## Annals of Functional Analysis

### Dominated operators from lattice-normed spaces to sequence Banach lattices

#### 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 $\ell_{p}(\Gamma)$ or $c_{0}(\Gamma)$ 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
Accepted: 11 May 2016
First available in Project Euclid: 5 October 2016

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
Secondary: 47B99: None of the above, but in this section

#### 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

#### 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.