Operator Integrals, Spectral Shift, and Spectral Flow
Printed: Apr 2009
N. A. Azamov
A. L. Carey
P. G. Dodds
F. A. Sukochev
We present a new and simple approach to the theory of multiple
operator integrals that applies to unbounded operators affiliated with general \vNa s.
For semifinite \vNa s 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.
47A56 - Functions whose values are linear operators (operator and matrix valued functions, etc., including analytic and meromorphic ones)
47B49 - Transformers, preservers (operators on spaces of operators)
47A55 - Perturbation theory [See also 47H14, 58J37, 70H09, 81Q15]
46L51 - Noncommutative measure and integration