Criteria for weak compactness of vector-valued integration maps

Susumu Okada; Werner J. Ricker

Commentationes Mathematicae Universitatis Carolinae (1994)

  • Volume: 35, Issue: 3, page 485-495
  • ISSN: 0010-2628

Abstract

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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 ℓ 1 , equipped with various weak topologies.

How to cite

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Okada, Susumu, and Ricker, Werner J.. "Criteria for weak compactness of vector-valued integration maps." Commentationes Mathematicae Universitatis Carolinae 35.3 (1994): 485-495. <http://eudml.org/doc/247612>.

@article{Okada1994,
abstract = {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.},
author = {Okada, Susumu, Ricker, Werner J.},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {weakly compact integration map; factorization of a vector measure; weakly compact integration map; factorization of a vector measure; weak compactness; integration map associated with a vector measure; space of integrable functions of a weakly compact integration map; mean convergence topology; bicontinuous isomorphism},
language = {eng},
number = {3},
pages = {485-495},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Criteria for weak compactness of vector-valued integration maps},
url = {http://eudml.org/doc/247612},
volume = {35},
year = {1994},
}

TY - JOUR
AU - Okada, Susumu
AU - Ricker, Werner J.
TI - Criteria for weak compactness of vector-valued integration maps
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 1994
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 35
IS - 3
SP - 485
EP - 495
AB - 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.
LA - eng
KW - weakly compact integration map; factorization of a vector measure; weakly compact integration map; factorization of a vector measure; weak compactness; integration map associated with a vector measure; space of integrable functions of a weakly compact integration map; mean convergence topology; bicontinuous isomorphism
UR - http://eudml.org/doc/247612
ER -

References

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  1. Aliprantis C.D., Burkinshaw O., Positive Operators, Academic Press, New York, 1985. Zbl1098.47001MR0809372
  2. Diestel J., Uhl J.J. Jr., Vector measures, Math. Surveys, No.15, Amer. Math. Soc., Providence, 1977. Zbl0521.46035MR0453964
  3. Dodds P.G., Ricker W.J., Spectral measures and the Bade reflexivity theorem, J. Funct. Anal. 61 (1985), 136-163. (1985) Zbl0577.46043MR0786620
  4. Kluvánek I., Knowles G., Vector measures and control systems, North Holland, Amsterdam, 1976. MR0499068
  5. Okada S., Ricker W.J., Compactness properties of the integration map associated with a vector measure, Colloq. Math., to appear. Zbl0884.28008MR1268062
  6. Okada S., Ricker W.J., Compactness properties of vector-valued integration maps in locally convex spaces, Colloq. Math., to appear. Zbl0821.46057MR1292938
  7. Ricker W.J., Spectral measures, boundedly σ -complete Boolean algebras and applications to operator theory, Trans. Amer. Math. Soc. 304 (1987), 819-838. (1987) Zbl0642.47029MR0911097
  8. Treves F., Topological Vector Spaces, Distributions and Kernels, Academic Press, New York, 1967. Zbl1111.46001MR0225131

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