Non-denseness of hyperbolicity for linear isomorphisms in Banach spaces

TL;DR

This paper constructs an example of an infinite dimensional Banach space where hyperbolic linear isomorphisms are not dense among all linear isomorphisms, highlighting limitations of hyperbolicity in such spaces.

Contribution

It provides the first known example demonstrating the non-denseness of hyperbolic isomorphisms in an infinite dimensional Banach space.

Findings

Hyperbolic isomorphisms are not dense in the set of all isomorphisms in a specific Banach space.

The example challenges assumptions about hyperbolicity's prevalence in infinite dimensions.

Abstract

We present an infinite dimensional Banach space in which the set of hyperbolic linear isomorphisms in that space is not dense (in the norm topology) in the set of linear isomorphisms.