Siberian Mathematical Journal

, Volume 49, Issue 2, pp 287–294

# Multiplicative representation of bilinear operators

• A. G. Kusraev
• 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

## 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).
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).
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).
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).
7. 7.
Boulabiar K. and Toumi M. A., “Lattice bimorphisms on f-algebras,” Algebra Universalis, 48, No. 1, 103–116 (2002).
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).
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).
11. 11.
Kusraev A. G. and Tabuev S. N., “On disjointness preserving bilinear operators,” Vladikavkazsk. Mat. Zh., 6, No. 1, 58–70 (2004).
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).
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).
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).
15. 15.
Scheffold E., “Über die Arens-Triadjungierte von Bimorphismen,” Rev. Roumaine Math. Pures Appl., 41, No. 9–10, 697–701 (1996).
16. 16.
Scheffold E., “Über symmetrische Operatoren auf Banachverbänden und Arens-Regularität,” Czechoslovak Math J., 48, No. 4, 747–753 (1998).
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.
18. 18.
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).
21. 21.
Kusraev A. G. and Kutateladze S. S., Introduction to Boolean Valued Analysis [in Russian], Nauka, Moscow (2005).
22. 22.
Aliprantis C. D. and Burkinshaw O., Positive Operators, Acad. Press, New York (1985).
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).
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