Let X he a Banach space with an unconditional finite-dimensional Schauder decomposition (En). We consider the general problem of characterizing conditions under which one can construct an unconditional basis for X by forming an unconditional basis for each En. For example, we show that if sup,, dim En < c~ and X has Gordon-Lewis local unconditional s t ructure then X has an unconditional basis ...

We prove that a Schauder frame for any separable Banach space is shrinking if and only if it has an associated space with a shrinking basis, and that a Schauder frame for any separable Banach space is shrinking and boundedly complete if and only if it has a reflexive associated space. To obtain these results, we prove that the upper and lower estimate theorems for finite dimensional decompositi...

In this article we suppose that E is an ordered Banach space the positive cone of which is defined by a countable familyF={fi|i ∈ N} of positive continuous linear functionals of E, i.e. E+ = {x ∈ E | fi(x) ≥ 0, for each i} and we study the existence of positive (Schauder) bases in the ordered subspaces X of E with the Riesz decomposition property. So we consider the elements x of E as sequences...

We construct a computable Banach space which possesses a Schauder basis, but does not possess any computable Schauder basis.

The Balian–Low Theorem is a strong form of the uncertainty principle for Gabor systems which form orthonormal or Riesz bases for L(R). In this paper we investigate the Balian–Low Theorem in the setting of Schauder bases. We prove that new weak versions of the Balian–Low Theorem hold for Gabor Schauder bases, but we constructively demonstrate that several variants of the BLT can fail for Gabor S...

We study conditions on a Banach frame that ensures the validity of a reconstruction formula. In particular, we show that any Banach frames for (a subspace of) Lp or Lp,q (1 ≤ p <∞) with respect to a solid sequence space always satisfies an unconditional reconstruction formula. The existence of reconstruction formulae allows us to prove some James-type results for atomic decompositions: an uncon...

Given a property of Banach spaces that is hereditary, it is natural to ask whether a Banach space has the property if each of its subspaces with a w Ž . Ž . particular structure such as a Schauder basis or a Schauder finite-dix mensional decomposition has the property. The motivation for such questions is that it is much easier to deal with Banach spaces with such an additional structure. A sub...

The original motivation for this paper is based on the natural question left open by the Gowers-Maurey solution of the unconditional basic sequence problem for Banach spaces ([12]). Recall that Gowers and Maurey have constructed a Banach space X with a Schauder basis (en)n but with no unconditional basic sequence. Thus, while every infinite dimensional Banach space contains a sequence (xn)n whi...