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

Matthew M Jones

doi:10.1186/1029-242x-2011-31

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 C φ : f ↦ f ∘ φ is a continuous operator that maps into itself. In this paper, we show that the compactness of C φ , as an operator on , can be modelled geometrically by its principal eigenfunction. In particular, under certain necessary conditions, we relate the compactness of C φ 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.

Highlights

Let D = {z ∈ C : |z| < 1} be the unit disk in the complex plane and T its boundary
We denote by B0 the little Bloch space of functions in B that satisfy lim|z|®1 (1 - |z|2)|f ’(z)| = 0
The purpose of this paper is to provide a similar result to this in the context of the Bloch space

Summary

Introduction

We denote by B0 the little Bloch space of functions in B that satisfy lim|z|®1 (1 - |z|2)|f ’(z)| = 0. If f : D → is a universal covering map and Ω is a hyperbolic domain in C, the Poincaré density on Ω is derived from the equation λ (f (z))|f (z)| = λD(z), which is independent of the choice of f. In [5], the Königs domain of a compact composition operator on the Hardy space was studied and the following result was proved.

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Conclusion