Abstract
Regular multilinear operators and regular homogeneous polynomials acting between Banach lattices are automatically continuous, but the converse, in general, is not true. The problem arises of characterizing Banach lattices for which the classes of continuous and regular multilinear operators (or homogeneous polynomials) coincide. The aim of this note is to extend two results in this direction, earlier obtained for linear operators, to the above-mentioned classes of operators and polynomials. The main method is linearization with the use of the Fremlin tensor product of Banach lattices.
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References
Z. A. Kusraeva, “Powers of quasi-Banach lattices and orthogonally additive polynomials,” J. Math. Anal. Appl. 458 (1), 767–780 (2018).
Hong Yun Xiong, “On whether or not \(\mathcal{L}(E,F)=\mathcal{L}^r(E,F)\) for some classical Banach lattices \(E\) and \(F\),” Indag. Math., No. 46, 267–282 (1984).
L. V. Kantorovich and B. Z. Vulikh, “Sur la représentation des opérations linéaires,” Compositio Math. 5, 119–165 (1937).
B. Z. Vulikh, Introduction to the Theory of Semi-Ordered Spaces (GIFML, Moscow, 1961) [in Russian].
C. D. Aliprantis and O. Burkinshaw, Positive Operators (Academic Press, New York, 1985).
P. Meyer-Nieberg, Banach Lattices (Springer- Verlag, Berlin, 1991).
Yu. Synnachke, “On an operator adjoint to a regular one and some of its applications to the question of complete continuity and weak complete continuity of regular operators,” Vestn. Leningradsk. Gos. Univ., Ser. Mat. Mekh., No. 1, 60–69 (1972).
Yu. A. Abramovich and A. W. Wickstead, “When each continuous operator is regular. II,” Indag. Math. (N. S.) 8 (3), 281–294 (1997).
Q. Bu and G. Buskes, “Polynomials on Banach lattices and positive tensor products,” J. Math. Anal. Appl. 388, 845–862 (2012).
J. Loane, Polynomials on Riesz Spaces, Thesis (Department of Mathematics, National University of Ireland, Galway, 2007).
S. Dineen, Complex Analysis on Infinite-Dimensional Spaces (Springer- Verlag, Berlin, 1999).
Y. A. Abramovich and C. D. Aliprantis, “Positive operators,” in Handbook of the Geometry of Banach Spaces (North-Holland, Amsterdam, 2001), Vol. I, pp. 85–122.
D. H. Fremlin, Measure Theory. Vol. 3. Measure Algebras (Cambridge Univ. Press, Cambridge, 2002).
Q Bu, G. Buskes and Y. Li, “Abstract \(L\)- and abstract \(M\)-spaces of polynomials on Banach lattices,” Proc. Edinb. Math. Soc. (2) 58 (3), 617–629 (2015).
R. A. Ryan, Introduction to Tensor Products of Banach Spaces (Springer- Verlag, London, 2002).
Z. A. Kusraeva, “Representation of orthogonally additive polynomials,” Siberian Math. J. 52 (2), 248–255 (2011).
Z. A. Kusraeva and S. N. Siukaev, “Some properties of orthogonally additive homogeneous polynomials on Banach lattices,” Vladikavkaz. Mat. Zh. 22 (4), 92–103 (2020).
D. H. Fremlin, “Tensor product of Banach lattices,” Math. Ann. 211, 87–106 (1974).
H. H. Schaefer, Banach Lattices and Positive Operators (Springer- Verlag, Berlin, 1974).
D. H. Fremlin, Topological Riesz Spaces and Measure Theory (Cambridge Univ. Press, London, 1974).
Yu. A. Abramovich and A. W. Wickstead, “The regularity of order bounded operators into \(C(K)\). II,” Quart. J. Math. Oxford Ser. (2) 44 (175), 257–270 (1993).
D. H. Fremlin, “Tensor product of Archimedean vector lattices,” Amer. J. Math. 94, 777–798 (1972).
A. W. Wickstead, “The regularity of order bounded operators into \(C(K)\),” Quart. J. Math. Oxford Ser. (2) 41 (163), 359–368 (1990).
K. Yu. Il’ina and Z. A. Kusraeva, “On the contunuation of positive operators,” Sibirsk. Mat. Zh. 61 (2), 330–336 (2020).
A. A. Gelieva and Z. A. Kusraeva, “On dominated extension of linear operators,” Math. Notes 108 (2), 171–178 (2020).
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Translated from Matematicheskie Zametki, 2021, Vol. 110, pp. 726–735 https://doi.org/10.4213/mzm13089.
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Kusraeva, Z.A. Regularity of Continuous Multilinear Operators and Homogeneous Polynomials. Math Notes 110, 718–725 (2021). https://doi.org/10.1134/S0001434621110080
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DOI: https://doi.org/10.1134/S0001434621110080