In this paper, we introduce and study the concept of L-Dunford-Pettis sets and L-Dunford-Pettis property in Banach spaces. Next, we give a characterization of the L-Dunford-Pettis property with respect to some well-known geometric properties of Banach spaces. Finally, some complementability of operators on Banach spaces with the L-Dunford-Pettis property are also investigated.
A. Retbi and B. El Wahbi, L-Dunford-Pettis property in Banach spaces, Methods Funct. Anal. Topology 22
(2016), no. 4, 387-392.
BibTex
@article {MFAT916,
AUTHOR = {Retbi, A. and El Wahbi, B.},
TITLE = {L-Dunford-Pettis property in Banach spaces},
JOURNAL = {Methods Funct. Anal. Topology},
FJOURNAL = {Methods of Functional Analysis and Topology},
VOLUME = {22},
YEAR = {2016},
NUMBER = {4},
PAGES = {387-392},
ISSN = {1029-3531},
MRNUMBER = {MR3591087},
ZBLNUMBER = {06742118},
URL = {http://mfat.imath.kiev.ua/article/?id=916},
}
References
C. D. Aliprantis and O. Burkinshaw, Positive operators, Springer, Dordrecht, 2006. MathSciNetCrossRef
M. Bahreini Esfahani, Complemented subspaces of bounded linear operators, Ph.D. thesis, University of North Texas, 2003. MathSciNet
P. Cembranos, $C(K,\,E)$ contains a complemented copy of $c_0$, Proc. Amer. Math. Soc. 91 (1984), no. 4, 556-558. MathSciNetCrossRef
G. Emmanuele, A dual characterization of Banach spaces not containing $l^ 1$, Bull. Polish Acad. Sci. Math. 34 (1986), no. 3-4, 155-160. MathSciNet
G. Emmanuele, Banach spaces in which Dunford-Pettis sets are relatively compact, Arch. Math. 58 (1992), no. 5, 477-485. MathSciNetCrossRef
I. Ghenciu and P. Lewis, The Dunford-Pettis property, the Gelfand-Phillips property, and $L$-sets, Colloq. Math. 106 (2006), no. 2, 311-324. MathSciNetCrossRef
N. J. Kalton, Spaces of compact operators, Math. Ann. 208 (1974), 267-278. MathSciNet
R. E. Megginson, An introduction to Banach space theory, Graduate Texts in Mathematics, vol. 183, Springer-Verlag, New York, 1998. MathSciNetCrossRef
P. Meyer-Nieberg, Banach lattices, Universitext, Springer-Verlag, Berlin, 1991. MathSciNetCrossRef
C. P. Niculescu, Weak compactness in Banach lattices, J. Operator Theory 6 (1981), no. 2, 217-231. MathSciNet
Y. Wen and J. Chen, Characterizations of Banach spaces with relatively compact Dunford-Pettis sets, Adv. in Math. (China) 45 (2016), no. 1, 122-132. CrossRef