Functions of perturbed operators

TL;DR

This paper establishes bounds on the differences of functions of perturbed self-adjoint operators within H"older and Zygmund classes, extending to Schatten classes and higher order differences, with applications to unitary operators and contractions.

Contribution

It provides new operator norm estimates for functions of perturbed operators in H"older and Zygmund classes, including Schatten class and higher order difference results.

Findings

Boundedness of $f(A)-f(B)$ with $ orm{f(A)-f(B)} o 0$ as $ orm{A-B} o 0$ for H"older functions.

Operator difference bounds for Zygmund class functions involving second differences.

Extension of results to Schatten classes, higher order differences, and classes of unitary operators and contractions.

Abstract

We prove that if $0<\a<1$ and $f$ is in the H\"older class $\L_\a(\R)$, then for arbitrary self-adjoint operators $A$ and $B$ with bounded $A-B$, the operator $f(A)-f(B)$ is bounded and $\|f(A)-f(B)\|\le\const\|A-B\|^\a$. We prove a similar result for functions $f$ of the Zygmund class $\L_1(\R)$: $\|f(A+K)-2f(A)+f(A-K)\|\le\const\|K\|$, where $A$ and $K$ are self-adjoint operators. Similar results also hold for all H\"older-Zygmund classes $\L_\a(\R)$, $\a>0$. We also study properties of the operators $f(A)-f(B)$ for $f\in\L_\a(\R)$ and self-adjoint operators $A$ and $B$ such that $A-B$ belongs to the Schatten--von Neumann class $\bS_p$. We consider the same problem for higher order differences. Similar results also hold for unitary operators and for contractions.