Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Banach spaces with biholomorphically equivalent unit balls are isomorphic

Authors: Wilhelm Kaup and Harald Upmeier
Journal: Proc. Amer. Math. Soc. 58 (1976), 129-133
MSC: Primary 32M05; Secondary 58C10, 46B05
MathSciNet review: 0422704
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: It is shown that in every complex Banach space $ E$ with open unit ball $ D \subset E$ there is a closed $ C$-linear subspace $ V \subset E$ such that $ V \cap D$ is the orbit of the origin $ 0 \in E$ under the group $ {\operatorname{Aut}}(D)$ of all biholomorphic automorphisms of $ D$. In particular two complex Banach spaces are isometrically equivalent if and only if their open unit balls are biholomorphically equivalent.

References [Enhancements On Off] (What's this?)

  • [1] J. Dieudonné, Foundations of modern analysis, Pure and Appl. Math. vol. 10, Academic Press, New York and London, 1969. (1960 ed., MR 22 #11074.) MR 0349288 (50:1782)
  • [2] L. A. Harris, A continuous form of Schwarz's Lemma in normed linear spaces, Pacific J. Math. 38 (1971), 635-639. MR 46 #4167. MR 0305037 (46:4167)
  • [3] -, Bounded symmetric homogeneous domains in infinite dimensional spaces, Proceedings on Infinite Dimensional Holomorphy, Lecture Notes in Math., vol. 364, Springer-Verlag, New York, 1973, pp. 13-40. MR 0407330 (53:11106)
  • [4] W. Kaup, Über das Randverhalten von holomorphen Automorphismen beschränkter Gebiete, Manuscripta Math. 3 (1970), 257-270. MR 43 #3502. MR 0277769 (43:3502)
  • [5] -, On the automorphisms of certain symmetric complex manifolds of infinite dimension. An. Acad. Brasil Ci. (to appear). MR 0457798 (56:16002)
  • [6] L. Nachbin, Topology on spaces of holomorphic mappings, Springer-Verlag, New York, 1969. MR 40 #7787. MR 0254579 (40:7787)
  • [7] H. Upmeier, Über die Automorphismengruppen beschränkter Gebiete in Banachräumen, Thesis, Tübingen, 1975. MR 0447644 (56:5954)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 32M05, 58C10, 46B05

Retrieve articles in all journals with MSC: 32M05, 58C10, 46B05

Additional Information

Article copyright: © Copyright 1976 American Mathematical Society

American Mathematical Society