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A note on a class of Banach algebra-valued polynomials. (English) Zbl 1024.46013
Let $$E$$ be a Banach space and let $$F$$ be a (unital) Banach algebra. The authors consider two classes of $$n$$-homogeneous polynomials of finite type from $$E$$ into $$F$$, the common finite polynomials \begin{aligned} P_f (^nE;F) &= \{f_1^n \otimes b_1 + \cdots + f_k^n \otimes b_k : f_j \in E',\;b_j \in F,\;1 \leq j \leq k,\;k \in \mathbb N\}\\ \text{and} P_{\text{fin}}(^nE;F)&= \{T_1^n + \cdots + T_k^n : T_j \in L(E;F),\;1 \leq j \leq k,\;k \in \mathbb N\}, \end{aligned} where $$T_j^n(x) := (T_j(x))^n$$.
The main results are the following.
Let $$F$$ be a Banach algebra. Then the following are equivalent.
(a) $$F$$ is a finite-dimensional space,
(b) $$P_{fin}(^nE;F) \subseteq P_f(^nE;F)$$ for every $$n \in \mathbb N$$ and every Banach space $$E$$,
(c) $$P_{fin}(^1E;F) \subseteq P_f(^1E;F)$$ for every Banach space $$E$$.
Using a result from M. L. Lourenço and L. A. Moraes [Publ. Res. Inst. Math. Sci. 37, 521-529 (2001; Zbl 1092.46032)] this is also true with equality in (b) and (c) for unital Banach algebras $$F$$.
Lemma 2.1 is a special case of the well-known polarization formula.
##### MSC:
 46G25 (Spaces of) multilinear mappings, polynomials 46H20 Structure, classification of topological algebras 47H60 Multilinear and polynomial operators
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