Cyclicity of composition operators on the Paley-Wiener spaces

Abstract

In this article we characterize the cyclicity of bounded composition operators $C_{\varphi }f=f\circ \varphi$ on the Paley-Wiener spaces of entire functions $B_{\sigma }^2$ for $\sigma >0$. We show that $C_{\varphi }$ is cyclic precisely when $\varphi (z)=z+b$ where either $b\in \mathbb{C}\setminus \mathbb{R}$ or $b\in \mathbb{R}$ with $0<|b|\le \pi /\sigma$. We also describe when the reproducing kernels of $B_{\sigma }^2$ are cyclic vectors for $C_{\varphi }$ 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.

References in corpus (2)

• Complex symmetric partial isometries
• Composition operators on reproducing kernel Hilbert spaces with analytic positive definite functions