On weakly compact operators on Banach lattices
HTML articles powered by AMS MathViewer
- by C. D. Aliprantis and O. Burkinshaw PDF
- Proc. Amer. Math. Soc. 83 (1981), 573-578 Request permission
Abstract:
Consider a Banach lattice $E$ and an order bounded weakly compact operator $T:E \to E$. The purpose of this note is to study the weak compactness of operators that are related with $T$ in some order sense. The main results are the following. (1) If $T$ is a positive weakly compact operator and an operator $S:E \to E$ satisfies $0 \leqslant S \leqslant T$, then ${S^2}$ is weakly compact. (Examples show that $S$ need not be weakly compact.) (2) If $T$ and $S$ are as in (1) and either $S$ is an orthomorphism or $E$ has an order continuous norm, then $S$ is weakly compact. (3) If $E$ is an abstract $L$-space and $T$ is weakly compact, then the modulus $|T|$ is weakly compact.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
- Owen Burkinshaw, Weak compactness in the order dual of a vector lattice, Trans. Amer. Math. Soc. 187 (1974), 183–201. MR 394098, DOI 10.1090/S0002-9947-1974-0394098-6 N. Dunford and J. T. Schwartz, Linear operators. I, Wiley (Interscience), New York, 1958. W. A. J. Luxemburg and A. C. Zaanen, Riesz spaces. I, North-Holland, Amsterdam, 1971.
- Helmut H. Schaefer, Banach lattices and positive operators, Die Grundlehren der mathematischen Wissenschaften, Band 215, Springer-Verlag, New York-Heidelberg, 1974. MR 0423039
- A. R. Schep, Compactness properties of an operator which imply that it is an integral operator, Trans. Amer. Math. Soc. 265 (1981), no. 1, 111–119. MR 607110, DOI 10.1090/S0002-9947-1981-0607110-7
- A. W. Wickstead, Representation and duality of multiplication operators on Archimedean Riesz spaces, Compositio Math. 35 (1977), no. 3, 225–238. MR 454728
- Anthony W. Wickstead, Extremal structure of cones of operators, Quart. J. Math. Oxford Ser. (2) 32 (1981), no. 126, 239–253. MR 615198, DOI 10.1093/qmath/32.2.239
Additional Information
- © Copyright 1981 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 83 (1981), 573-578
- MSC: Primary 47B55; Secondary 46B30
- DOI: https://doi.org/10.1090/S0002-9939-1981-0627695-X
- MathSciNet review: 627695