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 P( k E), k ∈ N, or the space of scalar holomorphic mappings of bounded type H b (E), are reflexive. In this case the dual space E

Three-Space Problems for Polynomial Properties in Banach Spaces

“…The purpose of this section is to give a partial solution to the 3-space problems for properties polynomial Grothendieck of González and Gutiérrez [18] ; polynomial V of González and Gutiérrez [19] and P-reflexivity of Farmer [16]. What we show is that to obtain a counter-example for their 3-space problem (if it exists) one first has to solve problem 3 in [12] : Does there exist a reflexive M-space without upper p-estimates ?…”

“…Extending the definition to polynomials, González and Gutiérrez [19] define a Dieudonné polynomial as a polynomial transforming weakly Cauchy sequences into weakly convergent sequences. Thus, a Banach space E is said to have the polynomial Dieudonné property (in short pD) if every Dieudonné polynomial is weakly compact.…”

“…Thus, a Banach space E is said to have the polynomial Dieudonné property (in short pD) if every Dieudonné polynomial is weakly compact. A result in [19] shows that property (pD) is equivalent to the property of not containing l 1 , which is a 3-space property (see [8]). A polynomial P 2 P( k E, F ) is said to be unconditionally convergent if, for every weakly unconditionally Cauchy series P O i=1 x i in E, the sequence {P( P n i=1 x i )} n converges in norm.…”

Quasi-completely continuous multilinear operators

“…A different notion of unconditionally converging multilinear operators and polynomials, giving rise to a strictly wider class, was used in [11,10]. However, the definition given above, used in [4,12], seems to be more appropriate for most applications.…”

Polynomials and limited sets

“…In view of Corollary 6, we may ask if polynomials map Grothendieck sets into Grothendieck sets. The negative answer to this question is provided by González and Gutiérrez in [10] who show the existence of a polynomial P : l ∞ → c 0 which maps the unit ball of l ∞ into a non-weakly relatively compact set. Our next proposition is a refinement of their result and its proof goes back to Proposition 4 of [3].…”