The compatible Grassmannian

Autores
Andruchow, Esteban; Chiumiento, Eduardo Hernan; Di Iorio y Lucero, María Eugenia
Año de publicación
2014
Idioma
inglés
Tipo de recurso
artículo
Estado
versión publicada
Descripción
Let A be a positive injective operator in a Hilbert space View the MathML source, and denote by View the MathML source the inner product defined by A : [f,g]=〈Af,g〉. A closed subspace S⊂H is called A -compatible if there exists a closed complement for S, which is orthogonal to S with respect to the inner product View the MathML source. Equivalently, if there exists a necessarily unique bounded idempotent operator QS such that R(QS)=S, which is symmetric for this inner product. The compatible Grassmannian GrA is the set of all A -compatible subspaces of H. By parametrizing it via the one to one correspondence S↔QS, this set is shown to be a differentiable submanifold of the Banach space of all bounded operators in H which are symmetric with respect to the form View the MathML source. A Banach–Lie group acts naturally on the compatible Grassmannian, the group of all invertible operators in H which preserve the form View the MathML source. Each connected component in GrA of a compatible subspace S of finite dimension, turns out to be a symplectic leaf in a Banach Lie–Poisson space. For 1⩽p⩽∞, in the presence of a fixed View the MathML source-orthogonal (direct sum) decomposition of H, H=S0+N0, we study the restricted compatible Grassmannian (an analogue of the restricted, or Sato Grassmannian). This restricted compatible Grassmannian is shown to be a submanifold of the Banach space of p -Schatten operators which are symmetric for the form View the MathML source. It carries the locally transitive action of the Banach–Lie group of invertible operators which preserve View the MathML source, and are of the form G=1+K, with K in the p-Schatten class. The connected components of this restricted Grassmannian are characterized by means of the Fredholm index of pairs of projections. Finsler metrics which are isometric for the group actions are introduced for both compatible Grassmannians, and minimality results for curves are proved.
Fil: Andruchow, Esteban. Universidad Nacional de General Sarmiento; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática; Argentina
Fil: Chiumiento, Eduardo Hernan. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática; Argentina. Universidad Nacional de La Plata. Facultad de Ciencias Exactas. Departamento de Matematicas; Argentina
Fil: Di Iorio y Lucero, María Eugenia. Universidad Nacional de General Sarmiento; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática; Argentina
Materia
Projection
Positive operator
Compatible subspace
Nivel de accesibilidad
acceso abierto
Condiciones de uso
https://creativecommons.org/licenses/by-nc-nd/2.5/ar/
Repositorio
CONICET Digital (CONICET)
Institución
Consejo Nacional de Investigaciones Científicas y Técnicas
OAI Identificador
oai:ri.conicet.gov.ar:11336/12165

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spelling The compatible GrassmannianAndruchow, EstebanChiumiento, Eduardo HernanDi Iorio y Lucero, María EugeniaProjectionPositive operatorCompatible subspacehttps://purl.org/becyt/ford/1.1https://purl.org/becyt/ford/1Let A be a positive injective operator in a Hilbert space View the MathML source, and denote by View the MathML source the inner product defined by A : [f,g]=〈Af,g〉. A closed subspace S⊂H is called A -compatible if there exists a closed complement for S, which is orthogonal to S with respect to the inner product View the MathML source. Equivalently, if there exists a necessarily unique bounded idempotent operator QS such that R(QS)=S, which is symmetric for this inner product. The compatible Grassmannian GrA is the set of all A -compatible subspaces of H. By parametrizing it via the one to one correspondence S↔QS, this set is shown to be a differentiable submanifold of the Banach space of all bounded operators in H which are symmetric with respect to the form View the MathML source. A Banach–Lie group acts naturally on the compatible Grassmannian, the group of all invertible operators in H which preserve the form View the MathML source. Each connected component in GrA of a compatible subspace S of finite dimension, turns out to be a symplectic leaf in a Banach Lie–Poisson space. For 1⩽p⩽∞, in the presence of a fixed View the MathML source-orthogonal (direct sum) decomposition of H, H=S0+N0, we study the restricted compatible Grassmannian (an analogue of the restricted, or Sato Grassmannian). This restricted compatible Grassmannian is shown to be a submanifold of the Banach space of p -Schatten operators which are symmetric for the form View the MathML source. It carries the locally transitive action of the Banach–Lie group of invertible operators which preserve View the MathML source, and are of the form G=1+K, with K in the p-Schatten class. The connected components of this restricted Grassmannian are characterized by means of the Fredholm index of pairs of projections. Finsler metrics which are isometric for the group actions are introduced for both compatible Grassmannians, and minimality results for curves are proved.Fil: Andruchow, Esteban. Universidad Nacional de General Sarmiento; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática; ArgentinaFil: Chiumiento, Eduardo Hernan. