On isometries and Tingley's problem for the spaces $T[\theta, S_{\alpha}], 1 \leq \alpha<\omega_{1}$

Abstract

We extend the existing results on surjective isometries of unit spheres in the Tsirelson space $T\left[\frac{1}{2}, S_1\right]$ to the class $T[\theta,S_{\alpha}]$ for any integer $\theta^{-1} \geq 2$ and $1 \leqslant \alpha < \omega_1$, where $S_{\alpha}$ denotes the Schreier family of order $\alpha$. This positively answers Tingley's problem for these spaces, which asks whether every surjective isometry between unit spheres can be extended to a surjective linear isometry of the entire space. Furthermore, we improve the result stating that every linear isometry on $T[\theta, S_1]$ ($\theta \in \left(0, \frac{1}{2}\right]$) 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 and a sign change of the remaining coordinates. Specifically, we prove that only the first $\lfloor \theta^{-1} \rfloor$ elements can be permuted. This finding enables us to establish a sufficient condition for being a linear isometry in these spaces.