Metric geometry in infinite dimensional Stiefel manifolds
- Autores
- Chiumiento, Eduardo Hernán
- Año de publicación
- 2010
- Idioma
- inglés
- Tipo de recurso
- artículo
- Estado
- versión publicada
- Descripción
- Let J be a separable Banach ideal in the space of bounded operators acting in a Hilbert space H and I the set of partial isometries in H. Fix v∈I. In this paper we study metric properties of the I-Stiefel manifold associated to v, namely. StI(v)={v0∈: v-v0∈I,j(v0*v0,v*v)=0}, where j(,) is the Fredholm index of a pair of projections. Let UI(H) be the Banach-Lie group of unitary operators which are perturbations of the identity by elements in I. Then StI(v) coincides with the orbit of v under the action of UI(H)×UI(H) on I given by (u,w)·v0=uv0w*, u,w∈UI(H) and v0∈StI(v). We endow StI(v) with a quotient Finsler metric by means of the Banach quotient norm of the Lie algebra of UI(H)×UI(H) by the Lie algebra of the isotropy group. We give a characterization of the rectifiable distance induced by this metric. In fact, we show that the rectifiable distance coincides with the quotient distance of UI(H)×UI(H) by the isotropy group. Hence this metric defines the quotient topology in StI(v).The other results concern with minimal curves in I-Stiefel manifolds when the ideal I is fixed as the compact operators in H. The initial value problem is solved when the partial isometry v has finite rank. In addition, we use a length-reducing map into the Grassmannian to find some special partial isometries that can be joined with a curve of minimal length.
Facultad de Ciencias Exactas - Materia
-
Matemática
Banach ideal
Finsler metric
Minimal curves
Partial isometry - Nivel de accesibilidad
- acceso abierto
- Condiciones de uso
- http://creativecommons.org/licenses/by-nc-sa/4.0/
- Repositorio
- Institución
- Universidad Nacional de La Plata
- OAI Identificador
- oai:sedici.unlp.edu.ar:10915/82430
Ver los metadatos del registro completo
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Metric geometry in infinite dimensional Stiefel manifoldsChiumiento, Eduardo HernánMatemáticaBanach idealFinsler metricMinimal curvesPartial isometryLet J be a separable Banach ideal in the space of bounded operators acting in a Hilbert space H and I the set of partial isometries in H. Fix v∈I. In this paper we study metric properties of the I-Stiefel manifold associated to v, namely. StI(v)={v0∈: v-v0∈I,j(v0*v0,v*v)=0}, where j(,) is the Fredholm index of a pair of projections. Let UI(H) be the Banach-Lie group of unitary operators which are perturbations of the identity by elements in I. Then StI(v) coincides with the orbit of v under the action of UI(H)×UI(H) on I given by (u,w)·v0=uv0w*, u,w∈UI(H) and v0∈StI(v). We endow StI(v) with a quotient Finsler metric by means of the Banach quotient norm of the Lie algebra of UI(H)×UI(H) by the Lie algebra of the isotropy group. We give a characterization of the rectifiable distance induced by this metric. In fact, we show that the rectifiable distance coincides with the quotient distance of UI(H)×UI(H) by the isotropy group. Hence this metric defines the quotient topology in StI(v).The other results concern with minimal curves in I-Stiefel manifolds when the ideal I is fixed as the compact operators in H. The initial value problem is solved when the partial isometry v has finite rank. In addition, we use a length-reducing map into the Grassmannian to find some special partial isometries that can be joined with a curve of minimal length.Facultad de Ciencias Exactas2010info:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersionArticulohttp://purl.org/coar/resource_type/c_6501info:ar-repo/semantics/articuloapplication/pdf469-479http://sedici.unlp.edu.ar/handle/10915/82430enginfo:eu-repo/semantics/altIdentifier/issn/0926-2245info:eu-repo/semantics/altIdentifier/doi/10.1016/j.difgeo.2009.12.003info: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:10915/82430Institucionalhttp://sedici.unlp.edu.ar/Universidad públicaNo correspondehttp://sedici.unlp.edu.ar/oai/snrdalira@sedici.unlp.edu.arArgentinaNo correspondeNo correspondeNo correspondeopendoar:13292025-09-29 11:15:31.293SEDICI (UNLP) - Universidad Nacional de La Platafalse |
dc.title.none.fl_str_mv |
Metric geometry in infinite dimensional Stiefel manifolds |
title |
Metric geometry in infinite dimensional Stiefel manifolds |
spellingShingle |
