Functions of triples of noncommuting self-adjoint operators under perturbations of class $\boldsymbol S_p$

Abstract

In this paper we study properties of functions of triples of not necessarily commuting self-adjoint operators. The main result of the paper shows that unlike in the case of functions of pairs of self-adjoint operators there is no Lipschitz type estimates in any Schatten--von Neumann norm $\boldsymbol S_p$, $1\le p\le\infty$, for arbitrary functions in the Besov class $B_{\infty,1}^1({\Bbb R}^3)$. In other words, we prove that for $p\in[1,\infty]$, there is no constant $K>0$ such that the inequality \begin{align*} \|f(A_1,B_1,C_1)&-f(A_2,B_2,C_2)\|_{\boldsymbol S_p}\\[.1cm] &\le K\|f\|_{B_{\infty,1}^1} \max\big\{\|A_1-A_2\|_{\boldsymbol S_p},\|B_1-B_2\|_{\boldsymbol S_p},\|C_1-C_2\|_{\boldsymbol S_p}\big\} \end{align*} holds for an arbitrary function $f$ in $B_{\infty,1}^1({\Bbb R}^3)$ and for arbitrary finite rank self-adjoint operators $A_1,\,B_1,\,C_1,\,A_2,\,B_2$ and $C_2$.