Continuity of operators commuting with translations for compact groups (Q1826063)

It is shown for certain Frechet spaces F(G) of functions (distributions) on a compact group G that if every translation invariant linear functional on F(G) is continuous then every linear operator T: F(G)\(\to F(G)\) commuting with translations is continuous. This solves partially a problem of \textit{G. H. Meisters} [Radical Banach algebras, Proc. Conf., Long Beach 1981, Lect. Notes Math. 975, 423-444 (1983; Zbl 0507.46022)]. An application for compact groups which do not have the mean zero weak containment property follows by a result of \textit{G. A. Willis} [Monatsh. Math. 105, 161-164 (1988; Zbl 0637.43003)].