Skip to main content
SpringerLink
Log in

Reflexive operators on analytic function spaces

  • Original Research
  • Published:
Indian Journal of Pure and Applied Mathematics Aims and scope Submit manuscript

Abstract

Following the recent work done in [27], we give various other conditions to ensure that the powers of the multiplication operator \(M_{z}\) are reflexive on a Banach space \({\mathcal {X}}\) of functions analytic on a plane domain. Also, some examples of function spaces satisfying the given conditions are considered.

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. H. Bercovici, C. Foias, J. Langsam, and C. Pearcy, (BCP)- operators are reflexive, Mich. Math. J., 29 (1982), 371-379.

    Article  MathSciNet  Google Scholar 

  2. P. S. Bourdon and J. H. Shapiro, Spectral and common cyclic vectors, Mich. Math. J., 37 (1990), 71-90.

    Article  MathSciNet  Google Scholar 

  3. B. Chevereau, C. Pearcy, and A. Shields, Finitely connected domains \(G\), representations of \(H^{\infty }(G)\), and invariant subspaces, J. Operator Theory, 6 (1981), 375-405.

    MathSciNet  MATH  Google Scholar 

  4. J. A. Deddens, Every isometry is reflexive, Proc. Amer. Math. Soc., 28 (1971), 509-512.

    Article  MathSciNet  Google Scholar 

  5. J. A. Deddens and P. A. Fillmore, Reflexive linear transformations, Linear Algebra Appl., 10 (1975), 89-93.

    Article  MathSciNet  Google Scholar 

  6. Y. N. Dehghan and B. Yousefi, Reflexivity on Hilbert spaces of analytic functions, Int. J. Appl. Math., 20 (6) (2007), 815-819.

    MathSciNet  MATH  Google Scholar 

  7. M. Faghih Ahmadi and K. Hedayatian, On the reflexivity of hyponormal and weighted shift operators, Acta Math. Sci., 30 B(4) (2010), 1100-1104.

  8. Z. Fattahi Vanani, A. I. Kashkooly, and B. Yousefi, Fixed points and compact weighted composition operators on Banach weighted Hardy spaces, Fixed Point Theory, 19 (1) (2018), 421-432.

    Article  MathSciNet  Google Scholar 

  9. T. Gamelin, Uniform Algebra, Chelsea, New York (1984).

    Google Scholar 

  10. Kh. Jahedi, On the multiplication operator on analytic function spaces, Ira. J. Sci. Technol. Trans. A Sci., 37 (4) (2013), 449-452.

    MathSciNet  Google Scholar 

  11. A. I. Kashkooly and Z. Fattahi, Reflexivity on Hilbert function spaces, Int. J. Pure Appl. Math., 111 (2) (2017), 253-262.

    Google Scholar 

  12. J. Lindenstrass and L. Tzafriri, Classical Banach Spaces I, Springer Verlag, New York (1977).

    Book  Google Scholar 

  13. R. Olin and J. Thomson, Algebras of subnormal operators, J. Funct. Anal., 37 (1980), 271-301.

    Article  MathSciNet  Google Scholar 

  14. H. Radjavi and P. Rosenthal, Invariant Subspaces, Springer-Verlag, New York (1971).

    MATH  Google Scholar 

  15. D. Sarason, Invariant subspaces and unstarred operator algebras, Pacific J. Math., 17 (1966), 511-517.

    Article  MathSciNet  Google Scholar 

  16. D. Sarason, Weak star generators of \(H^{\infty }\), Pacific J. Math., 17 (1966), 519-528.

    Article  MathSciNet  Google Scholar 

  17. K. Seddighi and B. Yousefi, On the reflexivity of operators on function spaces, Proc. Amer. Math. Soc., 116 (1992), 45-52.

    Article  MathSciNet  Google Scholar 

  18. A. Shields and L. Wallen, The commutants of certain Hilbert space operators, Indiana Univ. Math. J., 20 (1971), 777-788.

    Article  MathSciNet  Google Scholar 

  19. A. Shields, Weighted shift operators and analytic functions theory, Math. Surveys, A.M.S. Providence, 13 (1974), 49-128.

    MathSciNet  Google Scholar 

  20. B. Yousefi, K. Seddighi and K. Hedayatian, On reflexivity of the multiplication operators on Dirichlet spaces, Math. Japonica, 38 (1993), 1189-1194.

    MathSciNet  MATH  Google Scholar 

  21. B. Yousefi, On the space \(\ell ^{p}(\beta )\), Rend. Circ. Mat. Palermo, 49 (2000), 115-120.

    Article  MathSciNet  Google Scholar 

  22. B. Yousefi, Unicellularity of the multiplication operator on Banach spaces of formal power series, Studia Math., 147 (3) (2001), 201-209.

    Article  MathSciNet  Google Scholar 

  23. B. Yousefi, Bounded analytic structure of the Banach space of formal power series, Rend. Circ. Mat. Palermo, Serie II, Tomo LI (2002), 403-410.

  24. B. Yousefi, Multiplication operators on Hilbert spaces of analytic functions, Arch. der Math., 83 (6) (2004), 536-539.

    Article  MathSciNet  Google Scholar 

  25. B. Yousefi, On the eighteenth question of Allen Shields, Int. J. Math., 16 (1) (2005), 1-6.

    Article  MathSciNet  Google Scholar 

  26. B. Yousefi and Sh. Khoshdel, Reflexivity of powers of the multiplication operator on special function spaces, Acta Math. Sci., 32B (6) (2012), 2279-2284.

    Article  MathSciNet  Google Scholar 

  27. B. Yousefi and F. Zangeneh, On the property of reflexivity for multiplication operators on Banach function spaces, Proc. Indian Acad. Sci. (Math. Sci.), (2018) 128:38, https://doi.org/10.1007/s12044-018-0404-5

Download references

Acknowledgements

The authors would like to thank the referee for helpful suggestions. Also, we are grateful to Professor Gadadhar Misra for his valuable recommendations.

Author information

Authors and Affiliations

  1. Department of Mathematics, Payame Noor University, P.O. Box 19395-3697, Tehran, Iran

    F. Ershad, M. M. Khorami & B. Yousefi

Authors
  1. F. Ershad
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. M. M. Khorami
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. B. Yousefi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to F. Ershad.

Additional information

Communicated by Gadadhar Misra.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ershad, F., Khorami, M.M. & Yousefi, B. Reflexive operators on analytic function spaces. Indian J Pure Appl Math 52, 807–815 (2021). https://doi.org/10.1007/s13226-021-00140-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s13226-021-00140-6

Keywords

Mathematics Subject Classification

Search

Navigation