On spreading $c_0$-sequences in Banach spaces

Vassiliki Farmaki

ArticleOriginal scientific text

Title

On spreading $c_{0}$ -sequences in Banach spaces

Authors ¹

Affiliations

Department of Mathematics, University of Athens, Panepistimiopolis 15784, Athens, Greece

Abstract

We introduce and study the spreading-(s) and the spreading-(u) property of a Banach space and their relations. A space has the spreading-(s) property if every normalized weakly null sequence has a subsequence with a spreading model equivalent to the usual basis of

c_{0}

; while it has the spreading-(u) property if every weak Cauchy and non-weakly convergent sequence has a convex block subsequence with a spreading model equivalent to the summing basis of

c_{0}

. The main results proved are the following: (a) A Banach space X has the spreading-(s) property if and only if for every subspace Y of X and for every pair of sequences (x_n) in Y and

(x \cdot_{n})

in Y*, with(x_n) weakly null in Y and

(x_{n} \cdot)

uniformly weakly null in Y* (in the sense of Mercourakis), we have

x \cdot_{n} (x_{n}) \to 0

(i.e. X has a hereditary weak Dunford-Pettis property). (b) A Banach space X has the spreading-(u) property if and only if

B_{1} (X) \subseteq B_{\frac{1}{4}} (X)

in the sense of the classification of Baire-1 elements of X** according to Haydon-Odell-Rosenthal. (c) The spreading-(s) property implies the spreading-(u) property. Result (c), proved via infinite combinations, connects an internal condition on a Banach space with an external one.

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Title

On spreading c0-sequences in Banach spaces

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Abstract

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On spreading $c_{0}$ -sequences in Banach spaces