The endpoint problem for $\epsilon$-hypercyclicity

Abstract

For a fixed $0<\epsilon <1$, F. Bayart asked in 2024 whether there exists an operator $T$ such that, for every $0<\delta <1$, $T$ is $\delta$-hypercyclic if and only if $\delta \in [\epsilon ,1)$. We answer this question affirmatively by constructing a weighted backward shift on ${\ell }_2({\mathbb{N}}_0,{\ell }_2({\mathbb{N}}_0))$ with this property.

arXiv comment16 pages