Towards spectral descriptions of cyclic functions

Abstract

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 $H^2$ 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.