Spectra of Weighted Composition Operators with Quadratic Symbols

TL;DR

This paper determines the spectra of weighted composition operators with quadratic symbols on the Hardy Space, especially for parabolic type maps that converge to the Denjoy-Wolff point and are essentially linear fractional.

Contribution

It extends spectral analysis to quadratic self-maps of the disk of parabolic type, combining previous hypotheses into a unified framework.

Findings

Spectra can be explicitly computed for quadratic parabolic maps.

Most quadratic maps of this type exhibit both key properties for spectral analysis.

The results unify and extend previous spectral findings for composition operators.

Abstract

Previously, spectra of certain weighted composition operators on the Hardy Space were discovered under one of two hypotheses: either the compositional symbol converges under iteration to the Denjoy-Wolff point on all of the open disk rather than compact subsets, or the compositional symbol is "essentially linear fractional". We show that if the symbol is a quadratic self-map of the disk of parabolic type, then the spectrum of the weighted composition operators can be found when these maps exhibit both of the aforementioned properties, and most of them do.