Hypercyclic operators that commute with the Bergman backward shift

Bourdon, Paul; Shapiro, Joel

doi:10.1090/S0002-9947-00-02648-9

Hypercyclic operators that commute with the Bergman backward shift
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by Paul S. Bourdon and Joel H. Shapiro PDF

Trans. Amer. Math. Soc. 352 (2000), 5293-5316 Request permission

Abstract:

The backward shift $B$ on the Bergman space of the unit disc is known to be hypercyclic (meaning: it has a dense orbit). Here we ask: “Which operators that commute with $B$ inherit its hypercyclicity?” We show that the problem reduces to the study of operators of the form $\phi (B)$ where $\phi$ is a holomorphic self-map of the unit disc that multiplies the Dirichlet space into itself, and that the question of hypercyclicity for such an operator depends on how freely $\phi (z)$ is allowed to approach the unit circle as $|z|\to 1-$.

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