Hundertmark D., Simon B.
An Optimal L^p-Bound on the Krein Spectral Shift Function
(23K, AMS-LaTeX)

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the Krein spectral shift," which has been withdrawn.]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.