Connected components of the general linear group of a real hereditarily indecomposable Banach space

TL;DR

This paper characterizes the connected components of the general linear group of a real hereditarily indecomposable Banach space, revealing how complex structures influence their structure and showing they cannot coexist on the space and its hyperplanes.

Contribution

It provides a complete description of the connected components of the general linear group for these spaces, highlighting the role of complex structures.

Findings

Connected components depend on complex structures.

Complex structures cannot coexist on the space and hyperplanes.

Provides a full classification of the linear group structure.

Abstract

We give a complete description of the structure of the connected components of the general linear group of a real hereditarily indecomposable Banach space, depending on the existence of complex structures on the space itself and on its hyperplanes. A side result is the fact that complex structures cannot exist simultaneously on such a space and on its hyperplanes.