Composition Operators on Generalized Weighted Nevanlinna Class

Abstract

Let $\varphi$ be an analytic self-map of open unit disk $\mathbb{D}$. The operator given by $(C_{\varphi}f)(z)=f(\varphi(z))$, for $z \in \mathbb{D}$ and $f$ analytic on $\mathbb{D}$ is called a composition operator. Let $\omega$ be a weight function such that $\omega\in L^{1}(\mathbb{D}, dA)$. The space we consider is a generalized weighted Nevanlinna class $\mathcal{N}_{\omega}$, which consists of all analytic functions $f$ on $\mathbb{D}$ such that $\displaystyle \|f\|_{\omega}=\int_{\mathbb{D}}\log^{+}(|f(z)|) \omega(z) dA(z)$ is finite; that is, $\mathcal{N}_{\omega}$ is the space of all analytic functions belong to $L_{\log^+}(\mathbb{D}, \omega dA)$. In this paper we investigate, in terms of function-theoretic, composition operators on the space $\mathcal{N}_{\omega}$. We give sufficient conditions for the boundedness and compactness of these composition operators.