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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 \({\mathcal B}^1(U^n)\).

47B33 Linear composition operators
47B38 Linear operators on function spaces (general)
Full Text: DOI
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