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 {A ⁿ B ^k x : 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.