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The Lifshitz-Krein trace formula and operator Lipschitz functions


Author: V. V. Peller
Journal: Proc. Amer. Math. Soc. 144 (2016), 5207-5215
MSC (2010): Primary 47A55, 47B10; Secondary 47B15, 47B25, 47A60, 47B49
DOI: https://doi.org/10.1090/proc/13140
Published electronically: August 1, 2016
MathSciNet review: 3556265
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Abstract: We solve a problem by M.G. Krein and describe the maximal class of functions $ f$ on the real line, for which the Lifshitz-Krein trace formula $ \mathrm {trace}(f(A)-f(B))=\int _{\mathbb{R}} f'(s)\boldsymbol {\xi }(s)\,ds$ holds for arbitrary self-adjoint operators $ A$ and $ B$ with $ A-B$ in the trace class $ \boldsymbol {S}_1$. We prove that this class of functions coincideS with the class of operator Lipschitz functions.


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Additional Information

V. V. Peller
Affiliation: Department of Mathematics, Michigan State University, East Lansing, Michigan 48824

DOI: https://doi.org/10.1090/proc/13140
Received by editor(s): January 7, 2016
Received by editor(s) in revised form: February 5, 2016
Published electronically: August 1, 2016
Additional Notes: The author was partially supported by NSF grant DMS 1300924
Communicated by: Pamela B. Gorkin
Article copyright: © Copyright 2016 American Mathematical Society