Weighted composition operators from the minimal Möbius invariant space into the Bloch space
The authors study weighted composition operators \(uC_\varphi (f)= u(z)f(\varphi (z))\) for analytic functions \(u\) and analytic self-maps \(\varphi\) of the unit disc acting from the minimal Möbius invariant space, corresponding to the Besov space \(B_1\), into the maximal Möbius invariant space, corresponding to the Bloch space \(\mathcal B\). It is shown that boundedness can be described by the condition \(\sup_n \|u\varphi^n\|_{\mathcal B}<\infty\) or by the conditions \(u\in \mathcal B\) together with \(\sup_{|z|<1}\frac{(1-|z|^2)|u(z)\varphi'(z)|}{1-|\varphi (z)|^2}<\infty\). Regarding the compactness of \(uC_{\varphi}\), it is shown to be equivalent, among other things, to \(\|g_n\|_{\mathcal B}+\|u\varphi^n\|_{\mathcal B} \to 0\) as \(n\to \infty\), where \(g_n(z)=\int_0^z u'(\xi)\varphi(\xi)^n\,d\xi\).
scientific article; zbMATH DE number 6147195
scientific article