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Siberian Mathematical Journal

, Volume 49, Issue 2, pp 287–294 | Cite as

Multiplicative representation of bilinear operators

  • A. G. KusraevEmail author
  • S. N. Tabuev
Article

Abstract

We establish that each lattice bimorphism from the Cartesian product of two vector lattices into a universally complete vector lattice is representable as the product of two lattice homomorphisms defined on the factors. This fact makes it possible to reduce the problem to the linear case and obtain some results on representation of an order bounded disjointness preserving bilinear operator as a strongly disjoint sum of weighted shift or multiplicative operators.

Keywords

order bounded bilinear operator lattice bimorphism weighted shift operator multiplicative representation 

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References

  1. 1.
    Bernau S. J. and Huijsmans C. B., “The order bidual of almost f-algebras and d-algebras,” Trans. Amer. Math. Soc., 347, No. 11, 4259–4274 (1995).zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Buskes G. J. H. M., “Five theorems in the theory of Riesz spaces,” in: Circumspice, Katholieke Univ. Nijmegen, Nijmegen, 2001, pp. 3–10.Google Scholar
  3. 3.
    Buskes G. J. H. M. and van Rooij A. C. M., “Almost f-algebras: Commutativity and the Cauchy-Schwarz inequality,” Positivity, 4, No. 3, 233–243 (2000).zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Buskes G. J. H. M. and van Rooij A. C. M., “Almost f-algebras: Structure and the Dedekind completion,” Positivity, 4, No. 3, 227–231 (2000).zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Gaans O. W. van, “The Riesz part of a positive bilinear form,” in: Circumspice, Katholieke Univ. Nijmegen, Nijmegen, 2001, pp. 19–30.Google Scholar
  6. 6.
    Boulabiar K., “Some aspects of Riesz multimorphisms,” Indag. Math. (N.S.), 13, No. 4, 419–432 (2002).zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Boulabiar K. and Toumi M. A., “Lattice bimorphisms on f-algebras,” Algebra Universalis, 48, No. 1, 103–116 (2002).zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Buskes G. J. H. M. and van Rooij A. C. M., “Bounded variation and tensor products of Banach lattices,” Positivity, 7, No. 1, 47–59 (2003).zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Kusraev A. G. and Shotaev G. N., “Bilinear dominated operators,” in: Studies on Complex Analysis, Operator Theory, and Mathematical Modeling [in Russian], Izdat. VNTs RAN, Vladikavkaz, 2004, pp. 241–262.Google Scholar
  10. 10.
    Kusraev A. G., “Representation of orthosymmetric bilinear operators in vector lattices,” Vladikavkazsk. Mat. Zh., 7, No. 4, 30–34 (2005).MathSciNetGoogle Scholar
  11. 11.
    Kusraev A. G. and Tabuev S. N., “On disjointness preserving bilinear operators,” Vladikavkazsk. Mat. Zh., 6, No. 1, 58–70 (2004).MathSciNetGoogle Scholar
  12. 12.
    Boulabiar K., Buskes G., and Page R., “On some properties of bilinear maps of order bounded variation,” Positivity, 9, No. 3, 401–414 (2005).zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Buskes G. J. H. M. and van Rooij A. C. M., “Squares of Riesz spaces,” Rocky Mountain J. Math., 31, No. 1, 45–56 (2004).CrossRefGoogle Scholar
  14. 14.
    Scheffold E., “Über Bimorphismen und das Arens-Product bei kommutativen D-Banachverbands,” Rev. Roumaine Math. Pures Appl., 39, No. 3, 183–205 (1994).MathSciNetGoogle Scholar
  15. 15.
    Scheffold E., “Über die Arens-Triadjungierte von Bimorphismen,” Rev. Roumaine Math. Pures Appl., 41, No. 9–10, 697–701 (1996).zbMATHMathSciNetGoogle Scholar
  16. 16.
    Scheffold E., “Über symmetrische Operatoren auf Banachverbänden und Arens-Regularität,” Czechoslovak Math J., 48, No. 4, 747–753 (1998).zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Gutman A. E., “Banach bundles in the theory of lattice-normed spaces,” in: Linear Operators Compatible with Order [in Russian], Trudy Inst. Mat. (Novosibirsk). Vol. 29, Novosibirsk, 1995, pp. 63–211.MathSciNetGoogle Scholar
  18. 18.
    Kusraev A. G., Dominated Operators, Kluwer Academic Publishers, Dordrecht (2001).Google Scholar
  19. 19.
    Gutman A. E., “Disjointness preserving operators,” in: Vector Lattices and Integral Operators, Kluwer Acad. Publ., Dordrecht etc., 1996, pp. 361–454.Google Scholar
  20. 20.
    Abramovich Yu. A., “Multiplicative representation of disjointness preserving operators,” Indag. Math. (N.S.), 45, No. 3, 265–279 (1983).zbMATHMathSciNetGoogle Scholar
  21. 21.
    Kusraev A. G. and Kutateladze S. S., Introduction to Boolean Valued Analysis [in Russian], Nauka, Moscow (2005).zbMATHGoogle Scholar
  22. 22.
    Aliprantis C. D. and Burkinshaw O., Positive Operators, Acad. Press, New York (1985).zbMATHGoogle Scholar
  23. 23.
    Kusraev A. G., On a Property of the Base of K-Space of Regular Operators and Some of Their Applications [Preprint] [in Russian], Inst. Mat. (Novosibirsk), Novosibirsk (1977).Google Scholar
  24. 24.
    Fremlin D. H., “Tensor product of Archimedean vector lattices,” Amer. J. Math., 94, No. 3, 777–798 (1972).zbMATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Kusraev A. G. and Kutateladze S. S., “Nonstandard methods and Kantorovich spaces,” in: Nonstandard Analysis and Vector Lattices [in Russian], Izdat. Inst. Mat., Novosibirsk, 2005, pp. 1–123.Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2008

Authors and Affiliations

  1. 1.Institute of Applied Mathematics and InformaticsVladikavkazRussia

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