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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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