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
- Institución
- Consejo Nacional de Investigaciones Científicas y Técnicas
- OAI Identificador
- oai:ri.conicet.gov.ar:11336/12165
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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/ |
dc.format.none.fl_str_mv |
application/pdf application/pdf application/pdf application/pdf application/pdf application/pdf application/pdf |
dc.publisher.none.fl_str_mv |
Elsevier Science |
publisher.none.fl_str_mv |
Elsevier Science |
dc.source.none.fl_str_mv |
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Consejo Nacional de Investigaciones Científicas y Técnicas |
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CONICET Digital (CONICET) - Consejo Nacional de Investigaciones Científicas y Técnicas |
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dasensio@conicet.gov.ar; lcarlino@conicet.gov.ar |
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