Transactions of the American Mathematical Society

Author(s): Paul S. Bourdon; Joel H. Shapiro
Journal: Trans. Amer. Math. Soc. 352 (2000), 5293-5316.
MSC (2000): Primary 47B38
Posted: July 18, 2000
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The backward shift

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

inherit its hypercyclicity?'' We show that the problem reduces to the study of operators of the form $\varphi(B)$ where $\varphi$ 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 $\varphi(z)$ is allowed to approach the unit circle as $\vert z\vert\to 1-$ .

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Paul S. Bourdon
Affiliation: Department of Mathematics, Washington and Lee University, Lexington, Virginia 24450
Email: pbourdon@wlu.edu

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