Abstract
In this paper, we present some inequalities involving unital positive linear maps and extreme eigenvalues of Hermitian matrices. In addition, we discuss how unital positive linear maps can be used to obtain bounds for the extreme eigenvalues of Hermitian matrices.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Audenaert, K.M.R.: Variance bounds with an application to norm bounds for commutators. Linear Algebra Appl. 432, 1126–1143 (2010)
Bhatia, R.: Matrix Analysis. Springer, New York (1997)
Bhatia, R.: Positive Definite Matrices. Princeton University Press, Princeton (2007)
Bhatia, R., Davis, C.: A better bound on the variance. Am. Math. Mon. 107, 353–357 (2000)
Bhatia, R., Sharma, R.: Some inequalities for positive linear maps. Linear Algebra Appl. 436, 1562–1571 (2012)
Bhatia, R., Sharma, R.: Positive linear maps and spreads of matrices. Am. Math. Mon. 121, 619–624 (2014)
Bhatia, R., Sharma, R.: Positive linear maps and spreads of matrices-II. Linear Algebra Appl. 491, 30–40 (2016)
Dadkhah, A., Moslehian, M.S.: Grüss type inequalities for positive linear maps on \(\mathbb{C} ^{*}\)-algebras. Linear Multilinear Algebra 65, 1386–1401 (2017)
Kumar, R., Bhatia, V.: Some inequalities related to eigenvalues. Adv. Oper. Theory 7, 31 (2022). https://doi.org/10.1007/s43036-022-00193-2
Kumar, R., Sharma, R.: Some inequalities involving positive linear maps under certain conditions. Oper. Matrices 13, 843–854 (2019)
Rieffel, M.A.: Standard deviation is a strongly Leibniz seminorm. N. Y. J. Math. 20, 35–56 (2014)
Sharma, R., Thakur, A.: More inequalities for positive linear maps. J. Math. Inequal. 7, 1–9 (2013)
Sharma, R., Kumar, R., Garga, S.: On inequalities involving eigenvalues and traces of Hermitian matrices. Ann. Funct. Anal. 6, 78–90 (2015)
Wolkowicz, H., Styan, G.P.H.: Bounds for eigenvalues using traces. Linear Algebra Appl. 29, 471–506 (1980)
Acknowledgements
The authors would like to thank the anonymous referees for their valuable comments and successions, which improved the quality of the manuscript. This work was supported by the Science and Engineering Research Board, India, Grant no. EEQ/2019/000593. In addition, the authors are grateful to Prof. Rajendra Bhatia for useful discussions and suggestions, and to ISI Delhi for a visit in January 2015 when this work had begun.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by M. S. Moslehian.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Kumar, R., Sharma, R. Some inequalities involving eigenvalues and positive linear maps. Adv. Oper. Theory 8, 42 (2023). https://doi.org/10.1007/s43036-023-00271-z
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s43036-023-00271-z