Spectra of some composition operators and associated weighted composition operators

TL;DR

This paper characterizes the spectrum and essential spectrum of certain composition and weighted composition operators on the Hardy space, especially when their symbols have Denjoy-Wolff points on the unit circle.

Contribution

It provides a detailed spectral analysis of essentially linear fractional composition operators and their associated weighted operators on the Hardy space.

Findings

Spectrum and essential spectrum coincide when the Denjoy-Wolff point is on the unit circle.

Explicit descriptions of spectra for a class of weighted composition operators.

Characterization of spectra for operators with essentially linear fractional symbols.

Abstract

We characterize the spectrum and essential spectrum of "essentially linear fractional" composition operators acting on the Hardy space H-two of the open unit disc U. When the symbols of these composition operators have Denjoy-Wolff point on the unit circle, the spectrum and essential spectrum coincide. Our work permits us to describe the spectrum and essential spectrum of certain associated weighted composition operators on the Hardy space.