Unconditional bases and unconditional finite-dimensional decompositions in Banach spaces

Abstract

Let X be a Banach space with an unconditional finite-dimensional Schauder decomposition (E n ). 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 E n . For example, we show that if sup dim E n < ∞ and X has Gordon-Lewis local unconditional structure then X has an unconditional basis of this type. We also give an example of a non-Hilbertian space X with the property that whenever Y is a closed subspace of X with a UFDD (E n ) such that sup dim E n < ∞ then Y has an unconditional basis, showing that a recent result of Komorowski and Tomczak-Jaegermann cannot be improved.

Banach spaces of bounded Szlenk index II

“…We say that (v i ) is block stable if it is C-block stable for some constant C. This property has been considered before in various forms and under different names. In particular, it has been called the blocking principle [2] and the shift property [3] (see [6] for alternative forms).…”

Unconditional Decompositions in Subspaces of l2(X)

“…We present a different proof than the one from [2], which has the advantage of being also suitable for the context of local unconditional structures of higher order. In fact, the arguments rely on the calculations done already when proving Proposition 2.2.…”

Some equivalence relations which are borel reducible to isomorphism between separable banach spaces

“…It follows that (5) implies (1). That (4) implies (5), that is, Lispchitz isomorphism implies complemented biembeddability, comes from the fact that the spaces considered are separable dual spaces (Theorem 2.4 in [17]). (1) implies (2) by Lemma 2.1 and Corollary 2.2 and the rest is obvious.…”