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Operator Integrals, Spectral Shift, and Spectral Flow

  Published:2009-04-01
 Printed: Apr 2009
  • N. A. Azamov
  • A. L. Carey
  • P. G. Dodds
  • F. A. Sukochev
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Abstract

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.
MSC Classifications: 47A56, 47B49, 47A55, 46L51 show english descriptions Functions whose values are linear operators (operator and matrix valued functions, etc., including analytic and meromorphic ones)
Transformers, preservers (operators on spaces of operators)
Perturbation theory [See also 47H14, 58J37, 70H09, 81Q15]
Noncommutative measure and integration
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
 

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