Inequalities related to Bourin and Heinz means with a complex parameter

Autores
Bottazzi, Tamara Paula; Elencwajg, Rene; Larotonda, Gabriel; Varela, Alejandro
Año de publicación
2015
Idioma
inglés
Tipo de recurso
artículo
Estado
versión aceptada
Descripción
Fil: Bottazzi, Tamara P. Instituto Argentino de Matemática “Alberto P. Calderón”. Buenos Aires, Argentina.
Fil: Elencwajg, Rene. Instituto Argentino de Matemática “Alberto P. Calderón”. Buenos Aires, Argentina.
Fil: Larotonda, Gabriel. Instituto Argentino de Matemática “Alberto P. Calderón”. Buenos Aires, Argentina.
Fil: Varela, Alejandro. Instituto Argentino de Matemática “Alberto P. Calderón”. Buenos Aires, Argentina.
Fil: Larotonda, Gabriel. Universidad Nacional de General Sarmiento. Instituto de Ciencias. Buenos Aires, Argentina.
Fil: Varela, Alejandro. Universidad Nacional de General Sarmiento. Instituto de Ciencias. Buenos Aires, Argentina.
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⦀AtB1−t+BtA1−t⦀≤⦀AtB1−t+A1−tBt⦀. 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 sj(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).
Una conjetura hecha por Hayajneh y Kittaneh afirma que dados A y B matrices positivas, 0<=t<=1 y cualquier norma unitariamente invariante vale la siguiente desigualdad ⦀AtB1−t+BtA1−t⦀≤⦀AtB1−t+A1−tBt⦀. Recientemente, Bhatia probó la conjetura para el caso de la norma de Frobenius y t∈[1/4,3/4]. En este artículo, usando métodos compeljos, extenderemos este resultado a variables complejas t=z en la franja {z∈C:Re(z)∈[1/4,3/4]}. Damos una prueba elemantal de que la igualdad vale para algún z en la franja sii A y B conmutan. También mostramos un contraejemplo de la conjetura más general para la norma uniforme. Finalmente, también mostramos un contraejemplo para la desigualdad de valores singulares relacionada sj(AtB1−t+BtA1−t)≤sj(A+B), dando respuesta negativa a una pregunta hecha por K. Audenaert and F. Kittaneh.
Materia
Frobenius Norm
Heinz Mean
Norm Inequality
Complex Methods
Unitarily Invariant Norm
Tracial Algebra
Nivel de accesibilidad
acceso abierto
Condiciones de uso
https://creativecommons.org/licenses/by-nc-sa/4.0/
Repositorio
RID-UNRN (UNRN)
Institución
Universidad Nacional de Río Negro
OAI Identificador
oai:rid.unrn.edu.ar:20.500.12049/5360

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repository_id_str 4369
network_name_str RID-UNRN (UNRN)
spelling Inequalities related to Bourin and Heinz means with a complex parameterBottazzi, Tamara PaulaElencwajg, ReneLarotonda, GabrielVarela, AlejandroFrobenius NormHeinz MeanNorm InequalityComplex MethodsUnitarily Invariant NormTracial AlgebraFil: Bottazzi, Tamara P. Instituto Argentino de Matemática “Alberto P. Calderón”. Buenos Aires, Argentina.Fil: Elencwajg, Rene. Instituto Argentino de Matemática “Alberto P. Calderón”. Buenos Aires, Argentina.Fil: Larotonda, Gabriel. Instituto Argentino de Matemática “Alberto P. Calderón”. Buenos Aires, Argentina.Fil: Varela, Alejandro. Instituto Argentino de Matemática “Alberto P. Calderón”. Buenos Aires, Argentina.Fil: Larotonda, Gabriel. Universidad Nacional de General Sarmiento. Instituto de Ciencias. Buenos Aires, Argentina.Fil: Varela, Alejandro. Universidad Nacional de General Sarmiento. Instituto de Ciencias. Buenos Aires, Argentina.