Functions of triples of noncommuting self-adjoint operators under perturbations of class ${\mathbf{S}}_p$

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 ${\mathbf{S}}_p$, $1\le p\le \infty$, for arbitrary functions in the Besov class $B_{\infty ,1}^1({\mathbb{R}}^3)$. 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 $B_{\infty ,1}^1({\mathbb{R}}^3)$ and for arbitrary finite rank self-adjoint operators $A_1,B_1,C_1,A_2,B_2$ and $C_2$.

14 pages. arXiv admin note: substantial text overlap with arXiv:1606.08961

References in corpus (1)

• Functions of triples of noncommuting self-adjoint operators and their perturbations