The role of BMOA in the boundedness of weighted composition operators
Let \(\varphi\) be a non-constant analytic self map on the unit disc \(\mathbb{D}\) of the complex plane and let \(h\) be an analytic function on \(\mathbb{D}\). It is known that there are unbounded functions \(h\) such that the weighted composition operator \(W_{h,\varphi} f:= h(f \circ \varphi)\) is a bounded operator on the classical Hardy space \(H^p\), \(1\leq p < \infty\). The aim of this interesting article is to characterize boundedness (resp., compactness) of \(W_{h,\varphi}\) on the Hardy space \(H^2\) in terms of a Nevanlinna counting function associated the symbols \(h\) and \(\varphi\) whenever \(h\) belongs to BMOA (resp., to VMOA). Analogous results for the Hardy spaces \(H^p\) are presented. In general, boundedness and compactness of \(W_{h,\varphi}\) depends on \(p\). The methods used to prove the main theorems are applied by the authors with minor modifications to study weighted composition operators on weighted Bergman spaces \(A_{\alpha}^p\), \(1 \leq p < \infty\), \(\alpha > -1\). In the characterizations of boundedness and compactness in the case of Hilbert spaces, the role of BMOA and VMOA is replaced by the Bloch space \(\mathcal{B}\) and the little Bloch space \(\mathcal{B}_0\).
scientific article; zbMATH DE number 5708619