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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Hypercyclic operators that commute with the Bergman backward shift


Authors: Paul S. Bourdon and Joel H. Shapiro
Journal: Trans. Amer. Math. Soc. 352 (2000), 5293-5316
MSC (2000): Primary 47B38
Published electronically: July 18, 2000
MathSciNet review: 1778507
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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 $\varphi(B)$ 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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Additional Information

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

Joel H. Shapiro
Affiliation: Department of Mathematics, Michigan State University, East Lansing, Michigan 48824-1027
Email: shapiro@math.msu.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-00-02648-9
PII: S 0002-9947(00)02648-9
Keywords: Hypercyclic operator, Bergman space, backward shift
Received by editor(s): January 28, 1999
Received by editor(s) in revised form: September 13, 1999
Published electronically: July 18, 2000
Additional Notes: Both authors were supported in part by the National Science Foundation
Article copyright: © Copyright 2000 American Mathematical Society