Skip to main content
SpringerLink
Log in

a-Weyl’s theorem and hypercyclicity

  • Published:
Monatshefte für Mathematik Aims and scope Submit manuscript

Abstract

Let H be a complex infinite dimensional Hilbert space, B(H) be the algebra of all bounded linear operators acting on H, and \(\overline{HC(H)}\) \((\overline{SC(H)})\) be the norm closure of the class of all hypercyclic operators (supercyclic operators) in B(H). An operator \(T\in B(H)\) is said to be with hypercyclicity (supercyclicity) if T is in \(\overline{HC(H)}\) \((\overline{SC(H)})\). Using a new spectrum defined from “consistent in invertibility”, this paper gives necessary and sufficient conditions that T is with a-Browder’s theorem or with a-Weyl’s theorem. Further, this paper gives a necessary and sufficient condition that T is a-isoloid, with a-Weyl’s theorem and with hypercyclicity (supercyclicity) concurrently. Also, the relations between that T is with hypercyclicity (supercyclicity) and that T is both with a-Weyl’s theorem and a-isoloid are discussed by means of the new spectrum.

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

Data availability is not applicable to this article as no new data were created or analyzed in this work.

References

  1. Aiena, P.: Fredholm and Local Spectral Theory II: With Application to Weyl-Type Theorems. Lecture Notes of Mathematics, vol. 2235. Springer, Cham (2018)

    Book  Google Scholar 

  2. Aiena, P., Aponte, E., Balzan, E.: Weyl type theorems for left and right polaroid operators. Integr. Equ. Oper. Theory 66(1), 1–20 (2010). https://doi.org/10.1007/s00020-009-1738-2

    Article  MathSciNet  Google Scholar 

  3. Aiena, P., Guillén, J.R., Peña, P.: Property (\(R\)) for bounded linear operators. Mediterr. J. Math. 8(4), 491–508 (2011). https://doi.org/10.1007/s00009-011-0113-0

    Article  MathSciNet  Google Scholar 

  4. Aiena, P., Triolo, S.: Weyl-type theorems on Banach spaces under compact perturbations. Mediterr. J. Math. 15(3), 1–18 (2018). https://doi.org/10.1007/s00009-018-1176-y

    Article  MathSciNet  Google Scholar 

  5. Berkani, M., Kachad, M.: New Weyl-type theorems I. Funct. Anal. Approx. Comput. 4(2), 41–47 (2012)

    MathSciNet  Google Scholar 

  6. Berkani, M., Zariouh, H.: New extended Weyl type theorems. Mat. Vest. 62(240), 145–154 (2010)

    MathSciNet  Google Scholar 

  7. Cao, X.H.: Weyl type theorem and hypercyclic operators. J. Math. Anal. Appl. 323(1), 267–274 (2006). https://doi.org/10.1016/j.jmaa.2005.10.041

    Article  MathSciNet  Google Scholar 

  8. Cao, X.H., Liu, A.F.: Generalized Kato type operators and property (\(\omega \)) under perturbations. Linear Algebra Appl. 436(7), 2231–2239 (2012). https://doi.org/10.1016/j.laa.2011.10.027

    Article  MathSciNet  Google Scholar 

  9. Cao, X.H., Zhang, H.J., Zhang, Y.H.: Consistent invertibility and Weyl’s theorem. J. Math. Anal. Appl. 369(1), 258–264 (2010). https://doi.org/10.1016/j.jmaa.2010.03.023

    Article  MathSciNet  Google Scholar 

  10. Coburn, L.A.: Weyl’s theorem for nonnormal operators. Mich. Math. J. 13(3), 285–288 (1966). https://doi.org/10.1307/mmj/1031732778

    Article  MathSciNet  Google Scholar 

  11. Conway, J.B.: A Course in Functional Analysis, 2nd edn. Springer, Berlin (2007)

    Book  Google Scholar 

  12. Dai, L., Cao, X.H., Guo, Q.: Property (\(\omega \)) and the single-valued extension property. Acta. Math. Sin. 37(8), 1254–1266 (2021). https://doi.org/10.1007/s10114-021-0436-0

    Article  MathSciNet  Google Scholar 

  13. Djordjević, S.V., Han, Y.M.: Browder’s theorems and spectral continuity. Glasgow Math. J. 42(3), 479–486 (2000). https://doi.org/10.1017/s0017089500030147

    Article  MathSciNet  Google Scholar 

  14. Duggal, B.P.: Weyl’s theorem and hypercyclic/supercyclic operators. J. Math. Anal. Appl. 335(2), 990–995 (2007). https://doi.org/10.1016/j.jmaa.2007.01.086

