Hypercyclic Pairs of Coanalytic Toeplitz Operators

Abstract

A pair of commuting operators, (A,B), on a Hilbert space \({\mathcal{H}}\) is said to be hypercyclic if there exists a vector \(x \in {\mathcal{H}}\) such that {AnBkx : n, k ≥ 0} is dense in \({\mathcal{H}}\) . If f, g ∈H∞(G) where G is an open set with finitely many components in the complex plane, then we show that the pair (M*f, M*g) of adjoints of multiplcation operators on a Hilbert space of analytic functions on G is hypercyclic if and only if the semigroup they generate contains a hypercyclic operator. However, if G has infinitely many components, then we show that there exists f, g ∈H∞(G) such that the pair (M*f, M*g) is hypercyclic but the semigroup they generate does not contain a hypercyclic operator. We also consider hypercyclic n-tuples.