Quasi-Mixing C₀-Semigroups and Their Relations With Recurrent C₀-Semigroups

This article presents and investigates the concept of the quasi-mixing C0-semigroups. It is proved that the quasi-mixing of an invertible C0-semigroup is equivalent to the quasi-mixing of its inverse. The quasi-mixing vectors are defined and they are used to prove a sufficient condition for the quasi-mixing. It is established that the quasi-mixing of a C0-semigroup is equivalent to the quasi-mixing of any of its discretizations. The quasi-mixing of a C0-semigroup ( Tt )t ≥ 0, as it is demonstrated by this study, implies the quasi-mixing for each operator Tt, t ≥ 0. It is proved that the quasi-mixing of the direct sum of two C0-semigroups is equivalent to the quasi-mixing of both semigroups. Furthermore, some relations between the quasi-mixing and the recurrence are explored in this article. It is shown that the operators of a quasi-mixing C0-semigroup are recurrent. Also, it is demonstrated that the recurrence of every discretization of a C0-semigroup is equivalent to the quasi-mixing of the C0-semigroup.