarily invariant norms.

The generalized log-majorization concept applies to logarithmic integral averages.

The results unify and extend previous matrix inequality frameworks.

Abstract

We show that recent multivariate generalizations of the Araki-Lieb-Thirring inequality and the Golden-Thompson inequality [Sutter, Berta, and Tomamichel, Comm. Math. Phys. (2016)] for Schatten norms hold more generally for all unitarily invariant norms and certain variations thereof. The main technical contribution is a generalization of the concept of log-majorization which allows us to treat majorization with regards to logarithmic integral averages of vectors of singular values.