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Proceedings of the American Mathematical Society

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Compact composition operators
on the Smirnov class

Authors: Jun Soo Choa, Hong Oh Kim and Joel H. Shapiro
Journal: Proc. Amer. Math. Soc. 128 (2000), 2297-2308
MSC (1991): Primary 47B38; Secondary 30D55
Published electronically: December 8, 1999
MathSciNet review: 1653449
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Abstract: We show that a composition operator on the Smirnov class $N^+$ is compact if and only if it is compact on some (equivalently: every) Hardy space $H^p$ for $0<p<\infty$. Along the way we show that for composition operators on $N^+$ both the formally weaker notion of boundedness, and a formally stronger notion we call metric compactness, are equivalent to compactness.

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Additional Information

Jun Soo Choa
Affiliation: Department of Mathematics Education, Sung Kyun Kwan University, Jongro-Gu, Seoul 110–745, Korea

Hong Oh Kim
Affiliation: Department of Mathematics, KAIST, Taejon 305–701, Korea

Joel H. Shapiro
Affiliation: Department of Mathematics, Michigan State University, East Lansing, Michigan 48824-1027

Keywords: Composition operator, Smirnov class, compact operator
Received by editor(s): May 29, 1998
Received by editor(s) in revised form: September 10, 1998
Published electronically: December 8, 1999
Additional Notes: This research was supported in part by BSRI, KOSEF, and NSF
Communicated by: Albert Baernstein II
Article copyright: © Copyright 2000 American Mathematical Society