The concept and properties of quasi-rigid $C_{0}$-semigroups

In this paper, quasi-rigid $C_{0}$-semigroups are introduced and their properties are investigated. Also, some sufficient conditions for quasi-rigidity are stated. It is proved that quasi-rigid $C_{0}$-semigroups are recurrent. It is stated that if a $C_{0}$-semigroup contains a quasi-rigid operator, then it is quasi-rigid. The concept of topologically quasi-rigid $C_{0}$-semigroups is introduced. It is proved that topologically quasi-rigidity and quasi-rigidity are equivalent. The direct sum of quasi-rigid $C_{0}$-semigroups is investigated. It is demonstrated that a $C_{0}$-semigroup is quasi-rigid if and only if the direct sum of the $C_{0}$-semigroup with itself is quasi-rigid. Moreover, the finite direct sum of a quasi-rigid operator with itself is recurrent. So, quasi-rigidity with respect to the recurrency is similar to the weakly-mixing with respect to the hypercyclicity.