On the Koplienko spectral shift function, I. Basics

TL;DR

This paper explores the Koplienko Spectral Shift Function (KoSSF), comparing it with Krein's KrSSF, and presents new results including its range, boundary behavior, and invariance properties related to spectral theory.

Contribution

It introduces new properties of KoSSF, such as its range, boundary value issues, an alternative unitary case definition, and a novel proof of spectral invariance under Hilbert-Schmidt perturbations.

Findings

Any positive Riemann integrable function of compact support can be a KoSSF.

Existence of pairs with Hilbert-Schmidt difference where the determinant lacks boundary limits.

A new proof of absolute continuous spectrum invariance under trace class perturbations.

Abstract

We study the Koplienko Spectral Shift Function (KoSSF), which is distinct from the one of Krein (KrSSF). KoSSF is defined for pairs $A,B$ with $(A-B)\in\calI_2$, the Hilbert-Schmidt operators, while KrSSF is defined for pairs $A,B$ with $(A-B)\in\calI_1$, the trace class operators. We review various aspects of the construction of both KoSSF and KrSSF. Among our new results are: (i) that any positive Riemann integrable function of compact support occurs as a KoSSF; (ii) that there exist $A,B$ with $(A-B)\in\calI_2$ so $\det_2((A-z)(B-z)^{-1})$ does not have nontangential boundary values; (iii) an alternative definition of KoSSF in the unitary case; and (iv) a new proof of the invariance of the a.c. spectrum under $\calI_1$-perturbations that uses the KrSSF.