Dynamics and spectra of composition operators on the Schwartz space

2017/07/12 by Fernández, Carmen, Galbis, Antonio, Jordá, Enrique
#FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.1707.03627

Abstract

In this paper we study the dynamics of the composition operators defined in the Schwartz space S(ℝ) of rapidly decreasing functions. We prove that such an operator is never supercyclic and, for monotonic symbols, it is power bounded only in trivial cases. For a polynomial symbol φ of degree greater than one we show that the operator is mean ergodic if and only if it is power bounded and this is the case when φ has even degree and lacks fixed points. We also discuss the spectrum of composition operators.

Related

• Spectrum of composition operators on \mathcal S(\mathbb R) with polynomial symbols
• Composition operators on the Schwartz space
• Spectra and ergodic properties of multiplication and convolution operators on the space S(ℝ)
• Dynamics of composition operators on function spaces defined by local and global properties
• Expansivity and shadowing for composition operators on Paley-Wiener spaces