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Composition operators between $$H^\infty$$ and $$a$$-Bloch spaces on the polydisc. (English) Zbl 1118.47015
Let $$U^n$$ be the unit polydisc of $${\mathbb C}^n$$, $$\alpha>0$$, and let $$H^\infty(U^n)$$ and $${\mathcal B}^\alpha(U^n)$$ denote the space of holomorphic functions in the unit disc on $$U^n$$ which are bounded and belong to $$\alpha$$-Bloch space, i.e., $$\sup_{| z_k| <1}\sum_{k=1}^n(1-| z_k| ^2)^\alpha | \frac{\partial f}{\partial z_k}(z)| <\infty$$, respectively. Given a holomorphic self-map on $$U^n$$, $$\phi=(\phi_1,\dots,\phi_n)$$, the author shows that for $$\alpha\geq 1$$ the composition operator $$C_\phi:H^\infty(U^n)\to {\mathcal B}^\alpha(U^n)$$, defined by $$C_\phi(f)=f\circ \phi$$, is compact if and only if for every $$\varepsilon >0$$, there exists $$0<\delta<1$$ such that if dist$$(\phi(z),\partial U^n)<\delta$$, then
$\sum_{k,l=1}^n\frac{(1-| z_k| ^2)^\alpha}{1-| \phi_l(z)| ^2} \biggl| \frac{\partial \phi_l}{\partial z_k}(z)\biggr| <\varepsilon.$ He uses his result to produce noncompact composition operators form $$H^\infty(U^n)$$ to $${\mathcal B}^1(U^n)$$.

##### MSC:
 47B33 Linear composition operators 47B38 Linear operators on function spaces (general)
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