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Nuclear and integral polynomials

Part of: Nonlinear operators and their properties Normed linear spaces and Banach spaces; Banach lattices Measures, integration, derivative, holomorphy

Published online by Cambridge University Press:  09 April 2009

Raffaella Cilia  and
Joaquín M. Gutiérrez
Raffaella Cilia
Affiliation:
Dipartimento di Matematica Facoltà di Science, Università di Catania, Viale Andrea Doria, 6 95100 Catania, Itlay e-mail: [email protected]
Joaquín M. Gutiérrez
Affiliation:
Departamento de Matemática Aplicada ETS de Ingenieros IndustrialesUniversidad Politécnica de MadridC. José Gutiérrez Abascal 2 28006 Madrid, Spain e-mail: [email protected]
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Abstract

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Let E be a Banach space whose dual E* has the approximation property, and let m be an index. We show that E* has the Radon-Nikodým property if and only if every m-homogeneous integral polynomial from E into any Banach space is nuclear. We also obtain factorization and composition results for nuclear polynomials.

Keywords

1-dominated polynomial, integral polynonmialnuclear polynomialfactorization of polynomials

MSC classification

Secondary: 46G25: (Spaces of) multilinear mappings, polynomials 46B20: Geometry and structure of normed linear spaces 47H60: Multilinear and polynomial operators
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 76 , Issue 2 , April 2004 , pp. 269 - 280
Copyright
Copyright © Australian Mathematical Society 2004

References

[1]Alencar, R., ‘Multilinear mappings of nuclear and integral type’, Proc. Amer. Math. Soc. 94 (1985), 33–38.Google Scholar
[2]Alencar, R., ‘On reflexivity and basis for p(mE)’, Proc. Roy. Irish Acad. 85A (1985), 131–138.Google Scholar
[3]Carando, D., ‘Extendible polynomials on Banach spaces’, J. Math. Anal. Appl. 233 (1999), 359–372.Google Scholar
[4]Carando, D., ‘Extendibility of polynomials and analytic functions on lp’, Studia Math. 145 (2001), 63–73.Google Scholar
[5]Carando, D. and Dimant, V., ‘Duality in spaces of nuclear and integral polynomials’, J. Math. Anal. Appl. 241 (2000), 107–121.Google Scholar
[6] Cilia, R., D'Anna, M. and Gutiérrez, J. M., ‘Polynomials on Banach spaces whose duals are isomorphic to l1 (Γ)’, preprint.Google Scholar
[7]Cilia, R., D'Anna, M. and Gutiérrez, J. M., ‘Polynomial characterization of ℒ∞-spaces’, J. Math. Anal. Appl. 275 (2002), 900–912.CrossRefGoogle Scholar
[8] Cilia, R. and Gutiérrez, J. M., ‘Polynomial characterization of Asplund spaces’, Bull. Belgian Math. Soc. Simon Stevin, to appear.Google Scholar
[9]Defant, A. and Floret, K., Tensor norms and operator ideals, Math. Studies 176 (North-Holland, Amsterdam, 1993).Google Scholar
[10]Diestel, J., Jarchow, H. and Tonge, A., Absolutely summing operators, Cambridge Stud. Adv. Math. 43 (Cambridge University Press, Cambridge, 1995).Google Scholar
[11]Diestel, J. and Uhl, J. J. Jr, Vector measures, Math. Surveys Monographs 15 (Amer. Math. Soc., Providence, RI, 1977).CrossRefGoogle Scholar
[12]Dineen, S., Complex analysis on infinite dimensional spaces, Springer Monographs in Math. (Springer, Berlin, 1999).Google Scholar
[13]Floret, K., ‘Natural norms on symmetric tensor products of normed spaces’, Note Mat. 17 (1997), 153–188.Google Scholar
[14]Geiß, S., Ideale multilinearer Abbildungen (Diplomarbeit, Jena, 1984).Google Scholar
[15]Meléndez, Y. and Tonge, A., ‘Polynomials and the Pietsch domination theorem’, Math. Proc. Roy. Irish Acad. 99A (1999), 195–212.Google Scholar
[16]Mujica, J., Complex analysis in Banach spaces, Math. Studies 120 (North-Holland, Amsterdam, 1986).Google Scholar
[17]Pietsch, A., ‘Ideals of multilinear functionals (designs of a theory)’, in: Proceedings of the Second International Conference on Operator Algebras, Ideals, and their Applications in Theoretical Physics (Leipzig, 1983) (eds. Baumgärtel, H. et al. ), Teubner-Texte Math. 67 (Teubner, Leipzig, 1984) pp. 185–199.Google Scholar
[18]Rudin, W., Principles of mathematical analysis (McGraw-Hill, New York, 1976).Google Scholar
[19]Ruess, W. M. and Stegall, C. P., ‘Extreme points in duals of operator spaces’, Math. Ann. 261 (1982), 535–546.CrossRefGoogle Scholar
[20]Ryan, R. A., ‘Weakly compact holomorphic mappings on Banach spaces’, Pacific J. Math. 131 (1988), 179–190.Google Scholar
[21]Schneider, B., ‘On absolutely p-summing and related multilinear mappings’, Brandenburgische Landeshochschule Wissen. Z. 35 (1991), 105–117.Google Scholar
[22]Villanueva, I., ‘Integral mappings between Banach spaces’, J. Math. Anal. Appl. 279 (2003), 56–70.Google Scholar