A geometry for split operators
We study the set X of split operators acting in the Hilbert space H: X = {T ∈ B(H) : N(T) ∩ R(T) = {0} and N(T) + R(T) = H}. Inside X , we consider the set Y: Y = {T ∈ X : N(T) ⊥ R(T)}. Several characterizations of these sets are given. For instance T ∈ X if and only if there exists an oblique proje...
| Autores: | , , |
|---|---|
| Tipo de recurso: | artículo |
| Estado: | Versión publicada |
| Fecha de publicación: | 2013 |
| País: | Argentina |
| Institución: | Consejo Nacional de Investigaciones Científicas y Técnicas |
| Repositorio: | CONICET Digital (CONICET) |
| Idioma: | inglés |
| OAI Identifier: | oai:ri.conicet.gov.ar:11336/1691 |
| Acceso en línea: | http://hdl.handle.net/11336/1691 |
| Access Level: | acceso abierto |
| Palabra clave: | Split Operator Oblique Projection Projections Pseudo-Inverses Group Inverse Operators Ep Operators https://purl.org/becyt/ford/1.1 https://purl.org/becyt/ford/1 |
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A geometry for split operatorsAndruchow, EstebanCorach, GustavoMbekhta, MostafaSplit OperatorOblique ProjectionProjections Pseudo-InversesGroup Inverse OperatorsEp Operatorshttps://purl.org/becyt/ford/1.1https://purl.org/becyt/ford/1We study the set X of split operators acting in the Hilbert space H: X = {T ∈ B(H) : N(T) ∩ R(T) = {0} and N(T) + R(T) = H}. Inside X , we consider the set Y: Y = {T ∈ X : N(T) ⊥ R(T)}. Several characterizations of these sets are given. For instance T ∈ X if and only if there exists an oblique projection Q whose range is N(T) such that T + Q is invertible, if and only if T posseses a commuting (necessarilly unique) pseudo-inverse S (i.e. T S = ST,TST = T and STS = S). Analogous characterizations are given for Y. Two natural maps are considered: q : X → Q := {oblique projections in H}, q(T) = PR(T )//N(T ) and p : Y → P := {orthogonal projections in H}, p(T) = PR(T ), where PR(T )//N(T ) denotes the projection onto R(T) with nullspace N(T), and PR(T ) denotes the orthogonal projection onto R(T). These maps are in general non continuous, subsets of continuity are studied. For the map q these are: similarity orbits, and the subsets Xck ⊂ X of operators with rank k < ∞, and XFk ⊂ X of Fredholm operators with nullity k < ∞. For the map p there are analogous results. We show that the interior of X is XF0 ∪ XF1 , and that Xck and XFk are arc-wise connected differentiable manifolds.Fil: Andruchow, Esteban. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemáticas; Argentina. Universidad Nacional de General Sarmiento. Instituto de Ciencias; ArgentinaFil: Corach, Gustavo. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemáticas; Argentina. Universidad de Buenos Aires. Facultad de Ingeniería; ArgentinaFil: Mbekhta, Mostafa. Unité de Formation et de Recherche de Mathématiques. Université de Lille; Francia;Springer2013-12info:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersionhttp://purl.org/coar/resource_type/c_6501info:ar-repo/semantics/articuloapplication/pdfapplication/pdfhttp://hdl.handle.net/11336/1691Andruchow, Esteban; Corach, Gustavo; Mbekhta, Mostafa; A geometry for split operators; Springer; Integral Equations and Operator Theory; 77; 4; 12-2013; 559-5790378-620X1420-8989enginfo:eu-repo/semantics/altIdentifier/doi/info:eu-repo/semantics/altIdentifier/url/http://link.springer.com/article/10.1007%2Fs00020-013-2086-9info:eu-repo/semantics/altIdentifier/doi/10.1007/s00020-013-2086-9info:eu-repo/semantics/openAccesshttps://creativecommons.org/licenses/by-nc-sa/2.5/ar/reponame:CONICET Digital (CONICET)instname:Consejo Nacional de Investigaciones Científicas y Técnicas2024-05-08T13:56:17Zoai:ri.conicet.gov.ar:11336/1691instacron: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:34982024-05-08 13:56:17.639CONICET Digital (CONICET) - Consejo Nacional de Investigaciones Científicas y Técnicasfalse |
| dc.title.none.fl_str_mv |
A geometry for split operators |
| title |
A geometry for split operators |
| spellingShingle |
