Hypercyclic operators that
commute
with the Bergman backward shift
Paul S. Bourdon and Joel H. Shapiro
Trans. Amer. Math. Soc.
352 (2000), 52935316
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 selfmap of the unit disc, and that for such maps the
question of hypercyclicity for f(B) depends on how closely the
fimages 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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