  According to Kim, Peris and Song, a continuous linear operator $T$ on a
complex Banach space $X$ is called {\it numerically hypercyclic} if the
numerical orbit $\{f(T^nx):n\in\N\}$ is dense in $\C$ for some $x\in X$ and
$f\in X^*$ satisfying $\|x\|=\|f\|=f(x)=1$. They have characterized numerically
hypercyclic weighted shifts and provided an example of a numerically
hypercyclic operator on $\C^2$.
  We answer two questions of Kim, Peris and Song. Namely, we construct a
numerically hypercyclic operator, whose square is not numerically hypercyclic
as well as an operator which is not numerically hypercyclic but has two
numerical orbits whose union is dense in $\C$. We characterize numerically
hypercyclic operators on $\C^2$ as well as the operators similar to a
numerically hypercyclic one and those operators whose conjugacy class consists
entirely of numerically hypercyclic operators. We describe in spectral terms
the operator norm closure of the set of numerically hypercyclic operators on a
reflexive Banach space. Finally, we provide criteria for numeric hypercyclicity
and decide upon the numerical hypercyclicity of operators from various classes.
