Norm Inequalities for Commutators of Self-adjoint Operators

Abstract

Let A, B, and X be bounded linear operators on a complex separable Hilbert space. It is shown that if A and B are self-adjoint with \(a_{1} \leq A \leq a_{2}\) and \(b_{1} \leq B \leq b_{2}\) for some real numbers a1, a2, b1, and b2, then for every unitarily invariant norm|||·|||, $$|||AX - XB||| \leq {\rm max}(a_2 - b_1, b_2 - a_1) |||X||| $$ . If, in addition, X is positive, then $$|||AX - XA||| \leq \frac{1}{2} (a_2 - a_1) |||X \oplus X||| $$ . These norm inequalities generalize recent related inequalities due to Kittaneh, Bhatia-Kittaneh, and Wang-Du.