On set-valued cone absolutely summing maps

Coenraad Labuschagne; Valeria Marraffa

Open Mathematics (2010)

  • Volume: 8, Issue: 1, page 148-157
  • ISSN: 2391-5455

Abstract

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Spaces of cone absolutely summing maps are generalizations of Bochner spaces L p(μ, Y), where (Ω, Σ, μ) is some measure space, 1 ≤ p ≤ ∞ and Y is a Banach space. The Hiai-Umegaki space ℒ 1 ∑ , c b f ( X ) of integrably bounded functions F: Ω → cbf(X), where the latter denotes the set of all convex bounded closed subsets of a separable Banach space X, is a set-valued analogue of L 1(μ, X). The aim of this work is to introduce set-valued cone absolutely summing maps as a generalization of ℒ 1 ∑ , c b f ( X ) , and to derive necessary and sufficient conditions for a set-valued map to be such a set-valued cone absolutely summing map. We also describe these set-valued cone absolutely summing maps as those that map order-Pettis integrable functions to integrably bounded set-valued functions.

How to cite

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Coenraad Labuschagne, and Valeria Marraffa. "On set-valued cone absolutely summing maps." Open Mathematics 8.1 (2010): 148-157. <http://eudml.org/doc/269210>.

@article{CoenraadLabuschagne2010,
abstract = {Spaces of cone absolutely summing maps are generalizations of Bochner spaces L p(μ, Y), where (Ω, Σ, μ) is some measure space, 1 ≤ p ≤ ∞ and Y is a Banach space. The Hiai-Umegaki space \[ \mathcal \{L\}^1 \left[ \{\sum ,cbf(X)\} \right] \] of integrably bounded functions F: Ω → cbf(X), where the latter denotes the set of all convex bounded closed subsets of a separable Banach space X, is a set-valued analogue of L 1(μ, X). The aim of this work is to introduce set-valued cone absolutely summing maps as a generalization of \[ \mathcal \{L\}^1 \left[ \{\sum ,cbf(X)\} \right] \] , and to derive necessary and sufficient conditions for a set-valued map to be such a set-valued cone absolutely summing map. We also describe these set-valued cone absolutely summing maps as those that map order-Pettis integrable functions to integrably bounded set-valued functions.},
author = {Coenraad Labuschagne, Valeria Marraffa},
journal = {Open Mathematics},
keywords = {Banach lattice; Bochner space; Cone absolutely summing operator; Integrably bounded set-valued function; Set-valued operator; cone absolutely summing operator; integrably bounded set-valued function; set-valued operator},
language = {eng},
number = {1},
pages = {148-157},
title = {On set-valued cone absolutely summing maps},
url = {http://eudml.org/doc/269210},
volume = {8},
year = {2010},
}

TY - JOUR
AU - Coenraad Labuschagne
AU - Valeria Marraffa
TI - On set-valued cone absolutely summing maps
JO - Open Mathematics
PY - 2010
VL - 8
IS - 1
SP - 148
EP - 157
AB - Spaces of cone absolutely summing maps are generalizations of Bochner spaces L p(μ, Y), where (Ω, Σ, μ) is some measure space, 1 ≤ p ≤ ∞ and Y is a Banach space. The Hiai-Umegaki space \[ \mathcal {L}^1 \left[ {\sum ,cbf(X)} \right] \] of integrably bounded functions F: Ω → cbf(X), where the latter denotes the set of all convex bounded closed subsets of a separable Banach space X, is a set-valued analogue of L 1(μ, X). The aim of this work is to introduce set-valued cone absolutely summing maps as a generalization of \[ \mathcal {L}^1 \left[ {\sum ,cbf(X)} \right] \] , and to derive necessary and sufficient conditions for a set-valued map to be such a set-valued cone absolutely summing map. We also describe these set-valued cone absolutely summing maps as those that map order-Pettis integrable functions to integrably bounded set-valued functions.
LA - eng
KW - Banach lattice; Bochner space; Cone absolutely summing operator; Integrably bounded set-valued function; Set-valued operator; cone absolutely summing operator; integrably bounded set-valued function; set-valued operator
UR - http://eudml.org/doc/269210
ER -

References

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  10. [10] Marraffa V., A characterization of absolutely summing operators by means of McShane integrable functions, J. Math. Anal. Appl., 2004, 239, 71–78 http://dx.doi.org/10.1016/j.jmaa.2003.12.029 Zbl1087.47023
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