Functions of perturbed tuples of self-adjoint operators

TL;DR

This paper extends previous results to functions of n-tuples of commuting self-adjoint operators, establishing operator Lipschitz and Hölder continuity properties for functions in Besov spaces and related classes.

Contribution

It generalizes earlier operator function results to n-tuples, proving Lipschitz and Hölder estimates for functions in Besov spaces and Schatten classes.

Findings

Functions in Besov space B_{∞,1}^1 are operator Lipschitz.

Hölder continuity of functions implies norm estimates for operator differences.

Results apply to operators with differences in Schatten--von Neumann classes.

Abstract

We generalize earlier results of Peller, Aleksandrov - Peller, Aleksandrov - Peller - Potapov - Sukochev to the case of functions of $n$-tuples of commuting self-adjoint operators. In particular, we prove that if a function $f$ belongs to the Besov space $B_{\be,1}^1(\R^n)$, then $f$ is operator Lipschitz and we show that if $f$ satisfies a H\"older condition of order $\a$, then $\|f(A_1...,A_n)-f(B_1,...,B_n)\|\le\const\max_{1\le j\le n}\|A_j-B_j\|^\a$ for all $n$-tuples of commuting self-adjoint operators $(A_1,...,A_n)$ and $(B_1,...,B_n)$. We also consider the case of arbitrary moduli of continuity and the case when the operators $A_j-B_j$ belong to the Schatten--von Neumann class $\bS_p$.