Spectral shift for relative Schatten class perturbations

TL;DR

This paper proves the existence and uniqueness of a real-valued higher order spectral shift function for certain bounded perturbations of self-adjoint operators, extending previous results to broader classes of perturbations.

Contribution

It establishes the existence, trace formula, and uniqueness of the spectral shift function for bounded perturbations in Schatten classes, advancing previous partial results.

Findings

Spectral shift function exists for bounded Schatten class perturbations.

The function satisfies the same trace formula as in classical cases.

The spectral shift function is unique up to a polynomial of order n-1.

Abstract

We affirmatively settle the question on existence of a real-valued higher order spectral shift function for a pair of self-adjoint operators $H$ and $V$ such that $V$ is bounded and $V(H-iI)^{-1}$ belongs to a Schatten-von Neumann ideal $\mathcal{S}^n$ of compact operators in a separable Hilbert space. We also show that the function satisfies the same trace formula as in the known case of $V\in\mathcal{S}^n$ and that it is unique up to a polynomial summand of order $n-1$. Our result significantly advances earlier partial results where counterparts of the spectral shift function for noncompact perturbations lacked real-valuedness and aforementioned uniqueness as well as appeared in more complicated trace formulas for much more restrictive sets of functions. Our result applies to models arising in noncommutative geometry and mathematical physics.