Ordinal indices of Small Subspaces of Lp

We calculate the ordinal L _p index defined in [3] for Rosenthal’s space X _p , ({ell_p}) and ({ell_2}). We show that an infinite-dimensional subspace of L _p ({(2 < p < infty)}) non-isomorphic to ({ell_2}) embeds in ({ell_p}) if and only if its ordinal index is the minimal 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 .