Geometry of ℑ-Stiefel manifolds

Autores
Chiumiento, Eduardo Hernán
Año de publicación
2010
Idioma
inglés
Tipo de recurso
artículo
Estado
versión publicada
Descripción
Let ℑ be a separable Banach ideal in the space of bounded operators acting in a Hilbert space ℋ and U(ℋ) ℑ the Banach-Lie group of unitary operators which are perturbations of the identity by elements in ℑ. In this paper we study the geometry of the unitary orbits {UV : U ε U(ℋ) ℑ} and {UVW * : U,W ε U(ℋ) ℑ}, where V is a partial isometry. We give a spatial characterization of these orbits. It turns out that both are included in V + ℑ, and while the first one consists of partial isometries with the same kernel of V , the second is given by partial isometries such that their initial projections and V *V have null index as a pair of projections. We prove that they are smooth submanifolds of the affine Banach space V + ℑ and homogeneous reductive spaces of U(ℋ) ℑ and U(ℋ) ℑ ×U(ℋ) ℑ respectively. Then we endow these orbits with two equivalent Finsler metrics, one provided by the ambient norm of the ideal and the other given by the Banach quotient norm of the Lie algebra of U(ℋ) ℑ (or U(ℋ) ℑ × U(ℋ)I) by the Lie algebra of the isotropy group of the natural actions. We show that they are complete metric spaces with the geodesic distance of these metrics.
Facultad de Ciencias Exactas
Materia
Matemática
Banach ideal
Finsler metric
Partial isometry
Nivel de accesibilidad
acceso abierto
Condiciones de uso
http://creativecommons.org/licenses/by-nc-sa/4.0/
Repositorio
SEDICI (UNLP)
Institución
Universidad Nacional de La Plata
OAI Identificador
oai:sedici.unlp.edu.ar:10915/82501

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spelling Geometry of ℑ-Stiefel manifoldsChiumiento, Eduardo HernánMatemáticaBanach idealFinsler metricPartial isometryLet ℑ be a separable Banach ideal in the space of bounded operators acting in a Hilbert space ℋ and U(ℋ) ℑ the Banach-Lie group of unitary operators which are perturbations of the identity by elements in ℑ. In this paper we study the geometry of the unitary orbits {UV : U ε U(ℋ) ℑ} and {UVW * : U,W ε U(ℋ) ℑ}, where V is a partial isometry. We give a spatial characterization of these orbits. It turns out that both are included in V + ℑ, and while the first one consists of partial isometries with the same kernel of V , the second is given by partial isometries such that their initial projections and V *V have null index as a pair of projections. We prove that they are smooth submanifolds of the affine Banach space V + ℑ and homogeneous reductive spaces of U(ℋ) ℑ and U(ℋ) ℑ ×U(ℋ) ℑ respectively. Then we endow these orbits with two equivalent Finsler metrics, one provided by the ambient norm of the ideal and the other given by the Banach quotient norm of the Lie algebra of U(ℋ) ℑ (or U(ℋ) ℑ × U(ℋ)I) by the Lie algebra of the isotropy group of the natural actions. We show that they are complete metric spaces with the geodesic distance of these metrics.Facultad de Ciencias Exactas2010info:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersionArticulohttp://purl.org/coar/resource_type/c_6501info:ar-repo/semantics/articuloapplication/pdf341-353http://sedici.unlp.edu.ar/handle/10915/82501enginfo:eu-repo/semantics/altIdentifier/issn/0002-9939info:eu-repo/semantics/altIdentifier/doi/10.1090/S0002-9939-09-10080-1info:eu-repo/semantics/openAccesshttp://creativecommons.org/licenses/by-nc-sa/4.0/Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International (CC BY-NC-SA 4.0)reponame:SEDICI (UNLP)instname:Universidad Nacional de La Platainstacron:UNLP2025-09-29T11:15:31Zoai:sedici.unlp.edu.ar:10915/82501Institucionalhttp://sedici.unlp.edu.ar/Universidad públicaNo correspondehttp://sedici.unlp.edu.ar/oai/snrdalira@sedici.unlp.edu.arArgentinaNo correspondeNo correspondeNo correspondeopendoar:13292025-09-29 11:15:32.18SEDICI (UNLP) - Universidad Nacional de La Platafalse
dc.title.none.fl_str_mv Geometry of ℑ-Stiefel manifolds
title Geometry of ℑ-Stiefel manifolds
spellingShingle Geometry of ℑ-Stiefel manifolds
Chiumiento, Eduardo Hernán
Matemática
