Spectral shift function of higher order

TL;DR

This paper proves the existence and absolute continuity of higher order spectral shift functions, confirming Koplienko's conjecture and providing new estimates for their $L^1$-norm, extending previous results for lower orders.

Contribution

It establishes the existence, absolute continuity, and $L^1$-integrability of higher order spectral shift functions, solving a long-standing conjecture for all orders.

Findings

Proves the existence of higher order spectral shift functions.

Establishes absolute continuity of these measures.

Provides an $L^1$-norm estimate for the spectral shift functions.

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 $\eta_n$. 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 $\eta_1$ and $\eta_2$ 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.