Cyclicity of composition operators on the Paley-Wiener spaces

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 $0<|b|\leq \pi/\sigma$. 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.