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Proceedings of the American Mathematical Society

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

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The range of linear fractional maps on the unit ball
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by Alexander E. Richman PDF
Proc. Amer. Math. Soc. 131 (2003), 889-895 Request permission

Abstract:

In 1996, C. Cowen and B. MacCluer studied a class of maps on $\mathbb C^N$ that they called linear fractional maps. Using the tools of Kreĭn spaces, it can be shown that a linear fractional map is a self-map of the ball if and only if an associated matrix is a multiple of a Kreĭn contraction. In this paper, we extend this result by specifying this multiple in terms of eigenvalues and eigenvectors of this matrix, creating an easily verified condition in almost all cases. In the remaining cases, the best possible results depending on fixed point and boundary behavior are given.
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Additional Information
  • Alexander E. Richman
  • Affiliation: Department of Mathematics, Purdue University, West Lafayette, Indiana 47907
  • Address at time of publication: as of August 11, 2002: Department of Mathematics, Bucknell University, Lewisburg, Pennsylvania 17837
  • Email: richman@math.purdue.edu, arichman@bucknell.edu
  • Received by editor(s): September 12, 2001
  • Received by editor(s) in revised form: October 19, 2001
  • Published electronically: July 17, 2002
  • Communicated by: Joseph A. Ball
  • © Copyright 2002 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 131 (2003), 889-895
  • MSC (2000): Primary 32A10, 32A40, 47B50
  • DOI: https://doi.org/10.1090/S0002-9939-02-06596-6
  • MathSciNet review: 1937427