A Lidskii type formula for Dixmier traces

Abstract

We present a new formula to compute Dixmier traces τ ω ( T ) of pseudodifferential operators (respectively, almost periodic pseudodifferential operators) of order − n on n-dimensional compact Riemannian manifolds (respectively, R n ). Under a natural condition on the operator T, we show that τ ω ( T ) = ω - lim t → ∞ 1 log ( 1 + t ) ∫ λ ∉ 1 t G λ d μ T ( λ ) , where G is any bounded neighborhood of 0 ∈ C and μ T is the Brown spectral measure of T. If T is measurable, then the ω-limit may be replaced with the true (ordinary) limit. Our approach works equally well in both type I and II settings. To cite this article: N.A. Azamov, F.A. Sukochev, C. R. Acad. Sci. Paris, Ser. I 340 (2005).