00-370 Hundertmark D., Simon B.

An Optimal L^p-Bound on the Krein Spectral Shift Function (23K, AMS-LaTeX) Sep 18, 00

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Abstract. [{\it Note}: This paper supplants B. Simon's preprint, ``$L^p$ bounds on the Krein spectral shift," which has been withdrawn.] \medskip Let $\xi_{A,B}$ be the Krein spectral shift function for a pair of operators $A,B$, with $C=A-B$ trace class. We establish the bound \begin{displaymath} \int F(\abs{\xi_{A,B}(\lambda)})\, d\lambda \le \int F(\abs{\xi_{\abs{C},0}(\lambda)})\, d\lambda = \sum_{j=1}^\infty \big[F(j)-F(j-1)]\mu_j(C), \end{displaymath} where $F$ is any non-negative convex function on $[0,\infty)$ with $F(0)=0$ and $\mu_j(C)$ are the singular values of $C$. Specializing to $F(t)=t^p$, $p\ge 1$ this improves a recent bound of Combes, Hislop, and Nakamura.

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