On Read's type operators on Hilbert spaces

Abstract

Using Read's construction of operators without non-trivial invariant subspaces/subsets on ${\ell }_1$ or $c_0$, we construct examples of operators on a Hilbert space whose set of hypercyclic vectors is "large" in various senses. We give an example of an operator such that the closure of every orbit is a closed subspace, and then, answering a question of D. Preiss, an example of an operator such that the set of its non-hypercyclic vectors is Gauss null. This operator has the property that it is orbit-unicellular, i.e. the family of the closures of its orbits is totally ordered. We also exhibit an example of an operator on a Hilbert space which is not orbit-reflexive.

This is a preprint version of the article "On Read's type operators on Hilbert spaces", Int. Math. Res. Not., 2008 Art. ID rnn083, 42 pp