On finite dimensional and nuclear operators in Hardy spaces $H^2$ on compact Abelian groups

Compact and connected Abelian group $G$ with totally ordered dual is considered. It is shown that nontrivial finite rank Hankel operator exists on $G$ if and only if the dual group contains the first positive element. In this case the classical theorems by Kroneker, Hartman, and Peller are generalized to the case of Hankel operators on $G$.