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Uhl, J. Jerry, Jr. Girardi, Maria Kathryn 2011-05-07T13:45:00Z 2011-05-07T13:45:00Z 10000-01-01 1990 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. Made available in DSpace on 2011-05-07T13:45:00Z (GMT). No. of bitstreams: 2 license.txt: 4922 bytes, checksum: 910b249b4beec47e7ab768910c8f966f (MD5) 9026189.pdf: 1717764 bytes, checksum: cd709ca36296284fc644b7ad1fd942b4 (MD5) Previous issue date: 1990 Item marked as restricted to the 'UIUC Users [automated]' Group (id=2) by Howard Ding (hding2@illinois.edu) on 2011-05-07T14:58:42Z Item is restricted indefinitely. Restriction data tranferred 2014-07-01T11:27:37-05:00 Original Data Group with Access UIUC Users [automated] Release Date: none Reason: ETDs are only available to UIUC Users without author permission ETDs are only available to UIUC Users without author permission U of I Only AAI9026189 (UMI)AAI9026189 http://hdl.handle.net/2142/22597 eng Copyright 1990 Girardi, Maria Kathryn Mathematics Dunford-Pettis operators on L(,1) and the complete continuity property text Mathematics Mathematics University of Illinois at Urbana-Champaign Dissertation Ph.D.