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Generalized Cesàro operators in weighted Banach spaces of analytic functions with sup-norms

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

Abstract

An investigation is made of the generalized Cesàro operators $C_t$, for $t\in [0,1]$, when they act on the space $H(\mathbb{D})$ of holomorphic functions on the open unit disc $\mathbb{D}$, on the Banach space $H^\infty$ of bounded analytic functions and on the weighted Banach spaces $H_v^\infty$ and $H_v^0$ with their sup-norms. Of particular interest are the continuity, compactness, spectrum and point spectrum of $C_t$ as well as their linear dynamics and mean ergodicity.

Generalized Cesàro operators in weighted Banach spaces of analytic functions with sup-norms

Abstract

An investigation is made of the generalized Cesàro operators , for , when they act on the space of holomorphic functions on the open unit disc , on the Banach space of bounded analytic functions and on the weighted Banach spaces and with their sup-norms. Of particular interest are the continuity, compactness, spectrum and point spectrum of as well as their linear dynamics and mean ergodicity.
Paper Structure (3 sections, 18 theorems, 85 equations)

This paper contains 3 sections, 18 theorems, 85 equations.

Table of Contents

  1. Introduction and Preliminaries
  2. Continuity, Compactness and Spectrum of $C_t$
  3. Linear dynamics and Mean Ergodicity of $C_t$

Key Result

Lemma 1.1

Let $X$ be a lcHs. The compact operators are a 2-sided ideal in ${\mathcal{L}}(X)$.

Theorems & Definitions (36)

  • Lemma 1.1
  • Proposition 2.1
  • proof
  • Example 2.2
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • proof
  • Corollary 2.5
  • proof
  • ...and 26 more