Nearly representable operators

Among Bourgain's many remarkable theorems is one from 1980 which states that if $T$ is a non-Dunford-Pettis operator from ${L_1}[0,1]$ into an arbitrary Banach space, then there is a Dunford-Pettis operator $D$ from ${L_1}[0,1]$ into ${L_1}[0,1]$ such that the composition $T \circ D$ is not Bochner representable. This theorem sets up the following question: What are the operators $T$ from ${L_1}[0,1]$ into a Banach space $X$ such that $T \circ D$ is Bochner representable for all Dunford-Pettis operators $D:{L_1}[0,1] \to {L_1}[0,1]$ ? We call such an operator nearly representable. In view of Bourgain's theorem, all nearly representable operators are Dunford-Pettis. If $X$ is a Banach space such that all nearly representable operators are, in addition, Bochner representable, then we say $X$ has the near Radon-Nikodym property (NRNP) and ask which Banach spaces have the NRNP? This paper is an attempt to provide at least partial answers to these questions.

The first section collects terminology, gives the introductory results and shows that the NRNP is a three space property. The second section studies a continuity property that implies near representability. Finally, the third section contains the main result of the paper, Theorem 15, which states that if $T:{L_1}[0,1] \to {L_1}[0,1]$ is a nonrepresentable operator, there exists a Dunford-Pettis operator $D:{L_1}[0,1] \to {L_1}[0,1]$ such that $T \circ D$ is also nonrepresentable. This implies that the ${\text{NRNP}}$ is shared by ${L_1}[0,1]$, lattices not containing ${c_0}$, and ${L_1}({\mathbf{T}})/H_0^1$.