A geometry for split operators Andruchow, Esteban Split Operator Oblique Projection Projections Pseudo-Inverses Group Inverse Operators Ep Operators https://purl.org/becyt/ford/1.1 https://purl.org/becyt/ford/1 |
| title_short |
A geometry for split operators |
| title_full |
A geometry for split operators |
| title_fullStr |
A geometry for split operators |
| title_full_unstemmed |
A geometry for split operators |
| title_sort |
A geometry for split operators |
| dc.creator.none.fl_str_mv |
Andruchow, Esteban Corach, Gustavo Mbekhta, Mostafa |
| author |
Andruchow, Esteban |
| author_facet |
Andruchow, Esteban Corach, Gustavo Mbekhta, Mostafa |
| author_role |
author |
| author2 |
Corach, Gustavo Mbekhta, Mostafa |
| author2_role |
author author |
| dc.subject.none.fl_str_mv |
Split Operator Oblique Projection Projections Pseudo-Inverses Group Inverse Operators Ep Operators https://purl.org/becyt/ford/1.1 https://purl.org/becyt/ford/1 |
| topic |
Split Operator Oblique Projection Projections Pseudo-Inverses Group Inverse Operators Ep Operators https://purl.org/becyt/ford/1.1 https://purl.org/becyt/ford/1 |
| description |
We study the set X of split operators acting in the Hilbert space H: X = {T ∈ B(H) : N(T) ∩ R(T) = {0} and N(T) + R(T) = H}. Inside X , we consider the set Y: Y = {T ∈ X : N(T) ⊥ R(T)}. Several characterizations of these sets are given. For instance T ∈ X if and only if there exists an oblique projection Q whose range is N(T) such that T + Q is invertible, if and only if T posseses a commuting (necessarilly unique) pseudo-inverse S (i.e. T S = ST,TST = T and STS = S). Analogous characterizations are given for Y. Two natural maps are considered: q : X → Q := {oblique projections in H}, q(T) = PR(T )//N(T ) and p : Y → P := {orthogonal projections in H}, p(T) = PR(T ), where PR(T )//N(T ) denotes the projection onto R(T) with nullspace N(T), and PR(T ) denotes the orthogonal projection onto R(T). These maps are in general non continuous, subsets of continuity are studied. For the map q these are: similarity orbits, and the subsets Xck ⊂ X of operators with rank k < ∞, and XFk ⊂ X of Fredholm operators with nullity k < ∞. For the map p there are analogous results. We show that the interior of X is XF0 ∪ XF1 , and that Xck and XFk are arc-wise connected differentiable manifolds. |
| publishDate |
2013 |
| dc.date.none.fl_str_mv |
2013-12 |
| 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/11336/1691 Andruchow, Esteban; Corach, Gustavo; Mbekhta, Mostafa; A geometry for split operators; Springer; Integral Equations and Operator Theory; 77; 4; 12-2013; 559-579 0378-620X 1420-8989 |
| url |
http://hdl.handle.net/11336/1691 |
| identifier_str_mv |
Andruchow, Esteban; Corach, Gustavo; Mbekhta, Mostafa; A geometry for split operators; Springer; Integral Equations and Operator Theory; 77; 4; 12-2013; 559-579 0378-620X 1420-8989 |
| dc.language.none.fl_str_mv |
eng |
| language |
eng |
| dc.relation.none.fl_str_mv |
info:eu-repo/semantics/altIdentifier/doi/ info:eu-repo/semantics/altIdentifier/url/http://link.springer.com/article/10.1007%2Fs00020-013-2086-9 info:eu-repo/semantics/altIdentifier/doi/10.1007/s00020-013-2086-9 |
| dc.rights.none.fl_str_mv |
info:eu-repo/semantics/openAccess https://creativecommons.org/licenses/by-nc-sa/2.5/ar/ |
| eu_rights_str_mv |
openAccess |
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https://creativecommons.org/licenses/by-nc-sa/2.5/ar/ |
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application/pdf application/pdf |
| dc.publisher.none.fl_str_mv |
Springer |
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Springer |
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reponame:CONICET Digital (CONICET) instname:Consejo Nacional de Investigaciones Científicas y Técnicas |
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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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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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15.811543 |