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 $V=M_f^{\ast }{M}_f$ and study to what extent the cyclicity on weighted Hardy spaces $H_{\ddot{I}}^2‰$ of the function $z\mapsto a-z$ can be inferred from the spectral properties of analogous operators $V_a$. 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.