A note on the Königs domain of compact composition operators on the Bloch space

Abstract

Abstract Let be the unit disk in the complex plane. We define to be the little Bloch space of functions f analytic in which satisfy lim|z|→1 (1 - |z|2)|f'(z)| = 0. If is analytic then the composition operator Cφ : f ↦ f ∘ φ is a continuous operator that maps into itself. In this paper, we show that the compactness of Cφ , as an operator on , can be modelled geometrically by its principal eigenfunction. In particular, under certain necessary conditions, we relate the compactness of Cφ to the geometry of , where σ satisfies Schöder's functional equation σ ∘ φ = φ'(0)σ. 2000 Mathematics Subject Classification: Primary 30D05; 47B33 Secondary 30D45.