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Gantmacher Type Theorems for Holomorphic Mappings
Author(s) -
González Manuel,
Gutiérrez Joaquín M.
Publication year - 1997
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.3211860108
Subject(s) - mathematics , holomorphic function , banach space , bounded function , type (biology) , homomorphism , pure mathematics , class (philosophy) , discrete mathematics , combinatorics , mathematical analysis , ecology , biology , artificial intelligence , computer science
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.