Majorization in some symplectic weak supermajorizations

Symplectic eigenvalues are known to satisfy analogs of several classic eigenvalue inequalities. Of these is a set of weak supermajorization relations concerning symplectic eigenvalues that are weaker analogs of some majorization relations corresponding to eigenvalues. The aim of this letter is to establish necessary and sufficient conditions for the saturation of the symplectic weak supermajorization relations by majorization.
Comment: 8 pages, The old version had some typos that are corrected in the new version. The typos were in Eq.(1.14) and Eq.(1.23). Also, proof of Corollary 1.5 is added in the new version. Accepted in Linear Algebra and its Applications. June 2024