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The essential norms and spectra of composition operators on $$H^ \infty$$. (English) Zbl 1053.47022
Composition operators, acting on various analytic function spaces on the unit disk of the complex plane, have been extensively studied over the last three decades. Most of such studies have focused on well-known function spaces such as the Hardy and Bergman spaces. However, composition operators acting on $$H^\infty$$ have been barely studied so far.
In this paper, the author proves two results for composition operators on $$H^\infty$$. One is a result exhibiting a phenomenon quite different from the $$H^2$$ case and the other one is a result demonstrating a phenomenon quite similar to the $$H^2$$ case. More explicitly, the first result asserts that every noncompact composition operator on $$H^\infty$$ has essential norm 1. This result appears to be quite interesting, because it reveals a very different phenomenon from Shapiro’s essential norm formula on $$H^2$$.
The second result is a characterization of spectra of composition operators on $$H^\infty$$ in the case that symbols have interior fixed points. This result appears to be an $$H^\infty$$ analogue of the $$H^2$$ results obtained by H. Kamowitz [J. Funct. Anal. 18, 132–150 (1975; Zbl 0295.47003)] and C. C. Cowen B. D. MacCluer [J. Funct. Anal. 125, 223–251 (1994; Zbl 0814.47040)].

##### MSC:
 47B33 Linear composition operators 46E15 Banach spaces of continuous, differentiable or analytic functions 47B38 Linear operators on function spaces (general) 47A10 Spectrum, resolvent
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