{"id":{"repo_id":"uiuc","oai_identifier":"oai:www.ideals.illinois.edu:2142/22597"},"canonical_url":"https://search.dev.ndltd.org/etd/uiuc/oai:www.ideals.illinois.edu:2142/22597","repository":{"repo_id":"uiuc","name":"University of Illinois - Urbana-Champaign","base_url":"https://www.ideals.illinois.edu/oai-pmh"},"display":{"title":"Dunford-Pettis operators on L(,1) and the complete continuity property","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.","abstract_html":"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.","abstract_has_math":true,"creators":["Girardi, Maria Kathryn"],"institution":"University of Illinois at Urbana-Champaign","degree_name":"Ph.D.","degree_level":"Dissertation","degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":["Uhl, J. Jerry, Jr."],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2011,"date_issued":"2011-05-07T13:45:00Z","date_published":"2011-05-07T13:45:00Z","updated_at":"2026-07-22T22:25:20Z","subjects":["Mathematics"],"languages":["eng"],"rights":["Copyright 1990 Girardi, Maria Kathryn"],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["AAI9026189","(UMI)AAI9026189"],"render_values":[{"text":"AAI9026189","href":null,"code":true},{"text":"(UMI)AAI9026189","href":null,"code":true}]}]},"links":{"outbound_url":"http://hdl.handle.net/2142/22597","outbound_label":"Handle","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Uhl, J. Jerry, Jr."]},{"key":"dc:creator","label":"Author","values":["Girardi, Maria Kathryn"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2011-05-07T13:45:00Z","10000-01-01","1990"]},{"key":"dc:type","label":"Dc Type","values":["text"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Dissertation"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Ph.D."]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["University of Illinois at Urbana-Champaign"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Mathematics"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["eng"]},{"key":"dc:rights","label":"Dc Rights","values":["Copyright 1990 Girardi, Maria Kathryn"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["AAI9026189","(UMI)AAI9026189","http://hdl.handle.net/2142/22597"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["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"]},{"key":"dc:title","label":"Title","values":["Dunford-Pettis operators on L(,1) and the complete continuity property"]}]}],"canonical_facts":{"dc:contributor":["Uhl, J. Jerry, Jr."],"dc:creator":["Girardi, Maria Kathryn"],"dc:date":["2011-05-07T13:45:00Z","10000-01-01","1990"],"dc:description":["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"],"dc:identifier":["AAI9026189","(UMI)AAI9026189","http://hdl.handle.net/2142/22597"],"dc:language":["eng"],"dc:rights":["Copyright 1990 Girardi, Maria Kathryn"],"dc:subject":["Mathematics"],"dc:title":["Dunford-Pettis operators on L(,1) and the complete continuity property"],"dc:type":["text"],"thesis:degree_discipline":["Mathematics"],"thesis:degree_level":["Dissertation"],"thesis:degree_name":["Ph.D."],"thesis:institution_name":["University of Illinois at Urbana-Champaign"]},"updated_at":"2026-07-22T22:25:20Z"}