Operator and commutator moduli of continuity for normal operators

TL;DR

This paper investigates the properties of functions of perturbed normal operators, introducing operator moduli of continuity and establishing estimates for quasicommutators, especially for H"older functions, advancing the understanding of operator perturbations.

Contribution

It develops new estimates for operator and commutator moduli of continuity for normal operators, extending previous results and providing bounds for quasicommutator norms involving H"older functions.

Findings

Established bounds for quasicommutators involving H"older functions

Introduced operator and commutator moduli of continuity concepts

Provided lower estimates for constants in operator H"older estimates

Abstract

We study in this paper properties of functions of perturbed normal operators and develop earlier results obtained in \cite{APPS2}. We study operator Lipschitz and commutator Lipschitz functions on closed subsets of the plane. For such functions we introduce the notions of the operator modulus of continuity and of various commutator moduli of continuity. Our estimates lead to estimates of the norms of quasicommutators $f(N_1)R-Rf(N_2)$ in terms of $\|N_1R- RN_2\|$, where $N_1$ and $N_2$ are normal operator and $R$ is a bounded linear operator. In particular, we show that if $0<\a<1$ and $f$ is a H\"older function of order $\a$, then for normal operators $N_1$ and $N_2$, $$ \|f(N_1)R-Rf(N_2)\|\le\const(1-\a)^{-2}\|f\|_{\L_\a}\|N_1R-RN_2\|^\a\|R\|^{1-\a}. $$ In the last section we obtain lower estimates for constants in operator H\"older estimates.