Finite order spreading models

Abstract

Extending the classical notion of the spreading model, the kspreading models of a Banach space are introduced, for every k ∈ N. The definition, which is based on the k-sequences and plegma families, reveals a new class of spreading sequences associated to a Banach space. Most of the results of the classical theory are stated and proved in the higher order setting. Moreover, new phenomena like the universality of the class of the 2-spreading models of c 0 and the composition property are established. As consequence, a problem concerning the structure of the k-iterated spreading models is solved. ∞ l=1 Plm l ([M ] k ). Notice that for l = 1 and every k ∈ N, we have Plm 1 ([M ] k ) = [M ] k . Moreover, for k = 1 and every l ∈ N, Plm l ([M ] 1 ) = [M ] l . In the sequel the elements of Plm 2 ([M ] k ) will be called plegma pairs in [M ] k .

A study of conditional spreading sequences

“…Otherwise, if it had a weakly null subsequence, as it is well known, it would generates an unconditional spreading model. If it had a subsequence converging weakly to a non-zero element it would generate a singular spreading model or an ℓ 1 spreading model (see [AKT1,Theorem 38,page 592]), or if it had a norm convergent subsequence it would generate a trivial spreading model (i.e. a spreading sequence in seminormed space that is not a normed space).…”

“…The proof involves a construction similar to that of Schreier's space S from [S], a space that has a weakly null basis generating an ℓ 1 spreading model, yet the space S embeds into C(ω ω ). A noteworthy fact that was proved in [AKT1] is that c 0 admits every possible spreading sequence as a 2-spreading model. This is a notion of spreading models introduced and studied in [AKT1] and [AKT2] where there is developed an entire theory surrounding the so called ξ-spreading models.…”

“…A noteworthy fact that was proved in [AKT1] is that c 0 admits every possible spreading sequence as a 2-spreading model. This is a notion of spreading models introduced and studied in [AKT1] and [AKT2] where there is developed an entire theory surrounding the so called ξ-spreading models. This theory examines the idea of higher order spreading models in a direction that allows this notion to be taken with respect transfinite ordinal numbers.…”

On the structure of the set of higher order spreading models

“…In particular, the main results of the paper are Corollary 21 and Theorems 18,19,23 and 25. It is worth to mention that in general SM w ξ (X) does not coincide with SM w 1 (X), and therefore the transfinite hierarchy (SM w ξ (X)) ξ<ω1 is not trivial. In fact, for every k there exists a Banach space X such that SM w k (X) is a proper subset of SM w k+1 (X) [AKT1]. Moreover, there are reflexive Banach spaces X and Y which have, up to equivalence, the same set of spreading models of the first order but not of the second order.…”

On spaces admitting no ℓ p or c 0 spreading model

“…In [1], the authors construct a space not admitting an ℓ p , c 0 or reflexive spreading model. In paper [3] they show that a variant of the space X S does not admit any ℓ p or c 0 as a k-iterated spreading model for any k ∈ N.…”