Abstract. In this note we construct an operator $T$ on a (separable, complex) Hilbert space such that for every nonzero vector $x$, the sequence $\{\|T^nx\|\} _{n\in{\msbm N}}$ is dense in ${\msbm R}_{+}$, but despite this, $T$ is not hypercyclic (i.e., no vector in ${\cal H}$ has a dense orbit). In addition, this operator has the property that there are subsequences $\{r_{n}\} $ and $\{q_{n}\} $ of ${\msbm N}$ such that $T^{r_{n}}\rightarrow0$ and $T^{q_{n}}\rightarrow +\infty $ (properly defined) in the strong operator topology. Finally, neither $T$ nor $T^*$ has point spectrum. This partially answers a question in [5] and provides a counterexample to some reasonable conjectures.

AMS Subject Classification (1991): 47A16, 47A15; 47B37

Received December 12, 2007, and in revised form January 17, 2008. (Registered under 6043/2009.)