Geodesic
New characterizations for weighted composition operator from Zygmund type spaces to Bloch type spaces
Czechoslovak Mathematical Journal, Tome 65 (2015) no. 2, pp. 331-346
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Let $u$ be a holomorphic function and $\varphi $ a holomorphic self-map of the open unit disk $\mathbb {D}$ in the complex plane. We provide new characterizations for the boundedness of the weighted composition operators $uC_{\varphi }$ from Zygmund type spaces to Bloch type spaces in $\mathbb {D}$ in terms of $u$, $ \varphi $, their derivatives, and $\varphi ^n$, the $n$-th power of $\varphi $. Moreover, we obtain some similar estimates for the essential norms of the operators $uC_{\varphi }$, from which sufficient and necessary conditions of compactness of $uC_{\varphi }$ follows immediately.
Let $u$ be a holomorphic function and $\varphi $ a holomorphic self-map of the open unit disk $\mathbb {D}$ in the complex plane. We provide new characterizations for the boundedness of the weighted composition operators $uC_{\varphi }$ from Zygmund type spaces to Bloch type spaces in $\mathbb {D}$ in terms of $u$, $ \varphi $, their derivatives, and $\varphi ^n$, the $n$-th power of $\varphi $. Moreover, we obtain some similar estimates for the essential norms of the operators $uC_{\varphi }$, from which sufficient and necessary conditions of compactness of $uC_{\varphi }$ follows immediately.
DOI : 10.1007/s10587-015-0178-1
Classification : 26A24, 30H30, 47B33, 47B38
Keywords: weighted composition operator; Zygmund type space; Bloch type space; essential norm 
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Guo, Xin-Cui; Zhou, Ze-Hua. New characterizations for weighted composition operator from Zygmund type spaces to Bloch type spaces. Czechoslovak Mathematical Journal, Tome 65 (2015) no. 2, pp. 331-346. doi: 10.1007/s10587-015-0178-1

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