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Weighted differentiation composition operators from $H^{\infty}$ to Zygmund spaces. (English) Zbl 1225.47036
Summary: Let $\varphi$ be an analytic self-map of the unit disk $\Bbb D,\, H(\Bbb D)$ the space of analytic functions on $\Bbb D$ and $u\in H(\Bbb D)$. The boundedness and compactness of the weighted differentiation composition operators $D^n_{\varphi,u}$ [defined by $D^n_{\varphi,u}f(z)=u(z)f^{(n)}(\varphi(z))$, $f\in H(\Bbb D)$], where $n\in \Bbb N_{0}$, from $H ^{\infty }$ to Zygmund space are investigated in this paper.

47B38Operators on function spaces (general)
47B33Composition operators
30H05Bounded analytic functions
30D45Bloch functions, normal functions, normal families
46E15Banach spaces of continuous, differentiable or analytic functions
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