Characterizing Weighted Composition Operators on Weighted-Type High-Order Growth Spaces via the Component Function $\ddot{I}{†}_p$

Abstract

Let $\ddot{I}ˆ$ be a holomorphic function on the open unit ball $\BB \subset \C^N$, and let $\ddot{I}†$ be a holomorphic self-map of $\BB$, associated with normal weights $\hat{I}½$ and $\hat{I}¼$. We consider the weighted composition operator $W_{\ddot{I}ˆ,\ddot{I}†}:{\mathcal{H}}_{\hat{I}}{½}^{(n)}\to {\mathcal{H}}_{\hat{I}}{¼}^{(m)},n,m\in \mathbb{N},$ acting between weighted-type high-order growth spaces. Unlike previous studies that involve the full symbol $\ddot{I}†$, this paper establishes characterizations of the boundedness, compactness, and asymptotic norm estimates of $W_{\ddot{I}ˆ,\ddot{I}†}$ \emph{solely in terms of the symbol $\ddot{I}ˆ$ and a single component function $\ddot{I}{†}_p$ of $\ddot{I}†$}, offering a new approach to the analysis of such operators.

22 pages