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática; Argentina. Universidad Nacional de La Plata. Facultad de Ciencias Exactas. Departamento de Matematicas; ArgentinaFil: Di Iorio y Lucero, María Eugenia. Universidad Nacional de General Sarmiento; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática; ArgentinaElsevier Science2014-02info:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersionhttp://purl.org/coar/resource_type/c_6501info:ar-repo/semantics/articuloapplication/pdfapplication/pdfapplication/pdfapplication/pdfapplication/pdfapplication/pdfapplication/pdfhttp://hdl.handle.net/11336/12165Andruchow, Esteban; Chiumiento, Eduardo Hernan; Di Iorio y Lucero, María Eugenia; The compatible Grassmannian; Elsevier Science; Differential Geometry And Its Applications; 32; 2-2014; 1-270926-2245enginfo:eu-repo/semantics/altIdentifier/doi/10.1016/j.difgeo.2013.11.004info:eu-repo/semantics/altIdentifier/url/http://www.sciencedirect.com/science/article/pii/S0926224513001101info:eu-repo/semantics/altIdentifier/url/https://arxiv.org/abs/1208.6571info:eu-repo/semantics/openAccesshttps://creativecommons.org/licenses/by-nc-nd/2.5/ar/reponame:CONICET Digital (CONICET)instname:Consejo Nacional de Investigaciones Científicas y Técnicas2025-09-29T09:36:48Zoai:ri.conicet.gov.ar:11336/12165instacron: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 09:36:48.468CONICET Digital (CONICET) - Consejo Nacional de Investigaciones Científicas y Técnicasfalse
dc.title.none.fl_str_mv The compatible Grassmannian
title The compatible Grassmannian
spellingShingle The compatible Grassmannian
Andruchow, Esteban
Projection
Positive operator
Compatible subspace
title_short The compatible Grassmannian
title_full The compatible Grassmannian
title_fullStr The compatible Grassmannian
title_full_unstemmed The compatible Grassmannian
title_sort The compatible Grassmannian
dc.creator.none.fl_str_mv Andruchow, Esteban
Chiumiento, Eduardo Hernan
Di Iorio y Lucero, María Eugenia
author Andruchow, Esteban
author_facet Andruchow, Esteban
Chiumiento, Eduardo Hernan
Di Iorio y Lucero, María Eugenia
author_role author
author2 Chiumiento, Eduardo Hernan
Di Iorio y Lucero, María Eugenia
author2_role author
author
dc.subject.none.fl_str_mv Projection
Positive operator
Compatible subspace
topic Projection
Positive operator
Compatible subspace
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 A be a positive injective operator in a Hilbert space View the MathML source, and denote by View the MathML source the inner product defined by A : [f,g]=〈Af,g〉. A closed subspace S⊂H is called A -compatible if there exists a closed complement for S, which is orthogonal to S with respect to the inner product View the MathML source. Equivalently, if there exists a necessarily unique bounded idempotent operator QS such that R(QS)=S, which is symmetric for this inner product. The compatible Grassmannian GrA is the set of all A -compatible subspaces of H. By parametrizing it via the one to one correspondence S↔QS, this set is shown to be a differentiable submanifold of the Banach space of all bounded operators in H which are symmetric with respect to the form View the MathML source. A Banach–Lie group acts naturally on the compatible Grassmannian, the group of all invertible operators in H which preserve the form View the MathML source. Each connected component in GrA of a compatible subspace S of finite dimension, turns out to be a symplectic leaf in a Banach Lie–Poisson space. For 1⩽p⩽∞, in the presence of a fixed View the MathML source-orthogonal (direct sum) decomposition of H, H=S0+N0, we study the restricted compatible Grassmannian (an analogue of the restricted, or Sato Grassmannian). This restricted compatible Grassmannian is shown to be a submanifold of the Banach space of p -Schatten operators which are symmetric for the form View the MathML source. It carries the locally transitive action of the Banach–Lie group of invertible operators which preserve View the MathML source, and are of the form G=1+K, with K in the p-Schatten class. The connected components of this restricted Grassmannian are characterized by means of the Fredholm index of pairs of projections. Finsler metrics which are isometric for the group actions are introduced for both compatible Grassmannians, and minimality results for curves are proved.
