On the Koplienko Spectral Shift Function. I. Basics

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