Eigenvalues for Infinitesimal Generators of Semigroups of Composition Operators

TL;DR

This paper investigates the eigenvalues of infinitesimal generators of semigroups of composition operators on Hardy, Bergman, and Dirichlet spaces, revealing spectral properties based on the semigroup type and domain geometry.

Contribution

It characterizes the point spectrum of these generators for various semigroup types and geometries, extending previous work to new settings.

Findings

Identifies containment relations for eigenvalues based on semigroup and domain geometry.

Provides sufficient conditions for the point spectrum characterization.

Extends spectral analysis to Dirichlet spaces and non-elliptic semigroups.

Abstract

We study the eigenvalues for infinitesimal generators of semigroups of composition operators acting on Hardy spaces, Bergman spaces, and the Dirichlet space. Such semigroups are induced by semigroups of holomorphic functions. Depending on the type of the holomorphic semigroup and the Euclidean geometry of its Koenigs domain, we find containment relations as well as sufficient conditions for the characterization of the point spectrum of the induced infinitesimal generator. For the Dirichlet space we study all types of non-elliptic semigroups whereas for the Hardy and Bergman spaces we work on parabolic semigroups extending the work of Betsakos in the hyperbolic case.