Functions of operators under perturbations of class $\bS_p$
arXiv:0908.3623
Abstract
This is a continuation of our paper \cite{AP2}. We prove that for functions in the Hölder class $Ł_\a(\R)$ and $1<p<\be$, the operator belongs to $\bS_{p/\a}$, whenever and are self-adjoint operators with $A-B\in\bS_p$. We also obtain sharp estimates for the Schatten--von Neumann norms $\big\|f(A)-f(B)\big\|_{\bS_{p/\a}}$ in terms of $\|A-B\|_{\bS_p}$ and establish similar results for other operator ideals. We also estimate Schatten--von Neumann norms of higher order differences $\sum\limits_{j=0}^m(-1)^{m-j}(m\j)f\big(A+jK\big)$. 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 for to belong to $\bS_q$ under the assumption that $A-B\in\bS_p$. We also obtain Schatten--von Neumann estimates for quasicommutators , and introduce a spectral shift function and find a trace formula for operators of the form .
49 pages