The $\ell ^{1}$-indices of Tsirelson type spaces

Abstract:

If $\alpha$ and $\beta$ are countable ordinals such that $\beta \neq 0$, denote by $tilde{T}_{\alpha ,\beta }$ the completion of $c_{00}$ with respect to the implicitly defined norm \[ \Vert x\Vert = \max \{\Vert x\Vert _{\mathcal {S}_{\alpha }},\frac {1}{2}\sup \sum _{i=1}^{j}\Vert E_{i}x\Vert \}, \] where the supremum is taken over all finite subsets $E_{1}$, …, $E_{j}$ of $\mathbb {N}$ such that $E_{1}<\dots <E_{j}$ and $\{\min E_{1}, \dots , \min E_{j} \} \in \mathcal {S}$_{𝛽}$. It is shown that the Bourgain$ℓ^{1}$-index of$T̃_{𝛼,𝛽}$is$𝜔^{𝛼+𝛽⋅𝜔}$. In particular, if$𝜔_{1}>𝛼=𝜔^{𝛼_{1}}⋅m_{1}+…+𝜔^{𝛼_{n}}⋅m_{n}$in Cantor normal form and$𝛼_{n}$is not a limit ordinal, then there exists a Banach space whose$ℓ^{1}$-index is$𝜔^{𝛼}$.$