Singular value and norm inequalities involving the numerical radii of matrices

Abstract

It is shown that if A, B, X,  and Y are \(n\times n\) complex matrices, such that X and Y are positive semidefinite, then

$$\begin{aligned} s_{j}\left( AXB^{*}+BYA^{*}\right) \le \left( \left\| A\right\| \left\| B\right\| +\omega \left( A^{*}B\right) \right) s_{j}\left( X\oplus Y\right) \end{aligned}$$

for \(j=1,2,\ldots ,n\), and if A is accretive–dissipative, then

$$\begin{aligned} \left| \left| \left| A^{*}XA-AXA^{*}\right| \right| \right| \le 3\omega ^{2}\left( A\right) \ \left| \left| \left| X\right| \right| \right| \end{aligned}$$

for every unitarily invariant norm, where \(s_{j}\left( T\right) ,\left\| T\right\| \), and \(\omega \left( T\right) \) are the \(j^{th}\) largest singular value of T, the spectral norm of T, and the numerical radius of T, respectively.