A linear topological invariant for weighted spaces of holomorphic functions

TL;DR

This paper characterizes the linear topological invariant $( extOmega)$ for weighted spaces of holomorphic functions, linking it to weight functions, and explores the surjectivity of the Cauchy-Riemann operator on these spaces.

Contribution

It provides a complete characterization of the invariant $( extOmega)$ for weighted holomorphic function spaces using explicit weight conditions.

Findings

Characterization of $( extOmega)$ in terms of weight functions

Explicit conditions on weight systems for the invariant

Results on the surjectivity of the Cauchy-Riemann operator

Abstract

We study the linear topological invariant $(\Omega)$ for a class of Fr\'echet spaces of holomorphic functions of rapid decay on strip-like domains in the complex plane, defined via weight function systems. We obtain a complete characterization of the property $(\Omega)$ for such spaces in terms of an explicit condition on the defining weight function systems. As an application, we investigate the surjectivity of the Cauchy-Riemann operator on certain weighted spaces of vector-valued smooth functions.