Kernel and symbol criteria for Schatten classes and <i>r</i>-nuclearity on compact manifolds

Julio Delgado; Michael Ruzhansky

doi:10.1016/j.crma.2014.08.012

Mathematical analysis/Functional analysis

Kernel and symbol criteria for Schatten classes and r-nuclearity on compact manifolds
[Critères portant sur des symboles et noyaux pour les classes de Schatten et r-nucléarité sur les variétés compactes]

Présenté par : Jean-Michel Bony

Julio Delgado ¹ ; Michael Ruzhansky ¹

¹ Department of Mathematics, Imperial College London, 180 Queen's Gate, London SW7 2AZ, United Kingdom

Comptes Rendus. Mathématique, Volume 352 (2014) no. 10, pp. 779-784.

Abstract (VO)
Résumé

In this Note, we present criteria on both symbols and integral kernels ensuring that the corresponding operators on compact manifolds belong to Schatten classes. A specific test for nuclearity is established as well as the corresponding trace formulae. In the special case of compact Lie groups, kernel criteria in terms of (locally and globally) hypoelliptic operators are also given. A notion of invariant operator and its full symbol associated with an elliptic operator are introduced. Some applications to the study of r-nuclearity on $L^{p}$ spaces are also obtained.

Nous présentons dans cette Note des critères sur des symboles et noyaux pour s'assurer de ce que les opérateurs correspondants sur des variétés compactes appartiennent à une classe de Schatten. Les opérateurs à trace sont considérés comme un cas spécial. Nous introduisons aussi des notions d'opérateur invariant et de symbole global associés à un opérateur elliptique et les appliquons à l'etude de la nucléarité.

Reçu le : 2014-05-19
Accepté le : 2014-08-25
Publié le : 2014-09-23

DOI : 10.1016/j.crma.2014.08.012

Affiliations des auteurs :

Julio Delgado ¹ ; Michael Ruzhansky ¹

¹ Department of Mathematics, Imperial College London, 180 Queen's Gate, London SW7 2AZ, United Kingdom

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     title = {Kernel and symbol criteria for {Schatten} classes and \protect\emph{r}-nuclearity on compact manifolds},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {779--784},
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     year = {2014},
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Julio Delgado; Michael Ruzhansky. Kernel and symbol criteria for Schatten classes and r-nuclearity on compact manifolds. Comptes Rendus. Mathématique, Volume 352 (2014) no. 10, pp. 779-784. doi : 10.1016/j.crma.2014.08.012. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2014.08.012/

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