The polynomial property (V)

Abstract. Given Banach spaces E and F, we denote by ${\cal P}(^k\! E,F)$ the space of all k-homogeneous (continuous) polynomials from E into F, and by ${\cal P}_{\mathop {\rm wb}\nolimits }(^k\! E,F)$ 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) ${\cal P}(^k\! E,F) ={\cal P}_{\mathop {\rm wb}\nolimits }(^k\! E,F)$; (b) ${\cal P}_{\mathop {\rm wb}\nolimits}(^k\! E,F)$ contains no copy of c0; (c) ${\cal P}(^k\! E,F)$ contains no copy of $\ell _\infty$; (d) ${\cal P}_{\mathop {\rm wb}\nolimits }(^k\! E,F)$ is complemented in ${\cal P}(^k\! E,F)$. This result was obtained by Kalton for linear operators. As an application, we show that if E has Pełczyński's property (V) and satisfies ${\cal P}(^k\! E)={\cal P}_{\mathop {\rm wb}\nolimits }(^k\! E)$ then, for all F, every unconditionally converging $P\in {\cal P}(^k\! E,F)$ is weakly compact. If E has an unconditional finite dimensional expansion of the identity, then the converse is also true.