Generalized Ces\`aro operators in the disc algebra and in Hardy spaces

TL;DR

This paper explores the properties of generalized Cesàro operators acting on the disc algebra and Hardy spaces, analyzing their continuity, spectrum, and dynamic behavior to deepen understanding of their functional analysis characteristics.

Contribution

It provides a comprehensive analysis of the continuity, spectrum, and dynamics of generalized Cesàro operators on these classical function spaces, extending existing knowledge.

Findings

Characterization of when $C_t$ is continuous or compact

Determination of the spectrum and point spectrum of $C_t$

Analysis of linear dynamics and mean ergodicity of $C_t$

Abstract

Generalized Ces\`aro operators $C_t$, for $t\in [0,1)$, are investigated when they act on the disc algebra $A(\mathbb{D})$ and on the Hardy spaces $H^p$, for $1\leq p \leq \infty$. We study the continuity, compactness, spectrum and point spectrum of $C_t$ as well as their linear dynamics and mean ergodicity on these spaces.