Nearly representable operators

Authors:
R. Kaufman, Minos Petrakis, Lawrence H. Riddle and J. J. Uhl

Journal:
Trans. Amer. Math. Soc. **312** (1989), 315-333

MSC:
Primary 47B38; Secondary 46B22, 47A67

DOI:
https://doi.org/10.1090/S0002-9947-1989-0951887-5

MathSciNet review:
951887

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Abstract | References | Similar Articles | Additional Information

Abstract: Among Bourgain's many remarkable theorems is one from 1980 which states that if is a non-Dunford-Pettis operator from into an arbitrary Banach space, then there is a Dunford-Pettis operator from into such that the composition is not Bochner representable. This theorem sets up the following question: What are the operators from into a Banach space such that is Bochner representable for all Dunford-Pettis operators ? We call such an operator nearly representable. In view of Bourgain's theorem, all nearly representable operators are Dunford-Pettis. If is a Banach space such that all nearly representable operators are, in addition, Bochner representable, then we say 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 is a nonrepresentable operator, there exists a Dunford-Pettis operator such that is also nonrepresentable. This implies that the is shared by , lattices not containing , and .

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1989-0951887-5

Keywords:
Dunford-Pettis operators,
Bochner representable operators,
singular measures,
Radon-Nikodym property,
semi-embeddings

Article copyright:
© Copyright 1989
American Mathematical Society