Königs eigenfunction for composition operators on Bloch and H∞ type spaces

We discuss when the Königs eigenfunction associated with a non-automorphic selfmap of the complex unit disc that fixes the origin belongs to Banach spaces of holomorphic functions of Bloch and H∞H∞ type. In the latter case, our characterization answers a question of P. Bourdon. Some spectral properties of composition operators on H∞H∞ for unbounded Königs eigenfunction are obtained.