Operator ideal properties of vector measures with finite variation

Susumu Okada, Werner J. Ricker, and Luis Rodríguez-Piazza. "Operator ideal properties of vector measures with finite variation." Studia Mathematica 205.3 (2011): 215-249. <http://eudml.org/doc/285863>.

@article{SusumuOkada2011,
abstract = {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.},
author = {Susumu Okada, Werner J. Ricker, Luis Rodríguez-Piazza},
journal = {Studia Mathematica},
keywords = {operator ideal; vector measure; integration map},
language = {eng},
number = {3},
pages = {215-249},
title = {Operator ideal properties of vector measures with finite variation},
url = {http://eudml.org/doc/285863},
volume = {205},
year = {2011},
}

TY - JOUR
AU - Susumu Okada
AU - Werner J. Ricker
AU - Luis Rodríguez-Piazza
TI - Operator ideal properties of vector measures with finite variation
JO - Studia Mathematica
PY - 2011
VL - 205
IS - 3
SP - 215
EP - 249
AB - 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.
LA - eng
KW - operator ideal; vector measure; integration map
UR - http://eudml.org/doc/285863
ER -