A note on completeness of weighted normed spaces of analytic functions

TL;DR

This paper investigates the conditions under which weighted spaces of holomorphic functions are complete, providing necessary and sufficient criteria based on the properties of the weight function.

Contribution

It offers new characterizations of completeness for weighted spaces of analytic functions, especially with radial weights on balanced domains.

Findings

Established necessary and sufficient conditions for completeness.

Provided examples illustrating the criteria.

Characterized completeness for spaces with radial weights on balanced domains.

Abstract

Given a non-negative weight $v$, not necessarily bounded or strictly positive, defined on a domain $G$ in the complex plane, we consider the weighted space $H_v^\infty(G)$ of all holomorphic functions on $G$ such that the product $v|f|$ is bounded in $G$ and study the question of when is such a space complete under the canonical sup-seminorm. We obtain both some necessary and some sufficient conditions in terms of the weight $v$, exhibit several relevant examples, and characterize completeness in the case of spaces with radial weights on balanced domains.