We build on a characterization of inner functions f due to Le, in terms of the spectral properties of the operator \(V=M_f^*M_f\) and study to what extent the cyclicity on weighted Hardy spaces \(H^2_\omega \) of the function \(z \mapsto a-z\) can be similarly inferred from the spectral properties of the corresponding operator V. We describe several properties of the spectra that hold in a large class of spaces and then, we focus on the particular case of Bergman-type spaces, for which we describe completely the spectrum of such operators and find all eigenfunctions.
