Bloch-to-BMOA pullbacks on the disk

Choe, Boo; Ramey, Wade; Ullrich, David

doi:10.1090/S0002-9939-97-03873-2

Bloch-to-BMOA pullbacks on the disk
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by Boo Rim Choe, Wade Ramey and David Ullrich PDF

Proc. Amer. Math. Soc. 125 (1997), 2987-2996 Request permission

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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