Let T be a continuous linear operator on a Hausdorff topological vector space 𝓧 over the field ℂ. We show that if T is N-supercyclic, i.e., if 𝓧 has an N-dimensional subspace whose orbit under T is dense in 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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm165-2-4,
author = {P. S. Bourdon and N. S. Feldman and J. H. Shapiro},
title = {Some properties of N-supercyclic operators},
journal = {Studia Mathematica},
volume = {162},
year = {2004},
pages = {135-157},
zbl = {1056.47008},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm165-2-4}
}
P. S. Bourdon; N. S. Feldman; J. H. Shapiro. Some properties of N-supercyclic operators. Studia Mathematica, Tome 162 (2004) pp. 135-157. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm165-2-4/