Orthosymmetric bilinear map on Riesz spaces
2015, Commentationes Mathematicae Universitatis Carolinae
https://doi.org/10.14712/1213-7243.2015.132Abstract
Let E be a Riesz space, F a Hausdorff topological vector space (t.v.s.). We prove, under a certain separation condition, that any orthosymmetric bilinear map T : E × E → F is automatically symmetric. This generalizes in certain way an earlier result by F. Ben Amor [On orthosymmetric bilinear maps, Positivity 14 (2010), [123][124][125][126][127][128][129][130][131][132][133][134]. As an application, we show that under a certain separation condition, any orthogonally additive homogeneous polynomial P : E → F is linearly represented. This fits in the type of results by Y.
References (20)
- Aliprantis C.D., Burkinshaw O., Positive Operators, Springer, Dordrecht, 2006.
- Ben Amor F., On orthosymmetric bilinear maps, Positivity 14 (2010), 123-134.
- Benyamini Y., Lassalle S., Llavona J.L.G., Homogeneous orthogonally additive polynomials on Banach lattices, Bull. London Math. Soc. 38 (2006), no. 3, 459-469.
- Beukers F., Huijsmans C.B., Calculus in f-algebras, J. Austral. Math. Soc. Ser. A 37 (1984), no. 1, 110-116.
- Boulabiar K., On products in lattice-ordered algebras, J. Austral. Math. Soc. 75 (2003), no. 1, 1435-1442.
- Bu Q., Buskes G., Polynomials on Banach lattices and positive tensor products, J. Math. Anal. Appl. 388 (2012), 845-862.
- Buskes G., van Rooij A., Almost f -algebras: commutativity and the Cauchy-Schwarz in- equality, Positivity 4 (2000), no. 3, 227-231.
- Carando D., Lassalle S., Zalduendo I., Orthogonally additive polynomials over C(K) are measures -a short proof , Integral Equations Operator Theory, 56 (2006), no. 4, 597-602.
- Carando D., Lassalle S., Zalduendo I., Orthogonally additive holomorphic functions of bounded type over C(K), Proc. Edinb. Math. Soc. (2) 53 (2010), no. 3, 609-618.
- Chil E., Order bounded orthosymmetric bilinear operator, Czechoslovak Math. J. 61 (2011), no. 4, 873-880.
- Chil E., Meyer M., Mokaddem M., On orthosymmetric multilinear maps, Positivity (preprint).
- de Pagter B., f -algebras and Orthomorphisms, thesis, Leiden, 1981.
- Ibort A., Linares P., Llavona J.G., On the representation of orthogonally additive polyno- mials in ℓp, Publ. RIMS Kyoto Univ. 45 (2009), 519-524.
- Jaramillo J.A., Prieto A., Zalduendo I., Orthogonally additive holomorphic functions on open subsets of C(K), Rev. Mat. Complut. 25 (2012), no. 1, 31-41.
- Luxemburg W.A., Zaanen A.C., Riesz Spaces I , North-Holland, Amsterdam, 1971.
- Meyer-Nieberg P., Banach Lattices, Springer, Berlin, 1991.
- Palazuelos C., Peralta A.M., Villanueva I., Orthogonally additive polynomials on C * - algebras, Quart. J. Math. 59 (2008), 363-374.
- Perez-García D., Villanueva I., Orthogonally additive polynomials on space of continuous functions, J. Math. Anal. Appl. 306 (2005), no. 1, 97-105.
- Sundaresan K., Geometry of spaces of homogeneous polynomials on Banach lattices, Ap- plied geometry and discrete mathematics, 571-586, DIMACS Series in Discrete Mathemat- ics and Theoretical computer Science no. 4, Amer. Math. Soc., Providence, RI, 1991.
- Chil E., Hassen B.: Institut prparatoire aux tudes d'ingenieurs de Tunis, 2 Rue jawaher lel Nehrou Monflery 1008 Tunisia E-mail: Elmiloud.chil@ipeit.rnu.tn bourokba.hassen@gmail.com