Functions of operators under perturbations of class $\bS_p$

TL;DR

This paper investigates how functions of operators behave under perturbations within Schatten classes, establishing sharp norm estimates, extending results to unitary and analytic functions, and exploring spectral shift functions.

Contribution

It extends the theory of operator functions under perturbations by providing sharp Schatten norm estimates, including higher order differences and quasicommutators, for a broad class of functions and operators.

Findings

Proved that $f(A)-f(B)$ belongs to $S_{p/eta}$ under certain conditions.

Established sharp Schatten norm estimates for operator differences and higher order differences.

Extended results to unitary operators, functions on the unit circle, and analytic functions in the disk.

Abstract

This is a continuation of our paper \cite{AP2}. We prove that for functions $f$ in the H\"older class $\L_\a(\R)$ and $1<p<\be$, the operator $f(A)-f(B)$ belongs to $\bS_{p/\a}$, whenever $A$ and $B$ 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 $f$ for $f(A)-f(B)$ to belong to $\bS_q$ under the assumption that $A-B\in\bS_p$. 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 $f(A-K)-2f(A)+f(A+K)$.