Skip to main content
Log in

On generalized powers of operators

  • Published:
Rendiconti del Circolo Matematico di Palermo Series 2 Aims and scope Submit manuscript

Abstract

In this note, we introduce generalized powers of linear operators. More precisely, operators are not raised to numbers but to other operators. We discuss several properties as regards this notion.

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. Chaban, A., Mortad, M.H.: Exponentials of bounded normal operators. Colloq. Math. 133(2), 237–244 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  2. Conway, J.B.: A Course in Functional Analysis, 2nd edn. Springer (1990)

  3. Dehimi, S., Mortad, M.H.: Right (or left) invertibility of bounded and unbounded operators and applications to the spectrum of products. Complex Anal. Oper. Theory 12(3), 589–597 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  4. Furuta, T.: Invitation to Linear Operators: From Matrices to Bounded Linear Operators on a Hilbert Space. Taylor & Francis Group, London (2001)

    Book  MATH  Google Scholar 

  5. Gustafson, K., Mortad, M.H.: Unbounded products of operators and connections to Dirac-type operators. Bull. Sci. Math. 138(5), 626–642 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  6. Gustafson, K., Mortad, M.H.: Conditions implying commutativity of unbounded self-adjoint operators and related topics. J. Oper. Theory 76(1), 159–169 (2016)

    Article  MathSciNet  MATH  Google Scholar 

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

  8. Jung, I.B., Mortad, M.H., Stochel, J.: On normal products of selfadjoint operators. Kyungpook Math. J. 57(2017), 457–471 (2017)

    MathSciNet  MATH  Google Scholar 

  9. Kubrusly, C.S.: Hilbert Space Operators, A Problem Solving Approach. Birkhäuser. Boston Inc, Boston, MA (2003)

    Book  MATH  Google Scholar 

  10. Kubrusly, C.S.: The Elements of Operator Theory, 2nd edn. Birkhäuser/Springer, New York (2011)

    Book  MATH  Google Scholar 

  11. Kurepa, S.: A note on logarithms of normal operators. Proc. Am. Math. Soc. 13, 307–311 (1962)

    Article  MathSciNet  MATH  Google Scholar 

  12. Meziane, M., Mortad, M.H.: Maximality of linear operators. Rend. Circ. Mat. Palermo Ser. II 68(3), 441–451 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  13. Mortad, M.H.: An application of the Putnam-Fuglede theorem to normal products of self-adjoint operators. Proc. Am. Math. Soc. 131(10), 3135–3141 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  14. Mortad, M.H.: Explicit formulae for the wave operators of perturbed self-adjoint operators. J. Math. Anal. Appl. 356(2), 704–710 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  15. Mortad, M.H.: Exponentials of normal operators and commutativity of operators: a new approach. Colloq. Math. 125(1), 1–6 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  16. Mortad, M.H.: On the normality of the sum of two normal operators. Complex Anal. Oper. Theory 6(1), 105–112 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  17. Mortad, M.H.: Commutativity of unbounded normal and self-adjoint operators and applications. Oper. Matrices 8(2), 563–571 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  18. Mortad, M.H.: A criterion for the normality of unbounded operators and applications to self-adjointness. Rend. Circ. Mat. Palermo (2) 64(1), 149–156 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  19. Mortad, M.H.: An Operator Theory Problem Book. World Scientific Publishing Co. (2018)

  20. Mortad, M.H.: Counterexamples in operator theory. Birkhäuser/Springer, Cham (2022)

    Book  MATH  Google Scholar 

  21. Mortad, M.H. Unbounded operators: (square) roots, nilpotence, closability and some related invertibility results. arXiv:2007.12027

  22. Putnam, C.R.: On square roots and logarithms of self-adjoint operators. Proc. Glasgow Math. Assoc. 4(1958), 1–2 (1958)

    Article  MathSciNet  MATH  Google Scholar 

  23. Reed, M., Simon, B.: Methods of modern mathematical physics, Vol. 1: Functional analysis, Academic Press (1972)

  24. Schmoeger, Ch.: On logarithms of linear operators on Hilbert spaces. Demonstratio Math. 35(2), 375–384 (2002)

    MathSciNet  MATH  Google Scholar 

  25. Schmüdgen, K.: Unbounded self-adjoint operators on Hilbert space, Springer. GTM 265 (2012)

  26. Wermuth, E.M.E.: A remark on commuting operator exponentials. Proc. Am. Math. Soc. 125(6), 1685–1688 (1997)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Mohammed Hichem Mortad.

Ethics declarations

Conflict of interest

The authors declare that there is no conflict of interests regarding the publication of this paper.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor 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

Bachir, A., Mortad, M.H. & Ali Sayyaf, N. On generalized powers of operators. Rend. Circ. Mat. Palermo, II. Ser 72, 2761–2769 (2023). https://doi.org/10.1007/s12215-022-00823-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12215-022-00823-x

Keywords

Mathematics Subject Classification

Navigation