Compact Composition operators on L^1


Joel H. Shapiro and Carl Sundberg


Proc. Amer. Math. Soc. 108 (1990), 443--449.
Abstract: The composition operator induced by a holomorphic selfmap of the unit disc is compact on L^1 of the unit circle if and only if it is compact on the Hardy space H^2 of the disc. This shows that an integral condition of Sarason characterizing compactness on L^1 is equivalent to Shapiro's asymptotic condition on the Nevanlinna counting function that characterizes compactness on H^2 (The essential norm of a composition operator, Annals of Mathematics125 (1987), 375--404).


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