An optimal \(L^p\)-bound on the Krein spectral shift function.
Let \(A,B\) be bounded self-adjoint operators such that \(C=A-B\) is a trace class operator and \(\xi_{A,B}\) be the Krein spectral shift function for the pair \((A,B)\). The authors give two different proofs of the inequality \[ \int F(| \xi_{A,B}(\lambda )| )\,d\lambda \leq \int F(| \xi_{| C| ,0}(\lambda )| )\,d\lambda =\sum\limits_{j=1}^\infty [F(j)-F(j-1)]\mu_j(C), \] where \(\mu_j(C)\) are the singular values of \(C\) and \(F\) is an arbitrary non-negative convex function on \([0,\infty )\) such that \(F(0)=0\). For the case \(F(t)=t^p\), \(p\geq 1\), this result improves a bound found by \textit{J. M. Combes}, \textit{P. D. Hislop} and \textit{S. Nakamura} [Commun. Math. Phys. 218, 113--130 (2001; Zbl 1042.82024)].
scientific article; zbMATH DE number 1931228