Estimations des Solutions de l'équation de Bezout dans les Algèbres de Beurling analytiques
Résumé
Let $A$ be a unitary commutative Banach algebra with unit $e$. For $f\in A$ we denote by $\hat f$ the Gelfand transform of $f$ defined on $\hat A$, the set of maximal ideals of $A$. Let $(f_1, ... , f_n)\in A^n$ be such that $\displaystyle \sum _{i=1}^n \|f_i\|^2 \leq 1$. We study here the existence of solutions $(g_1, ... , g_n)\in A^n$ to the Bezout equation $f_1g_1+ ... +f_ng_n=e$, whose norm is controlled by a function of $n$ and $\delta=\inf_{\chi\in\hat A}\left(|\hat f_1(\chi)|^2+...+|\hat f_n(\chi)|^2\right)^{1/2}$. \par We treat this problem for the analytic Beurling algebras and their quotient by closed ideals. The general Banach algebras with compact Gelfand transform are also considered.