ON THE NORM INEQUALITIES FOR COMMUTATOR OF OPERATORS

Let A, B, and X be bounded linear operators on a complex separable Hilbert space. We show that if a · A + A¤ · b for some real numbers a and b, then for every unitarily invariant norm jjj:jjj, jjjAX ¡ XA + A¤X ¡ XA¤jjj · (b ¡ a)jjjXjjj: