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 Goo (U) and a bounded holomorphic mapping gu: U -G? (U) with the following universal property: For each complex Banach space F and each bounded holomorphic mapping f: U -* F , there is a unique continuous linear operator Tf: Goo (U) -F such that Tf ? gu = f . The correspondence f Tf is an isometric isomorphism between the space Ho (U; F) of all bounded holomorphic mappings from U into F , and the space L(G?? (U); F) of all continuous linear operators from G? (U) into F. These properties characterize G??'(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?O(U;F) and L(G?(U);F). This paper consists of five sections. In ? 1 we establish our notation and terminology. In ?2 we prove the aforementioned linearization theorem. In ?3 we translate certain properties of a mapping f E H' (U; F) into properties of the corresponding operator Tf E L(G'(U); F). We show, for instance, that f has a relatively compact range if and only if Tf is a compact operator. In ?4 we give a seminorm characterization of the unique locally convex topology T y on H?(U; F) such that the correspondence f --* Tf is a topological isomorphism between the spaces (H (U; F), TY) and (L(G ( U); F), T,), where TC denotes the compact-open topology. Finally, in ? 5 we use the preceding results to establish necessary and sufficient conditions for the spaces G? (U) and H? (U) to have the approximation property. These are holomorphic analogues of classical results of A. Grothendieck [8], and complement results of R. Aron and M. Schottenloher [2]. We show, in particular, that if U is a balanced, bounded, open set in a complex Banach space E, then G? (U) has the approximation property if and only if E has the approximation property. We also show that if U is an arbitrary open set in a complex Banach space E, then H? (U) has the approximation property if and only if, for each complex Banach space F, each mapping in H?(U; F) with a relatively compact range can be uniformly approximated on U by mappings in H?(U; F) with finite-dimensional range. Since it is still unknown whether Received by the editors April 10, 1989. 1980 Mathematics Subject Classification (1985 Revision). Primary 46G20, 46E15; Secondary 46E10. (D 1991 American Mathematical Society 0002-9947/91 $1.00+ $.25 per page