## Spectral averaging and the Krein spectral shift

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- by Barry Simon PDF
- Proc. Amer. Math. Soc.
**126**(1998), 1409-1413

## 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 $\frac {d}{ds} \text {Tr}(f(A(s))=\text {Tr}(Bf’(A(s)))$.## References

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## Additional Information

**Barry Simon**- Affiliation: Division of Physics, Mathematics, and Astronomy, California Institute of Technology, Pasadena, California 91125
- MR Author ID: 189013
- Email: bsimon@cco.caltech.edu
- 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
- © Copyright 1998 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