Some properties of $N$-supercyclic operators

P. S. Bourdon¹ ; N. S. Feldman¹ ; J. H. Shapiro²

¹Washington and Lee University Lexington, VA 24450, U.S.A.
²Michigan State University East Lansing, MI 48824, U.S.A.

Studia Mathematica, Tome 165 (2004) no. 2, pp. 135-157

Résumé

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.

Détail

DOI : 10.4064/sm165-2-4

Keywords: continuous linear operator hausdorff topological vector space mathcal field mathbb n supercyclic mathcal has n dimensional subspace whose orbit under dense cal * has eigenvalues counting geometric multiplicity n supercyclicity cannot occur nontrivially finite dimensional setting orbit n dimensional subspace cannot dense dimensional space finally subnormal operator infinite dimensional hilbert space never n supercyclic