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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 $$B^{\alpha} = \{f \in H(D): \sup_{z \in D} (1 - \vert z\vert ^{2})^{\alpha} \vert f'(z)\vert < \infty \},$$ where the space $B^{\alpha}$ is given the norm $$\Vert f\Vert _{B^{\alpha}} = \vert f(0)\vert + \sup_{z \in D} (1 - \vert z\vert ^{2})^{\alpha} \vert f'(z)\vert.$$ The Zygmund space $Z$ is the space $$Z = \left\{f \in H(D) \cap C(\overline{D}): \sup_{\theta \in [0, 2\pi], h > 0} \frac {\vert f(e^{i \theta + h}) + f(e^{i \theta - h}) - 2 f(e^{i \theta})\vert } {h} < \infty \right\},$$ with the norm given by $$\Vert f\Vert _{Z} = \vert f(0)\vert + \vert f'(0)\vert + \sup_{z \in D} (1 - \vert z\vert ^{2}) \vert f''(z)\vert.$$ 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 $$B_{0}^{\alpha} = \left\{f \in B^{\alpha}: \lim_{\vert z\vert \to 1} \ (1 - \vert z\vert ^{2})^{\alpha}\vert f'(z) = 0 \right\},$$ and letting $$Z_{0} = \left\{f \in Z: \lim_{\vert z\vert \to 1} (1 - \vert z\vert ^{2})\vert f''(z)\vert = 0 \right\},$$ 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 $$\sup_{z\in D}\,(1-\vert z\vert^{2})^{\alpha}\vert g(z)\vert \log\frac{1}{1-\vert\varphi(z)\vert^{2}}<\infty.$$ In addition, this operator is compact if and only it is bounded and $$\lim_{\vert \varphi(z)\vert \to 1} \ (1 - \vert z\vert ^{2})^{\alpha} \vert g(z)\vert \log \frac {1} {1 - \vert \varphi(z)\vert ^{2}} = 0.$$ 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 $$\lim_{\vert \varphi(z)\vert \to 1} \frac {(1 - \vert z\vert ^{2}) \vert \varphi'(z)\vert\,\vert g(z)\vert} {(1 - \vert z\vert^{2})^{\alpha + 1}} = 0\quad\text{and}\quad\lim_{\vert \varphi(z)\vert \to 1} \frac {(1-\vert z\vert^{2})\vert g'(z)\vert}{(1-\vert \varphi(z)\vert ^{2})^{\alpha}} = 0.$$

47B33Composition operators
30D45Bloch functions, normal functions, normal families
47B38Operators on function spaces (general)
46E15Banach spaces of continuous, differentiable or analytic functions
Full Text: DOI
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