On weak positive supercyclicity (Q1001412)

Let \(X\) be a complex topological vector space and let \({\mathcal M}'\) be the commutant of a semigroup \({\mathcal M}\) of continuous linear operators on \(X\). The main result of this paper states that if there is an operator \(T\in{\mathcal M}\cap{\mathcal M}'\) which is not a multiple of the identity and for which \(p(T)\) has dense range for all polynomials \(p\) such that \(p(T)\not=0\), and if for some \(x\in X\), the set \(\{\lambda Sx:|\lambda|=1\text{ and }S\in{\mathcal M}\}\) is dense in \(X\), then the set \(\{Sx:S\in{\mathcal M}\}\) is dense in \(X\). From this result, the authors derive several consequences regarding hypercyclic, weakly hypercyclic, supercyclic and weakly supercyclic operators.