Hypercyclic behaviour of operators in a hypercyclic C0-semigroup

Let {Tt }t 0 be a hypercyclic strongly continuous semigroup of operators. Then each Tt (t > 0) is hypercyclic as a single operator, and it shares the set of hypercyclic vectors with the semigroup. This answers in the affirmative a natural question concerning hypercyclic C0-semigroups. The analogous result for frequent hypercyclicity is also obtained. © 2006 Elsevier Inc. All rights reserved.