On the Birman–Krein Theorem
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 $UX$ (or $XU$), for each unitary operator $U$, then $X$ is the identity operator. In other words, there is no nontrivial generalization of Birman–Krein Theorem that incl...
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| Format: | Article |
| Language: | English |
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Académie des sciences
2023-10-01
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| Series: | Comptes Rendus. Mathématique |
| Online Access: | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.473/ |
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| author | Bazao, Vanderléa R. de Oliveira, César R. Diaz, Pablo A. |
| author_facet | Bazao, Vanderléa R. de Oliveira, César R. Diaz, Pablo A. |
| author_sort | Bazao, Vanderléa R. |
| collection | DOAJ |
| description | 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 $UX$ (or $XU$), 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. |
| format | Article |
| id | doaj-art-3cb0cd706a9342fea0a18ae1c80b991e |
| institution | Kabale University |
| issn | 1778-3569 |
| language | English |
| publishDate | 2023-10-01 |
| publisher | Académie des sciences |
| record_format | Article |
| series | Comptes Rendus. Mathématique |
| spelling | doaj-art-3cb0cd706a9342fea0a18ae1c80b991e2025-02-07T11:09:55ZengAcadémie des sciencesComptes Rendus. Mathématique1778-35692023-10-01361G71081108610.5802/crmath.47310.5802/crmath.473On the Birman–Krein TheoremBazao, Vanderléa R.0de Oliveira, César R.1Diaz, Pablo A.2Faculdade de Ciências Exatas e Tecnologias, UFGD, Dourados, MS, 79804-970 BrazilDepartamento de Matemática, UFSCar, São Carlos, SP, 13560-970 BrazilDepartamento de Matemática, UFSCar, São Carlos, SP, 13560-970 BrazilIt is shown that if $X$ is a unitary operator so that a singular subspace of $U$ is unitarily equivalent to a singular subspace of $UX$ (or $XU$), 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.https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.473/ |
| spellingShingle | Bazao, Vanderléa R. de Oliveira, César R. Diaz, Pablo A. On the Birman–Krein Theorem Comptes Rendus. Mathématique |
| title | On the Birman–Krein Theorem |
| title_full | On the Birman–Krein Theorem |
| title_fullStr | On the Birman–Krein Theorem |
| title_full_unstemmed | On the Birman–Krein Theorem |
| title_short | On the Birman–Krein Theorem |
| title_sort | on the birman krein theorem |
| url | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.473/ |
| work_keys_str_mv | AT bazaovanderlear onthebirmankreintheorem AT deoliveiracesarr onthebirmankreintheorem AT diazpabloa onthebirmankreintheorem |