Journal of Operator Theory

Volume 93, Issue 1, Winter 2025 pp. 3-36.

Higher-order spectral shift function for resolvent comparable perturbations

Authors: T.D.H. van Nuland (1), A. Skripka (2)
Author institution:(1) School of Mathematics and Statistics, University of New South Wales, Kensington, NSW, 2052, Australia
(2) Department of Mathematics and Statistics, University of New Mexico, Albuquerque, NM, 87106, U.S.A

Summary: Given a pair of self-adjoint operators $H$ and $V$ such that $H$ is possibly unbounded, $V$ is bounded, and $(H+V-\mathrm i)^{-1}-(H-\mathrm i)^{-1}$ belongs to the Schatten ideal $\mathcal{S}^n$ , $n\geqslant 2$ , of operators on a separable Hilbert space, we establish the existence, regularity, and uniqueness --- up to polynomial summands --- of the respective locally integrable real-valued spectral shift functions, for a broad set of functions. We develop a change of variables method for multiple operator integrals, unifying and extending earlier results. The proof is moreover a stepping stone towards unbounded perturbations.

DOI: http://dx.doi.org/10.7900/jot.2022nov15.2413
Keywords: resolvent comparable, perturbation theory, trace formula, multiple operator integral, spectral shift

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