Metric geometry in infinite dimensional Stiefel manifolds
- Autores
- Chiumiento, Eduardo Hernan
- 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 in I. In this paper we study metric properties of the J-Stiefel manifold associated to v, namely SG = { v_0 in I , : , v- v_0 in J, , j(v_0*v_0,v*v)=0 }, where j( , ) is the Fredholm index of a pair of projections. Let UJ(H) be the Banach-Lie group of unitary operators which are perturbations of the identity by elements in J. Then SG coincides with the orbit of v under the action of UJ(H) on I given by (u,w) v_0=uv_0w*, u,w in UJ(H) and v_0 in SG. We endow SG with a quotient Finsler metric by means of the Banach quotient norm of the Lie algebra of UJ(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 coincide with the quotient distance of UJ(H) by the isotropy group. Hence this metric defines the quotient topology in SG. The other results concern with minimal curves in J-Stiefel manifolds when the ideal J 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 join with a curve of minimal length.
Fil: Chiumiento, Eduardo Hernan. Universidad Nacional de La Plata. Facultad de Ciencias Exactas. Departamento de Matemáticas; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - La Plata; Argentina - Materia
-
PARTIAL ISOMETRY
BANACH IDEAL
FINSLER METRIC
MINIMAL CURVES - Nivel de accesibilidad
- acceso abierto
- Condiciones de uso
- https://creativecommons.org/licenses/by-nc-sa/2.5/ar/
- Repositorio
- Institución
- Consejo Nacional de Investigaciones Científicas y Técnicas
- OAI Identificador
- oai:ri.conicet.gov.ar:11336/242788
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Metric geometry in infinite dimensional Stiefel manifoldsChiumiento, Eduardo HernanPARTIAL ISOMETRYBANACH IDEALFINSLER METRICMINIMAL CURVEShttps://purl.org/becyt/ford/1.1https://purl.org/becyt/ford/1Let 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 in I. In this paper we study metric properties of the J-Stiefel manifold associated to v, namely SG = { v_0 in I , : , v- v_0 in J, , j(v_0*v_0,v*v)=0 }, where j( , ) is the Fredholm index of a pair of projections. Let UJ(H) be the Banach-Lie group of unitary operators which are perturbations of the identity by elements in J. Then SG coincides with the orbit of v under the action of UJ(H) on I given by (u,w) v_0=uv_0w*, u,w in UJ(H) and v_0 in SG. We endow SG with a quotient Finsler metric by means of the Banach quotient norm of the Lie algebra of UJ(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 coincide with the quotient distance of UJ(H) by the isotropy group. Hence this metric defines the quotient topology in SG. The other results concern with minimal curves in J-Stiefel manifolds when the ideal J 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 join with a curve of minimal length.Fil: Chiumiento, Eduardo Hernan. Universidad Nacional de La Plata. Facultad de Ciencias Exactas. Departamento de Matemáticas; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - La Plata; ArgentinaElsevier Science2010-08info:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersionhttp://purl.org/coar/resource_type/c_6501info:ar-repo/semantics/articuloapplication/pdfapplication/octet-streamapplication/pdfhttp://hdl.handle.net/11336/242788Chiumiento, Eduardo Hernan; Metric geometry in infinite dimensional Stiefel manifolds; Elsevier Science; Differential Geometry and its Applications; 28; 4; 8-2010; 469-4790926-2245CONICET DigitalCONICETenginfo:eu-repo/semantics/altIdentifier/url/https://www.sciencedirect.com/science/article/pii/S0926224509001259info:eu-repo/semantics/altIdentifier/doi/10.1016/j.difgeo.2009.12.003info:eu-repo/semantics/openAccesshttps://creativecommons.org/licenses/by-nc-sa/2.5/ar/reponame:CONICET Digital (CONICET)instname:Consejo Nacional de Investigaciones Científicas y Técnicas2025-09-29T09:54:00Zoai:ri.conicet.gov.ar:11336/242788instacron:CONICETInstitucionalhttp://ri.conicet.gov.ar/Organismo científico-tecnológicoNo correspondehttp://ri.conicet.gov.ar/oai/requestdasensio@conicet.gov.ar; lcarlino@conicet.gov.arArgentinaNo correspondeNo correspondeNo correspondeopendoar:34982025-09-29 09:54:00.591CONICET Digital (CONICET) - Consejo Nacional de Investigaciones Científicas y Técnicasfalse |
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 Hernan PARTIAL ISOMETRY BANACH IDEAL FINSLER METRIC MINIMAL CURVES |
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 Hernan |
author |
Chiumiento, Eduardo Hernan |
