Abstract. Let $G$ be a nondiscrete compact abelian group, and $M(p,q)$ the algebra of translation invariant operators of $L^p(G)$ to $L^q(G)$. If $1< p< q< \infty $, the closure of the range of Gelfand transform of $T$ in $M(p,q)$ coincides with the closure of range of Fourier transform of $T$.

AMS Subject Classification (1991): 42A45, 43A22

Received December 4, 1991. (Registered under 5555/2009.)