Skip to main content
SpringerLink
Log in

Invariant Subspaces of Idempotents on Hilbert Spaces

  • Published:
Integral Equations and Operator Theory Aims and scope Submit manuscript

Abstract

In the setting of operators on Hilbert spaces, we prove that every quasinilpotent operator has a non-trivial closed invariant subspace if and only if every pair of idempotents with a quasinilpotent commutator has a non-trivial common closed invariant subspace. We also present a geometric characterization of invariant subspaces of idempotents and classify operators that are essentially idempotent.

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

Data Availability Statemet

Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.

References

  1. Allan, G., Zemánek, J.: Invariant subspaces for pairs of projections. J. London Math. Soc. 57, 449–468 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  2. Ando, T.: Unbounded or bounded idempotent operators in Hilbert space. Linear Algebra Appl. 438, 3769–3775 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  3. Apostol, C., Voiculescu, D.: On a problem of Halmos. Rev. Roumaine Math. Pures Appl. 19, 283–284 (1974)

    MathSciNet  MATH  Google Scholar 

  4. Barraa, M., Boumazgour, M.: Spectra of the difference, sum and product of idempotents. Studia Math. 148, 1–3 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bernik, J., Radjavi, H.: Invariant and almost-invariant subspaces for pairs of idempotents. Integral Equs. Operat. Theory 84, 283–288 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  6. Davis, C.: Generators of the ring of bounded operators. Proc. Am. Math. Soc. 6, 970–972 (1955)

    Article  MathSciNet  MATH  Google Scholar 

  7. Foiaş, C., Pearcy, C.: A model for quasinilpotent operators. Michigan Math. J. 21, 399–404 (1974)

    MathSciNet  MATH  Google Scholar 

  8. Gohberg, I., Goldberg, S., Kaashoek, M.A.: Classes of linear operators. Vol. I, Operator Theory: Advances and Applications, 63, Birkhäuser Verlag, Basel (1993)

  9. Hartwig, R., Putcha, M.: When is a matrix a difference of two idempotents? Linear and Multilinear Algebra 26, 267–277 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  10. Herrero, D.: Almost every quasinilpotent Hilbert space operator is a universal quasinilpotent. Proc. Am. Math. Soc. 13, 212–216 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  11. Nordgren, E., Radjavi, H., Rosenthal, P.: A geometric equivalent of the invariant subspace problem. Proc. Am. Math. Soc. 61, 66–68 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  12. Nordgren, E., Radjabalipour, M., Radjavi, H., Rosenthal, P.: Quadratic operators and invariant subspaces. Studia Math. 88, 263–268 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  13. Olsen, C.: A structure theorem for polynomially compact operators. Am. J. Math. 93, 686–698 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  14. Radjavi, H., Rosenthal, P.: Simultaneous Triangularization. Universitext, Springer-Verlag, New York (2000)

    Book  MATH  Google Scholar 

  15. Radjavi, H., Rosenthal, P.: Invariant Subspaces, 2nd edn. Dover Publications Inc, Mineola, NY (2003)

    MATH  Google Scholar 

  16. Wang, J., Wu, P.: Difference and similarity models of two idempotent operators. Linear Algebra Appl. 208(209), 257–282 (1994)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

We are grateful to the referee for a careful reading of the paper and for useful suggestions. The research of first author is supported by the Theoretical Statistics and Mathematics Unit, Indian Statistical Institute, Bangalore, India, and the research of the second author is supported by the NBHM post-doctoral fellowship, Department of Atomic Energy (DAE), Government of India (File No: 0204/16(20)/2020/R &D-II/10). The third named author is supported in part by the Core Research Grant (CRG/2019/000908), by SERB (DST), Government of India.

Author information

Authors and Affiliations

  1. Statistics and Mathematics Unit, Indian Statistical Institute, 8th Mile, Mysore Road, Bangalore, 560 059, India

    Neeru Bala, Nirupam Ghosh & Jaydeb Sarkar

Authors
  1. Neeru Bala
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Nirupam Ghosh
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jaydeb Sarkar
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jaydeb Sarkar.

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 (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

Bala, N., Ghosh, N. & Sarkar, J. Invariant Subspaces of Idempotents on Hilbert Spaces. Integr. Equ. Oper. Theory 95, 4 (2023). https://doi.org/10.1007/s00020-022-02723-2

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00020-022-02723-2

Keywords

Mathematics Subject Classification

Search

Navigation