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Generalized composition operators on Zygmund spaces and Bloch type spaces. (English) Zbl 1135.47021
Let $$D$$ denote the unit disc in the complex plane, let $$H(D)$$ denote the set of all functions holomorphic on $$D$$, and let $$C(\overline{D})$$ denote the set of all functions continuous on the closure of $$D$$. A Bloch type space (or $$\alpha$$-Bloch space) is a space of the form $B^{\alpha} = \{f \in H(D): \sup_{z \in D} (1 - | z| ^{2})^{\alpha} | f'(z)| < \infty \},$ where the space $$B^{\alpha}$$ is given the norm $\| f\| _{B^{\alpha}} = | f(0)| + \sup_{z \in D} (1 - | z| ^{2})^{\alpha} | f'(z)|.$ The Zygmund space $$Z$$ is the space $Z = \left\{f \in H(D) \cap C(\overline{D}): \sup_{\theta \in [0, 2\pi], h > 0} \frac {| f(e^{i \theta + h}) + f(e^{i \theta - h}) - 2 f(e^{i \theta})| } {h} < \infty \right\},$ with the norm given by $\| f\| _{Z} = | f(0)| + | f'(0)| + \sup_{z \in D} (1 - | z| ^{2}) | f''(z)|.$ Throughout, $$\varphi$$ denotes a non-constant analytic self-map of $$D$$. A basic composition operator is given by $$C_{\varphi}f = f \circ \varphi$$ for $$f \in H(D)$$. Let $$g \in H(D)$$ and define the linear operator $(C_{\varphi}^{g}f)(z) = \int_{0}^{z} f'(\varphi(\zeta))g(\zeta)\,d\zeta.$ The authors give criteria under which the general composition operator $$C_{\varphi}^{g}:Z \to B^{\alpha}$$ is a bounded operator, and also when it is a compact operator. Also considered are the cases when $$C_{\varphi}^{g}: Z \to Z$$ and when $$C_{\varphi}^{g}: B^{\alpha} \to Z$$ is a bounded operator, and when it is a compact operator. Letting $B_{0}^{\alpha} = \left\{f \in B^{\alpha}: \lim_{| z| \to 1} \;(1 - | z| ^{2})^{\alpha}| f'(z) = 0 \right\},$ and letting $Z_{0} = \left\{f \in Z: \lim_{| z| \to 1} (1 - | z| ^{2})| f''(z)| = 0 \right\},$ corresponding results are obtained using $$B_{0}^{\alpha}$$ in place of $$B^{\alpha}$$ and using $$Z_{0}$$ in place of $$Z$$. Two typical results are as follows. Theorem. If $$0 < \alpha < \infty$$, if $$g \in H(D)$$ and $$\varphi$$ is an analytic self-map of $$D$$, then $$C_{\varphi}^{g}: Z \to B^{\alpha}$$ is bounded if and only if $\sup_{z\in D}\,(1-| z|^{2})^{\alpha}| g(z)| \log\frac{1}{1-|\varphi(z)|^{2}}<\infty.$ In addition, this operator is compact if and only it is bounded and $\lim_{| \varphi(z)| \to 1} \;(1 - | z| ^{2})^{\alpha} | g(z)| \log \frac {1} {1 - | \varphi(z)| ^{2}} = 0.$ Theorem. If $$0 < \alpha < \infty$$, if $$g \in H(D)$$, and if $$\varphi$$ is an analytic self-map of $$D$$, then the following statements are equivalent: (i) $$C_{\varphi}^{g}: B^{\alpha} \to Z$$ is compact; (ii) $$C_{\varphi}^{g}: B_{0}^{\alpha} \to Z$$ is compact; (iii) $$C_{\varphi}^{g}: B^{\alpha} \to Z$$ is bounded and both $\lim_{| \varphi(z)| \to 1} \frac {(1 - | z| ^{2}) | \varphi'(z)|\,| g(z)|} {(1 - | z|^{2})^{\alpha + 1}} = 0\quad\text{and}\quad\lim_{| \varphi(z)| \to 1} \frac {(1-| z|^{2})| g'(z)|}{(1-| \varphi(z)| ^{2})^{\alpha}} = 0.$

##### MSC:
 47B33 Linear composition operators 30D45 Normal functions of one complex variable, normal families 47B38 Linear operators on function spaces (general) 46E15 Banach spaces of continuous, differentiable or analytic functions
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