Functions of triples of noncommuting self-adjoint operators under perturbations of class 𝑆_{𝑝}

Peller, V.

doi:10.1090/proc/13854

Functions of triples of noncommuting self-adjoint operators under perturbations of class $\boldsymbol {S}_p$
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Proc. Amer. Math. Soc. 146 (2018), 1699-1711 Request permission

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 $B_{\infty ,1}^1(\mathbb {R}^3)$. 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 $C_2$.

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