C0-semigroups and mean ergodic operators in a class of Fréchet spaces

It is shown that the generator of every exponentially equicontinuous, uniformly continuous C0-semigroup of operators in the class of quojection Fréchet spaces X (which includes properly all countable products of Banach spaces) is necessarily everywhere defined and continuous. If, in addition, X is a Grothendieck space with the Dunford–Pettis property, then uniform continuity can be relaxed to strong continuity. Two results, one of M. Lin and one of H.P. Lotz, both concerned with uniformly mean ergodic operators in Banach spaces, are also extended to the class of Fréchet spaces mentioned above. They fail to hold for arbitrary Fréchet spaces.