(untitled) - Viet Hai, Pham; Noor, Waleed; Reis Severiano, Osmar (Comptes Rendus. Mathématique, 2025)
In this article we characterize the cyclicity of bounded composition operators $C_\phi f=f\circ \phi $ on the Paley–Wiener spaces of entire functions $B^2_\sigma $ for $\sigma >0$. We show that $C_\phi $ is cyclic precisely when $\phi (z)=z+b$ where either $b\in \mathbb{C}\setminus \mathbb{R}$ or $b\in \mathbb{R}$ with $0<\vert b \vert \le \pi /\sigma $. We also describe when the reproducing kernels of $B^2_\sigma $ are cyclic vectors for $C_\phi $ and see that this is related to a question of completeness of exponential sequences in $L^2[-\sigma ,\sigma ]$. The interplay between cyclicity and complex symmetry plays a key role in this work.