Recurrent and Subspace-recurrent C0-semigroups and Their Properties

Abstract

In this chapter, we are familiar with recurrence and subspace-recurrence of the \(C_0\)- semigroups. Recurrent \(C_0\)-semigroups and subspace-recurrent \(C_0\)-semigroups are introduced and their properties are investigated. It is established that recurrent \(C_0\)-semigroups and subspace-recurrent \(C_0\)-semigroups can be constru-cted on finite-dimensional Banach spaces. Some criteria for recurrency are stated in this chapter. Some of them are based on open sets, neighborhoods of zero, and some of them are based on special eigenvectors. Recurrent vectors are introduced. It is proved that if a \(C_0\)-semigroup has a dense set of recurrent vectors, then it is recurrent and it is proved that a recurrent \(C_0\)-semigroup has a dense set of recurrent vectors. It is stated that the recurrence of the direct sum of two \(C_0\)-semigroups implies that any of them is a recurrent \(C_0\)-semigroup. It is shown that the direct sum of two mixing \(C_0\)-semigroups is recurrent. Also, if a \(C_0\)-semigroup satisfies the Hypercyclicity Criterion, it is recurrent.