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An optimalLp-bound on the Krein spectral shift function

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Abstract

Let ξA,B be the Krein spectral shift function for a pair of operatorsA, B, with C =A-B trace class. We establish the bound

$$\int {F(|\xi _{A,B} (\lambda )|)} d\lambda \leqslant \int {F(|\xi _{|C|,0} (\lambda )|)} d\lambda = \sum\limits_{j = 1}^\infty {[F(j) - F(j - 1)]\mu _j (C),} $$

whereF is any non-negative convex function on [0, ∞) with F(0) = 0 and Ώj (C) are the singular values ofC. The choice F(t) =tp,p ≥ 1, improves a recent bound of Combes, Hislop and Nakamura.

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Authors and Affiliations

  1. Department of Mathematics 253-37, California Institute of Technology, 91125, Pasadena, CA, USA

    Dirk Hundertmark & Barry Simon

Authors
  1. Dirk Hundertmark
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  2. Barry Simon
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Correspondence to Dirk Hundertmark.

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Supported in part by NSF grant DMS-9707661.

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Hundertmark, D., Simon, B. An optimalLp-bound on the Krein spectral shift function. J. Anal. Math. 87, 199–208 (2002). https://doi.org/10.1007/BF02868474

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