International Journal of Nonlinear Analysis and Applications

Almost order-weakly compact operators on Banach lattices

Document Type : Research Paper

Authors

Department of Mathematics and Applications, Faculty of Sciences, University of Mohaghegh Ardabili, Ardabil, Iran

Abstract

A continuous operator T between two Banach lattices E and     F is called almost order-weakly compact, whenever for each almost order bounded subset A of E,  T(A) is a relatively weakly compact subset of F. We show that the positive operator T from  E into a  Dedekind complete Banach lattice F  is almost order-weakly compact iff  T(xn)→‖.‖0 in F for each disjoint almost order bounded sequence {xn} in E. In this manuscript, we study some properties of this class of operators and its relationships with the others known classes of operators.

Keywords

[1] C.D. Aliprantis and O. Burkinshaw, Positive Operators, Springer, Berlin, 2006.
[2] B. Aqzzouz, A. Elbour and J. Hmichane, The duality problem for the class of b-compact operators, Positivity. 13 (2009), no. 4, 683–692.
[3] B. Aqzzouz, A. Elbour and J. Hmichane, Some results on order weakly compact operators, Math. Bohemica 134 (2009), no. 4, 359–367.
[4] J.B. Conway, A course in functional analysis, 2nd edition, Springer-Verlag, New York, 1990.
[5] Y.Deng, M.O’Brien and V.G.Troitsky. Unbounded norm convergence in Banach lattices, Positivity. 21 (2017), 963–974.
[6] N. Gao, Unbounded order convergence in dual spaces, J. Math. Anal. Appl. 419 (2014), 347–354.
[7] N. Gao and F. Xanthos, Unbounded order convergence and application to martingales without probability, J. Math. Anal. Appl. 415 (2014), 931–947.
[8] K. Haghnejad Azar, M. Matin and R. Alavizadeh, Unbounded order-norm continuous and unbounded norm continuous operators, Filomat 35 (2021), no. 13, 4417—4426.
 [9] K.D. Schmidt, On the modulus of weakly compact operators and strongly additive vector measures, Proc. Amer. Math. 102 (1988), no. 4, 862–866.
[10] H.H. Schaefer, Banach Lattices and Positive Operators, Springer, Berlin, 1974.
[11] W. Wnuk, Some characterizations of Banach lattices with the Schur property, Cong. Funct. Anal. Rev. Mat. Univ. Complut. Madrid 2 (suppl.), 1989, pp. 217–224.
International Journal of Nonlinear Analysis and Applications
Volume 15, Issue 1
January 2024
Pages 353-360
  • Receive Date: 23 April 2022
  • Revise Date: 04 July 2022
  • Accept Date: 06 July 2022