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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]
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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

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