Nabil MACHRAFI, Birol ALTIN

Some characterizations of KB-operators on Banach lattices and ordered Banach spaces

We determine that two recent classes of KB-operators and weak KB-operators and the well-known class of b -weakly compact operators, from a Banach lattice into a Banach space, are the same. We extend our study to the ordered Banach space setting by showing that a weak chain-preserving operator between two ordered Banach spaces is a KB-operator if and only if it is a weak KB-operator.

PDF

___

[1] Aliprantis CD, Burkinshaw O. Positive Operators. Dordrecht, Netherlands: Springer, 2006.
[2] Aliprantis CD, Tourky R. Cones and Duality. Providence, RI, USA: American Mathematical Society, 2007.
[3] Alpay S, Altin B, Tonyali C. On property (b) of vector lattices. Positivity 2003; 7: 135-139. doi: 10.1023/A:1025840528211
[4] Altin B. On b-weakly compact operators on Banach lattices. Taiwanese Journal of Mathematics 2007; 11 (1): 143-150. doi: 10.11650/twjm/1500404641
[5] Bahramnezhad A, Azar KH. KB-operators on Banach lattices and their relationships with Dunford-Pettis and order weakly compact operators. University Politehnica of Bucharest Scientific Bulletin 2018; 80 (2): 91-98.
[6] Groenewegen GLM, Van Rooij ACM. Spaces of Continuous Functions. Amsterdam, Netherlands: Atlantis Press, 2016.
[7] Guo D, Cho YJ, Zhu J. Partial Ordering Methods in Nonlinear Problems. New York, NY, USA: Nova Science Publishers Inc., 2004.
[8] Hansell RW. Monotone subnets in partially ordered sets. Proceedings of the American Mathematical Society 1967; 18: 854-858. doi: 10.1090/S0002-9939-1967-0215759-7
[9] Machrafi N. Common coupled fixed point results for operators without mixed monotone type properties and application to nonlinear integral equations. Fixed Point Theory and Applications 2016; 2017 (2): 1-20. doi: 10.1186/s13663-017-0595-y
[10] Meyer-Nieberg P. Banach Lattices. New York, NY, USA: Springer, 1991.
[11] Turan B, Altin B. The relation between b -weakly compact operator and KB-operator. Turkish Journal of Mathematics 2019; 43: 2818-2820. doi: 10.3906/mat-1908-11