Estimations des solutions de l'équation de Bezout dans les algèbres de Beurling analytiques - El-Fallah, O.; Zarrabi, M. (MATHEMATICA SCANDINAVICA, 2005)
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 $\sum_{i=1}^n\|f_i\|^2 ≤ 1$. 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 $δ=\inf_{χ\in\hat A}(|\hat f_1(χ)|^2+…+|\hat f_n(χ)|^2)^{1/2}$. 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.