Inequalities for commutators of positive operators

Abstract

It is shown that if A , B , and X are operators on a complex separable Hilbert space such that A and B are compact and positive, then the singular values of the generalized commutator A X − X B are dominated by those of ‖ X ‖ ( A ⊕ B ) , where ‖ . ‖ is the usual operator norm. Consequently, for every unitarily invariant norm ⦀ . ⦀ , we have ⦀ A X − X B ⦀ ⩽ ‖ X ‖ ⦀ A ⊕ B ⦀ . It is also shown that if A and B are positive and X is compact, then ⦀ A X − X B ⦀ ⩽ max ( ‖ A ‖ , ‖ B ‖ ) ⦀ X ⦀ for every unitarily invariant norm. Moreover, if X is positive, then the singular values of the commutator A X − X A are dominated by those of 1 2 ‖ A ‖ ( X ⊕ X ) . Consequently, ⦀ A X − X A ⦀ ⩽ 1 2 ‖ A ‖ ⦀ X ⊕ X ⦀ for every unitarily invariant norm. For the usual operator norm, these norm inequalities hold without the compactness conditions, and in this case the first two norm inequalities are the same. Our inequalities include and improve upon earlier inequalities proved in this context, and they seem natural enough and applicable to be widely useful.