On certain subspaces of $${\ell _p}$$ for $${0<p\le 1}$$ and their applications to conditional quasi-greedy bases in p-Banach spaces

Abstract

We construct for each 0 < p ≤ 1 an infinite collection of subspaces of ℓ p that extend the example of Lindenstrauss from [21] of a subspace of ℓ 1 with no unconditional basis. The structure of this new class of p-Banach spaces is analyzed and some applications to the general theory of L p -spaces for 0 < p < 1 are provided. The introduction of these spaces serves the purpose to develop the theory of conditional quasi-greedy bases in p-Banach spaces for p < 1. Among the topics we consider are the existence of infinitely many conditional quasi-greedy bases in the spaces ℓ p for p ≤ 1 and the careful examination of the conditionality constants of the "natural basis" of these spaces.

New parameters and Lebesgue-type estimates in greedy approximation

“…Let Γ : N → N ∪ {0} be the left inverse of the function defined by n → σ (n) (1), n ∈ N ∪ {0}. In [8] it was constructed an almost greedy basis X δ of a subspace X δ of ℓ 1 with 1…”

“…The dual space of X δ is isomorphic to ℓ ∞ , and the dual basis X * δ spans a space isomorphic to c 0 . In [8] it is also proved that for each increasing concave function φ : [0, ∞) → [0, ∞) with φ(0) = 0, we can choose δ so that Γ grows as (φ(log(m))) ∞ m=2 . By [8, Proposition 4.4 and Lemma 7…”

“…Another important property of quasi-greedy bases is that their unconditionality constants grow slowly. Indeed, if X is a p-Banach space, the contribution of k m to L m is at most of the order of (log m) 1/p (see [19,Lemma 8.2], [30,Theorem 5.1] and [8,Theorem 5.1]). Hence, if X is quasi-greedy and democratic there is a constant C such that…”

Weaker forms of unconditionality of bases in greedy approximation

“…An asymptotic upper bound for the unconditionality constants of truncation quasi-greedy bases in p-Banach spaces was estimated in [5], where the following theorem was proved. Theorem 3.6 ([5, Theorem 5.1]).…”

Weak forms of unconditionality of bases in greedy approximation

“…An asymptotic upper bound for the unconditionality constants of truncation quasi-greedy bases in p-Banach spaces was estimated in [5].…”