On large indecomposable Banach spaces

TL;DR

This paper constructs the first example of an indecomposable Banach space with density between two and continuum, showing such spaces can be larger than hereditarily indecomposable ones, and explores their operator structure.

Contribution

It provides the first known example of an indecomposable Banach space with density between two and continuum, expanding understanding of their possible sizes.

Findings

Existence of an indecomposable Banach space of density continuum

Such spaces can be constructed consistently

Operators on these spaces have a specific form involving weakly compact operators

Abstract

Hereditarily indecomposable Banach spaces may have density at most continuum (Plichko-Yost, Argyros-Tolias). In this paper we show that this cannot be proved for indecomposable Banach spaces. We provide the first example of an indecomposable Banach space of density two to continuum. The space exists consistently, is of the form C(K) and it has few operators in the sense that any bounded linear operator T on C(K) satisfies T(f)=gf+S(f) for every f in C(K), where g is in C(K) and S is weakly compact (strictly singular).