Some generalizations for singular value inequalities of compact operators

Audeh and Kittaneh have proved the following. Let X, Y and Z be compact operators on a complex separable Hilbert space such that XZZ∗Y≥0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left[ \begin{array}{cc} X &{} Z \\ Z^{*} &{} Y \end{array} \right] \ge 0$$\end{document}. Then sj(Z)≤sj(X⊕Y)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} s_{j}(Z)\le s_{j}(X\oplus Y) \end{aligned}$$\end{document}for j=1,2,…\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$j=1,2,\ldots $$\end{document} In this paper, we provide a considerable generalization of this singular value inequality, which states that: Let X, Y and Z be compact operators on a complex separable Hilbert space such that XZZ∗Y≥0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left[ \begin{array}{cc} X &{} Z \\ Z^{*} &{} Y \end{array} \right] \ge 0$$\end{document} and let A, B be bounded linear operators on a complex separable Hilbert space. Then sj(AZB∗)≤maxA2,B2sj(X⊕Y)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} s_{j}(AZB^{*})\le \max \left\{ \left\| A\right\| ^{2},\left\| B\right\| ^{2}\right\} s_{j}(X\oplus Y) \end{aligned}$$\end{document}for j=1,2,…\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$j=1,2,\ldots $$\end{document} Several generalizations for singular value inequalities of compact operators are also given.