Abstract
In this survey, we shall present the characterizations of some distinguished classes of bounded linear operators acting on a complex separable Hilbert space in terms of operator inequalities related to the arithmetic–geometric mean inequality.
Similar content being viewed by others
References
Al-khlyleh, M., Alrimawi, F.: Generalizations of the unitarily invariant norm version of the arithmetic–geometric mean inequality. Adv. Oper. Theory 6, 2 (2021)
Ando, T.: Operators with norm condition. Acta Sci. Math. (Szeged) 33, 169–178 (1972)
Bouldin, R.: The pseudo-inverse of a product. SIAM J. Appl. Math. 24, 489–495 (1973)
Bouraya, C., Seddik, A.: On the characterizations of some distinguished subclasses of Hilbert space operators. Acta Sci. Math. (Szeged) 84, 611–627 (2018)
Conde, C., Moslehian, M.S., Seddik, A.: Operator inequalities related to the Corach–Porta–Recht inequality. Linear Algebra Appl. 436, 3008–3017 (2012)
Corach, G., Porta, R., Recht, L.: An operator inequality. Linear Algebra Appl. 142, 153–158 (1990)
Fujii, J., Fujii, M., Furuta, T., Nakamoto, R.: Norm inequalities equivalent to Heinz inequality. Proc. Am. Math. Soc. 118(3), 827–830 (1993)
Furuta, T., Yanagida, M.: On powers of p-hyponormal operators. Sci. Math. 2, 279–284 (1999)
Halmos, P.R.: A Hilbert Space Problem Book, 2nd edn. Springer, Berlin (1982)
Heinz, E.: Beiträge zur Störungstheorie der Spectralzerlegung. Math. Ann. 123, 415–438 (1951)
Izumino, S.: The product of operators with closed range and an extension of the reverse order law. Tôhoku Math. J. (2) 34(1), 43–52 (1982)
Magajna, B., Petkovsek, M., Turnsek, A.: Linear Algebra Appl. 377, 181–194 (2004)
McIntosh, A.: Heinz inequalities and perturbation of spectral families. Macquarie Mathematical Reports, Macquarie University (1979)
Seddik, A.: Some results related to Corach–Porta–Recht inequality. Proc. Am. Math. Soc. 129, 3009–3015 (2001)
Seddik, A.: On the numerical range and norm of elementary operators. Linear Multilinear Algebra 52, 293–302 (2004)
Seddik, A.: Rank one operators and norm of elementary operators. Linear Algebra Appl. 424, 177–183 (2007)
Seddik, A.: On the injective norm of \(\sum \limits _{i=1}^{n}A_{_{i}}\otimes B_{_{i}}\) and characterization of normaloid operators. Oper. Matrices 2, 67–77 (2008)
Seddik, A.: On the injective norm and characterization of some subclasses of normal operators by inequalities or equalities. J. Math. Anal. Appl. 351, 277–284 (2009)
Seddik, A.: Characterization of the class of unitary operators by operator inequalities. Linear Multilinear Algebra 59, 1069–1074 (2011)
Seddik, A.: Closed operator inequalities and open problems. Math. Inequal. Appl. 14, 147–157 (2011)
Seddik, A.: Moore–Penrose inverse and operator inequalities. Extr. Math. 30, 29–39 (2015)
Seddik, A.: Corrigendum to Moore–Penrose inverse and operator inequalities. Extr. Math. 30, 29–39 (2015). Extr. Math. 32, 209–211 (2017)
Seddik, A.: Selfadjoint operators, normal operators, and characterizations. Oper. Matrices 13, 835–842 (2019)
Stampfli, J.G.: Normality and the numerical range of an operator. Bull. Am. Math. Soc. 72, 1021–1022 (1966)
Williams, J.P.: Finite operators. Proc. Am. Math. Soc. 26, 129–136 (1970)
Acknowledgements
The author is grateful to the reviewers for their careful reading, constructive criticisms, and helpful comments, which significantly improved the final version of the paper.
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
Seddik, A. Operator inequalities related to the arithmetic–geometric mean inequality and characterizations. Adv. Oper. Theory 8, 8 (2023). https://doi.org/10.1007/s43036-022-00234-w
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s43036-022-00234-w