Spaces of absolutely summing polynomials

Abstract

This paper has a twofold purpose: to prove a much more general Dvoretzky-Rogers type theorem for absolutely summing polynomials and to introduce a more convenient norm on the space of everywhere summing polynomials.

On multi-ideals and polynomial ideals of Banach spaces: a new approach to coherence and compatibility

“…ev (2) (p;q) ) ∞ n=1 has indeed good properties (see [53] for details). It was also proved in [3,Proposition 4.4] that one can also show that P n : K → K; P n (λ) = λ n ev (2) (p;q) = 1 for all p ≥ q ≥ 1 and it is also not difficult to show that (P n,ev as(p;q) , . ev (2) (p;q) ) ∞ n=1 is a Banach polynomial ideal.…”

“…Matos also proved that this norm is complete and that (P n,ev as(p;q) , · ev (1) (p;q) ) is a global holomorphy type. From [3] it is known that lim n→∞ P n : K → K; P n (λ) = λ n ev (1) (p;q) = ∞ and this estimate will allow us to conclude that (P n,ev as(p;q) , · ev (1) (p;q) ) ∞ n=1 is "compatible, in the sense of Carando et al" with no operator ideal; here we have used the term "compatible , in the sense of Carando et al" in a more general form, since the sequence (P n,ev as(p;q) , · ev (1) (p;q) ) ∞ n=1 is not exactly a normed polynomial ideal (since it fails (P2)), but just a global holomorphy type.…”

“…Given n ∈ N, let P n : K −→ K denote the trivial n-homogeneous polynomial given by P n (λ) = λ n . From [3,Proposition 4.4] it is also known that lim n→∞ P n ev (1) (p;q) = ∞ for all p ≥ q > 1. So, by considering u = γ = P 1 , we conclude that γ n−1 u belongs to P n,ev as(p;q) ( n E; K) and lim n→∞ γ n−1 u ev (1) (p;q) = lim n→∞ P n ev (1) (p;q) = ∞ and, on the other hand, for every operator ideal I, we have…”

On almost summing polynomials and multilinear mappings

“…The next result is inspired on [3] and [21], but some parts (specially the proof of (d) ) (a)) need a different approach: THEOREM 3.7 The following assertions are equivalent:…”

“…, E n ; F Þ: A similar definition holds for polynomials and the respective spaces are represented by P ðaÞ al, p ð n E; F Þ and P ev al, p ð n E; F Þ, with 15p 2. Some of the arguments used in the forthcoming proofs related to multilinear/ polynomial notions of almost summing operators can be obtained, mutatis mutandis, from similar arguments used for absolutely summing operators in [3,21]. For this reason we will omit the more simple adaptations and concentrate on the parts that need a different treatment.…”