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Li--Yorke chaotic weighted composition operators on Hardy and Bergman spaces over the unit disk

Carlos F. Álvarez, João R. Carmo, Juan Manzur

Abstract

We study Li--Yorke and mean Li--Yorke chaos for weighted composition operators $C_{w,\varphi}$ on Banach spaces of analytic functions on the unit disk $\mathbb{D}$. Under natural conditions on the space, we show that $C_{w,\varphi}$ is (densely) Li--Yorke chaotic if and only if it is not power-bounded, and (densely) mean Li--Yorke chaotic if and only if it is not absolutely Cesàro bounded. These results are applied to Hardy spaces $H^p(\mathbb{D})$, $1 \le p \le \infty$, and weighted Bergman spaces $A^2_β(\mathbb{D})$, $-1 < β< \infty$.

Li--Yorke chaotic weighted composition operators on Hardy and Bergman spaces over the unit disk

Abstract

We study Li--Yorke and mean Li--Yorke chaos for weighted composition operators on Banach spaces of analytic functions on the unit disk . Under natural conditions on the space, we show that is (densely) Li--Yorke chaotic if and only if it is not power-bounded, and (densely) mean Li--Yorke chaotic if and only if it is not absolutely Cesàro bounded. These results are applied to Hardy spaces , , and weighted Bergman spaces , .
Paper Structure (11 sections, 19 theorems, 33 equations)

This paper contains 11 sections, 19 theorems, 33 equations.

Table of Contents

  1. Introduction
  2. Li--Yorke Chaotic, Power Bounded, and Absolutely Cesàro Bounded Operators
  3. Notions of Li--Yorke chaos
  4. Power-bounded and absolutely Cesàro bounded operators
  5. Li--Yorke chaotic operators are not power-bounded
  6. Li--Yorke chaos for weighted composition operators
  7. Setup and structural conditions
  8. The Hardy spaces Hp for 1<p<infinity
  9. The weighted Bergman spaces A2beta for -1<beta<infinity
  10. The Hinfinity space
  11. Example

Key Result

Proposition 1

Let $T \in \mathcal{L}(X)$, where $X$ is an infinte-dimensional separable Banach space. If the set of all points $x \in X$ such that the sequence $(T^n x)$ has a subsequence converging to zero is dense in $X$, then it is residual in $X$.

Theorems & Definitions (39)

  • Definition 1
  • Remark 1
  • Proposition 1: See Bernardes2015
  • Definition 2
  • Theorem 1: See Bermudez2011
  • Definition 3
  • Definition 4
  • Definition 5
  • Definition 6
  • Theorem 2: See BernardesPerisSanchez
  • ...and 29 more