Singular value inequalities for compact operators (Q1758455)

In [SIAM J. Matrix Anal. Appl. 11, No. 2, 272--277 (1990; Zbl 0704.47014)], \textit{R. Bhatia} and \textit{F. Kittaneh} established the arithmetic-geometric mean inequality for compact operators \(A,B\) on a separable Hilbert space as \[ 2S_j(AB^*)\leq S_j(A^*A+B^*B). \] Some inequalities equivalent to this one can be found in the literature. In the paper under review, the authors prove another equivalent inequality, namely, \[ 2s_j(A)\leq S_j((B+A)\oplus(B-A)), \] where \(A\) is selfadjoint operator, \(B\geq0\) and \(\pm A\leq B\). As an application, they are able to prove a Cauchy-Schwarz type inequality and a triangle-type inequality.