On the equivalence between the uniform exponential stability of a \(C_0\)-semigroup and the boundedness of the resolvent

We consider a \(C_0\) $C_0$ -semigroup on a Banach space such that the resolvent is uniformly bounded on the right half-plane. In this paper we provide a condition on the resolvent which is sufficient and necessary for the uniform exponential stability of such a semigroup. As a consequence, we give an alternative proof of Gearhart’s theorem (Trans. Amer. Math. Soc. 236, 385–394 (1978)). The approach lies on a complex inversion formula and tempered distributions.