    Article  MathSciNet  Google Scholar 

  15. Gong, W.B., Han, D.G.: Spectrum of the products of operators and compact perturbations. Proc. Am. Math. Soc. 120(3), 755–760 (1994). https://doi.org/10.1090/s0002-9939-1994-1197538-6

    Article  MathSciNet  Google Scholar 

  16. Han, Y.M., Djordjević, S.V.: A note on a-Weyl’s theorem. J. Math. Anal. Appl. 260, 200–213 (2001). https://doi.org/10.1006/jmaa.2001.7448

    Article  MathSciNet  Google Scholar 

  17. Harte, R., Lee, W.: Another note on Weyl’s theorem. Trans. Am. Math. Soc. 349(5), 2115–2124 (1997)

    Article  MathSciNet  Google Scholar 

  18. Herrero, D.A.: Limits of hypercyclic and supercyclic operators. J. Funct. Anal. 99(1), 179–190 (1991). https://doi.org/10.1016/0022-1236(91)90058-d

    Article  MathSciNet  Google Scholar 

  19. Mecheri, S.: On a new class of operators and Weyl type theorems. Filomat 27(4), 629–636 (2013). https://doi.org/10.2298/fil1304629m

    Article  MathSciNet  Google Scholar 

  20. Mecheri, S., Seddik, M.: Weyl Type theorems for posinormal operators. Math. Proc. R. Irish. Acad. 108A(1), 69–79 (2008). https://doi.org/10.3318/pria.2008.108.1.69

    Article  MathSciNet  Google Scholar 

  21. Müller, V.: Spectral Theory of Linear Operators and Spectral Systems in Banach Algebras. Springer, New York (2007)

    Google Scholar 

  22. Rakočević, V.: Operators obeying \(\alpha \)-Weyl’s theorem. Rev. Roum. Math. Pures 34(10), 915–919 (1989). https://doi.org/10.5486/pmd.1999.2051

    Article  MathSciNet  Google Scholar 

  23. Rashid, M.H.M.: Weyl’s type theorems and hypercyclic operators. Acta Math. Sci. 32(2), 539–551 (2012). https://doi.org/10.1016/S0252-9602(12)60036-8

    Article  MathSciNet  Google Scholar 

  24. Ren, Y.X., Jiang, L.N., Kong, Y.Y.: Property (WE) and topological uniform descent. Bull. Belg. Math. Soc. 29(1), 1–17 (2022). https://doi.org/10.1007/s11464-014-0373-7

    Article  MathSciNet  Google Scholar 

  25. Sanabria, J., Vasquez, L., Carpintero, C., et al.: On strong variations of Weyl type theorems. Acta Math. Univ. Comen. 86(2), 345–356 (2017)

    MathSciNet  Google Scholar 

  26. Shen, J., Chen, A.: A new variation of Weyl type theorems and perturbations. Filomat 28(9), 1899–1906 (2014). https://doi.org/10.2298/fil1409899s

    Article  MathSciNet  Google Scholar 

  27. Weyl, H.: Über beschränkte quadratische Formen, deren Differenz vollstetig ist. Rend. Circ. Mat. Palerm. 27, 373–392 (1909). https://doi.org/10.1007/bf03019655

    Article  Google Scholar 

Download references

Acknowledgements

The authors thank Lining Jiang for his help on revising the paper. The anonymous reviewer provided helpful and constructive comments that improved the manuscript substantially.

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Beijing Institute of Technology, Beijing, People’s Republic of China

    Ying Liu

  2. School of Mathematics and Statistics, Shaanxi Normal University, Xi’an, People’s Republic of China

    Xiaohong Cao

Authors
  1. Ying Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Xiaohong Cao
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

All authors contributed to the study conception and design. The first draft of the manuscript was written by [YL]. [XC] conceived of the study, and participated in its design and coordination. All authors read and approved the final manuscript.

Corresponding author

Correspondence to Ying Liu.

Ethics declarations

Conflict of interest

This work has not accepted any financial support from any individual, funding or institution. The authors declared no potential conflicts of interest with respect to the research, author-ship, or publication of this article.

Additional information

Communicated by Gerald Teschl.

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

Liu, Y., Cao, X. a-Weyl’s theorem and hypercyclicity. Monatsh Math 204, 107–125 (2024). https://doi.org/10.1007/s00605-024-01951-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00605-024-01951-5

Keywords

Mathematics Subject Classification

Search

Navigation