Inequalities related to Bourin and Heinz means with a complex parameter

Autores
Bottazzi, Tamara Paula; Elencwajg, René; Larotonda, Gabriel Andrés; Varela, Alejandro
Año de publicación
2015
Idioma
inglés
Tipo de recurso
artículo
Estado
versión publicada
Descripción
A conjecture posed by S. Hayajneh and F. Kittaneh claims that given A, B positive matrices, 0≤t≤1, and any unitarily invariant norm the following inequality holds{triple vertical-rule fence}AtB1-t+BtA1-t{triple vertical-rule fence}≤{triple vertical-rule fence}AtB1-t+A1-tBt{triple vertical-rule fence}. Recently, R. Bhatia proved the inequality for the case of the Frobenius norm and for t∈[14,34]. In this paper, using complex methods we extend this result to complex values of the parameter t=z in the strip {z∈C:Re(z)∈[14,34]}. We give an elementary proof of the fact that equality holds for some z in the strip if and only if A and B commute. We also show a counterexample to the general conjecture by exhibiting a pair of positive matrices such that the claim does not hold for the uniform norm. Finally, we give a counterexample for a related singular value inequality given by s(AtB1-t+BtA1-t)≤sj(A+B), answering in the negative a question made by K. Audenaert and F. Kittaneh. The methods of proof and examples can be adapted with no modifications to operator algebras (infinite dimensional setting), for instance it follows that the inequality above holds for Hilbert-Schmidt operators with their Banach algebra norm derived from the infinite trace of B(H).
Fil: Bottazzi, Tamara Paula. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática Alberto Calderón; Argentina
Fil: Elencwajg, René. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática Alberto Calderón; Argentina
Fil: Larotonda, Gabriel Andrés. Universidad Nacional de General Sarmiento; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática Alberto Calderón; Argentina
Fil: Varela, Alejandro. Universidad Nacional de General Sarmiento; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática Alberto Calderón; Argentina
Materia
COMPLEX METHODS
FROBENIUS NORM
HEINZ MEAN
NORM INEQUALITY
TRACIAL ALGEBRA
UNITARILY INVARIANT NORM
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:11336/98215
Acceder
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oai_identifier_str oai:ri.conicet.gov.ar:11336/98215
network_acronym_str CONICETDig
repository_id_str 3498
network_name_str CONICET Digital (CONICET)
spelling Inequalities related to Bourin and Heinz means with a complex parameterBottazzi, Tamara PaulaElencwajg, RenéLarotonda, Gabriel AndrésVarela, AlejandroCOMPLEX METHODSFROBENIUS NORMHEINZ MEANNORM INEQUALITYTRACIAL ALGEBRAUNITARILY INVARIANT NORMhttps://purl.org/becyt/ford/1.1https://purl.org/becyt/ford/1A conjecture posed by S. Hayajneh and F. Kittaneh claims that given A, B positive matrices, 0≤t≤1, and any unitarily invariant norm the following inequality holds{triple vertical-rule fence}AtB1-t+BtA1-t{triple vertical-rule fence}≤{triple vertical-rule fence}AtB1-t+A1-tBt{triple vertical-rule fence}. Recently, R. Bhatia proved the inequality for the case of the Frobenius norm and for t∈[14,34]. In this paper, using complex methods we extend this result to complex values of the parameter t=z in the strip {z∈C:Re(z)∈[14,34]}. We give an elementary proof of the fact that equality holds for some z in the strip if and only if A and B commute. We also show a counterexample to the general conjecture by exhibiting a pair of positive matrices such that the claim does not hold for the uniform norm. Finally, we give a counterexample for a related singular value inequality given by s(AtB1-t+BtA1-t)≤sj(A+B), answering in the negative a question made by K. Audenaert and F. Kittaneh. The methods of proof and examples can be adapted with no modifications to operator algebras (infinite dimensional setting), for instance it follows that the inequality above holds for Hilbert-Schmidt operators with their Banach algebra norm derived from the infinite trace of B(H).Fil: Bottazzi, Tamara Paula. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática Alberto Calderón; ArgentinaFil: Elencwajg, René. