Singular values, norms, and commutators

Abstract

Let A i , B i , and X i , i = 1 , … , n , be bounded linear operators on a separable Hilbert space such that X i is compact for i = 1 , … , n . It is shown that the singular values of ∑ i = 1 n A i X i B i are dominated by those of ∑ i = 1 n ‖ A i ‖ ‖ B i ‖ ( ⊕ i = 1 n X i ) , where ‖ · ‖ is the usual operator norm. Among other applications of this inequality, we prove that if A and B are self-adjoint operators such that a 1 ⩽ A ⩽ a 2 and b 1 ⩽ B ⩽ b 2 for some real numbers a 1 , a 2 , b 1 , and b 2 , and if X is compact, then the singular values of the generalized commutator AX - XB are dominated by those of max ( b 2 - a 1 , a 2 - b 1 ) ( X ⊕ X ) . This inequality proves a recent conjecture concerning the singular values of commutators. Several inequalities for norms of commutators are also given.