On some matrix inequalities

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Inequalities for commutators of positive operators (2007, background): “33] and [24]) that if A,B ∈ B(H) are compact and positive, then sj (A − B) sj (A ⊕ B) (6) for j = 1,2, .”
The matrix arithmetic-geometric mean inequality revisited (2008, background): “Zhan [38] showed that the Level 3 inequality (3.5) can be enhanced to a Level 2 version, sj (A − B) sj (A ⊕ B), 1 j n (3.6) and further, this statement is equivalent to the AGMI (3.1).”
More results on singular value inequalities of matrices (2006, methodology): “We employ the arguments similar to the ones in [2] as follows: 0 .”
Matrix Mathematics: Theory, Facts, and Formulas with Application to Linear Systems Theory (2005, unknown): “(Proof: See [623].”