International Journal of Mathematics and Mathematical Sciences
Volume 25 (2001), Issue 4, Pages 239-252

Kreĭn's trace formula and the spectral shift function

Khristo N. Boyadzhiev

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.


Let A,B be two selfadjoint operators whose difference BA is trace class. Kreĭn proved the existence of a certain function ξL1() such that tr[f(B)f(A)]=f(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.