On recurrent sets of operators

Abstract

In this paper, we introduce and study the notion of recurrent sets of operators and some of its variations on Banach spaces. As application, we study the recurrence of \(C\)-regularized group of operators. We show that there exists recurrent \(C\)-regularized group in each Banach space with finite or infinite dimensional. Moreover, we prove that if \((S(z))_{z\in\mathbb{C}}\) is a recurrent \(C\)-regularized group and \(S(z_0)\) is an operator in this \(C\)-regularized group, then \(S(z_0)\) is not necessarily recurrent.