Some generalizations for singular value inequalities of compact operators

Abstract

Audeh and Kittaneh have proved the following. Let X, Y and Z be compact operators on a complex separable Hilbert space such that $$\left[ \begin{array}{cc} X &{} Z \\ Z^{*} &{} Y \end{array} \right] \ge 0$$ . Then $$\begin{aligned} s_{j}(Z)\le s_{j}(X\oplus Y) \end{aligned}$$ for $$j=1,2,\ldots $$ 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 $$\left[ \begin{array}{cc} X &{} Z \\ Z^{*} &{} Y \end{array} \right] \ge 0$$ and let A, B be bounded linear operators on a complex separable Hilbert space. Then $$\begin{aligned} s_{j}(AZB^{*})\le \max \left\{ \left\| A\right\| ^{2},\left\| B\right\| ^{2}\right\} s_{j}(X\oplus Y) \end{aligned}$$ for $$j=1,2,\ldots $$ Several generalizations for singular value inequalities of compact operators are also given.