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Zbl 0826.47023
Madigan, Kevin; Matheson, Alec
Compact composition operators on the Bloch space. (English)
[J] Trans. Am. Math. Soc. 347, No.7, 2679-2687 (1995). ISSN 0002-9947; ISSN 1088-6850/e

Summary: Necessary and sufficient conditions are given for a composition operator $C_\phi f= f\circ \phi$ to be compact on the Bloch space $\cal B$ and on the little Bloch space ${\cal B}_0$. Weakly compact composition operators on ${\cal B}_0$ are shown to be compact. If $\phi\in {\cal B}_0$ is a conformal mapping of the unit disk $\bbfD$ into itself whose image $\phi(\bbfD)$ approaches the unit circle $\bbfT$ only in a finite number of nontangential cusps, then $C_\phi$ is compact on ${\cal B}_0$. On the other hand if there is a point of $\bbfT\cap \overline{\phi(\bbfD)}$ at which $\phi(\bbfD)$ does not have a cusp, then $C_\phi$ is not compact.

MSC 2000:: *47B38 Operators on function spaces
47B07 Operators defined by compactness properties
30D55 H (sup p)-classes
46J15 Banach algebras of differentiable functions

Keywords: composition operator; Bloch space; nontangential cusps

Cited in: Zbl 1058.47024 Zbl 1024.47010 Zbl 1013.47010 Zbl 1049.47026 Zbl 0976.30018

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