Gantmacher Type Theorems for Holomorphic Mappings

Given a holomorphic mapping of bounded type g ∈ H_b(U, F), where U ? E is a balanced open subset, and E, F are complex Banach spaces, let A : H_b(F) ∈ H_b(U) be the homomorphism defined by A(f) = fog for all f ∈ H_b(F). We prove that: (a) for F having the Dunford-Pettis property, A is weakly compact if and only if 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 ? U. To obtain these results, we prove that the class of Dunford - Pettis sets is stable under projecti ve tensor products. Moreover, we diaracterize 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. As an application, we obtain some results on linearization of holomorphic mappings.