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


Author: V. V. Peller
Journal: Proc. Amer. Math. Soc. 146 (2018), 1699-1711
MSC (2010): Primary 47A55; Secondary 47B10, 46E35
DOI: https://doi.org/10.1090/proc/13854
Published electronically: January 12, 2018
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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

$\displaystyle \Vert f(A_1,B_1,C_1)$ $\displaystyle -f(A_2,B_2,C_2)\Vert _{\boldsymbol S_p}$    
  $\displaystyle \le K\Vert f\Vert _{B_{\infty ,1}^1} \max \big \{\Vert A_1\!-\!A_... ...-\!B_2\Vert _{\boldsymbol {S}_p},\Vert C_1-C_2\Vert _{\boldsymbol {S}_p}\big \}$    

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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Additional Information

V. V. Peller
Affiliation: Department of Mathematics, Michigan State University, East Lansing, Michigan 48824 – and – Peoples’ Friendship University of Russia (RUDN University), 6 Miklukho-Maklay Street, Moscow, 117198, Russia
Email: peller@math.msu.edu

DOI: https://doi.org/10.1090/proc/13854
Received by editor(s): May 9, 2017
Received by editor(s) in revised form: June 18, 2017
Published electronically: January 12, 2018
Additional Notes: The author was partially supported by NSF grant DMS 1300924; the publication was financially supported by the Ministry of Education and Science of the Russian Federation (the Agreement number 02.A03.21.0008).
Communicated by: Stephan Ramon Garcia
Article copyright: © Copyright 2018 American Mathematical Society

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