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Two new properties of ideals of polynomials and applications. (English) Zbl 1089.46027
An ideal of polynomials is a natural generalization of the notion of an operator ideal. The authors consider two properties of such ideals of polynomials.
Let \(\mathcal Q\) be an ideal of polynomials. 1. \({\mathcal Q}\) is closed under differentiation (\(\mathcal Q\) is cud) if \(\widehat{d} P(a)\in {\mathcal Q} (E;F)\) for every \(a\in E\) and every \(n\)-homogeneous continuous polynomial \(P\in {\mathcal Q }(^n E;F)\), 2. \(\mathcal Q\) is closed under scalar multiplication (\(\mathcal Q\) is csm) if \(\varphi P\in {\mathcal Q} (^{n+1} E;F)\) for every \(\varphi \in E'\) and \(P\in {\mathcal Q}(^n E;F)\) for all \(n\in \mathbb N\) and Banach spaces \(E\) and \(F\).
It is shown that the ideals \({\mathcal P}_{L_{(\mathcal I )}}\) and \({\mathcal P}_{[\mathcal I ]}\) share these properties. Here, \(\mathcal I\) is an operator ideal and \({\mathcal P}_{L_{(\mathcal I )}}, {\mathcal P}_{[\mathcal I ]}\) go back to ideals of multilinear mappings defined in [A. Pietsch, in: Operator algebras, ideals, and their applications in theoretical physics, Proc. int. Conf., Leipzig 1983, Teubner-Texte Math. 67, 185–199 (1984; Zbl 0562.47037)]. In this way, the authors give, among other results, a lot of new polynomial characterisations of \({\mathcal L}_{\infty}\)-spaces and spaces whose duals are isomorphic to \(\ell_1 (\Gamma )\), extending classical results.

46G25 (Spaces of) multilinear mappings, polynomials
46G20 Infinite-dimensional holomorphy
47B10 Linear operators belonging to operator ideals (nuclear, \(p\)-summing, in the Schatten-von Neumann classes, etc.)
Full Text: DOI
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