Hopf-Rinow theorem in the Sato Grassmannian
- Autores
- Andruchow, Esteban; Larotonda, Gabriel Andrés
- Año de publicación
- 2008
- Idioma
- inglés
- Tipo de recurso
- artículo
- Estado
- versión publicada
- Descripción
- Let U2 (H) be the Banach-Lie group of unitary operators in the Hilbert space H which are Hilbert-Schmidt perturbations of the identity 1. In this paper we study the geometry of the unitary orbit{u p u* : u ∈ U2 (H)}, of an infinite projection p in H. This orbit coincides with the connected component of p in the Hilbert-Schmidt restricted Grassmannian Grres (p) (also known in the literature as the Sato Grassmannian) corresponding to the polarization H = p (H) ⊕ p (H)⊥. It is known that the components of Grres (p) are differentiable manifolds. Here we give a simple proof of the fact that Grres0 (p) is a smooth submanifold of the affine Hilbert space p + B2 (H), where B2 (H) denotes the space of Hilbert-Schmidt operators of H. Also we show that Grres0 (p) is a homogeneous reductive space. We introduce a natural metric, which consists in endowing every tangent space with the trace inner product, and consider its Levi-Civita connection. This connection has been considered before, for instance its sectional curvature has been computed. We show that the Levi-Civita connection coincides with a linear connection induced by the reductive structure, a fact which allows for the easy computation of the geodesic curves. We prove that the geodesics of the connection, which are of the form γ (t) = et z p e- t z, for z a p-co-diagonal anti-hermitic element of B2 (H), have minimal length provided that {norm of matrix} z {norm of matrix} ≤ π / 2. Note that the condition is given in terms of the usual operator norm, a fact which implies that there exist minimal geodesics of arbitrary length. Also we show that any two points p1, p2 ∈ Grres0 (p) are joined by a minimal geodesic. If moreover {norm of matrix} p1 - p2 {norm of matrix} < 1, the minimal geodesic is unique. Finally, we replace the 2-norm by the k-Schatten norm (k > 2), and prove that the geodesics are also minimal for these norms, up to a critical value of t, which is estimated also in terms of the usual operator norm. In the process, minimality results in the k-norms are also obtained for the group U2 (H).
Fil: Andruchow, Esteban. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática Alberto Calderón; Argentina. Universidad Nacional de General Sarmiento; Argentina
Fil: Larotonda, Gabriel Andrés. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática Alberto Calderón; Argentina - Materia
-
HILBERT-SCHMIDT OPERATORS
INFINITE PROJECTIONS
SATO GRASSMANNIAN - 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:11336/93027
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Hopf-Rinow theorem in the Sato GrassmannianAndruchow, EstebanLarotonda, Gabriel AndrésHILBERT-SCHMIDT OPERATORSINFINITE PROJECTIONSSATO GRASSMANNIANhttps://purl.org/becyt/ford/1.1https://purl.org/becyt/ford/1Let U2 (H) be the Banach-Lie group of unitary operators in the Hilbert space H which are Hilbert-Schmidt perturbations of the identity 1. In this paper we study the geometry of the unitary orbit{u p u* : u ∈ U2 (H)}, of an infinite projection p in H. This orbit coincides with the connected component of p in the Hilbert-Schmidt restricted Grassmannian Grres (p) (also known in the literature as the Sato Grassmannian) corresponding to the polarization H = p (H) ⊕ p (H)⊥. It is known that the components of Grres (p) are differentiable manifolds. Here we give a simple proof of the fact that Grres0 (p) is a smooth submanifold of the affine Hilbert space p + B2 (H), where B2 (H) denotes the space of Hilbert-Schmidt operators of H. Also we show that Grres0 (p) is a homogeneous reductive space. We introduce a natural metric, which consists in endowing every tangent space with the trace inner product, and consider its Levi-Civita connection. This connection has been considered before, for instance its sectional curvature has been computed. We show that the Levi-Civita connection coincides with a linear connection induced by the reductive structure, a fact which allows for the easy computation of the geodesic curves. We prove that the geodesics of the connection, which are of the form γ (t) = et z p e- t z, for z a p-co-diagonal anti-hermitic element of B2 (H), have minimal length provided that {norm of matrix} z {norm of matrix} ≤ π / 2. Note that the condition is given in terms of the usual operator norm, a fact which implies that there exist minimal geodesics of arbitrary length. Also we show that any two points p1, p2 ∈ Grres0 (p) are joined by a minimal geodesic. If moreover {norm of matrix} p1 - p2 {norm of matrix} < 1, the minimal geodesic is unique. Finally, we replace the 2-norm by the k-Schatten norm (k > 2), and prove that the geodesics are also minimal for these norms, up to a critical value of t, which is estimated also in terms of the usual operator norm. In the process, minimality results in the k-norms are also obtained for the group U2 (H).Fil: Andruchow, Esteban. