The Biduality Problem and M-Ideals in Weighted Spaces of Holomorphic Functions

Given a weight $v$ on an open subset $U$ of ${\bf C}^n$, ${\cal H}_v(U)$ (resp. ${\cal H}_{v_o}(U)$) denotes the Banach space of holomorphic functions $f$ on $U$ such that $vf$ is bounded on $U$ (resp. converges to $0$ on the boundary of $U$). We show that ${\cal H}_v(U)$ is canonically isometrically isomorphic to the bidual of ${\cal H}_{v_o}(U)$ if and only if ${\cal H}_{v_o}(U)$ is an M-ideal in ${\cal H}_v(U)$ and the associated weights $\tilde v_o$ and $\tilde v$ coincide