An operator T is hypercyclic if there is a vector with dense orbit; so if T acts on H, then T is hypercyclic if there is a vector x in H such that {x, Tx, T^2x, ...} is dense in H. We say that a pair of operators (A,B) is hypercyclic if there is a vector x such that {A^nB^kx : n,k \geq 0} is dense in H. The semigroup generated by A and B is the set of operators {A^nB^k : n,k \geq 0}. Clearly, if the semigroup generated by A and B contains a hypercyclic operator, then the pair (A,B) will be hypercyclic.

In this paper we consider operators which are adjoints of multiplication operators on Hilbert spaces of analytic functions. If A and B are two such operators on a Hilbert space H of analytic functions on an open set G, then we prove that

1) If G has only finitely many components, then:

The pair (A,B) is hypercyclic if and only if the semigroup they generate contains a hypercyclic operator.

2) If G has infinitely many components, then there exists two bounded analytic functions on G such that if A and B are the adjoints of multiplication by these two functions, then (A,B) is hypercyclic, yet the semigroup generated by A and B does not contain a hypercyclic operator.