Cyclicity of composition operators on the Paley-Wiener spaces

TL;DR

This paper characterizes when bounded composition operators on Paley-Wiener spaces are cyclic, linking their properties to the form of the symbol function and the completeness of exponential sequences.

Contribution

It provides a precise characterization of cyclicity for composition operators on Paley-Wiener spaces, including conditions on the symbol and the relation to exponential sequence completeness.

Findings

Cyclic composition operators occur only for specific affine symbols.

Reproducing kernels are cyclic vectors under certain conditions.

Cyclicity relates to the completeness of exponential sequences in L^2 spaces.

Abstract

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.