A Lidskii type formula for Dixmier traces

Nurulla A. Azamov; Fyodor A. Sukochev

doi:10.1016/j.crma.2004.12.005

Mathematical Analysis

A Lidskii type formula for Dixmier traces
[Une nouvelle formule pour calculer les traces de Dixmier]

Présenté par : Alain Connes

Nurulla A. Azamov ¹ ; Fyodor A. Sukochev ¹

¹ School of Informatics and Engineering, Flinders University of South Australia, Bedford Park, 5042, SA, Australia

Comptes Rendus. Mathématique, Volume 340 (2005) no. 2, pp. 107-112.

Abstract (VO)
Résumé

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 $τ_{ω} (T) = ω - \lim_{t \to \infty} \frac{1}{\log (1 + t)} \int_{λ \notin \frac{1}{t} G} λ d μ_{T} (λ)$ , where G is any bounded neighborhood of $0 \in C$ 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.

Nous présentons une nouvelle formule pour calculer les traces de Dixmier $τ_{ω} (T)$ des opérateurs pseudodifférentiels (respectivement, des opérateurs pseudodifférentiels presque périodiques) d'ordre −n sur des variétés compactes de dimension n (respectivement, $R^{n}$ ). Lorsque T satisfait une condition naturelle, nous montrons que $τ_{ω} (T) = ω - \lim_{t \to \infty} \frac{1}{\log (1 + t)} \int_{λ \notin \frac{1}{t} G} λ d μ_{T} (λ)$ , où G est un voisinage borné de 0 dans $C$ et $μ_{T}$ 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.

Reçu le : 2004-09-15
Accepté le : 2004-11-30
Publié le : 2005-01-05

DOI : 10.1016/j.crma.2004.12.005

Affiliations des auteurs :

Nurulla A. Azamov ¹ ; Fyodor A. Sukochev ¹

¹ School of Informatics and Engineering, Flinders University of South Australia, Bedford Park, 5042, SA, Australia

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     author = {Nurulla A. Azamov and Fyodor A. Sukochev},
     title = {A {Lidskii} type formula for {Dixmier} traces},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {107--112},
     publisher = {Elsevier},
     volume = {340},
     number = {2},
     year = {2005},
     doi = {10.1016/j.crma.2004.12.005},
     language = {en},
}

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Nurulla A. Azamov; Fyodor A. Sukochev. A Lidskii type formula for Dixmier traces. Comptes Rendus. Mathématique, Volume 340 (2005) no. 2, pp. 107-112. doi : 10.1016/j.crma.2004.12.005. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2004.12.005/

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