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Transactions of the American Mathematical Society

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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
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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  • [1] José M. Ansemil and Seán Dineen, Locally determining sequences in infinite-dimensional spaces, Note Mat. 7 (1987), no. 1, 41–45 (English, with Italian summary). MR 1014837
  • [2] Richard M. Aron and M. Schottenloher, Compact holomorphic mappings on Banach spaces and the approximation property, J. Functional Analysis 21 (1976), no. 1, 7–30. MR 0402504
  • [3] A. O. Chiacchio, M. C. Matos, and M. S. M. Roversi, On best approximation by rational and holomorphic mappings between Banach spaces, J. Approx. Theory 58 (1989), no. 3, 334–351. MR 1012681, 10.1016/0021-9045(89)90033-6
  • [4] J. B. Cooper, Saks spaces and applications to functional analysis, 2nd ed., North-Holland Mathematics Studies, vol. 139, North-Holland Publishing Co., Amsterdam, 1987. Notas de Matemática [Mathematical Notes], 116. MR 886477
  • [5] Joseph Diestel, Sequences and series in Banach spaces, Graduate Texts in Mathematics, vol. 92, Springer-Verlag, New York, 1984. MR 737004
  • [6] Seán Dineen, Complex analysis in locally convex spaces, North-Holland Mathematics Studies, vol. 57, North-Holland Publishing Co., Amsterdam-New York, 1981. Notas de Matemática [Mathematical Notes], 83. MR 640093
  • [7] K. Floret, Elementos de posto finito em produtos tensoriais topológicos, Atas 24$ ^\circ$ Seminário Brasileiro de Análise, Sociedade Brasileira de Matematica, 1986, pp. 189-195.
  • [8] Alexandre Grothendieck, Produits tensoriels topologiques et espaces nucléaires, Mem. Amer. Math. Soc. No. 16 (1955), 140 (French). MR 0075539
  • [9] John Horváth, Topological vector spaces and distributions. Vol. I, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1966. MR 0205028
  • [10] Joram Lindenstrauss and Lior Tzafriri, Classical Banach spaces. I, Springer-Verlag, Berlin-New York, 1977. Sequence spaces; Ergebnisse der Mathematik und ihrer Grenzgebiete, Vol. 92. MR 0500056
  • [11] Jorge Mujica, Complex homomorphisms of the algebras of holomorphic functions on Fréchet spaces, Math. Ann. 241 (1979), no. 1, 73–82. MR 531152, 10.1007/BF01406710
  • [12] Jorge Mujica, A Banach-Dieudonné theorem for germs of holomorphic functions, J. Funct. Anal. 57 (1984), no. 1, 31–48. MR 744918, 10.1016/0022-1236(84)90099-5
  • [13] Jorge Mujica, Complex analysis in Banach spaces, North-Holland Mathematics Studies, vol. 120, North-Holland Publishing Co., Amsterdam, 1986. Holomorphic functions and domains of holomorphy in finite and infinite dimensions; Notas de Matemática [Mathematical Notes], 107. MR 842435
  • [14] Kung Fu Ng, On a theorem of Dixmier, Math. Scand. 29 (1971), 279–280 (1972). MR 0338740
  • [15] R. Ryan, Applications of topological tensor products to infinite dimensional holomorphy, Ph.D. Thesis, Trinity College, Dublin, 1980.
  • [16] Helmut H. Schaefer, Topological vector spaces, Springer-Verlag, New York-Berlin, 1971. Third printing corrected; Graduate Texts in Mathematics, Vol. 3. MR 0342978
  • [17] Antoni Zygmund, Trigonometrical series, Dover Publications, New York, 1955. MR 0072976

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