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

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.

arXiv comment28 p

References in corpus (1)

• Interpolation of subspaces and applications to exponential bases