Ces\`aro operators on the space of analytic functions with logarithmic growth

TL;DR

This paper studies the properties of Cesàro operators acting on the space of analytic functions with logarithmic growth, focusing on aspects like continuity, compactness, spectrum, and ergodic behavior.

Contribution

It provides a detailed analysis of Cesàro operators on the space $VH(D)$, a space characterized by logarithmic growth, extending previous work by Taskinen and Jasiczak.

Findings

Characterization of the spectrum of Cesàro operators

Conditions for the operators' compactness and continuity

Analysis of ergodic properties of the operators

Abstract

Continuity, compactness, the spectrum and ergodic properties of Ces\`aro operators are investigated when they act on the space $VH(\mathbb{D})$ of analytic functions with logarithmic growth on the open unit disc $\mathbb{D}$ of the complex plane. The space $VH(\mathbb{D})$ is a countable inductive limit of weighted Banach spaces of analytic functions with compact linking maps. It was introduced and studied by Taskinen and also by Jasiczak.