Spectra of infinitesimal generators of composition semigroups on weighted Bergman spaces induced by doubling weights

TL;DR

This paper analyzes the spectrum of the infinitesimal generator of composition semigroups on weighted Bergman spaces with doubling weights, focusing on elliptic semigroups and employing spectral mapping and operator characterization techniques.

Contribution

It provides a spectral analysis of the generator on weighted Bergman spaces induced by doubling weights, including characterizations of spectra and compact operators, extending previous results.

Findings

Spectral characterization of the infinitesimal generator.

Conditions for norm-continuity of the semigroup.

Identification of related compact integral operators.

Abstract

Suppose $(C_t)_{t\geq0}$ is the composition semigroup induced by a one-parameter semigroup $(\varphi_t)_{t\geq0}$ of analytic self-maps of the unit disk. The main purpose of the paper is to investigate the spectrum of the infinitesimal generator of $(C_t)_{t\geq0}$ acting on the weighted Bergman space induced by doubling weights, provided $(\varphi_t)_{t\geq0}$ is elliptic. The method applied is a certain spectral mapping theorem and a characterization of the spectra of certain composition operators. Eventual norm-continuity of $(C_t)_{t\geq0}$ also plays an important role, which can be depicted in terms of studying the difference of two distinct composition operators. As a byproduct, we also characterize a certain compact integral operator that is closely related to the resolvent of the infinitesimal generator of $(C_t)_{t\geq0}$.