Abstract
The “Up-and-down” theorem which describes the structure of the Boolean algebra of fragments of a linear positive operator is the well known result in operator theory. We prove an analog of this theorem for a positive abstract Uryson operator defined on a vector lattice and taking values in a Dedekind complete vector lattice. This result is used to prove a theorem of domination for order narrow positive abstract Uryson operators from a vector lattice E to a Banach lattice F with an order continuous norm.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
The research was supported by Russian Foundation for Basic Research, the grant number 15-51-53119
\((C_{1})\) and \((C_{2})\) are called the Carathéodory conditions.
References
Popov, M., Randrianantoanina, B.: Narrow Operators on Function Spaces and Vector Lattices, De Gruyter Studies in Mathematics 45, De Gruyter (2013)
Pliev, M., Popov, M.: Narrow orthogonally additive operators. Positivity 18(4), 641–667 (2014)
Aliprantis, C.D., Burkinshaw, O.: Positive Operators. Springer, Dordrecht (2006)
Jech, T.: Set Theory. Springer, Berlin (2003)
Kusraev, A.G.: Dominated Operators. Kluwer Acad. Publ, Dordrecht-Boston-London (2000)
Luxemburg, W.A.J., Zaanen, A.C.: Riesz Spaces, vol. 1. North Holland Publ. Comp, Amsterdam-London (1971)
Mazón, J.M., Segura de León, S.: Order bounded ortogonally additive operators. Rev. Roumane Math. Pures Appl. 35(4), 329–353 (1990)
Mazón, J.M., Segura de León, S.: Uryson operators. Rev. Roumane Math. Pures Appl. 35(5), 431–449 (1990)
Kusraev, A.G., Pliev, M.A.: Orthogonally additive operators on lattice-normed spaces. Vladikavkaz Math. J. 1(3), 33–43 (1999)
Kusraev, A.G., Pliev, M.A.: Weak integral representation of the dominated orthogonally additive operators. Vladikavkaz Math. J. 1(4), 22–39 (1999)
Ben Amor, M.A., Pliev, M.: Laterally continuous part of an abstract Uryson operator. Int. J. Math. Anal. 758, 2853–2860 (2013)
Getoeva, A., Pliev, M.: Domination problem for orthgonally additive operators in lattice-normed spaces. Int. J. Math. Anal. 9(27), 1341–1352 (2015)
Gumenchuk, A.V., Pliev, M.A., Popov, M.M.: Extensions of orthogonally additive operators. Mat. Stud. 41(2), 214–219 (2014)
Pliev, M., Popov, M.: Dominated Uryson operators. Int. J. Math. Anal. 8(22), 1051–1059 (2014)
Maslyuchenko, O.V., Mykhaylyuk, V.V., Popov, M.M.: A lattice approach to narrow operators. Positivity 13(3), 459–495 (2009)
Pliev, M.: Narrow operators on lattice-normed spaces. Open Math. 9(6), 1276–1287 (2011)
Pliev, M.A., Weber, M.R.: Disjointness and order projections in the vector lattices of abstract Uryson operators. Positivity. (2015). doi:10.1007/s11117-015-0381-1
Aliprantis, C.D., Burkinshaw, O.: The components of a positive operator. Math. Z. 184(2), 245–257 (1983)
Kusraev, A.G., Strizhevski, M.A.: Lattice-normed spaces and dominated operators. Studies on Geometry and Functional Analysis. Vol. 7. Trudy Inst. Mat. (Novosibirsk), Novosibirsk, pp. 132–158 (1987)
de Pagter, B.: The components of a positive operator. Indag. Math. 48, 229–241 (1983)
Flores, J., Ruiz, C.: Domination by positive narrow operators. Positivity 7(4), 303–321 (2003)
Acknowledgments
I would like to thank Mikhail Popov for valuable suggestions and discussions. I would also like to thank the anonymous referee for the useful remarks and for the careful reading of the text.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Pliev, M. Domination problem for narrow orthogonally additive operators. Positivity 21, 23–33 (2017). https://doi.org/10.1007/s11117-016-0401-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11117-016-0401-9