The author considers operators T acting on complex separable Banach spaces. The set {x, T x, T2x, . . . , T nx, . . .} is called the orbit of x under T. If dense orbits exist T is called hypercyclic. If the spectrum of a hypercyclic operator can be represented as the union of two nonvoid, compact, mutually disjoint sets then each of these sets must have nonvoid intersection with the unit circle. No nonzero reducing subspace of a hypercyclic operator T reduces T to a normal, respectively to a compact operator. For Hilbert space contractions A if λA is hypercyclic for some complex λ then A is in C0. \ C0 .