The spectrum of composition operators induced by a rotation in the space of all analytic functions on the disc

TL;DR

This paper characterizes the spectrum of rotation-induced composition operators on a space of analytic functions, revealing how spectral points depend on Diophantine approximation and extending recent spectral theory results.

Contribution

It provides a detailed spectral characterization for composition operators induced by rotations on a specific analytic function space, connecting spectral properties with number-theoretic approximation.

Findings

The spectrum depends on whether the rotation angle satisfies Diophantine conditions.

The point 1 may or may not be in the spectrum depending on the rotation.

Results extend and complement existing spectral operator theories.

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.