Unconditionally converging polynomials on Banach spaces

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 P(E), k ∈ N, or the space of scalar holomorphic mappings of bounded type Hb(E), are reflexive. In this case the dual space E ∗, like the dual of Tsirelson’s space, is reflexive and contains no copies of lp. 1991 Mathematics Subject Classification. Primary: 46B20, 46G20 Supported in part by DGICYT Grant PB 91-0307 (Spain) Supported in part by DGICYT Grant PB 90-0044 (Spain) 1 In the study of polynomials acting on Banach spaces, the weak topology is not such a good tool as in the case of linear operators, due to the bad behaviour of the polynomials with respect to the weak convergence. For example, Q : (xn) ∈ l2 −→ (x 2 n) ∈ l1 is a continuous polynomial taking a weakly null sequence into a sequence having no weakly Cauchy subsequences. In this paper we show that the situation is not so bad for unconditional series. Recall that ∑∞ i=1 xi is a weakly unconditionally Cauchy series (in short a w.u.C. series) in a Banach space E if for every f ∈ E we have that ∑∞ i=1 |f(xi)| < ∞; and ∑∞ i=1 xi is an unconditionally converging series (in short an u.c. series) if every subseries is norm convergent. We prove that a continuous polynomial takes w.u.C. (u.c.) series into w.u.C. (u.c.) series. We derive this result from an estimate of the unconditional norm of the image of a sequence by a homogeneous polynomial, which is also a fundamental tool in other parts of the paper, and could be of some interest in itself. In view of the preservation of w.u.C. (u.c.) series by polynomials, it is natural to introduce the class Puc of unconditionally converging polynomials as those taking w.u.C. series into u.c. series. It turns out that most of the classes of polynomials that have been considered in the literature are contained in Puc. By means of the class of unconditionally converging polynomials we introduce the polynomial property (V ) for Banach spaces, and we show that spaces with this property share some of the properties of Tsirelson’s space T . In fact, these spaces are reflexive, and their dual spaces cannot contain copies of lp, 1 < p < ∞. This is a consequence of the following characterization: A Banach space E has the polynomial property (V ) if and only if the space of scalar polynomials P(E) is reflexive for every positive integer k. We also apply Puc to characterize the polynomial counterpart of other isomorphic properties of Banach spaces, like the Dieudonné property, the Schur property, and property (V ), obtaining remarkable differences with the corresponding linear (usual) properties. This is in contrast with the results of [14], where it is proved that the polynomial Dunford-Pettis property coincides with the Dunford-Pettis property. Throughout the paper, E and F will be real or complex Banach spaces, BE the unit ball of E, and E its dual space. The scalar field will be always R or C, the real or the complex field, and we will write N for the set of all natural numbers. Moreover, P(E,F ) will stand for the space of all (continuous) polynomials from E into F. Any P ∈ P(E,F ) can be decomposed as a sum of homogeneous polynomials: P = ∑n k=0 Pk, with Pk ∈ P( E,F ), the space of all k-homogeneous polynomials from E into F. 1 Unconditionally converging polynomials In this section we obtain an estimate for the unconditional norm of the image of a sequence by a homogeneous polynomial, and we apply it to prove the preservation of w.u.C. series and u.c. series by homogeneous polynomials. Then we introduce the 2 class of unconditionally converging polynomials, and compare it with other classes of polynomials that have appeared in the literature. In the proof of the estimate, we will need the generalized Rademacher functions, denoted by sn(t), n ∈ N, which were introduced in [4]. These functions are defined as follows: Fix 2 ≤ k ∈ N, and let α1 = 1, α2, . . . , αk denote the k th roots of unity. Let s1 : [0, 1] → C be the step function taking the value αj on ((j − 1)/k, j/k) for j = 1, . . . , k. Then, assuming that sn−1 has been defined, define sn as follows. Fix any of the k subintervals I of [0, 1] used in the definition of sn−1. Divide I into k equal intervals I1, . . . , Ik, and set sn(t) = αj if t ∈ Ij . The generalized Rademacher functions are orthogonal [4, Lemma 1.2] in the sense that, for any choice of integers i1, . . . , ik; k ≥ 2, we have ∫ 1 0 si1(t)...sik(t)dt = { 1, if i1 = · · · = ik; 0, otherwise. Lemma 1 Let E and F be Banach spaces. Given k ∈ N there exists a constant Ck such that for every P ∈ P(E,F ), and x1, ..., xn ∈ E we have sup |ǫj |≤1 ∥ ∥ ∥ ∥ ∥ ∥ n ∑ j=1 ǫjPxj ∥ ∥ ∥ ∥ ∥ ∥ ≤ Ck sup |νj |≤1 ∥ ∥ ∥ ∥ ∥ ∥ P 