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática Alberto Calderón; Argentina. Universidad Nacional de General Sarmiento; ArgentinaFil: Larotonda, Gabriel Andrés. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática Alberto Calderón; ArgentinaAcademic Press Inc Elsevier Science2008-10info:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersionhttp://purl.org/coar/resource_type/c_6501info:ar-repo/semantics/articuloapplication/pdfapplication/pdfapplication/pdfapplication/pdfhttp://hdl.handle.net/11336/93027Andruchow, Esteban; Larotonda, Gabriel Andrés; Hopf-Rinow theorem in the Sato Grassmannian; Academic Press Inc Elsevier Science; Journal of Functional Analysis; 255; 7; 10-2008; 1692-17120022-1236CONICET DigitalCONICETenginfo:eu-repo/semantics/altIdentifier/url/https://www.sciencedirect.com/science/article/pii/S0022123608003005info:eu-repo/semantics/altIdentifier/arxiv/https://arxiv.org/abs/0808.2525info:eu-repo/semantics/altIdentifier/doi/10.1016/j.jfa.2008.07.027info:eu-repo/semantics/openAccesshttps://creativecommons.org/licenses/by-nc-nd/2.5/ar/reponame:CONICET Digital (CONICET)instname:Consejo Nacional de Investigaciones Científicas y Técnicas2025-09-03T10:02:08Zoai:ri.conicet.gov.ar:11336/93027instacron: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:34982025-09-03 10:02:09.192CONICET Digital (CONICET) - Consejo Nacional de Investigaciones Científicas y Técnicasfalse |
dc.title.none.fl_str_mv |
Hopf-Rinow theorem in the Sato Grassmannian |
title |
Hopf-Rinow theorem in the Sato Grassmannian |
spellingShingle |
Hopf-Rinow theorem in the Sato Grassmannian Andruchow, Esteban HILBERT-SCHMIDT OPERATORS INFINITE PROJECTIONS SATO GRASSMANNIAN |
title_short |
Hopf-Rinow theorem in the Sato Grassmannian |
title_full |
Hopf-Rinow theorem in the Sato Grassmannian |
title_fullStr |
Hopf-Rinow theorem in the Sato Grassmannian |
title_full_unstemmed |
Hopf-Rinow theorem in the Sato Grassmannian |
title_sort |
Hopf-Rinow theorem in the Sato Grassmannian |
dc.creator.none.fl_str_mv |
Andruchow, Esteban Larotonda, Gabriel Andrés |
author |
Andruchow, Esteban |
author_facet |
Andruchow, Esteban Larotonda, Gabriel Andrés |
author_role |
author |
author2 |
Larotonda, Gabriel Andrés |
author2_role |
author |
dc.subject.none.fl_str_mv |
HILBERT-SCHMIDT OPERATORS INFINITE PROJECTIONS SATO GRASSMANNIAN |
topic |
HILBERT-SCHMIDT OPERATORS INFINITE PROJECTIONS SATO GRASSMANNIAN |
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 U2 (H) be the Banach-Lie group of unitary operators in the Hilbert space H which are Hilbert-Schmidt perturbations of the identity 1. In this paper we study the geometry of the unitary orbit{u p u* : u ∈ U2 (H)}, of an infinite projection p in H. This orbit coincides with the connected component of p in the Hilbert-Schmidt restricted Grassmannian Grres (p) (also known in the literature as the Sato Grassmannian) corresponding to the polarization H = p (H) ⊕ p (H)⊥. It is known that the components of Grres (p) are differentiable manifolds. Here we give a simple proof of the fact that Grres0 (p) is a smooth submanifold of the affine Hilbert space p + B2 (H), where B2 (H) denotes the space of Hilbert-Schmidt operators of H. Also we show that Grres0 (p) is a homogeneous reductive space. We introduce a natural metric, which consists in endowing every tangent space with the trace inner product, and consider its Levi-Civita connection. This connection has been considered before, for instance its sectional curvature has been computed. We show that the Levi-Civita connection coincides with a linear connection induced by the reductive structure, a fact which allows for the easy computation of the geodesic curves. We prove that the geodesics of the connection, which are of the form γ (t) = et z p e- t z, for z a p-co-diagonal anti-hermitic element of B2 (H), have minimal length provided that {norm of matrix} z {norm of matrix} ≤ π / 2. Note that the condition is given in terms of the usual operator norm, a fact which implies that there exist minimal geodesics of arbitrary length. Also we show that any two points p1, p2 ∈ Grres0 (p) are joined by a minimal geodesic. If moreover {norm of matrix} p1 - p2 {norm of matrix} < 1, the minimal geodesic is unique. Finally, we replace the 2-norm by the k-Schatten norm (k > 2), and prove that the geodesics are also minimal for these norms, up to a critical value of t, which is estimated also in terms of the usual operator norm. In the process, minimality results in the k-norms are also obtained for the group U2 (H). Fil: Andruchow, Esteban. