A note about the spectrum of composition operators induced by a rotation

Abstract

A characterization of those points in the unit disc which belong to the spectrum of a composition operator $C_{\varphi}$, defined by a rotation $\varphi(z)=rz$ with $|r|=1$, on the space $H_0(\mathbb{D})$ of all analytic functions on the unit disc which vanish at $0$, is given. Examples show that the point $1$ may or may not belong to the spectrum of $C_{\varphi}$, and this is related to Diophantine approximation. Our results complement recent work by Arendt, Celari\`es and Chalendar.