On the Koplienko Spectral Shift Function. I. Basics - Gesztesy, F.; Pushnitski, A.; Simon, B. (Журнал математичної фізики, аналізу, геометрії, 2009)
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)I_2, the Hilbert-Schmidt operators, while KrSSF is defined for pairs A, B with (A-B)I_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)I_2, so _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 I_1-perturbations that uses the KrSSF. Mathematics Subject Classification: 47A10, 81Q10, 34B27, 47A40, 81Uxx.