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 Pełczyński'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.

9 pages