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Isometries of the disc algebra


Authors: Mohamad El-Gebeily and John Wolfe
Journal: Proc. Amer. Math. Soc. 93 (1985), 697-702
MSC: Primary 46J15; Secondary 30H05
DOI: https://doi.org/10.1090/S0002-9939-1985-0776205-9
MathSciNet review: 776205
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Abstract: The linear isometries $ u:A \to A$ of the disc algebra $ A$ into itself are completely described. Such isometries $ u$ must be one of two distinct types. The first type is $ uf = \psi \cdot f(\phi )$, where $ \psi \in A$ and $ \phi \in {H^\infty }$ satisfy certain described conditions. The second type is $ uf = E(\psi \cdot f(\phi ))$, where $ \phi :Q \to T$ is any continuous function from a closed zero measure subset $ Q$ of the unit circle $ T$ onto itself, $ \psi \in C(Q)$ is unimodular, and $ E:Y \to A$ is a norm 1 extension operator, where $ Y = \left\{ {\psi \cdot f(\phi ):f \in A} \right\} \subset C(Q)$. Isometries of $ C(K)$ spaces into the disc algebra are also described.


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DOI: https://doi.org/10.1090/S0002-9939-1985-0776205-9
Article copyright: © Copyright 1985 American Mathematical Society

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