On the Koplienko Spectral Shift Function. I. Basics
Анотація
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 $(A-B)\in \mathcal{I}_2$, the Hilbert-Schmidt operators, while KrSSF is defined for pairs $A$, $B$ with $(A-B)\in \mathcal{I}_1$, 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 $(A-B)\in \mathcal{I}_2$, 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 $\mathcal{I}_1$-perturbations that uses the KrSSF.
Mathematics Subject Classification: 47A10, 81Q10, 34B27, 47A40, 81Uxx.
Ключові слова:
Krein's spectral shift function, Koplienko's spectral shift function, selfadjoint operators, trace class and Hilbert-Schmidt perturbations, convexity properties, boundary values of (modified) Fredholm determinants
