Hypercyclicity on invariant subspaces

Journal article, 2008

A continuous linear operator T : X -> X is called hypercyclic if there exists an x is an element of X such that the orbit {T(n)x}(n >= 0) is dense. We consider the problem: given an operator T : X -> X, hypercylic or not, is the restriction T vertical bar y to some closed invariant subspace Y subset of X hypercyclic? In particular, it is well-known that any non-constant partial differential operator p(D) on H(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) : kerq(D) -> ker q(D) is hypercyclic. We also study hypercyclicity for other types of operators on subspaces of H (Cd).