Skip to main content
SpringerLink
Log in

Order Unbounded Orthogonally Additive Operators in Vector Lattices

  • Published:
Mediterranean Journal of Mathematics Aims and scope Submit manuscript

Abstract

In this paper, we introduce a new class of operators in vector lattices. We say that orthogonally additive operator T from vector lattice E to vector lattice F is laterally-to-order bounded if for any element x of E an operator T maps the set \(\mathcal {F}_{x}\) of all fragments of x onto an order bounded subset of F. We get a lattice calculus of orthogonally additive laterally-to-order bounded operators defined on a vector lattice and taking values in a Dedekind complete vector lattice. It turns out that these operators, in general, are not order bounded. We investigate the band of laterally continuous orthogonally additive operators and obtain formulas for the order projection onto this band. We consider the procedure of the extension of an orthogonally additive operator from a lateral ideal to the whole space. Finally we obtain conditions on the integral representability for a laterally-to-order bounded orthogonally additive operator.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Abasov, N., Pliev, M.: Some remarks about abstract Uryson operators. Int. J. Math. Anal. 9(53), 2635–2642 (2015)

    Article  Google Scholar 

  2. Abasov, N., Pliev, M.: On extensions of some nonlinear maps in vector lattices. J. Math. Anal. Appl. 455, 516–527 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  3. Abramovich, Y.A., Aliprantis, C.D.: An Invitation to Operator Theory. AMS, (2002)

  4. Aliprantis, C.D., Burkinshaw, O.: Positive Operators. Springer, Dordrecht (2006)

    Book  MATH  Google Scholar 

  5. Appel, J.A., Zabrejko, P.P.: Nonlinear Superposition Operators. Cambridge University Press, Cambridge (2008)

    Google Scholar 

  6. Ben Amor, M.A., Pliev, M.: Laterally continuous part of an abstract Uryson operator. Int. J. Math. Anal. 7(58), 2853–2860 (2013)

    Article  MathSciNet  Google Scholar 

  7. Bukhvalov, A.V.: Application of methods of the theory of order-bounded operators to the theory of operators in \(L_{p}\)-spaces. Russ. Math. Surv. 38(6), 43–98 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  8. Feldman, W.A.: A characterization of non-linear maps satisfying orthogonality properties. Positivity 21(1), 85–97 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  9. Feldman, W.A.: Lattice preserving maps on lattices of continuous functions. J. Math. Anal. Appl. 404, 310–316 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  10. Feldman, W.A., Pramod, S.: A characterization of positively decomposable non-linear maps between Banach lattices. Positivity 12, 495–502 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  11. A. V, Gumenchuk: On the sum of narrow and finite-rank orthogonally addiitive operators. Ukr. Math. J. 67(12), 1831–1837 (2016)

    Article  Google Scholar 

  12. Kolmogorov, A.N., Fomin, C.V.: Introductory Real Analysis. Dover Publicaton, New York (1975)

    MATH  Google Scholar 

  13. Mazón, J.M., Segura de León, S.: Order bounded orthogonally additive operators. Rev. Roum. Math. Pures Appl. 35(4), 329–353 (1990)

    MathSciNet  MATH  Google Scholar 

  14. Mazón, J.M., Segura de León, S.: Uryson operators. Rev. Roum. Math. Pures Appl. 35(5), 431–449 (1990)

    MATH  Google Scholar 

  15. Mykhaylyuk, V., Pliev, M., Popov, M., Sobchuk, O.: Dividing measures and narrow operators. Stud. Math. 231, 97–116 (2015)

    MathSciNet  MATH  Google Scholar 

  16. Orlov, V., Pliev, M., Rode, D.: Domination problem for \(AM\)-compact abstract Uryson operators. Arch. Math. 107(5), 543–552 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  17. Pliev, M.: Domination problem for narrow orthogonally additive operators. Positivity 21(1), 23–33 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  18. Pliev, M., Fang, X.: Narrow orthogonally operators in lattice-normed spaces. Sib. Math. J. 58(1), 134–141 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  19. Pliev, M., Popov, M.: Narrow orthogonally additive operators. Positivity 18(4), 641–667 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  20. Pliev, M., Popov, M.: On extension of abstract Urysohn operators. Sib. Math. J. 57(3), 552–557 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  21. Pliev, M.A., Weber, M.R.: Disjointness and order projections in the vector lattices of abstract Uryson operators. Positivity 20(3), 695–707 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  22. Pliev, M.A., Weber, M.R.: Finite elements in some vector lattices of nonlinear operators. Positivity 22(1), 245–260 (2018)

    Article  MathSciNet  Google Scholar 

  23. Segura de León, S.: Bukhvalov type characterization of Urysohn operators. Stud. Math. 99(3), 199–220 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  24. Zaanen, A.G.: Riesz Spaces II. North Holland, Amsterdam (1983)

    MATH  Google Scholar 

Download references

Acknowledgements

We are very grateful to the anonymous referee for the careful reading of the text and valuable remarks.  M. Pliev was supported by the Russian Foundation for Basic Research (the grant number 17-51-12064).

Author information

Authors and Affiliations

  1. Southern Mathematical Institute of the Russian Academy of Sciences, str. Markusa 22, Vladikavkaz, Russia, 362027

    Marat Pliev

  2. Research Laboratory of Algebra, Topology, Arithmetic, and Order, Department of Mathematics, Faculty of Mathematical Physical and Natural Sciences of Tunis, University of Tunis El Manar, Tunis, Tunisia

    Kawtar Ramdane

Authors
  1. Marat Pliev
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Kawtar Ramdane
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Marat Pliev.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Pliev, M., Ramdane, K. Order Unbounded Orthogonally Additive Operators in Vector Lattices. Mediterr. J. Math. 15, 55 (2018). https://doi.org/10.1007/s00009-018-1100-5

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00009-018-1100-5

Keywords

Mathematics Subject Classification

Search

Navigation