This question is related to Ask some matrix eigenvalue inequalities.

Let $\begin{bmatrix} A& B \\\\ B^* &A \end{bmatrix}$ be positive semidefinite. Is it true $\lambda_i^{1/2}(B^*B)\le \lambda_i(A)$? Here, $λ_i(⋅)$ means the ith largest eigenvalue of ⋅.