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|Title:||Uniform Convergence of Operators and Grothendieck Spaces With the Dunford-Pettis Property|
|Author(s):||Leung, Denny Ho-Hon|
|Department / Program:||Mathematics|
|Degree Granting Institution:||University of Illinois at Urbana-Champaign|
|Abstract:||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.
Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 1987.
|Date Available in IDEALS:||2014-12-16|