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Proceedings of the American Mathematical Society

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

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Compact composition operators on $B(D).$
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by Donald W. Swanton PDF
Proc. Amer. Math. Soc. 56 (1976), 152-156 Request permission

Abstract:

Let $D$ be a domain in the complex plane, $\phi :D \to D$ be analytic, and $B(D)$ be the uniform algebra of bounded analytic functions on $D$ with maximal ideal space $M$. The composition operator ${C_\phi }(f) = f \circ \phi$ is compact if and only if the weak* and norm closures of $\phi (D)$ coincide if and only if whenever the Euclidean closure of $\phi (D)$ contains a point $\lambda$ of the boundary of $D$ then each $f \in B(D)$ extends continuously from $\phi (D)$ to $\lambda$. If ${C_\phi }$ is compact, then either $\phi$ fixes a point of $D$ or else the adjoint of ${C_\phi }$ fixes a point of $M$.
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Additional Information
  • © Copyright 1976 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 56 (1976), 152-156
  • MSC: Primary 47B37; Secondary 46J15
  • DOI: https://doi.org/10.1090/S0002-9939-1976-0407648-5
  • MathSciNet review: 0407648