## Files in this item

FilesDescriptionFormat

application/pdf

8711827.pdf (2MB)
(no description provided)PDF

## Description

 Title: Uniform Convergence of Operators and Grothendieck Spaces With the Dunford-Pettis Property Author(s): Leung, Denny Ho-Hon Department / Program: Mathematics Discipline: Mathematics Degree Granting Institution: University of Illinois at Urbana-Champaign Degree: Ph.D. Genre: Dissertation Subject(s): Mathematics 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. Issue Date: 1987 Type: Text Description: 89 p.Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 1987. URI: http://hdl.handle.net/2142/71250 Other Identifier(s): (UMI)AAI8711827 Date Available in IDEALS: 2014-12-16 Date Deposited: 1987
﻿