Isometries of combinatorial Tsirelson spaces

TL;DR

This paper characterizes the structure of isometries on a class of Tsirelson-type spaces, showing how they are determined by permutations and sign-changes, with more rigidity in higher-order Schreier families.

Contribution

It extends the characterization of isometries from classical Tsirelson spaces to a broader class involving Schreier families of arbitrary countable order.

Findings

Isometries on $T[ heta, ext{S}_1]$ are determined by permutations and sign-changes.

Isometries on $T[ heta, ext{S}_eta]$ for $eta eq 1$ are solely sign-change operations.

The results generalize previous characterizations to a wider class of Tsirelson-type spaces.

Abstract

We extend existing results that characterize isometries on the Tsirelson-type spaces $T\big[\frac{1}{n}, \mathcal{S}_1\big]$ ($n\in \mathbb{N}, n\geq 2$) to the class $T[\theta, \mathcal{S}_{\alpha}]$ ($\theta \in \big(0, \frac{1}{2}\big]$, $1\leqslant \alpha < \omega_1$), where $\mathcal{S}_{\alpha}$ denote the Schreier families of order $\alpha$. We prove that every isometry on $T[\theta, \mathcal{S}_1]$ ($\theta \in \big(0, \frac{1}{2}\big]$) is determined by a permutation of the first $\lceil {\theta}^{-1} \rceil$ elements of the canonical unit basis followed by a possible sign-change of the corresponding coordinates together with a sign-change of the remaining coordinates. Moreover, we show that for the spaces $T[\theta, \mathcal{S}_{\alpha}]$ ($\theta \in \big(0, \frac{1}{2}\big]$, $2\leqslant \alpha < \omega_1$) the isometries exhibit a more rigid character, namely, they are all implemented by a sign-change operation of the vector coordinates.