Some singular value inequalities for matrices (Q6618757)

Let \(s_j(A)\) be the \(j\)-th largest singular value of \(A\). A featured result in the abstract states that if \(A,B,C,D,X,Y\) are \(n\times n\) complex matrices such that such that \(X\) and \(Y\) are positive semidefinite, then \N\[ \Ns_j(AXB^*+CYD^*)\le \sqrt{\left\||A^*|^2+|C^*|^2\right\|\left\||B^*|^2+|D^*|^2\right\|}s_j(X\oplus Y) ,\N\] \Nfor \(j=1,2,\ldots,n\).\N\NThe motivation for this result is inequality (3) from [\textit{O. Hirzallah} and \textit{F. Kittaneh}, Linear Algebra Appl. 424, No. 1, 71--82 (2007; Zbl 1116.47012)].\NIndeed, the condition ``\(X\) and \(Y\) are positive semidefinite'' is superfluous. Therefore the main result of the present paper turns out to be not new, as it is a special case of Theorem 2.1 in [loc. cit.].