×

A Radon-Nikodým type theorem for orthosymmetric bilinear operators. (English) Zbl 1219.47058

The author proves some new facts concerning the structure of the square of a vector lattice, characterizes orthoregular bilinear operators that may be presented as differences of symmetric lattice bimorphisms, and provide a Radon-Nikodým type theorem for orthosymmetric order continuous order interval preserving bilinear operators. At the end of the paper, the author makes many interesting remarks.

MSC:

47B65 Positive linear operators and order-bounded operators
46A40 Ordered topological linear spaces, vector lattices
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Aliprantis C.D., Burkinshaw O.: Positive Operators. Academic Press, London (1985)
[2] Azouzi Y., Boulabiar K., Buskes G.: The de Schipper formula and squares of Riesz spaces. Indag. Math. (N.S.) 17(4), 265–279 (2006) · Zbl 1131.46003 · doi:10.1016/S0019-3577(07)00003-1
[3] Boulabiar K., Buskes G.: Vector lattice powers: f-algebras and functional calculus. Commun. Algebra 34(4), 1435–1442 (2006) · Zbl 1100.46001 · doi:10.1080/00927870500454885
[4] de Buskes G., Pagter B., van Rooij A.: Functional calculus in Riesz spaces. Indag. Math. (N.S.) 4(2), 423–436 (1991) · Zbl 0781.46008 · doi:10.1016/0019-3577(91)90028-6
[5] Buskes G., Kusraev A.G.: Representation and extension of orthoregular bilinear operators. Vladikavkaz Math. J. 9(1), 16–29 (2007) · Zbl 1324.46011
[6] Buskes G., van Rooij A.: Almost f-algebras: commutativity and the Cauchy–Schwarz inequality. Positivity 4(3), 227–231 (2000) · Zbl 0987.46002 · doi:10.1023/A:1009826510957
[7] Buskes G., van Rooij A.: Squares of Riesz spaces. Rocky M. J. Math. 31(1), 45–56 (2001) · Zbl 0987.46003 · doi:10.1216/rmjm/1008959667
[8] Fremlin D.H.: Tensor product of Archimedean vector lattices. Am. J. Math. 94(3), 777–798 (1972) · Zbl 0252.46094 · doi:10.2307/2373758
[9] van Gaans, O.W.: The Riesz part of a positive bilinear form, In: Circumspice, Katholieke Universiteit Nijmegen, Nijmegen, pp. 19–30 (2001)
[10] Krivine, J.L.: Théorèmes de factorisation dans les espaces réticulés, Seminar Maurey–Schwartz, (1973–1974), École Politech., Exposé 22–23
[11] Kusraev, A.G.: On a Property of the Base of the K-space of Regular Operators and Some of Its Applications, Sobolev Inst. of Math., Novosibirsk (1977)
[12] Kusraev A.G.: General desintegration formulas. Dokl. Akad. Nauk SSSR 265(6), 1312–1316 (1982)
[13] Kusraev A.G.: Dominated Operators. Kluwer, Dordrecht (2000)
[14] Kusraev A.G.: On the representation of orthosymmetric bilinear operators in vector lattices. Vladikavkaz Math. J. 7(4), 30–34 (2005) · Zbl 1299.47078
[15] Kusraev A.G.: On the structure of orthosymmetric bilinear operators in vector lattices. Dokl. RAS 408(1), 25–27 (2006) · Zbl 1327.47033
[16] Kusraev A.G.: Hölder type inequalities for orthosymmetric bilinear operators. Vladikavkaz Math. J. 9(3), 36–46 (2007) · Zbl 1324.47035
[17] Kusraev A.G., Kutateladze S.S.: Subdifferentials: Theory and Applications. Kluwer, Dordrecht (1995) · Zbl 0832.49012
[18] Kutateladze S.S.: On differences of Riesz homomorphisms. Siberian Math. J. 46(2), 305–307 (2005) · doi:10.1007/s11202-005-0031-0
[19] Lindenstrauss J., Tzafriri L.: Classical Banach spaces. Vol. 2. Function Spaces. Springer, Berlin (1979) · Zbl 0403.46022
[20] Lozanovskiĭ G.Ya.: On topologically reflexive KB-spaces. Dokl. Akad. Nauk SSSR 158(3), 516–519 (1964)
[21] Lozanovskiĭ, G.Ya.: Certain Banach lattices, I–IV, Sibirsk. Mat. Zh., 10(3) (1969), 584–599, 12(3) (1971), 552–567, 13(6) (1972), 1304–1313, 14(1) (1973), 140–155
[22] Lozanovskiĭ, G.Ya.: The functions of elements of vector lattices, Izv. Vyssh. Uchebn. Zaved. Mat., (4), 45–54 (1973)
[23] Luxemburg W.A.J., Schep A.: A Radon–Nikodým type theorem for positive operators and a dual. Indag. Math. 40, 357–375 (1978) · Zbl 0389.47018
[24] Maharam, D.: The representation of abstract integrals, Trans. Am. Math. Soc., 75(1), 154–184 (1953) 79(1), 229–255 (1955) · Zbl 0051.29203
[25] Maharam D.: On positive operators. Contemporary Math. 26, 263–277 (1984) · Zbl 0583.47042
[26] Schwarz H.-U.: Banach Lattices and Operators. Teubner, Leipzig (1984)
[27] Shotaev G.N.: Some properties of regular bilinear operators. Vladikavkaz Math. J. 1(2), 44–47 (1999) · Zbl 1050.47508
[28] Szulga J.: (p,r)-convex functions on vector lattices. Proc. Edinb. Math. Soc. 37(2), 207–226 (1994) · Zbl 0805.46006 · doi:10.1017/S0013091500006027
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.