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Compact composition operators on BMOA and the Bloch space. (English) Zbl 1194.47038
Let $\Delta$ denote the unit disc in the complex plane and let $H(\Delta)$ denote the set of all functions holomorphic on $\Delta$. For a point $a \in \Delta$, let $$\sigma_{a}(z) = \frac {a - z} {1 - \bar{a}z}.$$ The spaces BMOA and the Bloch space ${\cal B}$ are defined by $$\text{BMOA} = \left\{f \in H(\Delta): \|f\|_{*}^{2} = \sup_{a \in \Delta} \left\{ \lim_{r \to 1} \frac {1} {2 \pi} \int_{0}^{2 \pi} |f(\sigma_{a}(r e^{i \theta})) - f(a)|^{2}\, d \theta \right\} < \infty \right\}$$ and $${\cal B} = \Big\{ f \in H(\Delta): \|f\|_{**} = \sup_{z \in \Delta} { \{ |f'(z)| (1 - |z|^{2}) \} } < \infty\Big\}.$$ Then $\|f\|_{\text{BMOA}} = \|f\|_{*} + |f(0)|$ and $\|f\|_{\cal B} = \|f\|_{**} + |f(0)|$. Let $\varphi \in H(\Delta)$ such that $\varphi(\Delta) \subset \Delta$, and, for $f \in H(\Delta)$, let $C_{\varphi}(f) = f \circ \varphi$. Let $X$ denote either of the spaces BMOA or ${\cal B}$. The authors prove that the composition operator $C_{\varphi}$ is a compact operator on the space $X$ if and only if $\lim_{n \to \infty} \|\varphi^{n}\|_{X} = 0$. For the space BMOA, this improves a result of the first author [Sci. China, Ser. A 50, No.  7, 997--1004 (2007; Zbl 1126.30023)].

47B38Operators on function spaces (general)
30H30Bloch spaces
30J99Function theory on the disc
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