Isomorphic spectrum and isomorphic length of a Banach space

Abstract

We prove that, given any ordinal $\delta < \omega_2$, there exists a transfinite $\delta$-sequence of separable Banach spaces $(X_\alpha)_{\alpha < \delta}$ such that $X_\alpha$ embeds isomorphically into $X_\beta$ and contains no subspace isomorphic to $X_\beta$ for all $\alpha < \beta < \delta$. All these spaces are subspaces of the Banach space $E_p = \bigl( \bigoplus_{n=1}^\infty \ell_p \bigr)_2$, where $1 \leq p < 2$. Moreover, assuming Martin's axiom, we prove the same for all ordinals $\delta$ of continuum cardinality.