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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 Discipline: Mathematics Degree Granting Institution: University of Illinois at Urbana-Champaign Degree: Ph.D. Genre: Dissertation Subject(s): Mathematics 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 Type: Text Language: English URI: http://hdl.handle.net/2142/22597 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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