A continuous linear operator $T;:;X{ ightarrow}X$ is called hypercyclic if there exists an $x;{in};X$ such that the orbit ${T^nx}_{n{geq}0}$ is dense. We consider the problem: given an operator $T;:;X{ ightarrow}X$, hypercyclic or not, is the restriction $T|y$ to some closed invariant subspace $y{subset}X$ hypercyclic? In particular, it is well-known that any non-constant partial differential operator p(D) on $H({mathbb{C}}^d)$ (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) $ ightarrow$ ker q(D) is hypercyclic. We also study hypercyclicity for other types of operators on subspaces of $H({mathbb{C}}^d)$.