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

We consider Toeplitz operators on the spaces $H^p(G)$, $1< p<\infty$, associated with a compact connected Abelian group $G$ whose character group is ordered and, in the case of total order, prove a theorem on the Fredholm index for those operators which have continuous symbols which generalizes the classical Gohberg-Krein theorem. The results thus obtained are applied to the spectral theory of Toeplitz operators and examples where the index is evaluated explicitly are considered.
Bibliography: 22 titles.