In Koenigs' footsteps: Diagonalization of composition operators

Abstract

Let φ:D→D be a holomorphic map with a fixed point α∈D such that 0≤|φ′(α)|<1. We show that the spectrum of the composition operator Cφ on the Fréchet space Hol(D) is {0}∪{φ′(α)n:n=0,1,⋯} and its essential spectrum is reduced to {0}. This contrasts the situation where a restriction of Cφ to Banach spaces such as H2(D) is considered. Our proofs are based on explicit formulae for the spectral projections associated with the point spectrum found by Koenigs. Finally, as a byproduct, we obtain information on the spectrum for bounded composition operators induced by a Schröder symbol on arbitrary Banach spaces of holomorphic functions.