Angular Derivatives
and Compact Composition Operators
on the Hardy and Bergman Spaces

Barbara D. MacCluer and Joel H. Shapiro


Canadian J. Math. 38 (1986), 878--906
Abstract: In this paper it is shown that for "normally weighted" Bergman spaces of the unit disc, a composition operator is compact if and only if its inducing map has a finite angular derivative at no point of the unit circle. The same is true for the Hardy space H^p (0 < p < infinity) provided the inducing map is univalent. These results are completed for Hardy space, and unified for Bergman spaces, in my subsequent paper: The essential norm of a composition operator (Annals of Mathematics 125 (1987), 375--404),



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