A bounded linear operator T defined on a Hilbert space H is said to be supercyclic if there exists a vector x ∈ H such that the set {λT n x : n ∈ N, λ ∈ C} is dense in H. In the present work, two open questions posed by N. H. Salas and J. Zemánek respectively, are solved. Namely, we will exhibit that the classical Volterra operator V and the identity plus Volterra operator I + V are not supercyclic.

A bounded linear operator T on a separable complex Banach space X is called weakly supercyclic if there exists a vector x ∈ X such that the projective orbit {λT n x : n ∈ N λ ∈ C} is weakly dense in X. Among other results, it is proved that an operator T such that σp(T ) = ∅, is weakly supercyclic if and only if T is positive weakly supercyclic, that is, for every supercyclic vector x ∈ X, only considering the positive projective orbit: {rT n x : n ∈ N, r ∈ R + } we obtain a weakly dense subset in X. As a consequence it is established the existence of non-weakly supercyclic vectors (non-trivial) for positive operators defined on an infinite dimensional separable complex Banach space. The paper is closed with concluding remarks and further directions.

Abstract. A bounded linear operator T defined on a Banach space X is said to be supercyclic if there exists a vector x ∈ X such that the projective orbit {λT n x : λ ∈ C, n ∈ N} is dense in X. The aim of this survey is to show the relationship between positivity and supercyclicity. This relationship comes from the so called Positive Supercyclicity Theorem. Throughout this exposition, interesting new directions and open problems will appear.

No abstract