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Composition operators on uniform algebras, essential norms, and hyperbolically bounded sets. (English) Zbl 1118.47014
Let $$M_A$$ be the spectrum of a uniform algebra $$A$$ and let $$\Phi$$ be a self-map of $$M_A$$ that induces a composition operator $$C_\Phi:A\to A$$. The authors relate their earlier notion of hyperbolic boundedness [J. Korean Math. Soc. 41, No. 1, 1–20 (2004; Zbl 1049.46031)] to properties of the essential spectrum of $$C_\Phi$$. For example, they show that the spectral radius of $$C_\Phi$$ is $$<1$$ iff the image of $$M_A$$ under some iterate of $$\Phi$$ is hyperbolically bounded. They also partition the set of all composition operators into so-called hyperbolic vicinities each of which is clopen with respect to the essential norm. They show that this partition is related to an analogous partition with respect to the uniform operator norm.

MSC:
 47B33 Linear composition operators 47B48 Linear operators on Banach algebras 46J15 Banach algebras of differentiable or analytic functions, $$H^p$$-spaces 46J10 Banach algebras of continuous functions, function algebras
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References:
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