Weighted composition operators between weighted-type spaces and the Bloch space and BMOA

Abstract

Let ${\mathbb {D}}$ denote the open unit disk in ${\mathbb {C}}$. For an integer $n\ge 0$, let $V_n$ be the space defined recursively by

$$\begin{aligned} V_0=\big \{f:{\mathbb {D}}\rightarrow {\mathbb {C}}: f \text { analytic}, |f(z)|=O\big ((1-|z|)^{-1}\big )\big \}, \end{aligned}$$

and for $n\ge 1$, $f\in V_n$ if and only if $f'\in V_{n-1}$. In this work, we characterize the bounded and the compact weighted composition operators between $V_n$ and the Bloch space and the space BMOA of analytic functions of bounded mean oscillation, respectively. We also show that the bounded (respectively, compact) weighted composition operators mapping the space BMOA into $V_n$ are precisely the same as the bounded (respectively, compact) weighted composition operators mapping the Bloch space into $V_n$.