07-127 Fritz Gesztesy, Alexander Pushnitski, and Barry Simon

On the Koplienko Spectral Shift Function, I. Basics (432K, pdf) May 24, 07

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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 $(A-B)\in\calI_2$, the Hilbert-Schmidt operators, while KrSSF is defined for pairs $A,B$ with $(A-B)\in\calI_1$, 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 $(A-B)\in\calI_2$ 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 $\calI_1$-perturbations that uses the KrSSF.

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