Ordinal Indices of small subspaces of $L_p$

TL;DR

This paper computes the ordinal $L_p$ index for specific Banach spaces and characterizes when subspaces of $L_p$ embed into $ ext{ell}_p$, providing new insights into their structure.

Contribution

It calculates the ordinal $L_p$ index for key Banach spaces and establishes criteria for embeddings of subspaces into $ ext{ell}_p$, advancing understanding of their structure.

Findings

Subspace of $L_p$ with minimal ordinal index embeds into $ ext{ell}_p$ if not isomorphic to $ ext{ell}_2$.

Provides a sufficient condition for $ ext{L}_p$ subspaces of $ ext{ell}_p igoplus ext{ell}_2$ to be isomorphic to $X_p$.

Calculates the ordinal $L_p$ index for $X_p$, $ ext{ell}_p$, and $ ext{ell}_2$.

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 $X_p$.