# Unconditional structures of weakly null sequences

@article{Argyros1999UnconditionalSO, title={Unconditional structures of weakly null sequences}, author={Spiros A. Argyros and Ioannis Gasparis}, journal={Transactions of the American Mathematical Society}, year={1999}, volume={353}, pages={2019-2058} }

The following dichotomy is established for a normalized weakly null sequence in a Banach space: Either every subsequence admits a convex block subsequence equivalent to the unit vector basis of c, the Banach space of null sequences under the supremum norm, or there exists a subsequence which is boundedly convexly complete. This result generalizes J. Elton's dichotomy on weakly null sequences.

## 36 Citations

WEAKLY NULL SEQUENCES IN THE BANACH

- 2006

The hierarchy of the block bases of transfinite normalized averages of a normalized Schauder basic sequence is introduced and a criterion is given for a normalized weakly null sequence in C(K), the…

Weakly null sequences in the Banach space C(K)

- Mathematics
- 2004

The hierarchy of the block bases of transfinite normalized averages of a normalized Schauder basic sequence is introduced and a criterion is given for a normalized weakly null sequence in C(K), the…

New examples of c0-saturated Banach spaces

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For every p ∈ (1, ∞), an isomorphically polyhedral Banach space Ep is constructed which has an unconditional basis and does not embed isomorphically into a C(K) space for any countable and compact…

Strictly singular non-compact operators on hereditarily indecomposable Banach spaces

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- 2002

An example is given of a strictly singular non-compact operator on a Hereditarily Indecomposable, reflexive, asymptotic l 1 Banach space. The construction of this operator relies on the existence of…

Extracting long basic sequences from systems of dispersed vectors

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- 2010

Abstract We study Banach spaces satisfying some geometric or structural properties involving tightness of transfinite sequences of nested linear subspaces. These properties are much weaker than WCG…

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- 2002

We present partial results to the following question:
Does every infinite dimensional Banach space have an infinite dimensional subspace on which one can define an operator which is not a compact…

New examples of $c_0$-saturated Banach spaces II

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- 2008

For every Banach space $Z$ with a shrinking unconditional basis satisfying upper $p$-estimates for some $p > 1$, an isomorphically polyhedral Banach space is constructed having an unconditional basis…

Operators on C[0,1] preserving copies of asymptotic ℓ1 spaces

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- 2005

Given separable Banach spaces X, Y, Z and a bounded linear operator T:X→Y, then T is said to preserve a copy of Z provided that there exists a closed linear subspace E of X isomorphic to Z and such…

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- 2002

Abstract. A proof of Rosenthal's co- index conjecture is given. Our approach uses optimal sequences of continuous functions converging to a Baire-1 function. Their existence is obtained by the…

On Zippin's Embedding Theorem of Banach spaces into Banach spaces with bases

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We present a new proof of Zippin's Embedding Theorem, that every separable reflexive Banach space embeds into one with shrinking and boundedly complete basis, and every Banach space with a separable…

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