A note on free divergence-free vector fields

Hyuga Ito; Akihiro Miyagawa

doi:10.5802/crmath.631

Article de recherche - Analyse fonctionnelle

A note on free divergence-free vector fields
[Note sur les champs de vecteurs libres à divergence nulle]

Hyuga Ito ¹ ; Akihiro Miyagawa ^2,³

¹ Graduate School of Mathematics, Nagoya University, Furocho, Chikusaku, Nagoya, 464-8602, Japan
² Department of Mathematics, Kyoto University, Kitashirakawa Oiwake-cho, Sakyo-ku, 606-8502, Japan
³ Department of Mathematics, University of California San Diego, La Jolla, CA 92093, USA

Comptes Rendus. Mathématique, Volume 362 (2024), pp. 1545-1554.

Résumé
Abstract

Nous présentons une base orthonormée de gradients cycliques et une base (non orthogonale) du champ de vecteurs libre homogène à divergence nulle sur l’espace de Fock plein et déterminons la dimension du champ de vecteurs libre au sens de Voiculescu à divergence nulle de degré k ou moins.

En outre, nous donnons une formule concrète pour la projection orthogonale sur l’espace des gradients cycliques ainsi que pour la version libre de la projection de Leray.

We exhibit an orthonormal basis of cyclic gradients and a (non-orthogonal) basis of the homogeneous free divergence-free vector field on the full Fock space and determine the dimension of Voiculescu’s free divergence-free vector field of degree $k$ or less. Moreover, we also give a concrete formula for the orthogonal projection onto the space of cyclic gradients as well as the free Leray projection.

Reçu le : 2024-01-22
Révisé le : 2024-03-05
Accepté le : 2024-03-18
Publié le : 2024-11-25

DOI : 10.5802/crmath.631

Classification : 46L54, 46L57
Keywords: Free probability, Free semi-circular system, Free divergence-free vector fields, Cyclic gradients
Mots-clés : Probabilité libre, système semi-circulaire libre, champ de vecteurs libre à divergence nulle, gradients cycliques.

Affiliations des auteurs :

Hyuga Ito ¹ ; Akihiro Miyagawa ^{2, 3}

¹ Graduate School of Mathematics, Nagoya University, Furocho, Chikusaku, Nagoya, 464-8602, Japan
² Department of Mathematics, Kyoto University, Kitashirakawa Oiwake-cho, Sakyo-ku, 606-8502, Japan
³ Department of Mathematics, University of California San Diego, La Jolla, CA 92093, USA

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     author = {Hyuga Ito and Akihiro Miyagawa},
     title = {A note on free divergence-free vector fields},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1545--1554},
     publisher = {Acad\'emie des sciences, Paris},
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     year = {2024},
     doi = {10.5802/crmath.631},
     language = {en},
}

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Hyuga Ito; Akihiro Miyagawa. A note on free divergence-free vector fields. Comptes Rendus. Mathématique, Volume 362 (2024), pp. 1545-1554. doi : 10.5802/crmath.631. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.631/

Bibliographie
Cité par

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