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
- Institución
- Universidad Nacional de Río Negro
- OAI Identificador
- oai:rid.unrn.edu.ar:20.500.12049/5360
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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/ |
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application/pdf |
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Elservier |
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Universidad Nacional de Río Negro |
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