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Linearization of bounded holomorphic mappings on Banach spaces


Author: Jorge Mujica
Journal: Trans. Amer. Math. Soc. 324 (1991), 867-887
MSC: Primary 46G20; Secondary 32A10, 46E15
DOI: https://doi.org/10.1090/S0002-9947-1991-1000146-2
MathSciNet review: 1000146
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Abstract: 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^\infty }(U)$ and a bounded holomorphic mapping $ {g_U}:U \to {G^\infty }(U)$ with the following universal property: For each complex Banach space $ F$ and each bounded holomorphic mapping $ f:U \to F$, there is a unique continuous linear operator $ {T_f}:{G^\infty }(U) \to F$ such that $ {T_f} \circ {g_U} = f$. The correspondence $ f \to {T_f}$ is an isometric isomorphism between the space $ {H^\infty }(U;F)$ of all bounded holomorphic mappings from $ U$ into $ F$, and the space $ L({G^\infty }(U);F)$ of all continuous linear operators from $ {G^\infty }(U)$ into $ F$. These properties characterize $ {G^\infty }(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^\infty }(U;F)$ and $ L({G^\infty }(U);F)$.


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DOI: https://doi.org/10.1090/S0002-9947-1991-1000146-2
Article copyright: © Copyright 1991 American Mathematical Society

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