Compact holomorphic mappings on Banach spaces and the approximation property

AbstractLet H(E) be the space of complex valued holomorphic functions on a complex Banach space E. The approximation property for H(E), endowed with various natural locally convex topologies, is studied. For example, H(E) with the compact-open topology has the approximation property if and only if E has the approximation property. In order to characterize when H(E) has the approximation property for topologies other than the compact-open, the notion of a compact holomorphic map between Banach spaces in introduced and studied