On the commutator modulus of continuity for operator monotone functions

Abstract

Let $f\ge 0$ be operator monotone on $[0,\infty )$. In this paper we prove that for any unitarily-invariant norm $|||-|||$ on $M_n(\mathbb{C})$ and matrices $A,B,X\in M_n(\mathbb{C})$ with $A,B\ge 0$ and $|||X|||\le 1$, \[|||f(A)X-Xf(B)||| \leq C f(|||AX-XB|||)\] for $C<1.01975$. We do this by reducing this inequality to a function approximation problem and we choose approximate minimizers. This is much progress toward the conjecturally optimal value of $C=1$ which is known only in the case of the Hilbert-Schmidt norm. When $|||-|||$ is the the operator norm $||-||$, we obtain a great reduction of the previously known estimate of $C=1.25$. We further prove that for $|||X|||\le 1$, \[|||A^{1/2}X-XB^{1/2}||| \leq 1.00891 |||AX-XB|||.\] This is a great improvement toward the conjecture of G. Pedersen that this inequality for $|||-|||$ being the operator norm holds with $C=1$. We discuss other related inequalities, including some sharp commutator inequalities. We also prove a sharp equivalence inequality between the operator modulus of continuity and the commutator modulus of continuity for continuous functions on $\mathbb{R}$.

arXiv comment41 pages, 2 figures, email author for supplemental files