Let (G,X) be a compact transformation semigroup. Suppose that (G,X) is topologically transitive, i.e. Gx is dense in X for some \(x\in X\). A function f in C(X) is called an eigenfunction if there is a function \(\lambda\) on G such that \(f(gx)=\lambda (g) f(x)\) for all \(g\in G\), \(x\in X\). If the closed linear span of the set of eigenfunctions is C(X) then (G,X) is said to have a topological discrete spectrum.
The aim of this paper is to obtain characterizations of such transformation semigroups with topological discrete spectra.