Montel resolvents and uniformly mean ergodic semigroups of operators

For $C_0$-semigroups of continuous linear operators acting in a Banach space criteria are available which are equivalent to uniform mean ergodicity of the semigroup, meaning the existence of the limit (in the operator norm) of the Cesàro or Abel averages of the semigroup. Best known, perhaps, are criteria due to Lin, in terms of the range of the infinitesimal generator A being a closed subspace or, whether 0 belongs to the resolvent set of A or is a simple pole of the resolvent map $\lambda\mapsto (\lambda − A)^{-1}$. It is shown in the setting of locally convex spaces (even in Fréchet spaces), that neither of these criteria remain equivalent to uniform ergodicity of the semigroup (i.e., the averages should now converge for the topology of uniform convergence on bounded sets). Our aim is to exhibit new results dealing with uniform mean ergodicity of $C_0$-semigroups in more general spaces. A characterization of when a complete, barrelled space with a basis is Montel, in terms of uniform mean ergodicity of certain C0-semigroups acting in the space, is also presented.