The \(\bar\partial\)-problem for holomorphic (0,2)-forms on pseudoconvex domains in separable Hilbert spaces and D. F. N. spaces (Q1600020)

The following result is shown: Let \(E\) be a DFN space and \(\Omega \) a pseudoconvex domain in \(E,\) let \(f:\Omega \rightarrow \Lambda(0,2)(E)\) be a holomorphic \((0,2)\)-form. Then there exists a \(\mathcal C^{\infty}\) \((0,1)\)-form \(g\) on \(\Omega \) such that \(\overline \partial g=f.\) The authors first prove the corresponding result for pseudoconvex open subsets in a separable Hilbert space.