Relatively bounded and relatively trace class perturbations

Aleksei B. Aleksandrov; Vladimir V. Peller

doi:10.5802/crmath.722

Article de recherche - Analyse fonctionnelle, Analyse harmonique

Relatively bounded and relatively trace class perturbations
[Perturbations relativement bornées et relativement de classe trace]

Aleksei B. Aleksandrov ^1,² ; Vladimir V. Peller ^1,²

¹ St. Petersburg State University, Universitetskaya nab., 7/9, 199034 St. Petersburg, Russia
² St. Petersburg Department, Steklov Institute of Mathematics, Russian Academy of Sciences, Fontanka 27, 191023 St. Petersburg, Russia

Comptes Rendus. Mathématique, Volume 363 (2025), pp. 377-382.

Abstract (VO)
Résumé

In this note we study the behaviour of functions of self-adjoint operators under relatively bounded and relatively trace class perturbations. We introduce and study the class of relatively operator Lipschitz functions. We obtain a trace formula in the case of relatively trace class perturbations and show that this class of functions is the maximal class of functions for which the trace formula holds. Our method also gives us a new approach to the inequality $\int | ξ (t) | (1 + | t |)^{- 1} d t < \infty$ for the spectral shift function $ξ$ in the case of relatively trace class perturbations.

Dans cette note nous étudions le comportement des fonctions d’opérateurs auto-adjoints sous des perturbations relativement dans la classe des opérateurs à trace et relativement bornées. Nous introduisons et étudions la classe de fonctions relativement lipschitziennes opératorielles. Nous obtenons une formule de trace dans le cas de perturbations relativement à trace et montrons que cette classe de fonctions est maximale pour la validité de la formule de trace. Notre méthode nous donne une nouvelle approche de l’inégalité $\int | ξ (t) | (1 + | t |)^{- 1} d t < \infty$ pour la fonction $ξ$ de décalage spectral.

Reçu le : 2024-12-19
Révisé le : 2025-01-23
Accepté le : 2025-01-24
Publié le : 2025-05-20

DOI : 10.5802/crmath.722

Classification : 47A55
Keywords: Relatively bounded perturbation, relatively trace class perturbation, relatively operator Lipschitz class, trace formula, self-adjoint operators, double operator integrals
Mots-clés : Perturbations relativement bornées, perturbations relativement à trace, fonctions relativement lipschitziennes opératorielles, formules des traces, opérateurs auto-adjoints, intégrales d’opérateurs doubles

Affiliations des auteurs :

Aleksei B. Aleksandrov ^{1, 2} ; Vladimir V. Peller ^{1, 2}

¹ St. Petersburg State University, Universitetskaya nab., 7/9, 199034 St. Petersburg, Russia
² St. Petersburg Department, Steklov Institute of Mathematics, Russian Academy of Sciences, Fontanka 27, 191023 St. Petersburg, Russia

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

@article{CRMATH_2025__363_G4_377_0,
     author = {Aleksei B. Aleksandrov and Vladimir V. Peller},
     title = {Relatively bounded and relatively trace class perturbations},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {377--382},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {363},
     year = {2025},
     doi = {10.5802/crmath.722},
     language = {en},
}

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Aleksei B. Aleksandrov; Vladimir V. Peller. Relatively bounded and relatively trace class perturbations. Comptes Rendus. Mathématique, Volume 363 (2025), pp. 377-382. doi : 10.5802/crmath.722. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.722/

Bibliographie
Cité par

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