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) ∈ I2, the Hilbert–Schmidt operators, while KrSSF is defined for pairs A, B with (A−B) ∈ I1, 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) ∈ I2 so det2((A − z)(B − z)) 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 I1-perturbations that uses the KrSSF.