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 $Z$ $\setminus$ $H$ 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.