Kreǐn's trace formula and the spectral shift function (Q5931286)

M. G. Krejn proved, for two selfadjoint operators \(A\) and \(B\) whose difference \(B-A\) is of trace class, the trace formula NEWLINE\[NEWLINE\text{Tr}[f(B)- f(A)]= \int_{\mathbb{R}} f'(x) \xi(x) dxNEWLINE\]NEWLINE for a certain class of functions \(f\). Here \(\xi(x)\) is a function in \(L^1(\mathbb{R})\), which is called the spectral shift function. In the paper under review, the author provides another proof based on the integral representation of harmonic functions on the upper half plane, emphasizing to use the Baker-Campbell-Hausdorff formula, and gives some examples of admissible functions \(f\).