On the Birman–Krein Theorem

Vanderléa R. Bazao; César R. de Oliveira; Pablo A. Diaz

doi:10.5802/crmath.473

Functional analysis

On the Birman–Krein Theorem

Vanderléa R. Bazao ¹; César R. de Oliveira ² ; Pablo A. Diaz ²

¹ Faculdade de Ciências Exatas e Tecnologias, UFGD, Dourados, MS, 79804-970 Brazil
² Departamento de Matemática, UFSCar, São Carlos, SP, 13560-970 Brazil

Comptes Rendus. Mathématique, Volume 361 (2023), pp. 1081-1086.

Abstract
Résumé

It is shown that if $X$ is a unitary operator so that a singular subspace of $U$ is unitarily equivalent to a singular subspace of $U X$ (or $X U$ ), for each unitary operator $U$ , then $X$ is the identity operator. In other words, there is no nontrivial generalization of Birman–Krein Theorem that includes the preservation of a singular spectral subspace in this context.

On montre que si $X$ est un opérateur unitaire tel qu’un sous-espace singulier de $U$ est unitairement équivalent à un sous-espace singulier de $U X$ (ou $X U$ ), pour chaque opérateur unitaire $U$ , alors $X$ est l’opérateur d’identité. En d’autres termes, il n’y a pas de généralisation non triviale du théorème de Birman–Krein qui inclut la préservation d’un sous-espace spectral singulier dans ce contexte.

Received: 2022-06-14
Accepted: 2023-01-31
Published online: 2023-10-24

DOI: 10.5802/crmath.473

Classification: 47A55, 47A10, 81Q10

Author's affiliations:

Vanderléa R. Bazao ¹; César R. de Oliveira ²; Pablo A. Diaz ²

¹ Faculdade de Ciências Exatas e Tecnologias, UFGD, Dourados, MS, 79804-970 Brazil
² Departamento de Matemática, UFSCar, São Carlos, SP, 13560-970 Brazil

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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     title = {On the {Birman{\textendash}Krein} {Theorem}},
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Vanderléa R. Bazao; César R. de Oliveira; Pablo A. Diaz. On the Birman–Krein Theorem. Comptes Rendus. Mathématique, Volume 361 (2023), pp. 1081-1086. doi : 10.5802/crmath.473. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.473/

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