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⦀AtB1−t+BtA1−t⦀≤⦀AtB1−t+A1−tBt⦀. 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 sj(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).Una conjetura hecha por Hayajneh y Kittaneh afirma que dados A y B matrices positivas, 0<=t<=1 y cualquier norma unitariamente invariante vale la siguiente desigualdad ⦀AtB1−t+BtA1−t⦀≤⦀AtB1−t+A1−tBt⦀. Recientemente, Bhatia probó la conjetura para el caso de la norma de Frobenius y t∈[1/4,3/4]. En este artículo, usando métodos compeljos, extenderemos este resultado a variables complejas t=z en la franja {z∈C:Re(z)∈[1/4,3/4]}. Damos una prueba elemantal de que la igualdad vale para algún z en la franja sii A y B conmutan. También mostramos un contraejemplo de la conjetura más general para la norma uniforme. Finalmente, también mostramos un contraejemplo para la desigualdad de valores singulares relacionada sj(AtB1−t+BtA1−t)≤sj(A+B), dando respuesta negativa a una pregunta hecha por K. Audenaert and F. Kittaneh.Elservier2015info:eu-repo/semantics/articleinfo:eu-repo/semantics/acceptedVersionhttp://purl.org/coar/resource_type/c_6501info:ar-repo/semantics/articuloapplication/pdfBottazzi, Tamara P., Elencwajg, Rene., Larotonda, Gabriel y Varela, Alejandro (2015). Inequalities related to Bourin and Heinz means with a complex parameter. Elservier; Journal of Mathematical Analysis and its Applications; 426 (2); 765-7730022-247Xhttps://www.sciencedirect.com/science/article/pii/S0022247X15000657?via%3Dihubhttps://rid.unrn.edu.ar/jspui/handle/20.500.12049/5360https://doi.org/10.1016/j.jmaa.2015.01.046eng426 (2)Journal of Mathematical Analysis and its Applicationsinfo:eu-repo/semantics/openAccesshttps://creativecommons.org/licenses/by-nc-sa/4.0/reponame:RID-UNRN (UNRN)instname:Universidad Nacional de Río Negro2025-09-29T14:29:11Zoai:rid.unrn.edu.ar:20.500.12049/5360instacron:UNRNInstitucionalhttps://rid.unrn.edu.ar/jspui/Universidad públicaNo correspondehttps://rid.unrn.edu.ar/oai/snrdrid@unrn.edu.arArgentinaNo correspondeNo correspondeNo correspondeopendoar:43692025-09-29 14:29:11.542RID-UNRN (UNRN) - Universidad Nacional de Río Negrofalse
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
Frobenius Norm
Heinz Mean
Norm Inequality
Complex Methods
Unitarily Invariant Norm
Tracial Algebra
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, Rene
Larotonda, Gabriel
Varela, Alejandro
author Bottazzi, Tamara Paula
author_facet Bottazzi, Tamara Paula
Elencwajg, Rene
Larotonda, Gabriel
Varela, Alejandro
author_role author
author2 Elencwajg, Rene
Larotonda, Gabriel
Varela, Alejandro
author2_role author
author
author
dc.subject.none.fl_str_mv Frobenius Norm
Heinz Mean
Norm Inequality
Complex Methods
Unitarily Invariant Norm
Tracial Algebra
topic Frobenius Norm
Heinz Mean
Norm Inequality
Complex Methods
Unitarily Invariant Norm
Tracial Algebra
dc.description.none.fl_txt_mv Fil: Bottazzi, Tamara P. Instituto Argentino de Matemática “Alberto P. Calderón”. Buenos Aires, Argentina.
Fil: Elencwajg, Rene. Instituto Argentino de Matemática “Alberto P. Calderón”. Buenos Aires, Argentina.
Fil: Larotonda, Gabriel. Instituto Argentino de Matemática “Alberto P. Calderón”. Buenos Aires, Argentina.
Fil: Varela, Alejandro. Instituto Argentino de Matemática “Alberto P. Calderón”. Buenos Aires, Argentina.