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática Alberto Calderón; Argentina. Universidad Nacional de General Sarmiento; Argentina Fil: Larotonda, Gabriel Andrés. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática Alberto Calderón; Argentina |
description |
Let U2 (H) be the Banach-Lie group of unitary operators in the Hilbert space H which are Hilbert-Schmidt perturbations of the identity 1. In this paper we study the geometry of the unitary orbit{u p u* : u ∈ U2 (H)}, of an infinite projection p in H. This orbit coincides with the connected component of p in the Hilbert-Schmidt restricted Grassmannian Grres (p) (also known in the literature as the Sato Grassmannian) corresponding to the polarization H = p (H) ⊕ p (H)⊥. It is known that the components of Grres (p) are differentiable manifolds. Here we give a simple proof of the fact that Grres0 (p) is a smooth submanifold of the affine Hilbert space p + B2 (H), where B2 (H) denotes the space of Hilbert-Schmidt operators of H. Also we show that Grres0 (p) is a homogeneous reductive space. We introduce a natural metric, which consists in endowing every tangent space with the trace inner product, and consider its Levi-Civita connection. This connection has been considered before, for instance its sectional curvature has been computed. We show that the Levi-Civita connection coincides with a linear connection induced by the reductive structure, a fact which allows for the easy computation of the geodesic curves. We prove that the geodesics of the connection, which are of the form γ (t) = et z p e- t z, for z a p-co-diagonal anti-hermitic element of B2 (H), have minimal length provided that {norm of matrix} z {norm of matrix} ≤ π / 2. Note that the condition is given in terms of the usual operator norm, a fact which implies that there exist minimal geodesics of arbitrary length. Also we show that any two points p1, p2 ∈ Grres0 (p) are joined by a minimal geodesic. If moreover {norm of matrix} p1 - p2 {norm of matrix} < 1, the minimal geodesic is unique. Finally, we replace the 2-norm by the k-Schatten norm (k > 2), and prove that the geodesics are also minimal for these norms, up to a critical value of t, which is estimated also in terms of the usual operator norm. In the process, minimality results in the k-norms are also obtained for the group U2 (H). |
publishDate |
2008 |
dc.date.none.fl_str_mv |
2008-10 |
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/11336/93027 Andruchow, Esteban; Larotonda, Gabriel Andrés; Hopf-Rinow theorem in the Sato Grassmannian; Academic Press Inc Elsevier Science; Journal of Functional Analysis; 255; 7; 10-2008; 1692-1712 0022-1236 CONICET Digital CONICET |
url |
http://hdl.handle.net/11336/93027 |
identifier_str_mv |
Andruchow, Esteban; Larotonda, Gabriel Andrés; Hopf-Rinow theorem in the Sato Grassmannian; Academic Press Inc Elsevier Science; Journal of Functional Analysis; 255; 7; 10-2008; 1692-1712 0022-1236 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/S0022123608003005 info:eu-repo/semantics/altIdentifier/arxiv/https://arxiv.org/abs/0808.2525 info:eu-repo/semantics/altIdentifier/doi/10.1016/j.jfa.2008.07.027 |
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 |
dc.publisher.none.fl_str_mv |
Academic Press Inc Elsevier Science |
publisher.none.fl_str_mv |
Academic Press Inc Elsevier Science |
dc.source.none.fl_str_mv |
reponame:CONICET Digital (CONICET) instname:Consejo Nacional de Investigaciones Científicas y Técnicas |
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CONICET Digital (CONICET) |
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CONICET Digital (CONICET) |
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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 |
repository.mail.fl_str_mv |
dasensio@conicet.gov.ar; lcarlino@conicet.gov.ar |
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1842269740451495936 |
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13.13397 |