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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Compact composition operators on the Bloch space
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by Kevin Madigan and Alec Matheson PDF
Trans. Amer. Math. Soc. 347 (1995), 2679-2687 Request permission

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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Additional Information
  • © Copyright 1995 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 347 (1995), 2679-2687
  • MSC: Primary 47B38; Secondary 30D45, 47B07
  • DOI: https://doi.org/10.1090/S0002-9947-1995-1273508-X
  • MathSciNet review: 1273508