Operator integrals, spectral shift and spectral flow

TL;DR

This paper introduces a simplified approach to multiple operator integrals for unbounded operators in von Neumann algebras, with applications to spectral shift functions and spectral flow, advancing the mathematical understanding of operator theory.

Contribution

It develops a new method for operator integrals applicable to unbounded operators and links spectral shift functions with spectral flow in von Neumann algebra contexts.

Findings

New approach to multiple operator integrals for unbounded operators.

Representation of spectral shift function via spectral measures in type II algebras.

Connection established between spectral shift function and spectral flow.

Abstract

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 Fr\'echet 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.