On the Oka principle in a Banach space. II
This is the second paper of \textit{I. Patyi} on the Oka principle in a Banach space, and like the first one [Math. Ann. 326, No. 3, 417--441 (2003; Zbl 1044.32018)] it is carefully written. This paper studies cohomology of bundles over a pseudoconvex domain \(\Omega \subset X\) where \(X\) is one of the Banach spaces \(c_0\) or \(\ell_p,\) \(1 \leq p\leq \infty.\) Firstly, if \(E \rightarrow \Omega\) is a holomorphic Banach vector bundle, the author shows that a suitable Runge type approximation hypothesis (on \(X\) and on the structure group) implies that \(H^q(\Omega, {\mathcal O}^E)=0\) for \(q \geq 1.\) Secondly, if \(\Gamma \rightarrow \Omega\) is a Banach Lie bundle, the author shows that a suitable Runge type approximation hypothesis (on \(X\) and on both the structure group and the fiber group) implies the injectivity (and for \(X=\ell_1\) also the surjectivity) of the Oka--Grauert map \(H^1(\Omega, {\mathcal O}^{\Gamma}) \rightarrow H^1(\Omega, {\mathcal C}^{\Gamma}).\) In either case, the hypothesis is shown to hold true when the Lie groups in question are solvable, thus providing many examples for his theorems.
scientific article; zbMATH DE number 1982577