# $$\Gamma$$-Supercyclicity for Strongly Continuous Semigroups

Research paper by Arafat Abbar

Indexed on: 28 Apr '20Published on: 03 Jul '19Published in: Complex Analysis and Operator Theory

#### Abstract

We characterize the subsets $$\Gamma$$ of the complex plane $$\mathbb {C}$$ for which the notion of $$\Gamma$$-supercyclicity for strongly continuous semigroups coincides with that of hypercyclicity. In addition, we characterize the sets $$\Gamma \subset {\mathbb {C}}$$ such that for every strongly continuous semigroup of operators $${\mathcal {T}}=(T_t)_{t\geqslant 0}$$ on a Banach space X, a vector $$x\in X$$ is hypercyclic for $${\mathcal {T}}$$ if and only if $${\mathrm {Orb}}(\Gamma x,\mathcal {T})$$ is somewhere dense in X. We derive a characterization of $$C_0$$-semigroup versions of 1-dimensional hypercyclic subsets and Bourdon–Feldman subsets. We finally prove a multi-$$\Gamma$$-supercyclicity result for $$C_0$$-semigroups.