Generalizations for singular value inequalities of operators

Abstract

This paper proves singular value inequality, from which well-known singular value and norm inequalities are special cases: Let A, B, and X are positive operators on a complex separable Hilbert space. Then $$\begin{aligned}&s_{j} \left( A^{1/2}XA^{1/2}+B^{1/2}XB^{1/2}\right) \\&\quad \le s_{j}\left( \left( A^{1/2}XA^{1/2}+\left| B^{1/2}XA^{1/2}\right| \right) \oplus \left( B^{1/2}XB^{1/2}+\left| A^{1/2}XB^{1/2}\right| \right) \right) \end{aligned}$$for $$j=1,2,...$$. In particular, $$\begin{aligned} s_{j}\left( A+B\right) \le s_{j}\left( \left( A+\left| B^{1/2}A^{1/2}\right| \right) \oplus \left( B+\left| A^{1/2}B^{1/2}\right| \right) \right) \end{aligned}$$for $$j=1,2,...$$. Moreover, we give singular value inequalities for sums and products of Hilbert space operators which are sharper than several singular value inequalities.