Proposition 2

Let $\varrho_1, . . . , \varrho_n$ be real numbers $>0$ and let $U\subset \mathbf{C}^n$ be the polydisc product of the open discs with center 0 and radius $\varrho_i$. Let

$(k_1,...,k_n)\in \mathbf{N}^n$

be a formal series with coefficients in A. Suppose that this series converges in U, and let $f$ denote the holomorphic function in U of which it is the sum.

Let $\boldsymbol{a}= (a_1, . . . , a_n)\in A^n$ be such that $\varrho (a_i)< \varrho_i$ for $1\leqslant i\leqslant n$. Then Sp$^n(\boldsymbol{a})\subset U$, the family $(c(k_1, . . . , k_n)a^{k_1}_1\cdots a^{k_n}_n)$ of elements of A is absolutely summable, and

$(k_1,...,k_n)\in \mathbf{N}^n$

For every character $\chi$ of A and every integer $i$ such that $1\leqslant i\leqslant n$, one has $|\chi (a_i)|\leqslant \varrho (a_i)< \varrho_i$, hence Sp$^n(\boldsymbol{a})\subset U$ by definition of the simultaneous spectrum. Let $z_1, . . . , z_n$ be the restrictions to U of the coordinate functions on $\mathbf{C}^n$. Then the family $(c(k_1, . . . , k_n)z_1^{k_1}\cdots z^{k_n}_n)$ is summable in $\mathscr{O}(U; A)$ and has sum $f$. In view of lemma 10 and the continuity of the mapping $\Theta^U_{\boldsymbol{a}}$, the family $(c(k_1, . . . , k_n)a^{k_1}_1. . . a^{k_n}_n)$ is therefore summable in A and has sum $\Theta_{\boldsymbol{a}}(f)$. For $1\leqslant i\leqslant n$, let $\lambda_i$ be a real number such that $\varrho (a_i)< \lambda_i< \varrho_i$. There exists $M_i<+\infty$ such that $\|a^k_i\|\leqslant M_i\lambda^k_i$ for every integer $k\geqslant 0$. One then has

$(k_1,...,k_n)\in \mathbf{N}^n$ $M_1\cdots M_n\sum\|c(k_1, . . . , k_n)\|\lambda^{k_1}_1\cdots \lambda^{k_n}_n$

$(k_1,...,k_n)\in \mathbf{N}^n$ which is finite by hypothesis, hence the family $(c(k_1, . . . , k_n)a^{k_1}_1\cdots a^{k_n}_n)$ is absolutely summable.