Greedy approximation for biorthogonal systems in quasi-Banach spaces

Abstract

The general problem addressed in this work is the development of a systematic study of the thresholding greedy algorithm for general biorthogonal systems (also known as Markushevich bases) in quasi-Banach spaces from a functional-analytic point of view. We revisit the concepts of greedy, quasi-greedy, and almost greedy bases in this comprehensive framework and provide the (nontrivial) extensions of the corresponding characterizations of those types of bases. As a by-product of our work, new properties arise, and the relations amongst them are carefully discussed.

On the permutative equivalence of squares of unconditional bases

“…They showed that the spaces c 0 (ℓ 1 ), c 0 (ℓ 2 ), ℓ 1 (c 0 ), ℓ 1 (ℓ 2 ) and their complemented subspaces with unconditional basis all have a (UTAP) unconditional basis, while ℓ 2 (ℓ 1 ) and ℓ 2 (c 0 ) do not. However, the hopes of attaining a satisfactory classification were shattered when they found a nonclassical Banach space, namely the 2-convexification T (2) of Tsirelson's space having a (UTAP) unconditional basis. Their work also left many open questions, most of which remain unsolved as of today.…”

“…To prove (ii), we pick a semi-normalized well complemented block basic sequence (u m ) m∈M with good projecting functionals (u * m ) m∈M . By Lemma 3.6, we can suppose that u * m = 1 * supp(um) so that sup ), who had proved the uniqueness of unconditional basis up to permutation of the 2-convexifyed Tsirelson's space T (2) of T (see Example 5.10 in § 5 for the definition). Unlike T (2) , which is "highly" Euclidean, the space T is anti-Euclidean.…”

“…By Lemma 3.6, we can suppose that u * m = 1 * supp(um) so that sup ), who had proved the uniqueness of unconditional basis up to permutation of the 2-convexifyed Tsirelson's space T (2) of T (see Example 5.10 in § 5 for the definition). Unlike T (2) , which is "highly" Euclidean, the space T is anti-Euclidean. To see the latter requires the notion of dominance, introduced in [13].…”

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

“…From the general point of view of approximation theory, and more specifically the practical implementation of the greedy algorithm for general biorthogonal systems, it is very natural to ask about the existence of conditional quasi-greedy bases in the context of nonlocally convex quasi-Banach spaces. Since L p ([0, 1]) for 0 < p < 1 has trivial dual (making it therefore impossible for L p to have a basis), the first nonlocally convex spaces that come to mind as objects of study for having conditional quasi-greedy bases are the spaces ℓ p for 0 < p < 1 (see [1,Problem 12.8]). However, the tools that have been developed for building conditional quasi-greedy bases in Banach spaces break down when local convexity is lifted.…”

On a 'philosophical' question about Banach envelopes

“…They also proved that, in general, the Banach envelope of ℓ ϕ is represented by the closure of J(ℓ ϕ ) in ℓ ψ ⊕ ℓ ϕ /m ϕ , where the envelope map J is given by J(f ) = (f, f + m ϕ ) for all f ∈ ℓ ϕ . Other more recent articles working specifically on Banach envelopes include, e.g., [3,11,13,[25][26][27].…”