On isometries and Tingley’s problem for the spaces $T[\theta , \mathcal{S}_{\alpha }]$, $1 \leqslant\alpha \lt \omega _{1}$
Volume 273 / 2023
Abstract
We extend the existing results on surjective isometries of unit spheres in the Tsirelson space $T[1/2, \mathcal S_1]$ to the class $T[\theta , \mathcal S_{\alpha}]$ for any integer $\theta^{─1} \geq 2$ and $1 \leqslant \alpha \lt \omega _1$, where $\mathcal {S}_{\alpha }$ 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 $T[\theta , \mathcal S_1]$ ($\theta \in (0, {1}/{2}]$) 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 enables us to establish a sufficient condition for being a linear isometry in these spaces.
