Skip to main content
SpringerLink
Log in

Fredholm Weighted Generalized Composition Operators

  • Published:
Complex Analysis and Operator Theory Aims and scope Submit manuscript

Abstract

In the present paper, properties of the self map on \(\mathbb {D}\), the open unit disc, and complex valued mapping on \(\mathbb {D}\) are obtained when the induced k th-order weighted generalized composition operator on weighted Hardy space is Fredholm. We also describe the action of the adjoint of generalized multiplication, generalized composition and \(k{th}\)-order weighted generalized composition operators on the derivative evaluation kernel functions, which in turn provide some invariant subspaces for these operators.

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

Availability of Data and Materials

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

References

  1. Bourdon, P.S.: Fredholm multiplication and composition operators on the Hardy space. Integr. Eqn. Oper. Theory 13, 607–610 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bourdon, P.S.: Spectra of some composition operators and associated weighted composition operators. J. Oper. Theory 67(2), 537–560 (2009)

    MathSciNet  MATH  Google Scholar 

  3. Contreras, M.D., Hernańdez-Diaz, A.G.: Weighted composition operators on Hardy spaces. J. Math. Anal. Appl. 263(1), 224–233 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  4. Cowen, C.C., MacCluer, B.D.: Composition Operators on Spaces of Analytic Functions. CRC Press, Boca Raton (1995)

    MATH  Google Scholar 

  5. Datt, G., Jain, M.: Generalized composition operators of higher order on weighted Hardy spaces. Chamchuri J. Math. 9, 52–60 (2017)

    MathSciNet  Google Scholar 

  6. Datt, G., Jain, M.: Generalized multiplication operators of higher order on weighted Hardy spaces. Asian Eur. J. Math. 13(5), 2050102 (2020). https://doi.org/10.1142/S1793557120501028

    Article  MathSciNet  MATH  Google Scholar 

  7. Datt, G., Jain, M., Ohri, N.: On weighted generalized composition operators on weighted Hardy spaces. Filomat 34(5), 1689–1700 (2020)

    Article  MathSciNet  Google Scholar 

  8. Gunatillake, G.K.: Weighted composition operators. Purdue University Graduate School, Thesis (2005)

  9. Gunatillake, G.K.: Invertible weighted composition operators. J. Funct. Anal. 261(3), 831–860 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  10. Hatori, O.: Fredholm composition operators on spaces of holomorphic functions. Integral Equ. Oper. Theory 18, 202–210 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  11. Johnson, W.P.: The curious history of Faà De Bruno’s Formula. Am. Math. Mon. 109(3), 217–234 (2002)

    MATH  Google Scholar 

  12. Kelley, R.L.: Weighted shifts on Hilbert space. Dissertation, University of Michigan, Mich, Ann Arbor (1966)

  13. MacCluer, B.D.: Fredholm composition operators. Proc. Am. Math. Soc. 125(1), 163–166 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  14. Nordgren, E.A.: Composition operators. Can. J. Math. 20, 442–449 (1968)

    Article  MATH  Google Scholar 

  15. Ridge, W.C.: Composition operators. Indiana University, Thesis (1969)

  16. Schwartz, H.J.: Composition operators on \(H^p\). University of Toledo, Thesis (1969)

  17. Shields, A.L.: Weighted Shift Operators and Analytic Function Theory, Topics in Operator Theory, Math. Surveys, vol. 13, pp. 49–128. American Mathematical Society, Rhode Ireland (1974)

    Google Scholar 

  18. Singh, R.K.: Composition operators. University of New Hampshire, Thesis (1972)

  19. Zhao, L.: Fredholm weighted composition operator on weighted Hardy spaces. J. Funct. Spaces Appl. (2013). https://doi.org/10.1155/2013/327692

    Article  MathSciNet  MATH  Google Scholar 

  20. Zorboska, N.: Compact composition operators on some weighted Hardy spaces. J. Oper. Theory 22, 233–241 (1989)

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

We acknowledge the suggestions and observations provided by Prof. Raúl Curto, wich helped to shape the paper in present form.

Funding

Not applicable.

Author information

Authors and Affiliations

  1. Department of Mathematics, PGDAV College, University of Delhi, Delhi, India

    Gopal Datt & Mukta Jain

Authors
  1. Gopal Datt
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mukta Jain
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Gopal Datt.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Additional information

Communicated by Dan Volok.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This article is part of the topical collection “Reproducing kernel spaces and applications” edited by Daniel Alpay.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Datt, G., Jain, M. Fredholm Weighted Generalized Composition Operators. Complex Anal. Oper. Theory 15, 40 (2021). https://doi.org/10.1007/s11785-021-01087-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s11785-021-01087-6

Keywords

Mathematics Subject Classification

Search

Navigation