Singular value inequalities for positive semidefinite matrices (Q464553)

Using some block matrix techniques, the authors generalize the singular value inequality \[ 2s_j(A^{1/2}(A+B)^rB^{1/2})\leq s_j((A+B)^{r+1}) \] for all \(r\geq0\) and positive semidefinite matrices \(A\) and \(B\), obtained by \textit{R. Bhatia} and \textit{F. Kittaneh} [Linear Algebra Appl. 428, No. 8--9, 2177--2191 (2008; Zbl 1148.15014)]. They prove that if \(A,B\) are two positive semidefinite matrices and \(0\leq q\leq1\), we have \[ s_j(A^{q/2(q+1)}K(A+B)^rKB^{q/2(q+1)})\leq s_j((A+B)^{1+r}),\quad r\geq0, \] where \(K=(A^{1/(q+1)}+B^{1/(q+1)})^{1/2}\).