
Joel H. Shapiro 

Abstract: We find natural hypothesis on a Banach space X of functions holomorphic in the unit disc which insure that: if a holomorphic selfmap of the disc induces a compact composition operator on X then it must take the disc into a relatively compct subset. Spaces X that satisfy our hypothesis include the disc algebra, "heavily weighted" Dirichlet spaces, and spaces with a derivative in a Hardy space H^p (p >=1). It is well known that ``large'' spaces such as Hardy and Bergman spaces do not have this property, but surprisingly it also fails in very small spaces. The property of Mobiiusinvariance place a crucial and mysterious role in these matters. 
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