Using Read's construction of operators without non-trivial invariant subspaces/subsets on ℓ1 or c0, 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. 2000 Mathematics Subject Classification. -47A15, 47A16. Key words and phrases. -Cyclic and hypercyclic vectors, orbits of linear operators, invariant subspaces, Haar and Gauss null sets, orbit-unicellular operators, orbit-reflexive operators.The first author was partially supported by ANR-Projet Blanc DYNOP.

Let T be an operator on a separable Banach space, and denote by p (T ) its point spectrum. We answer a question left open in (Israel J. Math. 146 (2005) 93-110) by showing that it is possible that p (T ) ∩ T be uncountable, yet T n ∞. We further investigate the relationship between the growth of sequences (n k ) such that sup k T n k < ∞ and the possible size of p (T ) ∩ T. Analogous results are also derived for continuous operator semigroups (T t ) t 0 .

We study how the singularity (in the sense of Hausdorff dimension) of a vector valued measure can be affected by certain restrictions imposed on its Fourier transform. The restrictions, we are interested in, concern the direction of the (vector) values of the Fourier transform. The results obtained could be considered as a generalizations of F. and M. Riesz theorem, however a phenomenon, which have no analogy in the scalar case, arise in the vector valued case. As an example of application, we show that every measure from µ = (µ 1 , . . . , µ d ) ∈ M(R d , R d ) annihilating gradients of C (1) 0 (R d ) embedded in the natural way into C 0 (R d , R d ), i.e., such that i ∂ i f dµ i = 0 for f ∈ C (1) 0 (R d ), has Hausdorff dimension at least one. We provide examples which show both completeness and incompleteness of our results.

We present a general method for constructing operators without non-trivial invariant closed subsets on a large class of non-reflexive Banach spaces. In particular, our approach unifies and generalizes several constructions due to Read of operators without non-trivial invariant subspaces on the spaces 1, c0 or 2 J, and without non-trivial invariant subsets on 1. We also investigate how far our methods can be extended to the Hilbertian setting, and construct an operator on a quasi-reflexive dual Banach space which has no non-trivial w * -closed invariant subspace.We retrieve in particular the existence of an operator on 1 with no non-trivial invariant closed subset [22], and improve the results of [23] by showing that c 0 and 2 J support operators without non-trivial invariant closed subsets.An interesting consequence of Theorem 1.1 is the following corollary.Corollary 1.2. Let X be an infinite-dimensional non-reflexive separable space having an unconditional basis. Then there exists a bounded operator on X with no non-trivial invariant closed subset.