Skip to main content
SpringerLink
Log in

On compressions and generalized spectra of operators over \(C^{*}\)-algebras

  • Original Paper
  • Published:
Annals of Functional Analysis Aims and scope Submit manuscript

Abstract

In the classical operator theory, there are several versions of spectra, related to special classes of operators (Fredholm, semi-Fredholm, upper/lower semi-Fredholm, etc.). We generalize these notions for adjointable operators on Hilbert \(C^*\)-modules replacing scalars by the center of the algebra, and show that most relations between these spectra are still true for these generalized versions. The relation between these spectra of an operator and those of its compressions is also transferred to the case of Hilbert \(C^*\)-modules.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Aghasizadeh, T., Hejazian, S.: Maps preserving semi-Fredholm operators on Hilbert C*-modules. J. Math. Anal. Appl. 354, 625–629 (2009)

    Article  MathSciNet  Google Scholar 

  2. Fredholm, E.I.: Sur une classe d’equations fontionnelles. Acta Math. 27, 365–390 (1903)

    Article  MathSciNet  Google Scholar 

  3. Ivković, S.: Semi-Fredholm theory on Hilbert C*-modules. Banach J. Math. Anal. 13(4), 989–1016 (2019)

    Article  MathSciNet  Google Scholar 

  4. Lazović, Z.: Compact and “compact” operators on standard Hilbert modules over C*-algebras. Adv. Oper. Theory 3(4), 829–836 (2018)

    Article  MathSciNet  Google Scholar 

  5. Manuilov, V. M., Troitsky, E. V.: Hilbert C*-modules. In: Translations of Mathematical Monographs. 226, American Mathematical Society, Providence, RI (2005)

  6. Miličić, D., Veselić, K.: On the boundary of essential spectra. Glasnik Mat. Ser. III 6 (26) No 1, 73–78 (1971)

  7. Mishchenko, A.S., Fomenko, A.T.: The index of eliptic operators over C*-algebras. Izv. Akad. Nauk SSSR Ser. Mat. 43, 831–859 (1979). English transl., Math. USSR-Izv. 15, 87–112 (1980)

  8. Moslehian, M.S., Sharifi, K., Forough, M., Chakoshi, M.: Moore-Penrose inverses of Gram operator on Hilbert C*-modules. Stud. Math. 210, 189–196 (2012)

    Article  MathSciNet  Google Scholar 

  9. Pokrzywa, A.: A characterizations of the Weyl spectum. Proc. Am. Math. Soc. 92, 215–218 (1984)

    Article  MathSciNet  Google Scholar 

  10. Sharifi, K.: The product of operators with closed range in Hilbert C*-modules. Linear Algebra Appl. 435(5), 1122–1130 (2011)

    Article  MathSciNet  Google Scholar 

  11. Wegge-Olsen, N.E.: K-theory and C*-algebras. Oxford Univ. Press, Oxford (1993)

    MATH  Google Scholar 

  12. Zemanek, J.: Compressions and the Weyl–Browder spectra. Proc. R. Irish Acad. Sec. A 86, 57–62 (1986)

    MathSciNet  MATH  Google Scholar 

  13. Zlatanović, S., Živković, Rakočević, V., Djordjević, D.S.: Fredholm theory. University of Niš, Faculty of Sciences and Mathematics, Niš (2019)

Download references

Acknowledgements

I am especially grateful to my supervisor Professor Vladimir M. Manuilov for careful reading of my paper and for inspiring comments that led to the improved presentation of the paper. Also I am grateful to Professor Dragan S. Djordjevic for suggesting the research topic of this paper and for introducing to me the relevant reference books. Finally, I am grateful to the Referee for the comments and suggestions that led to the improved presentation of the paper.

Author information

Authors and Affiliations

  1. The Mathematical Institute of the Serbian Academy of Sciences and Arts, p.p. 367, Kneza Mihaila 36, Belgrade, 11000, Serbia

    Stefan Ivković

Authors
  1. Stefan Ivković
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Stefan Ivković.

Additional information

Communicated by Baruch Solel.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ivković, S. On compressions and generalized spectra of operators over \(C^{*}\)-algebras. Ann. Funct. Anal. 11, 505–522 (2020). https://doi.org/10.1007/s43034-019-00034-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s43034-019-00034-z

Keywords

Mathematics Subject Classification

Search

Navigation