Regularity of continuous multilinear operators and homogeneous polynomials. (English. Russian original) Zbl 1487.46048

Kusraeva, Z. A.

Math. Notes 110, No. 5, 718-725 (2021); translation from Mat. Zametki 110, No. 5, 726-735 (2021).

The author extends to the case of multilinear operators and homogeneous polynomials the results about the regularity of continuous operators of [J. Synnatzschke, Vestn. Leningr. Univ., Mat. Mekh. Astron. 1972, No. 1, 60–69 (1972; Zbl 0234.47035); Y. A. Abramovich and A. W. Wickstead, Indag. Math., New Ser. 8, No. 3, 281–294 (1997; Zbl 0908.47031)].
The research utilizes the technique of the Fremlin tensor product.

Reviewer: S. S. Kutateladze (Novosibirsk)

MSC:

46G25	(Spaces of) multilinear mappings, polynomials
47B65	Positive linear operators and order-bounded operators
46B42	Banach lattices

Keywords:

Banach lattice; Levi property; multilinear operator; homogeneous polynomial; Fremlin tensor product; linearization

Citations:

Zbl 0234.47035; Zbl 0908.47031

References:

[1]	Kusraeva, Z. A., Powers of quasi-Banach lattices and orthogonally additive polynomials, J. Math. Anal. Appl., 458, 1, 767-780 (2018) · Zbl 1465.46022 · doi:10.1016/j.jmaa.2017.09.019
[2]	Xiong, Hong Yun, On whether or not \(\mathcal{L}(E,F)=\mathcal{L}^r(E,F)\) for some classical Banach lattices \(E\) and \(F\), Indag. Math., 0, 46, 267-282 (1984) · Zbl 0562.46017 · doi:10.1016/1385-7258(84)90027-1
[3]	Kantorovich, L. V.; Vulikh, B. Z., Sur la représentation des opérations linéaires, Compositio Math., 5, 119-165 (1937) · Zbl 0017.21502
[4]	Vulikh, B. Z., Introduction to the Theory of Semi-Ordered Spaces (1961), Moscow: GIFML, Moscow · Zbl 0101.08501
[5]	Aliprantis, C. D.; Burkinshaw, O., Positive Operators (1985), New York: Academic Press, New York · Zbl 0608.47039
[6]	Meyer-Nieberg, P., Banach Lattices (1991), Berlin: Springer- Verlag, Berlin · Zbl 0743.46015 · doi:10.1007/978-3-642-76724-1
[7]	Synnachke, Yu., 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., 0, 1, 60-69 (1972) · Zbl 0234.47035
[8]	Abramovich, Yu. A.; Wickstead, A. W., When each continuous operator is regular. II, Indag. Math. (N. S.), 8, 3, 281-294 (1997) · Zbl 0908.47031 · doi:10.1016/S0019-3577(97)81812-5
[9]	Bu, Q.; Buskes, G., Polynomials on Banach lattices and positive tensor products, J. Math. Anal. Appl., 388, 845-862 (2012) · Zbl 1245.46035 · doi:10.1016/j.jmaa.2011.10.001
[10]	Loane, J., Polynomials on Riesz Spaces (2007), Galway: Department of Mathematics, National University of Ireland, Galway · Zbl 1187.46035
[11]	Dineen, S., Complex Analysis on Infinite-Dimensional Spaces (1999), Berlin: Springer- Verlag, Berlin · Zbl 1034.46504 · doi:10.1007/978-1-4471-0869-6
[12]	Abramovich, Y. A.; Aliprantis, C. D., Positive operators, Handbook of the Geometry of Banach Spaces, I, 85-122 (2001) · Zbl 1202.47042 · doi:10.1016/S1874-5849(01)80004-8
[13]	Fremlin, D. H., Measure Theory. Vol. 3. Measure Algebras (2002), Cambridge: Cambridge Univ. Press, Cambridge · Zbl 1165.28002
[14]	Bu, Q.; Buskes, G.; Li, Y., Abstract \(L\)- and abstract \(M\)-spaces of polynomials on Banach lattices, Proc. Edinb. Math. Soc. (2), 58, 3, 617-629 (2015) · Zbl 1336.46042 · doi:10.1017/S0013091514000297
[15]	Ryan, R. A., Introduction to Tensor Products of Banach Spaces (2002), London: Springer- Verlag, London · Zbl 1090.46001 · doi:10.1007/978-1-4471-3903-4
[16]	Kusraeva, Z. A., Representation of orthogonally additive polynomials, Siberian Math. J., 52, 2, 248-255 (2011) · Zbl 1226.46043 · doi:10.1134/S003744661102008X
[17]	Kusraeva, Z. A.; Siukaev, S. N., Some properties of orthogonally additive homogeneous polynomials on Banach lattices, Vladikavkaz. Mat. Zh., 22, 4, 92-103 (2020) · Zbl 1474.46082
[18]	Fremlin, D. H., Tensor product of Banach lattices, Math. Ann., 211, 87-106 (1974) · Zbl 0272.46050 · doi:10.1007/BF01344164
[19]	Schaefer, H. H., Banach Lattices and Positive Operators (1974), Berlin: Springer- Verlag, Berlin · Zbl 0296.47023 · doi:10.1007/978-3-642-65970-6
[20]	Fremlin, D. H., Topological Riesz Spaces and Measure Theory (1974), London: Cambridge Univ. Press, London · Zbl 0273.46035 · doi:10.1017/CBO9780511897207
[21]	Abramovich, Yu. A.; Wickstead, A. W., The regularity of order bounded operators into \(C(K)\). II, Quart. J. Math. Oxford Ser. (2), 44, 175, 257-270 (1993) · Zbl 0991.47023 · doi:10.1093/qmath/44.3.257
[22]	Fremlin, D. H., Tensor product of Archimedean vector lattices, Amer. J. Math., 94, 777-798 (1972) · Zbl 0252.46094 · doi:10.2307/2373758
[23]	Wickstead, A. W., The regularity of order bounded operators into \(C(K)\), Quart. J. Math. Oxford Ser. (2), 41, 163, 359-368 (1990) · Zbl 0716.47022 · doi:10.1093/qmath/41.3.359
[24]	Il’ina, K. Yu.; Kusraeva, Z. A., On the contunuation of positive operators, Sibirsk. Mat. Zh., 61, 2, 330-336 (2020) · Zbl 1445.47031 · doi:10.33048/smzh.2020.61.208
[25]	Gelieva, A. A.; Kusraeva, Z. A., On dominated extension of linear operators, Math. Notes, 108, 2, 171-178 (2020) · Zbl 1518.47068 · doi:10.1134/S0001434620070184