Spectral-free methods in the theory of hereditarily indecomposable Banach spaces

TL;DR

This paper introduces spectral-free, simplified proofs of key properties of hereditarily indecomposable Banach spaces, extending classical results and deriving new insights without spectral theory, applicable to both real and complex spaces.

Contribution

The paper provides new, spectral-free proofs of classical properties and introduces novel results, including a quantitative version of a known theorem and analysis of homotopy relations.

Findings

Hereditary indecomposability prevents a space from being isomorphic to its proper subspace.

Banach spaces isometric to all their subspaces have bases with unconditional constants close to 1.

New proofs are applicable to both real and complex Banach spaces.

Abstract

We give new and simple proofs of some classical properties of hereditarily indecomposable Banach spaces, including the result by W. T. Gowers and B. Maurey that a hereditarily indecomposable Banach space cannot be isomorphic to a proper subspace of itself. These proofs do not make use of spectral theory and therefore, they work in real spaces as well as in complex spaces. We use our method to prove some new results. For example, we give a quantitative version of the latter result by Gowers and Maurey and deduce that Banach spaces that are isometric to all of their subspaces should have an unconditional basis with unconditional constant arbitrarily close to $1$. We also study the homotopy relation between into isomorphisms in hereditarily indecomposable spaces.