International Journal of Mathematics and Mathematical Sciences
Volume 25 (2001), Issue 4, Pages 239-252
doi:10.1155/S0161171201004318

Kreǐn's trace formula and the spectral shift function

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.