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Weighted composition operators on the Bloch space. (English) Zbl 0985.47022

Let $D$ be the open disk of the complex plane. Then the Bloch space $ℬ$ on $D$ and the little Bloch space ${ℬ}_{0}$ on $D$ are defined by

$ℬ=\left\{f:f\phantom{\rule{4.pt}{0ex}}\text{is}\phantom{\rule{4.pt}{0ex}}\text{analytic}\phantom{\rule{4.pt}{0ex}}\text{on}\phantom{\rule{4.pt}{0ex}}D\phantom{\rule{4.pt}{0ex}}\text{and}\phantom{\rule{4.pt}{0ex}}\text{sup}\left\{\left(1-{|z|}^{2}\right)|{f}^{\text{'}}\left(z\right)|,z\in D\left\{<\infty \right\},$

where ${f}^{\text{'}}$ is the derivative of $f$;

${ℬ}_{0}=\left\{f:f\phantom{\rule{4.pt}{0ex}}\text{is}\phantom{\rule{4.pt}{0ex}}\text{analytic}\phantom{\rule{4.pt}{0ex}}\text{on}\phantom{\rule{4.pt}{0ex}}D\phantom{\rule{4.pt}{0ex}}\text{and}\phantom{\rule{4.pt}{0ex}}\left(1-{|z|}^{2}\right)|{f}^{\text{'}}\left(z\right)|\to 0\phantom{\rule{4.pt}{0ex}}\text{as}\phantom{\rule{4.pt}{0ex}}|z|\to 1\right\}·$

With norm ${|·|}_{ℬ}$ defined by ${|f|}_{ℬ}=|f\left(0\right)|+sup\left\{\left(1-{|z|}^{2}\right)|{f}^{\text{'}}\left(z\right)|,z\in D\right\}$, $ℬ$ is a Banach space and ${ℬ}_{0}$ is a closed subspace of $ℬ$. If $u$ is an analytic function on $D$ and $\phi :D\to D$, then the linear weighted composition operator $uC\phi$ is defined by

$uC\phi \left(f\right)\left(z\right)=\left(uf\circ \phi \right)\left(z\right)=u\left(z\right)f\left(\phi \left(z\right)\right)·$

In the results of this paper, the authors derive characterizations for bounded and compact weighted composition operators. In particular, it is shown that

(1) $uC\phi$ is bounded on the Bloch space $ℬ$ if and only if

(i) $sup\left\{\left(1-{|z|}^{2}\right)|{u}^{\text{'}}{\left(z\right)|log\left(2/\left(1-|\phi \left(z\right)|}^{2}\right)\right),z\in D\left\{<\infty$;

(ii) $sup\left\{\left(1-{|z|}^{2}{\right)/\left(1-|\phi \left(z\right)|}^{2}\right)\right\}|{u}^{\text{'}}\left(z\right){\phi }^{\text{'}}\left(z\right)|,z\in D\right\}<\infty$;

and

(2) $uC\phi$ is compact on the Bloch space $ℬ$ if and only if expressions of (1) converge to 0 as $|\phi \left(z\right)|\to 1$; and $uC\phi$ is compact on ${ℬ}_{0}$ if and only if the expressions of (1) converge to 0 as $|z|\to 1$.

(3) $uC\phi$ is bounded on ${ℬ}_{0}$ if the conditions of (1) are satisfied and $|u\left(z\right){\phi }^{\text{'}}\left(z\right)|\left(1-{|z|}^{2}\right)\to 0$ as $|z|\to 1$.

##### MSC:
 47B33 Composition operators 47B38 Operators on function spaces (general) 47B07 Operators defined by compactness properties