$ \mathbb{C} $-supercyclic versus $ \mathbb{R}^+ $-supercyclic operators

Abstract.

An operator T on a complex, separable, infinite dimensional Banach space X is supercyclic ($\mathbb{C}$-supercyclic) if there is a vector $x \in X$ such that the set of complex scalar multiples of the orbit {x, Tx, T ² x,...} is dense. We study different definitions of supercyclicity with real numbers ($\mathbb{R}$-supercyclic) and positive real numbers ($\mathbb{R}^+$-supercyclic). In particular, we show that T is $\mathbb{R}$-supercyclic if and only if T is $\mathbb{R}^+$-supercyclic, and we give examples of $\mathbb{C}$-supercyclic operators which are not $\mathbb{R}^+$-supercyclic.