Operator H\"older--Zygmund functions

Abstract

It is well known that a Lipschitz function on the real line does not have to be operator Lipschitz. We show that the situation changes dramatically if we pass to H\"older classes. Namely, we prove that if $f$ belongs to the H\"older class $\L_\a(\R)$ with $0<\a<1$, then $\|f(A)-f(B)\|\le\const\|A-B\|^\a$ for arbitrary self-adjoint operators $A$ and $B$. We prove a similar result for functions $f$ in the Zygmund class $\L_1(\R)$: for arbitrary self-adjoint operators $A$ and $K$ we have $\|f(A-K)-2f(A)+f(A+K)\|\le\const\|K\|$. We also obtain analogs of this result for all H\"older--Zygmund classes $\L_\a(\R)$, $\a>0$. Then we find a sharp estimate for $\|f(A)-f(B)\|$ for functions $f$ of class $\L_\o\df\{f: \o_f(\d)\le\const\o(\d)\}$ for an arbitrary modulus of continuity $\o$. In particular, we study moduly of continuity, for which $\|f(A)-f(B)\|\le\const\o(\|A-B\|)$ for self-adjoint $A$ and $B$, and for an arbitrary function $f$ in $\L_\o$. We obtain similar estimates for commutators $f(A)Q-Qf(A)$ and quasicommutators $f(A)Q-Qf(B)$. Finally, we estimate the norms of finite differences $\sum\limits_{j=0}^m(-1)^{m-j}(m j)f\big(A+jK\big)$ for $f$ in the class $\L_{\o,m}$ that is defined in terms of finite differences and a modulus continuity $\o$ of order $m$. We also obtaine similar results for unitary operators and for contractions.