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Hypercyclic operators that commute with the Bergman backward shift. (English) Zbl 0960.47003

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 \(|z|\to 1-\).

MSC:

47A16 Cyclic vectors, hypercyclic and chaotic operators
47B38 Linear operators on function spaces (general)
47A60 Functional calculus for linear operators
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