Some singular value inequalities on majorisation for positive semi-definite matrices

Abstract

In this note, some singular value inequalities on majorisation for positive semi-definite matrices are given. First, the weak log-majorisation of the quantity $f(A)g(B)\pm f(B)g(A)$ for certain functions is obtained, where A, $B\ge 0$. As a consequence, the weak log-majorisation of the quantity $A^{p}B^{q}\pm B^{p}A^{q}$ for p, $q>0$ is also obtained. Finally, different estimates for the singular values of the quantity $A^{p}B^{q}\pm B^{p}A^{q}$ are deduced for A, $B>0$ and $0<p\le q$.