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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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##### References:
 [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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