Königs eigenfunction for composition operators on analytic function spaces

In 1884 G. Königs solved Schröder’s functional equation $Cφ(f) = foφ =\lambda f$, where φ is a given non-automorphic selfmap of the complex unit disc D that fixes the origin and 0 |φ′(0)| 1. The solution to Schröder’s equation is called the Königs eigenfunction of the composition operator C_{φ}$ and its eigenvalue is the multiplier $\lambda= φ′(0)$. In this talk we discuss when the Königs$ eigenfunction belongs to Banach spaces of holomorphic functions on D, like Bloch type and H1 .type spaces