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

DOI10.1007/S00208-021-02277-5arXiv2104.13077MaRDI QIDQ6366225

Publication date: 27 April 2021

Abstract: For a separable rearrangement invariant space

X

on

(0, i n f t y)

of fundamental type we identify the set of all

p i n [1, i n f t y]

such that

e l l^{p}

is finitely represented in

X

in such a way that the unit basis vectors of

e l l^{p}

(

c_{0}

if

p = i n f t y

) 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) m a p s t o 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.

Mathematics Subject Classification ID

Local theory of Banach spaces (46B07) Banach lattices (46B42) Dilations, extensions, compressions of linear operators (47A20)

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