# Article

 Title: Criteria for weak compactness of vector-valued integration maps (English) Author: Okada, S. Author: Ricker, W. J. Language: English Journal: Commentationes Mathematicae Universitatis Carolinae ISSN: 0010-2628 (print) ISSN: 1213-7243 (online) Volume: 35 Issue: 3 Year: 1994 Pages: 485-495 . Category: math . Summary: 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. (English) Keyword: weakly compact integration map Keyword: factorization of a vector measure MSC: 28B05 MSC: 46A05 MSC: 46E30 MSC: 46G10 MSC: 47B07 MSC: 47B38 idZBL: Zbl 0805.46040 idMR: MR1307275 . Date available: 2009-01-08T18:12:41Z Last updated: 2012-04-30 Stable URL: http://hdl.handle.net/10338.dmlcz/118688 . Reference: [1] Aliprantis C.D., Burkinshaw O.: Positive Operators.Academic Press, New York, 1985. Zbl 1098.47001, MR 0809372 Reference: [2] Diestel J., Uhl J.J. Jr.: Vector measures.Math. Surveys, No.15, Amer. Math. Soc., Providence, 1977. Zbl 0521.46035, MR 0453964 Reference: [3] Dodds P.G., Ricker W.J.: Spectral measures and the Bade reflexivity theorem.J. Funct. Anal. 61 (1985), 136-163. Zbl 0577.46043, MR 0786620 Reference: [4] Kluvánek I., Knowles G.: Vector measures and control systems.North Holland, Amsterdam, 1976. MR 0499068 Reference: [5] Okada S., Ricker W.J.: Compactness properties of the integration map associated with a vector measure.Colloq. Math., to appear. Zbl 0884.28008, MR 1268062 Reference: [6] Okada S., Ricker W.J.: Compactness properties of vector-valued integration maps in locally convex spaces.Colloq. Math., to appear. Zbl 0821.46057, MR 1292938 Reference: [7] Ricker W.J.: Spectral measures, boundedly $\sigma$-complete Boolean algebras and applications to operator theory.Trans. Amer. Math. Soc. 304 (1987), 819-838. Zbl 0642.47029, MR 0911097 Reference: [8] Treves F.: Topological Vector Spaces, Distributions and Kernels.Academic Press, New York, 1967. Zbl 1111.46001, MR 0225131 .

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