Characteristic of left invertible semigroups and admissibility of observation operators

In this paper we discuss the characteristic property of the left invertible semigroups on general Banach spaces and admissibility of the observation operators for such semigroups. We obtain a sufficient and necessary condition about their generators. Further, for the left invertible and exponentially stable semigroup in Hilbert space we show that there is an equivalent norm under which it is contractive. Based on these results we prove that for any observation operator satisfying the resolvent condition is admissible for the left invertible semigroup if its range is finite-dimensional. In addition we prove that any observation operator satisfying the resolvent condition can be approximated by the admissible observation operators. Finally we give a sufficient condition of exact observability of the left invertible semigroup.