Linearization of bounded holomorphic mappings on Banach spaces

Abstract:

The main result in this paper is the following linearization theorem. For each open set $U$ in a complex Banach space $E$, there is a complex Banach space ${G^\infty }(U)$ and a bounded holomorphic mapping ${g_U}:U \to {G^\infty }(U)$ with the following universal property: For each complex Banach space $F$ and each bounded holomorphic mapping $f:U \to F$, there is a unique continuous linear operator ${T_f}:{G^\infty }(U) \to F$ such that ${T_f} \circ {g_U} = f$. The correspondence $f \to {T_f}$ is an isometric isomorphism between the space ${H^\infty }(U;F)$ of all bounded holomorphic mappings from $U$ into $F$, and the space $L({G^\infty }(U);F)$ of all continuous linear operators from ${G^\infty }(U)$ into $F$. These properties characterize ${G^\infty }(U)$ uniquely up to an isometric isomorphism. The rest of the paper is devoted to the study of some aspects of the interplay between the spaces ${H^\infty }(U;F)$ and $L({G^\infty }(U);F)$.