Singular Value Inequalities for Compact Operators

A singular value inequality due to Bhatia and Kittaneh says that if A and B are compact operators on a complex separable Hilbert space such that A is self-adjoint, B 0, and A B; then sj(A) sj(B B) for j = 1; 2; :::We give an equivalent inequality, which states that if A;B; and C are compact operators such that A B B C 0; then sj(B) sj(A C) for j = 1; 2; :::Moreover, we give a sharper inequality and we prove that this inequality is equivalent to three equivalent inequalities considered by Tao. In particular, we show that if A and B are compact operators such that A is self-adjoint, B 0, and A B; then 2sj(A) sj((B +A) (B A)) for j = 1; 2; ::: Some applications of these results will be given.