Norm of weighted composition operators from \(\alpha \)-Bloch spaces to weighted-type spaces
For \(\alpha>0\), let \(\mathcal B^\alpha\) be the \(\alpha\)-Bloch space in the unit ball \(\mathbb B_n\) of \(\mathbb C^n\). The author endows this space with the norm \[ \|f\|_{\mathcal B^\alpha}:=|f(0)|+\sup_{z\in \mathbb B^n} (1-|z|)^\alpha |\nabla f(z)|. \] Also, for a positive continuous function \(\mu\) on \(\mathbb B^n\), called a weight, let \(H^\infty_\mu\) be the space of all \(f\) holomorphic in \(\mathbb B^n\) for which \(\|f\|_{H^\infty_\mu}:=\sup_{z\in \mathbb B^n} \mu(z)|f(z)|<\infty\). The paper contains a single result: the explicite determination of the norm of the weighted composition operator \(T=uC_\varphi: \mathcal B^\alpha\to H^\infty_\mu\), \(f\mapsto u(f\circ \varphi)\), where \(u:\mathbb B_n\to \mathbb C\) is a holomorphic function and \(\varphi\) a holomorphic selfmap of \(\mathbb B_n\). It is shown that, if \(\alpha\not=1\) and \(T\) is bounded, then \[ \|T\|_{\text{op}}=\max\left\{ \|u\|_{H^\infty_\mu} , \sup_{z\in\mathbb B_n} \frac{\mu(z)|u(z)|}{\alpha-1} \left( \frac{1}{(1-|\varphi(z)|)^{\alpha-1}}-1 \right) \right\}. \] The case \(\alpha=1\) was done in [\textit{S. Stevic}, Ars. Combin. 88, 125--127 (2008)].
scientific article; zbMATH DE number 5612859
scientific article