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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2024 MCQ for Transactions of the American Mathematical Society is 1.48 .

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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

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 $|z|\to 1-$.
References
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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