International Journal of Mathematics and Mathematical Sciences
Volume 25 (2001), Issue 4, Pages 239-252
doi:10.1155/S0161171201004318
Kreĭn's trace formula and the spectral shift function
Department of Mathematics, Ohio Northern University, Ada 45810, Ohio, USA
Received 15 October 1999; Revised 3 January 2000
Copyright © 2001 Khristo N. Boyadzhiev. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
Let A,B be two selfadjoint operators whose difference B−A is
trace class. Kreĭn proved the existence of a certain function ξ∈L1(ℝ) such that tr[f(B)−f(A)]=∫ℝf′(x)ξ(x)dx for a large set of functions f. We give here a new proof of this
result and discuss the class of admissible functions. Our proof is
based on the integral representation of harmonic functions on the
upper half plane and also uses the Baker-Campbell-Hausdorff
formula.