Operational Calculus in Several Variables

Abstract

We wish to extend the operational calculus established in Section 3 to functions of several variables. Let 𝔄 be a Banach algebra and x1,…, x n ∈ 𝔄. If P is a polynomial in n variables $$ p\left( {{z_1}, \ldots ,{z_n}} \right) = \sum\limits_v {{A_v}} z_1^{{v_1}} \cdots z_n^{{v_n}} $$ it is natural to define $$ P\left( {{x_1}, \cdots ,{x_n}} \right) = \sum\limits_v {{A_v}} x_1^{{v_1}} \cdots x_n^{{v_n}} \in $$