Functions of perturbed operators

We prove that if $0<\text{a}<1$ and $f$ is in the H"older class ${\text{L}}_{\text{a}}(\mathbb{R})$, then for arbitrary self-adjoint operators $A$ and $B$ with bounded $A-B$, the operator $f(A)-f(B)$ is bounded and $\mathrm{\parallel }f(A)-f(B)\mathrm{\parallel }\le \text{const}\mathrm{\parallel }A-B{\mathrm{\parallel }}^{\text{a}}$. We prove a similar result for functions $f$ of the Zygmund class ${\text{L}}_1(\mathbb{R})$: $\mathrm{\parallel }f(A+K)-2f(A)+f(A-K)\mathrm{\parallel }\le \text{const}\mathrm{\parallel }K\mathrm{\parallel }$, where $A$ and $K$ are self-adjoint operators. Similar results also hold for all H"older-Zygmund classes ${\text{L}}_{\text{a}}(\mathbb{R})$, $\text{a}>0$. We also study properties of the operators $f(A)-f(B)$ for $f\in {\text{L}}_{\text{a}}(\mathbb{R})$ and self-adjoint operators $A$ and $B$ such that $A-B$ belongs to the Schatten–von Neumann class ${\text{bS}}_p$. We consider the same problem for higher order differences. Similar results also hold for unitary operators and for contractions.