Singular value inequalities for commutators of Hilbert space operators

We prove several singular value inequalities for commutators of Hilbert space operators. It is shown, among other inequalities, that if A, B, and X are operators on a complex separable Hilbert space such that A and B are positive, and X is compact, then the singular values of AX-XB are dominated by those of max(∥A∥,∥B∥)(X⊕X), where ∥·∥ is the usual operator norm.