University of Illinois Urbana-Champaign

Dunford-Pettis operators on L(,1) and the complete continuity property

Girardi, Maria Kathryn

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Description

Title
Dunford-Pettis operators on L(,1) and the complete continuity property
Author(s)
Girardi, Maria Kathryn
Issue Date
1990
Doctoral Committee Chair(s)
Uhl, J. Jerry, Jr.
Department of Study
Mathematics
Discipline
Mathematics
Degree Granting Institution
University of Illinois at Urbana-Champaign
Degree Name
Ph.D.
Degree Level
Dissertation
Keyword(s)
Mathematics
Language
eng
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.
Type of Resource
text
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http://hdl.handle.net/2142/22597
Copyright and License Information
Copyright 1990 Girardi, Maria Kathryn

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