Functions of perturbed normal operators

TL;DR

This paper extends sharp estimates for functions of self-adjoint operators to normal operators, establishing operator Lipschitz continuity for functions in certain classes and exploring perturbation properties in Schatten classes.

Contribution

It generalizes existing results to normal operators and introduces new bounds for functions in H"older, modulus of continuity, and Besov classes, including Schatten class perturbations.

Findings

Established Lipschitz-type bounds for functions of normal operators.

Proved operator Lipschitz property for Besov class functions.

Analyzed perturbation effects in Schatten-von Neumann classes.

Abstract

In \cite{Pe1}, \cite{Pe2}, \cite{AP1}, \cite{AP2}, and \cite{AP3} sharp estimates for $f(A)-f(B)$ were obtained for self-adjoint operators $A$ and $B$ and for various classes of functions $f$ on the real line $\R$. In this note we extend those results to the case of functions of normal operators. We show that if $f$ belongs to the H\"older class $\L_\a(\R^2)$, $0<\a<1$, of functions of two variables, and $N_1$ and $N_2$ are normal operators, then $\|f(N_1)-f(N_2)\|\le\const\|f\|_{\L_\a}\|N_1-N_2\|^\a$. We obtain a more general result for functions in the space $\L_\o(\R^2)=\big\{f: |f(\z_1)-f(\z_2)|\le\const\o(|\z_1-\z_2|)\big\}$ for an arbitrary modulus of continuity $\o$. We prove that if $f$ belongs to the Besov class $B_{\be1}^1(\R^2)$, then it is operator Lipschitz, i.e., $\|f(N_1)-f(N_2)\|\le\const\|f\|_{B_{\be1}^1}\|N_1-N_2\|$. We also study properties of $f(N_1)-f(N_2)$ in the case when $f\in\L_\a(\R^2)$ and $N_1-N_2$ belongs to the Schatten-von Neuman class $\bS_p$.