Functions of perturbed $n$-tuples of commuting self-adjoint operators

TL;DR

This paper establishes sharp operator norm estimates for functions of perturbed commuting self-adjoint operators in multiple variables, introducing a new method applicable for any number of operators and extending previous results.

Contribution

It introduces a novel approach for estimating functions of perturbed commuting self-adjoint operators for any number of variables, overcoming limitations of earlier methods.

Findings

Sharp bounds for operator differences in terms of perturbations.

Extension of estimates to functions of multiple commuting operators.

New method applicable for arbitrary n, including n≥3.

Abstract

Let $(A_1,\cdots,A_n)$ and $(B_1,\cdots,B_n)$ be $n$-tuples of commuting self-adjoint operators on Hilbert space. For functions $f$ on $\R^n$ satisfying certain conditions, we obtain sharp estimates of the operator norms (or norms in operator ideals) of $f(A_1,\cdots,A_n)-f(B_1,\cdots,B_n)$ in terms of the corresponding norms of $A_j-B_j$, $1\le j\le n$. We obtain analogs of earlier results on estimates for functions of perturbed self-adjoint and normal operators. It turns out that for $n\ge3$, the methods that were used for self-adjoint and normal operators do not work. We propose a new method that works for arbitrary $n$. We also get sharp estimates for quasicommutators $f(A_1,\cdots,A_n)R-Rf(B_1,\cdots,B_n)$ in terms of norms of $A_jR-RB_j$, $1\le j\le n$, for a bounded linear operator $R$.