Compact composition operators with symbol a universal covering map

Jones, Matthew ORCID: https://orcid.org/0000-0002-5252-5234 (2015) Compact composition operators with symbol a universal covering map. Journal of Functional Analysis, 268 (4) . pp. 887-901. ISSN 0022-1236 [Article] (doi:10.1016/j.jfa.2014.11.003)

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Abstract

In this paper we study composition operators, Cϕ, acting on the Hardy spaces that have symbol, ϕ , a universal covering map of the disk onto a finitely connected domain of the form D0\{p1,…,pn}, where D0 is simply connected and pi, i=1,…,ni=1,…,n, are distinct points in the interior of D0. We consider, in particular, conditions that determine compactness of such operators and demonstrate a link with the Poincare series of the uniformizing Fuchsian group. We show that Cϕ is compact if, and only if ϕ does not have a finite angular derivative at any point of the unit circle, thereby extending the result for univalent and finitely multivalent ϕ.

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