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

Let $A, B$ be $n\times n$ positive semidefinite matrices. Bhatia and Kittaneh asked whether it is true $$ \sqrt{σ_j(AB)}\le \frac{1}{2} λ_j(A+B), \qquad j=1, \ldots, n$$ where $σ_j(\cdot)$, $λ_j(\cdot)$, 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.