Skip to main content
SpringerLink
Log in

On the Aluthge Transforms of Weighted Shifts with Moments of Fibonacci Type. Application to Subnormality

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

Abstract

In this paper, we investigate the properties of the Aluthge transforms of weighted shifts whose moment sequence is a generalized Fibonacci sequence of order r (or of Fibonacci type). We provide sufficient conditions under which the Aluthge transform has also a moment sequence of Fibonacci type. Notably, we detail the case when the moment sequence is a generalized Fibonacci sequence of order r ≤ 4. This allows us to recover the main result given by S. H. Lee et al. concerning the subnormality of Aluthge transforms of weighted shifts with two atomic Berger measure. We end this paper by considering the extended case of the subnormal weighted shift of a finite atomic Berger measure, with a view to whether its Aluthge transforms is again subnormal. As a result, we give a negative answer to a recent conjecture posed by S. H. Lee et al. in its general setting. Even so, under some mild assumptions we give a positive answer to this conjecture.

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. Aluthge A.: On p-hyponormal operators for 0 < p < 1. Integral Equ. Oper. Theory 13, 307–315 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  2. Ben Taher R., Rachidi M., Zerouali H.: Recursive subnormal completion and truncated moment problem. Bull. Lond. Math. Soc. 33, 425–432 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  3. Bernoussi B., Rachidi M., Saeki O.: Factorial Binet formula and distributional moment formulation of generalized Fibonacci sequences. Fibonacci Q. 42(4), 320–329 (2004)

    MATH  MathSciNet  Google Scholar 

  4. Chidume E.E., Rachidi M., Zerouali E.H.: Solving the general truncated moment problem by r-generalized Fibonacci sequences method. J. Math. Anal. Appl. 256, 625–635 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  5. Curto R.E., Fialkow L.A.: Recursiveness, positivity and truncated moment problem. Houston J. Math. 17(4), 603–635 (1991)

    MATH  MathSciNet  Google Scholar 

  6. Curto R.E., Fialkow L.A.: Solution of the truncated complex moment problem for flat data. Mem. Am. Math. Soc. 119, 568 (1996)

    MathSciNet  Google Scholar 

  7. Curto R.E, Fialkow L.A.: Recursively generated weighted shifts and the subnormal completion. Integral Equ. Oper. Theory 17, 202–246 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  8. Dubeau F., Motta W., Rachidi M., Saeki O.: On weighted r-generalized Fibonacci sequences. Fibonacci Q. 35, 102–110 (1997)

    MATH  MathSciNet  Google Scholar 

  9. Lee S.H., Lee W.Y., Yoon J.: Subnormality of Aluthge transforms of weighted shifts. Integral Equ. Oper. Theory 72, 241–251 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  10. Schwartz, L.: Théorie des Distributions. Editions Hermann, Paris (1966)

  11. Stanley R.P.: Enumerative Combinatorics, Vol. I. Cambridge University Press, Cambridge (1997)

    Book  Google Scholar 

  12. Stochel J., Stochel J.B.: On the kth root of a Stieltjes moment sequence. J. Math. Anal. Appl. 396, 786–800 (2012)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Equip of DEFA, Département de Mathématiques et Informatique, Faculté des Sciences, Université Moulay Ismail, B.P. 4010, Beni M’hamed, Meknes, Morocco

    R. Ben Taher & M. Rachidi

  2. Laboratoire d’Analyse et Applications, Département de Mathématiques et Informatique, Faculté des Sciences de Rabat, Université Mohamed V-Agdal, Rabat, Morocco

    E. H. Zerouali

Authors
  1. R. Ben Taher
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. M. Rachidi
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. E. H. Zerouali
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to R. Ben Taher.

Additional information

M. Rachidi is associate with Equip DEFA, DMI, Faculty of Sciences, Meknes, Morocco.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ben Taher, R., Rachidi, M. & Zerouali, E.H. On the Aluthge Transforms of Weighted Shifts with Moments of Fibonacci Type. Application to Subnormality. Integr. Equ. Oper. Theory 82, 287–299 (2015). https://doi.org/10.1007/s00020-015-2229-2

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00020-015-2229-2

Mathematics Subject Classification

Keywords

Search

Navigation