Fil: Andruchow, Esteban. Universidad Nacional de General Sarmiento; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática; Argentina
Fil: Chiumiento, Eduardo Hernan. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática; Argentina. Universidad Nacional de La Plata. Facultad de Ciencias Exactas. Departamento de Matematicas; Argentina
Fil: Di Iorio y Lucero, María Eugenia. Universidad Nacional de General Sarmiento; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática; Argentina
description Let A be a positive injective operator in a Hilbert space View the MathML source, and denote by View the MathML source the inner product defined by A : [f,g]=〈Af,g〉. A closed subspace S⊂H is called A -compatible if there exists a closed complement for S, which is orthogonal to S with respect to the inner product View the MathML source. Equivalently, if there exists a necessarily unique bounded idempotent operator QS such that R(QS)=S, which is symmetric for this inner product. The compatible Grassmannian GrA is the set of all A -compatible subspaces of H. By parametrizing it via the one to one correspondence S↔QS, this set is shown to be a differentiable submanifold of the Banach space of all bounded operators in H which are symmetric with respect to the form View the MathML source. A Banach–Lie group acts naturally on the compatible Grassmannian, the group of all invertible operators in H which preserve the form View the MathML source. Each connected component in GrA of a compatible subspace S of finite dimension, turns out to be a symplectic leaf in a Banach Lie–Poisson space. For 1⩽p⩽∞, in the presence of a fixed View the MathML source-orthogonal (direct sum) decomposition of H, H=S0+N0, we study the restricted compatible Grassmannian (an analogue of the restricted, or Sato Grassmannian). This restricted compatible Grassmannian is shown to be a submanifold of the Banach space of p -Schatten operators which are symmetric for the form View the MathML source. It carries the locally transitive action of the Banach–Lie group of invertible operators which preserve View the MathML source, and are of the form G=1+K, with K in the p-Schatten class. The connected components of this restricted Grassmannian are characterized by means of the Fredholm index of pairs of projections. Finsler metrics which are isometric for the group actions are introduced for both compatible Grassmannians, and minimality results for curves are proved.
publishDate 2014
dc.date.none.fl_str_mv 2014-02
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/11336/12165
Andruchow, Esteban; Chiumiento, Eduardo Hernan; Di Iorio y Lucero, María Eugenia; The compatible Grassmannian; Elsevier Science; Differential Geometry And Its Applications; 32; 2-2014; 1-27
0926-2245
url http://hdl.handle.net/11336/12165
identifier_str_mv Andruchow, Esteban; Chiumiento, Eduardo Hernan; Di Iorio y Lucero, María Eugenia; The compatible Grassmannian; Elsevier Science; Differential Geometry And Its Applications; 32; 2-2014; 1-27
0926-2245
dc.language.none.fl_str_mv eng
language eng
dc.relation.none.fl_str_mv info:eu-repo/semantics/altIdentifier/doi/10.1016/j.difgeo.2013.11.004
info:eu-repo/semantics/altIdentifier/url/http://www.sciencedirect.com/science/article/pii/S0926224513001101
info:eu-repo/semantics/altIdentifier/url/https://arxiv.org/abs/1208.6571
dc.rights.none.fl_str_mv info:eu-repo/semantics/openAccess
https://creativecommons.org/licenses/by-nc-nd/2.5/ar/
eu_rights_str_mv openAccess
rights_invalid_str_mv https://creativecommons.org/licenses/by-nc-nd/2.5/ar/
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application/pdf
application/pdf
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dc.publisher.none.fl_str_mv Elsevier Science
publisher.none.fl_str_mv Elsevier Science
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repository.mail.fl_str_mv dasensio@conicet.gov.ar; lcarlino@conicet.gov.ar
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