ON A CLASS OF OPERATORS FROM WEIGHTED BERGMAN SPACES TO SOME SPACES OF ANALYTIC FUNCTIONS

Abstract

Let $\mathbb D = \{ z \in \mathbb C: |z| \lt 1\}$ be the open unit disk in the complex plane $\mathbb C$, $H(\mathbb D)$ be the space of all analytic functions on $\mathbb D$, $\varphi$ be an analytic self-map of $\mathbb D$ and $u \in H(\mathbb D)$. Define operators by $DW_{\varphi,u}f = (u \cdot f \circ \varphi)'$ and $W_{\varphi,u}Df = (u \cdot f' \circ \varphi)$ for $f \in H(\mathbb D)$. In this paper we characterize bounded operators $DW_{\varphi,u}$ and $W_{\varphi,u}D$ from weighted Bergman space to Zygmund-type space, Bloch-type space and Bers-type space on the open unit disk. We also give some sufficient and necessary conditions for these operators to be compact operators in terms of inducing maps $\varphi$ and $u$.