Estimates of operator moduli of continuity

Abstract

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 $$\Vert |S|-|T|\Vert \le C\Vert S-T\Vert \log (2+\log \frac{\Vert S\Vert +\Vert T\Vert }{\Vert S-T\Vert })$$ 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 $\mathbb{R}$ that vanishes on $(-\be,0]$ and is concave on $[0,\be)$, then its operator modulus of continuity ${Ø}_f$ admits the estimate $$ Ø_f(\d)\le\const\int_e^\be\frac{f(\d t)\,dt}{t^2\log t},\quad\d>0. $$ We also study the problem of sharpness of estimates obtained in \cite{AP2} and \cite{AP4}. We construct a $C^\be$ function $f$ on $\mathbb{R}$ such that $\|f\|_{L^\be}\le1$, $\|f\|_{\Li}\le1$, and $$ Ø_f(\d)\ge\const\,\d\sqrt{\log\frac2\d},\quad\d\in(0,1]. $$ In the last section of the paper we obtain sharp estimates of $\Vert f(A)-f(B)\Vert$ 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}.

50 pages