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The regularity of Dunford-Pettis operators


Author: James R. Holub
Journal: Proc. Amer. Math. Soc. 104 (1988), 89-95
MSC: Primary 47B37; Secondary 47B38
DOI: https://doi.org/10.1090/S0002-9939-1988-0958049-0
MathSciNet review: 958049
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Abstract: Let $ \lambda $ denote a symmetric, solid Banach sequence space having $ \left\{ {{e_i}} \right\}_{i = 1}^\infty $ as a symmetric basis and considered as a Banach lattice with order defined coordinatewise. A complete description of the relationship between regular and Dunford-Pettis operators $ T:{L^1}[0,1] \to \lambda $ is given. The results obtained complete earlier work of Gretsky and Ostroy and of the author in this area.


References [Enhancements On Off] (What's this?)

  • [1] J. Bourgain, Dunford-Pettis operators on $ {L^1}$ and the Radon-Nikodym property, Israel J. Math. 37 (1980), 34-47. MR 599300 (82k:47047a)
  • [2] -, A characterization of non-Dunford-Pettis operators on $ {L^1}$, Israel J. Math. 37 (1980), 48-53. MR 599301 (82k:47047b)
  • [3] N. Dunford and B. J. Pettis, Linear operators on summable functions, Trans. Amer. Math. Soc. 47 (1940), 323-392. MR 0002020 (1:338b)
  • [4] N. Dunford and J. Schwartz, Linear operators. I, Interscience, New York, 1963. MR 0188745 (32:6181)
  • [5] N. Gretsky and J. Ostroy, Thick and thin market non-atomic exchange economies, Advances in Equilibrium Theory, Lecture Notes in Economics and Mathematical Systems, No. 244, 1985, pp. 107-130. MR 873762 (88h:90041)
  • [6] -, The compact range property and $ {c_0}$, Glasgow Math. J. 28 (1986), 113-114. MR 826634 (87f:47053)
  • [7] -, Dunford-Pettis operators and average range (preprint).
  • [8] A. Grothendieck, Sur les applications lineaires faiblement compactes d'espaces du type $ C(K)$, Canad. J. Math. 5 (1953), 129-173. MR 0058866 (15:438b)
  • [9] J. Holub, A note on Dunford-Pettis operators, Glasgow Math. J. 29 (1987), 271-273. MR 901675 (88m:47054)
  • [10] J. Lindenstrauss and L. Tzafriri, Classical Banach spaces, Springer-Verlag, Berlin and New York, 1973. MR 0415253 (54:3344)
  • [11] K. Musial, An operator characterization of Banach spaces which do not contain any isomorphic copy of $ {l^1}$, Aarhus Universität Math. Inst. Preprint Series, No. 28, 1976/77.
  • [12] H. Rosenthal, Convolution by a biased coin, The Altgeld Book 1975/76, Univ. of Illinois, Functional Analysis Seminar. MR 1017038 (91i:46018)
  • [13] E. Saab, On Dunford-Pettis operators, Canad. Math. Bull. 25 (1982), 207-209. MR 663615 (84e:46016)
  • [14] -, On Dunford-Pettis operators that are Pettis representable, Proc. Amer. Math. Soc. 85 (1982), 363-365. MR 656103 (83m:47035)
  • [15] I. Singer, Bases in Banach spaces. I, Springer-Verlag, New York-Heidelberg-Berlin, 1970. MR 0298399 (45:7451)
  • [16] M. Talagrand, Pettis integral and measure theory, Mem. Amer. Math. Soc. no. 307, Providence, R.I., 1984. MR 756174 (86j:46042)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1988-0958049-0
Keywords: Dunford-Pettis operator, regular operator
Article copyright: © Copyright 1988 American Mathematical Society

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