Non-superreflexivity of Garling sequence spaces and applications to the existence of special types of conditional bases

Albiac, Fernando; Ansorena, José L.; Dilworth, S. J.; Kutzarova, Denka

doi:10.4064/sm180910-1-2

Cited by 3 publications

(2 citation statements)

References 17 publications

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“…Lemma 11.5 (cf. [10,Lemma 2.3]). Let 0 < p < ∞ and w ∈ W. Given 0 < ε < 1 and tuples f and g with f g(w,p) ≤ 1, there is a tuple h such that h f g(w,p) ≤ 1 and g h ≥ ( g p g(w,p) + 1 − ε) 1/p .…”

Section: Banach Envelopesmentioning

confidence: 99%

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Greedy approximation for biorthogonal systems in quasi-Banach spaces

Albiac

Ansorena

Berná

et al. 2021

Dissertationes Math.

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116

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The general problem addressed in this work is the development of a systematic study of the thresholding greedy algorithm for general biorthogonal systems in quasi-Banach spaces from a functional-analytic point of view. If (xn, x * n ) ∞ n=1 is a biorthogonal system in X then for each x ∈ X we have a formal expansionxn. The thresholding greedy algorithm (with threshold ε > 0) applied to x is formally defined asxn. The properties of this operator give rise to the different classes of greedy-type bases. We revisit the concepts of greedy, quasi-greedy, and almost greedy bases in this comprehensive framework and provide the (non-trivial) extensions of the corresponding characterizations of those types of bases. As a by-product of our work, new properties arise, and the relations among them are carefully discussed.

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Section: Banach Envelopesmentioning

confidence: 99%

“…10 (see[26, Theorems 1.3.7 and 1.3.8]). Suppose 1 < q ≤ ∞ and let w be a non-increasing weight with primitive weight s. Then the discrete Hardy operator is bounded from d 1,q (w) into d 1,∞ (w) if and only if s −1 is a regular weight.…”

mentioning

confidence: 99%

Greedy approximation for biorthogonal systems in quasi-Banach spaces

Albiac

Ansorena

Berná

et al. 2021

Dissertationes Math.

Self Cite

116

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

2020

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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.

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A dichotomy for subsymmetric basic sequences with applications to Garling spaces

Albiac¹,

Ansorena²,

Dilworth³

et al. 2021

Trans. Amer. Math. Soc.

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Our aim in this article is to contribute to the study of the structure of subsymmetric basic sequences in Banach spaces (even, more generally, in quasi-Banach spaces). For that we introduce the notion of positionings and develop new tools which lead to a dichotomy theorem that holds for general spaces with subsymmetric bases. As an illustration of how to use this dichotomy theorem we obtain the classification of all subsymmetric sequences in certain types of spaces. To be more specific, we show that Garling sequence spaces have a unique symmetric basic sequence but no symmetric basis and that these spaces have a continuum of subsymmetric basic sequences.

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Non-superreflexivity of Garling sequence spaces and applications to the existence of special types of conditional bases

Cited by 3 publications

References 17 publications

Greedy approximation for biorthogonal systems in quasi-Banach spaces

Greedy approximation for biorthogonal systems in quasi-Banach spaces

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

A dichotomy for subsymmetric basic sequences with applications to Garling spaces

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