Some properties of $N$-supercyclic operators
Let $T$ be a continuous linear operator on a Hausdorff topological vector space $\mathcal X$ over the field $\mathbb C$. We show that if $T$ is $N$-supercyclic, i.e., if $\mathcal X$ has an $N$-dimensional subspace whose orbit under $T$ is dense in $\cal X$, then $T^*$ has at most $N$ eigenvalues (counting geometric multiplicity). We then show that $N$-supercyclicity cannot occur nontrivially in the finite-dimensional setting: the orbit of an $N$-dimensional subspace cannot be dense in an $(N+1)$-dimensional space. Finally, we show that a subnormal operator on an infinite-dimensional Hilbert space can never be $N$-supercyclic.