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\(\Gamma \)-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.