Types of tightness in spaces with unconditional basis

TL;DR

This paper constructs a reflexive Banach space with an unconditional basis that is quasi-minimal and tight by range, extending known examples in Gowers' classification program by also achieving tightness with constants.

Contribution

It introduces a new reflexive Banach space with an unconditional basis that is tight with constants, expanding the classification of such spaces.

Findings

The space is reflexive with an unconditional basis.

It is quasi-minimal and tight by range.

It is tight with constants, unlike previous examples.

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, but contrary to the recently constructed space of type (4) also tight with constants, thus essentially extending the list of known examples in Gowers classification program. The space is defined on the base on a boundedly modified mixed Tsirelson space with use of a special coding function.