Publication: On Pełczyński's property (V∗) in vector sequence spaces
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Publication Date
1988
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Springer
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Let E be a Banach space and A⊂E a (V∗) set, i.e. a (bounded) set such that for every weakly unconditionally converging series ∑x∗n in E∗ one has limnsupA|x∗n(x)|=0. The space E is said to possess the property (V∗) of Pełczyński if any (V∗) set in E is relatively weakly compact, and the weak property (V∗) if any (V∗) set is conditionally weakly compact. The main result of the paper is the following. Theorem 9: Let (En) be a sequence of Banach spaces, 1≤p<∞, and E=(∑⊕En)p. Then E has the property (V∗) [resp. the weak property (V∗)] if and only if so does any En.
The result is based upon the following characterizations of (V∗) sets in vector sequence spaces. Proposition 4: Let (En) be a sequence of Banach spaces and A⊂E=(∑⊕En)1 a bounded subset. The following assertions are equivalent: (a) A is a (V∗) set; (b) Πn(A)={xn:x=(xk)k∈A} is a (V∗) set in En, for every natural number n and lim supn→∞{∑∞k=n∥xk∥: x∈A}=0. Proposition 5: Let (En) be a sequence of Banach spaces and 1<p<∞ or p=0. For a bounded subset A⊂E=(∑⊕En)p, the following assertions are equivalent: (1) A is a (V∗) set; (b) Πn(A) is a (V∗) set in En, for every natural number n. Furthermore, well-known results about relatively (conditionally) weakly compact subsets of (∑⊕En)p are used in the proof of the main result.
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F. Bombal, Operators on vector sequence spltces, in Geometric Aspects oi Banach Spaces pp. 94-106, London Mathematical Society Lecture Note Series 140, Cambridge Universit) Press, Cambridge, 1989.
F. Bombal, On (V*) sets and Pelzynski's property (V*), Glasgow Math. J., to appear.
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A. Pelczynski, On Banach spaces on which every unconditionally convergent operator il weakly compact, Bull. Acad. Po. Sci. 10 91962), 641-648.
E. Saab and P. Saab, On Pelczynski's properties (V) and (V*), Pacif. J. Math. 125 (1986) 205-210.