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Leung, Denny Ho-Hon 2014-12-16T06:18:17Z 2014-12-16T06:18:17Z 10000-01-01 1987 1987 A theorem of D. W. Dean states that all Grothendieck spaces with the Dunford-Pettis property fail to admit Schauder decompositions. Similar results of H. P. Lotz concern the uniform convergence of ergodic operators and semi-groups of operators. This thesis considers extensions of these results to wider classes of Banach spaces. We say that a Banach space E has the surjective Dunford-Pettis property if every operator from E onto a reflexive Banach space maps weakly compact sets onto norm compact sets. The main result can now be stated. Theorem. Let E be a Grothendieck space with the surjective Dunford-Pettis property, then (1) The space E admits no Schauder decompositions; (2) Every strongly ergodic operator T on E with (VBAR)(VBAR)T('n)/n(VBAR)(VBAR) (--->) 0 is uniformly ergodic; and (3) Every strongly continuous semi-group of operators (T(,t))(,t(GREATERTHEQ)0) on E is uniformly continuous. Examples are presented to distinguish the surjective Dunford-Pettis property from the closely related Dunford-Pettis property. Made available in DSpace on 2014-12-16T06:18:17Z (GMT). No. of bitstreams: 1 8711827.pdf: 2402542 bytes, checksum: 65d828d7b411ddbc74ef602599d6825f (MD5) Previous issue date: 1987 Embargo set by: Seth Robbins for item 71416 Lift date: Forever Reason: Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs Restricted to the U of I community idenfinitely during batch ingest of legacy ETDs U of I Only 89 p. Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 1987. http://hdl.handle.net/2142/71250 (UMI)AAI8711827 Mathematics Uniform Convergence of Operators and Grothendieck Spaces With the Dunford-Pettis Property text Mathematics Mathematics University of Illinois at Urbana-Champaign Dissertation Ph.D.