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Narrow and C-compact Orthogonally Additive Operators in Lattice-Normed Spaces

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Abstract

In this article we consider orthogonally additive operators on lattice-normed spaces. In the first part of the article we present some examples of narrow, laterally-to-norm continuous and C-compact operators defined on a lattice-normed space and taking value in a Banach space. We show that any laterally-to-norm continuous narrow orthogonally additive operator defined on a decomposable lattice-normed space (V, E) over an atomic vector lattice E with the projection property is equal to zero. In the second part we prove that the sum of two orthogonally additive operators \(T+S\) defined on a order complete, decomposable lattice-normed space V and taking value in Banach space X, where \(T:V\rightarrow X\) is a laterally-to-norm continuous C-compact operator and \(S:V\rightarrow X\) is a narrow operator, is a narrow operator as well.

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References

  1. Abasov, N., Megahed, A.M., Pliev, M.: Dominated operators from lattice-normed spaces to sequence Banach lattices. Ann. Funct. Anal. 7(4), 646–655 (2016)

    Article  MathSciNet  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  Google Scholar 

  3. Abasov, N., Pliev, M.: Dominated orthogonally additive operators in lattice-normed spaces. Adv. Oper. Theory 4(1), 251–264 (2019)

    Article  MathSciNet  Google Scholar 

  4. Abasov, N., Pliev, M.: Two definitions of narrow operators on Köthe–Bochner spaces. Arch. Math. 111(2), 167–176 (2018)

    Article  MathSciNet  Google Scholar 

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

    Book  Google Scholar 

  6. Aydin, A., Emelyanov, E.Yu., Erkursun Ozcan, N., Marabeh, M.A.A.: Compact-like operators in lattice-normed spaces. Indag. Math. 29(2), 633–656 (2018)

  7. Buskes, G., Dever, J.: The complexification of a lattice seminormed space. Indag. Math. 25(2), 170–185 (2014)

    Article  MathSciNet  Google Scholar 

  8. Diestel, J., Uhl, J.J.: Vector Measures. AMS, Providence (1977)

    Book  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  11. Feldman, W.A.: A factorization for orthogonally additive operators on Banach lattices. J. Math. Anal. Appl. 472(1), 238–245 (2019)

    Article  MathSciNet  Google Scholar 

  12. Humenchuk, H.I.: On the sum of narrow and finite-rank orthogonally additive operator. Ukr. Math. J. 67(12), 1831–1837 (2016)

    Article  Google Scholar 

  13. Kantorovich, L.V.: On a class of functional equations. Dokl. Akad. Nauk SSSR 4(5), 211–216 (1936)

    Google Scholar 

  14. Kusraev, A.G.: Dominated Operators. Kluwer Academic Publishers, Dordrecht (2000)

    Book  Google Scholar 

  15. Luxemburg, W.A.J., Zaanen, A.C.: Riesz Spaces, vol. 1. North Holland Publ. Comp, Amsterdam (1971)

    MATH  Google Scholar 

  16. Maslyuchenko, O., Mykhaylyuk, V., Popov, M.: A lattice approach to narrow operators. Positivity 13, 459–495 (2009)

    Article  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  18. Mykhaylyuk, V., Popov, M.: On sums of narrow operators on Köthe function space. J. Math. Anal. Appl. 404, 554–561 (2013)

    Article  MathSciNet  Google Scholar 

  19. Mykhaylyuk, V.: On the sum of narrow and a compact operators. J. Funct. Anal. 266, 5912–5920 (2014)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  21. Plichko, A., Popov, M.: Symmetric function spaces on atomless probability spaces. Diss. Math. (Rozpr. Mat.) 306, 1–85 (1990)

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  23. Pliev, M.: Narrow operators on lattice-normed spaces. Open Math. 9(6), 1276–1287 (2011)

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  26. Popov, M., Randrianantoanina, B.: Narrow Operators on Function Spaces and Vector Lattices. De Gruyter Studies in Mathematics, vol. 45. De Gruyter, Berlin (2013)

    Book  Google Scholar 

Download references

Acknowledgements

Marat Pliev was supported by the Russian Foundation for Basic Research (Grant No. 17-51-12064). Martin Weber was supported by the Deutsche Forschungsgemeinschaft (Grant No. CH 1285/5-1, Order preserving operators in problems of optimal control and in the theory of partial differential equations).

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Authors and Affiliations

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

    Marat Pliev

  2. Department of Mathematics, Faculty of Science, Cankiri Karatekin University, Cankiri, Turkey

    Faruk Polat

  3. Department of Mathematics, Institute of Analysis, Technical University Dresden, 01062, Dresden, Germany

    Martin Weber

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  1. Marat Pliev
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  2. Faruk Polat
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Correspondence to Marat Pliev.

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Pliev, M., Polat, F. & Weber, M. Narrow and C-compact Orthogonally Additive Operators in Lattice-Normed Spaces. Results Math 74, 157 (2019). https://doi.org/10.1007/s00025-019-1075-y

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