On Drury's solution of Bhatia \& Kittaneh's question

Abstract

Let $A,B$ be $n\times n$ positive semidefinite matrices. Bhatia and Kittaneh asked whether it is true $$\sqrt{{\sigma }_j(AB)}\le \frac{1}{2}{\lambda }_j(A+B),j=1,\dots ,n$$ where ${\sigma }_j(\cdot )$, ${\lambda }_j(\cdot )$, are the $j$-th largest singular value, eigenvalue, respectively. The question was recently solved by Drury in the affirmative. This article revisits Drury's solution. In particular, we simplify the proof for a key auxiliary result in his solution.

8 pages