On the set of supercyclic operators

Abstract

In this article, we address a problem posed by 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 fully 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 ${\ell }_1$.