## Angular derivatives and compact composition operators on the Hardy and Bergman spaces.(English)Zbl 0608.30050

With $$\alpha >-1$$ and $$d\lambda$$ Lebesgue measure in the unit disc U, define the Bergman $$A^ p_{\alpha}$$ space by $A^ p_{\alpha}=\{f\quad holomorphic\quad U| \int_{U}| f|^ p(1-| z|)^{\alpha}d\lambda <\infty \}.$ If $$\phi$$ is a holomorphic map of U into U the composition operator $$C\phi f=f\circ \phi$$ is a bounded operator on $$A^ p_{\alpha}$$, and on the Hardy spaces $$H^ p$$. The principal result of the paper is to characterize the compact composition operators on $$A^ p_{\alpha}$$ with $$\alpha >-1$$, $$0<p<\infty$$. The key to their characterization is the existence (or nonexistence) of the angular derivative $\lim_{z\to \zeta}\frac{(\phi (z)-\omega)}{(z-\zeta)}$ where $$| \zeta | =1=| \omega |$$ and the limit is taken as z in U tends non-tangentially to $$\zeta$$.
Suppose $$0<p<\infty$$ and $$\alpha >-1$$ are given and $$\phi$$ : $$U\to U$$. Then $$C_{\phi}$$ is a compact operator on $$A^ p_{\alpha}$$ if and only if $$\phi$$ has no angular derivative at any point of $$\partial U$$. The paper contains a unifying approach to other results in this area. Also the last section studies the problem for spaces of functions holomorphic on the ball in $${\mathbb{C}}^ M$$. The latter situation is more complicated and the results less sharp.
Reviewer: J.A.Cima

### MSC:

 30H05 Spaces of bounded analytic functions of one complex variable
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