Fredholm and spectral properties of Toeplitz operators on $H^p$ spaces over ordered groups

Toeplitz operators on spaces $H^p(G)(1<p<\mathrm{\infty })$ associated with compact connected Abelian group $G$ with ordered dual are considered and the generalization of the classical Gohberg-Krein theorem on the Fredholm index of such operators with continuous symbols is proved. Applications to spectral theory of Toeplitz operators are given and examples of evident computation of index have been considered.