On uniqueness of higher order spectral shift functions
Volume 251 / 2020
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.
