Vector spaces of non-extendable holomorphic functions

Abstract

In this paper, we analyze the linear structure of the family H e (G) of holomorphic functions on a domain G of the complex plane that are not analytically continuable beyond the boundary of G. 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. We also consider the case in which G is a domain of existence in a complex Banach space. The results obtained complete or extend a number of previous results by several authors.