Vanishing cohomology for holomorphic vector bundles in a Banach setting (Q1769369)

The author proves: for a complex Banach space \(X\) with countable unconditional basis, all open pseudoconvex \(\Omega\subset X\), and for a locally trivial holomorphic Banach bundle \(E\to \Omega\), the cohomology groups \(H^q(\Omega,E)\) vanish for \(q\geq 1\). More generally, the same vanishing theorem is established for \(X\) with a Schauder basis and a certain approximation hypothesis (which is fulfilled in the case above). Also, an infinite dimensional extension of H. Cartan's Theorem A is given.