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A class of polynomials from Banach spaces into Banach algebras. (English) Zbl 1092.46032
Let \(E\) be a complex Banach space and let \(F\) be a complex Banach algebra. The space of all \(n\)-homogeneous continuous polynomials from \(E\) into \(F\) is denoted by \({\mathcal P}(^nE;F)\). \({\mathcal P}_f (^n E;F)= \{\sum_{i=1}^k q_i^n \otimes b_i : q_1 ,\ldots ,q_k \in E' ,b_1,\ldots ,b_k \in F, k\in \mathbb N\}\) the well-known subspace of finite type polynomials.
The authors define a new subspace \({\mathbb P}_{f} (^n E;F)= \{ \sum_{i=1}^k T_i^n :T_1,\ldots ,T_k \in L(E;F),k\in \mathbb N\}\), where \(T^n (x):=(Tx)^n\). In section 2 the relations between \(P_f (^n E;F)\) and \({\mathbb P}_{f} (^n E;F)\) and their closures in \({\mathcal P}(^n E;F)\) are studied and several examples are considered.
In section 3 the space \({\mathbb P}(^n E;F):=\left\{ \sum_{i=1}^{\infty} T_i^n , T_i \in L(E;F), ~ \sum_{i=1}^{\infty} \| T_i \| ^n <\infty \right\}\) provided with the norm \(||| P ||| :=\inf \left\{ \sum_{i=1}^{\infty} \| T_i \|^n : P= \sum_{i=1}^{\infty} T_i^n \right\}\) is introduced. The authors establish an isometric isomorphism between the spaces \({\mathbb P} (^n E;F)'\) and \(\{ Q\in {\mathcal P}(^n L(E;F);{\mathbb C} ): \sum_{i=1}^{\infty} Q(T_i)=0\) if \(\sum_{i=1}^{\infty} T_i^n=0 \}\).

MSC:
46G25 (Spaces of) multilinear mappings, polynomials
47H60 Multilinear and polynomial operators
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[1] Aron, R. M. and Prolla, J. B., Polynomial approximation of differentiable functions on Banach spaces, J. Reine Angew. Math., 313 (1980), 195-216. · Zbl 0413.41022 · doi:10.1515/crll.1980.313.195 · crelle:GDZPPN002196956 · eudml:152204
[2] Mujica, J., Complex Analysis in Banach Spaces, North-Holland Math. Stud., 120, Notas de Matematica, 107 North Holland, 1986. i i i i · Zbl 0586.46040
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