Krein's trace theorem revisited

M. G. Krein’s celebrated Trace Theorem” states that if A ,B are self-adjoint operators in a separable Hilbert space such that A − B is a trace class operator, then, for any function f of a real variable, whose derivative f ′ in distributional sense has Fourier transform belonging to L1(R) , the difference f(A)− f(B) is again a trace class operator and the formula Trace ( f(A)− f(B) ) = ∫ R f ′(s)ξ(s)ds holds where the function ξ ∈ L1(R) depends only on A and B and is uniquely determined by the above formula. The function ξ is called the spectral shift function of the pair A ,B and is an important ingredient in the perturbation theory of self-adjoint operators. The original proof of Krein uses complex analysis and is quite involved. We supply a new proof which does not use complex analysis. Our proof works also for σ-finite von Neumann algebrasM of type II and unbounded perturbations from the predual ofM. The exposition is based on joint work with D. Potapov and D. Zanin. E-mail address: f.sukochev@unsw.edu.au School of Mathematics & Statistics, University of NSW, Kensington NSW 2052 AUSTRALIA