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Proceedings of the American Mathematical Society

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

 
 

 

Compact composition operators on $B(D).$


Author: Donald W. Swanton
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
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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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Keywords: Bounded analytic functions, composition operators, compact operators, fixed points, distinguished homomorphisms
Article copyright: © Copyright 1976 American Mathematical Society