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática Alberto Calderón; ArgentinaFil: Larotonda, Gabriel Andrés. Universidad Nacional de General Sarmiento; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática Alberto Calderón; ArgentinaFil: Varela, Alejandro. Universidad Nacional de General Sarmiento; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática Alberto Calderón; ArgentinaAcademic Press Inc Elsevier Science2015-06-15info: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/11336/98215Bottazzi, Tamara Paula; Elencwajg, René; Larotonda, Gabriel Andrés; Varela, Alejandro; Inequalities related to Bourin and Heinz means with a complex parameter; Academic Press Inc Elsevier Science; Journal of Mathematical Analysis and Applications; 426; 2; 15-6-2015; 765-7730022-247X1096-0813CONICET DigitalCONICETenginfo:eu-repo/semantics/altIdentifier/url/http://www.sciencedirect.com/science/article/pii/S0022247X15000657#info:eu-repo/semantics/altIdentifier/doi/10.1016/j.jmaa.2015.01.046info:eu-repo/semantics/altIdentifier/arxiv/https://arxiv.org/abs/1403.7472info:eu-repo/semantics/openAccesshttps://creativecommons.org/licenses/by-nc-nd/2.5/ar/reponame:CONICET Digital (CONICET)instname:Consejo Nacional de Investigaciones Científicas y Técnicas2025-09-29T10:09:49Zoai:ri.conicet.gov.ar:11336/98215instacron: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 10:09:49.984CONICET Digital (CONICET) - Consejo Nacional de Investigaciones Científicas y Técnicasfalse
dc.title.none.fl_str_mv Inequalities related to Bourin and Heinz means with a complex parameter
title Inequalities related to Bourin and Heinz means with a complex parameter
spellingShingle Inequalities related to Bourin and Heinz means with a complex parameter
Bottazzi, Tamara Paula
COMPLEX METHODS
FROBENIUS NORM
HEINZ MEAN
NORM INEQUALITY
TRACIAL ALGEBRA
UNITARILY INVARIANT NORM
title_short Inequalities related to Bourin and Heinz means with a complex parameter
title_full Inequalities related to Bourin and Heinz means with a complex parameter
title_fullStr Inequalities related to Bourin and Heinz means with a complex parameter
title_full_unstemmed Inequalities related to Bourin and Heinz means with a complex parameter
title_sort Inequalities related to Bourin and Heinz means with a complex parameter
dc.creator.none.fl_str_mv Bottazzi, Tamara Paula
Elencwajg, René
Larotonda, Gabriel Andrés
Varela, Alejandro
author Bottazzi, Tamara Paula
author_facet Bottazzi, Tamara Paula
Elencwajg, René
Larotonda, Gabriel Andrés
Varela, Alejandro
author_role author
author2 Elencwajg, René
Larotonda, Gabriel Andrés
Varela, Alejandro
author2_role author
author
author
dc.subject.none.fl_str_mv COMPLEX METHODS
FROBENIUS NORM
HEINZ MEAN
NORM INEQUALITY
TRACIAL ALGEBRA
UNITARILY INVARIANT NORM
topic COMPLEX METHODS
FROBENIUS NORM
HEINZ MEAN
NORM INEQUALITY
TRACIAL ALGEBRA
UNITARILY INVARIANT NORM
purl_subject.fl_str_mv https://purl.org/becyt/ford/1.1
https://purl.org/becyt/ford/1
dc.description.none.fl_txt_mv A conjecture posed by S. Hayajneh and F. Kittaneh claims that given A, B positive matrices, 0≤t≤1, and any unitarily invariant norm the following inequality holds{triple vertical-rule fence}AtB1-t+BtA1-t{triple vertical-rule fence}≤{triple vertical-rule fence}AtB1-t+A1-tBt{triple vertical-rule fence}. Recently, R. Bhatia proved the inequality for the case of the Frobenius norm and for t∈[14,34]. In this paper, using complex methods we extend this result to complex values of the parameter t=z in the strip {z∈C:Re(z)∈[14,34]}. We give an elementary proof of the fact that equality holds for some z in the strip if and only if A and B commute. We also show a counterexample to the general conjecture by exhibiting a pair of positive matrices such that the claim does not hold for the uniform norm. Finally, we give a counterexample for a related singular value inequality given by s(AtB1-t+BtA1-t)≤sj(A+B), answering in the negative a question made by K. Audenaert and F. Kittaneh. The methods of proof and examples can be adapted with no modifications to operator algebras (infinite dimensional setting), for instance it follows that the inequality above holds for Hilbert-Schmidt operators with their Banach algebra norm derived from the infinite trace of B(H).
