Hypercyclic operators that commute with the Bergman backward shift

Abstract

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 ϕ(B) where ϕ 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) is allowed to approach the unit circle as |z| → 1−.

Hypercyclic Toeplitz Operators

“…A novel feature of these conditions is the role of univalence or N-valence (where N is the degree of p) of the symbol. It seems that such conditions did not appear in the linear dynamics before, with one notable exception: in [3] Bourdon and Shapiro studied Bergman space Toeplitz operators with antianalytic symbols and in some of their results the univalence of the symbol plays a role.…”

Hypercyclicity of special operators on Hilbert function spaces

“…It was developed independently by Kitai [10], Gethner and Shapiro [6]. This criterion has been used to show that hypercyclic operators arise within the classes of composition operators [4], weighted shifts [13], adjoints of multiplication operators [5], and adjoints of subnormal and hyponormal operators [3].…”

Ten questions in linear dynamics

“…This was first discovered by Godefroy and Shapiro, who produced in [19] their well-known criterion that if the eigenvectors of an operator T associated to eigenvalues of modulus greater than 1 and smaller than 1, respectively, span a dense subspace of the space, then T is hypercyclic. This was developed by Bourdon and Shapiro in [13], and then in the works [4], [5], [6]. The notion of frequent hypercyclicity was introduced and investigated there, and the study of operators from the ergodic-theoretic point of view was also developed there, building on early work of Flytzanis [17].…”