Composition operators acting on weighted Hilbert spaces of analytic functions (Q2793876)

For a positive integrable function (weight) \(\omega \in C^2[0,1)\) which is radial, \(\omega(z)=\omega(|z|)\), the weighted Hilbert space of analytic functions \({\mathcal{H}}_{\omega}\) on the unit disk \(\mathbb{D}\) is given by NEWLINE\[NEWLINE {\mathcal{H}}_{\omega} = \{ f \in H(\mathbb{D}) : ||f'||_{\omega}^2 = \int\limits_{\mathbb{D}} |f'(z)|^2\omega(z)\,dA(z) <\infty \},NEWLINE\]NEWLINE where \(dA\) is the normalized area measure. \({\mathcal{H}}_{\omega}\) equipped with the norm \(||f||_{\omega}^2 = |f(0)|^2+||f'||_{\omega}^2\) is indeed a Hilbert space. Likewise, the weighted Bergmann space \({\mathcal{A}}_{\omega}^2\) is defined by NEWLINE\[NEWLINE {\mathcal{A}}_{\omega}^2 = \{ f \in H(\mathbb{D}) : ||f||_{\omega}^2 = \int\limits_{\mathbb{D}} |f(z)|^2\omega(z)\,dA(z)< \infty \}.NEWLINE\]NEWLINE For \(\alpha > -1\), \(\omega_{\alpha}(r) = (1-r^2)^{\alpha}\) (standard weight), if \(0 \leq \alpha <1\), then \({\mathcal{H}}_{\omega_{\alpha}} = D_{\alpha}\) (the weighted Dirichlet space) and \({\mathcal{H}}_{\omega_1} = H^2\) (the Hardy space).NEWLINENEWLINE The paper under review investigates composition operators on various weighted Hilbert spaces of analytic functions generalizing the results of \textit{J. Pau} and \textit{P. A. Pérez} [J. Math. Anal. Appl. 401, No. 2, 682--694 (2013; Zbl 1293.47022)]. For an analytic self map \(\varphi\) of \(\mathbb{D}\) (\(\varphi(D) \subseteq \mathbb{D}\)), the generalized Nevanlinna counting function of \(\varphi\) associated to a weight \(\omega\) is defined as NEWLINE\[NEWLINEN_{\omega,\omega}(z)=\sum\limits_{a \in \varphi^{-1}(z)} \omega(a), \;\;z \in \mathbb{D} \setminus \{ \varphi(0) \}.NEWLINE\]NEWLINE The author's first result gives a characterization for the essential norm (and hence compactness) of composition operators \(C_{\varphi}\) on \({\mathcal{H}}_{\omega}\), under certain condition (admissibility) on the weight \(\omega\), in terms of the generalized Nevanlinna counting function \(N_{\omega,\omega}\). Under a certain extra condition on the weight \(\omega\), the author also proves a characterization for the membership in the Schatten-class of closed composition operators in terms of \(N_{\omega,\omega}\). The author concludes the presentation by giving a Fredholm composition operator as an illustrative example of a closed range operator satisfying the characteristic condition (as established in the paper) for the closed range property of composition operators.