author_facet |
Chiumiento, Eduardo Hernan |
author_role |
author |
dc.subject.none.fl_str_mv |
PARTIAL ISOMETRY BANACH IDEAL FINSLER METRIC MINIMAL CURVES |
topic |
PARTIAL ISOMETRY BANACH IDEAL FINSLER METRIC MINIMAL CURVES |
purl_subject.fl_str_mv |
https://purl.org/becyt/ford/1.1 https://purl.org/becyt/ford/1 |
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 in I. In this paper we study metric properties of the J-Stiefel manifold associated to v, namely SG = { v_0 in I , : , v- v_0 in J, , j(v_0*v_0,v*v)=0 }, where j( , ) is the Fredholm index of a pair of projections. Let UJ(H) be the Banach-Lie group of unitary operators which are perturbations of the identity by elements in J. Then SG coincides with the orbit of v under the action of UJ(H) on I given by (u,w) v_0=uv_0w*, u,w in UJ(H) and v_0 in SG. We endow SG with a quotient Finsler metric by means of the Banach quotient norm of the Lie algebra of UJ(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 coincide with the quotient distance of UJ(H) by the isotropy group. Hence this metric defines the quotient topology in SG. The other results concern with minimal curves in J-Stiefel manifolds when the ideal J 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 join with a curve of minimal length. Fil: Chiumiento, Eduardo Hernan. Universidad Nacional de La Plata. Facultad de Ciencias Exactas. Departamento de Matemáticas; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - La Plata; Argentina |
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 in I. In this paper we study metric properties of the J-Stiefel manifold associated to v, namely SG = { v_0 in I , : , v- v_0 in J, , j(v_0*v_0,v*v)=0 }, where j( , ) is the Fredholm index of a pair of projections. Let UJ(H) be the Banach-Lie group of unitary operators which are perturbations of the identity by elements in J. Then SG coincides with the orbit of v under the action of UJ(H) on I given by (u,w) v_0=uv_0w*, u,w in UJ(H) and v_0 in SG. We endow SG with a quotient Finsler metric by means of the Banach quotient norm of the Lie algebra of UJ(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 coincide with the quotient distance of UJ(H) by the isotropy group. Hence this metric defines the quotient topology in SG. The other results concern with minimal curves in J-Stiefel manifolds when the ideal J 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 join with a curve of minimal length. |
publishDate |
2010 |
dc.date.none.fl_str_mv |
2010-08 |
dc.type.none.fl_str_mv |
info:eu-repo/semantics/article info:eu-repo/semantics/publishedVersion http://purl.org/coar/resource_type/c_6501 info:ar-repo/semantics/articulo |
format |
article |
status_str |
publishedVersion |
dc.identifier.none.fl_str_mv |
http://hdl.handle.net/11336/242788 Chiumiento, Eduardo Hernan; Metric geometry in infinite dimensional Stiefel manifolds; Elsevier Science; Differential Geometry and its Applications; 28; 4; 8-2010; 469-479 0926-2245 CONICET Digital CONICET |
url |
http://hdl.handle.net/11336/242788 |
identifier_str_mv |
Chiumiento, Eduardo Hernan; Metric geometry in infinite dimensional Stiefel manifolds; Elsevier Science; Differential Geometry and its Applications; 28; 4; 8-2010; 469-479 0926-2245 CONICET Digital CONICET |
dc.language.none.fl_str_mv |
eng |
language |
eng |
dc.relation.none.fl_str_mv |
info:eu-repo/semantics/altIdentifier/url/https://www.sciencedirect.com/science/article/pii/S0926224509001259 info:eu-repo/semantics/altIdentifier/doi/10.1016/j.difgeo.2009.12.003 |
dc.rights.none.fl_str_mv |
info:eu-repo/semantics/openAccess https://creativecommons.org/licenses/by-nc-sa/2.5/ar/ |
eu_rights_str_mv |
openAccess |
rights_invalid_str_mv |
https://creativecommons.org/licenses/by-nc-sa/2.5/ar/ |
dc.format.none.fl_str_mv |
application/pdf application/octet-stream application/pdf |
dc.publisher.none.fl_str_mv |
Elsevier Science |
publisher.none.fl_str_mv |
Elsevier Science |
dc.source.none.fl_str_mv |
reponame:CONICET Digital (CONICET) instname:Consejo Nacional de Investigaciones Científicas y Técnicas |
reponame_str |
CONICET Digital (CONICET) |
collection |
CONICET Digital (CONICET) |
instname_str |
Consejo Nacional de Investigaciones Científicas y Técnicas |
repository.name.fl_str_mv |
CONICET Digital (CONICET) - Consejo Nacional de Investigaciones Científicas y Técnicas |
repository.mail.fl_str_mv |
dasensio@conicet.gov.ar; lcarlino@conicet.gov.ar |
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