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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 $H\left(D\right)$ denote the set of all functions holomorphic on $D$, and let $C\left(\overline{D}\right)$ 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

${B}^{\alpha }=\left\{f\in H\left(D\right):\underset{z\in D}{sup}{\left(1-|z|}^{2}{\right)}^{\alpha }|{f}^{\text{'}}\left(z\right)|<\infty \right\},$

where the space ${B}^{\alpha }$ is given the norm

${\parallel f\parallel }_{{B}^{\alpha }}=|f\left(0\right)|+\underset{z\in D}{sup}{\left(1-|z|}^{2}{\right)}^{\alpha }|{f}^{\text{'}}\left(z\right)|·$

The Zygmund space $Z$ is the space

$Z=\left\{f\in H\left(D\right)\cap C\left(\overline{D}\right):\underset{\theta \in \left[0,2\pi \right],h>0}{sup}\frac{|f\left({e}^{i\theta +h}\right)+f\left({e}^{i\theta -h}\right)-2f\left({e}^{i\theta }\right)|}{h}<\infty \right\},$

with the norm given by

${\parallel f\parallel }_{Z}=|f\left(0\right)|+|{f}^{\text{'}}\left(0\right)|+\underset{z\in D}{sup}{\left(1-|z|}^{2}\right)|{f}^{\text{'}\text{'}}\left(z\right)|·$

Throughout, $\phi$ denotes a non-constant analytic self-map of $D$. A basic composition operator is given by ${C}_{\phi }f=f\circ \phi$ for $f\in H\left(D\right)$. Let $g\in H\left(D\right)$ and define the linear operator

$\left({C}_{\phi }^{g}f\right)\left(z\right)={\int }_{0}^{z}{f}^{\text{'}}\left(\phi \left(\zeta \right)\right)g\left(\zeta \right)\phantom{\rule{0.166667em}{0ex}}d\zeta ·$

The authors give criteria under which the general composition operator ${C}_{\phi }^{g}:Z\to {B}^{\alpha }$ is a bounded operator, and also when it is a compact operator. Also considered are the cases when ${C}_{\phi }^{g}:Z\to Z$ and when ${C}_{\phi }^{g}:{B}^{\alpha }\to Z$ is a bounded operator, and when it is a compact operator. Letting

${B}_{0}^{\alpha }=\left\{f\in {B}^{\alpha }:\underset{|z|\to 1}{lim}{\phantom{\rule{4pt}{0ex}}\left(1-|z|}^{2}{\right)}^{\alpha }|{f}^{\text{'}}\left(z\right)=0\right\},$

and letting

${Z}_{0}=\left\{f\in Z:\underset{|z|\to 1}{lim}{\left(1-|z|}^{2}\right)|{f}^{\text{'}\text{'}}\left(z\right)|=0\right\},$

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 $g\in H\left(D\right)$ and $\phi$ is an analytic self-map of $D$, then ${C}_{\phi }^{g}:Z\to {B}^{\alpha }$ is bounded if and only if

$\underset{z\in D}{sup}{\phantom{\rule{0.166667em}{0ex}}\left(1-|z|}^{2}{\right)}^{\alpha }|g\left(z\right)|log\frac{1}{1-{|\phi \left(z\right)|}^{2}}<\infty ·$

In addition, this operator is compact if and only it is bounded and

$\underset{|\phi \left(z\right)|\to 1}{lim}{\phantom{\rule{4pt}{0ex}}\left(1-|z|}^{2}{\right)}^{\alpha }|g\left(z\right)|log\frac{1}{1-{|\phi \left(z\right)|}^{2}}=0·$

Theorem. If $0<\alpha <\infty$, if $g\in H\left(D\right)$, and if $\phi$ is an analytic self-map of $D$, then the following statements are equivalent: (i) ${C}_{\phi }^{g}:{B}^{\alpha }\to Z$ is compact; (ii) ${C}_{\phi }^{g}:{B}_{0}^{\alpha }\to Z$ is compact; (iii) ${C}_{\phi }^{g}:{B}^{\alpha }\to Z$ is bounded and both

$\underset{|\phi \left(z\right)|\to 1}{lim}\frac{{\left(1-|z|}^{2}\right)|{\phi }^{\text{'}}\left(z\right)|\phantom{\rule{0.166667em}{0ex}}|g\left(z\right)|}{{\left(1-|z|}^{2}{\right)}^{\alpha +1}}=0\phantom{\rule{1.em}{0ex}}\text{and}\phantom{\rule{1.em}{0ex}}\underset{|\phi \left(z\right)|\to 1}{lim}\frac{{\left(1-|z|}^{2}\right)|{g}^{\text{'}}\left(z\right)|}{{\left(1-|\phi \left(z\right)|}^{2}{\right)}^{\alpha }}=0·$

##### MSC:
 47B33 Composition operators 30D45 Bloch functions, normal functions, normal families 47B38 Operators on function spaces (general) 46E15 Banach spaces of continuous, differentiable or analytic functions
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