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
SEDICI (UNLP)
Institución
Universidad Nacional de La Plata
OAI Identificador
oai:sedici.unlp.edu.ar:10915/82430

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network_name_str SEDICI (UNLP)
spelling 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
format article
status_str publishedVersion
dc.identifier.none.fl_str_mv http://sedici.unlp.edu.ar/handle/10915/82430
url http://sedici.unlp.edu.ar/handle/10915/82430
dc.language.none.fl_str_mv eng
language eng
dc.relation.none.fl_str_mv 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)
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)
dc.format.none.fl_str_mv application/pdf
469-479
dc.source.none.fl_str_mv reponame:SEDICI (UNLP)
instname:Universidad Nacional de La Plata
instacron:UNLP
reponame_str SEDICI (UNLP)
collection SEDICI (UNLP)
instname_str Universidad Nacional de La Plata
instacron_str UNLP
institution UNLP
repository.name.fl_str_mv SEDICI (UNLP) - Universidad Nacional de La Plata
repository.mail.fl_str_mv alira@sedici.unlp.edu.ar
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