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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

   
 

 

Spectral averaging and the Krein spectral shift


Author: Barry Simon
Journal: Proc. Amer. Math. Soc. 126 (1998), 1409-1413
MSC (1991): Primary 47B10, 47A60
DOI: https://doi.org/10.1090/S0002-9939-98-04261-0
MathSciNet review: 1443857
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Abstract: We provide a new proof of a theorem of Birman and Solomyak that if $A(s) = A_{0} + sB$ with $B\geq 0$ trace class and $d\mu _{s} (\cdot ) = \text{Tr}(B^{1/2} E_{A(s)}(\cdot ) B^{1/2})$, then $\int ^{1}_{0} [d\mu _{s} (\lambda )]\, ds = \xi (\lambda )\, d\lambda $, where $\xi $ is the Krein spectral shift from $A(0)$ to $A(1)$. Our main point is that this is a simple consequence of the formula $\frac{d}{ds} \text{Tr}(f(A(s))=\text{Tr}(Bf'(A(s)))$.


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Additional Information

Barry Simon
Affiliation: Division of Physics, Mathematics, and Astronomy, California Institute of Technology, Pasadena, California 91125
Email: bsimon@cco.caltech.edu

DOI: https://doi.org/10.1090/S0002-9939-98-04261-0
Received by editor(s): October 14, 1996
Additional Notes: This material is based upon work supported by the National Science Foundation under Grant No. DMS-9401491. The government has certain rights in this material.
Communicated by: Palle E. T. Jorgensen
Article copyright: © Copyright 1998 Barry Simon