Estimates of operator moduli of continuity

TL;DR

This paper improves estimates of operator moduli of continuity for specific classes of functions and operators, providing sharp bounds and exploring the relationship between function properties and operator differences.

Contribution

It refines previous bounds on operator moduli of continuity, introduces new sharp estimates for specific function classes, and analyzes the case of operators with finite spectra.

Findings

Improved estimate for |S|-|T| in terms of operator norm and logarithmic factors.

Established sharpness of certain inequalities related to operator moduli.

Derived bounds for functions with specific monotonicity and concavity properties.

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 $$ \big\|\,|S|-|T|\,\big\|\le C\|S-T\|\log(2+\log\frac{\|S\|+\|T\|}{\|S-T\|}) $$ 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}.