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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Compact composition operators on $ L\sp 1$


Authors: Joel H. Shapiro and Carl Sundberg
Journal: Proc. Amer. Math. Soc. 108 (1990), 443-449
MSC: Primary 47B38; Secondary 30D55, 47B05
MathSciNet review: 994787
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Abstract: The composition operator induced by a holomorphic self-map 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 answers a question posed by Donald Sarason: it proves that Sarason's integral condition characterizing compactness on $ {L^1}$ is equivalent to the asymptotic condition on the Nevanlinna counting function which characterizes compactness on $ {H^2}$.


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DOI: https://doi.org/10.1090/S0002-9939-1990-0994787-0
Keywords: Compact composition operator, Nevanlinna counting function, Riesz mass
Article copyright: © Copyright 1990 American Mathematical Society