A Lieb-Thirring inequality for singular values

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 $\sigma (B^p{A}^p){\succ }_w\sigma ((BA{)}^p)$.

arXiv comment8 pages; a Mathematica notebook is available on request