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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

Compact composition operators on the Bloch space


Authors: Kevin Madigan and Alec Matheson
Journal: Trans. Amer. Math. Soc. 347 (1995), 2679-2687
MSC: Primary 47B38; Secondary 30D45, 47B07
MathSciNet review: 1273508
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Abstract: Necessary and sufficient conditions are given for a composition operator $ {C_\phi }f = f{\text{o}}\phi $ to be compact on the Bloch space $ \mathcal{B}$ and on the little Bloch space $ {\mathcal{B}_0}$. Weakly compact composition operators on $ {\mathcal{B}_0}$ are shown to be compact. If $ \phi \in {\mathcal{B}_0}$ is a conformal mapping of the unit disk $ \mathbb{D}$ into itself whose image $ \phi (\mathbb{D})$ approaches the unit circle $ \mathbb{T}$ only in a finite number of nontangential cusps, then $ {C_\phi }$ is compact on $ {\mathcal{B}_0}$. On the other hand if there is a point of $ \mathbb{T} \cap \phi (\mathbb{D})$ at which $ \phi (\mathbb{D})$ does not have a cusp, then $ {C_\phi }$ is not compact.


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DOI: http://dx.doi.org/10.1090/S0002-9947-1995-1273508-X
PII: S 0002-9947(1995)1273508-X
Keywords: Composition operator, compact operator, Bloch space, cusp
Article copyright: © Copyright 1995 American Mathematical Society