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On the set of supercyclic operators

Thiago R. Alves, Gustavo C. Souza

Abstract

In this article, we address a problem posed by F. Bayart regarding the existence of an infinite-dimensional closed vector subspace (excluding the null operator) within the set of supercyclic operators on Banach spaces. We resolve this problem by establishing the existence of the closed subspace. Furthermore, we prove that the set of supercyclic operators on $\ell_1$ contains, up to the null operator, an isometric copy of $\ell_1$.

On the set of supercyclic operators

Abstract

In this article, we address a problem posed by F. Bayart regarding the existence of an infinite-dimensional closed vector subspace (excluding the null operator) within the set of supercyclic operators on Banach spaces. We resolve this problem by establishing the existence of the closed subspace. Furthermore, we prove that the set of supercyclic operators on contains, up to the null operator, an isometric copy of .
Paper Structure (4 sections, 5 theorems, 49 equations)

This paper contains 4 sections, 5 theorems, 49 equations.

Table of Contents

  1. Introduction and main result
  2. Background and notations
  3. Proof of the main result
  4. Technical lemmas

Key Result

Theorem 1.1

The set of supercyclic operators in $\mathcal{L}(X)$ is spaceable. Furthermore, the set of supercyclic operators in $\mathcal{L}(\ell_1)$ contains (up to the null operator) an isometric copy of $\ell_1$.

Theorems & Definitions (9)

  • Theorem 1.1
  • Lemma 4.1
  • proof
  • Lemma 4.2
  • proof
  • Lemma 4.3
  • proof
  • Lemma 4.4
  • proof