Vector spaces on non-extendable holomorphic functions

TL;DR

This paper investigates the linear and algebraic structures within the family of non-extendable holomorphic functions on a domain, revealing dense algebras and large vector spaces, including in Banach space contexts.

Contribution

It establishes the existence of dense algebras and large vector spaces within non-extendable holomorphic functions, extending previous results and covering Banach space domains.

Findings

Existence of dense algebra of non-extendable holomorphic functions

Presence of dense vector spaces of boundary-regular functions

Extension of results to domains in complex Banach spaces

Abstract

In this paper, the linear structure of the family $H_e(G)$ of holomorphic functions in a domain $G$ of the complex plane that are not analytically continuable beyond the boundary of $G$ is analyzed. We prove that $H_e(G)$ contains, except for zero, a dense algebra; and, under appropriate conditions, the subfamily of $H_e(G)$ consisting of boundary-regular functions contains dense vector spaces with maximal dimension, as well as infinite dimensional closed vector spaces and large algebras. The case in which $G$ is a domain of existence in a complex Banach space is also considered. The results obtained complete or extend a number of previous ones by several authors.