Given a vector measure m with values in a Banach space X, a desirable property (when available) of the associated Banach function space L¹(m) of all m-integrable functions is that L¹(m) = L¹(|m|), where |m| is the [0,∞]-valued variation measure of m. Closely connected to m is its X-valued integration map Iₘ: f ↦ ∫f dm for f ∈ L¹(m). Many traditional operators from analysis arise as integration maps in this way. A detailed study is made of the connection between the property L¹(m) = L¹(|m|) and the membership of Iₘ in various classical operator ideals(e.g., the compact, p-summing, completely continuous operators). Depending on which operator ideal is under consideration, the geometric nature of the Banach space X may also play a crucial role. Of particular importance in this regard is whether or not X contains an isomorphic copy of the classical sequence space ℓ¹. The compact range property of X is also relevant.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm205-3-2,
author = {Susumu Okada and Werner J. Ricker and Luis Rodr\'\i guez-Piazza},
title = {Operator ideal properties of vector measures with finite variation},
journal = {Studia Mathematica},
volume = {204},
year = {2011},
pages = {215-249},
zbl = {1236.28009},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm205-3-2}
}
Susumu Okada; Werner J. Ricker; Luis Rodríguez-Piazza. Operator ideal properties of vector measures with finite variation. Studia Mathematica, Tome 204 (2011) pp. 215-249. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm205-3-2/