Nurulla Azamov

Operator integrals, spectral shift and spectral flow

We present a new and simple approach to the theory of multiple operator integrals that ap- plies to unbounded operators affiliated with general von Neumann algebras. For semifinite von Neu- mann algebras we give applications to the Frechet 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 measuresin the type II setting. We also exhibit a surpris- ing connection between the spectral shift function and spectral flow.

Spectral averaging for trace compatible operators

In this note the notions of trace compatible operators and infin- itesimal spectral flow are introduced. We define the spectral shift function as the integral of infinitesimal spectral flow. It is proved that the spectral shift function thus defined is absolutely continuous and Kreins formula is es- tablished. Some examples of trace compatible affine spaces ofoperators are given.