Cohomology vanishing in Hilbert spaces (Q1430001)

Here the authors give two cohomology vanishing theorems (one of Scheja's type and one of Andreotti-Grauert's type) for non-pseudoconvex domain of a separable Hilbert space \(H\), and characterize the holomorphy of domains with smooth boundaries in \(H\). Here is their Scheja's type theorem. Let \(U\) be a pseudoconvex domain in \(H\) and \(A\) an analytic subset of \(U\) such that there exists an integer \(n_0\) such that for all finite dimensional affine subspaces \(M\) of \(H\) with \(\dim (M) \geq n_0\), \(A\cap M\) has codimension \(\geq 2\) in \(M\). Then \(H^1(U\backslash A,\mathcal {O}_{U\backslash A}) = 0\). Their proof uses \(L^2\) estimates for solutions of a \(\bar{\partial }\)-equation, following \textit{L. Gruman} [Ill. J. Math. 18, 20--26 (1974; Zbl 0276.32017)].