On weak positive supercyclicity

Abstract

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 {λTnx: n ∈ ℕ λ ∈ ℂ} is weakly dense in X. Among other results, it is proved that an operator T such that σp(T*) = 0, 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: {rTnx: n ∈ ℂ, 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.