Functions of noncommuting operators under perturbation of class ${\mathbf{S}}_p$

Abstract

In this article we prove that for $p>2$, there exist pairs of self-adjoint operators $(A_1,B_1)$ and $(A_2,B_2)$ and a function $f$ on the real line in the homogeneous Besov class $B_{\infty ,1}^1({\mathbb{R}}^2)$ such that the differences $A_2-A_1$ and $B_2-B_1$ belong to the Schatten--von Neumann class ${\mathbf{S}}_p$ but $f(A_2,B_2)-f(A_1,B_1)\notin {\mathbf{S}}_p$. A similar result holds for functions of contractions. We also obtain an analog of this result in the case of triples of self-adjoint operators for any $p\ge 1$

17 pages