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

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 $R$.