GANTMACHER TYPE THEOREMS FOR HOLOMORPHIC MAPPINGS

Given a holomorphic mapping of bounded type g epsilon H-b (U,F), where U subset of or equal to E is a balanced open subset, and E, F are com plex Banach spaces, let A : H-b(F) --> H-b(U) be the homomorphism defi ned by A(f) = f o g for all f epsilon H-b(F). We prove that: (a) for F having the Dunford-Pettis property, A is weakly compact if and only i f g is weakly compact; (b) A is completely continuous if and only if g (W) is a Dunford-Pettis set for every U-bounded subset W subset of U. To obtain these results, we prove that the class of Dunford-Pettis set s is stable under projective tensor products. Moreover, we characteriz e the reflexivity of the space H-b(U, F) and prove that E- and F have the Schur property if and only if H-b(U, F) has the Schur property. A s an application, we obtain some results on linearization of holomorph ic mappings.