Sequences of Weighted Composition Operators on the Hardy–Hilbert Space

Abstract

Let $$\varphi $$ be an analytic self map of the open unit disc $$\mathbb {D}$$ . Assume that $$\psi $$ is an analytic map of $$\mathbb {D}$$ . Suppose that f is in the Hardy–Hilbert space of the open unit disc $$H^2$$ . The operator that takes f into $$\psi \cdot f \circ \varphi $$ is a weighted composition operator, and is denoted by $$C_{\psi ,\varphi }$$ . In this paper we relate the convergence of the sequence $$\{ C_{\psi _n,\varphi _n}\}$$ in different operator topologies to the convergence of the two sequences of maps $$\{\varphi _n \}$$ and $$\{ \psi _n\}$$ .