On the Koplienko Spectral Shift Function. I. Basics


  • F. Gesztesy Department of Mathematics, University of Missouri Columbia, MO 65211, USA
  • A. Pushnitski Department of Mathematics, King's College London Strand, London WC2R 2LS, England, UK
  • B. Simon Mathematics 253-37, California Institute of Technology Pasadena, CA 91125, USA

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.


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) Î I 2, the Hilbert-Schmidt operators, while KrSSF is defined for pairs A, B with (A - B) Î 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) Î 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.

Mathematics Subject Classification: 47A10, 81Q10 (primary); 34B27, 47A40, 81Uxx (secondary).


Як цитувати

Gesztesy, F.; Pushnitski, A.; Simon, B. On the Koplienko Spectral Shift Function. I. Basics. Журн. мат. фіз. анал. геом. 2009, 4, 63-107.





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