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Operator integrals, spectral shift, and spectral flow. (English) Zbl 1163.47008

This paper presents a new approach to the theory of multiple operator integrals, which provides a coherent path to the theory of differentiation of operator functions, the spectral shift function, and the theory of the spectral flow, in the setting of type II von Neumann algebras. An extension of the Birman–Solomyak formula, concerning spectral averaging, is established. While the theory of the spectral shift function is a part of operator theory, the theory of the spectral flow finds its proper analytic setting in the framework of non-commutative geometry. In the paper under review, a connection between the spectral shift function and the spectral flow is established, showing that these two theories coincide in the case of trace class perturbation.

MSC:

47A56 Functions whose values are linear operators (operator- and matrix-valued functions, etc., including analytic and meromorphic ones)
47B49 Transformers, preservers (linear operators on spaces of linear operators)
47A55 Perturbation theory of linear operators
46L51 Noncommutative measure and integration
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