Continuity and Holomorphicity of Symbols of Weighted Composition Operators

TL;DR

This paper investigates conditions under which the symbols of weighted composition operators are continuous or holomorphic, extending known results to general function spaces and complex manifolds.

Contribution

It generalizes the continuity and holomorphicity properties of weighted composition operators to abstract normed spaces of functions and complex manifolds.

Findings

Characterization of when symbols are continuous or holomorphic

Extension of properties to general function spaces

Analysis of basic properties of weighted composition operators

Abstract

The main problem considered in this article is the following: if $\mathbf{F}$, $\mathbf{E}$ are normed spaces of continuous functions over topological spaces $X$ and $Y$ respectively, and $\omega:Y\to\mathbb{C}$ and $\Phi:Y\to X$ are such that the weighted composition operator $W_{\Phi,\omega}$ is continuous, when can we guarantee that both $\Phi$ and $\omega$ are continuous? An analogous problem is also considered in the context of spaces of holomorphic functions over (connected) complex manifolds. Additionally, we consider the most basic properties of the weighted composition operators, which only have been proven before for more concrete function spaces.