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
CONICET Digital (CONICET)
Institución
Consejo Nacional de Investigaciones Científicas y Técnicas
OAI Identificador
oai:ri.conicet.gov.ar:11336/242788

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spelling 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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