Functions of operators under perturbations of class ${\text{bS}}_p$

This is a continuation of our paper \cite{AP2}. We prove that for functions $f$ in the H"older class ${\text{L}}_{\text{a}}(\mathbb{R})$ and $1<p<\text{be}$, the operator $f(A)-f(B)$ belongs to ${\text{bS}}_{p\mathrm{/}\text{a}}$, whenever $A$ and $B$ are self-adjoint operators with $A-B\in {\text{bS}}_p$. We also obtain sharp estimates for the Schatten–von Neumann norms $\parallel f(A)-f(B){\parallel }_{{\text{bS}}_{p\mathrm{/}\text{a}}}$ in terms of $\mathrm{\parallel }A-B{\mathrm{\parallel }}_{{\text{bS}}_p}$ and establish similar results for other operator ideals. We also estimate Schatten–von Neumann norms of higher order differences ${\sum }_{j=0}^m(-1{)}^{m-j}(m\text{j})f(A+jK)$. We prove that analogous results hold for functions on the unit circle and unitary operators and for analytic functions in the unit disk and contractions. Then we find necessary conditions on $f$ for $f(A)-f(B)$ to belong to ${\text{bS}}_q$ under the assumption that $A-B\in {\text{bS}}_p$. We also obtain Schatten–von Neumann estimates for quasicommutators $f(A)Q-Qf(B)$, and introduce a spectral shift function and find a trace formula for operators of the form $f(A-K)-2f(A)+f(A+K)$.