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Weighted composition operators from Zygmund spaces into Bloch spaces. (English) Zbl 1215.47022

Let \(\mathbb{D}=\{ z\in \mathbb{C}:| z| <1\} \) be the open unit disk in the complex plane \(\mathbb{C}\) and \(H(\mathbb{D})\) the class of all analytic functions on \(\mathbb{D}\), and let \(\varphi \) be an analytic self-map, that is, \(\varphi :\mathbb{D}\rightarrow \mathbb{D}\). Then \( \varphi \) induces the composition operator \(C_{\varphi }\) on \(H(\mathbb{D}),\) defined by \((C_{\varphi })(z)=f(\varphi (z))\) for \(z\in \mathbb{D}\) and \( f\in H(\mathbb{D})\). Let \(u\) be a fixed analytic function on \(\mathbb{D}\). The weighted composition operator \(uC_{\varphi }\) is defined on \(H(\mathbb{D} )\) by \(((uC_{\varphi })f)(z)=u(z)f(\varphi (z)),\) \(z\in \mathbb{D},\) \(f\in H( \mathbb{D}).\)
Let \(\mathcal{Z}\) denote the space of all \(f\in H(\mathbb{D})\cap C( \overline{\mathbb{D}})\) such that
\[ \| f\| _{\mathcal{Z}}=\sup \frac{| f(e^{i(\theta+h))}+f(e^{i(\theta -h)})-2f(e^{i\theta })| }{h}<\infty, \]
where the supremum is taken over all \(e^{i\theta }\in \partial \mathbb{D}\) and \(h>0\). The little Zygmund space \(\mathcal{Z}_{0}\) is defined by \(f\in \mathcal{Z}_{0}\) if and only if \(\lim_{| z|\rightarrow 1}(1-| z|)| f''(z)|=0\). An analytic function \(f\) is said to belong to the Bloch space \(B\) if \(B(f)=\sup_{z\in \mathbb{D}}(1-| z|^{2})| f'(z)| <\infty\). Finally, the closed subspace \(B_{0}\) (the little Bloch space) of \(B\) consists of the functions \(f\in B\) satisfying \(\lim_{| z| \rightarrow 1}(1-| z| ^{2})| f'(z)| =0\). Let \(X_{1}\) and \(X_{2}\) be the Banach spaces. For a given operator \(A:X_{1}\rightarrow X_{2}\), the problem of investigating its boundedness and compactness is a natural and important problem of operator theory.
In this paper, the authors study the boundedness and compactness of weighted composition operators from Zygmund space into Bloch space and little Bloch space, based mainly on the classical methods of function theory.

MSC:

47B33 Linear composition operators
46E15 Banach spaces of continuous, differentiable or analytic functions
30H05 Spaces of bounded analytic functions of one complex variable
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