In Koenigs' footsteps: Diagonalization of composition operators

Abstract

Let $\phi :\mathbb{D}\to \mathbb{D}$ be a holomorphic map with a fixed point $\alpha \in \mathbb{D}$ such that $0\le |{\phi }^{\prime }(\alpha )|<1$. We show that the spectrum of the composition operator $C_{\phi }$ on the Fréchet space $\text{Hol}(\mathbb{D})$ is $\{0\}\cup \{{\phi }^{\prime }(\alpha {)}^n:n=0,1,\cdots \}$ and its essential spectrum is reduced to $\{0\}$. This contrasts the situation where a restriction of $C_{\phi }$ to Banach spaces such as $H^2(\mathbb{D})$ 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.