Convex unconditionality and summability of weakly null sequences

Abstract

It is proved that every normalized weakly null sequence has a subsequence which is convexly unconditional. Further, an Hierarchy of summability methods is introduced and with this we give a complete classification of the complexity of weakly null sequences.

Strictly singular non-compact operators on hereditarily indecomposable Banach spaces

“…A standard property of the repeated averages hierarchy [9] (Lemma 2.3 of [13]) allows us to find F k ∈ S p k with max{k, l} < min F k , positive scalars (a i ) i∈F k with i∈F k a i = 1 and such that i∈G a i < 1 m k for every G ∈ S p k −1 . It follows readily from Corollary 4.4 that for every x * ∈ N, the set {i ∈ F k : |x * (e i )| ≥ 2/m k } belongs to S p k −1 .…”

Weakly null sequences with an unconditional subsequence

“…Namely J. Elton's near unconditionality ( [4]) and Argyros-Mercourakis-Tsarpalias convex unconditionality [2]. It extends an earlier result in [2] which was used for an alternative proof of the above-mentioned Rosenthal's Theorem. This proof also can be found in [1].…”

Some remarks on spreading models and mixed Tsirelson spaces

“…We next pass to the definition of the repeated averages hierarchy introduced in [7]. We let (e n ) denote the standard basis of c 00 .…”

“…Recent results [6], [7], [15], have shown the necessity of studying the higher ordinal structure of an asymptotic 1 Banach space in order to obtain results on the global structure of its infinite dimensional subspaces. A normalized sequence (x n ) n in a Banach space X is said to be a c − ξ 1 spreading model if n∈F α n x n ≥ c n∈F |α n | ∀F ∈ S ξ , (a n ) n∈F ⊂ R, where S ξ , ξ < ω 1 , are the generalized Schreier families defined in [1].…”

“…A complete classification of normalized weakly null sequences, in connection with spreading models, has been provided in [7].…”