The polynomial property (V)

Abstract

Given Banach spaces E and F, we denote by ${\mathcal P}(^k!E,F)$ the space of all k-homogeneous (continuous) polynomials from E into F, and by ${\mathcal P}_{wb}(^k!E,F)$ the subspace of polynomials which are weak-to-norm continuous on bounded sets. It is shown that if E has an unconditional finite dimensional expansion of the identity, the following assertions are equivalent: (a) ${\mathcal P}(^k!E,F)={\mathcal P}_{wb}(^k!E,F)$; (b) ${\mathcal P}_{wb}(^k!E,F)$ contains no copy of $c_0$; (c) ${\mathcal P}(^k!E,F)$ contains no copy of $\ell_\infty$; (d) ${\mathcal P}_{wb}(^k!E,F)$ is complemented in ${\mathcal P}(^k!E,F)$. This result was obtained by Kalton for linear operators. As an application, we show that if E has Pe\l czy\'nski's property (V) and satisfies ${\mathcal P}(^k!E) ={\mathcal P}_{wb}(^k!E)$ then, for all F, every unconditionally converging $P\in{\mathcal P}(^k!E,F)$ is weakly compact. If E has an unconditional finite dimensional expansion of the identity, then the converse is also true.