Maximal Norms of Orthogonal Projections and Closed-Range Operators

Autores
Aljawi, Salma; Conde, Cristian Marcelo; Feki, Kais; Furuichi, Shigeru
Año de publicación
2025
Idioma
inglés
Tipo de recurso
artículo
Estado
versión publicada
Descripción
Using the Dixmier angle between two closed subspaces of a complex Hilbert space H,we establish the necessary and sufficient conditions for the operator norm of the sum oftwo orthogonal projections, PW1 and PW2 , onto closed subspaces W1 and W2, to attainits maximum, namely ∥PW1 + PW2∥ = 2. These conditions are expressed in terms of thegeometric relationship and symmetry between the ranges of the projections. We applythese results to orthogonal projections associated with a closed-range operator via itsMoore–Penrose inverse. Additionally, for any bounded operator T with closed range in H,we derive sufficient conditions ensuring ∥TT† + T†T∥ = 2, where T† denotes the Moore–Penrose inverse of T. This work highlights how symmetry between operator ranges andtheir algebraic structure governs norm extremality and extends a recent finite-dimensionalresult to the general Hilbert space setting.
Fil: Aljawi, Salma. Princess Nourah Bint Abdulrahman University; Arabia Saudita
Fil: Conde, Cristian Marcelo. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Area de Matematica (area de Matematica) ; Instituto de Ciencias ; Universidad Nacional de General Sarmiento;
Fil: Feki, Kais. Najran University; Arabia Saudita
Fil: Furuichi, Shigeru. Nihon University; Japón
Materia
orthogonal projections
norm of projection sums
Dixmier angle
closed-range operators
Nivel de accesibilidad
acceso abierto
Condiciones de uso
https://creativecommons.org/licenses/by-nc-sa/2.5/ar/
Repositorio
CONICET Digital (CONICET)
Institución
Consejo Nacional de Investigaciones Científicas y Técnicas
OAI Identificador
oai:ri.conicet.gov.ar:11336/279219
Acceder
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oai_identifier_str oai:ri.conicet.gov.ar:11336/279219
network_acronym_str CONICETDig
repository_id_str 3498
network_name_str CONICET Digital (CONICET)
spelling Maximal Norms of Orthogonal Projections and Closed-Range OperatorsAljawi, SalmaConde, Cristian MarceloFeki, KaisFuruichi, Shigeruorthogonal projectionsnorm of projection sumsDixmier angleclosed-range operatorshttps://purl.org/becyt/ford/1.1https://purl.org/becyt/ford/1Using the Dixmier angle between two closed subspaces of a complex Hilbert space H,we establish the necessary and sufficient conditions for the operator norm of the sum oftwo orthogonal projections, PW1 and PW2 , onto closed subspaces W1 and W2, to attainits maximum, namely ∥PW1 + PW2∥ = 2. These conditions are expressed in terms of thegeometric relationship and symmetry between the ranges of the projections. We applythese results to orthogonal projections associated with a closed-range operator via itsMoore–Penrose inverse. Additionally, for any bounded operator T with closed range in H,we derive sufficient conditions ensuring ∥TT† + T†T∥ = 2, where T† denotes the Moore–Penrose inverse of T. This work highlights how symmetry between operator ranges andtheir algebraic structure governs norm extremality and extends a recent finite-dimensionalresult to the general Hilbert space setting.Fil: Aljawi, Salma. Princess Nourah Bint Abdulrahman University; Arabia SauditaFil: Conde, Cristian Marcelo. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Area de Matematica (area de Matematica) ; Instituto de Ciencias ; Universidad Nacional de General Sarmiento;Fil: Feki, Kais. Najran University; Arabia SauditaFil: Furuichi, Shigeru. Nihon University; JapónMultidisciplinary Digital Publishing Institute2025-07info:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersionhttp://purl.org/coar/resource_type/c_6501info:ar-repo/semantics/articuloapplication/pdfapplication/pdfhttp://hdl.handle.net/11336/279219Aljawi, Salma; Conde, Cristian Marcelo; Feki, Kais; Furuichi, Shigeru; Maximal Norms of Orthogonal Projections and Closed-Range Operators; Multidisciplinary Digital Publishing Institute; Symmetry; 17; 7; 7-2025; 1-162073-8994CONICET DigitalCONICETenginfo:eu-repo/semantics/altIdentifier/url/https://www.mdpi.com/2073-8994/17/7/1157info:eu-repo/semantics/altIdentifier/doi/10.3390/sym17071157info:eu-repo/semantics/openAccesshttps://creativecommons.org/licenses/by-nc-sa/2.5/ar/reponame:CONICET Digital (CONICET)instname:Consejo Nacional de Investigaciones Científicas y Técnicas2026-02-26T10:22:07Zoai:ri.conicet.gov.ar:11336/279219instacron: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:34982026-02-26 10:22:08.143CONICET Digital (CONICET) - Consejo Nacional de Investigaciones Científicas y Técnicasfalse
dc.title.none.fl_str_mv Maximal Norms of Orthogonal Projections and Closed-Range Operators
