Non-Hilbert Banach spaces with the self-extension property

TL;DR

This paper explores the self-extension property in Banach spaces, relating it to other classical properties, analyzing specific low-dimensional spaces, and showing that higher-dimensional spaces can be renormed to lack this property.

Contribution

It extends the understanding of the self-extension property in Banach spaces, introduces the concept of $k$-self-extensible spaces, and analyzes its stability and prevalence in various spaces.

Findings

The self-extension property relates to 1-injective and 1-projective spaces.

Certain low-dimensional spaces like $ ext{R}_1^3$ and $ ext{R}_1^4$ are analyzed.

Higher-dimensional spaces can be renormed to lack the self-extension property.

Abstract

In 1992, Kiendi, Adamy and Stelzner investigated under which conditions a certain type of function constituted a Lyapunov function for some time-invariant linear system. Six years later, it was obtained that this property holds if and only if the Banach space enjoys the self-extension property. However, the knowledge of these spaces needed to be extended in order to make useful this characterization, since there were little information on which classic Banach spaces satisfy this property and its relations with other classic properties. We present the self-extension property in a wider frame, relating it to other well-known spaces as the $1$-injective or $1$-projective ones. We investigate the property in two important low-dimensional classic Banach spaces: $\mathbb{R}_1^3$ and $\mathbb{R}_1^4$. We introduce the concept of $k-$self-extensible spaces and a discussion of the stability of the property. We also show that every real Banach space of dimension greater than or equal to $3$ can be equivalently renormed to fail the self-extension property. Finally, we summarize some consequences of our study and mention some open questions which appear naturally.