Annals of Functional Analysis

Dominated operators from lattice-normed spaces to sequence Banach lattices

Nariman Abasov, Abd El Monem Megahed, and Marat Pliev

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

Abstract

We show that every dominated linear operator from a Banach–Kantorovich space over an atomless Dedekind-complete vector lattice to a sequence Banach lattice p(Γ) or c0(Γ) is narrow. As a consequence, we obtain that an atomless Banach lattice cannot have a finite-dimensional decomposition of a certain kind. Finally, we show that the order-narrowness of a linear dominated operator T from a lattice-normed space V to the Banach space with a mixed norm (W,F) over an order-continuous Banach lattice F implies the order-narrowness of its exact dominant |T|.

Article information

Source
Ann. Funct. Anal. Volume 7, Number 4 (2016), 646-655.

Dates
Received: 8 November 2015
Accepted: 11 May 2016
First available in Project Euclid: 5 October 2016

Permanent link to this document
https://projecteuclid.org/euclid.afa/1475685111

Digital Object Identifier
doi:10.1215/20088752-3660990

Mathematical Reviews number (MathSciNet)
MR3555756

Zentralblatt MATH identifier
1365.46018

Subjects
Primary: 46B42: Banach lattices [See also 46A40, 46B40]
Secondary: 47B99: None of the above, but in this section

Keywords
narrow operators dominated operators lattice-normed spaces Banach lattices

Citation

Abasov, Nariman; Megahed, Abd El Monem; Pliev, Marat. Dominated operators from lattice-normed spaces to sequence Banach lattices. Ann. Funct. Anal. 7 (2016), no. 4, 646--655. doi:10.1215/20088752-3660990. https://projecteuclid.org/euclid.afa/1475685111


Export citation

References

  • [1] F. Albiac and N. Kalton, Topics in Banach Space Theory, Grad. Texts in Math. 233, Springer, Berlin, 2006.
  • [2] C. D. Aliprantis and O. Burkinshaw, Positive Operators, Springer, Berlin, 2006.
  • [3] J. Flores, F. L. Hernández, and P. Tradacete, Domination problems for strictly singular operators and other related classes, Positivity 15 (2011), no. 4, 595–616.
  • [4] A. G. Kusraev, Dominated Operators, Math. Appl. 519, Kluwer, Dordrecht, 2000.
  • [5] J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces, Vol. 1: Sequence Spaces, Ergeb. Math. Grenzgeb. (3) 92, Springer, Berlin, 1977.
  • [6] J. Lindenstrauss and L. Tzafriri, Classical Banach Spaces, Vol. 2: Function Spaces, Ergeb. Math. Grenzgeb. (3) 97, Springer, Berlin, 1979.
  • [7] O. V. Maslyuchenko, V. V. Mykhaylyuk, and M. M. Popov, A lattice approach to narrow operators, Positivity 13 (2009), no. 3, 459–495.
  • [8] V. Mykhaylyuk, M. Pliev, M. Popov, and O. Sobchuk, Dividing measures and narrow operators, Studia Math. 231 (2015), no. 2, 97-116.
  • [9] M. Pliev, Narrow operators on lattice-normed spaces, Cent. Eur. J. Math. 9 (2011), no. 6, 1276–1287.
  • [10] M. Pliev and M. Popov, Narrow orthogonally additive operators, Positivity 18 (2014), no. 4, 641–667.
  • [11] A. M. Plichko and M. M. Popov, Symmetric function spaces on atomless probability spaces, Dissertationes Math. (Rozprawy Mat.) 306 (1990), 1–85.
  • [12] M. Popov and B. Randrianantoanina, Narrow Operators on Function Spaces and Vector Lattices, de Gruyter Stud. Math. 45, de Gruyter, Berlin, 2013.