A Lieb-Thirring inequality for singular values

TL;DR

This paper explores the extension of Lieb-Thirring inequalities to singular values of positive semidefinite matrices, identifying conditions on the parameter p for majorization relations.

Contribution

It establishes new conditions under which Lieb-Thirring inequalities can be extended to singular values, including cases of both majorization and reversed inequalities.

Findings

Identifies values of p for which the majorization holds.

Determines conditions for the reversed inequality.

Provides insights into singular value inequalities for positive semidefinite matrices.

Abstract

Let $A$ and $B$ be positive semidefinite matrices. We investigate the conditions under which the Lieb-Thirring inequality can be extended to singular values. That is, for which values of $p$ does the majorisation $\sigma(B^p A^p) \prec_w \sigma((BA)^p)$ hold, and for which values its reversed inequality $\sigma(B^p A^p) \succ_w \sigma((BA)^p)$.