Abstract. The author considers operators $T$ acting on complex separable Banach spaces. The set $\{x,Tx,T^2x,\ldots,T^nx,\ldots\} $ is called the orbit of $x$ under $T$. If dense orbits exist $T$ is called {\it 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 $\lambda A$ is hypercyclic for some complex $\lambda $ then $A$ is in $C_0.\setminus C_0 $.

AMS Subject Classification (1991): 47A15, 47A20

Received January 31, 1992 and in revised form March 30, 1992. (Registered under 5556/2009.)