On Read\u27s type operators on Hilbert spaces

Using Read\u27s construction of operators without nontrivial invariant subspaces/subsets on l 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 nonhypercyclic 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. \ua9 The Author 2008. Published by Oxford University Press. All rights reserved