We show mean ergodic theorems for vector-valued weakly almost periodic functions (in the sense of Eberlein) defined on a semigroup which take values in a locally convex topological vector space. Next, motivated by Fréchet , we study the relationship between almost periodicity of semigroups of mappings and their equicontinuity, and also prove mean ergodic theorems for equicontinuous semigroups.
"MEAN ERGODIC THEOREMS FOR ALMOST PERIODIC SEMIGROUPS." Taiwanese J. Math. 14 (3B) 1079 - 1091, 2010. https://doi.org/10.11650/twjm/1500405906