Spectrum of a composition operator with automorphic symbol

TL;DR

This paper characterizes the spectrum of composition operators with automorphic symbols on various Banach spaces of analytic functions, revealing different spectral structures depending on the automorphism type.

Contribution

It provides a complete spectral description for composition operators induced by automorphisms on spaces like the Bloch space and BMOA, including parabolic, hyperbolic, and elliptic cases.

Findings

Spectrum is the unit circle for parabolic and hyperbolic automorphisms.

For elliptic automorphisms, spectrum is either the unit circle or a finite cyclic subgroup.

The results apply to a broad class of Banach spaces of analytic functions.

Abstract

We give a complete characterization of the spectrum of composition operators, induced by an automorphism of the open unit disk, acting on a family of Banach spaces of analytic functions that includes the Bloch space and BMOA. We show that for parabolic and hyperbolic automorphisms, the spectrum is the unit circle. For the case of elliptic automorphisms, the spectrum is either the unit circle or a finite cyclic subgroup of the unit circle.