Hypercyclic operators that commute
with the Bergman backward shift

Paul S. Bourdon and Joel H. Shapiro

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

Abstract: A hypercyclic operator  is one that has a dense orbit. The backward shift B on the Bergman space A^2 of the unit disc has this property, and we ask here: ``Which operators that commute with B also have it?'' It is known that each operator on A^2 that commutes with B has a natural representation of the form f(B) where f is a multiplier of the Dirichlet space. In this setting we show that our problem reduces to the case where f is a self-map of the unit disc, and that for such maps the question of hypercyclicity for f(B) depends on how closely the f-images of points in the unit disc are allowed to approach the boundary. This contrasts sharply with what is known for the Hardy space H^2, where the backward shift is not hypercyclic (it is a contraction), and the hypercyclic operators that commute with it are easily described by previous work of Godefroy and Shapiro. In further contrast with the H^2 setting our present work leads into diverse issues concerning multipliers of the Dirichlet space, Carleson sets, and regularity of outer functions. The results we obtain bear an intriguing resemblance to certain phenomena involving composition operators.

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