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

Jones, Matthew ORCID: https://orcid.org/0000-0002-5252-5234 (2011) A note on the Königs domain of compact composition operators on the Bloch space. Journal of Inequalities and Applications (31) . ISSN 1029-242X [Article] (doi:10.1186/1029-242X-2011-31)

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Abstract

Let D be the unit disk in the complex plane. We define B0 to be the little Bloch space of functions f analytic in D which satisfy lim┬(|z|→1)⁡〖(1- |z|^2 )|f^' (z) |=0.〗 If φ:D→D is analytic then the composition operator C_φ:f→f∘φ is a continuous operator that maps B0 into itself. In this paper, we show that the compactness of C_φ, as and operator on B0, can be modelled geometrically by its principle eigenfunction. In particular, under certain necessary conditions, we relate the compactness of C_φto the geometry of Ω=σ(D) where σ satisfies Schroder’s functional equation σ∘φ=φ^' (0)σ.

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