On a criterion for continuity and compactness of composition operators on the weighted Bloch space
The authors deal with the weighted Bloch space \(B^{\mathrm{log}}\) of all holomorphic functions on the complex unit disc \(\mathbb{D}\) satisfying the growth condition \[ \sup_{z\in \mathbb{D}} (1-|z|^2) \log\left( \frac{e}{(1-|z|^2)} \right) |f'(z)| < +\infty , \] and characterize the continuity and compactness of the composition operators \( \mathcal{C}_\psi(f) := f \circ \psi \) acting on \(B^{\mathrm{log}}\), for given holomorphic self-maps \(\psi\) of \(\mathbb{D}\). The characterization is given in terms of the quantity \({\left\| \sigma_a \circ \psi \right\|}\), where \(\sigma_a \), \(1/2 < |a| < 1\), is a specific class of functions on \(\mathbb{D}\).
scientific article; zbMATH DE number 6478193
scientific article