Compactness in 𝐿₁, Dunford-Pettis operators, geometry of Banach spaces

Girardi, Maria

doi:10.1090/S0002-9939-1991-1039256-8

Compactness in $L_ 1$, Dunford-Pettis operators, geometry of Banach spaces
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by Maria Girardi

Proc. Amer. Math. Soc. 111 (1991), 767-777

DOI: https://doi.org/10.1090/S0002-9939-1991-1039256-8

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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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