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Proceedings of the American Mathematical Society

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Compactness in $ L\sb 1$, Dunford-Pettis operators, geometry of Banach spaces


Author: Maria Girardi
Journal: Proc. Amer. Math. Soc. 111 (1991), 767-777
MSC: Primary 46B20; Secondary 46A50, 46E30, 47B38
DOI: https://doi.org/10.1090/S0002-9939-1991-1039256-8
MathSciNet review: 1039256
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Abstract: A type of oscillation modeled on BMO is introduced to characterize norm compactness in $ {L_1}$. This result is used to characterize the bounded linear operators from $ {L_1}$ into a Banach space $ \mathfrak{X}$ that map weakly convergent sequences onto norm convergent sequences (i.e., are Dunford-Pettis). This characterization is used to study the geometry of Banach spaces $ \mathfrak{X}$ with the property that all bounded linear operators from $ {L_1}$ into $ \mathfrak{X}$ are Dunford-Pettis.


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DOI: https://doi.org/10.1090/S0002-9939-1991-1039256-8
Article copyright: © Copyright 1991 American Mathematical Society