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Operators Commuting with the Volterra Operator are not Weakly Supercyclic

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Abstract

We prove that any bounded linear operator on L p [0, 1] for 1 ≤ p < ∞, commuting with the Volterra operator V, is not weakly supercyclic, which answers affirmatively a question raised by Léon-Saavedra and Piqueras-Lerena. It is achieved by providing an algebraic type condition on an operator which prevents it from being weakly supercyclic and is satisfied for any operator commuting with V.

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Authors and Affiliations

  1. Department of Pure Mathematics, Queen’s University Belfast, University Road, Belfast, BT7 1NN, UK

    Stanislav Shkarin

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  1. Stanislav Shkarin
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Correspondence to Stanislav Shkarin.

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The author would like to thank A. Montes and F. Leon for their interest and comments.

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Shkarin, S. Operators Commuting with the Volterra Operator are not Weakly Supercyclic. Integr. Equ. Oper. Theory 68, 229–241 (2010). https://doi.org/10.1007/s00020-010-1790-y

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  • DOI: https://doi.org/10.1007/s00020-010-1790-y

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