# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
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, $\varphi$ denotes a non-constant analytic self-map of $D$. A basic composition operator is given by ${C}_{\varphi }f=f\circ \varphi$ for $f\in H\left(D\right)$. Let $g\in H\left(D\right)$ and define the linear operator

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

The authors give criteria under which the general composition operator ${C}_{\varphi }^{g}:Z\to {B}^{\alpha }$ 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

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

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 $\varphi$ is an analytic self-map of $D$, then ${C}_{\varphi }^{g}:Z\to {B}^{\alpha }$ is bounded if and only if

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

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

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

Theorem. If $0<\alpha <\infty$, if $g\in H\left(D\right)$, 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

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

##### 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