Metric geometry in infinite dimensional Stiefel manifolds Chiumiento, Eduardo Hernán Matemática Banach ideal Finsler metric Minimal curves Partial isometry |
title_short |
Metric geometry in infinite dimensional Stiefel manifolds |
title_full |
Metric geometry in infinite dimensional Stiefel manifolds |
title_fullStr |
Metric geometry in infinite dimensional Stiefel manifolds |
title_full_unstemmed |
Metric geometry in infinite dimensional Stiefel manifolds |
title_sort |
Metric geometry in infinite dimensional 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 Minimal curves Partial isometry |
topic |
Matemática Banach ideal Finsler metric Minimal curves Partial isometry |
dc.description.none.fl_txt_mv |
Let J be a separable Banach ideal in the space of bounded operators acting in a Hilbert space H and I the set of partial isometries in H. Fix v∈I. In this paper we study metric properties of the I-Stiefel manifold associated to v, namely. StI(v)={v0∈: v-v0∈I,j(v0*v0,v*v)=0}, where j(,) is the Fredholm index of a pair of projections. Let UI(H) be the Banach-Lie group of unitary operators which are perturbations of the identity by elements in I. Then StI(v) coincides with the orbit of v under the action of UI(H)×UI(H) on I given by (u,w)·v0=uv0w*, u,w∈UI(H) and v0∈StI(v). We endow StI(v) with a quotient Finsler metric by means of the Banach quotient norm of the Lie algebra of UI(H)×UI(H) by the Lie algebra of the isotropy group. We give a characterization of the rectifiable distance induced by this metric. In fact, we show that the rectifiable distance coincides with the quotient distance of UI(H)×UI(H) by the isotropy group. Hence this metric defines the quotient topology in StI(v).The other results concern with minimal curves in I-Stiefel manifolds when the ideal I is fixed as the compact operators in H. The initial value problem is solved when the partial isometry v has finite rank. In addition, we use a length-reducing map into the Grassmannian to find some special partial isometries that can be joined with a curve of minimal length. Facultad de Ciencias Exactas |
description |
Let J be a separable Banach ideal in the space of bounded operators acting in a Hilbert space H and I the set of partial isometries in H. Fix v∈I. In this paper we study metric properties of the I-Stiefel manifold associated to v, namely. StI(v)={v0∈: v-v0∈I,j(v0*v0,v*v)=0}, where j(,) is the Fredholm index of a pair of projections. Let UI(H) be the Banach-Lie group of unitary operators which are perturbations of the identity by elements in I. Then StI(v) coincides with the orbit of v under the action of UI(H)×UI(H) on I given by (u,w)·v0=uv0w*, u,w∈UI(H) and v0∈StI(v). We endow StI(v) with a quotient Finsler metric by means of the Banach quotient norm of the Lie algebra of UI(H)×UI(H) by the Lie algebra of the isotropy group. We give a characterization of the rectifiable distance induced by this metric. In fact, we show that the rectifiable distance coincides with the quotient distance of UI(H)×UI(H) by the isotropy group. Hence this metric defines the quotient topology in StI(v).The other results concern with minimal curves in I-Stiefel manifolds when the ideal I is fixed as the compact operators in H. The initial value problem is solved when the partial isometry v has finite rank. In addition, we use a length-reducing map into the Grassmannian to find some special partial isometries that can be joined with a curve of minimal length. |
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 http://purl.org/coar/resource_type/c_6501 info:ar-repo/semantics/articulo |
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http://sedici.unlp.edu.ar/handle/10915/82430 |
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http://sedici.unlp.edu.ar/handle/10915/82430 |
dc.language.none.fl_str_mv |
eng |
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eng |
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info:eu-repo/semantics/altIdentifier/issn/0926-2245 info:eu-repo/semantics/altIdentifier/doi/10.1016/j.difgeo.2009.12.003 |
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) |
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openAccess |
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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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application/pdf 469-479 |
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