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
CONICET Digital (CONICET)
Institución
Consejo Nacional de Investigaciones Científicas y Técnicas
OAI Identificador
oai:ri.conicet.gov.ar:11336/93027

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spelling 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
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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