Linearization of Bounded Holomorphic Mappings on Banach Spaces

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°°(Í7) and a bounded holomorphic mapping gv: U —► G°°(U) with the following universal property: For each complex Banach space F and each bounded holomorphic mapping /: U —> F , there is a unique continuous linear operator T,\ G°°(U) —» F such that T, o gv = f. The correspondence / —► Tr is an isometric isomorphism between the space H°°(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°°(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 / e H°°(U ; F) into properties of the corresponding operator T, £ L(G°°(U) ; F). We show, for instance, that / has a relatively compact range if and only if 7\ is a compact operator. In §4 we give a seminorm characterization of the unique locally convex topology t on H°°(U; F) such that the correspondence / —> T, is a topological isomorphism between the spaces (H°°(U; F), r ) and (L(G°°(U); F), xf, where xc 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 77°° (L7; 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. ©1991 American Mathematical Societv 0002-9947/91 $1.00+ $.25 per page