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Polynomial Grothendieck properties

Published online by Cambridge University Press:  18 May 2009

Manuel González  and
Joaquí M. Gutiérrez
Manuel González
Affiliation:
Departamento de Matemáticas, Facultad de Ciendcias, Universidad de Cantabria, 39071 Santander, Spain
Joaquí M. Gutiérrez
Affiliation:
Departamento de Matemática Aplicada, ETS de Ingenieros Industriaiales, Universidad Politécnica de Nadrid, C. José Gutiérrez Abascal 2, 28006 Madrid, Spain
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Abstract

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A Banach space sE has the Grothendieck property if every (linear bounded) operator from E into c0 is weakly compact. It is proved that, for an integer k > 1, every k-homogeneous polynomial from E into c0 is weakly compact if and only if the space (kE) of scalar valued polynomials on E is reflexive. This is equivalent to the symmetric A>fold projective tensor product of £(i.e., the predual of (kE)) having the Grothendieck property. The Grothendieck property of the projective tensor product EF is also characterized. Moreover, the Grothendieck property of E is described in terms of sequences of polynomials. Finally, it is shown that if every operator from E into c0 is completely continuous, then so is every polynomial between these spaces.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 37 , Issue 2 , May 1995 , pp. 211 - 219
Copyright
Copyright © Glasgow Mathematical Journal Trust 1995

References

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