An optimalL p -bound on the Krein spectral shift function

Let ξA,B be the Krein spectral shift function for a pair of operatorsA, B, with C =A-B trace class. We establish the bound% MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfKttLearuqr1ngBPrgarmqr1ngBPrgitL% xBI9gBamXvP5wqSXMqHnxAJn0BKvguHDwzZbqegm0B1jxALjhiov2D% aeHbuLwBLnhiov2DGi1BTfMBaebbfv3ySLgzGueE0jxyaibaiiYdf9% irVeeu0dXdh9vqqj-hEeeu0xXdbba9frFj0-OqFfea0dXdd9vqaq-J% frVkFHe9pgea0dXdar-Jb9hs0dXdbPYxe9vr0-vr0-vqpWqaaeaabi% GaciaacaqabeaadaabauaaaOqaamaapeaabaGaemOrayKaeiikaGIa% eiiFaWNaeqOVdG3aaSbaaSqaaiabdgeabjabcYcaSiabdkeacbqaba% GccqGGOaakcqaH7oaBcqGGPaqkcqGG8baFcqGGPaqkaSqabeqaniab% gUIiYdqegeKCPfgBaGGbcOGaa8hiaiabdsgaKjabeU7aSjabgsMiJo% aapeaabaGaemOrayKaeiikaGIaeiiFaWNaeqOVdG3aaSbaaSqaaiab% cYha8jabdoeadjabcYha8jabcYcaSiabicdaWaqabaGccqGGOaakcq% aH7oaBcqGGPaqkcqGG8baFcqGGPaqkaSqabeqaniabgUIiYdGccaWF% GaGaemizaqMaeq4UdWMaeyypa0ZaaabCaeaacqGGBbWwcqWGgbGrcq% GGOaakcqWGQbGAcqGGPaqkcqGHsislcqWGgbGrcqGGOaakcqWGQbGA% cqGHsislcqaIXaqmcqGGPaqkcqGGDbqxcqaH8oqBdaWgaaWcbaGaem% OAaOgabeaakiabcIcaOiabdoeadjabcMcaPiabcYcaSaWcbaGaemOA% aOMaeyypa0JaeGymaedabaGaeyOhIukaniabggHiLdaaaa!899B!$$\int {F(|\xi _{A,B} (\lambda )|)} d\lambda \leqslant \int {F(|\xi _{|C|,0} (\lambda )|)} d\lambda = \sum\limits_{j = 1}^\infty {[F(j) - F(j - 1)]\mu _j (C),} $$ whereF is any non-negative convex function on [0, ∞) with F(0) = 0 and Ώj (C) are the singular values ofC. The choice F(t) =tp,p ≥ 1, improves a recent bound of Combes, Hislop and Nakamura.