Some singular value inequalities on commutators (Q6662081)

Let \(\sigma_j(A)\) be the \(j\)-th largest singular value of a compact operator \(A\). A special case of Theorem 2.1 in [\textit{O. Hirzallah} and \textit{F. Kittaneh}, Linear Algebra Appl. 424, No. 1, 71--82 (2007; Zbl 1116.47012)] states that \N\[ \N\sigma_j(A_1X_1B_1+A_2X_2B_2)\le \sqrt{\left\||A_1^*|^2+|A_2^*|^2\right\|\left\||B_1|^2+|B_2|^2\right\|}\sigma_j(X_1\oplus X_2), \N\] \Nwhere \(X_1, X_2\) are compact and others are bounded.\N\NBy taking \(A_1=S, X_1=X, B_1=T, A_2=tI, X_2=I, B_2=\frac{1}{t}Y\), where \(t>0\), it follows from the previous inequality that \N\begin{align*} \N\sigma_j(SXT+Y) &\le \sqrt{(\|S\|^2+t^2)\left\||T|^2+\frac{1}{t^2}|Y|^2\right\|}\sigma_j(X\oplus I) \\\N&\le \sqrt{(\|S\|^2+t^2)(\|T\|^2+\frac{1}{t^2}\|Y\|^2)}\sigma_j(X\oplus I).\N\end{align*}\NThe featured result of the paper under review is obtained by taking \(t=\sqrt{\|S\|\|T\|/\|Y\|}\).