An optimal Lᵖ-bound on the Krein spectral shift function

Let ξ_(A,B) be the Krein spectral shift function for a pair of operatorsA, B, with C =A-B trace class. We establish the bound ∫F(|ξA,B(λ)|)dλ⩽∫F(|ξ|C|,0(λ)|)dλ=∑_(j=1)^∞[F(j)−F(j−1)]μj(C), where F is any non-negative convex function on [0, ∞) with F(0) = 0 and μj (C) are the singular values of C. The choice F(t) = t^p,p ≥ 1, improves a recent bound of Combes, Hislop and Nakamura.