Abstract
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.
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Acknowledgements
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.
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J. Bonet was partially supported by the research projects MTM2016-76647-P and GV Prometeo 2017/102 (Spain). M. Worku is supported by ISP project, Addis Ababa University, Ethiopia.
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Bonet, J., Mengestie, T. & Worku, M. Dynamics of the Volterra-Type Integral and Differentiation Operators on Generalized Fock Spaces. Results Math 74, 197 (2019). https://doi.org/10.1007/s00025-019-1123-7
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DOI: https://doi.org/10.1007/s00025-019-1123-7
Keywords
- Generalized Fock spaces
- power bounded
- uniformly mean ergodic
- Volterra-type integral operator
- differential operator
- Hardy operator
- supercyclic
- hypercyclic
- cyclic
- Ritt’s resolvent condition