On the Koplienko spectral shift function, I. Basics

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 $(A-B)\in {\text{calI}}_2$, the Hilbert-Schmidt operators, while KrSSF is defined for pairs $A,B$ with $(A-B)\in {\text{calI}}_1$, 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 $(A-B)\in {\text{calI}}_2$ 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 ${\text{calI}}_1$-perturbations that uses the KrSSF.