Functions of perturbed operators

Abstract

We prove that if $0<\a<1$ and $f$ is in the H\"older class $\L_\a(\R)$, then for arbitrary self-adjoint operators $A$ and $B$ with bounded $A-B$, the operator $f(A)-f(B)$ is bounded and $\|f(A)-f(B)\|\le\const\|A-B\|^\a$. We prove a similar result for functions $f$ of the Zygmund class $\L_1(\R)$: $\|f(A+K)-2f(A)+f(A-K)\|\le\const\|K\|$, where $A$ and $K$ are self-adjoint operators. Similar results also hold for all H\"older-Zygmund classes $\L_\a(\R)$, $\a>0$. We also study properties of the operators $f(A)-f(B)$ for $f\in\L_\a(\R)$ and self-adjoint operators $A$ and $B$ such that $A-B$ belongs to the Schatten--von Neumann class $\bS_p$. We consider the same problem for higher order differences. Similar results also hold for unitary operators and for contractions.