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⩾0B⩾0, and ±A⩽B±A⩽B, thensj(A)⩽sj(B⊕B)sj(A)⩽sj(B⊕B)for j=1,2,…j=1,2,…. We give an equivalent inequality, which states that if A,BA,B, and C   are compact operators such that ABB∗C⩾0, thensj(B)⩽sj(A⊕C)sj(B)⩽sj(A⊕C)for j=1,2,…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⩾0B⩾0, and ±A⩽B±A⩽B, then2sj(A)⩽sj((B+A)⊕(B-A))2sj(A)⩽sj((B+A)⊕(B-A))for j=1,2,…j=1,2,…. Some applications of these results will be given.