Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Automorphism groups of bounded domains in Banach spaces

Authors: Stephen J. Greenfield and Nolan R. Wallach
Journal: Trans. Amer. Math. Soc. 166 (1972), 45-57
MSC: Primary 32K05; Secondary 32N15, 46B99, 58B10
MathSciNet review: 0296359
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We prove a weak Schwarz lemma in Banach space and use it to show that in Hilbert space a Siegel domain of type II is not necessarily biholomorphic to a bounded domain. We use a strong Schwarz lemma of L. Harris to find the full group of automorphisms of the infinite dimensional versions of the Cartan domains of type I. We then show that all domains of type I are holomorphically inequivalent, and are different from k-fold products of unit balls $ (k \geqq 2)$. Other generalizations and comments are given.

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

  • [1] R. Abraham and S. Smale, Lectures of Smale on differential topology, Notes, Columbia University, New York, 1962/63.
  • [2] Stefan Bergman, The Kernel Function and Conformal Mapping, Mathematical Surveys, No. 5, American Mathematical Society, New York, N. Y., 1950. MR 0038439
  • [3] J. Dieudonné, Foundations of modern analysis, Pure and Applied Mathematics, Vol. X, Academic Press, New York-London, 1960. MR 0120319
  • [4] Clifford J. Earle and Richard S. Hamilton, A fixed point theorem for holomorphic mappings, Global Analysis (Proc. Sympos. Pure Math., Vol. XVI, Berkeley, Calif., (1968), Amer. Math. Soc., Providence, R.I., 1970, pp. 61–65. MR 0266009
  • [5] B. A. Fuks, Spetsialnye glavy teorii analiticheskikh funktsiimnogikh kompleksnykh peremennykh, Gosudarstv. Izdat. Fiz.-Mat. Lit., Moscow, 1963 (Russian). MR 0174786
  • [6] Lawrence A. Harris, Schwarz’s lemma in normed linear spaces, Proc. Nat. Acad. Sci. U.S.A. 62 (1969), 1014–1017. MR 0275179
  • [7] L. K. Hua, Harmonic analysis of functions of several complex variables in the classical domains, Translated from the Russian by Leo Ebner and Adam Korányi, American Mathematical Society, Providence, R.I., 1963. MR 0171936
  • [8] T. L. Hayden and T. J. Suffridge, Biholomorphic maps have a fixed point, Notices Amer. Math. Soc. 17 (1970), 170. Abstract #672-308.
  • [9] Shizuo Kakutani, Topological properties of the unit sphere of a Hilbert space, Proc. Imp. Acad. Tokyo 19 (1943), 269–271. MR 0014203
  • [10] Nicolaas H. Kuiper, The homotopy type of the unitary group of Hilbert space, Topology 3 (1965), 19–30. MR 0179792
  • [11] R. S. Phillips, On symplectic mappings of contraction operators, Studia Math. 31 (1968), 15–27. MR 0236754
  • [12] I. I. Pjateckiĭ-Šapiro, On bounded homogeneous domains in an 𝑛-dimensional complex space, Izv. Akad. Nauk SSSR Ser. Mat. 26 (1962), 107–124 (Russian). MR 0141141
  • [13] Carl Ludwig Siegel, Symplectic geometry, Academic Press, New York-London, 1964. MR 0164063

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 32K05, 32N15, 46B99, 58B10

Retrieve articles in all journals with MSC: 32K05, 32N15, 46B99, 58B10

Additional Information

Keywords: Hilbert space, Banach space, unit balls, inequivalence of polyballs, Cartan domains, Siegel domains, Schwarz lemma
Article copyright: © Copyright 1972 American Mathematical Society