Products of differentiation and composition operators on the Bloch space
Let \(H({\mathbb{D}})\) be the space of all analytic functions defined on the unit disk \({\mathbb{D}}\) in the complex plane \(\mathbb C\). Let \(\sigma_a(z)=\frac{a-z}{1-\bar{a}z}\) be the Möbius transformation on \({\mathbb{D}}\). For a nonconstant analytic self-map \(\phi\) of \({\mathbb{D}}\), the composition operator \(C_\phi\) on \(H({\mathbb{D}})\) is defined by \(C_\phi f=f\circ \phi\). In the paper under review the authors consider linear product operators \[ DC_\phi f=(f\circ\phi)'\quad\text{and}\quad C_\phi D^m f=f^{(m)}\circ \phi \] acting on the well known Bloch space \(\mathcal{B}\), where \(m\in \mathbb{Z}\). The authors give new and simple compactness criteria for the operator \(DC_\phi\) on \(\mathcal{B}\). The paper contains six new theorems, three lemmas and one interesting example. Theorem 2.1 says that \(DC_\phi\) is compact on \(\mathcal{B}\) if and only if \(DC_\phi\) is bounded on \(\mathcal{B}\) and \[ \lim_{n\to\infty}\|D\phi ^n\|_\mathcal{B} =0. \] In the next section they prove that \(C_\phi D^m\) is compact on \(\mathcal{B}\) if and only if \[ \lim_{|a|\to 1} \|C_\phi D^m\sigma_a\|_\mathcal{B} = 0. \]
scientific article; zbMATH DE number 6112444
scientific article