Towards spectral descriptions of cyclic functions

TL;DR

This paper explores the spectral properties of operators associated with cyclic functions in weighted Hardy and Bergman-type spaces, linking spectral characteristics to function cyclicity.

Contribution

It extends Le's characterization of inner functions to weighted Hardy spaces and fully describes spectra and eigenfunctions in Bergman-type spaces.

Findings

Spectral properties of operators relate to cyclicity in weighted Hardy spaces.

Complete spectral description and eigenfunctions for operators in Bergman-type spaces.

Several spectral properties hold broadly across classes of spaces.

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 $H^2_\omega$ 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.