Hypercyclic operators that commute with the Bergman backward shift

Abstract

The backward shift B 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 B inherit its hypercyclicity?” We show that the problem reduces to the study of operators of the form ϕ ( B ) \phi (B) where ϕ \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 ϕ ( z ) \phi (z) is allowed to approach the unit circle as | z | → 1 − |z|\to 1- .