Composition operators on weighted Hardy spaces of polynomial growth

TL;DR

This paper investigates the properties of composition operators on weighted Hardy spaces of polynomial growth, focusing on norms, spectra, and Fredholmness, and characterizes these operators based on their symbols.

Contribution

It provides new estimates for norms, characterizes spectra for automorphism symbols, and establishes criteria for Fredholmness of composition operators on these spaces.

Findings

Norm estimates for composition operators with disk automorphism symbols.

Spectral characterization showing spectrum is the unit circle for parabolic automorphisms.

Fredholmness criteria linking the property to the symbol being a finite Blaschke product or automorphism.

Abstract

In the present paper, we study the composition operators acting on weighted Hardy spaces of polynomial growth, which are concerned with norms, spectra and (semi-)Fredholmness. Firstly, we estimate the norms of the composition operators with symbols of disk automorphisms. Secondly, we discuss the spectra of the composition operators with symbols of disk automorphisms. In particular, it is proven of that the spectrum of a composition operator with symbol of any parabolic disk automorphism is always the unit circle. Thirdly, we consider the Fredholmness of the composition operator $C_{\varphi}$ with symbol $\varphi$ which is an analytic self-map on the closed unit disk. We prove that $C_{\varphi}$ acting on a weighted Hardy space of polynomial growth has closed range (semi-Fredholmness) if and only if $\varphi$ is a finite Blaschke product. Furthermore, it is obtained that $C_{\varphi}$ is Fredholm if and only if $\varphi$ is a disk automorphism.