Banach Journal of Mathematical Analysis

The strong Gelfand–Phillips property in Banach lattices

Halimeh Ardakani, S. Mohammad Moshtaghioun, S. M. Sadegh Modarres Mosadegh, and Manijeh Salimi

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


We introduce the concept of the strong Gelfand–Phillips (GP) property in Banach lattices, and we characterize Banach lattices with the strong GP property. Next, by introducing the class of almost limited completely continuous operators from an arbitrary Banach lattice E to another F, we give some properties of them related to some well-known classes of operators and, especially, to the strong GP property of the Banach lattice E.

Article information

Banach J. Math. Anal., Volume 10, Number 1 (2016), 15-26.

Received: 6 December 2014
Accepted: 25 March 2015
First available in Project Euclid: 15 October 2015

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 46B42: Banach lattices [See also 46A40, 46B40]
Secondary: 46B50: Compactness in Banach (or normed) spaces 47B65: Positive operators and order-bounded operators

almost limited set limited completely continuous operator Gelfand–Phillips property


Ardakani, Halimeh; Moshtaghioun, S. Mohammad; Modarres Mosadegh, S. M. Sadegh; Salimi, Manijeh. The strong Gelfand–Phillips property in Banach lattices. Banach J. Math. Anal. 10 (2016), no. 1, 15--26. doi:10.1215/17358787-3158354.

Export citation


  • [1] C. D. Aliprantis and O. Burkishaw, Locally Solid Riesz Spaces, Pure Appl. Math. 76, Academic Press, New York, 1978.
  • [2] C. D. Aliprantis and O. Burkishaw, Positive Operators, Academic Press, New York, 1978.
  • [3] J. Borwein, M. Fabian, and J. Vanderwerff, Characterizations of Banach spaces via convex and other locally Lipschitz functions, Acta Math. Vietnam 22 (1997), no. 1, 53–69.
  • [4] J. Bourgain and J. Diestel, Limited operators and strict cosingularity, Math. Nachr. 119 (1984), 55–58.
  • [5] A. V. Buhvalov, Locally convex spaces that are generated by weakly compact sets (in Russian), Vestnik Leningrad Univ. Mat. Meh. Astronom. 7 (1973), 11–17.
  • [6] J. X. Chen, Z. L.Chen, and G. X. Ji, Almost limited sets in Banach lattices, J. Math. Anal. Appl. 412 (2014), no. 1, 547–563.
  • [7] L. Drewnowski, On Banach spaces with the Gelfand–Phillips property, Math. Z. 193 (1986), no. 3, 405–411.
  • [8] A. El Kaddouri and M. Moussa, About the class of ordered limited operators, Acta Univ. Carolin. Math. Phys. 54 (2013), no. 1, 37–43.
  • [9] G. Emmanuele, On Banach spaces with the Gelfand–Phillips property, III, J. Math. Pures Appl. (9) 72 (1993), no. 3, 327–333.
  • [10] N. Machrafi, A. Elbour, and M. Moussa, Some characterizations of almost limited sets and applications, preprint, arXiv:1312.2770v1 [math.FA].
  • [11] P. Meyer-Nieberg, Banach Lattices, Springer, Berlin, 1991.
  • [12] M. Salimi and S. M. Moshtaghioun, The Gelfand–Phillips property in closed subspaces of some operator spaces, Banach J. Math. Anal. 5 (2011), no. 2, 84–92.
  • [13] J. A. Sánchez Henríquez, The positive Schur property in Banach lattices, Extracta Math. 7, (1992), no. 2–3, 161–163.
  • [14] W. Wnuk, A note on the positive Schur property, Glasgow Math. J. 31 (1989), no. 2, 169–172.
  • [15] W. Wnuk, Banach Lattices with Order Continuous Norms, Polish Scientific Publishers PWN, Warsaw, 1999.
  • [16] W. Wnuk, On the dual positive Schur property in Banach lattices, Positivity 17 (2013), no. 3, 759–773.