title Maximal Norms of Orthogonal Projections and Closed-Range Operators
spellingShingle Maximal Norms of Orthogonal Projections and Closed-Range Operators
Aljawi, Salma
orthogonal projections
norm of projection sums
Dixmier angle
closed-range operators
title_short Maximal Norms of Orthogonal Projections and Closed-Range Operators
title_full Maximal Norms of Orthogonal Projections and Closed-Range Operators
title_fullStr Maximal Norms of Orthogonal Projections and Closed-Range Operators
title_full_unstemmed Maximal Norms of Orthogonal Projections and Closed-Range Operators
title_sort Maximal Norms of Orthogonal Projections and Closed-Range Operators
dc.creator.none.fl_str_mv Aljawi, Salma
Conde, Cristian Marcelo
Feki, Kais
Furuichi, Shigeru
author Aljawi, Salma
author_facet Aljawi, Salma
Conde, Cristian Marcelo
Feki, Kais
Furuichi, Shigeru
author_role author
author2 Conde, Cristian Marcelo
Feki, Kais
Furuichi, Shigeru
author2_role author
author
author
dc.subject.none.fl_str_mv orthogonal projections
norm of projection sums
Dixmier angle
closed-range operators
topic orthogonal projections
norm of projection sums
Dixmier angle
closed-range operators
purl_subject.fl_str_mv https://purl.org/becyt/ford/1.1
https://purl.org/becyt/ford/1
dc.description.none.fl_txt_mv Using the Dixmier angle between two closed subspaces of a complex Hilbert space H,we establish the necessary and sufficient conditions for the operator norm of the sum oftwo orthogonal projections, PW1 and PW2 , onto closed subspaces W1 and W2, to attainits maximum, namely ∥PW1 + PW2∥ = 2. These conditions are expressed in terms of thegeometric relationship and symmetry between the ranges of the projections. We applythese results to orthogonal projections associated with a closed-range operator via itsMoore–Penrose inverse. Additionally, for any bounded operator T with closed range in H,we derive sufficient conditions ensuring ∥TT† + T†T∥ = 2, where T† denotes the Moore–Penrose inverse of T. This work highlights how symmetry between operator ranges andtheir algebraic structure governs norm extremality and extends a recent finite-dimensionalresult to the general Hilbert space setting.
Fil: Aljawi, Salma. Princess Nourah Bint Abdulrahman University; Arabia Saudita
Fil: Conde, Cristian Marcelo. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Area de Matematica (area de Matematica) ; Instituto de Ciencias ; Universidad Nacional de General Sarmiento;
Fil: Feki, Kais. Najran University; Arabia Saudita
Fil: Furuichi, Shigeru. Nihon University; Japón
description Using the Dixmier angle between two closed subspaces of a complex Hilbert space H,we establish the necessary and sufficient conditions for the operator norm of the sum oftwo orthogonal projections, PW1 and PW2 , onto closed subspaces W1 and W2, to attainits maximum, namely ∥PW1 + PW2∥ = 2. These conditions are expressed in terms of thegeometric relationship and symmetry between the ranges of the projections. We applythese results to orthogonal projections associated with a closed-range operator via itsMoore–Penrose inverse. Additionally, for any bounded operator T with closed range in H,we derive sufficient conditions ensuring ∥TT† + T†T∥ = 2, where T† denotes the Moore–Penrose inverse of T. This work highlights how symmetry between operator ranges andtheir algebraic structure governs norm extremality and extends a recent finite-dimensionalresult to the general Hilbert space setting.
publishDate 2025
dc.date.none.fl_str_mv 2025-07
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/11336/279219
Aljawi, Salma; Conde, Cristian Marcelo; Feki, Kais; Furuichi, Shigeru; Maximal Norms of Orthogonal Projections and Closed-Range Operators; Multidisciplinary Digital Publishing Institute; Symmetry; 17; 7; 7-2025; 1-16
2073-8994
CONICET Digital
CONICET
url http://hdl.handle.net/11336/279219
identifier_str_mv Aljawi, Salma; Conde, Cristian Marcelo; Feki, Kais; Furuichi, Shigeru; Maximal Norms of Orthogonal Projections and Closed-Range Operators; Multidisciplinary Digital Publishing Institute; Symmetry; 17; 7; 7-2025; 1-16
2073-8994
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.mdpi.com/2073-8994/17/7/1157
info:eu-repo/semantics/altIdentifier/doi/10.3390/sym17071157
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
rights_invalid_str_mv https://creativecommons.org/licenses/by-nc-sa/2.5/ar/
dc.format.none.fl_str_mv application/pdf
application/pdf
dc.publisher.none.fl_str_mv Multidisciplinary Digital Publishing Institute
publisher.none.fl_str_mv Multidisciplinary Digital Publishing Institute
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 13.176822

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