A converse to the Oka principle over certain complex Banach manifolds

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First Kajiwara then Leiterer gave geometric or cohomological criteria in the spirit of the Grauert–Oka principle for an open subset D of a Stein manifold M to be itself Stein. We give here criteria analogous to Leiterer's, e.g., for a relatively open subset D of a closed complex Hilbert submanifold M of separable Hilbert space to be itself biholomorphic to a closed complex Hilbert submanifold of separable Hilbert space.