Regular subspaces of a Bourgain-Delbaen space $\mathscr B_{mT}$

TL;DR

This paper investigates the structure of a specific Bourgain-Delbaen space modeled on a mixed Tsirelson space, demonstrating that every infinite dimensional subspace contains a basic sequence similar to weighted basis averages from the model space.

Contribution

It establishes that all infinite dimensional subspaces of the constructed Bourgain-Delbaen space contain sequences equivalent to weighted basis averages, extending known results to this new space.

Findings

Every infinite dimensional subspace contains a basic sequence equivalent to weighted basis averages.

The same property holds for the original space defined by Argyros and Haydon.

The space is modeled on a mixed Tsirelson space with specific structural properties.

Abstract

The space $\mathscr B_{mt}[(m_j)_j,(n_j)_j]$ is a Bourgain-Delbaen space modelled on a mixed Tsirelson space $T[(m_j)_j,(n_j)_j]$ and is a slight modification of $\mathfrak B_{\text{mt}}[(m_j)_j,(n_j)_j]$, a space defined by S. Argyros and R. Haydon. We prove that in every infinite dimensional subspace of $\mathscr B_{mt}[(m_j)_j,(n_j)_j]$ there exists a basic sequence equivalent to a sequence of weighted basis averages of increasing length from $T[(m_j)_j,(n_j)_j]$. We remark that the same is true for the original space $\mathfrak B_{\text{mt}}[(m_j)_j,(n_j)_j]$.