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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.

##### MSC:
 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.)
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