Results in Mathematics

, 74:197 | Cite as

Dynamics of the Volterra-Type Integral and Differentiation Operators on Generalized Fock Spaces

  • José BonetEmail author
  • Tesfa Mengestie
  • Mafuz Worku


Various dynamical properties of the differentiation and Volterra-type integral operators on generalized Fock spaces are studied. We show that the differentiation operator is always supercyclic on these spaces. We further characterize when it is hypercyclic, power bounded and uniformly mean ergodic. We prove that the operator satisfies the Ritt’s resolvent condition if and only if it is power bounded and uniformly mean ergodic. Some similar results are obtained for the Volterra-type and Hardy integral operators.


Generalized Fock spaces power bounded uniformly mean ergodic Volterra-type integral operator differential operator Hardy operator supercyclic hypercyclic cyclic Ritt’s resolvent condition 

Mathematics Subject Classification

Primary 47B38 30H20 Secondary 46E15 47A16 47A35 



The authors are very thankful to the referee for the careful reading of our paper and many suggestions which corrected and improved our manuscript. Part of this work was done during the third-named author’s stay at the Instituto Universitario de Matemática Pura y Aplicada of the Universitat Politècnica de València. He would like to thank Prof. José Bonet, Prof. Alfred Peris and all other members of the institute for their hospitality and kindness during his stay in Valencia, Spain.


  1. 1.
    Abanin, A.V., Tien, P.T.: Differentiation and integration operators on weighted Banach spaces of holomorphic functions. Math. Nachr. 290(8–9), 1144–1162 (2017)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Atzmon, A., Brive, B.: Surjectivity and invariant subspaces of differential operators on weighted Bergman spaces of entire functions, Bergman spaces and related topics in complex analysis, Contemp. Math., vol. 404, Amer. Math. Soc., Providence, RI, pp. 27–39 (2006)Google Scholar
  3. 3.
    Bayart, F., Matheron, E.: Dynamics of Linear Operators, Cambridge Tracts in Math, vol. 179. Cambridge Univ. Press, Cambridge (2009)CrossRefGoogle Scholar
  4. 4.
    Bermúdez, T., Bonilla, A., Peris, A.: On hypercyclicity and supercyclicity criteria. Bull. Austral. Math. Soc. 70, 45–54 (2004)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Beltrán, M.J.: Dynamics of differentiation and integration operators on weighted space of entire functions. Studia Math. 221, 35–60 (2014)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Beltrán, M.J., Bonet, J., Fernández, C.: Classical operators on weighted Banach spaces of entire functions. Proc. Am. Math. Soc. 141, 4293–4303 (2013)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Bès, J., Peris, A.: Hereditarily hypercyclic operators. J. Funct. Anal. 167, 94–112 (1999)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Bonet, J.: Dynamics of the differentiation operator on weighted spaces of entire functions. Math. Z. 26, 649–657 (2009)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Bonet, J.: The spectrum of Volterra operators on weighted Banach spaces of entire functions. Q. J. Math. 66, 799–807 (2015)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Bonet, J., Bonilla, A.: Chaos of the differentiation operator on weighted Banach spaces of entire functions. Complex Anal. Oper. Theory 7, 33–42 (2013)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Bonet, J., Taskinen, J.: A note about Volterra operators on weighted Banach spaces of entire functions. Math. Nachr. 288, 1216–1225 (2015)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Constantin, O., Persson, A.-M.: The spectrum of Volterra-type integration operators on generalized Fock spaces. Bull. Lond. Math. Soc. 47, 958–963 (2015)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Constantin, O., Peláez, J.-Á.: Integral operators, embedding theorems and a Littlewood–Paley formula on weighted Fock spaces. J. Geom. Anal. 26, 1109–1154 (2016)MathSciNetCrossRefGoogle Scholar
  14. 14.
    De La Rosa, M., Read, C.: A hypercyclic operator whose direct sum is not hypercyclic. J. Oper. Theory 61, 369–380 (2009)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Dunford, N.: Spectral theory. I. Convergence to projections. Trans. Am. Math. Soc. 54, 185–217 (1943)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Grosse-Erdmann, K.G., Peris Manguillot, A.: Linear Chaos. Springer, New York (2011)CrossRefGoogle Scholar
  17. 17.
    Harutyunyan, A., Lusky, W.: On the boundedness of the differentiation operator between weighted spaces of holomorphic functions. Studia Math. 184, 233–247 (2008)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Krengel, U.: Ergodic Theorems. Walter de Gruyter, Berlin (1985)CrossRefGoogle Scholar
  19. 19.
    Lyubich, Yu.: Spectral localization, power boundedness and invariant subspaces under Ritt’s type condition. Studia Mathematica 143(2), 153–167 (1999)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Mengestie, T.: A note on the differential operator on generalized Fock spaces. J. Math. Anal. Appl. 458(2), 937–948 (2018)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Mengestie, T.: Spectral properties of Volterra-type integral operators on Fock–Sobolev spaces. J. Kor. Math. Soc. 54(6), 1801–1816 (2017)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Mengestie, T.: On the spectrum of volterra-type integral operators on Fock–Sobolev spaces. Complex Anal. Oper. Theory 11(6), 1451–1461 (2017)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Mengestie, T., Ueki, S.: Integral, differential and multiplication operators on weighted Fock spaces. Complex Anal. Oper. Theory 13, 935–95 (2019)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Mengestie, T., Worku, M.: Isolated and essentially isolated Volterra-type integral operators on generalized Fock spaces. Integr. Transf. Spec. Funct. 30, 41–54 (2019)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Nagy, B., Zemanek, J.A.: A resolvent condition implying power boundedness. Studia Math. 134, 143–151 (1999)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Nevanlinna, O.: Convergence of iterations for linear equations. Lecture Notes in Mathematics. ETH Zürich, Birkhäuser, Basel (1993)CrossRefGoogle Scholar
  27. 27.
    Ritt, R.K.: A condition that \(\lim _{n\rightarrow \infty } n^{-1}T^n =0\). Proc. Am. Math. Soc. 4, 898–899 (1953)Google Scholar
  28. 28.
    Ueki, S.: Characterization for Fock-type space via higher order derivatives and its application. Complex Anal. Oper. Theory 8, 1475–1486 (2014)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Yosida, K.: Functional Analysis. Springer, Berlin (1978)CrossRefGoogle Scholar
  30. 30.
    Yosida, K., Kakutani, S.: Operator-theoretical treatment of Marko’s process and mean ergodic theorem. Ann. Math. 42(1), 188–228 (1941)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Instituto Universitario de Matemática Pura y Aplicada IUMPAUniversitat Politècnica de ValènciaValenciaSpain
  2. 2.Department of Mathematical SciencesWestern Norway University of Applied SciencesStordNorway
  3. 3.Department of MathematicsAddis Ababa UniversityAddis AbabaEthiopia

Personalised recommendations