CONTINUITY WITH RESPECT TO SYMBOLS OF COMPOSITION OPERATORS ON THE WEIGHTED BERGMAN SPACE

Abstract

Let $\alpha \gt -1$, $U$ be the open unit disk in $\mathbb C$ and denote by $H(U)$ the set of all holomorphic functions on $U$. Let $C_\varphi$ be a composition operator induced by an analytic self-map $\varphi$ of $U$. Composition operators $C_\varphi$ on the weighted Hilbert Bergman space ${\mathcal A}^2_\alpha(U) = \big\{f \in H(U) \;|\; \int_U |f(z)|^2(1-|z|^2)^\alpha dm(z) \lt \infty \big\}$ are considered. We investigate when convergence of sequences $(\varphi_n)$ of symbols to a given symbol $\varphi$, implies the convergence of the induced composition operators. We give a necessary and sufficient condition for a sequence of Hilbert-Schmidt composition operators $(C_{\varphi_n})$ to converge in Hilbert-Schmidt norm to $C_\varphi$, and we obtain a sufficient condition for convergence in operator norm.