ON Γ-RECURRENT C₀-SEMIGROUPS AND THEIR PROPERTIES

Abstract

This paper introduces the concept of Γ-recurrent C₀-semigroups and providesvarious illustrative examples. We demonstrate that Γ-recurrence is preserved under inverse.Surprisingly, we establish the existence of Γ-recurrent C₀-semigroups on finite-dimensionalBanach spaces. Additionally, we explore the notion of Γ-recurrent vectors and show that if a C₀-semigroup possesses a Γ-recurrent vector, it necessarily has many such vectors. Finally,we prove that when Γ is closed under multiplication, the Γ-recurrence of a C₀-semigroup isequivalent to having a dense set of Γ-recurrent vectors.