A note on weighted composition operators on the weighted Bergman space
For the unit disk \(\mathbb{D}\) and the space \(H(\mathbb{D})\) of all analytic functions on \(\mathbb{D}\), \(1 < p < \infty\), \(\alpha > -1\), the weighted Bergman space \({\mathcal{L}}_a^p(dA_{\alpha})\) is defined as \({\mathcal{L}}_a^p(dA_{\alpha}) = H(\mathbb{D}) \cap {\mathcal{L}}^p(\mathbb{D},dA_{\alpha})\), where \(dA_{\alpha}(z)=(\alpha+1)(1-|z|^2)^{\alpha}dA(z)\) is the Möbius invariant measure, and \(dA\) denotes the normalized area measure on \(\mathbb{D}\). For an analytic self-map \(\varphi\) on \(\mathbb{D}\) and \(\psi \in H(\mathbb{D})\), the weighted composition operator \(C_{\psi,\varphi}\) on \(H(\mathbb{D})\) is defined as \(C_{\psi,\varphi}=M_{\psi}\circ C_{\varphi} : f \mapsto \psi\cdot (f\circ\varphi)\) for \(f \in H(\mathbb{D})\), where \(M_{\psi}\) and \(C_{\varphi}\) are the constituent multiplication and composition operators with the respective symbols \(\psi\) (weighted symbol) and \(\varphi\) (composition symbol). An analytic self-map \(\varphi\) is a finite Blaschke product if \[ \varphi =z^m \prod\limits_{k=1}^n\frac{\overline{a}_k(a_k-z)}{a_k(1-\overline{a}_kz)}, \] where \(a_k \in \mathbb{D} \setminus\{0\}\), \(k=1,2,\dots{},n\). It is well known that \(C_{\varphi}\) is bounded on \({\mathcal{L}}_a^p(dA_{\alpha})\) for any self-map \(\varphi\) of \(\mathbb{D}\) and \(M_{\psi}\) is bounded on \({\mathcal{L}}_a^p(dA_{\alpha})\) if and only if \(\psi \in H^{\infty}(\mathbb{D})\), the space of all bounded analytic functions on \(\mathbb{D}\). Clearly, bounded multiplicative and composition operators yield a bounded weighted composition operator \(C_{\psi,\varphi}\). However, the converse is not true, as shown in [\textit{M. D. Contreras} and \textit{A. G. Hernández-Díaz}, Integral Equations Oper. Theory 46, No. 2, 165--188 (2003; Zbl 1042.47017)]. For the Hardy space \(H^p\), \(1 < p < \infty\), it was shown in [loc. cit.] that \(\{ \psi \in H^p : C_{\psi,\varphi}H^p \subset H^p \} = H^{\infty}(\mathbb{D})\) if and only if \(\varphi\) is a finite Blaschke product. The paper under review extends these results to the case of weighted Bergman spaces \({\mathcal{L}}_a^p(dA_{\alpha})\). Indeed, for \(1 < p < \infty\), \(\alpha > -1\), an analytic self-map \(\varphi\) of \(\mathbb{D}\), the authors prove that \(\{ \psi \in {\mathcal{L}}_a^p(dA_{\alpha}) : C_{\psi,\varphi}{\mathcal{L}}_a^p(dA_{\alpha}) \subset {\mathcal{L}}_a^p(dA_{\alpha}) \} = H^{\infty}(\mathbb{D})\) if and only if \(\varphi\) is a finite Blaschke product. In connection with this result, it is interesting to ponder on the extended, yet open, question of whether or not the above equality holds (for the Hardy and/or weighted Bergman spaces) when the composition symbol \(\varphi\) is an infinite Blaschke product.
scientific article; zbMATH DE number 6498080
scientific article