Archiv der Mathematik

, Volume 110, Issue 5, pp 477–486 | Cite as

The lattices of invariant subspaces of a class of operators on the Hardy space

  • Željko Čučković
  • Bhupendra Paudyal


In the authors’ first paper, a Beurling-Rudin-Korenbljum type characterization of the closed ideals in a certain algebra of holomorphic functions was used to describe the lattice of invariant subspaces of the shift plus a complex Volterra operator. The current work is an extension of the previous work and it describes the lattice of invariant subspaces of the shift plus a positive integer multiple of the complex Volterra operator on the Hardy space. Our work was motivated by a paper by Ong who studied the real version of the same operator.


Complex analysis Invariant subspaces Hardy space 

Mathematics Subject Classification



Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    A. Aleman and B. Korenblum, Volterra invariant subspaces of \(H^p\), Bull. Sci. Math. 132 (2008), 510-528MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    A. Beurling, On two problems concerning linear transformations in Hilbert space, Acta Math. 81 (1948), 17 ppGoogle Scholar
  3. 3.
    Ž. Čučković and B. Paudyal, Invariant subspaces of the shift plus complex Volterra operator, J. Math. Anal. Appl. 426 (2015), 1174-1181MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Y. Katznelson, An Introduction to Harmonic Analysis, Third edition, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 2004.CrossRefzbMATHGoogle Scholar
  5. 5.
    B. I. Korenbljum, Invariant subspaces of the shift operator in a weighted Hilbert space, (Russian) Mat. Sb. (N.S.) 89 (131) (1972), 110-137Google Scholar
  6. 6.
    B. I. Korenbljum, Invariant subspaces of the shift operator in certain weighted Hilbert spaces of sequences, (Russian) Dokl. Akad. Nauk SSSR 202 (1972), 1258-1260MathSciNetGoogle Scholar
  7. 7.
    A. Montes-Rodriguez, M. Ponce-Escudero, and S. A. Shkarin, Invariant subspaces of parabolic self-maps in the Hardy space, Math. Res. Lett. 17 (2010), 99-107MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    B. Ong, Invariant subspace lattices for a class of operators, Pacific J. Math. 94 (1981), 385-405MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    W. Rudin, The closed ideals in an algebra of analytic functions, Canad. J. Math. 9 (1957), 426-434MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Edited and with a foreword by S.M. Nikol’ ski, Translated from the 1987 Russian original, Revised by the authors, Gordon and Breach Science Publishers, Yverdon, 1993.Google Scholar
  11. 11.
    D. Sarason, A remark on the Volterra operator, J. Math. Anal. Appl. 12 (1965), 244-246MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    D. Sarason, Invariant subspaces, In: Topics in Operator Theory, 1-47, Math. Surveys, No. 13, Amer. Math. Soc., Providence, RI, 1974.Google Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsThe University of ToledoToledoUSA
  2. 2.Mathematics and Computer Science DepartmentCentral State UniversityWilberforceUSA

Personalised recommendations