A characterization is given of those Banach-space-valued vector measures m with finite variation whose associated integration operator Iₘ: f ↦ ∫fdm is compact as a linear map from L¹(m) into the Banach space. Moreover, in every infinite-dimensional Banach space there exist nontrivial vector measures m (with finite variation) such that Iₘ is compact, and other m (still with finite variation) such that Iₘ is not compact. If m has infinite variation, then Iₘ is never compact.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm150-2-3,
author = {S. Okada and W. J. Ricker and L. Rodr\'\i guez-Piazza},
title = {Compactness of the integration operator associated with a vector measure},
journal = {Studia Mathematica},
volume = {151},
year = {2002},
pages = {133-149},
zbl = {0998.28009},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm150-2-3}
}
S. Okada; W. J. Ricker; L. Rodríguez-Piazza. Compactness of the integration operator associated with a vector measure. Studia Mathematica, Tome 151 (2002) pp. 133-149. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm150-2-3/