On the modulus of cone absolutely summing operators and vector measures of bounded variation
HTML articles powered by AMS MathViewer
- by Boris Lavrič PDF
- Proc. Amer. Math. Soc. 108 (1990), 479-481 Request permission
Abstract:
Let $E$ and $F$ be Banach lattices. It is shown that if $F$ has the Levi and the Fatou property, then the ordered Banach space ${\mathcal {L}^l}\left ( {E,F} \right )$ of cone absolutely summing operators is a Banach lattice and an order ideal of the Riesz space ${\mathcal {L}^r}\left ( {E,F} \right )$ of regular operators. The same argument yields a Jordan decomposition of $F$-valued vector measures of bounded variation.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
- J. Diestel and B. Faires, On vector measures, Trans. Amer. Math. Soc. 198 (1974), 253–271. MR 350420, DOI 10.1090/S0002-9947-1974-0350420-8
- B. Faires and T. J. Morrison, The Jordan decomposition of vector-valued measures, Proc. Amer. Math. Soc. 60 (1976), 139–143. MR 419723, DOI 10.1090/S0002-9939-1976-0419723-X
- Helmut H. Schaefer, Banach lattices and positive operators, Die Grundlehren der mathematischen Wissenschaften, Band 215, Springer-Verlag, New York-Heidelberg, 1974. MR 0423039, DOI 10.1007/978-3-642-65970-6
- Ulf Schlotterbeck, Über Klassen majorisierbarer Operatoren auf Banachverbänden, Rev. Acad. Cienc. Zaragoza (2) 26 (1971), 585–614 (German, with English summary). MR 320809
- Klaus D. Schmidt, Decompositions of vector measures in Riesz spaces and Banach lattices, Proc. Edinburgh Math. Soc. (2) 29 (1986), no. 1, 23–39. MR 829177, DOI 10.1017/S0013091500017375
- Klaus D. Schmidt, On the modulus of weakly compact operators and strongly additive vector measures, Proc. Amer. Math. Soc. 102 (1988), no. 4, 862–866. MR 934857, DOI 10.1090/S0002-9939-1988-0934857-7
- A. C. Zaanen, Riesz spaces. II, North-Holland Mathematical Library, vol. 30, North-Holland Publishing Co., Amsterdam, 1983. MR 704021, DOI 10.1016/S0924-6509(08)70234-4
Additional Information
- © Copyright 1990 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 108 (1990), 479-481
- MSC: Primary 47B10; Secondary 28B05, 47B55, 47D15
- DOI: https://doi.org/10.1090/S0002-9939-1990-0993756-4
- MathSciNet review: 993756