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Title:Dunford-Pettis operators on L(,1) and the complete continuity property
Author(s):Girardi, Maria Kathryn
Doctoral Committee Chair(s):Uhl, J. Jerry, Jr.
Department / Program:Mathematics
Degree Granting Institution:University of Illinois at Urbana-Champaign
Abstract:The interplay between the behavior of bounded linear operators from $L\sb1$ into a Banach space ${\cal X}$ and the internal geometry of ${\cal X}$ has long been evident. The Radon-Nikodym property (RNP) and strong regularity arose as operator theoretic properties but were later realized as geometric properties.
Another operator theoretic property, the complete continuity property (CCP), is a weakening of both the RNP and strong regularity. A Banach space ${\cal X}$ has the CCP if all bounded linear operators from $L\sb1$ into ${\cal X}$ are Dunford-Pettis (i.e. take weakly convergent sequences to norm convergent sequences). There are motivating partial results suggesting that the CCP also can be realized as a geometric property. This thesis provides such a realization.
Our first step is to derive an oscillation characterization of Dunford-Pettis operators. Using this oscillation characterization, we obtain a geometric description of the CCP; namely, we show that ${\cal X}$ has the CCP if and only if all bounded subsets of ${\cal X}$ are Bocce dentable, or equivalently, all bounded subsets of ${\cal X}$ are weak-norm-one dentable. This geometric description leads to yet another; ${\cal X}$ has the CCP if and only if no bounded separated $\delta$-trees grow in ${\cal X}$, or equivalently, no bounded $\delta$-Rademacher trees grow in ${\cal X}$. We also localize these results. We motivate these characterizations by the corresponding (known) characterizations of the RNP and of strong regularity.
Issue Date:1990
Rights Information:Copyright 1990 Girardi, Maria Kathryn
Date Available in IDEALS:2011-05-07
Identifier in Online Catalog:AAI9026189
OCLC Identifier:(UMI)AAI9026189

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