Some Inequalities for Norms of Commutators

Abstract

Let $A, B$ be positive operators and let f be any operator monotone function. We obtain inequalities for $|||f(A)X - Xf(B)|||$ in terms of $|||f\left(|AX - XB| \right) |||$ for every unitarily invariant norm. The case $X = I$ was considered by T. Ando [Math. Z., 197 (1988), pp. 403--409], and some of our results reduce to his results in this special case. Some related inequalities are obtained.