Operator integrals, spectral shift, and spectral flow

We present a new and simple approach to the theory of multiple operator integrals that applies to unbounded operators affiliated with general von Neumann algebras. For semifinite von Neumann algebras we give applications to the Fréchet differentiation of operator functions that sharpen existing results, and establish the Birman-Solomyak representation of the spectral shift function of M. G. Krein in terms of an average of spectral measures in the type II setting. We also exhibit a surprising connection between the spectral shift function and spectral flow.