The group of L^2 - isometries on H_0^1

Autores
Andruchow Colombo, Ana; Chiumiento, Eduardo Hernan; Larotonda, Gabriel Andrés
Año de publicación
2013
Idioma
inglés
Tipo de recurso
artículo
Estado
versión publicada
Descripción
Let be an open subset of Rn. Let L2 = L2( ; dx) and H1 0 = H1 0 ( ) be the standard Lebesgue and Sobolev spaces of complex-valued functions. The aim of this paper is to study the group G of invertible operators on H1 0 which preserve the L2-inner product. When is bounded and @ is smooth, this group acts as the intertwiner of the H1 0 solutions of the non-homogeneous Helmholtz equation u u = f, uj@ = 0. We show that G is a real Banach{Lie group, whose Lie algebra is (i times) the space of symmetrizable operators. We discuss the spectrum of operators belonging to G by means of examples. In particular, we give an example of an operator in G whose spectrum is not contained in the unit circle. We also study the one-parameter subgroups of G. Curves of minimal length in G are considered. We introduce the subgroups Gp := G(I Bp(H1 0 )), where Bp(H1 0 ) is the Schatten ideal of operators on H1 0 . An invariant (weak) Finsler metric is dened by the p-norm of the Schatten ideal of operators on L2. We prove that any pair of operators G1;G2 2 Gp can be joined by a minimal curve of the form (t) = G1eitX , where X is a symmetrizable operator in Bp(H1 0 ).
Fil: Andruchow Colombo, Ana. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemáticas; Argentina
Fil: Chiumiento, Eduardo Hernan. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática; Argentina
Fil: Larotonda, Gabriel Andrés. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática; Argentina
Materia
Banach Lie Group
Sobolev Space
Symmetrizable Operator
One Parameter Subgroup
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:11336/3274
Acceder
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oai_identifier_str oai:ri.conicet.gov.ar:11336/3274
network_acronym_str CONICETDig
repository_id_str 3498
network_name_str CONICET Digital (CONICET)
spelling The group of L^2 - isometries on H_0^1Andruchow Colombo, AnaChiumiento, Eduardo HernanLarotonda, Gabriel AndrésBanach Lie GroupSobolev SpaceSymmetrizable OperatorOne Parameter Subgrouphttps://purl.org/becyt/ford/1.1https://purl.org/becyt/ford/1Let be an open subset of Rn. Let L2 = L2( ; dx) and H1 0 = H1 0 ( ) be the standard Lebesgue and Sobolev spaces of complex-valued functions. The aim of this paper is to study the group G of invertible operators on H1 0 which preserve the L2-inner product. When is bounded and @ is smooth, this group acts as the intertwiner of the H1 0 solutions of the non-homogeneous Helmholtz equation u u = f, uj@ = 0. We show that G is a real Banach{Lie group, whose Lie algebra is (i times) the space of symmetrizable operators. We discuss the spectrum of operators belonging to G by means of examples. In particular, we give an example of an operator in G whose spectrum is not contained in the unit circle. We also study the one-parameter subgroups of G. Curves of minimal length in G are considered. We introduce the subgroups Gp := G(I Bp(H1 0 )), where Bp(H1 0 ) is the Schatten ideal of operators on H1 0 . An invariant (weak) Finsler metric is dened by the p-norm of the Schatten ideal of operators on L2. We prove that any pair of operators G1;G2 2 Gp can be joined by a minimal curve of the form (t) = G1eitX , where X is a symmetrizable operator in Bp(H1 0 ).Fil: Andruchow Colombo, Ana. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemáticas; ArgentinaFil: Chiumiento, Eduardo Hernan. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática; ArgentinaFil: Larotonda, Gabriel Andrés. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática; ArgentinaPolish Acad Sciences Inst Mathematics2013-10info:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersionhttp://purl.org/coar/resource_type/c_6501info:ar-repo/semantics/articuloapplication/pdfapplication/octet-streamapplication/pdfhttp://hdl.handle.net/11336/3274Andruchow Colombo, Ana; Chiumiento, Eduardo Hernan; Larotonda, Gabriel Andrés; The group of L^2 - isometries on H_0^1; Polish Acad Sciences Inst Mathematics; Studia Mathematica; 217; 3; 10-2013; 193-2170039-3223enginfo:eu-repo/semantics/altIdentifier/url/https://www.impan.pl/en/publishing-house/journals-and-series/studia-mathematica/all/217/3info:eu-repo/semantics/openAccesshttps://creativecommons.org/licenses/by-nc-nd/2.5/ar/reponame:CONICET Digital (CONICET)instname:Consejo Nacional de Investigaciones Científicas y Técnicas2025-10-22T11:12:45Zoai:ri.conicet.gov.ar:11336/3274instacron: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:34982025-10-22 11:12:45.44CONICET Digital (CONICET) - Consejo Nacional de Investigaciones Científicas y Técnicasfalse
dc.title.none.fl_str_mv The group of L^2 - isometries on H_0^1
