no. 2
Mathematical Analysis
A Lidskii type formula for Dixmier traces
[Une nouvelle formule pour calculer les traces de Dixmier]

Présenté par : Alain Connes

Azamov, Nurulla A.1 ; Sukochev, Fyodor A.1
1 School of Informatics and Engineering, Flinders University of South Australia, Bedford Park, 5042, SA, Australia
Comptes Rendus. Mathématique, Tome 340 (2005) no. 2, pp. 107-112.

Nous présentons une nouvelle formule pour calculer les traces de Dixmier τω(T) des opérateurs pseudodifférentiels (respectivement, des opérateurs pseudodifférentiels presque périodiques) d'ordre −n sur des variétés compactes de dimension n (respectivement, Rn). Lorsque T satisfait une condition naturelle, nous montrons que τω(T)=ω-limt→∞1log(1+t)∫λ∉1tGλdμT(λ), où G est un voisinage borné de 0 dans C et μT est la mesure spectrale de Brown de T. Si T est mesurable, on peut remplacer la limite faible par la limite au sens usuel. Notre approche s'applique aux types I et II.

We present a new formula to compute Dixmier traces τω(T) of pseudodifferential operators (respectively, almost periodic pseudodifferential operators) of order −n on n-dimensional compact Riemannian manifolds (respectively, Rn). Under a natural condition on the operator T, we show that τω(T)=ω-limt→∞1log(1+t)∫λ∉1tGλdμT(λ), where G is any bounded neighborhood of 0∈C and μT 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.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2004.12.005
Azamov, Nurulla A. 1 ; Sukochev, Fyodor A. 1

1 School of Informatics and Engineering, Flinders University of South Australia, Bedford Park, 5042, SA, Australia 
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Azamov, Nurulla A.; Sukochev, Fyodor A. A Lidskii type formula for Dixmier traces. Comptes Rendus. Mathématique, Tome 340 (2005) no. 2, pp. 107-112. doi : 10.1016/j.crma.2004.12.005. http://www.numdam.org/articles/10.1016/j.crma.2004.12.005/

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