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

Abstract

Let $A = \begin{bmatrix} E & F \\ F^T & G \end{bmatrix}$ 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: $ \lambda \left( \left(\mathscr{C}(G)^{1/2} \mathscr{C}(E) \mathscr{C}(G)^{1/2}\right)^{1/2} \right) \prec^w \lambda \left( \left(G^{1/2} E G^{1/2} \right)^{1/2} \right)$. Here $\lambda \left( \left( G^{1/2}E G^{1/2} \right)^{1/2} \right)$ 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.