Operators with hypercyclic Cesaro means

Volume 152 / 2002

Fernando León-Saavedra Studia Mathematica 152 (2002), 201-215 MSC: 47B37, 47B38, 47B99. DOI: 10.4064/sm152-3-1

Abstract

An operator $T$ on a Banach space ${\cal B}$ is said to be hypercyclic if there exists a vector $x$ such that the orbit $\{ T^nx\} _{n\geq 1}$ is dense in ${\cal B}$. Hypercyclicity is a strong kind of cyclicity which requires that the linear span of the orbit is dense in ${\cal B}$. If the arithmetic means of the orbit of $x$ are dense in ${\cal B}$ then the operator $T$ is said to be Cesàro-hypercyclic. Apparently Cesàro-hypercyclicity is a strong version of hypercyclicity. We prove that an operator is Cesàro-hypercyclic if and only if there exists a vector $x\in {\cal B}$ such that the orbit $\{ n^{-1}T^nx\} _{n\geq 1}$ is dense in ${\cal B}$. This allows us to characterize the unilateral and bilateral weighted shifts whose arithmetic means are hypercyclic. As a consequence we show that there are hypercyclic operators which are not Cesàro-hypercyclic, and more surprisingly, there are non-hypercyclic operators for which the Cesàro means of some orbit are dense. However, we show that both classes, the class of hypercyclic operators and the class of Cesàro-hypercyclic operators, have the same norm-closure spectral characterization.

Authors

  • Fernando León-SaavedraDepartamento de Matemáticas
    Escuela Superior de Ingeniería
    Universidad de Cádiz
    C//Sacramento 82
    11003 Cádiz, Spain
    e-mail

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