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Bloch-to-BMOA pullbacks on the disk

Authors: Boo Rim Choe, Wade Ramey and David Ullrich
Journal: Proc. Amer. Math. Soc. 125 (1997), 2987-2996
MSC (1991): Primary 30D45, 47B38
MathSciNet review: 1396971
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Abstract: For a given holomorphic self map $\varphi $ of the unit disk, we consider the Bloch-to-$BMOA$ composition property (pullback property) of $\varphi $. Our results are $(1)$ $\varphi $ cannot have the pullback property if $\varphi $ touches the boundary too smoothly, $(2)$ while $\varphi $ has the pullback property if $\varphi $ touches the boundary rather sharply. One of these results yields an interesting consequence completely contrary to a higher dimensional result which has been known. These results resemble known results concerning the compactness of composition operators on the Hardy spaces. Some remarks in that direction are included.

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Boo Rim Choe
Affiliation: Department of Mathematics, Korea University, Seoul, Korea

Wade Ramey
Affiliation: Department of Mathematics, Michigan State University, East Lansing, Michigan

David Ullrich
Affiliation: Department of Mathematics, Oklahoma State University, Stillwater, Oklahoma

Keywords: Bloch space, BMOA, pullback property
Received by editor(s): September 22, 1995
Received by editor(s) in revised form: May 17, 1996
Additional Notes: The first author is supported in part by BSRI (96-1407) and GARC (96) of Korea.
Communicated by: Theodore Gamelin
Article copyright: © Copyright 1997 American Mathematical Society