A continuous linear operator

is called hypercyclic if there exists an

such that the orbit

is dense. We consider the problem: given an operator

, hypercyclic or not, is the restriction

to some closed invariant subspace

hypercyclic? In particular, it is well-known that any non-constant partial differential operator p(D) on

(entire functions) is hypercyclic. Now, if q(D) is another such operator, p(D) maps ker q(D) invariantly (by commutativity), and we obtain a necessary and sufficient condition on p and q in order that the restriction p(D) : ker q(D)

ker q(D) is hypercyclic. We also study hypercyclicity for other types of operators on subspaces of

.