Continuity of operators commuting with translations for compact groups

In this note we show for certain Frechet spacesF(G) of functions (distributions) on a compact groupG that if every translation invariant linear functional onF(G) is continuous then every linear operatorT:F(G)F(G) commuting with translations is continuous. This solves partially a problem in 7] ofG. H. Meisters and improves the result 5] ofC. J. Lester. An application for compact groups which do not have the mean zero weak containment property follows by the result 10] ofG. A. Willis.