Polynomial Grothendieck Properties *

A Banach space E 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 P(kE) of scalar valued polynomials on E is reflexive. This is equivalent to the symmetric k-fold projective tensor product of E (i.e., the predual of P(kE)) having the Grothendieck property. The Grothendieck property of the projective tensor product E ⊗̂ F 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. Throughout, E, F will be Banach spaces, and E the dual of E. We denote by L(E,F ) the space of all (linear bounded) operators from E to F , and by Co (E,F ) (WCo (E,F )) the subspace of all (weakly) compact operators. We say that T ∈ L(E,F ) is completely continuous if it takes weakly convergent sequences into norm convergent sequences, and we write T ∈ CC(E,F ). For an integer k, we shall consider the following classes of polynomials: 1991 AMS Subject Classification: Primary 46B20, 46E99 Supported in part by DGICYT Grant PB 91–0307 (Spain) Supported in part by DGICYT Grants PB 90–0044 and PB 91-0307 (Spain)