Estimating Norms of Commutators
We find estimates on the norm of a commutator of the form [f(x), y] in terms of the norm of [x, y], assuming that x and y are bounded linear operators on Hilbert space, with x normal and with spectrum within the domain of f. In particular, we discuss ‖[x2, y]‖ and ‖[x1/2, y]‖ for 0 ≤ x ≤ 1. For larger values of δ = ‖[x, y]‖, we can rigorously calculate the best possible upper bound ‖[f(x), y]‖ ≤ ηf(δ) for many f. In other cases, we have conducted numerical experiments that strongly suggest that we have in many cases found the correct formula for the best upper bound.
Estimates of operator moduli of continuity
In \cite{AP2} we obtained general estimates of the operator moduli of continuity of functions on the real line. In this paper we improve the estimates obtained in \cite{AP2} for certain special classes of functions. 
In particular, we improve estimates of Kato \cite{Ka} and show that $$ \big\|\,|S|-|T|\,\big\|\le C\|S-T\|\log(2+\log\frac{\|S\|+\|T\|}{\|S-T\|}) $$ for every bounded operators $S$ and $T$ on Hilbert space. Here $|S|\df(S^*S)^{1/2}$. Moreover, we show that this inequality is sharp. 
We prove in this paper that if $f$ is a nondecreasing continuous function on $\R$ that vanishes on $(-\be,0]$ and is concave on $[0,\be)$, then its operator modulus of continuity $\O_f$ admits the estimate $$ \O_f(\d)\le\const\int_e^\be\frac{f(\d t)\,dt}{t^2\log t},\quad\d>0. $$ 
We also study the problem of sharpness of estimates obtained in \cite{AP2} and \cite{AP4}. We construct a $C^\be$ function $f$ on $\R$ such that $\|f\|_{L^\be}\le1$, $\|f\|_{\Li}\le1$, and $$ \O_f(\d)\ge\const\,\d\sqrt{\log\frac2\d},\quad\d\in(0,1]. $$ 
In the last section of the paper we obtain sharp estimates of $\|f(A)-f(B)\|$ in the case when the spectrum of $A$ has $n$ points. Moreover, we obtain a more general result in terms of the $\e$-entropy of the spectrum that also improves the estimate of the operator moduli of continuity of Lipschitz functions on finite intervals, which was obtained in \cite{AP2}.
On a problem of J. P. Williams
Let B(H) be the algebra of all bounded operators on a Hilbert space H. Let g be a continuous function on the closed disk D and let ∥g(A)X - Xg(A)∥ ≤ C∥AX - XA∥, for some C > 0, for all X ∈ B(H) and all A ∈ B(H) with ∥A∥ < 1. Then g is differentiable on D. The paper shows that the function g may have a discontinuous derivative.