A note on a class of Banach algebra-valued polynomials (Q1854142)

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 \textit{M. L. Lourenço} and \textit{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.