Dunford-Pettis operators on Banach lattices
HTML articles powered by AMS MathViewer
- by C. D. Aliprantis and O. Burkinshaw PDF
- Trans. Amer. Math. Soc. 274 (1982), 227-238 Request permission
Abstract:
Consider a Banach lattice $E$ and two positive operators $S,T:E \to E$ that satisfy $0 \leqslant S \leqslant T$. In $[{\mathbf {2,3}}]$ we examined the case when $T$ is a compact (or weakly compact) operator and studied what effect this had on an operator (such as $S$) dominated by $T$. In this paper, we extend these techniques and study similar questions regarding Dunford-Pettis operators. In particular, conditions will be given on the operator $T$, to ensure that $S$ (or some power of $S$) is a Dunford-Pettis operator. As a sample, the following is one of the major results dealing with these matters. Theorem. Let $E$ be a Banach lattice, and let $S,T:E \to E$ be two positive operators such that $0 \leqslant S \leqslant T$. If $T$ is compact then (1) ${S^3}$ is a compact operator (although ${S^2}$ need not be compact); (2) ${S^2}$ is a Dunford-Pettis and weakly compact operator ( although $S$ need not be ); (3) $S$ is a weak Dunford-Pettis operator. In another direction, our techniques and results will be related to the lattice stracture of the Dunford-Pettis operators. For instance, it will be shown that under certain conditions the Dunford-Pettis operators form a band.References
- Charalambos D. Aliprantis and Owen Burkinshaw, Locally solid Riesz spaces, Pure and Applied Mathematics, Vol. 76, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. MR 0493242
- Charalambos D. Aliprantis and Owen Burkinshaw, Positive compact operators on Banach lattices, Math. Z. 174 (1980), no. 3, 289–298. MR 593826, DOI 10.1007/BF01161416
- C. D. Aliprantis and O. Burkinshaw, On weakly compact operators on Banach lattices, Proc. Amer. Math. Soc. 83 (1981), no. 3, 573–578. MR 627695, DOI 10.1090/S0002-9939-1981-0627695-X
- Kevin T. Andrews, Dunford-Pettis sets in the space of Bochner integrable functions, Math. Ann. 241 (1979), no. 1, 35–41. MR 531148, DOI 10.1007/BF01406706
- J. Bourgain, Dunford-Pettis operators on $L^{1}$ and the Radon-Nikodým property, Israel J. Math. 37 (1980), no. 1-2, 34–47. MR 599300, DOI 10.1007/BF02762866 O. Burkinshaw, Weak compactness in the order dual of a vector lattice, Ph. D. thesis, Purdue University, 1972.
- J. Diestel and J. J. Uhl Jr., Vector measures, Mathematical Surveys, No. 15, American Mathematical Society, Providence, R.I., 1977. With a foreword by B. J. Pettis. MR 0453964
- P. G. Dodds, $o$-weakly compact mappings of Riesz spaces, Trans. Amer. Math. Soc. 214 (1975), 389–402. MR 385629, DOI 10.1090/S0002-9947-1975-0385629-1
- P. G. Dodds and D. H. Fremlin, Compact operators in Banach lattices, Israel J. Math. 34 (1979), no. 4, 287–320 (1980). MR 570888, DOI 10.1007/BF02760610
- Michel Duhoux, $\textrm {o}$-weakly compact mappings from a Riesz space to a locally convex space, Bull. Math. Soc. Sci. Math. R. S. Roumanie (N.S.) 22(70) (1978), no. 4, 371–378. MR 522533
- Helmut H. Schaefer, Banach lattices and positive operators, Die Grundlehren der mathematischen Wissenschaften, Band 215, Springer-Verlag, New York-Heidelberg, 1974. MR 0423039
Additional Information
- © Copyright 1982 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 274 (1982), 227-238
- MSC: Primary 47B55; Secondary 46B30, 47D15
- DOI: https://doi.org/10.1090/S0002-9947-1982-0670929-1
- MathSciNet review: 670929