Fil: Bottazzi, Tamara Paula. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática Alberto Calderón; Argentina
Fil: Elencwajg, René. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática Alberto Calderón; Argentina
Fil: Larotonda, Gabriel Andrés. Universidad Nacional de General Sarmiento; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática Alberto Calderón; Argentina
Fil: Varela, Alejandro. Universidad Nacional de General Sarmiento; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática Alberto Calderón; Argentina
description A conjecture posed by S. Hayajneh and F. Kittaneh claims that given A, B positive matrices, 0≤t≤1, and any unitarily invariant norm the following inequality holds{triple vertical-rule fence}AtB1-t+BtA1-t{triple vertical-rule fence}≤{triple vertical-rule fence}AtB1-t+A1-tBt{triple vertical-rule fence}. Recently, R. Bhatia proved the inequality for the case of the Frobenius norm and for t∈[14,34]. In this paper, using complex methods we extend this result to complex values of the parameter t=z in the strip {z∈C:Re(z)∈[14,34]}. We give an elementary proof of the fact that equality holds for some z in the strip if and only if A and B commute. We also show a counterexample to the general conjecture by exhibiting a pair of positive matrices such that the claim does not hold for the uniform norm. Finally, we give a counterexample for a related singular value inequality given by s(AtB1-t+BtA1-t)≤sj(A+B), answering in the negative a question made by K. Audenaert and F. Kittaneh. The methods of proof and examples can be adapted with no modifications to operator algebras (infinite dimensional setting), for instance it follows that the inequality above holds for Hilbert-Schmidt operators with their Banach algebra norm derived from the infinite trace of B(H).
publishDate 2015
dc.date.none.fl_str_mv 2015-06-15
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/11336/98215
Bottazzi, Tamara Paula; Elencwajg, René; Larotonda, Gabriel Andrés; Varela, Alejandro; Inequalities related to Bourin and Heinz means with a complex parameter; Academic Press Inc Elsevier Science; Journal of Mathematical Analysis and Applications; 426; 2; 15-6-2015; 765-773
0022-247X
1096-0813
CONICET Digital
CONICET
url http://hdl.handle.net/11336/98215
identifier_str_mv Bottazzi, Tamara Paula; Elencwajg, René; Larotonda, Gabriel Andrés; Varela, Alejandro; Inequalities related to Bourin and Heinz means with a complex parameter; Academic Press Inc Elsevier Science; Journal of Mathematical Analysis and Applications; 426; 2; 15-6-2015; 765-773
0022-247X
1096-0813
CONICET Digital
CONICET
dc.language.none.fl_str_mv eng
language eng
dc.relation.none.fl_str_mv info:eu-repo/semantics/altIdentifier/url/http://www.sciencedirect.com/science/article/pii/S0022247X15000657#
info:eu-repo/semantics/altIdentifier/doi/10.1016/j.jmaa.2015.01.046
info:eu-repo/semantics/altIdentifier/arxiv/https://arxiv.org/abs/1403.7472
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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