Multiplication operators, integration operators and companion operators on weighted Bloch space
For a function \(g\), analytic in the unit disc \({\mathcal D}=\{z:| z| <1\}\), the operator \(J_g\) is defined by the formula \(J_g(h)(z)=\int_0^zh(\zeta)g'(\zeta)\,d\zeta\). The companion operator \(I_g\) and the multiplication operator \(M_g\) are defined by \(I_g(h)(z)=\int_0^zg(\zeta)h'(\zeta)\,d\zeta\) and \(M_g(h)(z)=g(z)h(z)\), respectively. In this paper the author studies the boundedness and compactness of \(I_g\), \(g\) analytic on the unit disk~\(\mathcal D\), defined on the \(\alpha\)-Bloch space \({\mathcal B}^\alpha\), \(\alpha\geq1\). For a certain kind of weight function \(\omega\) the same problem is considered for \(I_g\) defined on the weighted Bloch space \({\mathcal B}_\omega\). Further, the relationship between multiplication operator \(M_g\) and the operators \(I_g, J_g\) defined on \({\mathcal B}_\omega\) is given. Finally, the criteria for the boundedness and compactness of the product \(I_fI_g\), \(f\) and \(g\) analytic on~\(\mathcal D\), defined on \({\mathcal B}^\alpha\), \(\alpha\geq1\), are established.
scientific article; zbMATH DE number 2166198