A characterization of ℓ^p-spaces symmetrically finitely represented in symmetric sequence spaces

2021/05/14 by С. В. Асташкин, Astashkin, Sergey
Mathematics · #Advanced Banach Space Theory #Approximation Theory and Sequence Spaces #Advanced Harmonic Analysis Research

paper · pdf · doi:10.48550/arxiv.2105.06871

Abstract

For a separable symmetric sequence space X of fundamental type we identify the set \mathcal F(X) of all p∈ [1,∞] such that ℓ^p is block finitely represented in the unit vector basis \e_k\_k=1^∞ of X in such a way that the unit basis vectors of ℓ^p (c₀ if p=∞) correspond to pairwise disjoint blocks of \e_k\ with the same ordered distribution. It turns out that \mathcal F(X) coincides with the set of approximate eigenvalues of the operator (x_k)↦ ∑_k=2^∞ x_[k/2]e_k in X. In turn, we establish that the latter set is the interval [2^α_X,2^β_X], where α_X and β_X are the Boyd indices of X. As an application, we find the set \mathcal F(X) for arbitrary Lorentz and separable sequence Orlicz spaces.

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