Spectral properties of weighted composition operators on the Bloch and Dirichlet spaces

TL;DR

This paper investigates the spectral characteristics of invertible weighted composition operators on the Bloch and Dirichlet spaces, providing explicit descriptions and formulas for spectra and spectral radii under various automorphism conditions.

Contribution

It offers a comprehensive analysis of spectra for these operators, including complete descriptions for parabolic and elliptic automorphisms and explicit spectral radius formulas for hyperbolic automorphisms.

Findings

Complete spectrum description for parabolic and elliptic automorphisms in the Bloch space.

Explicit spectral radius formulas for hyperbolic automorphisms.

Spectral properties differ based on the type of automorphism of the unit disc.

Abstract

The spectra of invertible weighted composition operators $uC_\varphi$ on the Bloch and Dirichlet spaces are studied. In the Bloch case we obtain a complete description of the spectrum when $\varphi$ is a parabolic or elliptic automorphism of the unit disc. In the case of a hyperbolic automorphism $\varphi$, exact expressions for the spectral radii of invertible weighted composition operators acting on the Bloch and Dirichlet spaces are derived.