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In this thesis we introduce the class (DELTA)(X,Y) of nearly represent- able operators from a Banach space X to a Banach space Y. These are the operators that map X-valued uniformly bounded martingales that are Cauchy in the Pettis norm into Y-valued martingales that converge almost everywhere.
We will see that (DELTA)(L('1),X) contains all representable operators and is contained in the class of Dunford-Pettis operators from L('1) to X. We prove that these inclusions are strict in the case X is the space c(,0). We also prove every nearly representable operator from L('1) to a Banach lattice not containing a copy of c(,0) is representable.
We study the class of M(,0)-continuous operators. (An operator T : L('1) (--->) X is called M(,0)-continuous if
(DIAGRAM, TABLE OR GRAPHIC OMITTED...PLEASE SEE DAI)
where the supremum is taken over all subintervals I of 0,1 ).
In the last chapter we give geometric conditions on a Banach space X that imply that all operators from L('1) to X are Dunford-Pettis.
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