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Composition operators on analytic Lipschitz spaces. (English) Zbl 0793.47037

Let $D$ denote the unit disk $\left\{z:|z|\le 1\right\}$ of the complex plane. If $f$ is an analytic function on $D$ and $0, then ${\parallel f\parallel }_{q}$ is defined by

${\parallel f\parallel }_{q}=sup\left\{{|f\left(z\right)-f\left(w\right)|/|z-w|}^{q};\phantom{\rule{4pt}{0ex}}z\in D,\phantom{\rule{4pt}{0ex}}w\in D\right\},$

and the space ${A}_{q}$ is defined by ${A}_{q}={\left\{f:\parallel f\parallel }_{q}<\infty \right\}$. If $\varphi :D\to D$ is analytic, then the composition operator ${C}_{\varphi }$ is defined by ${C}_{\varphi }\left(f\right)\left(x\right)=f\circ \varphi \left(x\right)=f\left(\varphi \left(x\right)\right)$.

The main statements of this paper indicate that

(i) ${C}_{\varphi }:{A}_{q}\to {A}_{q}$ is bounded if and only if

$sup\left\{{\left(\frac{1-{|z|}^{2}}{1-{|\varphi \left(z\right)|}^{2}}\right)}^{1-q}|{\varphi }^{\text{'}}\left(z\right)|:|z|<1\right\}<\infty ;$

and

(ii) if ${lim}_{|z|\to 1}{\left(\frac{1-{|z|}^{2}}{1-{|\varphi \left(z\right)|}^{2}}\right)}^{1-q}|{\varphi }^{\text{'}}\left(z\right)|=0$, then ${C}_{\varphi }:{A}_{q}\to {A}_{q}$ is $w$-compact.

Notes indicated as added in the proof of the paper state that similar conclusions have been derived by different procedures in papers of R. C. Roan [Rocky Mt. J. Math., 10, 371-379 (1980; Zbl 0433.46023)] and J. H. Shapiro [Proc. Am. Math. Soc. 100, 49-57 (1987; Zbl 0622.47028)].

##### MSC:
 47B38 Operators on function spaces (general) 46E15 Banach spaces of continuous, differentiable or analytic functions 47B07 Operators defined by compactness properties 30C20 Conformal mappings of special domains
##### Keywords:
Lipschitz spaces; composition operator