A dual method of constructing hereditarily indecomposable Banach spaces

TL;DR

This paper introduces a novel dual method for constructing hereditarily indecomposable Banach spaces, unifying the creation of reflexive and non-reflexive HI spaces, and reveals new operator properties including invariant subspaces.

Contribution

It presents a new unified approach to constructing hereditarily indecomposable Banach spaces, including reflexive and non-reflexive examples, with unique operator properties.

Findings

All constructed spaces satisfy that the composition of two strictly singular operators is compact.

First example of a Banach space with no reflexive subspace where every operator has a non-trivial invariant subspace.

Provides a new framework for constructing and analyzing hereditarily indecomposable Banach spaces.

Abstract

A new method of defining hereditarily indecomposable Banach spaces is presented. This method provides a unified approach for constructing reflexive HI spaces and also HI spaces with no reflexive subspace. All the spaces presented here satisfy the property that the composition of any two strictly singular operators is a compact one. This yields the first known example of a Banach space with no reflexive subspace such that every operator has a non-trivial closed invariant subspace.