# Mathematics Colloquia and Seminars

### An optimal $L^p$ bound on the Krein spectral shift function.
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 egin{displaymath} int F(abs{xi_{A,B}(lambda)}), dlambda le int F(abs{xi_{abs{C},0}(lambda)}), dlambda = sum_{j=1}^infty ig[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 $F(t)=t^p$, $pge 1$ this improves a recent bound of Combes, Hislop, and Nakamura.