Spectra of composition operators on Paley-Wiener spaces and some consequences
Carlos F. Álvarez, O. R. Severiano
TL;DR
This work characterizes bounded composition operators on the Paley–Wiener spaces $PW_a$ as those induced by affine symbols $\phi(z)=cz+d$ with $c\in\mathbb{R}$ and $0<|c|\le 1$, and determines their spectra, spectral radii, and several dynamical properties. By leveraging an isometric similarity to weighted composition operators on $L^2([-a,a])$ and reproducing-kernel techniques, the authors derive explicit spectral descriptions: for $|c|<1$, $\sigma(C_{\phi})=\{\lambda:|\lambda|\le|c|^{-1/2}\}$; for $c=1$, $\sigma(C_{\phi})=\{e^{i dt}: t\in[-a,a]\}$; for $c=-1$, $\sigma(C_{\phi})=\{-1,1\}$; and the spectral radius is $|c|^{-1/2}$ (or $e^{|\mathrm{Im}(d)|a}$ when $c=1$). The paper shows there are no compact operators on any $PW_a$, no Li–Yorke chaos, and no positive shadowing property for these operators; absolutely Cesàro boundedness occurs only in the cases $c=-1$ or $c=1$ with $d\in\mathbb{R}$. These results complete the affine-symbol landscape for $PW_a$ and contribute precise dynamical-characterization results for composition operators on Paley–Wiener spaces.
Abstract
Bounded composition operators in Paley-Wiener spaces have simple forms, and they are just operators composed through affine mappings of the complex plane. The purpose of this article is to explore some notions about bounded operators and linear dynamics and provide complete answers for composition operators in Paley-Wiener spaces concerning compactness, spectrum, spectral radius, Li-Yorke chaos, positive expansivity, positive shadowing property, and absolute Cesàro boundedness.