Ordinal Indices of small subspaces of $L_p$

Abstract

We calculate ordinal $L_p$ index defined in "An ordinal L_p index for Banach spaces with an application to complemented subspaces of L_p" authored by J. Bourgain, H. P. Rosenthal and G. Schechtman, for Rosenthal's space $X_p$, ${\ell }_p$ and ${\ell }_2$. We show a subspace of $L_p$ $(2<p<\infty )$ non isomorphic to ${\ell }_2$ embeds in ${\ell }_p$ if and only if its ordinal index is minimum possible. We also give a sufficient condition for a ${\mathcal{L}}_p$ subspace of ${\ell }_p\oplus {\ell }_2$ to be isomorphic to $X_p$.