On compactness of composition operators on Bergman spaces over planar domains. (English) Zbl 1515.47033
Summary: In this paper we consider compact composition operators acting between Bergman spaces over planar domains. When those two domains equal the open unit disc, it is well known that any analytic selfmap of that disc induces a bounded composition operator, and that operator is compact if and only if the inducing selfmap has no finite angular derivative on the unit circle. We reformulate that condition in geometric terms and prove that it characterizes compact composition operators between Bergman spaces over planar domains if the boundaries of those domains are \(C^2\)-smooth. We give examples showing that the smoothness condition cannot be dropped. We also discuss some interesting phenomena related to domains with cusps on the boundary.
MSC:
| 47B33 | Linear composition operators |
| 30H05 | Spaces of bounded analytic functions of one complex variable |
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