# On set-valued cone absolutely summing maps

Coenraad Labuschagne; Valeria Marraffa

Open Mathematics (2010)

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

## Access Full Article

top## Abstract

top## How to cite

topCoenraad 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

top- [1] Belanger A., Dowling P.N., Two remarks on absolutely summing operators, Math. Nachr., 1988, 136, 229–232 http://dx.doi.org/10.1002/mana.19881360115 Zbl0654.47009
- [2] Chaney J., Banach lattices of compact maps, Math. Z., 1972, 129, 1–19 http://dx.doi.org/10.1007/BF01229536 Zbl0231.46020
- [3] Diestel J., An elementary characterization of absolutely summing operators, Math. Ann., 1972, 196, 101–105 http://dx.doi.org/10.1007/BF01419607 Zbl0221.46040
- [4] Diestel J., Uhl J.J., Vector measures, A.M.S. Surveys, Vol. 15, Providence, Rhode Island, 1977 Zbl0369.46039
- [5] Dinculeanu N., Integral representation of linear operators, Stud. Cerc. Mat., 1966, 18, 349–385, 483–536 Zbl0156.14803
- [6] Hiai F., Umegaki H., Integrals, conditional expectations, and martingales of multivalued functiones, Journal of Multivariate Analysis, 1977, 7, 149–182 http://dx.doi.org/10.1016/0047-259X(77)90037-9 Zbl0368.60006
- [7] Jeurnink G.A.M., Integration of functions with values in a Banach lattice, PhD Thesis, University of Nijmegen, The Netherlands, 1982
- [8] Labuschagne C.C.A., Pinchuck A.L., van Alten C.J., A vector lattice version of Rådström’s embedding theorem, Quaestiones Mathematicae, 2007, 30, 285–308 Zbl1144.06009
- [9] Li S., Ogura Y., Kreinovich V., Limit theorems and applications of set-valued and fuzzy set-valued random variables, Kluwer Acadenic Press, Dordrecht-Boston-London, 2002
- [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
- [11] Meyer-Nieberg P., Banach lattices, Springer Verlag, Berlin-Heidelberg-New York, 1991 Zbl0743.46015
- [12] Rådström H., An embedding theorem for spaces of convex sets, Proc. Amer. Math. Soc., 1952, 3, 165–169 http://dx.doi.org/10.2307/2032477 Zbl0046.33304
- [13] Rodríguez J., Absolutely summing operators and integration of vector-valued functions, J. Math. Anal. Appl., 2006, 316, 579–600 http://dx.doi.org/10.1016/j.jmaa.2005.05.001 Zbl1097.46028
- [14] Schaefer H.H., Banach lattices and positive operators, Springer-Verlag, Berlin-Heidelberg-New York, 1974 Zbl0296.47023
- [15] Zaanen A.C., Introduction to operator theory in Riesz spaces, Springer-Verlag, Berlin Heidelberg New York, 1997 Zbl0878.47022

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.