Log in

Operator inequalities related to the arithmetic–geometric mean inequality and characterizations

  • Survey Article
  • Published:
Advances in Operator Theory Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Al-khlyleh, M., Alrimawi, F.: Generalizations of the unitarily invariant norm version of the arithmetic–geometric mean inequality. Adv. Oper. Theory 6, 2 (2021)

    Article  MATH  Google Scholar 

  2. Ando, T.: Operators with norm condition. Acta Sci. Math. (Szeged) 33, 169–178 (1972)

    MATH  Google Scholar 

  3. Bouldin, R.: The pseudo-inverse of a product. SIAM J. Appl. Math. 24, 489–495 (1973)

    Article  MATH  Google Scholar 

  4. Bouraya, C., Seddik, A.: On the characterizations of some distinguished subclasses of Hilbert space operators. Acta Sci. Math. (Szeged) 84, 611–627 (2018)

    Article  MATH  Google Scholar 

  5. Conde, C., Moslehian, M.S., Seddik, A.: Operator inequalities related to the Corach–Porta–Recht inequality. Linear Algebra Appl. 436, 3008–3017 (2012)

    Article  MATH  Google Scholar 

  6. Corach, G., Porta, R., Recht, L.: An operator inequality. Linear Algebra Appl. 142, 153–158 (1990)

    Article  MATH  Google Scholar 

  7. Fujii, J., Fujii, M., Furuta, T., Nakamoto, R.: Norm inequalities equivalent to Heinz inequality. Proc. Am. Math. Soc. 118(3), 827–830 (1993)

    Article  MATH  Google Scholar 

  8. Furuta, T., Yanagida, M.: On powers of p-hyponormal operators. Sci. Math. 2, 279–284 (1999)

    MATH  Google Scholar 

  9. Halmos, P.R.: A Hilbert Space Problem Book, 2nd edn. Springer, Berlin (1982)

    Book  MATH  Google Scholar 

  10. Heinz, E.: Beiträge zur Störungstheorie der Spectralzerlegung. Math. Ann. 123, 415–438 (1951)

    Article  MATH  Google Scholar 

  11. 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)

    Article  MATH  Google Scholar 

  12. Magajna, B., Petkovsek, M., Turnsek, A.: Linear Algebra Appl. 377, 181–194 (2004)

    Article  Google Scholar 

  13. McIntosh, A.: Heinz inequalities and perturbation of spectral families. Macquarie Mathematical Reports, Macquarie University (1979)

  14. Seddik, A.: Some results related to Corach–Porta–Recht inequality. Proc. Am. Math. Soc. 129, 3009–3015 (2001)

    Article  MATH  Google Scholar 

  15. Seddik, A.: On the numerical range and norm of elementary operators. Linear Multilinear Algebra 52, 293–302 (2004)

    Article  MATH  Google Scholar 

  16. Seddik, A.: Rank one operators and norm of elementary operators. Linear Algebra Appl. 424, 177–183 (2007)

    Article  MATH  Google Scholar 

  17. 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)

    Article  Google Scholar 

  18. 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)

    Article  MATH  Google Scholar 

  19. Seddik, A.: Characterization of the class of unitary operators by operator inequalities. Linear Multilinear Algebra 59, 1069–1074 (2011)

    Article  MATH  Google Scholar 

  20. Seddik, A.: Closed operator inequalities and open problems. Math. Inequal. Appl. 14, 147–157 (2011)

    MATH  Google Scholar 

  21. Seddik, A.: Moore–Penrose inverse and operator inequalities. Extr. Math. 30, 29–39 (2015)

    MATH  Google Scholar 

  22. Seddik, A.: Corrigendum to Moore–Penrose inverse and operator inequalities. Extr. Math. 30, 29–39 (2015). Extr. Math. 32, 209–211 (2017)

  23. Seddik, A.: Selfadjoint operators, normal operators, and characterizations. Oper. Matrices 13, 835–842 (2019)

    Article  MATH  Google Scholar 

  24. Stampfli, J.G.: Normality and the numerical range of an operator. Bull. Am. Math. Soc. 72, 1021–1022 (1966)

    Article  MATH  Google Scholar 

  25. Williams, J.P.: Finite operators. Proc. Am. Math. Soc. 26, 129–136 (1970)

    Article  MATH  Google Scholar 

Download references

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

Authors

Corresponding author

Correspondence to Ameur Seddik.

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.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

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

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s43036-022-00234-w

Keywords

Mathematics Subject Classification

Navigation