Gantmacher Type Theorems for Holomorphic Mappings

✍ Scribed by Manuel González; Joaquín M. Gutiérrez

• Publisher: John Wiley and Sons
• Year: 2009
• Tongue: English
• Weight: 821 KB
• Volume: 186
• Category: Article
• ISSN: 0025-584X
• DOI: 10.1002/mana.3211860108

✦ Synopsis

Givena holomorphicmapping of bounded typeg E Hb(u, F), where U E is a balanced open subset, and E , F are complex B a n d spaces, let A : Hb(F) 4 Hb(U) be the homomorphism defined by A ( f ) = f o g for all f E Hb(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 C U. To obtain these results, we prove that the class of Dunford-Pettis sets is stable under projective tensor products.

Moreover, we diaracterize the reflexivity of the space Hb(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.