5 research outputs found
On the commutator modulus of continuity for operator monotone functions
Let be operator monotone on . In this paper we prove
that for any unitarily-invariant norm on and
matrices with and ,
for . 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 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 .
We further prove that for , This is a great improvement toward the conjecture of G.
Pedersen that this inequality for being the operator norm holds with
.
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 .Comment: 41 pages, 2 figures, email author for supplemental file