title The group of L^2 - isometries on H_0^1
spellingShingle The group of L^2 - isometries on H_0^1
Andruchow Colombo, Ana
Banach Lie Group
Sobolev Space
Symmetrizable Operator
One Parameter Subgroup
title_short The group of L^2 - isometries on H_0^1
title_full The group of L^2 - isometries on H_0^1
title_fullStr The group of L^2 - isometries on H_0^1
title_full_unstemmed The group of L^2 - isometries on H_0^1
title_sort The group of L^2 - isometries on H_0^1
dc.creator.none.fl_str_mv Andruchow Colombo, Ana
Chiumiento, Eduardo Hernan
Larotonda, Gabriel Andrés
author Andruchow Colombo, Ana
author_facet Andruchow Colombo, Ana
Chiumiento, Eduardo Hernan
Larotonda, Gabriel Andrés
author_role author
author2 Chiumiento, Eduardo Hernan
Larotonda, Gabriel Andrés
author2_role author
author
dc.subject.none.fl_str_mv Banach Lie Group
Sobolev Space
Symmetrizable Operator
One Parameter Subgroup
topic Banach Lie Group
Sobolev Space
Symmetrizable Operator
One Parameter Subgroup
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 be an open subset of Rn. Let L2 = L2( ; dx) and H1 0 = H1 0 ( ) be the standard Lebesgue and Sobolev spaces of complex-valued functions. The aim of this paper is to study the group G of invertible operators on H1 0 which preserve the L2-inner product. When is bounded and @ is smooth, this group acts as the intertwiner of the H1 0 solutions of the non-homogeneous Helmholtz equation u u = f, uj@ = 0. We show that G is a real Banach{Lie group, whose Lie algebra is (i times) the space of symmetrizable operators. We discuss the spectrum of operators belonging to G by means of examples. In particular, we give an example of an operator in G whose spectrum is not contained in the unit circle. We also study the one-parameter subgroups of G. Curves of minimal length in G are considered. We introduce the subgroups Gp := G(I Bp(H1 0 )), where Bp(H1 0 ) is the Schatten ideal of operators on H1 0 . An invariant (weak) Finsler metric is dened by the p-norm of the Schatten ideal of operators on L2. We prove that any pair of operators G1;G2 2 Gp can be joined by a minimal curve of the form (t) = G1eitX , where X is a symmetrizable operator in Bp(H1 0 ).
Fil: Andruchow Colombo, Ana. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemáticas; Argentina
Fil: Chiumiento, Eduardo Hernan. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática; Argentina
Fil: Larotonda, Gabriel Andrés. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática; Argentina
description Let be an open subset of Rn. Let L2 = L2( ; dx) and H1 0 = H1 0 ( ) be the standard Lebesgue and Sobolev spaces of complex-valued functions. The aim of this paper is to study the group G of invertible operators on H1 0 which preserve the L2-inner product. When is bounded and @ is smooth, this group acts as the intertwiner of the H1 0 solutions of the non-homogeneous Helmholtz equation u u = f, uj@ = 0. We show that G is a real Banach{Lie group, whose Lie algebra is (i times) the space of symmetrizable operators. We discuss the spectrum of operators belonging to G by means of examples. In particular, we give an example of an operator in G whose spectrum is not contained in the unit circle. We also study the one-parameter subgroups of G. Curves of minimal length in G are considered. We introduce the subgroups Gp := G(I Bp(H1 0 )), where Bp(H1 0 ) is the Schatten ideal of operators on H1 0 . An invariant (weak) Finsler metric is dened by the p-norm of the Schatten ideal of operators on L2. We prove that any pair of operators G1;G2 2 Gp can be joined by a minimal curve of the form (t) = G1eitX , where X is a symmetrizable operator in Bp(H1 0 ).
publishDate 2013
dc.date.none.fl_str_mv 2013-10
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/11336/3274
Andruchow Colombo, Ana; Chiumiento, Eduardo Hernan; Larotonda, Gabriel Andrés; The group of L^2 - isometries on H_0^1; Polish Acad Sciences Inst Mathematics; Studia Mathematica; 217; 3; 10-2013; 193-217
0039-3223
url http://hdl.handle.net/11336/3274
identifier_str_mv Andruchow Colombo, Ana; Chiumiento, Eduardo Hernan; Larotonda, Gabriel Andrés; The group of L^2 - isometries on H_0^1; Polish Acad Sciences Inst Mathematics; Studia Mathematica; 217; 3; 10-2013; 193-217
0039-3223
dc.language.none.fl_str_mv eng
language eng
dc.relation.none.fl_str_mv info:eu-repo/semantics/altIdentifier/url/https://www.impan.pl/en/publishing-house/journals-and-series/studia-mathematica/all/217/3
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/octet-stream
application/pdf
dc.publisher.none.fl_str_mv Polish Acad Sciences Inst Mathematics
publisher.none.fl_str_mv Polish Acad Sciences Inst Mathematics
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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score 13.229304

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