## Hypercyclic operators that commute with the Bergman backward shift

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- by Paul S. Bourdon and Joel H. Shapiro
- Trans. Amer. Math. Soc.
**352**(2000), 5293-5316 - DOI: https://doi.org/10.1090/S0002-9947-00-02648-9
- Published electronically: July 18, 2000
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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 $\phi (B)$ 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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## Bibliographic 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
- 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
- © Copyright 2000 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**352**(2000), 5293-5316 - MSC (2000): Primary 47B38
- DOI: https://doi.org/10.1090/S0002-9947-00-02648-9
- MathSciNet review: 1778507