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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 $H\left({\Delta }\right)$ denote the set of all functions holomorphic on ${\Delta }$. For a point $a\in {\Delta }$, let

${\sigma }_{a}\left(z\right)=\frac{a-z}{1-\overline{a}z}·$

The spaces BMOA and the Bloch space $ℬ$ are defined by

$\text{BMOA}=\left\{f\in {H\left({\Delta }\right):\parallel f\parallel }_{*}^{2}=\underset{a\in {\Delta }}{sup}\left\{\underset{r\to 1}{lim}\frac{1}{2\pi }{\int }_{0}^{2\pi }{|f\left({\sigma }_{a}\left(r{e}^{i\theta }\right)\right)-f\left(a\right)|}^{2}\phantom{\rule{0.166667em}{0ex}}d\theta \right\}<\infty \right\}$

and

$ℬ=\left\{f\in {H\left({\Delta }\right):\parallel f\parallel }_{**}=\underset{z\in {\Delta }}{sup}\left\{|{f}^{\text{'}}\left(z\right)|\left(1-{|z|}^{2}\right)\right\}<\infty \right\}·$

Then ${\parallel f\parallel }_{\text{BMOA}}={\parallel f\parallel }_{*}+|f\left(0\right)|$ and ${\parallel f\parallel }_{ℬ}={\parallel f\parallel }_{**}+|f\left(0\right)|$. Let $\varphi \in H\left({\Delta }\right)$ such that $\varphi \left({\Delta }\right)\subset {\Delta }$, and, for $f\in H\left({\Delta }\right)$, let ${C}_{\varphi }\left(f\right)=f\circ \varphi$. Let $X$ denote either of the spaces BMOA or $ℬ$. The authors prove that the composition operator ${C}_{\varphi }$ is a compact operator on the space $X$ if and only if ${lim}_{n\to \infty }{\parallel {\varphi }^{n}\parallel }_{X}=0$. 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)].

##### MSC:
 47B38 Operators on function spaces (general) 30H30 Bloch spaces 30H35 BMO-spaces 30J99 Function theory on the disc
##### Keywords:
composition operator; BMOA; Bloch space