Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Generalized Cesàro operators in the disc algebra and in Hardy spaces

Angela A. Albanese, José Bonet, Werner J. Ricker

Abstract

Generalized Cesàro 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.

Generalized Cesàro operators in the disc algebra and in Hardy spaces

Abstract

Generalized Cesàro operators , for , are investigated when they act on the disc algebra and on the Hardy spaces , for . We study the continuity, compactness, spectrum and point spectrum of as well as their linear dynamics and mean ergodicity on these spaces.

Paper Structure

This paper contains 3 sections, 16 theorems, 62 equations.

Table of Contents

  1. Introduction and Preliminaries
  2. Continuity, Compactness and Spectrum of $C_t$ in $H^\infty$ and in $A({\mathbb D})$
  3. Continuity, Compactness and Spectrum of $C_t$ acting in $H^p$

Key Result

Theorem 1.1

Let $X$ be a Banach space and let $T\in {\mathcal{L}}(X)$ be a compact operator such that $\sigma(T;X)\subseteq \overline{{\mathbb D}}$ and $\sigma(T;X)\cap {\mathbb T}=\{1\}$ and which satisfies $\mathop{\rm Ker} (I-T)\cap {\rm Im} (I-T)=\{0\}$. Then $T$ is power bounded and uniformly mean ergodic.

Theorems & Definitions (31)

  • Theorem 1.1
  • Remark 1.2
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • proof
  • ...and 21 more