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On uniqueness of higher order spectral shift functions

Volume 251 / 2020

Anna Skripka, Maxim Zinchenko Studia Mathematica 251 (2020), 207-218 MSC: Primary 47A55; Secondary 15A29, 41A15. DOI: 10.4064/sm181007-1-1 Published online: 19 September 2019

Abstract

We prove that the Taylor remainder of order $n\geq 2$ vanishes on all admissible functions if and only if the respective trace class self-adjoint perturbation of a self-adjoint operator with a countable spectrum is zero. Alternatively, the result can be stated as follows: the spectral shift function of order $n\geq 2$ is zero if and only if the perturbation is zero. This uniqueness is in contrast to nonuniqueness of the first order Taylor remainder and first order spectral shift function.

Authors

  • Anna SkripkaDepartment of Mathematics and Statistics
    University of New Mexico
    311 Terrace Street NE, MSC01 1115
    Albuquerque, NM 87131, U.S.A.
    e-mail
  • Maxim ZinchenkoDepartment of Mathematics and Statistics
    University of New Mexico
    311 Terrace Street NE, MSC01 1115
    Albuquerque, NM 87131, U.S.A.
    e-mail

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