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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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DOI: https://doi.org/10.1090/S0002-9939-1976-0407648-5
Keywords: Bounded analytic functions, composition operators, compact operators, fixed points, distinguished homomorphisms
Article copyright: © Copyright 1976 American Mathematical Society