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
- Institución
- Consejo Nacional de Investigaciones Científicas y Técnicas
- OAI Identificador
- oai:ri.conicet.gov.ar:11336/3274
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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 Técnicas2025-09-29T09:44:01Zoai: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:34982025-09-29 09:44:01.406CONICET 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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13.070432 |