Operational Calculus in Several Variables

Abstract

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