Skip to main content
SpringerLink
Log in

Invariant Subspaces for Commuting Operators on a Real Banach Space

  • Brief Communications
  • Published:
Functional Analysis and Its Applications Aims and scope

Abstract

It is proved that the commutative algebra A of operators on a reflexive real Banach space has an invariant subspace if each operator T ∈ A satisfies the condition

$${\left\| {1 - \varepsilon {T^2}} \right\|_e} \leqslant 1 + o\left( \varepsilon \right)as\varepsilon \searrow 0,$$

where ║ · ║ e denotes the essential norm. This implies the existence of an invariant subspace for any commutative family of essentially self-adjoint operators on a real Hilbert space.

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

References

  1. M. S. Livshits, Mat. Sb., N. Ser., 34(76):1 (1954), 145–199.

    Google Scholar 

  2. L. A. Sakhnovich, Izv. Vyssh. Uchebn. Zaved., Mat., 1959, No. 4, 141–149.

    Google Scholar 

  3. I. Ts. Gokhberg and M. G. Krein, Dokl. Akad. Nauk SSSR, 128:2 (1959), 227–230.

    MathSciNet  Google Scholar 

  4. V. I. Matsaev, Dokl. Akad. Nauk SSSR, 139:4 (1961), 810–814.

    MathSciNet  Google Scholar 

  5. J. Schwartz, Comm. Pure Appl. Math., 15 (1962), 159–172.

    Article  MathSciNet  Google Scholar 

  6. V. I. Matsaev, Dokl. Akad. Nauk SSSR, 139:3 (1961), 548–552.

    MathSciNet  Google Scholar 

  7. V. Lomonosov, Proc. Amer. Math. Soc., 115:3 (1992), 775–777.

    Article  MathSciNet  Google Scholar 

  8. V. Lomonosov, Israel J. Math., 75:2–3 (1991), 329–339.

    Article  MathSciNet  Google Scholar 

  9. A. Simonič, Trans. Amer. Math. Soc., 348:3 (1996), 975–995.

    Article  MathSciNet  Google Scholar 

  10. A. Atzmon, Ann. Inst. Fourier, 51:5 (2001), 1407–1418.

    Article  MathSciNet  Google Scholar 

  11. A. Atzmon, G. Godefroy, and N. J. Kalton, Positivity, 8:2 (2004), 101–107.

    Article  MathSciNet  Google Scholar 

  12. S. Grivaux, Bull. Sci. Math., 126:8 (2002), 681–691.

    Article  MathSciNet  Google Scholar 

  13. V. I. Lomonosov, J. Math. Anal. Appl., 245:1 (2000), 221–224.

    Article  MathSciNet  Google Scholar 

  14. V. I. Lomonosov, Funkts. Anal. Prilozhen., 7:3 (1973), 55–56; English transl.: Functional Anal. Appl., 7:3 (1973), 213–214.

    Google Scholar 

  15. M. Lindström and G. Shlüchtermann, Canad. Math. Bull., 43:1 (2000), 87–89.

    Article  MathSciNet  Google Scholar 

  16. H. Radjavi and P. Rosenthal, Invariant Subspaces, Springer-Verlag, New York–Heidelberg, 1973.

    Book  MATH  Google Scholar 

  17. E. Bishop and R. R. Phelps, in: Proc. Symp. PureMath., vol. 7, Amer. Math. Soc., Providence, RI, 1962, 27–35.

    Article  Google Scholar 

  18. R. R. Phelps, in: Lecture Notes in Pure and Applied Math., vol. 136, 1992, 337–340.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Kent State University, Kent, USA

    V. I. Lomonosov

  2. Department of Higher Mathematics, Vologda State University, Vologda, Russia

    V. S. Shul’man

Authors
  1. V. I. Lomonosov
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. V. S. Shul’man
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to V. I. Lomonosov.

Additional information

Translated from Funktsional’nyi Analiz i Ego Prilozheniya, Vol. 52, No. 1, pp. 65–69, 2018

Original Russian Text Copyright © by V. I. Lomonosov and V. S. Shul’man

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Lomonosov, V.I., Shul’man, V.S. Invariant Subspaces for Commuting Operators on a Real Banach Space. Funct Anal Its Appl 52, 53–56 (2018). https://doi.org/10.1007/s10688-018-0207-6

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10688-018-0207-6

Key words

Search

Navigation