Banach ideal
Finsler metric
Partial isometry
title_short Geometry of ℑ-Stiefel manifolds
title_full Geometry of ℑ-Stiefel manifolds
title_fullStr Geometry of ℑ-Stiefel manifolds
title_full_unstemmed Geometry of ℑ-Stiefel manifolds
title_sort Geometry of ℑ-Stiefel manifolds
dc.creator.none.fl_str_mv Chiumiento, Eduardo Hernán
author Chiumiento, Eduardo Hernán
author_facet Chiumiento, Eduardo Hernán
author_role author
dc.subject.none.fl_str_mv Matemática
Banach ideal
Finsler metric
Partial isometry
topic Matemática
Banach ideal
Finsler metric
Partial isometry
dc.description.none.fl_txt_mv Let ℑ be a separable Banach ideal in the space of bounded operators acting in a Hilbert space ℋ and U(ℋ) ℑ the Banach-Lie group of unitary operators which are perturbations of the identity by elements in ℑ. In this paper we study the geometry of the unitary orbits {UV : U ε U(ℋ) ℑ} and {UVW * : U,W ε U(ℋ) ℑ}, where V is a partial isometry. We give a spatial characterization of these orbits. It turns out that both are included in V + ℑ, and while the first one consists of partial isometries with the same kernel of V , the second is given by partial isometries such that their initial projections and V *V have null index as a pair of projections. We prove that they are smooth submanifolds of the affine Banach space V + ℑ and homogeneous reductive spaces of U(ℋ) ℑ and U(ℋ) ℑ ×U(ℋ) ℑ respectively. Then we endow these orbits with two equivalent Finsler metrics, one provided by the ambient norm of the ideal and the other given by the Banach quotient norm of the Lie algebra of U(ℋ) ℑ (or U(ℋ) ℑ × U(ℋ)I) by the Lie algebra of the isotropy group of the natural actions. We show that they are complete metric spaces with the geodesic distance of these metrics.
Facultad de Ciencias Exactas
description Let ℑ be a separable Banach ideal in the space of bounded operators acting in a Hilbert space ℋ and U(ℋ) ℑ the Banach-Lie group of unitary operators which are perturbations of the identity by elements in ℑ. In this paper we study the geometry of the unitary orbits {UV : U ε U(ℋ) ℑ} and {UVW * : U,W ε U(ℋ) ℑ}, where V is a partial isometry. We give a spatial characterization of these orbits. It turns out that both are included in V + ℑ, and while the first one consists of partial isometries with the same kernel of V , the second is given by partial isometries such that their initial projections and V *V have null index as a pair of projections. We prove that they are smooth submanifolds of the affine Banach space V + ℑ and homogeneous reductive spaces of U(ℋ) ℑ and U(ℋ) ℑ ×U(ℋ) ℑ respectively. Then we endow these orbits with two equivalent Finsler metrics, one provided by the ambient norm of the ideal and the other given by the Banach quotient norm of the Lie algebra of U(ℋ) ℑ (or U(ℋ) ℑ × U(ℋ)I) by the Lie algebra of the isotropy group of the natural actions. We show that they are complete metric spaces with the geodesic distance of these metrics.
publishDate 2010
dc.date.none.fl_str_mv 2010
dc.type.none.fl_str_mv info:eu-repo/semantics/article
info:eu-repo/semantics/publishedVersion
Articulo
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info:ar-repo/semantics/articulo
format article
status_str publishedVersion
dc.identifier.none.fl_str_mv http://sedici.unlp.edu.ar/handle/10915/82501
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dc.language.none.fl_str_mv eng
language eng
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info:eu-repo/semantics/altIdentifier/doi/10.1090/S0002-9939-09-10080-1
dc.rights.none.fl_str_mv info:eu-repo/semantics/openAccess
http://creativecommons.org/licenses/by-nc-sa/4.0/
Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International (CC BY-NC-SA 4.0)
eu_rights_str_mv openAccess
rights_invalid_str_mv http://creativecommons.org/licenses/by-nc-sa/4.0/
Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International (CC BY-NC-SA 4.0)
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repository.name.fl_str_mv SEDICI (UNLP) - Universidad Nacional de La Plata
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