Isometries of combinatorial Tsirelson spaces

Abstract

We extend existing results that characterize isometries on the Tsirelson-type spaces T [ 1 n , S 1 ] T\big [\frac {1}{n}, \mathcal {S}_1\big ] ( n ∈ N , n ⩾ 2 n\in \mathbb {N}, n\geqslant 2 ) to the class T [ θ , S α ] T[\theta , \mathcal {S}_{\alpha }] ( θ ∈ ( 0 , 1 2 ] \big (\theta \in \big (0, \frac {1}{2}\big ] , 1 ⩽ α > ω 1 1\leqslant \alpha > \omega _1 \big), where S α \mathcal {S}_{\alpha } denote the Schreier families of order α \alpha . We prove that every isometry on T [ θ , S 1 ] T[\theta , \mathcal {S}_1] \big( θ ∈ ( 0 , 1 2 ] \theta \in \big (0, \frac {1}{2}\big ] \big) is determined by a permutation of the first ⌈ θ − 1 ⌉ \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 [ θ , S α ] T[\theta , \mathcal {S}_{\alpha }] \big( θ ∈ ( 0 , 1 2 ] \theta \in \big (0, \frac {1}{2}\big ] , 2 ⩽ α > ω 1 2\leqslant \alpha > \omega _1 \big) the isometries exhibit a more rigid character, namely, they are all implemented by a sign-change operation of the vector coordinates.