Fil: Larotonda, Gabriel. Universidad Nacional de General Sarmiento. Instituto de Ciencias. Buenos Aires, Argentina.
Fil: Varela, Alejandro. Universidad Nacional de General Sarmiento. Instituto de Ciencias. Buenos Aires, Argentina.
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⦀AtB1−t+BtA1−t⦀≤⦀AtB1−t+A1−tBt⦀. 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 sj(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).
Una conjetura hecha por Hayajneh y Kittaneh afirma que dados A y B matrices positivas, 0<=t<=1 y cualquier norma unitariamente invariante vale la siguiente desigualdad ⦀AtB1−t+BtA1−t⦀≤⦀AtB1−t+A1−tBt⦀. Recientemente, Bhatia probó la conjetura para el caso de la norma de Frobenius y t∈[1/4,3/4]. En este artículo, usando métodos compeljos, extenderemos este resultado a variables complejas t=z en la franja {z∈C:Re(z)∈[1/4,3/4]}. Damos una prueba elemantal de que la igualdad vale para algún z en la franja sii A y B conmutan. También mostramos un contraejemplo de la conjetura más general para la norma uniforme. Finalmente, también mostramos un contraejemplo para la desigualdad de valores singulares relacionada sj(AtB1−t+BtA1−t)≤sj(A+B), dando respuesta negativa a una pregunta hecha por K. Audenaert and F. Kittaneh.
description Fil: Bottazzi, Tamara P. Instituto Argentino de Matemática “Alberto P. Calderón”. Buenos Aires, Argentina.
publishDate 2015
dc.date.none.fl_str_mv 2015
dc.type.none.fl_str_mv info:eu-repo/semantics/article
info:eu-repo/semantics/acceptedVersion
http://purl.org/coar/resource_type/c_6501
info:ar-repo/semantics/articulo
format article
status_str acceptedVersion
dc.identifier.none.fl_str_mv Bottazzi, Tamara P., Elencwajg, Rene., Larotonda, Gabriel y Varela, Alejandro (2015). Inequalities related to Bourin and Heinz means with a complex parameter. Elservier; Journal of Mathematical Analysis and its Applications; 426 (2); 765-773
0022-247X
https://www.sciencedirect.com/science/article/pii/S0022247X15000657?via%3Dihub
https://rid.unrn.edu.ar/jspui/handle/20.500.12049/5360
https://doi.org/10.1016/j.jmaa.2015.01.046
identifier_str_mv Bottazzi, Tamara P., Elencwajg, Rene., Larotonda, Gabriel y Varela, Alejandro (2015). Inequalities related to Bourin and Heinz means with a complex parameter. Elservier; Journal of Mathematical Analysis and its Applications; 426 (2); 765-773
0022-247X
url https://www.sciencedirect.com/science/article/pii/S0022247X15000657?via%3Dihub
https://rid.unrn.edu.ar/jspui/handle/20.500.12049/5360
https://doi.org/10.1016/j.jmaa.2015.01.046
dc.language.none.fl_str_mv eng
language eng
dc.relation.none.fl_str_mv 426 (2)
Journal of Mathematical Analysis and its Applications
dc.rights.none.fl_str_mv info:eu-repo/semantics/openAccess
https://creativecommons.org/licenses/by-nc-sa/4.0/
eu_rights_str_mv openAccess
rights_invalid_str_mv https://creativecommons.org/licenses/by-nc-sa/4.0/
dc.format.none.fl_str_mv application/pdf
dc.publisher.none.fl_str_mv Elservier
publisher.none.fl_str_mv Elservier
dc.source.none.fl_str_mv reponame:RID-UNRN (UNRN)
instname:Universidad Nacional de Río Negro
reponame_str RID-UNRN (UNRN)
collection RID-UNRN (UNRN)
instname_str Universidad Nacional de Río Negro
repository.name.fl_str_mv RID-UNRN (UNRN) - Universidad Nacional de Río Negro
repository.mail.fl_str_mv rid@unrn.edu.ar
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