Unconditionally converging polynomials on Banach spaces

Abstract

We prove that weakly unconditionally Cauchy (w.u.C.) series and unconditionally converging (u.c.) series are preserved under the action of polynomials or holomorphic functions on Banach spaces, with natural restrictions in the latter case. Thus it is natural to introduce the unconditionally converging polynomials, defined as polynomials taking w.u.C. series into u.c.\ series, and analogously, the unconditionally converging holomorphic functions. We show that most of the classes of polynomials which have been considered in the literature consist of unconditionally converging polynomials. Then we study several ``polynomial properties'' of Banach spaces, defined in terms of relations of inclusion between classes of polynomials, and also some ``holomorphic properties''. We find remarkable differences with the corresponding ``linear properties''. For example, we show that a Banach space $E$ has the polynomial property (V) if and only if the spaces of homogeneous scalar polynomials ${\cal P}(^k\!E)$, $k\in{\bf N}$, or the space of scalar holomorphic mappings of bounded type ${\cal H}_b(E),$ are reflexive. In this case the dual space $E^*$, like the dual of Tsirelson's space, is reflexive and contains no copies of $\ell_p$.