Estimates of generalized Nevanlinna counting function and applications to composition operators (Q2825917)

Let \(\mathcal{D}_{\alpha}\), \(0 \leq \alpha \leq 1\), be the weighted Dirichlet space of holomorphic functions on the unit disc, that is, a Hilbert space with its natural norm. Given a holomorphic self map \(\varphi\) in the unit disc, the authors obtain several estimates of the generalized Nevanlinna counting function \(N_{\varphi,\alpha}\) in terms of the norms of \(\varphi^n\) in the space \(\mathcal{D}_{\alpha}\). As a consequence, new characterizations and examples of bounded, compact and Hilbert-Schmidt composition operators \(C_{\varphi} f= f \circ \varphi\) on \(\mathcal{D}_{\alpha}\) are obtained. They complement work by \textit{K. Kellay} and \textit{P. Lefèvre} [J. Math. Anal. Appl. 386, No. 2, 718--727 (2012; Zbl 1231.47024)].