Symmetric finite representability of $\ell^p$-spaces in rearrangement invariant spaces on $(0,\infty)$

TL;DR

This paper characterizes the set of all p for which ll^p spaces are finitely represented in rearrangement invariant spaces on (0,0) by analyzing eigenvalues of a doubling operator, with applications to Lorentz and Orlicz spaces.

Contribution

It provides a complete description of finite representability of ll^p in rearrangement invariant spaces using eigenvalue analysis of a doubling operator.

Findings

The set of approximate eigenvalues is either an interval or a union of two intervals.

The characterization depends on dilation indices of the space.

Applications are demonstrated for Lorentz and Orlicz spaces.

Abstract

For a separable rearrangement invariant space $X$ on $(0,\infty)$ of fundamental type we identify the set of all $p\in [1,\infty]$ such that $\ell^p$ is finitely represented in $X$ in such a way that the unit basis vectors of $\ell^p$ ($c_0$ if $p=\infty$) correspond to pairwise disjoint and equimeasurable functions. This characterization hinges upon a description of the set of approximate eigenvalues of the doubling operator $x(t)\mapsto x(t/2)$ in $X$. We prove that this set is surprisingly simple: depending on the values of some dilation indices of such a space, it is either an interval or a union of two intervals. We apply these results to the Lorentz and Orlicz spaces.