A type (4) space in (FR)-classification

TL;DR

This paper constructs a reflexive Banach space with an unconditional basis that is quasi-minimal and tight by range, fitting into the type (4) classification in Gowers' program, extending the understanding of Banach space structures.

Contribution

It introduces a new unconditional variant of the Gowers Hereditarily Indecomposable space, classified as type (4) in Ferenczi-Rosendal's list, within Gowers' classification framework.

Findings

The space is reflexive with an unconditional basis.

It is quasi-minimal and tight by range.

It extends the class of known Banach spaces with these properties.

Abstract

We present a reflexive Banach space with an unconditional basis which is quasi-minimal and tight by range, i.e. of type (4) in Ferenczi-Rosendal list within the framework of Gowers' classification program of Banach spaces. The space is an unconditional variant of the Gowers Hereditarily Indecomposable space with asymptotically unconditional basis.