Siberian Mathematical Journal

, Volume 60, Issue 1, pp 114–123 | Cite as

Sums of Order Bounded Disjointness Preserving Linear Operators

  • A. G. KusraevEmail author
  • Z. A. KusraevaEmail author


Necessary and sufficient conditions are found under which the sum of N order bounded disjointness preserving operators is n-disjoint with n and N naturals. It is shown that the decomposition of an order bounded n-disjoint operator into a sum of disjointness preserving operators is unique up to “Boolean permutation,” the meaning of which is clarified in the course of the presentation.


vector lattice purely n-disjoint operator Boolean permutation factorization 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Carothers D. C. and Feldman W. A., “Sums of homomorphisms on Banach lattices,” J. Operator Theory, vol. 24, No. 2, 337–349 (1990).MathSciNetzbMATHGoogle Scholar
  2. 2.
    Huijsmans C. B. and de Pagter B., “Disjointness preserving and diffuse operators,” Comp. Math., vol. 79, No. 3, 351–374 (1991).MathSciNetzbMATHGoogle Scholar
  3. 3.
    Bernau S. J., Huijsmans C. B., and de Pagter B., “Sums of lattice homomorphisms,” Proc. Amer. Math. Soc., vol. 115, No. 1, 151–156 (1992).MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Radnaev V. A., “On n–disjoint operators,” Siberian Adv. Math., vol. 7, No. 4, 44–78 (1997).MathSciNetzbMATHGoogle Scholar
  5. 5.
    Radnaev V. A. On Metric n–Indecomposability in Ordered Lattice Normed Spaces and Its Applications, [Russian], PhD Thesis, Sobolev Institute Press, Novosibirsk (1997).Google Scholar
  6. 6.
    De Pagter B. and Schep A. R., “Band decomposition for disjointness preserving operators,” Positivity, vol. 4, No. 3, 259–288 (2000).MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Kusraev A. G. and Kutateladze S. S., Boolean Valued Analysis: Selected Topics, Vladikavkaz Sci. Center Press, Vladikavkaz (2014).zbMATHGoogle Scholar
  8. 8.
    Abramovich Y. A. and Kitover A. K., Inverses of Disjointness Preserving Operators, Amer. Math. Soc., Providence (2000) (Mem. Amer. Math. Soc., vol. 143, no. 679).Google Scholar
  9. 9.
    Boulabiar K., “Recent trends on order bounded disjointness preserving operators,” Irish Math. Soc. Bull., no. 62, 43–69 (2008).MathSciNetzbMATHGoogle Scholar
  10. 10.
    Boulabiar K., Buskes G., and Triki A., “Results in f–algebras,” in: Positivity (Eds. Boulabiar K., Buskes G., Triki A.), Birkhäuser, Basel, 2000, 73–96.Google Scholar
  11. 11.
    Gutman A. E., “Disjointness preserving operators,” in: Vector Lattices and Integral Operators (Ed. Kutateladze S. S.), Kluwer, Dordrecht, 1996, 361–454.Google Scholar
  12. 12.
    Huijsmans C. B., “Disjointness preserving operators on Banach lattices,” Operator Theory Adv. Appl., vol. 75, 173–189 (1999).MathSciNetzbMATHGoogle Scholar
  13. 13.
    Aliprantis C. D. and Burkinshaw O., Positive Operators, Academic Press, London (1985).zbMATHGoogle Scholar
  14. 14.
    Meyer M., “Le stabilisateur d’un espace vectoriel réticulé,” C. R. Acad. Sci., vol. 283, 249–250 (1976).MathSciNetzbMATHGoogle Scholar
  15. 15.
    Huijsmans C. B. and de Pagter B., “Invertible disjointness preserving operators,” Proc. Edinb. Math. Soc., vol. 37, no. 1, 125–132 (1993).MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Kutateladze S. S., “Subdifferentials of convex operators,” Sib. Math. J., vol. 18, No. 5, 747–752 (1977).MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Buskes G. and van Rooij A., “Small Riesz spaces,” Math. Proc. Camb. Phil. Soc., vol. 105, No. 3, 523–536 (1989).MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Pagter B., “The space of extended orthomorphisms in Riesz space,” Pacific J. Math., vol. 1, No. 112, 193–210 (1984).MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Kusraev A. G., Dominated Operators, Kluwer Academic Publishers, Dordrecht (2001).zbMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2019

Authors and Affiliations

  1. 1.Southern Mathematical InstituteNorth Ossetian State University named after K. L. KhetagurovVladikavkazRussia
  2. 2.Regional Mathematical Center of Southern Federal UniversityRostov-on-DonRussia
  3. 3.Southern Mathematical InstituteVladikavkazRussia

Personalised recommendations