On holomorphic Banach vector bundles over Banach spaces

Let X be a Banach space with a countable unconditional basis (e.g., X = ℓ ₂ Hilbert space), Ω ⊂ X pseudoconvex open, E → Ω a locally trivial holomorphic vector bundle with a Banach space Z for fiber type, the sheaf of germs of holomorphic sections of E → Ω, and Z ₁ the Banach space . Then and Ω × Z ₁ are holomorphically isomorphic, is acyclic and E is so to speak stably trivial over Ω in a generalized sense. We also show that if E is continuously trivial over Ω, then E is holomorphically trivial over Ω. In particular, if Z = ℓ ₂ or Ω is contractible, then E is holomorphically trivial over Ω. Some applications are also given. To my dear little daughter, Sári Mangala, on her third birthday. Supported in part by a Research Initiation Grant from Georgia State University.