On isometries of hardy spaces on compact abelian groups

Let H^p(m),<0 p ≦ ∞, be the Hardy spaces on a quotient K of the Bohr group. In this paper we completely determine the isometries of H^p(m), p ≠ 2, onto itself. Our result is a generalization of a recent work of Muhly who determined the isometries of H^p(m) onto itself under the assumption that the dual group of K is countable, and it may be regarded as a partial answer to a question posed by Muhly.