Weak supermajorization between symplectic spectra of positive definite matrix and its pinching

Abstract

Let $A=\left[\begin{array}{cc}E& F\\ F^T& G\end{array}\right]$ be a $2n\times 2n$ real positive definite matrix, where $E,F,$ and $G$ are $n\times n$ blocks. It is shown that $d(E\oplus G){\prec }^{w}d(A)$. Here $d(A)$ denotes the $n$-vector consisting of the symplectic eigenvalues of $A$ arranged in the non-decreasing order. We also observe the following weak supermajorization relation, which is interesting on its own: $\hat{I}»\left({\left(\mathcal{C}(G{)}^{1/2}\mathcal{C}(E)\mathcal{C}(G{)}^{1/2}\right)}^{1/2}\right){\prec }^w\hat{I}»\left({\left(G^{1/2}E{G}^{1/2}\right)}^{1/2}\right)$. Here $\hat{I}»\left({\left(G^{1/2}E{G}^{1/2}\right)}^{1/2}\right)$ denotes the $n$-vector with entries given by the eigenvalues of ${\left(G^{1/2}E{G}^{1/2}\right)}^{1/2}$ in the non-decreasing order.