Hostname: page-component-77c89778f8-cnmwb Total loading time: 0 Render date: 2024-07-17T05:48:07.603Z Has data issue: false hasContentIssue false
  • English
  • Français

Polynomials on banach spaces whose duals are isomorphic to ℓ1 (Γ)

Published online by Cambridge University Press:  17 April 2009

Raffaella Cilia ,
Maria D'Anna  and
Joaquín M. Gutiérrez
Raffaella Cilia
Affiliation:
Dipartimento di Matematica, Facolt` di Scienze, Università di Catania, Viale Andrea Doria 6, 95100 Catania, Italy e-mail: [email protected]
Maria D'Anna
Affiliation:
Dipartimento di Matematica, Facoltà di Scienze, Università di Catania, Viale Andrea Doria 6, 95100 Catania, Italy e-mail: [email protected]
Joaquín M. Gutiérrez
Affiliation:
Departamento de Matemática Aplicada, ETS de Ingenieros Industriales, Universidad Politécnica de Madrid, C. José Gutiérrez Abascal 2, 28006 Madrid, Spain e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We prove that the dual of a Banach space E is isomorphic to an ℓ1(Γ) space if and only if, for a fixed integer m, every m-homogeneous 1-dominated polynomial on E is nuclear. This extends a result for linear operators due to Lewis and Stegall. The same techniques used for this result allow us to prove that, if every m-homogeneous integral polynomial between two Banach spaces is nuclear, then every integral (linear) operator between the same spaces is nuclear.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 70 , Issue 1 , August 2004 , pp. 117 - 124
Copyright
Copyright © Australian Mathematical Society 2004

References

[1]Blasco, F., ‘Complementation in spaces of symmetric tensor products and polynomials’, Studia Math. 123 (1997), 165–173.CrossRefGoogle Scholar
[2]Cilia, R., D'Anna, M. and Gutiérrez, J.M., ‘Polynomial characterization of ℒ∞-spaces’, J. Math. Anal. Appl. 275 (2002), 900–912.CrossRefGoogle Scholar
[3]Diestel, J., Jarchow, H. and Tonge, A., Absolutely summing operators, Cambridge Stud. Adv. Math. 43 (Cambridge University Press, Cambridge, 1995).CrossRefGoogle Scholar
[4]Diestel, J. and Uhl, J.J. Jr, Vector measures, Math. Surveys Monographs 15 (American Mathematical Society, Providence, R.I., 1977).CrossRefGoogle Scholar
[5]Dineen, S., Complex analysis on infinite dimensional apaces, Springer Monographs in Math. (Springer-Verlag, Berlin, 1999).CrossRefGoogle Scholar
[6]Floret, K., ‘Natural norms on symmetric tensor products of normed spaces’, Note Mat. 17 (1997), 153–188.Google Scholar
[7]Geiℬ, S., Ideale multilinearer Abbildungen, (Diplomarbeit, Jena, 1984).Google Scholar
[8]Lewis, D. R. and Stegall, C., ‘Banach spaces whose duals are isomorphic to ℓ1(Γ)’, J. Funct. Anal. 12 (1973), 177–187.CrossRefGoogle Scholar
[9]Matos, M.C., ‘Absolutely summing holomorphic mappings,’, An. Acad. Brasil Ciênc 68 (1996), 1–13.Google Scholar
[10]Meléndez, Y. and Tonge, A., ‘Polynomials and the Pietsch domination theorem’, Math. Proc. Roy. Irish Acad. 99A (1999), 195–212.Google Scholar
[11]Mujica, J., Complex Analysis in Banach Spaces, Math. Studies 120, North-Holland, Amsterdam 1986.Google Scholar
[12]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), (Baumgärtel, H., Lasner, G., Pietsck, A. and Uhlmann, A., Editors), Teubner-Texte Math. 67 (Teubner, Leipzing, 1984), pp. 185–199.Google Scholar
[13]Schneider, B., ‘On absolutely p-summing and related multilinear mappings’, Wiss. Z. Brandenburg. Landeshochsch. 35 (1991), 105–117.Google Scholar
[14]Stegall, C., ‘Banach spaces whose duals contain ℓ1(Γ) with applications to the study of dual L 1(μ) spaces’, Trans. Amer. Math. Soc. 176 (1973), 463–477.Google Scholar
[15]Villanueva, I., ‘Integral mappings between Banach spaces’, J. Math. Anal. Appl. 279 (2003), 56–70.CrossRefGoogle Scholar