Previous |  Up |  Next


weakly compact integration map; factorization of a vector measure
Criteria are given for determining the weak compactness, or otherwise, of the integration map associated with a vector measure. For instance, the space of integrable functions of a weakly compact integration map is necessarily normable for the mean convergence topology. Results are presented which relate weak compactness of the integration map with the property of being a bicontinuous isomorphism onto its range. Finally, a detailed description is given of the compactness properties for the integration maps of a class of measures taking their values in $\ell^1$, equipped with various weak topologies.
[1] Aliprantis C.D., Burkinshaw O.: Positive Operators. Academic Press, New York, 1985. MR 0809372 | Zbl 1098.47001
[2] Diestel J., Uhl J.J. Jr.: Vector measures. Math. Surveys, No.15, Amer. Math. Soc., Providence, 1977. MR 0453964 | Zbl 0521.46035
[3] Dodds P.G., Ricker W.J.: Spectral measures and the Bade reflexivity theorem. J. Funct. Anal. 61 (1985), 136-163. MR 0786620 | Zbl 0577.46043
[4] Kluvánek I., Knowles G.: Vector measures and control systems. North Holland, Amsterdam, 1976. MR 0499068
[5] Okada S., Ricker W.J.: Compactness properties of the integration map associated with a vector measure. Colloq. Math., to appear. MR 1268062 | Zbl 0884.28008
[6] Okada S., Ricker W.J.: Compactness properties of vector-valued integration maps in locally convex spaces. Colloq. Math., to appear. MR 1292938 | Zbl 0821.46057
[7] Ricker W.J.: Spectral measures, boundedly $\sigma$-complete Boolean algebras and applications to operator theory. Trans. Amer. Math. Soc. 304 (1987), 819-838. MR 0911097 | Zbl 0642.47029
[8] Treves F.: Topological Vector Spaces, Distributions and Kernels. Academic Press, New York, 1967. MR 0225131 | Zbl 1111.46001
Partner of
EuDML logo