Extensions of Holomorphic Maps through Hypersurfaces and Relations to the Hartogs Extensions in Infinite Dimension

A generalization of Kwack’s theorem to the infinite dimensional case is obtained. We consider a holomorphic map f from Z \ H into Y , where H is a hypersurface in a complex Banach manifold Z and Y is a hyperbolic Banach space. Under various assumptions on Z, H and Y we show that f can be extended to a holomorphic map from Z into Y . Moreover, it is proved that an increasing union of pseudoconvex domains containing no complex lines has the Hartogs extension property.