Year: 2009
Type: preprint
Abstract: This paper resolves affirmatively Koplienko's conjecture of 1984 on existence of higher order spectral shift measures. Moreover, the paper establishes absolute continuity of these measures and, thus, existence of the higher order spectral shift functions $η_n$. We show the higher order spectral shift function is a $L^1$-function and prove an estimate on its $L^1$-norm. Existence and summability of $η_1$ and $η_2$ were established by Krein in 1953 and Koplienko in 1984, respectively, whereas for $n > 2$ the problem was unresolved. Our method is derived from [arXiv:0904.4095]; it also applies to the general semi-finite von Neumann algebra setting of the perturbation theory.
Source: arXiv (Cornell University)
Authors: Potapov, Denis, Skripka, Anna, Sukochev, Fedor
Institutions: University of New Mexico, UNSW Sydney
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Funder: National Science Foundation
OpenAlex ID: W2951331971