On the number of permutatively inequivalent basic sequences in a Banach space

Abstract

Let X be a Banach space with a Schauder basis (e n ) n∈N . The relation E 0 is Borel reducible to permutative equivalence between normalized block-sequences of (e n ) n∈N or X is c 0 or p saturated for some 1 p < +∞. If (e n ) n∈N is shrinking unconditional then either it is equivalent to the canonical basis of c 0 or p , 1 < p < +∞, or the relation E 0 is Borel reducible to permutative equivalence between sequences of normalized disjoint blocks of X or of X * . If (e n ) n∈N is unconditional, then either X is isomorphic to 2 , or X contains 2 ω subspaces or 2 ω quotients which are spanned by pairwise permutatively inequivalent normalized unconditional bases.

The complexity of classifying separable Banach spaces up to isomorphism

“…Also Rosendal [20] shows that the relation of equivalence between basic sequences is Borel bireducible with a complete K σ equivalence relation. And finally, Ferenczi [7] proves that if (e i ) is an unconditional basic sequence, then either E 0 Borel reduces to the relation of permutative equivalence between the normalised blockbases of (e i ), or, for some ℓ p or c 0 , any normalised blockbasis has a subsequence equivalent with this ℓ p or c 0 .…”

Complexity and homogeneity in Banach spaces

“…Schlumprecht's space is a relevant example by its minimality and the fact that E 0 is Borel reducible to permutative equivalence between its normalised block-sequences [13].…”

“…Ferenczi and Rosendal proved that if a normalised Schauder basis is not equivalent to the canonical basis of c 0 or p , 1 p < +∞, then E 0 reduces to equivalence between its normalised block-sequences [19]. Ferenczi [13] obtained that if X has an unconditional basis, then E 0 is Borel reducible to permutative equivalence on bb(X) or every normalised block-sequence has a subsequence equivalent to the unit vector basis of some fixed p or c 0 . Some apparently weaker properties turn out to be equivalent to block homogeneity.…”

On Strictly Singular Operators Between Separable Banach Spaces

“…The space Z p,q is a variant of James tree space JT [19]. We notice that spaces of this form have found significant applications and have been extensively studied by several authors (see, for instance, [5,9,10,13,15,16]). We gather, below, some elementary properties of the space Z p,q .…”

“…For every vector z in Z p,q the support supp(z) of z is defined to be the set {t ∈ N <N : z * t (z) = 0}. For every A ⊆ N <N non-empty let (16) Z A p,q = span{z t : t ∈ A}. The subspace Z A p,q of Z p,q is complemented via the natural projection (17) P A : Z p,q → Z A p,q .…”