On the Recurrent C 0 -Semigroups, Their Existence, and Some Criteria

Abstract

In this paper, recurrent $C_0$ -semigroups are introduced and investigated. It is proved that, despite hypercyclic $C_0$ -semigroups, recurrent $C_0$ -semigroups can be found on finite-dimensional Banach spaces. Some criteria are stated for recurrence, which is based on open sets, neighborhoods of zero, and special eigenvectors. It is established that having a dense set of recurrent vectors is a sufficient and necessary condition for a $C_0$ -semigroup to be recurrent. Moreover, the direct sum of recurrent $C_0$ -semigroups is investigated.