Abstract. Let $T$ be a bounded linear operator on a complex Banach space $X$. We establish that if $T$ is supercyclic and $(\|T^n\|)$ is a bounded sequence then $(T^nx)$ converges to $0$ for each $x\in X$, which generalizes a result by V.~Matache. As an application, we show that a composition operator on the Hardy space of the open unit disk $U$ is not supercyclic whenever it is induced by a mapping fixing a point in $U$. We then turn our attention to powers of cyclic operators. We prove that if $T^n$ is cyclic for each positive integer $n$, then the set of cyclic vectors for $T$ is dense in $X$. We also discuss simultaneous cyclicity of $T$ and $T^{-1}$.

AMS Subject Classification (1991): 47A15, 47B38

Received September 27, 1996. (Registered under 6110/2009.)